High Energy Astrophysics T. J.

High Energy Astrophysics
T. J.-L. Courvoisier
1
Chapter 1
Introduction
High energy astrophysics is a very poorly defined field. The energy of the photons
emitted by a system is not necessary nor is it sufficient to determine whether the
study of this type of systems is part of it or not. Indeed many topics studied with
radio astronomy techniques are part of the domain, while the interior of stars, where
the temperatures are very high is excluded. The domain is therefore as defined by
the traditions and the work that has been done by people who are high energy
astrophysicists.
At present high energy astrophysics is a very lively part of astrophysics. This is
due to the fact that the subject did only really start after the beginning of the
space age in the 1960s, to a steady progress in the instrumentation available and
to an unprecedented set of instruments in orbit now. These instruments include
XMM-Newton, Chandra and INTEGRAL. The first two are large X-ray instruments,
one specialised in imaging (Chandra) the other in spectroscopy (XMM), because of
its very large collecting area. INTEGRAL is sensitive above few keV and up to
some MeV. However, other observation tools in all domains of the electro-magnetic
spectrum are used in high energy astrophysics, including optical, infrared and radio
telescopes. Since 2005 or so very high energy gamma ray astrophysics in the GeVTeV parts of the photon spectrum has obtained some remarkable success with the
discovery of about 70 sources (2007). This period follows a very long time during
which progress had been very slow.
High energy astrophysics has unveiled a Universe very different from that known
from sole optical observations. Objects emitting most of their radiation in the
optical domain are dominated by thermal emission with temperatures of few to
several thousand degrees. These are stars and collections of them mainly in the
form of galaxies. The evolution of these objects happen on timescales given by E/L,
where E is the energy available in the form of nuclear fuel and L is their luminosities.
The typical timescales resulting are of millions to billions of years. In contrast high
energy astrophysics work has revealed many type of objects which typical variability
timescales are as short as years, months, days hours (quasars, X-ray binaries, etc)
and down to milli-seconds (gamma ray bursts). The sources of energy that are met
are only very seldom nuclear fusion, and most of the time gravitation, a paradox
when one thinks that gravitation is by many orders of magnitude the weakest of the
fundamental interactions.
2
Knowledge of the objects revealed by high energy astrophysics in the last decades
and of the physical conditions met in these objects an d associated processes are
nowadays part of the culture of astrophysicists, also of those active in other domains
of astronomy. This course aims at giving this scientific culture and at providing
those intending to be active in high energy astrophysics a broad basis on which they
should be able to build the more specific knowledge they will need and to place this
knowledge in an appropriately broad frame. It is also hoped that the course will help
students in recognising physical processes when they are revealed by observational
signatures in contexts that may differ widely from those presented here.
The course has two main parts. In the first part we start from the physical process,
e.g. a an emission process, discuss it and try to lay the physics involved down and
then proceed to present one example in which the process is at work in nature. In
the second part, we take an opposite view and start from a type of object (e.g. X-ray
binaries) and proceed to understand their nature as far as possible.
1.1
The Parameter Space of high energy Astrophysics
Deep gravitational fields and temperature The temperature that corresponds
to a random velocity is
2
T = 4 × 10−5 v[m/s]
(1.1)
for a gas of Hydrogen. The gravitational field around the Earth is such that the
escape velocity is of 11 km/s. When one isotropises this velocity, one obtains
temperatures of the order of 5000 K with this typical velocity. Indeed were the
atmosphere temperature of that order, it would evaporate. Around a neutron
star, the typical velocities associated with the gravitational field is of the order
of 1/3 × c. The corresponding temperatures are of some 1011 K or 10 MeV.
The emission of gas in such regions are therefore expected to be in the X-rays
(keV) up to gamma ray regions of the spectrum.
It follows from these considerations that matter in a deep gravitational field
emits predominantly in the high energy domain. Conversely, X- and gammaray astrophysics is the predominant tool to study compact objects. Figure 1.1
illustrates this by showing the very broad emission line that is seen from a
fluorescence line of Fe at 6.4 keV in the central regions of an active galaxy, i.e.
in matter surrounding a massive black hole in the nucleus of the galaxy. This
illustrates the very large velocities (width of the line) and large gravitational
fields (asymmetry in the profile) that are directly observable from gas that
emits in the X-ray domain.
Extreme magnetic fields Magnetically induced electron transitions (cyclotron lines)
occur at the Larmor frequency. The line energy is given by
EkeV = 12 × B12 ,
(1.2)
where B12 is the magnetic field in units of 1012 Gauss and the energy is given
in keV. Figure 1.2 shows the spectrum of a X-ray binary obtained by the
INTEGRAL satellite. The absorption lines in this spectrum directly show the
existence of a magnetic field of few 1012 G in the binary system.
3
Figure 1.1: The line profile of iron Kα from MCG-6-30-15 observed by the ASCA
satellite (Tanaka et al. 1995, Nature, 375, 659).The emission line is extremely broad,
with a width indicating velocities of order 1/3 × c. The marked asymmetry towards
energies lower than the rest-energy of the emission line (6.4 keV) is most likely caused
by gravitational and relativistic-Doppler shifts near the black hole at the center of
the active galaxy. The solid line shows the model profile expected from a disk of
matter orbiting the hole, extending between 3 and 10 Schwarzschild radii.
It is also now apparent that decaying magnetic fields up to 1015 G are at
the origin of the emission of so-called magnetars (soft gamma repeaters and
anomalous X-ray pulsars).
Nucleosynthesis Figure 1.3 shows a map of the Galaxy obtained in the light of
a nuclear transition corresponding to the decay of 26 Al. This shows a direct
observation of a nuclear reaction. The halflife of Al is of about 1 million year.
The figure therefore shows convincingly that Aluminium has been produced
in the Galaxy during the last million years, and therefore, that the creation of
the Universe is an on-going process and not an act of once in the past.
Note that in this case the nuclear process at work is radio-active decay rather
than fusion.
1.2
Instruments
Since the photon energy is not the defining criterion of the high energy astrophysics,
many instruments are used, that cover most of the electro-magnetic spectrum. However, high energy astrophysics does live a very peculiar period with a host of outstanding space missions now in operations in the (photon) high energy domain.
Results from several of them will be used in these lectures. A schematic list of
recent and flying missions is:
Compton Gamma Ray Observatory, CGRO This US mission flew from 1991
to 2000. It included instruments that had no imaging capability and were sensitive between some 100 keV to GeVs (Figure 1.4). One of the most important
4
instruments on board was BATSE that registered gamma ray bursts on 2π of
the sky.
ASCA A Japanese X-ray telescope that provided images up to some 10 keV.
Beppo-SAX An italian X-ray satellite that provided the first location of a gamma
ray burst with a precision sufficient lead to make optical follow-up observations,
and finally to find the counter parts of GRBs and establish their extra-galactic
nature.
Chandra Launched in 1999 is a US X-ray telescope with an excellent (less than
1") angular resolution.
XMM-Newton Another X-ray telescope with a very high throuput. It is a European mission also launched in 1999 (Figure 1.5).
INTEGRAL A high energy X-ray and gamma ray instrument with imaging capability launched in 2002 (Figure 1.6). This is the instrument for which we are
providing the science data centre (ISDC).
normalized counts/sec/keV
10.00
V0332+53
a)
1.00
0.10
0.01
χ
20
b)
0
χ
χ
χ
−20
15
10
5
0
−5
−10
−15
4
2
0
−2
−4
4
2
0
−2
−4
c)
d)
e)
10
100
Channel Energy [keV]
Figure 1.2: Spectrum of the High Mass X-ray binary V0332+53 during an outburst
observed by INTEGRAL on 2005, Jan 7-10. a: the raw spectra taken with the
JEM-X (red) and IBIS (blue) instruments where two (or perhaps three) cyclotron
absorption lines are clearly visible. b: residuals for the model on the upper panel
without, c: with one cyclotron line at 24.9 keV, d: with a second cyclotron line at
50.5 keV and e: with a third cyclotron line at 71.7 keV (Kreykenbohm et al. 2005,
A&A 433L, 45).
5
Figure 1.3: The instrument COMPTEL, on the Compton Gamma-Ray Observatory,
has mapped the sky in 1.809 MeV gamma-ray line emission attributed to radioactive
26
Al (Oberlack U. et al., 1996, A&AS, 120, 3110). With its mean life time of about
1 million year, 26 Al directly traces recent nucleosynthesis in the Galaxy.
SWIFT launched in November 2004 to make multi-wavelength observations of
gamma ray bursts.
SUZAKU A Japanese multi purpose X-ray instrument launched in July 2005.
HESS and MAGIC are two facilities that observe the interaction of TeV photons
with the atmosphere. These two facilities are the last of a long series of early
instruments that maesure the Cerenkov radiation emitted as charged particles
created by the diffusion of a high energy gamma ray on atmospheric nuclei
travel faster than the speed of light in the air.
1.3
Sources studied
There are a number of types of sources that are traditionally part of high energy
astrophysics. They are:
Neutron stars They come in many different guises. Some emit bursts of X-rays
(they are then called bursters), others regular pulsations in the radio domain
(and are then called (radio) pulsars) or in the X-rays (and are called X-ray
pulsars). Some only emit dimly and thermally from their surface (isolated
neutron stars).
Black holes In this case one actually observes matter in the surrounding of the
black hole rather than the black hole itself. They come either with masses
typical of stars (stellar black holes) or with masses of millions of solar masses.
In the latter case they live in the centers of galaxies, they can be very bright
6
Figure 1.4: The Compton Gamma Ray Observatory was launched in April 1991
and then safely deorbited and re-entered the Earth’s atmosphere in June 2000.
The satellite had four instruments (BATSE, OSSE, COMPTEL and EGRET) that
covered six decades of the electromagnetic spectrum, from 30 keV to 30 GeV.
Figure 1.5: The XMM-Newton observatory was launched on December 10, 1999
to study the soft X-ray emission from the sky (0.1-12 keV). The three main scientific instruments on board this satellite are the photon imaging cameras EPIC, the
reflection grating spectrometers RGS and the optical monitor OM (Credit: ESA).
Figure 1.6: Launched on 17 October, 2002, INTEGRAL is dedicated to spectroscopy
and fine imaging in the energy range 3 keV – 8 MeV. The payload consists of two main
gamma-ray instruments the imager IBIS and the spectrometer SPI. Simultaneous
observations are performed by the X-ray monitors JEM-X and the optical monitor
OMC (Credit: ESA).
7
Figure 1.7: Amount of absorption at different wavelengths in the atmosphere. The
half-absorption altitude is defined as the altitude in the atmosphere (from the Earth’s
surface) where 1/2 of the radiation at a given wavelength incident on the upper
atmosphere has been absorbed. Except for visible and radio ranges, the atmosphere
absorbs very strongly and measurements at other wavelengths require observations
from orbiting instruments above the atmosphere.
(and are called Active Galactic Nuclei (AGN)) or very quiet as in the centre
of our Galaxy.
Clusters of galaxies host very large quantities of hot gas that emits in the X-rays.
Supernova remnants The rest of stellar explosions that form shocks in which
gas is heated to high temperatures and that are most likely the source of non
thermal distributions of particles observed in the Earth vicinity as cosmic rays.
1.4
Historical remarks
Figure 1.7 shows the radiation that reaches the Earth as a function of wavelength.
Clearly most of the "light" does not reach the ground and is therefore not available
to do astronomical observations from there. This is particularly true for high energy
radiation that must be captured above the atmosphere in order to be studied. As a
consequence, high energy astrophysics developed only in the space age.
It also should be remarked that if one needs to go out of the atmosphere to observe
in the X-rays, the region between the UV (longward of 1Ryd) and the X-rays at
about 0.1 keV is inaccessible even from space as the interstellar matter is opaque.
Figure 1.8 shows the absorption cross section of matter with cosmic abundances.
This has a peak at the photoionisation of H (1 Ryd) and decreases shortward with
roughly the third power of the frequency. This means that this region will remain
unexplored for a long time to come.
8
Figure 1.8: The effective cross-section of the interstellar medium (cross-section per
hydrogen atom or proton of the IM). Solid line - gaseous component with normal
composition and temperature; dot-dash - hydrogen in its molecular form; long dash
- HII region about a B star; long dash-dash-dash - HII region about an O star; short
dash - dust (Cruddace R., Paresce F., Bowyer S. and Lampton M. 1974, ApJ., 187,
497).
It must be added that if you extrapolate the X-ray flux from the Sun to that we
would expect from even the closest stars you find extremely weak fluxes that were
not expected to be observable with the instrumentation of the 50’s or 60’s. There
was therefore not much on which one could build in order to start a new set of
research activities in the 50’s. Despite this, R. Giacconi and colleagues started
a program to observe the sky in the X-rays in a series of rocket flights. They
observed unexpectedly a bright X-ray source now called Sco X-1 and the bright
X-ray background (Giacconi R.et al., 1962, Physical Review Letters 9, 439). This
earned Giacconi the 2002 Nobel Prize.
The main steps in X-ray astrophysics have been
1962 Unexpected discovery of Sco X-1 by Giacconi et al. (Nobel prize 2002) on a
rocket flight during which the diffuse background was also measured.
1963 The discovery of quasars by associating their optical and radio observations
and by understanding that the lines observed in emission are highly redshifted
H lines, proving that the objects were much more luminous than whole galaxies
(M. Schmidt; 1963, Nature 197, 1040).
1967 The discovery of radio pulsars by Jocelyn Bell and Hewish (the latter got a
Nobel prize in 1974) while measuring solar wind induced fluctuations of radio
9
fluxes.
1970-1973 The first survey of the X-ray sky by the non imaging UHURU satellite.
1978-1981 The first X-ray images by the Einstein satellite. This provided an immense increase in sensitivity over previous detectors.
1981-1983 Long observations by the EXOSAT satellite showed the importance of
variability studies.
1990-1999 The first imaging survey of the X-ray sky (soft X-rays) with ROSAT
provided upwards of 105 sources.
1993-2001 The first X-ray sensitive CCD on the ASCA satellite.
1996-2001 Beppo-SAX and the localisation of gamma ray bursts.
1999 Launches of Chandra and XMM-Newton.
In the gamma rays there are two additional difficulties, the most fundamental is that
the energy flux of most sources is fν ∝ ν −1 and therefore the photon flux is ∝ ν −2 .
Since the quality of the information obtained from a source is given by the number
of photons registered, gamma ray observations at 1 MeV will be considerably more
difficult that X-ray observations at 1 keV. The second difficulty is that gamma rays
(and to date X-rays above about 10 keV) cannot be focused. This implies that the
detectors are as large as the pupil and the signal to noise (that is given by the size
of the detector) is very poor.
The main milestones are therefore much fewer and far apart:
up to now Many balloon flights.
1975-1982 First survey of the sky by COS-B. This satellite produced a catalogue
of registered photons....
1991-2000 CGRO. Wide band instruments on board. Not imaging.
1989-1998 SIGMA, a French instrument on the GRANAT satellite of the Soviet
Union. This was the first instrument on a satellite with which images of the
γ-ray sky could be made. It used a coded mask.
2002 INTEGRAL launch. Large increase of the sensitivity with imaging capabilities also based on the coded mask technology.
???? start of the operations of HESS and MAGIC in the TeV domain.
1.5
Content of the lectures
There are two main parts in this course, the first discusses physical processes including some applications and the second starts the discussion at the objects level
and develops the models that are currently used to understand them.
The first part includes:
10
1. Radiation from an accelerated charge
2. Bremsstrahlung and the emission of clusters of galaxies
3. Synchrotron radiation and radio galaxies
4. Cyclotron emission and their signatures in X-ray pulsars
5. Compton emission and the Sunyaev Zeldovich effect
6. Accretion disks
7. Particle acceleration and cosmic rays
The second part includes:
1. Neutron star structure
2. Pulsars (radio pulsars) and some particular objects
3. X-ray binaries with either neutron stars or black holes as compact objects
4. Magnetars
5. Gamma ray bursts
6. Active Galactic Nuclei
Some information on the current instrumentation will be interleaved within the
different chapters.
1.6
Useful books
Radiative Processes in Astrophysics, Rybicki B. and Lightman A.P., John Wiley
and Sons, New York, 1979
Accretion power in astrophysics, Frank J., King A. and Raine D., Cambridge University Press, 3rd edition 2002
High Energy Astrophysics, Vols 1 and 2, Longair M., CUP, 2nd edition 1991
11
Chapter 2
Radiation of an accelerated charge
We will follow in this presentation an argument of J.J. Thomson as rendered in
Longair (High energy astrophysics vol.1). This presentation gives the essentials of
the discussion, while replacing a full discussion using the retarded potentials as given
e.g. in Jackson’s electro-dynamics.
2.1
Energy loss
Consider a charge at the origin of an inertial system at t=0. Imagine then that
the source is accelerated to a small velocity (compared with the velocity of light c,
this discussion in non-relativistic) ∆v in a time interval ∆t. Draw the electric field
lines that result from this arrangement at a time t. At a large distance the lines
are radial, centered at the origin of the inertial system, because the signal that a
perturbation has occured to the charge has not yet had the time to reach there. At
small distances, however, the lines are radial around the new position of the source
(remember that the velocity disturbance is small compared to the velocity of light).
In between, the lines are connected in a non radial way in a small zone of width
c · ∆t.
Figures 2.1 and 2.2 give the large picture and the detail of the perturbed field lines.
You can read from Figure 2.2 that the ratio of the tangential to the radial field lines
in the perturbed zone is
∆v · t sin θ
Eθ
=
Er
c∆t
(2.1)
Since the radial field is given by the Coulomb law
Er =
e
,
r2
e in e.s.u., r = ct,
you can deduce the tangential field and find
12
(2.2)
Figure 2.1: Schematical view of the electric field lines at time t due to a charged
particle accelerated to a velocity ∆v c in a time interval ∆t (from High Energy
Astrophysics, Vols 1, Longair M.).
∆v
1
sin θ 2 · t
∆t
cr
r̈ sin θ
= e 2 .
cr
Eθ = e ·
(2.3)
(2.4)
Figure 2.2: Expanded version of Figure 2.1 used to evaluate the strength of the
tangential component of the electric field due to the acceleration of an electron
(from High Energy Astrophysics, Vols 1, Longair M.).
13
Note that this field depends on the distance to the centre as 1/r rather than 1/r2 .
Introducing the electrical dipole moment p = e · r, we write
p̈ sin θ
.
c2 r
Eθ =
(2.5)
We may now calculate the energy flux that corresponds to this disturbance. Indeed
the disturbance moves outward with the velocity of light and carries therefore some
energy away from the system. The energy flux is given by the Poynting vector S:
c
S=
E × B with B = n × E.
¯ 4π
¯
¯ ¯
(2.6)
The energy loss in the direction θ in a solid angle dΩ is then
dE
c |p̈|2 sin2 θ 2
dΩ =
· r dΩ
dt
4π c4 r2
(2.7)
In order to find the energy loss from the charge, one needs to integrate (2.7) over the
solid angle dΩ remembering that in a problem that is symmetrical around the second
angle (here the symmetry axis is the direction of the acceleration) dΩ = 2π sin θdθ.
The result is
2 Z π
dE 2 |p̈|2
3
= c |p̈|
2π
sin
θ
dθ
=
dt 4π c4
3 c3
0
(2.8)
This is the so-called Larmor formula, it is given here in Gaussian units and gives the
energy carried by the electromagnetic radiation emitted by a charge in acceleration
as a function of this acceleration. The radiation is dipolar (see the sin2 θ in 2.7).
The absolute value is there to remind us that the sign will be different whether one
considers the loss of energy from the charge or the gain in the radiation.
2.2
Spectrum of the radiation
One may use the results we have deduced to calculate the spectrum of the emitted
radiation. This is done by considering the Fourier transform of the dipol p(t)
Z
∞
e−iωt p̂(ω) dω
p(t) =
(2.9)
−∞
remembering that
Z
∞
ω 2 e−iωt p̂(ω) dω
p̈(t) = −
−∞
Taking the transform of the electric field and inserting (2.5) one has
14
(2.10)
∞
Z
Eθ (t)
e−iωt Ê(ω) dω
=
∞
∞
Z
(2.5,2.10)
ω 2 e−iωt p̂(ω)
−
===
(2.11)
−∞
sin θ
dω
c2 r
(2.12)
and therefore
Ê(ω) = −ω 2 p̂(ω)
sin θ
c2 r
(2.13)
Integrating the energy loss (2.8) over the time one finds the energy that crosses a
surface per surface element dA
dE
=
dA
Z
∞
Z
∞
energy flux · dt =
−∞
−∞
c 2
E (t) dt
4π
(2.14)
where we have used that the energy flux is given by the Poynting vector (2.6).
From the theory of Fourier transforms we use
Z
∞
Z
2
∞
2
E (t) dt = 2π
−∞
Z
∞
|Ê(ω)| dω = 4π
−∞
|Ê(ω)|2 dω
(2.15)
0
and therefore
dE
=c
dA
∞
Z
|Ê(ω)|2 dω
(2.16)
0
giving finally the emitted spectrum:
dE
dω
Z
=
(2.13)
==
=
c|Ê(ω)|2 dA
ω 4 |p̂(ω) sin θ|2
dA
c4 r 2
8π ω 4
|p̂(ω)|2
3 c3
(2.17)
Z
c
(2.18)
(2.19)
This shows that in a non relativistic approximation (remember that we assumed ∆v
to be small compared to the velocity of light) the spectrum is given by the square
of the Fourier transform of the dipol moment.
In the following we will deduce the properties of the radiation emitted by different
processes by estimating or calculating the dipol moment and deducing the efficiency
of the process through the energy loss formula of Larmor (2.8) and the emitted
spectrum through (2.19).
(First part)
15
Chapter 3
Bremsstrahlung
This is also called free-free emission. It is the emission that electrons produce when
accelerated in the vicinity of ions.
The classical description of this process starts from the emission of an accelerated
charge as we have derived it in chapter 1. Omitting the constants, we had there for
the energy loss of the charge:
dE
∝ |p̈|2 ,
dt
(3.1)
where p is the electric dipole moment.
Remark: For a collection of charges with identical e/m ratios:
p=
X
ei ri ∼
i
X
mi ri
(3.2)
i
one sees that the total electrical dipole is the same as the center of mass. It follows
that p̈ vanishes in the absence of external forces and therefore that such a system
will not radiate.
One should also note that here (and elsewhere) in a plasma of electrons and ions,
the electrons are accelerated a factor mp /me more than the ions. The radiation
is therefore predominantly emitted by the electrons. This is naturally true of ionelectron acceleration, it is also true when the same force acts on both electrons and
ions, which is the case in all electrodynamic contexts. We will therefore concentrate
in the following on electrons.
The following derivation of the bremsstrahlung spectrum and emissivity is largely
based on Rybicky and Lightman, Radiation processes in Astrophysics.
3.1
Isolated electrons
We can calculate the spectrum emitted during the collision between a single electron
of charge e− and a single ion of charge Ze with trajectories such that the collision
16
Figure 3.1: Bremsstrahlung radiation is emitted by an electron accelerated due to
its Coulomb interaction with another charged particle, usually an ion. The impact
parameter b is the distance of closest approach between the two particles.
impact parameter is b (Figure 3.1). Since the ion is negligibly accelerated in the
process we will consider it fixed.
We derived in eq. 2.19 the emitted spectrum as a function of the Fourier transform
of the dipole:
8π ω 4
dE
=
|p̂(ω)|2 .
dω
3 c3
(3.3)
We must therefore calculate |p̂(ω)|2 . The electric dipole is as usual p = −e r, where
¯
¯
the underscore indicates a 3-vector. Its second derivative is
p̈ = −ev̇,
¯
¯
(3.4)
from which we can calculate the Fourier transform of the dipole. The Fourier transform of eq. 3.4 can be written as:
e
−ω p̂(ω) = −
2π
2
Z
∞
v̇eiωt dt
(3.5)
−∞
which we can now estimate knowing electrostatic forces and the characteristics of
the collision. We introduce τ , the characteristic time of the collision:
b
τ := ,
v
(3.6)
where v is the velocity of the electron. For ω τ1 , i.e. for frequencies that are
large compared to the inverse of the characteristic time, the term exp(iωt) oscillates
rapidly and the integral in 3.5 vanishes. In the other
limit: ω τ1 , ωt vanishes and
R
the exponential is 1 and the integral reduces to v̇ dt ' ∆v. We therefore obtain:
17
e
∆v,
2πω 2
p̂(ω) ∼
¯
0,
¯
if ωτ 1
,
if ωτ 1
(3.7)
which we can insert in 3.3 for the spectrum to obtain:
dE
=
dω
2 e2
|∆v|2 ,
3 c3 π
0,
¯
if ωτ 1
.
if ωτ 1
(3.8)
In order to estimate ∆v we take the case of a large impact parameter. In this case
the acceleration is predominantly perpendicular to the velocity and is given by the
electric force felt by the electron:
∆v⊥
Z ∞
e
Ez dt
= −
me −∞
Z
Ze2 ∞
b
= −
dt
2
m −∞ (b + v 2 t2 )3/2
2Ze2
.
= −
mbv
(3.9)
(3.10)
(3.11)
We can now use 3.6 to express τ in terms of b (ωτ = ω b/v) and insert 3.11 in the
expression for spectrum 3.3 to give
dE
=
dω
3.2
8
Z 2 e6
,
3 πc3 m2 b2 v 2
0
if b ωv
if b ωv .
(3.12)
Electron distribution: the impact parameter
The result obtained in the previous subsection is possibly interesting, it is, however, very far from any physical reality. Indeed in nature we observe macroscopic
plasmas in which the electrons and ions do not come isolated but in large populations described by distributions. The first ensemble we want to consider is one in
which the relative velocity of the electrons has always the same module, but where
a distribution of impact parameters is considered.
The energy emission per unit frequency and per unit time in a volume element dV
is given by
18
dE
dωdV dt
Z
ion density ·
=
dE(b)
2πb db · electron
|
{z flux} · dω
(3.13)
ne v
Z
=
(3.12)
==
=
∞
dE
ni ne v2π
db · b
dω
bmin
Z
16 Z 2 e6 ni ne bmax db
3 c2 m2 v bmin b
16e6 Z 2
bmax
ne ni ln
,
3c3 m2 v
bmin
(3.14)
(3.15)
(3.16)
where we have used 3.12 to express the spectrum emitted in a single interaction of
impact parameter b. bmin and bmax are the boundaries of the integral. bmax is limited
by the condition ω v/b for which the integral in 3.12 vanishes. We therefore use
bmax = ωv .
For very small bmin , the approximation we made of a large impact parameter is not
valid. We will therefore leave this as a parameter and write
√
bmin
16πe6
3
dE
2
= √
ne ni Z gf f (v, ω), where gf f (v, ω) =
ln
. (3.17)
dωdV dt
π
bmax
3 3c3 m2 v
gf f is of the order 1 and cannot be calculated with the method we described here.
It is called the Gaunt factor.
It is important to note that, quite expectedly, the emissivity is proportional to the
square of the density. This process will therefore play a role whenever the densities
are high.
3.3
Electron distributions: Thermal Bremsstrahlung
The next step in estimating the bremsstrahlung of a plasma is to consider a distribution of the velocities of the electrons. We must therefore integrate equation
3.17 over the velocity distribution of the electrons. This distribution can have many
shapes that will depend on the origin of the electrons in the plasma. One particularly relevant distribution is that describing a thermal plasma. The probability that
an electron has a velocity v in a thermal non relativistic plasma of temperature T is
mv 2
dP ∼ e−E/kT d3 v ∼ v 2 e− 2kT dv,
(3.18)
where k is the Boltzmann constant.
We can now integrate equation 3.17 over the velocities and normalise with the integral of the probability distribution of equation 3.18 to obtain
19
dE
(T, ω) =
dV dtdω
R∞
vmin
(v, ω) v 2 e−mv
dv dVdE
dtdω
R∞
v 2 e−mv2 /2kT dv
0
2 /2kT
.
(3.19)
The integration limit vmin is given by the condition 21 mv 2 > ~ω. When this condition is not satisfied, the collision cannot give rise to a photon of energy ~ω. The
integral cannot be solved analytically, be it only because we have in it the function
gf f (v, ω) for which we have no analytical form. We can, however, establish the main
dependencies of the spectrum from an examination of the terms of equation 3.19.
First we note frm equation 3.17 that dVdE
(v, ω) ∝ v1 . The integration will therefore
dtdω
1
∝ T −1/2 . Second, we cannot expect to create
have some term proportional to <v>
photons of energy larger than that of the particles. The integration will therefore
be proportional to exp( −hν
), where ν is the cyclic frequency rather than the angular
kT
frequency. We therefore expect that the integration will lead to
dE
∼ ne ni T −1/2 · e−hν/kT
dV dtdν
(ν =
ω
).
2π
(3.20)
When all the algebra is carried out, one gets
25 πe6
dE
=
dV dtdν
3mc2
2π
3km
−1/2
T −1/2 Z 2 ne ni e−hν/kT ḡf f ,
(3.21)
where ḡf f is the Gaunt factor averaged over velocity, it is a function of the temperature T and frequency ν. The resulting emissivity in c.g.s. units is
fν f = 6.8 · 10−38 Z 2 ne ni T −1/2 e−hν/kT ḡf f
ergs
.
s · cm3 · Hz
(3.22)
hν
The numerical value of ḡf f is 1 < ḡf f < 5 for 10−4 < kT
< 1. More precise values
can be found in the literature. Integrated over the spectrum the emissivity is
ergs
.
(3.23)
s · cm3 ·
ḡB is the integrated Gaunt factor, its value is between 1.1 and 1.5, adopting a value
of 1.2 leads to results precise to about 20%.
f f = 1.410−27 T̄ 1/2 ne ni Z 2 ḡB
The same reasoning that we made here for a thermal electron distribution can naturally be made using other electron distributions that might result from non thermal
physical processes.
3.4
3.4.1
Applications
Clusters of galaxies
Thermal X-ray emission is observed in clusters of galaxies. The temperature of the
gas, expressed in units of energy is of the order of 1-10 keV. The emission is more
20
Figure 3.2: Optical emission from the Coma Cluster of galaxies (Credit: Kitt Peak).
or less regular in the clusters, depending on how virialised or relaxed the cluster
actually is. The mass of the gas far exceeds that of the sum of the individual
galaxies. Figure 3.2 shows the Coma cluster as seen in the optical domain. Clearly,
the emission is dominated by that of the galaxies in the cluster. In Figure 3.3 we
show the same cluster but observed in the X-rays with XMM-Newton. The galaxies
are not seen anymore, their emission in this domain is negligible, the emission is
dominated by a smooth component that extends all over the cluster. The shape of
the spectrum can be used to deduce the temperature of the gas. In this case one
finds a temperature of kT = 8.25 keV (Arnaud M. et al. 2001, A&A 365, L67).
Figure 3.4 shows an early X-ray spectrum of the Perseus cluster obtained by the
HEAO-A1 instrument. The continuum is well described by a thermal emission
of kT ' 6.5 keV. Striking on this plot is the enhanced emission at two energies
compared to the smooth continuum. This enhanced emission is due to emission lines
created by the presence of Fe 25 times ionised (Fe XXVI). These lines correspond to
the Lyα and Lyβ lines of Fe in its one electron configuration. Indeed the line energy
is proportional to Z 2 and falls for Z = 26 where observed.
This observation leads immediately to the conclusion that the cluster gas is not
primordial. Only H and He were produced during the Big Bang nucleo-synthesis
in significant amounts. All other elements have been synthesised in the interior of
stars. Thus the presence of Fe in the cluster gas implies that the gas has been
processed by the stars of the galaxies.
Present day observations in particular with the XMM-Newton satellite show much
more details than shown in Figure 3.4. Figure 3.5 shows the observed low energy
spectrum of the cluster A 2052. From these data it is possible to deduce the abundance of the elements as a function of the distance to the centre of the cluster as
well as the temperature also as a function of the distance (Figures 3.7 and 3.6).
From these observations one sees that the central temperature is less than that of
the outskirts of the cluster. This is an immediate consequence of the dependence
21
Figure 3.3: XMM-Newton observations of the Coma Cluster (U. Briel, MPE Garching, Germany and ESA).
Figure 3.4: X-ray spectrum of the Perseus cluster from HEAO-A1 instrument. A
model of the emission as thermal bremsstrahlung emission of gas at about T =
6.5×107 K is shown. This high temperature is confirmed by the presence of emission
lines, due to highly ionised iron, F e+25 at energies of 6.7 and 7.9 keV. This high
temperatures are indeed required to ionise Fe so highly (Fabian et al 1981, ApJ,
248, 47).
22
Figure 3.5: XMM-Newton spectrum of A 2052 in the inner three shells (Kaastra J.S.
et al. 2004, A&A 413, 415). The spectra of the 0.5-1.00 and 1-20 shells have been
multiplied by factors of 5 and 25, respectively. The spectra are shown as energy
times counts/s/keV.
Figure 3.6: (a) Surface brightness, (b) electron density, (c) temperature, and (d)
pressure, as a function of radius deduced from Chandra observations of the A 2052
cluster (Blanton E.L. et al. 2001, ApJ 558, L15). The vertical dashed lines mark
the mean inner and outer radii of the bright X-ray ring.
23
Figure 3.7: Iron abundance profiles measured in the four nearby galaxy clusters, M
87/Virgo, Perseus, Centaurus, and A1795 as measured by XMM-Newton (Böhringer
et al. 2004, A&A 416, L21). The values are in solar units based on the solar abundance of iron quoted by Feldman 1992. The dashed line shows the iron abundance
with a value of ∼ 0.2 solar, observed on large scale in clusters and assumed to come
mostly from SN II enrichment before cluster formation.
24
Figure 3.8: The Perseus cluster as observed by Chandra. Modern images show that
the intra-cluster hot gas is not homogeneous, but has a very significant amount of
structure. The central galaxy of Perseus is an active galaxy that is probably injecting
very large amount of energy in the gas and stops it to cool through bremstrahlung
emission in the central dense regions.
on the square of the density of the emissivity. The central regions are denser and
thus cool faster through bremsstrahlung than the outside regions. This has led to
a long standing debate. The gas seems to cool with a characteristic time that is
considerably less than the age of the Universe. There should therefore be substancial amounts of cold gas in the central regions of clusters that should be observable
in some form. This gas has, however, never been seen, nor have stars that could
result from the presence of this gas been observed. This long standing "cooling
flow" problem is now still at the centre of the research. One has observed that the
structure of the clusters may be quite a bit more complex than early observations
led one to think. Shocks are observed that may well be created by the interaction
of the cluster gas with the active galaxies often in the central region of the clusters.
This could lead to additional heating of the gas and to the relief of the cooling flow
problem as shown in figure 3.8 which shows the perseus cluster seen by the Chandra
telesope and the VLA image of the central active galaxy NGC 1275.
These data may also be used to deduce the mass of the clusters. The mass of
the luminous matter in the galaxies is deduced from the optical luminosity of the
galaxies. The mass of the hot gas is deduced through the measurement of the gas
temperature and that of the luminosity. Equation 3.23, that gives the gas emissivity,
allows one to deduce the density. With the size given by the images one can deduce
the mass of the gas. A further mass can be deduced from the gas temperature.
Indeed in order to bind the gas of a given temperature, the gravitational field must
be so that the thermal velocity of the gas is less than the escape velocity. The mass
measured in this way is the total gravitational mass of the cluster. Figure 3.9 gives
as a function of the centre from the cluster these 3 mass estimates. One sees that
the mass of the galaxies is about an order of magnitude less than that of the hot gas,
25
Figure 3.9: Integrated radial mass profiles for the Perseus cluster of galaxies
(Boehringer, H. 1995, Reviews in Modern Astronomy, v.8, p.259-276). Shown are
the gravitational mass, the gas mass and the galaxy mass profile. For the first two
profiles the upper and lower limits are given. For the galaxy mass profile the luminosity profile was converted into a mass profile by assuming a mass to light ratio for
the galaxies of 5 in solar units.
which is then again much less than the total gravitational mass. This is a powerful
illustration of the dark matter problem. Indeed the largest fraction of the matter in
clusters is convincingly shown to be in forms that are unobserved.
3.4.2
Line emission
We have discussed at length free-free emission or bremsstrahlung. It is, however,
also important to note that at intermediate temperatures the ions are not completely ionised. Some electrons remain bound to the nuclei. Transitions between
bound states of the ion or between bound and unbound states are therefore possible.
These transitions lead to lines and edges in the emission of the plasmas. At low temperatures, few states can be excited, the lines are therefore not very important. At
high temperatures, the elements are completely ionised and the lines are again not
of overwhelming importance (see e.g. the spectrum of the cluster above, Figure 3.5).
However, in-between the contribution of the lines to the total emissivity of a plasma
may not be neglected at all. Figure 3.10 shows the spectrum of a plasma of T ∼ 106
K.
Figure 3.11 shows the emissivity of both the lines and the continuum, while Figure 3.12 shows the ratio of both components. It is easily seen that the lines may be
up to 25 times more efficient to cool a plasma than the continuum at temperature
of about 106 K. This is precisely the temperature of plasmas observed in the low
energy X-rays where now very high sensitivity can be achieved. Plasma diagnostics
is thus a domain that is very active now. Interestingly one of the greater difficulties
26
Figure 3.10: Simulated spectrum for a plasma with temperature of T = 106.2 K and
electron density of 1010 cm−3 .
Figure 3.11: Contribution of the thermal bremsstrahlung continuum and of the line
emission as a function of temperature in a plasma.
27
Figure 3.12: Ratio of the continuum and lines components shown in Figure 3.11.
is due to the lack of precise knowledge of the electronic structure of the very high
number of ions that contribute to this emission.
3.4.3
The diffuse X-ray background
The first rocket flight meant to observe the sky beyond the Sun in the X-ray domain has led to the discovery of a diffuse emission (in addition to the discovery
of the first bright source). This emission could be relatively well represented by a
bremsstrahlung fit of a 45 keV plasma, hence the place of this section in this chapter.
It was, however, immediately clear that it would be very difficult to understand how
a gas of this temperature could be heated and distributed throughout the space. A
ROSAT observation of the Moon shows a very vivid illustration of the extragalactic
background. One indeed immediately sees from figure 3.15 that the dark side of the
Moon is "darker" in soft X-rays than the outside regions, and thus that the diffuse
emission must come from beyond.
Figure 3.15 shows a ROSAT observation of the Moon in which one sees the clear
presence of the Sun lit Moon where fluorescence is observed, but also the fact that
the dark side of the Moon is darker in X-rays than the sky. This immediately shows
that there is an apparent diffuse emission that comes from beyond the Moon.
Figure 3.13 shows the spectrum as observed by a number of missions while Figure 3.14 shows a compilation of the background emission as observed over the entire electro-magnetic spectrum. From the latter one can see that the high energy
emission is an important component, much less, however, than the micro-wave background.
It has been suggested by Setti & Woltjer (1970, Ap&SS 9, 185) that the diffuse
extragalactic emission in the X-rays, might not be diffuse in the end at all, but rather
the sum of many very weak sources, individually not detected. This explanation
would considerably relieve the problem of the origin of a 45 keV plasma that would
prevade the complete Universe.Subsequent very long observations in the Lockman
hole (a region of very low absorption perpendicular to the plane of the Galaxy), first
28
Figure 3.13: Multiwavelength spectrum of the extragalactic background spectrum
from X-rays to high-energy gamma rays (Sreekumar P. et al. 1998, ApJ 494, 523).
Dot-dashed line is the estimated contribution from Seyfert I, dashed line from Seyfert
II, triple-dot-dashed line from steep-spectrum quasars, dotted line from Type Ia
supernovae, long-dashed line from blazars. The thick solid line indicates the sum of
all the components.
Figure 3.14: The overall cosmic energy density spectrum (νIν vs ν): a compilation of most recent datasets, from microwave to high energy gamma rays
(http://www.bo.astro.it/).
29
Figure 3.15: This image of the Moon was taken by the ROSAT PSPC on 29 June
1990. Black pixels denote no counts. The sunlit portion of the Moon is visible, as
well as a distinct X-ray shadow in the diffuse X-ray background cast by the dark
side of the Moon (Schmitt et al. 1991, Nature, 349, 583).
with ROSAT (Figure 3.16, left panel) and then with XMM-Newton (Figure 3.16,
right panel), have shown that indeed the "diffuse" background is the superposition
of many very weak sources in the soft X-rays. The sources are a number of Active
Galaxies. While this solves the question at soft X-rays, it does not yet solve it at the
harder energies where the diffuse emission is strongest. Indeed taking the spectral
energy distribution of active galaxies as we observe them in the nearby Universe
and superposing them leads to a completely different spectrum compared to that
observed. In order to describe the background asa superposition of weak objects
at high energies it is necessary to that there exists a population of weak active
galaxies in which the emission is considerably modified by absorption. Discovering
this population is a major challenge in which INTEGRAL observations are playing
an important role.
30
Figure 3.16: The Lockman Hole region seen by ROSAT (left panel, Hasinger G.
et al. 1998, A&A 329, 482) and XMM-Newton (right panel, credit ESA). Several
diffuse sources with red colours in XMM image are X-ray clusters of galaxies already
identified by ROSAT data. But XMM-Newton clearly reveals a number of green and
blue objects and these correspond to obscured faint sources.
31
Chapter 4
Cyclotron line emission
A charge in a magnetic field follows a movement which is circular perpendicular to
the axis of the magnetic field and free along the magnetic field (see Figure 4.1).
The charge is therefore accelerated and according to chapter 2 will radiate electromagnetic waves. When the movement is not relativistic, one speaks of cyclotron
emission.
The rotation frequency of the charge perpendicular to the field axis is described by
the angular frequency
ωB =
eB
,
γmc
(4.1)
Figure 4.1: The general path of a moving charge in a constant magnetic field is
that of a helix with its axis parallel to the direction of the magnetic field (Credit:
Richard Vawter).
32
Figure 4.2: Deconvolved X-ray spectrum of the Her X-1 pulsar (balloon observations
on 1976 May 3). Solid line, best-fitting exponential spectrum with a Gaussian line
to keep into account the line around 40 keV. For comparison, a total X-ray spectrum
of Her X-1 observed by OSO-8 during the 1975 August on-state is shown (Trümper
et al. 1978, ApJ 219, L105).
where m is the mass and e the charge of the particle. γ is the relativistic factor and
is 1 in the non relativistic case that we consider here. The frequency of eq. 4.1 is
called the Larmor frequency. A practical way to express eq. 4.1 in units that are
relevant for us is
Ec = ~ωB = 11.6 ·
B
1012 G
keV
(4.2)
During a balloon flight in 1976, Trümper et al. (1978, ApJ 219, L105) observed
the hard X-ray spectrum of Her X-1 in which a feature is clearly detected around
40 keV (Figure 4.2). Her X-1 is a well known X-ray pulsar (see lectures on X-ray
binaries for more explanations on these objects). Intepreted in terms of atomic
transitions, the line energy would necessarily imply elements way above Fe, because
the Lyman transitions of the Hydrogen like ion are around 7 keV. The energy of the
Hydrogen like transitions is proportional to Z 2 , a line around 40 keV might thus be
coming from Pt, a most unlikely element to be present in large quantities. A nuclear
transition is known at these energies from 241 Am, also a very unlikely element to
be present in large quantities in an optically thin environment necessary for us to
observe the transition. The most natural explanation for the feature observed in Her
X-1 is therefore in terms of a cyclotron transition in a B-field of some 3 × 1012 G as
33
JEM−X
ISGRI
−5
0
5 −5
0
5
Photon index = 0.95 (+/−0.2)
Cutoff energy= 25.2 (+/−1.0) keV
Fold energy = 8.8 (+/− 0.3) keV
Cycl. line centroid = 38.5 (+/−0.7) keV
Cycl. line width = 8.6 (+1.1/−1.2) keV
Cycl. line depth = 0.66 (+0.09/−0.10)
Iron line centroid = 6.31 (+/−0.2) kev
Iron line width = 0.8 (+/− 0.2) keV
5
10
20
Channel energy (keV)
50
Figure 4.3: INTEGRAL spectrum of Her X-1 with a complete spectral analysis from
Klochkov et al. 2007 (submitted)
given by eq. 4.2. The exact value of the field is not secure with the data presented
in Figure 4.2 as it is not possible to know from these data whether the line is an
absorption line or an emission line at a slightly higher energy. More recent studies
show that the line is in absorption (Fig./,4.3.
Although a field of the order of 1012 G may sound at first as improbable as atomic
transitions of Pt or nuclear transitions of Am, it is worth noting that neutron stars
(which are one of the component of an X-ray pulsar) are the product of stellar
collapse. Remembering that the magnetic flux is a conserved quantity, one can
estimate the field of the remnant of the collapse through
2
r 2
7 · 105
3
∼ 10 ·
= 5 · 1012 G,
B = B0 ·
10 km
10
(4.3)
where we have taken a field of 103 G as the initial field of the star and the size
of the Sun as the initial radius. This simple argument shows that fields of the
order of 1012 G are as plausible around compact objects as fields of 103 G in stellar
environments.
The power emitted by an electron in a magnetic field can be calculated from its
acceleration and the formula derived in eq. 2.8.
From the Figure 4.4 one sees that the acceleration of the electron in a helicoidal
= v · ω.
motion of angular frequency ω is dv
dt
The emitted power is
dE 2 e2 |v̇|2
P = =
¯ .
dt
3 c3
34
(4.4)
r1
v1
dθ
r2
v2 dθ
v1
dv
Figure 4.4: A particle moving with speed v (|v1 | = |v2 | = v) from r1 to r2 along a
¯
¯ |v − v | = ¯|dv| '
¯ vdθ and
circular orbit. By simple geometry, one can derive
that
2
1
¯
¯
¯
= v · ω.
then, using that dθ = ωdt, dv
dt
with ω = ωB , |v̇| = ωB · v one finds:
2
2 e2 eB
P =
· v2
3
3c
mc
4
2 e B 2v2
=
2 c4
3 |m{z
} c
(4.5)
(4.6)
r02
=
2 2 2 2
cr B β ,
3 0
(4.7)
where we have introduced the classical electron radius r0 .
A quantum mechanical treatment of the process starts from the Hamiltonian
e 2
1 p− A
where B = ∇ × A
H=
2m ¯ c ¯
¯
¯
in which the impulse p is replaced by the corresponding quantum operator:
p → p̂ = −i~∇
¯
¯
(4.8)
(4.9)
leading to
Ĥ =
e 2
1 p̂ − A .
2m ¯ c ¯
(4.10)
For a B field parallel to the z-axis the vector potential A is
Ax = −B · y, Ay = B · x, Az = 0
and the Hamilton operator becomes
35
(4.11)
Ĥ =
1 e 2
e 2
1 p̂2
p̂x − By +
p̂y + Bx + z .
2m
c
2m
c
2m
(4.12)
The trajectories corresponding to this Hamiltonian are made of a circular mouvement in the plane perpendicular to the magnetic field and constant velocity arallel
to the field. The energy levels are quantised and given by the Eigenvalues of the the
Schrödinger equation:
Ĥψ = Eψ
(4.13)
p2
1 e~B
+ z
En = n +
2 mc
2m
(4.14)
with the solutions
for the energy values. The transitions between the levels described by 4.14 occur
p2z
therefore at multiples of e~B
+ 2m
.
mc
When a cyclotron line is observed at a given energy E, one therefore expects to
see also a feature at 2E and possibly at higher multiples. Indeed this may be the
case already in the data presented in Figure 4.2. In more recent times further
observations of X-ray pulsars have been performed, e.g. with INTEGRAL in which
the cyclotron lines are very clearly seen, not only at the lowest energy but also at
one or more multiples of it, thus confirming the nature of the observed transitions
(see Figure 1.2).
Recent work tends to explain the observed transitions as absorption features. The
continuum radiation is produced close to the neutron star in the accretion flow. The
radiation field is then partially absorbed at the cyclotron frequency and its multiples. The structure of the magnetic field and the radiation transfer are, however,
extremely complex and modelisation work is still needed to fully understand the
geometry of the problem.
Very recently Bignami et al. (2003, Nature 423, 725) have reported XMM-Newton
spectra of the isolated neutron star 1E1207.4-5209 (Figure 4.5). This neutron star
is not in a binary system as the previously discussed X-ray pulsars, the observed
radiation is that of the hot surface of the star. The X-ray spectrum shows deep absorption features at 0.7, 1.4 and 2.1 keV which the authors interpret as the signature
of a magnetic field of 8 × 1010 G. This is the only isolated neutron star for which
this phenomenology has been observed.
Observations in the 1990’s by GINGA (a Japanese X-ray satellite) claimed detections
of cyclotron features in the emission of gamma ray bursts. These features could be
seen only during a short portion of the bursts. These observations have not yet been
confirmed by other satellite observations.
36
Figure 4.5: Spectra of 1E1207.4-5209 collected by two cameras on board of the
XMM-Newton satellite during August 2002. Data points and best fitting continuum
spectral models are shown, together with residuals in units of standard deviations
from the best fitting continuum. Three absorption features are visible at energies of
0.7, 1.4 and 2.1 keV (Bignami et al. 2003, Nature 423, 725).
37
Chapter 5
Synchrotron emission
One speaks of cyclotron radiation when the electron moving in a magnetic field is not
relativistic. When the electrons are relativistic and are moving in a magnetic field,
they are also accelerated and thus radiate. In this case one speaks of synchrotron
radiation.
This is a very important form of radiation in astrophysics. It is often observed in
the radio domain, but in some extreme cases one sees this form of radiation up to
the X- and gamma-ray domains. This is observed in active galactic nuclei (see later)
and in particular radio galaxies, in blazars and probably in gamma ray bursts. The
force acting on a particle is proportional to its charge, the acceleration therefore
inversely proportional to the mass of the particle. Electrons therefore dominate in
most situation the radiation and will be exclusively discussed here.
5.1
Radiation of a relativistic accelerated particle
We saw in the first lecture how a non relativistic accelerated charged particle radiates. We need here first to see how this result can be generalised to a relativistic
particle. We start with the special relativistic metrics
ds2 = c2 dτ 2 = c2 dt2 − dx2
¯
(5.1)
which describes the distance between two events in space time. This distance is
invariant under Lorentz transformations (left to the reader)
v 0
t = γ t − 2 x , x = γ(x − vt), y 0 = y, z 0 = z.
c
0
(5.2)
We next introduce the 4-velocity
uµ =
dxµ
,
dτ
(5.3)
which, as a small difference between coordinates is a vector. Written explicitly:
38
u0 =
dx0
dt
=c
= γ · c,
dτ
dτ
(5.4)
because
dτ 2
1 2
1
v2
2
2
( ) = (dt − 2 dx )/dt = 1 − 2 = 2 .
dt
c ¯
c
γ
(5.5)
dx
u = ¯ = γ · v,
¯
¯ dτ
(5.6)
Similarly
as
dτ 2
1
1
1 1
1
) = (dt2 − 2 dx2 )/dx2 = ( 2 − 2 )ev = 2 2 ev ,
(5.7)
dx
c ¯
v
c ¯
v γ ¯
¯
¯
where ev is a unit vector in the direction of the velocity. Note that we will write v
¯ 3-vectors. The acceleration is the second proper time derivative:
¯
to mean
(
aµ =
duµ
dτ
(5.8)
dγ
dτ
(5.9)
which from 5.4 and 5.6 gives
a0 = c ·
ai =
d(γ · v i )
.
dτ
(5.10)
In the system in which the particle is at rest, we have
γ = 1, dτ = dt ⇒ a0 = 0, ai =
dv i
.
dt
(5.11)
In this system, the non relativistic derivation we had made in the first lecture is
valid and we know the radiation of the particle:
2 |v̇|2
2 e2
P = e2 ¯3 =
~a · ~a
3 c
3 c3
(5.12)
To obtain the last equality in eq. 5.12, we have used eqs 5.8, 5.9 and 5.10. The
formulation
P =
2 e2
~a · ~a
3 c3
39
(5.13)
is, however, valid in all systems of reference. In order to convince yourself of this,
consider the transformations of the left part of the equality under Lorentz transformations:
Energy → γ · Energy
∆t → γ · ∆t
(5.14)
(5.15)
dE
dE
→
.
dt
dt
(5.16)
therefore
This is clearly a scalar. The right side of the equality is a scalar product and thus
also a scalar. The equality is therefore indeed valid in all systems of reference and
corresponds to the relativistic generalisation of the Larmor formula 2.8 that we were
seeking.
We can now write eq. 5.13 in the system of the observer in which the particle is
moving relativistically explicitly:
" 2 #
2
d(γ · v)
2 e2 2 dγ
c
−
P =
¯
3 c3
dτ
dτ
(5.17)
dγ
γ 3 dv
= 2v· ¯
dτ
c ¯ dτ
(5.18)
Doing the algebra one obtains:
2e2 6
P =
γ − v̇ ·
3c3
¯
v 2 |v̇|2
¯ − ¯2
c
γ
(5.19)
and
1 2
2e2 6 2
|P | =
γ ak + 2 a⊥
3c3
γ
(5.20)
where we have introduced (v · v̇) = v · ak and |v × v̇| = v · a⊥ , the components of
¯
¯ velocity,
¯
the acceleration parallel and¯perpendicular
to the
and used |v̇|2 = a2k + a2⊥ .
¯ particles or
The sign is naturally different if one considers the energy lost by the
that gained by the rotation field. It must be set accordingly. Thus
P =
2e2 4 2
γ (a⊥ + γ 2 a2k ).
3
3c
40
(5.21)
5.2
Power emitted by a single particle in a magnetic
field
In the specific case that we consider here, the electron has a helicoidal movement with
an angular frequency ωB in the magnetic field and correspondingly an acceleration
a⊥ = ωB · v⊥ , and ak = 0. Eq. 5.21 becomes then (with as usual β = vc ):
P =
2e2 4 e2 B 2 2 2
γ
β c.
3c3 γ 2 m2 c2 ⊥
(5.22)
For a distribution of velocities that is isotropic, we have
hβ⊥2 i
1
=
4π
Z
(β sin α)2 dΩ
hβ⊥2 i =
2β 2
3
(5.23)
(5.24)
and therefore
Psync
where uB =
B2
8π
4 e4 γ 2 β 2 B 2
=
9 c3 m2
1
=
σT cγ 2 β 2 B 2
6π
4
=
σT cγ 2 β 2 uB
3
(5.25)
(σT =
8π e4
)
3 m2 c4
(5.26)
(5.27)
is the energy density of the magnetic field (B in Gauss).
We have introduced the Thomson cross-section σT and the magnetic field density
uB to express the power emitted by the relativistic electron. We can use eq. 5.25 to
estimate the time an electron needs to lose a significant fraction of its initial energy:
tcool
5.3
Ee
γme c2
:=
=
≈ 5 · 10−8 B −2 γ −1 s
P
P
(5.28)
Synchrotron spectrum
In order to understand the shape of the spectrum emitted by a population of electrons which velocities are isotropically distributed we first consider the movement
of a single electron and the time during which the electron is observable along its
path. Indeed the radiation emitted by a charge moving at relativistic velocities is
bundled in a cone of half opening angle 1/γ (Figure 5.2, left). This relation giving
the opening angle of the cone in which the observer sees the radiation from the
41
moving lectrons is deduced from the Lorentz transformations in the following way
(see fig.):
x0
y0
z0
t0
γ(x − vt)
y
,
z
γ t − cv2 x
=
=
=
=
(5.29)
where the "’" system, the electron system, moves with a velocity v along the x-axis
with respect to the non-primed system, the observer system.
The transformation of the velocities is given by
γ(dx0 + vdt0 )
u0x + v
dx
=
=
0
x
dt
γdt0 (1 + cv2 u0x )
1 + vu
c2
u0y
dy
dy 0
uy =
=
=
0
x
dt
γdt0 (1 + cv2 u0x )
γ(1 + vu
)
c2
ux =
uz =
dz 0
u0z
dz
=
=
0
x
dt
γdt0 (1 + cv2 u0x )
γ(1 + vu
)
c2
(5.30)
(5.31)
(5.32)
which gives the following relations between the angles of the velocities in both systems:
tan θ :=
u0y
u0 sin θ0
uy
=
=
ux
γ(u0x + v)
γ(u0 cos θ0 + v)
(5.33)
If we now consider a photon emitted by the electron, u0 = c and then:
sin θ0
γ(cos θ0 + vc )
tan θ =
(5.34)
y'
photon
u' = uy' = c
e- at rest
observer
x'
vx = -v
Figure 5.1: The electron at rest in the unprimed system of reference emits a photon
along the y-axis, perpendicular to the velocity of the observer. The photon will be
seen at an angle 1/γ by the observer in his rest reference system.
42
and for photons emitted perpendicularly to v:
tan θ =
1
γβ
(5.35)
as expected.
The observer will therefore register a pulse of radiation while the electron covers the
section of arc ∆S = a · ∆θ with ∆θ = γ2 . Thus (see Figure 5.2, right)
∆S =
2a
.
γ
(5.36)
The equation of movement of the electron, needed to compute the acceleration (i.e.
∆v, and hence ∆θ here), in the magnetic field is given by
∆v
e
e
γm| ¯ | = |v × B| = v · B sin α,
∆t
c ¯ ¯
c
(5.37)
where we have used that since the force is perpendicular to the velocity γ is constant.
2 ∆θ
With |∆v| = |v∆θ| and ∆S = v∆t we have ∆v
= v∆S
. Inserting the equation of
∆t
¯
motion 5.37 then gives
γmcv
v
∆S
=
=
∆θ
eB sin α
ωB sin α
and with ∆θ =
(5.38)
2
γ
∆S =
2v
.
γωB sin α
(5.39)
∆θ
a
1/γ
Observer
∆s
Figure 5.2: Left: The electron moves helicoidally in a magnetic field emitting synchrotron radiation in a cone of half opening angle 1/γ in the direction of the motion.
Right: only the photons emitted while the electron in ∆t covers the section of arc
∆S will reach the observer who will register a pulse of radiation during a time ∆t0 .
43
The pulse thus lasts a time
∆t =
∆S
2
=
.
v
γωB sin α
(5.40)
This time is not the one during which an observer at rest will measure the light
pulse, because the photons emitted as the electron enters the arc during which it is
visible will travel while the electron progresses. The time during which the observer
"sees" the electron is given by (see Figure 5.3, left)
c∆t0 = c∆t − v∆t ⇒ ∆t0 = (1 − β)∆t '
1
∆t.
2γ 2
(5.41)
Inserting the expression for ∆t we found in eq. 5.40 gives finally
∆t0 =
1
.
sin α
(5.42)
γ 3 ωB
If we then look at the time dependence of the intensity of radiation registered by
the observer one finds that given in Figure 5.3 (right).
Fourier tranform theory tells us that the corresponding spectrum will include frequencies up to ∆t1 0 . We therefore introduce a characteristic frequency νc with
I
∆t0 =
A
B
1
γ3ωBsinα
O
t
v (tB- tA) = v ∆t
c (t'0- tB)
2π
ωB
c (t0- tA)
Figure 5.3: Left: The photons emitted by the electron in A and B, at time tA and
tB , reach the observer at time t0 and t00 , respectively. The duration of the pulse
seen by the observer (∆t0 = t00 − t0 ) is the result of the travel time of the photons
and of the contemporary motion of the electron. Right: Time dependence of the
intensity (arbitrary units) of the synchrotron emission from a single electron seen
by an observer. The pulses have duration ∆t0 = γ 3 ωB1sin α and period 2π/ωB .
44
Φν
arbitrary
units
ν/νC
Figure 5.4: Synchtrotron spectrum emitted by a single relativistic electron moving
in a magnetic field. The emission peaks around the characteristic frequency νc .
νc =
ωc
1 3
1
=
γ ωB sin α [∼
]
2π
2π
∆t0
1 2 eB
=
γ
sin α.
2π me c
(5.43)
(5.44)
The spectral shape of the emission by a single electron will therefore have a peak
around νc , as it is shown in Figure 5.4. We thus know at what characteristic frequencies an electron of given energy radiates (5.44) and we know with which luminosity
it does so from eq 5.25 and how long it takes to radiate a substancial fraction of its
energy, from 5.28. Combining these results we obtain
tcool =
γ≈
3γme c2
(using 5.28)
4σT cγ 2 β 2 uB
4πme cνc
3eB
(5.45)
1/2
(using 5.44)
(5.46)
or
tcool [s]
3me c2
=
4σT cβ 2 uB
"r
4πme cνc
3eB
#−1
−3/2 −1/2
≈ 6 · 108 B[G] ν[M Hz] .
(5.47)
which shows that the electrons emitting at higher energies cool faster.
5.4
Introduction to Active Galactic Nuclei
Active Galactic Nuclei (AGN) and in particular the quasar 3C 273 are often used in
the course of these lectures. It is therefore useful to spend a few lines at this point
to introduce them.
Activity in galaxies has been noted for a long time. Seyfert in the 1940’s noted that a
number of galaxies have a bright nucleus. These gaalxies are called Seyfert galaxies.
45
Figure 5.5: Early optical spectrum of 3C 273 showing the redshifted Balmer series.
It was also soon noted that the nuclei have bright emission lines (remember that
"normal" stars have absorption lines) that may be broad indicating that the line
emitting material has large velocities of few· 104 km/s or narrow indicating velocities
of the order of thousand km/s. The first were called Seyfert 1 galaxies, the second
Seyfert 2 galaxies.
Completely independently, in the 1960’s, it became possible to localise radio sources.
It was found that several bright radio sources were coincident on the sky with starlike objects. These objects were found to have bright emission lines of unknown
origin and were called quasi-stellar objects (QSOs) or quasars. M. schmidt in 1963
(Nature) identified the emission lines of 2 of these objects including the 273rd object
of the 3rd Cambridge catalogue of radio sources (3C 273) as very red-shifted lines of
the Hydrogen Balmer series. The redshift of 3C 273 was 0.158, the then largest ever
observed redshift. This immediately indicated that the objects were at cosmological
distances and therefore that their luminosities must be very large, indeed much
larger than that of whole galaxies.
It then took a long time to see that the two types of objects were of the same
nature. In Seyfert galaxies one sees a galaxy with a bright core, in quasars, the
core dominates so much that it is difficult to observe the underlying host galaxy in
the wings of the point spread function of the telescopes. Indeed the difference is
only semantic now and blow a luminosity of about 1044 ergs/s one speaks of Seyfert
galaxy and of quasars above.
The family of active galactic nuclei has many more members with properties that
can vary greatly but are all charaterised by large luminosities, most of the time very
46
Figure 5.6: One century of photographic observations of C 273 showing large amplitude variations on many timescales (from Angione and Smith, 1985, AJ 90, 2474).
rapid variability (see Fig. 5.6 and spectral energy distributions that span a much
larger part of the electromagnetic spectrum than stars or galaxies (see Fig.:5.7).
They also emit jets that were first seen as radio structures and then seen in the
optical (see Fig.: 5.8).
The basics of AGN can be understood very simply. The fact that objects vary
significantly on timescales of months (and even a lot shorter as we will see) indicated
very early on that the objects must be smaller than light-months, i.e. much smaller
than the distance between stars in a galaxy. The energy source cannot be due to
anything but gravity, as the luminosities are extremely large and we will see that
the energy that can be gained from accretion in a deep gravitational well exceeds
greatly the one that can be obtained from nuclear reactions. Knowing this, the mass
of the compact object may be estimated from the following argument. When matter
falls and liberates energy that is radiated in the form of electro-magnetic radiation
the matter is subjected to 2 forces, the gravitational attraction
Fg =
GM mp
r2
(5.48)
and the radiation pressure exerted by the luminosity:
Fl = σT · photonf lux · impuls transf er per interaction = σT ·
L
hν
·
. (5.49)
2
4πr hν c
Both forces have the same radial dependence. When the radiation pressure dominates at one radius it will therefore dominate at all radius and the matter will not be
accreted, rather it would be expelled. It follows that there is a limiting luminosity,
the Eddington luminosity LEdd that cannot be exceeded in accretion processes:
LEdd =
4πGM mp c
M
' 1.31038
ergs/s.
σT
M
47
(5.50)
Figure 5.7: Spectral energy distribution of 3C 273 as collected over 2 decades by
ISDC scientists.
Quasars can have luminosities all the way up to some 1048 ergs/s, implying that
the compact object onto which matter is accreted has a mass that can be of the
order of some 1010 M . The mass accretion rate Ṁ can also be estimated in a very
straightforward way assuming that a fraction η of the accreted rest mass is liberated.
In this case the luminosity of the object is:
L = η Ṁ c2 .
(5.51)
For accretion onto a black hole η is of the order of 10% (see later) and a healthy
quasar will have accretion rates of the order of Ṁ =' 1028 g/s ' 100 M /year.
48
Figure 5.8: An early observation of the jet of 3C 273 with the Kitt Peak telescope
(from wikipedia).
5.5
5.5.1
the infrared emission of the quasar 3C 273
the presence of hot dust
Observations of the quasar 3C 273 in 1986 showed very unexpected results in this
respect. Figure 5.9 shows this spectrum at several epochs in early 1986 compared
with previous observations. It can be seen that the near infrared emission remained
stable while the lower frequency emission was decreasing substancially. This cannot,
we have just seen, be understood in terms of synchrotron radiation. Therefore, if we
follow the commonly accepted view that the radio and mm emission of quasars (see
later) is due indeed to synchrotron emission, we have to admit that the near infrared
emission is not the high energy tail of that component, but of a very different origin,
which the authors of this result suggested to be due to thermal dust. A very recent
multi-wavelength spectrum of 3C 273 taken while the synchrotron emission was at
the lowest recorded level to date confirms this result very nicely (see Figure 5.10).
5.5.2
estimate of the magnetic field in the synchrotron emitting region
Using quantitatively eq. 5.47 during a synchrotron flare observed in the same quasar
in 1988 (see Figure 5.11) it is also possible to obtain a quantitative estimate of the
magnetic field. Observed are the frequency of the radiation and for this flare the
cooling time of the electrons, measured as the flux decrease time at that frequency.
It was thus possible using eq 5.45 and 5.46 to see that the electrons radiate with a
γ factor of about 1000 and that the magnetic field in the emission region is of the
order of 1 Gauss (note that this estimate assumes that the emission region is not
49
Figure 5.9: Millimetre to optical spectrum of 3C273 (Robson et al. 1986, Nature
323, 134): (a) the "quiescient" spectrum (reported as a dashed line in (b)-(d)); (b)
4-7 March 1983, flare spectrum; (c) 15-24 February 1986; (d) 3-6 March 1986.
moving with a bulk relativistic motion, see below).
5.6
Spectrum emitted by a population of electrons
Whereas we could already gain interesting physical insight in the nature of AGNs by
knowing the emission of single electrons, we do not yet have the tools to understand
the spectrum emitted by a source through synchrotron emission. In order to do this
we must consider how a population of electrons radiates.
We can always express the synchrotron spectrum emitted by a single electron as
4
Pν (γ) = β 2 γ 2 cσT uB φν (γ), with
3
Z
∞
φν (γ) dν = 1
(5.52)
0
(see the power emitted by a single electron). The shape of φν (γ) is in rough terms
generally known from our previous discussion (see Figure 5.4), we know that it peaks
at the characteristic frequency
50
Figure 5.10: Spectral energy distribution of 3C 273 in June 2004 as observed with
INTEGRAL, RXTE and XMM-Newton satellites and La Silla, IRAM, Metsähovi,
UMRAO and Effelsberg telescopes (Türler et al. in preparation). The June data are
compared with the historic average (grey line) and the observed range of variation,
showing that in June the quasar was at its weakest ever in the mm band.
Figure 5.11: Light curve at infrared energies (H band) for the quasar 3C 273 (left
panel) over more than ten years. In the right panel, a zoom of the flare occured in
1988 shows a variability of the emission on timescales of a day.
51
νc =
1 2 eB
γ
sin α
2π me c
(5.53)
from eq 5.44. We next consider a population of electrons distributed in energy
according to
n(γ) dγ = n0 γ −p dγ.
(5.54)
This distribution is a power law, it is a distribution often met in environments in
which particles are accelerated to very high energies through non thermal processes.
These are conditions often seen in synchrotron emitting regions. A different discussion could be made for other (e.g. thermal) distributions.
The spectrum emitted by the population of electrons is the superposition of the
emission of each electron (we consider an optically thin medium):
Z
fν ∼
∞
Pν (γ)n(γ) dγ
(5.55)
1
In order to perform the integration in eq.5.55 we need to know the function φν (γ),
which we only described in rough terms. However, if the electron distribution in
energy is very wide, we can use the fact that the emission of a single electron is
peaked at the charateristic frequency to approximate φν (γ) by a delta function:
φν (γ) → δ(ν − γ 2 νL ), with νL =
eB
.
2πme c
(5.56)
This form makes the γ dependence of the characteristic frequency explicit and allows
to perform the integration over γ:
Z
fν ∼
Z
∼
Z
∼
Pν (γ)n(γ) dγ
(5.57)
γ 2 φν (γ)n(γ) dγ
(5.58)
γ 2 δ(ν − γ 2 νL )γ −p dγ
(5.59)
dν 0
, with ν 0 = γ 2 νL
γ 2 δ(ν − ν 0 )γ −p
γ
Z 0 −(p−1)/2
ν
∼
δ(ν − ν 0 ) dν 0
νL
−(p−1)/2
ν
∼
.
νL
Z
∼
(5.60)
(5.61)
(5.62)
This is a power law, the index of which ( −(p−1)
) is dependent directly on that of the
2
energy dependence of the electron population (p). We will see that a power law is
indeed observed in several cases.
52
If you consider that the electron acceleration is perpendicular to the magnetic field,
you will realise that the synchrotron emission is bound to be polarised. A detailed
discussion (see e.g. Ribicky and Lightman) will yield that the linear polarisation is
of about 70% for an isotropic electron distribution in a homogeneous magnetic field.
Observations often yield considerably lower values, because the magnetic field is not
homogeneous in the region subtended by the telescope (at whatever wavelength)
point spread function. Polarisation is nonetheless an important diagnostic tool for
synchrotron emission.
A detailed discussion including the constant factors we have not considered in eq.5.62
gives the emissivity in ergs/cm2 (Rybicki and Lightman):
√
3 e3 N0 B sin α
Γ
Pf (ω) =
2π mc2 p + 1
p 19
+
4 12
p
1 mcω −(p−1)/2
Γ
−
4 12
3eB sin α
(5.63)
At low energies, we reach a level where the flux cannot increase without bounds.
The sources become optically thick and the emission is self absorbed. In order to see
how this influences the spectrum consider first the equation for radiative transfer:
dIν
= κν ρSν − κν ρIν ,
ds
(5.64)
where ρ is the mass density, κν the mass absorption coefficient, Iν the intensity
and Sν the source function. In an optically thick case, the left hand side vanishes,
because as much radiation is absorbed as is emitted. The right hand side then states
that the intensity is equal to the source function: Iν = Sν at all frequencies. In order
to see the shape of the source function, we make an analogy with another case of
well known optically thick emission, the black body, for which
Sν =
hν
2ν 2
.
2
hν/kT
c e
−1
(5.65)
The first term on the right hand side of eq.5.65 is proportional to the phase space
available for emission while the second term gives the mean energy of the oscillator
emitting at energy hν. At low energies, this latter factor is kT and the ν dependence
of the source function is ν 2 . In the case of the synchrotron emission, the first term
is the same (phase space is independent of the emission process), and the second
term is the energy γmc2 of the electron radiating at the frequency ν. We have seen
1
previously (and see eq.5.53) that this is proportional to ν 2 · B −1/2 . We therefore
have
Sν ∼ ν 2 · ν 1/2 = ν 5/2 · B −1/2 .
(5.66)
This result is independent of the shape of the electron distribution.
The general shape of the synchrotron emission of a power law distribution of electrons is therefore given in the Figure 5.12.
53
Log fν
fν
ν -(p -1)/2
m
ν
5/2
νm
Log ν
Figure 5.12: General shape of the synchrotron radiation emitted by a population of
electrons distibuted with a power law of index p.
The frequency at which the emission has a maximum, νm , holds information about
the magnetic field that can be useful. At this frequency, we can use either optically
thin or thick approximation with equal validity. A uniform sphere of intensity Iν
produces at a distance d a flux of
R
fν = πIν ( )2 ,
d
(5.67)
where R is the radius of the uniform sphere. For an optically thick source, Iν = Sν ,
which we have calculated in equation 5.66. Introducing the observed angular radius
of the source θs = R/d we see that the maximum flux is There is a maximum flux
that is given by
5/2 2
fνm ∼ B −1/2 νm
θs , ,
(5.68)
from which one may deduce that, having observed the spectrum and measured the
frequency and flux of the maximum emission as well as the angular size of the source,
one can estimate the magnetic field B:
−5 −2
B ∼ fν2m νm
θs .
(5.69)
Note, however, that the frequency of the maximum emission enters in eq.5.69 with
the fifth power, this expression will therefore hardly provide precise magnetic field
measurements.
5.7
Far infrared emission of active galaxies
The radio emission of radio loud quasars has been identified for a very long time to
be due to synchrotron processes. We have already encountered this identification
when discussing the emission of the radio loud quasar 3C 273. This emission extends
54
2.62
2.56
2.17
1.93
2.64
1.74
Figure 5.13: mm to X-ray spectra of 6 radio-quiet quasars (Chini R. et al. 1989, A&A
219, 87). 1.3 mm data were collected by the IRAM 30 m telescope. The spectral
indeces found between 100 µm and 1.3 mm suggest that the energy distributions
are dominated by thermal emission from dust.
55
to the mid infrared domain and, during flares, as we have seen, to the near infrared
and even optical domains. It was therefore natural to expect that the far infrared
emission of the radio quiet AGN would be due to the same process and that the
absence or weakness of radio emission in radio quiet objects would be due to self absorption effects. One therefore expected in the early 1980’s that as long wavelength
instrumentation become available, one would measure a spectrum proportional to
5
ν 2 . Observations turned out to give a quite different picture. Figure 5.13 shows
such early measurements from which it can be seen that the slope is steeper than
the 52 that was expected. Such slopes cannot be understood in terms of synchrotron
emission from a homogeneous electron population and only extremely constrained
models would provide these slopes in the context of synchrotron processes. A much
better interpretation of these results is that the far infrared emission of radio quiet
AGN is due to emission from cool dust. We have already seen that even in radio loud
objects as in 3C 273, dust emission plays an important role and dominates (at least
outside flares) the near infrared emission. Thus dust emission with a broad distribution of temperatures plays a very important role in the physics of active galactic
nuclei. The steepness of the spectrum is explained in this case by the emissivity of
dust that is a steep function of frequency.
5.8
Extended lobes of radio galaxies
The Figure 5.14 shows a Very Large Array (VLA, a radio telescope made of an array
of dishes in Arizona) image of the radio source Cygnus A, one of the brightest radio
sources in the sky. One sees two very large clouds of emission in this figure. The
size of the source is of about 120 kpc, very much larger than the optical size of a
galaxy. The emission is a power law (Figure 5.15) and is polarised. This emission
Figure 5.14: A radio map of the source Cygnus A at 5 GHz made with the VLA.
Its features include the compact core in the center of the galaxy, the jets emanating
from the core and carrying energy and particles to the lobes, and the radio lobes
themselves. Barely visible in the overexposed lobes are the hot spots where the jets
are terminated. (Courtesy of Richard A. Perley, John W. Dreher, and the National
Radio Astronomy Observatory).
56
Figure 5.15: Radio to optical spectrum of Cygnus A (Hobbs R. W. et al. 1978, ApJ
220, L77). The radio lobe spectrum shows a turnover near 20 MHz, then follows a
power law with index -0.8 up to 1 GHz and -1.2 up to 100 GHz. The radio data of
the central source are consistent with a rising spectrum (index 1/3), although a flat
spectrum is not excluded.
can therefore best be explained in terms of synchrotron emission.
We may then estimate the energy that is contained in the radio emitting lobes.
The energy density of the electron population is
Z
∞
ue− = u0
(γme c2 )γ −p dγ =
γmin
N0 me c2 −(p−2)
γ
.
2 − p min
(5.70)
where u0 and N0 give the normalisation of, respectively, the energy density and the
number density. For γmin one may use the lower observed frequency of the radio
emission (10 MHz), one can thus obtain at least a lower bound to the electron energy
density. The magnetic field energy density is
uB =
1 2
B
8π
(5.71)
The electron energy density is proportional to the normalisation of the electron
distribution N0 . For a given emissivity this is proportional to B −(p+1)/2 as one can
deduce from eq.5.63. Thus
ue− = C0 N0 = C1 · B −(p+1)/2 ,
(5.72)
C0 and C1 being constants. If we seek a lower limit to the energy contained in the
lobes, one can look for the magnetic field that will minimise the total energy density.
This is done by looking for the minimum of ue + uB :
57
due−
dB
p+1
= C1 −
B −(p−1)/2
2
p + 1 ue−
= −
.
2 B
(5.73)
(5.74)
and therefore at the minimum we have:
d
(p + 1)ue− 2uB
(uB + ue− )|B=B0 = 0 = −
+
dB
2B0
B0
(5.75)
I.e. ue ' uB for the observed spectral slope
(5.62)
fν ∼ ν −0.7 ⇒ 0.7 ==
p−1
⇒ p ≈ 2.5.
2
(5.76)
The total energy density is therefore about 2 × uB . The magnetic field can be
estimated in the lobes to be about 10−4 G, leading to
Z
uB dV =
B 2 4π 3
R ≈ 1060 ergs.
8π 3
(5.77)
This is a considerable amount of energy. It would e.g. require some 109 supernovae
to provide it. In absence of any observable stellar activity in the lobes, stellar
processes cannot give rise to this energy density.
It should also be noted that the cooling time for the electrons is rather short
tcool =
E
≈ 5 · 108 · B −2 γ −1 s ' 105 years.
P
(5.78)
The electrons need therefore be accelerated in situ. To see where the energy comes
from, a detailed look at Figure 5.14 is useful. In this figure one sees a thin emission
line that connects one of the lobes with the centre of the system. This is a jet which
is thought to channel energy from the core of the galaxy to the lobes. The jet energy
is dissipated there in shocks in which electrons are accelerated to then radiate and
cool through the synchrotron emission discussed here.
It is also very interesting to consider the X-ray emission of the area surrounding
Cygnus A (Figure 5.16). This shows the complex interactions between the lobes
and the surrounding medium and is directly relevant to the question of the cooling
flows mentioned in the discussion of the Bremstrahlung, as it shows that the nuclear
activity does contribute to the energy balance in the central regions of the clusters.
5.9
Crab nebula
The Crab nebula is seen over the complete electromagnetic spectrum, from radio
waves to gamma rays (see Figure 5.17). It has been associated since 1928 to the
58
Figure 5.16: X-ray image of Cygnus A obtained with Chandra on May 21, 2000
(NASA/UMD/A.Wilson et al.).
Figure 5.17: Images of the Crab Nebula at different wavelengths. Left upper : radio
image from the NRAO; right upper : infrared image from 2MASS; left lower : optical
image from VLT/ESO; right lower : X-ray image from Chandra.
59
Figure 5.18: Hubble Space Telescope image of the inner parts of the Crab. The
pulsar itself is visible as the left of the pair of stars near the center of the frame
(Credit: Jeff Hester and Paul Scowen, Arizona State University, and NASA).
Figure 5.19: The ISGRI 20-500 keV spectrum of the Crab obtained from a 130 ks
observation during revolution 239 of INTEGRAL. The best-fit photon index is 2.254
± 0.001 and the flux at 1 keV is 15.8 ph cm−2 s−1 keV−1 (Courtesy of Lubinski P.).
appearance of a guest star observed in 1054 by the Chinese (Figure 5.18). It is a
very strong X and gamma ray source and serves as calibration source and as flux
unit (the Crab) to high energy instrumentation, in a way somewhat similar to the
use of Vega in optical astronomy (1 Crab = 2.4 × 10−10 cmergs
2 s keV at 50 keV). The
luminosity of the nebula is of some 1038 ergs/s and the size of the nebula is of about
3.5’. The spectrum is given in Figure 5.19. One has to take some care to disentangle
the emission of the nebula from that, pulsed, of the central pulsar. The emission is
strongly polarised (40% in the optical domain). Depolarisation indicates a magnetic
field of about 5 × 10−4 G. The emission is therefore naturally interpreted in terms
of synchrotron emission.
Recalling the cooling time arguments made above, one concludes that electrons
emitting at 10 MHz have an energy of some 70 MeV while those emitting in the
gamma rays at 1022 Hz have an energy of 1015 eV, a very considerable energy indeed
for single electrons. Electrons radiating at 100 GHz cool in a comfortable 6000 years,
those emitting in the X-rays at 1020 Hz, however already cool in a matter of some 10
weeks, considerably less than the age of the nebula. One therefore concludes that
the electrons must be continuously provided and accelerated in the nebula. The
X-ray image above (right lower panel in Figure 5.17) shows indeed features that
indicate that energy may be fed from the central pulsar to the nebula, as in the
case of the radio lobes of the radio galaxies, and that the electrons are then locally
60
Figure 5.20: The jet of 3C 273 on all scales from arcminutes to mas and the superluminal motion (from a compilation in Camenzind 1997).
accelerated in shocks.
5.10
5.10.1
Relativistic jets
superluminal motion
We have already seen that jets convey energy over very large distances. Since they
are "seen" they must therefore radiate some of the energy they carry. These jets are
not static in space. Indeed Very Large Baseline Intereferometry (VLBI, a system
of linking radio dishes over the whole surface of the Earth to yield radio images of
very high angular resolution) observations at different epochs have shown that the
elements out of which the jets are made (often called blobs) are moving from epoch
to epoch. Such observations of the quasar 3C 273 (see figure 5.20) have shown for
example that the jet elements move away from the core at about 1 mas per year
(1 mas is a milli arcsecond). At the distance of 3C 273, 1 Gpc, this corresponds
to a linear velocity of some 10 times the velocity of light, at least without taking
relativistic effects into account. These apparently super luminal velocities may be
61
!"
!#
+
+
,#
$%&!%'()*
,"
Figure 5.21: Left: An electron moving from 0 to 1 along the jet emits two photons
γ0 and γ1 at t0 and t1 , respectively. If the jet velocity is close to c and the angle θ
between the jet direction and the line of sight is small, the apparent velocity seen
by the observer exceed the speed of light.
understood, however, if one considers jets moving at angles close to the line of sight
at relativistic velocities (see left panel in Figure 5.21). If one considers two photons
emitted in the jet, the first at t0 as one jet element moves away and the second from
the jet at t1 and if one considers that the second photon travels a shorter distance
than the first one, one will find that their arrival times at the observer differ by
∆ta
d − vt cos θ
d
+ t1 +
= − t0 +
c
c
vt cos θ
= −t0 + t1 −
c
v cos θ
= t 1−
c
(5.79)
(5.80)
(5.81)
(d being the distance to the observer which is absent from the end result!) and that
the apparent jet velocity on the plane of the sky is
v⊥ =
v sin θ
vt sin θ
=
θ
∆ta
1 − v cos
c
(5.82)
which can clearly exceed the speed of light for jet velocities close to c and small
angles θ, as shown in Fig. 5.22.
This clearly establishes that the jets are relativistic (unless the motions are not due
to matter in movement but to other effects).
5.10.2
relating the observed jet properties to the intrinsic
conditions
If we want to relate the observed properties of the radiation emitted by these jets to
the physical conditions of the jet, one has to relate both reference systems, as seen
62
8
Beta=0.9
Beta=0.95
Beta=0.99
Apparent perpendicular velocity [v/c]
7
6
5
4
3
2
1
0
0
10
20
30
40
50
Jet angle [deg]
60
70
80
90
Figure 5.22: The apparent jet velocity as a function of the angle to theline of sight
for different jet velocities
"
&
θ
#$%
!
Figure 5.23: Transformation of the absorption coefficient in a moving medium.
63
in section 5.3. First we remember that relativistic electrons emit in a cone of half
open angle 1/γ centered around the direction of the velocity. Jets will therefore be
much brighter when viewed along their axis. A second effect to take into account is
the Doppler effect that relates the emitted and observed frequencies:
ν0
ν=
,
γ(1 − β cos θ)
(5.83)
which takes into account time dilatation and the change in the way that the source
travels while radiating (1 − β cos θ).
The factor (γ(1 − β cos θ))−1 = D is called the Doppler factor.
Next we consider the transformation of the fluxes. To do this we consider n, the
number of particles per phase space element d3 xd3 p. This density is a Lorentz
invariant. The number of particles is an invariant on one side and
1 3 0
dx
γ
(5.84)
d3 p = γd3 p0 .
(5.85)
d3 x =
Thus
n=
number of particles
d3 xd3 p
(5.86)
is an invariant.
I c−1 dν dΩ = n · hν · d3 p = nhνp2 dp dΩ
}
|ν {z
energy density
(5.87)
p ∝ ν, therefore νIν3 ∝ n (for an isotropic distribution) is an invariant. We then know
how the intensity tranforms:
Iν =
ν3 0
Iν = D3 Iν0 (ν 0 ).
03
ν
(5.88)
The observed flux is given by
1
fν = 2
R
Z
jν dV
(5.89)
source
for an optically thin source with a source function jν . In order to know how to relate
the observed and the emitted fluxes we consider the transfer equation
dIν
jν
= −Iν + ,
dτ
µ
64
(5.90)
where τ is the optical depth and µ the absorption coefficient per unit length so that
dτ = µdx. τ gives the fraction of photons absorbed and is thus an invariant. Since
τ = µ∆ =
µν`
µ`
=
sin θ
ν sin θ
(5.91)
and ν sin θ ∝ py , the y component of the momentum, that is not changed by the
transformation, nor is ` which is the thickness of the jet (see right panel in Figure 5.21) and hence perpendicular to the x-axis, we conclude that µν is also not
affected by the transformation and hence µ = D−1 µ0 .
Using eq.5.88, the fact that equation 5.90 indicates that jν /µ transforms like Iν gives
jν (5.88) 3 jν0
j0
== D 0 = D2 ν ⇒ jν = D2 jν0 (ν 0 )
µ
µ
µ
(5.92)
and therefore for the flux
1
fν = 2
R
Z
1
jν dV = 2
R
source
Z
D2 jν0 (ν 0 ) dV
(5.93)
source
which gives the observed flux in the system of the observer as a function of the
emissivity in the jet system of reference (i.e. from the density etc as measured in
the jet system). Specialising for a source in which the emissivity is a power law of
the frequency, as we have seen to be the case in synchrotron radiation gives
jν0 (ν) =
ν −α
ν0
jν0 (ν 0 ) ⇒ jν0 (ν 0 ) = Dα jν0 (ν).
(5.94)
It follows that the observed flux at the observed frequency ν is
D2+α
fν (ν) =
R2
Z
jν0 (ν) dV,
(5.95)
where we consider the emissivity at the observed frequency.
A direct consequence of this result is that we can calculate the ratio of the observed
flux of a jet and a counter jet (the one symmetrical to the one observed) to be
fjet
fcounterjet
=
2+α
Djet
=
Dcounterjet
1 + β cos θ
1 − β cos θ
2+α
.
(5.96)
For small angles, one can approximate the cosines and obtain
fjet
fcounterjet
"
=
1 + (1 −
1 − (1 −
θ2
)
2
θ2
)
2
#2+α
2(2+α) 5
2
2
≈
∼
θ
θ
(5.97)
This ratio is shown in Fig. 5.24 explains in most cases why only one jet is observed
from a radio source, whereas two would be expected if the source were symmetrical.
65
10000
Beta=0.9
Beta=0.92
Beta=0.95
9000
8000
Jet/counterjet flux ratio
7000
6000
5000
4000
3000
2000
1000
0
0
5
10
15
Angle [deg]
20
25
30
Figure 5.24: Ratio between the jet and counter-jet as a function of the angle between
the jet axix and the direction of the observer for a number of jet velocities.
66
Chapter 6
Compton processes
(based on Rybicki and Lightman, Longair)
We study here the scattering of photons off electrons and the energy transfer between
the electron gas and the radiation field that results from these interactions. This
is one of the most important processes in high energy astrophysics, as it allows
to transfer an important fraction of the electron energy to the photons and thus
to radiate this energy away from the source. This is therefore an efficient cooling
mechanism. It is also an important process in the sense that the resulting radiation
appears predominantly in the high energy spectral domains, from the X-rays up to
TeV energies in some cases.
6.1
Thomson Cross section
A special case of the electron-photon interaction is the Thomson scattering. In this
case the photon energy hν is much less than the electron rest mass me c2 . The inertial
system of the centre of mass is therefore that in which the electron is at rest. The
derivation of the cross section is a direct application of the Larmor formula that we
derived in the first lecture:
2 |p̈|2
dE
= ¯3 .
dt
3 c
(6.1)
As before: p̈ = e · v̇. The electron acceleration in an interaction with the radiation
field is given by the electric field of the latter. Assuming a wave moving along the
z-axis (Figure 6.1) one has:
v̇x =
e
e
Ex , v̇y = Ey with Ex = E0x sin(ωt), Ey = E0y sin(ωt).
m
m
(6.2)
which gives:
2
dE
= 3
dt
3c
2 2
2 2
4
2
e
e
2
Ēx +
Ēy = 2 e |E0 |
m 3c3 m 3 m2 c3 2
67
(6.3)
x
Incident
radiation
Ex
θ
α
Ey
z
π/2
y
electron
Figure 6.1: Geometry of the Thomson scattering of a beam of radiation by a free
electron. We assume that the beam propagates in the positive z-direction and that
the scattering angle α lies in the x-z plane.
for the energy of the subsequent radiation. The factor 2 at the end of eq. 6.3 comes
from
Ēx2
E2
= 0x
2π
Z
sin2 (ωt) dt =
2
E0x
.
2
(6.4)
If we remember that the cross section of a diffusion σ is the ratio between the
radiated energy and the incoming energy flux and that the incoming energy flux is
2
c E0
c
E 2 , S̄ = 4π
of the incoming radiation, we can write
the Poynting vector S = 4π
2
the cross section for the Thomson diffusion as
σT =
r0 =
e2
me c2
2 e4
dĒ/dt
8π |E02 |
=
·
3 m2 c3 c 2|E02 |
S̄
8π e4
=
3 m2 c4
8π 2
=
r = 6.65 · 10−25 cm2
3 0
(6.5)
(6.6)
(6.7)
is the classical electron radius.
dσ
To understand the angular dependence of the diffusion and thus obtain dΩ
we consider again what we obtained in the first lecture for the energy radiated in the solid
angle dΩ:
dE
c |p̈|2 sin2 θ 2
dΩ =
· r dΩ,
dt
4π c4 r2
where θ is the angle between the electron acceleration (see Figure 6.1).
For a non polarised radiation field, S¯x = S¯y and
68
(6.8)
2
dEx
c e4 E0x
cos2 α
dΩ =
dΩ
dt
4π 2m2 c4
2
c e4 E0y
dEy
dΩ =
dΩ
dt
4π 2m2 c4
dE
e4
2
dΩ =
(E 2 + E0x
cos2 α)dΩ
dt
8πm2 c3 0y
S̄x,y =
2
c Ex,y
4π 2
dE
e4
dΩ = 2 4 (S̄y + S̄x cos2 α)dΩ,
dt
mc
(6.9)
(6.10)
(6.11)
(6.12)
(6.13)
which gives for a non polarised radiation field:
e4
S
dE
dΩ = 2 4 · (1 + cos2 α)dΩ.
dt
mc 2
(6.14)
Finally
dσT
dΩ
6.2
dE/dt
emitted energy in α
=
S
energy flux
4
1 e
=
(1 + cos2 α)
2
4
2m c
=
(6.15)
(6.16)
Compton scattering
In the preceeding subsection we have considered the case in which the incoming
photon energy hν1 (note how we move from a wave description of the radiation to a
particle description) is much less than the electron rest energy me c2 = 511 keV. The
frequency of the outgoing radiation is the same as that of the incoming radiation,
the photon energy is therefore also the same and the diffusion is elastic, hν2 = hν1 .
We relax here the hypothesis on the energy of the incoming photon energy and
consider the diffusion of X-ray photons (of energies up to few 100s keV). This is the
so called Compton scattering. We first consider the energy momentum conservation
for the four-vectors P~ . The subscript "i" will in the following indicate the incoming
particles and "f" the outgoing particles (electron e and photon γ ), will indicate
photon energies while E’s will indicate electron energies. We consider the scattering
in the frame in which the electron is initially at rest (P~ie = (mc, 0)).
¯
The direction 4-vectors of the photons are introduced as:
f
i
P~iγ = (1, ni ), P~f γ = (1, nf ).
c
c
¯
¯
69
(6.17)
Pγf
Pγi
θ
e-
Pef
Figure 6.2: Geometry for the scattering of a photon by an electron initially at rest.
where ni,f are unit vectors.
¯
The energy-momentum
conservation is
P~iγ + P~ie = P~f γ + P~f e
(6.18)
or
P~f e = P~iγ + P~ie − P~f γ .
This can be squared and transformed into an expression that gives the outgoing
photon energy (see Figure 6.2 for the definition of the angles):
|P~f e |2 = (P~iγ + P~ie − P~f γ )2
= P~iγ2 + P~ie2 + P~f2γ + 2P~ie P~iγ − 2P~f γ P~ iγ − 2P~f γ P~ie
(6.19)
m2e c2 = m2e c2 + 2P~ie P~iγ − 2P~f γ P~iγ − 2P~f γ P~ie
(6.21)
0 = me c
f
i i f
− 2 (1 − cos θ) − me c
c
c
c
f =
1+
i
i
(1
me c2
− cos θ)
.
(6.20)
(6.22)
(6.23)
This expression gives the change of energy of the photons in the frame in which the
electron is initially at rest. Clearly, when the incoming photon energy i << me c2 ,
we recover the elastic scattering described as Thomson scattering.
Note that when the photon energy becomes large compared with the electron rest
mass the cross section is modified. The resulting cross section is called Klein Nishina
cross section which is given by
dσ
r2
= 0
dΩ
2
f
i
i
f
2
+ − sin θ .
f
i
70
(6.24)
Molecular
cloud
X-ray
source
Flux
γi
θ = 180o
γf
511
170
Energy [keV]
Figure 6.3: Left: X-ray photons backscattered by a cloud of cold electrons. Right:
Spectrum of a monochromatic e+ -e− annihilation line and its reflection for the geometry shown in the left panel.
One illustration of this is given by the case in which a monochromatic energy source,
say emitting at 511 keV, the energy of the rest mass of the electrons, is situated in
front of a cloud of cold electrons, e.g. a molecular cloud (remember that for photon
energies of many keV the electron binding energy can be neglected and the electrons
considered free). In this case, eq. 6.23 implies that a line will also be observed at
170 keV, as the photons are backscattered and θ = π (see Figure 6.3).
Whereas this has not been observed to date, one case of Compton reflection has
been suggested to take place within our Galaxy. Revnivtsev et al. (2004, A&A 425,
L49) have suggested that the high energy radiation observed with INTEGRAL from
a molecular cloud (SgrB2) at some 100 pc from the centre of the Galaxy reflects light
that the latter source has emitted some 300 years ago and conclude that the centre
of the Galaxy was then some 104 times brighter than it is now (see figure 6.4).
We now consider the scattering in a frame in which the electron is not at rest initially
but moves with a relativistic speed and has an energy γ me c2 . The reasoning we
have made remains true in the electron rest frame, we must therefore transform the
incoming photon energy as measured in the observer frame to that of the electron.
This is done by considering the Doppler factor γ(1 − β cos θ). We then apply eq.
6.23. This means primarily that the photon direction is very significantly altered.
The outgoing photon energy must then be transferred back in the observer frame
(β changes sign). This is described by:
f = 0f γ(1 + β cos θ)
0i
γ(1 + β cos θ)
=
1 + meic2 (1 − cos θ)
=
i γ(1 − β cos θ)
γ(1 + β cos θ)
1 + meic2 (1 − cos θ)
≈ γ 2 i ,
(6.25)
(6.26)
(6.27)
(6.28)
71
where we see that in essence the photon energy was increased by a factor γ 2 . In
essence each factor γ comes from one of the reference frame changes. This factor
implies that when the electrons are relativistic, the photons can gain a very large
energy in the scattering. In this case the process is commonly called "inverse Compton". It explains why X-ray photons are often created by scattering of soft photons
Figure 6.4: 3.5◦ × 2.5◦ hard X-ray (18-60 keV) image of the Galactic Centre region
obtained with INTEGRAL/IBIS, with overplotted contours of brightness distribution in the 6.4 keV line as measured by ASCA/GIS. Largest molecular clouds (among
them, SgrB2) are indicated and the position of the Galactic Centre Sgr A? source
is marked with a cross. Spectrum in the lower panel. This spectrum is typical of a
reflexion component (Revnivtsev et al. 2004, A&A 425, L49).
72
on hot or relativistic electrons.
6.3
Power emitted by a single electron
Let us now calculate how much energy is radiated by a single electron in the Compton
process. In order to do this we first consider the density of the photons per unit
volume and energy:
V d = n d3 p,
(6.29)
where n is the photon density in the phase space. n is an invariant under the Lorentz
tranformation, as it is a number per element d3 xd3 p which is invariant.
It follows that
V
d0
d
= V0 0
(6.30)
is a Lorentz invariant. Remembering that the cross section of any interaction is the
ratio of the energy loss to the incoming energy flux and that the incoming energy
flux can be written as c0i Vi0 d0 in the electron rest frame, the emitted energy is
dEe0
= cσT
dt0
Z
0
0iγ Viγ
d0
(6.31)
With eq.6.30 and knowing that θ0 is the angle between the vectors x and p0f , we can
express the energy loss of the electrons in the observer rest frame:
0
Viγ
d0
0
iγ
Z
V
= cσT 02
d
iγ
dEe0
dEe
= 0 = cσT
dt
dt
dEe
= cσT γ 2
dt
Z
Z
02
iγ
(1 − β cos θ)2 V d.
(6.32)
(6.33)
(6.34)
For an isotropic distribution of velocities, one has
1
h(1 − β cos θ)2 i = 1 + β 2
3
(6.35)
1 2
dEe
2
= cσT γ 1 + β uγ ,
dt
3
(6.36)
and therefore
73
R
where uγ = Vd is the energy density of the incoming photon gas. Eq. 6.36 gives
the energy loss of the electron. In the energy budget for the photon gas, one must
also consider that lost by the incoming photon gas in the scattering:
dEγ
= −cσT
dt
Z
V d = −cσT uγ .
(6.37)
The energy budget for the radiation field is thus
dEe dEγ
dErad
=
+
dt
dt dt
1 2
2
= cσT uγ γ 1 + β − 1
3
(6.38)
(6.39)
Using the identity γ 2 − 1 = γ 2 β 2 , one finally obtains
PCompton =
dErad
4
= σT cγ 2 β 2 uγ .
dt
3
(6.40)
This results looks very similar to that obtained when describing the energy loss of
relativistic electrons in synchrotron emission (see eq. 5.27):
4
Psync = σT cγ 2 β 2 uB .
3
(6.41)
The ratio of the synchrotron to the Compton powers is thus
Psync
PCompton
=
uB
uγ
(6.42)
and is given by the ratio of the magnetic field energy density to that of the radiation
field. This expression also shows the very similar nature of both processes, which
both describe the interaction of the electron with the electromagnetic field, albeit
in very different guises.
6.4
Power emitted by a distribution of electrons
Eq. 6.40 gives the energy losses for a single electron. This expression can be generalised for a distribution of electrons by integrating over the distribution N (γ). For
a power law distribution:
Z
P =
PCompton (γ)N (γ) dγ
(6.43)
3−p
3−p
(γmax
− γmin
)
4
=
σT cuγ
,
3
3−p
(6.44)
74
and for a thermal distribution of non relativistic electrons (remember that Ecin =
3
kT = 12 mv 2 ):
2
2
hβ i =
v2
c2
≈
3kT
mc2
(6.45)
and
P =
4kT
mc2
cσT ne uγ ,
[P ] = ergs · s−1 cm−3 .
(6.46)
The emission spectrum can also be calculated in this case. This is done e.g. in
Rybicki and Lightman to which the reader is referred. For a power law distribution
of electrons in which the electron distribution is proportional to γe−p , the emission
p−1
spectrum is proportional to ν − 2 . This result is again identical to that obtained
when discussing the synchrotron radiation.
6.5
6.5.1
Energy gains per scattering
Non relativistic electrons
The energy budget of the radiation was found in eq. 6.40 to be
4
PCompton ≈ σT cβ 2 uγ
3
(6.47)
(γ ≈ 1 for non relativistic electrons). The rate of scattering per electron is
ND = σT nγ c, where nγ =
uγ
.
hν
(6.48)
Therefore
ND = σT c
uγ
.
hν
(6.49)
The average energy gain per scattering is therefore
PCompton
4
= β 2 hν
ND
3
(6.50)
∆
4
= β 2 , with = hν.
3
(6.51)
∆ ≈
For a thermal distribution of electrons of temperature Te , we have
75
3
3kTe
1
me hv 2 i = kTe ⇒ hβ 2 i =
2
2
m e c2
and we can therefore express
∆
with the temperature of the electron gas:
∆
4
4kTe
= hβ 2 i =
3
m e c2
6.5.2
(6.52)
(6.53)
Relativistic electrons
In the case of a relativistic electron population characterised by a (single) energy
γme c2 , we have
4
PCompton ≈ σT cγ 2 uγ
3
(β ≈ 1).
(6.54)
The rate of scattering is given as above and therefore
4
∆
= γ 2 , with = hν.
3
(6.55)
For a thermal (but relativistic) electron population we have
hγ 2 i =
hE 2 i
(mc2 )2
(6.56)
and
E
E 2 · E 2 exp − kT
dE
R
E
E 2 exp − kT dE
(6.57)
Γ(5)(kTe )5
= 12(kTe )2
hE i =
3
Γ(3)(kTe )
(6.58)
R
2
hE i =
which is given by the Γ function:
2
and thus finally the relative energy gain per scattering is
∆
= 16
6.6
kTe
mc2
2
.
(6.59)
Multiple scattering in an optically thin limit
It is interesting to consider the slope of the emitted Compton radiation when considering the multiple scattering that the photons undergo in an electron gas of well
76
defined energy that is optically thin. In this case we assume that the resulting
spectrum will be a power law of slope −α:
Iν ∝ ν −α .
(6.60)
At each scattering the photon energy is multiplied by a factor A which is a function
of the electron energy discussed in section 6.5. After K scattering the photon energy
has become
K = i AK .
(6.61)
The intensity at k is given by the intensity at and the probability to have K
scattering, which in an optically thin situation is given by τ K , where τ is the optical
depth of the medium. We therefore have
I(K ) = I(i )
K
i
−α
= τ K I(i ).
(6.62)
Using eq. 6.61 one obtains
I(i )(AK )−α = τ K I(i ) ⇒ AK·(−α) = τ K ⇒ −α ln A = ln τ
and then the following expression for the slope α:
α=−
ln τ
ln A
(6.63)
which is given by the optical depth of the electron gas and the energy gain factor A
which depends on the energy of the electrons.
6.6.1
example: X-ray emission of AGN
Consider a situation in which we measure the seed and the Comptonised photon
fluxes Cs and Ch . The optical depth is proportional to this ratio:
Cs
.
Ch
(6.64)
Cs
Cs
)/ln(A) = −ln(Const)/ln(A) − ln( ) · ln(A).
Ch
Ch
(6.65)
τ = Const ·
Inserting this inEq.6.63, one obtains
α = −ln(Const ·
A linear relationship between the slope of the X-ray spectrum and the ratio of the
count rates, both measurable quantities. The slope of this linear relation is
77
Figure 6.5: The X-ray slope α vs the log of the ratio of the UV (soft photon) count
rate and the X-ray (Comptonised) count rate for the quasar 3C 273. The slope of
theis relation indicates gives "A", the energy gain per scattering and indicates an
electron temperature of the order of 1 MeV (Walter and Courvoisier 1992, A&A 258,
255).
1
dα
=−
.
Cs
ln(A)
dln( Ch )
(6.66)
This relation can then be used to estimate the temperature of the Comptonising
medium, provided that indeed this process is at the origin of the X-ray emission.
This was done for few AGN, including 3C 273 for which the results are given in
Fig. 6.5.
78
Chapter 7
Comptonisation
We consider here the Compton process in a medium and will see how we can describe
statistically the changes in the photon distribution and apply the results to some
astrophysically relevant cases.
7.1
Compton Temperature
We first ask for which electron gas temperature there will be no net energy transfer
between the electron and the photon gas. We do this in a non relativistic case. The
resulting temperature is that of equilibrium between both distributions.
We have calculated in the last section the energy of the photon after the scattering
from its initial energy and the deflection angle:
f =
1+
i
i
(1
me c2
− cos θ)
.
(7.1)
From this we calculate the relative change of energy (energy loss, watch the sign)
for the scattering to be
i
hν
i − f
=
(1 − cos θ) =
,
2
f
me c
me c2
(7.2)
where we have used in the last equation the photon frequency to express the initial
photon energy. In the non relativistic limit, we have calculated also in the last
section the average energy gain per diffusion:
∆
4kTe
.
=
m e c2
(7.3)
There will therefore be no net energy transfer when the average change (loss equals
gains) of energy over the incoming photon distribution for:
79
R
hν
3
4kTc
2 n(p) d p
Rme c ¯
=
,
3
me c2
n(p) d p
¯
(7.4)
which means that
1
kTc = hhνi.
4
(7.5)
Taking an incoming photon power law of slope α = 1.7
fν ∼ ν −α
α−1
hνi =
2−α
between 1 and 100 keV
νmin
νmax
α−1
νmax ⇒ Tc ≈ 1 · 108 K.
(7.6)
(7.7)
The reader may make the same reasoning for a relativistic electron temperature.
Note that a null average energy exchange does not mean that the spectrum will be
unchanged.
7.2
The y parameter
The equilibrium temperature calculated in the previous subsection implies that there
is no net energy change between the two gases, it does not mean that the distributions are not modified. In order to assess whether a photon distribution is significantly modified by the Compton process in a given electron gas, one introduces the
so-called y parameter which is defined as
y = fractional energy change × number of scatterings.
The fractional energy change is the "A" calculated in the previous section, while the
number of scatterings for the photons in the electron gas is given by max(τ, τ 2 ). The
last point can easily be understood because for small optical depths τ , the number of
scatterings is Nscatt ' 1 − exp(−τ ) ' τ while for optically thick media, the photons
2
undergo a random walk for which Nscatt ' L`2 ' τ 2 , where L and ` are the size of
the region and the mean free path respectively.
For y 1: the incident spectrum will be significantly modified,
for y 1: the incident spectrum will not be markedly modified.
7.3
The Kompaneets Equation
The description of the spectral modifications is made through an equation called the
Kompaneets equation. This is derived from the general problem of the modification
80
of a distribution through collisions that is given by the Boltzmann equation. For a
photon distribution n(E):
∂n(E)
= diffusion into dE − diffusion out of dE.
∂t
(7.8)
This equation has the form
Z
∂n(E)
=c
∂t
3
Z
dp
dσ
dΩ [f (p1 )n(E1 )(1 + n(E)) − fe (p)n(E)(1 + n(E1 ))] (7.9)
dΩ
when one considers photons of energy E and electrons of momentum p scattering
into E1 and p1 . Energy and momentum conservations do have to be satisfied...
For non relativistic electrons the energy gain per scattering A is small and the
Boltzmann equation reduces to the Fokker-Planck equation. Introducing
∆ :=
E1 − E
kT
(7.10)
and expanding eq. 7.9 for small ∆ one has
n(E1 ) = n(E) + (E − E1 )
∂n
+ ···
∂E
∂f
+ ···
f (p1 ) = f (p) + (p − p1 )
¯
¯
¯ ¯ ∂p
¯
(7.11)
(7.12)
and therefore
1 ∂n
= [n0 + n(1 + n)]
c ∂t
Z
where n0 =
distribution
E
,
kTe
∂n
,
∂x
x =
3
Z
dp
Z
Z
dσ
1 00
1
dσ
0
3
dΩf ∆+ n + n (1 + n) + n(1 + n)
dp
dΩf ∆2 ,
dΩ
2
2
dΩ
(7.13)
and where we have specialised for a thermal electron
f (Ee ) =
Ee
ne
e− kT
3/2
(2πmkT )
(7.14)
for which f 0 = − kT1 e f . The two terms on the right hand side can be estimated (see
e.g. Rybicki and Lightman) to finally obtain the Kompaneets equation:
1 ∂n
=
ne σT c ∂t
kT
mc2
1 ∂ 4 0
2
x
(n
+
n
+
n
)
.
x2 ∂x
81
(7.15)
Figure 7.1: Strong Comptonization of a bremsstrahlung spectrum in an optically
thick, non-relativistic medium. The bremsstrahlung spectrum dominates at low
frequency and shows a characteristic self-absorption region and a flat region. At
higher frequency, photons have been multiply scattered via the Compton process so
hν
) (www.astro.utu.fi/ cflynn/).
that a Wien spectrum forms, I ∝ ν 3 exp(− kT
7.4
Solutions
7.4.1
Equilibrium solution
One may note that for E kTe , the term in n2 may be neglected compared to the
first order term. In this case one has an equilibrium solution that is given by
n ≈ e−x ,
(7.16)
n0 ≈ −n,
(7.17)
∂n
∼ 0.
∂t
(7.18)
for which
and therefore
7.4.2
y1
At high energies, the solution for the photon phase space density is given by eq.
7.18. The intensity I ∝ ν 3 exp(−x), the phase volume times the density. The high
energy spectrum will therefore have the shape given in the fig.7.1.
The maximum of the intensity is found through
82
hν
hν
h
d 3 − hν ν e kT = 3ν 2 e− kT − ν 3 e− kT
dν
kT
(7.19)
dh i
···
= 0 ⇒ hνmax = 3kT.
dν
ν=νmax
(7.20)
and
In the figure 7.1, the low energy photon spectrum is given by the source photon
spectrum (a bremsstrahlung spectrum in this case). The region of the maximum is
called the Wien peak.
7.4.3
Intermediary case, y ' 1
In this regime, the importance of the Wien peak decreases compared to above.
Numerical solutions show (fig. 7.2) that the importance of the peak increases with
increasing optical depth τ and hence with y. The figure is given for a power law
input photon distribution surrounded by a spherical 25 keV electron distribution.
7.4.4
Low optical depth
In this case the Wien peak is not seen, the spectrum curves down above ' kTe . This
is illustrated in fig. 7.3 in which the data points are from INTEGRAL and RXTE
satellites and the model, given by the full line, is a numerical representation of the
Comptonised spectrum with parameters kT ∼ 74 keV and τ ∼ 0.9.
7.5
The Sunyaev Zeldovich effect
Consider what happens when a cluster of galaxies lies along the path of the 3K
microwave background photons. The clusters of galaxies are, we have seen it, filled
with a hot (several keV) electron gas. The background photons scatter on the hot
electrons of the cluster through the Compton process, their spectrum is therefore
slightly modified. This is called the Sunyaev-Zeldovich effect.
hν
2
' 103K
In this case x = kT
7 K 1, not only the n term can be neglected, the n term
e
is also smaller than the n0 term :
n
1 (ex − 1)2
x
∝
'
'x1
0
x
x
n
e −1
e
1+x
(7.21)
The Kompaneets equation eq. 7.15 thus reduces to
∂n
1 ∂
∂n
= 2
x4
,
∂y
x ∂x
∂x
83
with x =
hν
,
kTe
(7.22)
Figure 7.2: The Comptonization of low frequency photons in a spherical plasma with
kTe = 25 keV. The solid curves are analytic solutions of the Kompaneets equation;
the histograms are the results of Monte Carlo simulations of the Compton scattering
process. These computations illustrate the development of the Wien peak at energy
hν ' kTe (Pozdniakov, Sobol and Sunyaev 1983, ASPRv 2, 189).
84
102 SPIROS counts s−1
normalized counts cm−2 s−1 keV−1
10 2
Rev. 11
Rev. 25
2002−11−16
2002−12−29
10 1
10 0
10−1
10−2
keV cm−2 s−1 keV−1
10−3
10 0
10−1
10−2
SPI
10−3
χ
PCA
SPI
HEXTE
PCA
ISGRI
HEXTE
ISGRI
4
2
0
−2
−4
10
Energy [keV]
100
10
Energy [keV]
100
Figure 7.3: Combined INTEGRAL-RXTE spectra of the Galactic source Cygnus
X-1 (Pottschmidt et al. 2004, Proc. 5th INTEGRAL Workshop ESA SP-552). The
counts spectra (upper panels) as well as the unfolded spectra (middle panels) for two
different data sets of observations (INTEGRAL revolutions 11 and 25) are shown
together with their best fit, a thermal Comptonization model. For rev. 11 they
+0.05
found a temperature of 82+16
−5 keV, and an optical depth of 0.71−0.07 and for rev. 25
+8
+0.08
kT = 65−5 keV and τ = 1.01−0.12 . The residuals are also displayed (lower panels).
85
where
y=
kT
τT
me c2
(7.23)
(beware of a factor 4!!)
In this case the optical depth is small, we can therefore assume that the deviations
from the Planck function n = ex1−1 will be small and solve for the deviation by
introducing the Planck function in the right hand side of eq. 7.22:
∂n
∂n
1 ∂
= 2
x4
∂y
x ∂x
∂x
ex
1 ∂
4
= 2
−x x
x ∂x
(e − 1)2
1 ex [−4x3 (ex − 1) + x4 (ex + 1)]
= 2
x
(ex − 1)3
x(ex + 1)
xex
−4
=
(ex − 1)2 (ex − 1)
(7.24)
(7.25)
(7.26)
(7.27)
In this case y is small and we may write:
∂n
∆n
∆n
'
=
.
∂y
∆y
y
(7.28)
Here y corresponds to the hot gas cloud crossed by the photons and ∆n is the
modification of the incident photon distribution as the gas crosses the cluster.
We may then write eq. 7.27 as
∆n
xex
=y x
n
e −1
x(ex + 1)
−4 ,
(ex − 1)
(7.29)
which in the limit of small x can be expanded to
∆n
xex
(x(ex + 1) − 4(ex − 1))
=
x
2
yn
(e − 1)
2
x(1 + x + x2 )
x2
x2
=
x 2+x+
−4 x+
2
2
2
(x + x2 )2
and
86
(7.30)
(7.31)
∆n
y(x + x2 )(−2x − x2 )
=
n
x2
2
2x
≈ y·− 2
x
∆I
= −2y =
I
(7.32)
(7.33)
(7.34)
where the last equality comes from I ∝ n and the proportionality is given by the
phase space volume which is not modified, as the photon energies are only slightly
changed.
Take for example the Coma cluster (e.g. Birkinshaw 1990): kTe = 7.9 keV (from
X-ray observations), the size R = 500 kpc and the electron density ne = 3 · 103 cm−3 .
It follows that
y=
kT
kT
τT =
ne σT R = 4.8 · 10−5 .
2
2
me c
me c
(7.35)
In the low frequency part of the black body spectrum, the Rayleigh-Jeans part:
Iν ∼ ν 2 kT
(7.36)
∆Iν (7.34)
∆T
∆Iν
⇒ ∆T =
=
T == −2.8 · 10−4 K.
Iν
T
Iν
(7.37)
and then
This effect is observable.
It must be noted that the temperature change is negative. This is not due to
a mistake in the signs. Rather it is due to the fact that the whole distribution is
somewhat shifted to the higher energies. In the Rayleigh-Jeans part of the spectrum,
this therefore corresponds to a shift towards lower fluxes at a given frequency and
therefore to a lower temperature (see fig. 7.4).
Figures 7.5 show the Sunyaev Zeldovich effect for a sample of clusters at different
redshifts.
It is interesting to consider a naive approach to the Sunyaev Zeldovich effect in which
one considers that in average each incoming photon is displaced to higher energies
by a factor τ × ∆E/E. This is suggested as an exercise.
The Sunyaev Zeldovich effect is important in cosmology, as it allows a measure of
distances that is completely independent of the classical distance scale based on the
properties of stars. Consider a cluster (i.e. a hot electron gas for our purpose).
The X-ray luminosity, assuming an isothermal sphere was calculated in chapter
Bremsstrahlung and found to be
4
LX = πR3 · 1.4 · 10−27 T 1/2 n2 ḡ.
3
87
(7.38)
We therefore have a system of equations for the size of the sphere R, its distance D,
the electron density ne :
LX = an2e R3 , y = bne R, R = D sin θ
⇒
LX
a
4πD2 fX
a
=
R,
=
D sin θ
y2
b2
y2
b2
in which we have as observables, the X-ray flux, the angular diameter, and the
temperature (given by the shape of the X-ray spectrum), hence the parameter y.
This system can thus be solved for the distance:
⇒D=
a
y2
sin
θ
b2
4πfX
(7.39)
The distances thus obtained can be used to compute the Hubble constant as shown
in fig. 7.6.
In practice the measurement of the distances through the Sunyaev Zeldovich effect
is rather complex. The clusters are not isothermal, nor are they perfect spheres and
the density is not constant within the cluster. Nonetheless, it is very impressive to
note that the value of the Hubble constant obtained in this way, while not necessarily
better constrained than that obtained through other methods, gives the same value.
This is a very strong point towards the coherence of the cosmological description
that we have developed over the last decades.
Figure 7.4: Compton scattering of a Planck distribution by hot electrons in the case
y = 0.15. The intensity decreases in the Rayleigh-jeans region of the spectrum and
increases in the Wien region (Sunyaev & Zeldovich 1980, ARA&A 18, 537).
88
Figure 7.5: Maps of the sky around six galaxy clusters (Carlstrom et al. 2002,
ARA&A 40, 643). The contours are 28.5 GHz radio data obtained using the BIMA
interferometer. They represent levels of constant negative intensity where there is
a deficit of cosmic microwave background photons caused by the Sunyaev Zeldovich
effect. The colored areas are X-ray emission from hot gas imaged by the ROSAT
PSPC instrument. The radio hole is deepest where the emission from the hot gas is
most intense.
89
Figure 7.6: Points are distances determined with the Sunyaev Zeldovich effect as a
function of redshift for a sample of galaxy clusters (Reese et al. 2002, ApJ 581, 53).
Also plotted are the theoretical angular diameter distance relations assuming the
Hubble constant H0 = 60 kms−1 Mpc−1 for three different cosmological models: the
currently favored Λ cosmology ΩM = 0.3, ΩΛ = 0.7 (solid line); an open ΩM = 0.3
(dashed line) universe; and a flat ΩM = 1.0 (dotted line) cosmology.
90
Chapter 8
Pair processes
A further form of interaction of photons with electrons is the creation and annihilation of electron-positron pairs.
8.1
Pair creation
We first show that it is not possible to create an electron positron pair out of a single
photon in vacuum. Let Eγ be the photon energy and Ep the pair energy:
Eγ = hν, Ep = 2γme c2 , Pγ =
hν
, Pp = 2γme vx ,
c
(8.1)
vx is the component of the velocity parallel to that of the incoming photon.
The conservation of the energy requires that hν = 2γme c2 while that of the momentum requires hν
= 2γme vx . Since vx 6= c both equations cannot be fulfilled
c
simultaneously. Pair creation from a single photon can only happen in the presence
of matter (nuclei) that can pick some momentum.
We next calculate the limiting energies of two photons ( of energies 1 and 2 ) to
create a pair of particles-antiparticles (also denoted 1,2). Let
2
P~γ2 =
, p2 .
c ¯
1
, p1 ,
P~γ1 =
c ¯
(8.2)
At the threshold, the particles are created with 0 momentum:
P~p1 = (mc, 0),
¯
P~p2 = (mc, 0).
¯
(8.3)
The conservation of energy momentum is written
P~γ1 + P~γ2 = P~p1 + P~p2
which we square and solve
91
(8.4)
2P~γ1 P~γ2 = 4m2 c2
1 2
−
p
p
= 4m2 c2
1 2
c2
¯ ¯
1 2
⇒ 2 (1 − cos θ) = 2m2 c2
c
⇒2
(8.5)
(8.6)
(8.7)
or
2 =
2m2 c4
,
1 (1 − cos θ)
(8.8)
where the angle is that between the 2 incoming photons.
The cross section for this process is close to the Thomson cross section, as we are
dealing with the interaction of photons with electrons. The cross section can be
√
expressed as a function of ω = 1 2 and is found to be (see e.g. Lang astrophysical
formulae)
πre2
1+β
2
2
4
σ(ω) =
(1 − β ) 2β(β − 2) + (3 − β ) ln
.
2
1−β
(8.9)
In the regime ω ' me c2 , this reduces to
σ=
πre2
1/2
m2 c4
≈ 0.2σT .
1− 2
ω
(8.10)
Thus from eq. 8.8 one can see that photons of energy 2 > 4 × 1014 eV interact with
microwave backround photons for which 1 ' 6 × 10−4 eV to create pairs of electron
positron. Note that lower energy photons will be able to interact with other sources
of low energy photons like the stellar infrared background for example.
One can use the cross section for this process and the density of low energy photons
(400 cm−3 for the microwave background) to calculate the distance ` at which the
optical depth τ equals 1, i.e. the distance at which the universe is opaque to the
high energy photons.
τ = nγ σ`,
(8.11)
and therefore
⇒ `τ ∼1 =
1
≈ 2 · 1022 cm.
nγ σ
(8.12)
We give in fig. 8.1 the distance at wich a photon of a given energy travels to reach an
optical depth of one, given densities of the low energy background photons (stellar
and microwave).
92
Figure 8.1: Interaction length of γ-rays on the various background radiation fields
(Wdowczyk et. 1972, J. Phys. A5, 1419). Unless otherwise stated the process
concerned is electron pair production. BB denotes the 2.7 K black body radiation
(∼ 400 photons cm−3 ). Several important distances are indicated on the right-hand
side; VIRGO denotes the distance to the important cluster of galaxies at the centre
of our supercluster.
Pair creation processes are also important where the density of ω ' 1 photons is
large. In this case the photons can interact within the region to create pairs, which
can considerably modify the emergent spectrum.
Consider an optical depth of one for the pair creation:
τγγ→e+ e− ∼ 1 ∼ nγ (ω ∼ 1)σT R.
(8.13)
The photon density may be calculated from the source luminosity and size R:
nγ (ω ∼ 1) =
1
L
.
2
4πR c me c2
(8.14)
and
τγγ→e+ e− ∼ 1 ∼
L
σT
4πme c3 R
(8.15)
The process is therefore important in the hard X-ray domain whenever the luminosity of a source is high and the size small. The latter is often measured from the
variability of the source.
One uses often a unitless number, the compactness l to characterise the ratio of
93
L
:
R
l :=
L σT
,
R me c3
(8.16)
Pair processes are therefore important for
l >∼ 4π ∼ 10, because then τγγ→e+ e− >∼ 1.
(8.17)
This process is of particular importance when discussing jets in active galactic nuclei
or in gamma ray bursts.
8.2
Pair annihilation
Clearly, the process inverse to pair creation, pair annihilation, is also possible. The
cross section is essentially given by the Thomson cros section (see the specialised
literature). For an electron at rest and a positron of energy γme c2 one has:
σe+ e− −−>2γ =
πre2
· [ln(2γ) − 1].
γ
(8.18)
When the electrons and positrons are cold (slow), they can recombine in "positronium", essentially a Hydrogen atom in which the proton is replaced by a positron.
25% of the positronium is in the form of a singlet 1 S0 , while 75% is in a triplet 3 S0 .
The wave functions of the positron and the electron overlap, which leads to unstable
configurations. The lifetime of singlet state is of 1.2510−10 s and the decomposition
results in 2 photons of 511 keV each. The lifetime of the triplet state is 1.510−7 s. It
decays in 3 photons forming a continuum.
This has been known to occur in the central regions of the Galaxy and has now been
mapped by INTEGRAL as shown in Fig.8.2. The spectrum includes the line from
the singlet state and the continuum from the triplet state. The intensity indicates
that there are some 1043 annihilations per second and the line profile indicates that
the positronium is at a temperature of about 7000K. While the origin of the electrons is not difficult to understand the presence of positrons cool enough to form
positronium is much more difficult to understand and the subject of numerous discussions. Possible positro sources include nucleosynthesis and decay of dark matter
particles.
94
Figure 8.2: The electron positron annihilation line at the centre of the Galaxy
observed by the spectrometer SPI (Knoedelseder et al., Churasov et al. 05).
95
Chapter 9
Spherically symmetric accretion
(following Frank, King and Reine, accretion power in astrophysics)
Accretion is central to he study of compact object. Indeed the binding energy of a
Particle at the surface of an object is
GM m
r
(9.1)
which leads to a fraction GM
of the rest mass of the object. For a a solar mass object
rc2
with a radius of 10ḱm, typical of a neutron star, this amounts to some 13%. This
is to be compared with the nuclear energy that can be gained from the fusion of
H,which is about 0.007mc2 . I.e. the fraction of the rest mass is 0.007, a factor 20 less
than the gravitational binding energy. Another way of expresing the same notion is
to say that a proton or a neutron is bound 20 times more strongly by gravitation
at the surface of a neutron star than this particle is bound to other nucleons in a
nucleus. We will see further in these lectures how this analysis is to be transposed
in the case in which the compact object is a black hole (which is known not to have
a solid surface).
We first consider the (somewhat academic) case in which a compact object of mass
M is at rest in a gas (interstellar medium) characterised by a density ρ∞ and T∞ .
In this case we only have a (negative) radial velocity v to consider.
The spherically symmetric continuity (mass conservation) equation is
1 d 2
(r ρv) = 0,
r2 dr
(9.2)
which leads to r2 ρv = const. This constant is essentially the mass accretion rate:
Ṁ = −4πρv.
(9.3)
This shows incidentally that the mass accretion rate is independent of the radius.
The Euler equation (conservation of momentum) in a spherically symmetric steady
96
(partial time derivatives vanish, no explicit time dependence) where the external
force is the gravitation becomes:
v
GM
dv 1 dP
+
+ 2 = 0.
dr ρ dr
r
(9.4)
We still need the equation of state for a perfect gas:
P =
ρkT
,
µmH
(9.5)
where µ is the mean molecular weight, 1 for neutral hydrogen and 1/2 for fully
ionised hydrogen, and the polytropic relation:
P = Kργ ,
(9.6)
where gamma is 5/3 for a non relativistic adiabatic gas. For an isothermal gas the
equation of state 9.5 indicates that γ = 1.
The aim is to find ρ(r), T (r), v(r) and P (r).
We need to transform the Euler equation in a form that can be integrated. To do
this we use the identity
dP
dp dρ
=
dr
dρ dr
(9.7)
in Eq. 9.4 to obtain
v
dv 1 2 dρ GM
+ c
+ 2 = 0,
dr ρ s dr
r
where we have used the fact that the sped of sound cs =
equation 9.2 allows us to express
(9.8)
q
dP
.
dρ
The continuity
ρ d 2
dρ
=− 2
(r v),
dr
r v dr
(9.9)
1
c2 d(v 2 )
GM
2c2 r
(1 − s2 )
= − 2 (1 − s ).
2
v
dr
r
GM
(9.10)
which we introduce to obtain
Although this equation can still not be formally integrated, as cs depends also on
the radius inan unknown way, it may be discussed and usefull insight gained.
The following points can be made:
• At large r, the right hand side is positive, as cs is finite and r may grow.
97
2
is negative, since the velocity increases as the
• At large r the left hand side, dv
dr
radius decreases (accretion and not wind solution). The left hand side is only
positive, therefore, if cs > v, i.e. if the flow is subsonic.
• As the gas approaches the star the factor (1 −
it starts negative). It reaches zero for
r = rs =
2c2s r
)
GM
increases (remember that
GM
T (rs ) M
' 7.5 1013 ( 4 −1
cm,
2cs (rs )
10 K) M
(9.11)
a size considerably larger tan the size of the compact object.
• At small radii, the right hand side must be negative and therefore the left
hand side must also be negative which requires that the flow is supersonic.
2
• At rs , the left hand side must also vanish, hence either dv
= 0, meaning no
dr
2
2
acceleration, or v = cs , the speed equals the speed of sound.
With this a hand we can characterize the solutions through their behaviour at rs .
There are 2 trans-sonic solutions (type 1 and 2 in figure ??), one accretion and one
2
= 0 at rs (type 3
wind solution. Thre are two more families of solutions with dv
dr
dv 2
and 4) and two more with dr = ∞ at rs (type 5 and 6). Only type 1 corresponds
to an accretion. Type 5 and 6 don’t go from large radii to small ones, Type 2 and 4
are supersonic at large distances, contrary to our starting point (and unphysical as
the velocities are small and the temperature (hence speed of sound) finite. Type 3
solutions are called stellar "breeze" and correspond to slowly settling atmospheres.
Performing now the integration and looking at the mass accretion at r = rs , v = cs
(i.e. everywhere), one finds
Ṁ = 4πrs2 ρ(rs )cs (rs ) ' 1.4 1011 (
M 2 ρ(∞) cs (∞) −3 −1
(
(
gs .
M 10−24 10kms−1
(9.12)
Remembering that the luminosity is the rate of loss of binding energy:
Lacc =
GM Ṁ
,
Rstar
(9.13)
one may see that the luminosity associated with the spherical mass accretion rate
calculated in eq. refeq:mrate is of the order of 1031 ergs s−1 , considerably less than
the Eddington luminosity and to the observed luminosities of compact objects.
98
Figure 9.1: Different types of solution to the spherical accretion problem. The types
are described in the text (from Frank, King and Raine).
99
Chapter 10
Accretion disks
Accretion disks form when matter with angular momentum falls towards the centre
of a deep gravitational potential well, i.e. onto a compact object or a black hole. In
order to fall, the matter must shed some angular momentum along the way. The
material follows nearly Keplerian orbits, so that the radial velocity is always very
small compared to the azimuthal velocity. As the matter falls towards the centre
of the disk, it loses more and more gravitational energy and gets, therefore hotter
and hotter. In the central regions around a neutron star or a black hole of stellar
masses, the temperatures reached are such that the material emits in the X-ray
domain. When the central object is less compact, as is the case for white dwarfs,
the disk is less hot and emits more in the UV domain.
We will describe here only some elements of the theory of accretion disks.
Accretion disks form when the infalling material has some angular momentum as is
expected when the material comes from a binary companion.
The sequence of figures show the equipotential curves in the equatorial plane of a
binary system (Figure ??). When one of the objects is so extended that it reaches
the inner Lagragian point L1, i.e. the point at which the attraction is the same
towards both objects of the binary system, the matter can flow from one object to
the other (Figure 10.1). The material then falls in the potential well of the other
object (in our case the compact object) and organises itself in a disk at which the
movement is at all radii nearly following Keplerian orbits (Figure 10.2).
Let us consider a disk with cylindrical coordinates r, Φ and z. Let the disk be
geometrically thin, so that zdisk << r and the mass can be described by its surface
density Σ. Let further the angular velocity be Ω(r). The compact object has a mass
M and radius Rstar . The Keplerian angular velocity is
r
ΩK =
GMstar
,
r3
(10.1)
and the azimuthal velocity is
vΦ ' vΦ,K = rΩK (r).
100
(10.2)
Figure 10.1: Flow of matter in a binary system in which the L1 point is at the
surface of the normal star (from Frank, King and Raine).
We write vr for the radial velocity.
The energy loss, and hence the radiation from the disk, is calculated from the laws
of conservation.
10.1
Conservation of Mass
The conservation of mass in a ring is found as usual by counting the material which
gets into the ring and that which gets out of it.
∂
(2πr∆rΣ(r)) = vr (r, t) · 2πrΣ(r, t) − vr (r + ∆r, t) · 2π(r + ∆r)Σ(r + ∆r, t). (10.3)
∂t
For matter with negative radial velocities, i.e. falling towards the centre, the first
term on the right hand side corresponds matter leaving the annulus of width ∆r at
r, while the second term corresponds to matter entering the same annulus at r + ∆r.
Noting that the right hand side is a partial derivative, one obtains:
Figure 10.2: Schematic structure of matter spiraling in an accretion disk formed in
a binary system (from Frank, King and Raine).
101
Figure 10.3: Schematics of particle transport across a boundary. The particles
transport angular momentum as described in the text.
r
∂Σ
∂
= − (rΣvr ),
∂t
∂r
(10.4)
where vr is the radial velocity of the material, assumed to be small compared with
the Keplerian velocity.
10.2
Conservation of Angular Momentum
Keplerian rotation implies a velocity profile that is not solid rotation, there exist
therefore shear and transfer of angular momentum from annulus to annulus. It is
this transport of angular momentum (towards the outside) that allows matter to
fall in and hence to be accreted.
We consider transport of angular momentum due to random motions of the material
in the disk. One source of such motion is the thermal movement of the particles
(we will see, however, that this is not sufficient to explain the observed properties
of disks, it may nonetheless be useful to keep this as an image in the following).
Consider random motions of velocity v̄ and of mean free path λ. λ is much smaller
than the size of the disk. Figure 10.3 shows two annuli and particles "A" that move
from the inside annulus towards the outside one and particles "B" that travel in the
opposite direction.
The azimuthal velocities at the different radii we consider are:
vφ (R) = Ω(R) · R
102
(10.5)
λ
λ
λ
vφ (R − ) = Ω(R − )(R − )
2
2
2
λ
> vφ (R + )
2
(10.6)
(10.7)
Particles "A" transport towards the outside ring a momentum mA vφ (R − λ2 ) larger
than the momentum transported in the other direction by the particles "B" which
is mB vφ (R + λ2 ) (see eq. 10.7).
The net transfer of angular momentum L = mvφ R across R per unit time is given
by
∆L
λ
λ
λ
λ
= G(R) = ṀA (R + )vφ (R − ) − ṀB (R − )vφ (R + )
∆t
2
2
2
2
λ
λ
λ
λ
λ
= ṀA (R + )Ω(R − )(R − ) − ṀB (R − )Ω(R + )(R +
2
2
2
2
2
2
2
λ
λ
λ
λ
= ṀA (R2 − )Ω(R − ) − ṀB (R2 − )Ω(R + )
4
2
4
2
λ
λ
' ṀA R2 Ω(R − ) − ṀB R2 Ω(R + )
2
2
(10.8)
λ
)(10.9)
2
(10.10)
(10.11)
Assuming that this process is random there is no net transport of matter :
ṀA = ṀB = v̄Σ2πR
(10.12)
and we obtain
G(R) = v̄Σ2πRλR2
dΩ
.
dR
(10.13)
We introduce the cinematic viscosity ν = λv̄ and write eq. 10.13 as
G(R) = 2πRνΣR2
dΩ(R)
dR
(10.14)
which gives us the transport of angular momentum across any R. Consider now an
annulus ∆R. The amount of angular momentum it contains is
2πR∆R · Σ · R · v(R) = 2πR∆RΣR2 Ω(R) = L(R).
(10.15)
This angular momentum will be changed by the transport at both boundaries (R +
∆R and R − ∆R) as we have just calculated (see Figure 10.4) and by the transport
linked with a general flow of matter. Thus we have
dL
d
∆R
∆R
=
(Ṁ R2 Ω)∆R + G(R +
) − G(R −
),
dt
2 {z
2 }
|dR
{z
}
|
dG
∆R
dR
d
2π∆R dR
(RΣ(−vR )R2 Ω)
103
(10.16)
!
!
!
!
!
!
Figure 10.4: Schematics of the conservation of angular momentum in an annulus.
The angular momentum is transferred at both boundaries of the annulus as described
in Figure 10.3.
the first term being given by the net radial velocity which carries a flux of angular
momentum
L(R) = 2πR∆RΣR2 Ω(R).
(10.17)
The net variation of the angular momentum of the annulus is therefore
d
∂
∂G
(2πRΣR2 Ω) = −2π
(RΣvR R2 Ω) +
,
dt
∂R
∂R
(10.18)
or, re-organising:
R
d
∂
1 ∂G
(ΣR2 Ω) +
(RΣvR R2 Ω) =
.
dt
∂R
2π ∂R
104
(10.19)
10.3
Static Disk
We now consider a disk which properties do not depend explicitly on the time. This
means that all temporal variations are small compared to R/vR . In this approxima∂
tion, ∂t
= 0 and mass conservation 10.4 becomes
RΣvR = const.
(10.20)
Ṁ = 2πRΣ(−vR ),
(10.21)
which we write
thus introducing the mass accretion rate Ṁ (remember that vR is negative). The
conservation of angular momentum (eq. 10.19) can also be trivially integrated to
become
RΣvR R2 Ω =
c
G
+
2π 2π
(10.22)
in which we explicitly introduce G as we calculated it in eq. 10.14 to obtain
ΣvR Ω = νΣ
dΩ
c
+
dR 2πR3
(10.23)
or, re-arranging slightly:
−νΣΩ0 = −ΣvR Ω +
c
.
2πR3
(10.24)
For a central star in slow rotation (actually for any stable star), the rotation velocity
of the star surface is less than the Keplerian rotation at the surface. On the other
side, Ω increases as the distance to the star decreases (Ω0 < 0). There exists therefore
a radius at which Ω ceases to increase towards decreasing distances and decreases
again. I.e. there is a radius R? at which Ω0 = 0. We can use this radius to deduce
the integration constant of eq. 10.24:
c
=
(10.21)
==
2πR?3 ΣvR Ω(R? )
(10.25)
−R?2 Ṁ Ω(R? ) = −Ṁ (GM R? )1/2 .
(10.26)
Writing explicitly
Ω0 = −
3√
GM R−5/2
2
in eq. 10.24 one finds
105
(10.27)
Ṁ
νΣ =
3π
1−
R?
R
1/2 !
.
(10.28)
We can use this result to calculate the rate of dissipation of energy in the disk due
is a force, hence ∂G
∆R is an energy and ∂G
∆RΩ is a power.
to the shears D(R): ∂G
∂r
∂r
∂r
It corresponds to the change of L which changes in its magnitude and not in its
direction. The rate of dissipation of energy is therefore:
Ω
∂G
∂
∂Ω
∆R =
(ΩG)∆R − G
∆R.
∂R
∂R
∂R
(10.29)
The first term on the right hand side, when integrated gives a term at the inner and
outer rims of the disk and the second one gives the local dissipation in any annulus.
When expressed per unit area it becomes
D(R) = G
dΩ 1 (10.14) νΣ
==
(RΩ0 )2 .
dR 4πR
2
(10.30)
For a Keplerian disk and with eq. 10.28 this becomes
"
1/2 #
3GM Ṁ
R?
D(R) =
1
−
.
8πR3
R
(10.31)
Note that this is not explicitly depending on the viscosity ν. This means that we
assume that there exists a viscosity such that, i.e. we assume that the micro-physics
is such that, the matter in the disk can have a Keplerian velocity. This assumption
is not trivial at all.
The luminosity of the disk is the integral of eq. 10.31 over the whole surface. Note
that in eq. 10.31 we have taken into account that the disk has 2 faces. Hence the
factor 2 below:
Z
∞
L = 2
D(R)2πR dR
(10.32)
R?
=
GM Ṁ
.
2R?
(10.33)
This corresponds to half of the binding energy at the surface of a star of radius
R? . This comes from the fact that the matter at the inner rim of the accretion disk
has still the kinetic energy corresponding to the Keplerian velocity at this distance.
When calculating the binding energy at the surface of a star one usually neglects
the rotation there, hence the factor 2. When matter is accreted onto a slow rotating
star, this means that there is a substancial amount of energy that is not dissipated
in the disk, but in a boundary layer in which the kinetic energy of the matter on
the last orbit in the disk is dissipated and radiated.
106
Figure 10.5: Emission spectrum of an accretion disk (from Frank, King and Raine).
10.4
Spectrum of the disk
The emission spectrum can be calculated if we assume that the disk is optically thick
and therefore that the dissipated energy at the radius r given by 10.31 is radiated
like a black body. In this case:
D(r) = σT 4 .
(10.34)
With 10.31 we can give the temperature profile of the disk:
T (r) = [
Rstar 1/2 1/4
Rstar 3/4
3GM Ṁ
(1 − (
) )] ∝ Tstar · (
) .
3
8πr σ
r
r
(10.35)
The last equation being valid at large r and
Tstar = (
3GM Ṁ 1/4
) .
3
8πRstar
σ
(10.36)
This temperature corresponds to that of the inner radius of the disk, that is the
maximum temperature of the accretion flow in this geometry.
For objects accreting at the Eddington luminosity (see below), i.e. for objects in
which the luminosity is proportional to the mass and for which Ṁ is proportional
to the mass, this results in a luminosity dependency of the maximum temperature
with the power of -1/4.
For white dwarfs, typical temperatures are of the order of few 104 K, while for
neutron stars or black holes, the temperatures correspond to some 107 K. The first
objects will therefore be observable in the UV, while the more compact objects will
mainly radiate in the X-rays.
107
To calculate the spectrum one just has to integrate the black body emissivity Bν (T )
of the proper temperatures:
Z
Rout
Sν ∝
Bν (T (r))2πrdr.
(10.37)
Rstar
The resulting spectrum has the exponential tail of the highest temperature in the
disk at high frequencies, the ν 2 dependency of the Rayleigh Jeans part of the lowest
temperature and in between a ν 1/3 dependency. The shape is given in the Figure 10.5.
10.5
The Viscosity
In eq. 10.14 we introduced the cinematic viscosity ν = λv̄ without specify on either
what λ the mean free path or v̄ the random velocity of the particles making up the
disk is. In molecular viscosity, λ = λD the deflection lenght and v̄ ' cs the speed of
sound. We can then compare viscosity to inertial terms using Reynolds number:
vφ2 /R
inertial
=
.
Re =
viscosity
λv̄vφ /R2
(10.38)
This shows that molecular viscosity is very far from being sufficient to transport
angular momentum outwards in the disk.
Another form of viscosity is given by turbulent motions in the disk. The characteristic size of this motion is λ ' H the height of the disk and the turbulent velocity
is a fraction of the speed of sound v̄ = αcs , α ≤ 1. This gives a viscosity:
ν = αcs H.
(10.39)
Although this is not a real physical explanation for the disk velocity, it allows a
parametrisation of the problem and puts the unknown physics in a single parameter.
Assuming this parameter to be a constant in the disk all the equations for the disk
structure can be expressed algebrically (see Frank, King and Raine).
10.6
Observational Evidence for Accretion Disks
The test case for the study of the temperature dependence is given by systems in
which the companion is a faint optical object, like in Low Mass X-Ray Binaries or
in Cataclysmic Variables (see chapter 17). Some binary systems with a white dwarf
are seen along the orbital plane. In such systems the companion star will occult
parts of the disk at each revolution, thus allowing some level of probing the disk
temperature structure. Figure 10.6 shows the schematic view of such an eclipse and
108
illustrates that the hot parts of the disk, small in extent, will produce deeper and
shorter eclipses than the cooler outer parts. Figure 10.7 shows measurements of this
effect on the Z Cha white dwarf binary system.
Figure 10.6: (a) Eclipse geometry for a binary system with an accretion disk. Because of the very different surface brightness distributions at short and long wavelengths, the light curves predicted are very different: deep and narrow at short
wavelength (b), shallow and broad at long wavelength (c) (from Frank, King and
Raine).
109
Figure 10.7: The eclipse mapping technique used to find the surface brightness
distribution of the accretion disk in the dwarf nova Z Cha. The observations were
made in an outburst, so that the disc dominates the optical light. (a) Eclipse
light curves (B, U, V from top). (b) Effective temperature distribution given by
maximum-entropy deconvolution, compared with eq. 10.35 for various values of Ṁ
(from Frank, King and Raine).
110
Chapter 11
Advection Dominated Accretion
Flows
After having studied the standard accretion disk model by Shakura & Sunyaev, one
can ask the following question: are there any different solutions to the hydrodynamical equations of an accreting flow, that were derived in a previous lecture? The
answer is yes. Several other solutions to these equations have been found, each
describing a different accreting situation. In this lecture, we focus on a solution
for low accretion rate flows, called optically thin ADAF (for Advection Dominated
Accretion Flow).
11.1
Introduction
The main idea of the Shakura & Sunyaev accretion disk is that the gravitational
energy gained by the electrons and the ions during accretion is fully radiated. The
luminosity is related to the accretion rate by the formula
LEdd = η ṀEdd c2 ,
(11.1)
where η ≈ 0.1 is the accretion efficiency. Thus, for a disk radiating at the Eddington
luminosity
38
LEdd ≈ 1.3 · 10
M
M
h
ergs i
,
s
(11.2)
the accretion rate is
−8
ṀEdd ≈ 1.3 · 10
M
M
M
.
yr
(11.3)
where M/M is the total mass of the accreting object, in solar units.
For a supermassive black hole of ∼ 109 M , e.g. in 3C 273, the deduced accretion
111
rate is of the order of 10 M /yr. This statement is based on the assumption that
all the gravitational energy gained by viscosity is radiated; but is it always the
case? To answer this question, we must have a look at the other solutions for the
hydrodynamical equations. To the present day, four different sets of solutions to
these equations have been found:
1. The standard accretion disk solution discussed in chapter 10.
2. The SLE (1978) solution (for Shapiro, Lightman and Eardley), which consists
in a 2-temperature plasma accreting at the Eddington rate, but is gravitationally unstable.
3. The optically thick ADAF, accreting at a super-Eddington rate, but radiating
less than expected, since most of the radiation is trapped inside the disk and
carried toward the central body.
4. The optically thin ADAF, which also consists in a 2-temperature plasma with
a sub-Eddington accretion rate.
In the following, we will only consider optically thin ADAFs, and we will just refer
to them as ADAFs.
11.2
The optically thin ADAF model
The main idea of the ADAF model is to consider an accretion flow in which most
of the gravitational energy gained by viscosity is stocked inside the ions as internal
energy instead of being radiated and is advected toward the central object. If this
object is a black hole, the energy is carried into the black hole and lost. Hence
the luminosity of such a model is much lower than the corresponding luminosity
of standard disk. In the accreting plasma, the temperature of the ions is much
higher than the one of the electrons (approximatively 2 orders of magnitude), and
as the system is optically thin, the plasma radiates with thermal and non-thermal
processes (bremsstrahlung, synchrotron radiation, inverse Compton processes). In
the following, we give a more detailed description of this model.
We consider a plasma of density ρ(R), mean temperature T (R) and radial velocity
v(R), accreting onto a central object.
The number of particles passing through a surface dσ per unit time is given by ρv dσ
(see fig. 11.1); consequently, the flux of particles is given by ρv. The first law of
thermodynamics gives us the internal energy carried per ion,
du = T ds,
(11.4)
s being the entropy of the ion. So the internal energy carried per ion and unit radius
is given by
ds
duadv
=T
.
(11.5)
dR
dR
Thus, the total energy transfer per unit volume is given by
112
Figure 11.1: Flux of particles through a surface dσ
Uadv = ρvT
ds
.
dR
(11.6)
Using (11.6) we can now write the energy conservation equation as
ρvT
ds
= Q+ − Q− ,
dR
(11.7)
where Q+ is the energy gained by the plasma due to viscosity (see the previous
lecture for details),
1/2
3GM Ṁ
R∗
+
Q = D(R) =
,
(11.8)
1−
8πR3
R
and Q− is the energy radiated by the plasma (thermal, synchrotron, bremsstrahlung,
etc.).
In the standard model, the energy gained by viscosity is fully radiated, so Q+ = Q− ,
and there is no gain of thermal energy; on the other hand, in the ADAF model,
Q+ Q− , so Uadv cannot be neglected any more, and the mean temperature of the
plasma increases.
11.3
Properties of ADAFs
In this section we describe a few important properties of the ADAF model (geometry,
density profile, etc.) without going into detail. For further details, see Narayan et
al. 1998, astro-ph/9803141.
• For a given viscosity coefficient α, there exists a critical accretion rate, Ṁcrit ,
such that if Ṁ > Ṁcrit , the system is well described by a standard accretion
disk, and if Ṁ < Ṁcrit , the system is in ADAF regime. Figure 11.2 shows the
113
Figure 11.2: The log Ṁ − log L diagram according to numerical simulations of the
hydrodynamical equations, with α = 0.3 (figure from Narayan et al. 1998). The
vertical dotted line corresponds to Ṁcrit , and the dashed line shows the Eddington
luminosity as a function of the accretion rate, L = η Ṁ c2 .
results of numerical simulations of the hydrodynamical equations for a given
α. The figure shows clearly a break in the log Ṁ − log L diagram around
the critical accretion rate. Hence, if an astrophysical system accretes matter
at a rate that is close to the critical accretion rate, a significant variability is
expected, since in this region the slope of the log Ṁ − log L function is very
high. This explanation has been used to describe the variability of several
systems, e.g. the black hole candidate source Cyg X-1.
• Unlike the standard solution, the geometrical form of an ADAF is not a disk,
since in such a model the height of the plasma is proportional to the radius.
Therefore, on the geometrical point of view an ADAF model is closer to a
spherical accretion model.
• The density profile deduced from the equations shows ρ ∼ R−3/2
• The ADAF model provides an interesting solution to the problem of weak
accreting systems, but it is however unstable on long time scales. The thermal
energy stocked in the ions can indeed increase to the threshold where the flow
is reversible, producing huge jets. The model predicts that ejected particles
would reach infinity with a positive energy, which is inconsistant. Work is still
in progress to solve this problem.
114
Figure 11.3: Schematic spectrum of an ADAF (figure from Narayan et al. 1998).
The three different lines in the optical-UV band show how the spectrum of inverse
Compton processes depends on the accretion rate. The lower line corresponds to
a very low Ṁ , the dashed line to an intermediate Ṁ and the dotted line to an
accretion rate that is close to the critical value.
11.4
Spectrum of an ADAF
To the present day, the spectrum of an ADAF has not been clearly determined, but
we can give some qualitative ideas about how such a model would radiate. The
schematic spectrum is shown in fig. 11.3.
In the radio-IR band, the emission is due to synchrotron radiation from different
parts of the plasma, since the velocities are higher closer to the central body (see
fig. 11.3). In the optical-UV band, the plasma radiates mainly through inverse
Compton processes, the soft synchrotron photons scattering off the hot electrons,
and producing a harder radiation. This effect is highly dependant on the accretion
rate, as shown in fig. 11.3, so the qualitative analysis is uncertain in this band.
In the X-ray band, the emission is due to bremsstrahlung, and shows a cut-off in
hard X-rays. The model predicts a γ-ray component coming from the decays of π 0
produced in proton-proton collisions, but to the present day there is no evidence
for this kind of emission. Finally, thermal emission from the electrons and the ions
needs to be added to all these processes, which makes the spectrum of an ADAF
even harder to describe. Work is currently in progress in order to have a more precise
description of the spectrum of an ADAF.
11.5
Application to astrophysical objects
ADAFs have been used to describe several different kinds of astrophysical objects.
In this section, we present briefly a few of these cases, and focus in the next section
on the Sgr A* source.
115
• ADAFs have been widely used to describe the large variability of black hole
candidates in X-ray binaries, e.g. for the Cyg X-1 source. If the orbit of the
companion star around the black hole is very eccentric, then the distance from
the star to the black hole changes, and one can expect some important changes
in the accretion rate when the star enters the roche lobe of the black hole. As
a consequence, if the system is in a quiet ADAF regime when the star is away
from the black hole, it might reach the critical rate when the star comes close
to the periastron, and therefore switch to the standard thin disk regime and
radiate at the Eddington luminosity. So we can expect this kind of object to
be highly variable, which is indeed the case.
• The ADAF model might provide some elements of explanation for the difference between active and non-active galaxies, which is hard to understand
since there is some evidence for the existence of supermassive black holes in
non-active galaxies like M87, where the luminosity of the nucleus is several
orders of magnitude lower than the Eddington luminosity. AGNs would then
be described by the standard accretion disk model, while non-active galactic
nuclei would be described by ADAF models.
• In the next section, we focus on the description of the source Sgr A*, which is
associated with the central object of our galaxy.
11.6
The galactic centre and the source Sgr A*
Sgr A* is a powerful radio-submm source which is invisible in the optical-UV band,
but has a counterpart in the X- and γ-ray bands. Fig. 11.4 shows 2 images of the
galactic center, a VLT image in the K band (near-IR) and an INTEGRAL image in
the 20-60 keV band.
To determine the nature of Sgr A*, people have studied the motion of visible stars
close to the source. Schoedel et al. 2003 report on 10 years (1992-2002) of NTT-VLT
observations of a 7 solar mass star (refered to as S2) close to Sgr A*. Their results
are shown in figure 11.5.
The best fit to their data is a keplerian orbit around a point mass, with an orbital
period PS2 = (15.2 ± 0.6) yr and a periastron speed of the order of 5’000 km/s. The
derived point mass is of Mcentral = (3.7 ± 1.1) · 106 M .
This amount of matter is concentrated inside a region of radius ' 0.8 light days
(=172 A.U.). Simulations have shown that a cluster of stars in such a small radius
would be unstable (see Schoedel et al. 2003 for further details). Hence, the central
object can only be a black hole of a few million solar masses.
The luminosity one can expect from matter accreting on the black hole at the Eddington rate is
ergs
ergs
M
38
' 1044
.
(11.9)
L = 1.3 · 10
M
s
s
116
Figure 11.4: The galactic centre. On the left, INTEGRAL ISGRI 20-60 keV image (1
over 2 degrees field). On the right, VLT NAOS image in the K band (λ = 2.18µm).
Figure from Scheodel et al. 2003.
Figure 11.5: On the left, the orbit of the S2 star around Sgr A* as determined by
Schoedel et al. 2003. On the right, the orbital parameters of the best fit to the data.
117
However, the observed bolometric luminosity of Sgr A* is
LSgr A∗ ' 1036
ergs
,
s
(11.10)
which is ∼ 8 orders of magnitude lower than the corresponding Eddington luminosity.
To explain this difference, people have claimed that the ADAF model could be used,
but further development is needed in order to have a better understanding of this
field of interest.
118
Chapter 12
Particle acceleration
We have seen in the discussion of the synchrotron radiation that there are high
energy electrons in different environments in nature. One also knows since the
beginning of the 20th century that there is a "penetrating" radiation that comes
from outer space and is measurable on the ground. This latter knowledge stems from
a long history of very ingenious experiments and deductions that started in 1900
(see e.g. the first chapter of M. Longair High Energy Astrophysics, CUP 1992). The
main steps in this history are first the remark that an electroscope (the apparatus
made of 2 thin sheets of metal that separate when both charged) discharges slowly
in time, also in the absence of any known source of ionisating radiation. This loss is
due to some, then unidentified ionising flux (remember that radioactivity was then
not known). However, in one experiment one of these electrometers was taken to
the top of the Eiffel tower by Wulf in 1910. It was seen that the ionisation flux
decreased by about a factor 2 from the ground to 330m. This decrease was much
less than would be expected if the ionisation was due to gamma rays originating at
the surface of the Earth. The next major step was due to Viktor Hess who flew
in an open air balloon to some 5 kilometers and measured the ionisation flux as he
ascended. He showed that the flux decreases for the first 1.5km and then increases
with altitude. From this result Hess concluded that there must be a very "high
penetration power" radiation coming from outer space.
It was then in 1929 that Bothe and Kohlhörster used the newly developed Geiger
counters to show that the cosmic radiation, as it was called, was made of charged
particles rather than gamma rays. This was shown by placing two counters on
either side of a high absorption medium and registering coincidences. Were the
cosmic radiation particles gamma rays, such coincidences would be very unlikely,
however, charged particles would be expected to trigger both counters, provided
they crossed the absorption medium. It was thus conclusively shown that there is
a flux of very energetic particles impinging onto the Earth from outer space. The
particle energy was indeed to be very high to explain the very important "penetration
power" necessary for the particles to cross the atmosphere and be observed at sea
level.
This work has continued over the following decades, yielding a wealth of data on
the cosmic rays. Figure 12.1 shows their energy spectrum from 108 to 1021 eV.
The low energy portion of this curve is modulated by solar activity. A strong solar
119
Figure 12.1: The all-particle spectrum of cosmic rays (from S. Swordy). The arrows
and values between parentheses indicate the integrated flux above the corresponding
energies.
activity increases the pressure in the inner solar system and pushes the low energy
cosmic rays out while having no effect on the higher energy particles. In the lower
energy domain, the composition of the cosmic rays can be measured and is given in
Figure 12.2.
The spectrum shown on Figure 12.1 is characterised by a power law of index 2.5-2.7
for energies below about 1015 eV and 3.08 for energies between 1015 eV and 1019 eV.
The break is commonly called the knee. Above 1019 eV, there is evidence for a hardening of the cosmic ray distribution. The break in the spectral energy distribution
here is called the ankle. The mere existence of these very high cosmic rays is a
puzzle to which we will come back at the end of this chapter.
The shape of the spectral energy distributions is thus described by power laws of the
form N (E)dE ∝ E −γ dE extending over very wide energy domains. The existence
of power law energy distributions for particles is also evident from the broad band
120
Figure 12.2: Abundances of cosmic rays nuclides at lower energies compared with
the solar system ones. The abundances are all normalized to 28 Si (K. Lodders 2003,
ApJ, 591, 1220).
synchrotron spectra and Compton emission spectra discussed in chapters 5 and 6.
These distributions differ very markedly from thermal distributions and are therefore
generated by non thermal phenomena.
Charge acceleration may be caused by very strong electric fields. Such fields are
found in the magnetosphere of fast rotating neutron stars where indeed electrons are
accelerated to high energies and then lose some of this energy in radio emission. This
will be briefly described in chapter 15. A very different type of particle acceleration
is found in multiple interactions of particles in which a small amount of energy is
gained at each interaction. This is called stochastic particle acceleration. We will
describe here one such process that is in some sense similar to the acceleration of a
ping pong ball that is being squeezed between the table and a bat.
12.1
Second order Fermi acceleration
We will consider the interaction of a population of light particles (of mass m) with a
number of massive "mirrors" (of mass M) with random motions (Figure 12.3). Let
the particles be relativistic and the mirrors non relativistic and the interaction be
an elastic collision. This treatment is due to Fermi (1949 and 1954).
The 4-impuls of any particle is pµ = m0 · umu , u0 = γ · c and uµ = γv i . The energy
conservation during an interaction from an initial state "i" and a final state "f" reads
1
2
2
∆E = mc2 (γp,f − γp,i ) = M (vM,i
− vM,f
).
2
The impuls conservation reads:
121
(12.1)
v
m
V
M
Figure 12.3: Interaction geometry for the second order Fermi acceleration process,
illustrating the collision between a particle of mass m and a cloud of mass M .
M vM,f + mγp,f vp,f = M vM,i + mγp,i vp,i ,
(12.2)
For collisions during which the particle energy changes by only a small amount
γi ' γf ' γ, vp,f ' −vp,i and vM,f ' vM,i . With these approximations and some
algebra Eq. 12.2 becomes
2
2
M 2 (vM,i
− vM,f
) ' 2M (mγc2 )(−2
v p,i v M,i
),
c2
(12.3)
which can be inserted into eq.12.1 to obtain
∆E = −2E
vp,i vM i
.
c2
(12.4)
The probability of collisions is proportional to the relative velocity of the particles
(and their densities). Thus "head on" collision probability is given by vp + vM , while
"rear" collisions are proportional to vp − vM . Thus the average energy gain per
collision is
< ∆E >=
vp,x vM,x vp − vM
vp,x vM,x
vp + vM
· 2E
−
· 2E
,
2
2vp
c
2vp
c2
(12.5)
which simplifies to
< ∆E >= 2
2
vM
E.
c2
(12.6)
Let the characteristic time elapsed between 2 collisions be tF . The average energy
gain per unit time is then
dE
< ∆E >
2v 2
E
∝
= M2 E =:
,
dt
tF
tF c
τacc
122
(12.7)
2
with τacc = t2vF 2c , the acceleration time. We may therefore relate the time a particle
M
has spent in the region with its energy:
t
E = e τacc , or t = τacc ln E.
(12.8)
(The time at which the process started is 0.)
In the case where the rate with which accelerated particles leave the acceleration
region is
1
dn
=
· n(t),
dt
τesc
(12.9)
with n(t) the density of particles and τesc independent of the particle’s energy, we
have
t
n(t) = n0 · e− τesc .
(12.10)
Since eq. 12.8 gives a one to one relationship between time and energy, we can
substitute it into eq. 12.10 and obtain the energy distribution of the accelerated
particles:
n(E(t)) = n0 e−(
ln E·τacc
τesc
τ
τacc
) = n eln E − τacc
esc
= n0 · E − τesc .
0
(12.11)
The differential distribution, i.e. the number of particles between E and E+dE is
trivially obtained from eq. 12.11:
dn(E) ∝ E −γ ,
(12.12)
2
with γ = 1 + ττacc
= 1 + 2vt2F cτesc , where we have re-introduced the acceleration time
esc
M
from the properties of the collisions.
This has already one of the properties we want to see in the distribution of particles, namely the shape of a power law. This result was obtained by Fermi in
1949 (Phys.Rev. B, 75, 1169). A paper well worth reading for the elegance of the
derivation.
The result has, however, a number of weaknesses. For one it is a slow process of
second order in vM
, as is easily seen from eq. 12.6. Remember also that vM is
c
the velocity of the massive "mirrors" and is non relativistic. The process is called
of second order Fermi process and is thought to be very ineffective. The second
order nature of the process is to be found in the fact that only the probability
for the "head on" and "rear" collisions differ, not the energy exchanged. A first
order process would imply that both interactions lead to different energy exchanges
between the "mirrors" and the accelerated particles.
A second difficulty lies in the fact that many observations lead to very similar indices
of particle distributions and that the cosmic ray energy distribution is well described
123
by power laws of constant indeces over very wide energy ranges. It can be seen from
eq. 12.12 that this implies that the acceleration and escape times are in some sense
"universal" (or at least that their ratio is). This is very unlikely in different physical
environments.
A third difficulty lies therein that for protons of E ≤ 100 MeV, ionisation is a very
important process in any medium in which the protons lose a considerable amount
of energy. It therefore seems impossible to take thermal particles of energies of MeV
or less and accelerate them beyond 100 MeV, where the Fermi process would begin
to dominate over the losses. This is known as the injection problem.
12.2
Diffusive shock acceleration
It was observed in so called collisionless plasmas heated by electromagnetic radiation, for example in thermonuclear fusion experiments, that there was also a small
fraction of high energy particles that formed a high energy tail of the thermal particle distributions. The theory of turbulent plasmas was then developed (in the
1960’s). In these plasmas one finds a turbulent plasma and E and B fields as well as
quasi particles (plasmons). It was shown that this combination can lead to particle
acceleration. This then leads to the development of the diffusive shock acceleration.
The ingredients of this theory are a collisionless shock in a plasma. By collisionless
shock one understands a shock of width less than the mean free path of the particles.
On both sides of the shock, interactions between the particles and fluctuations of
the electromagnetic field isotropise the distributions. See Figure 12.4.
This configuration leads to a first order particle acceleration process. In order to see
this consider a non relativistic particle that crosses the shock twice, after having been
"isotropised" on either sides of the shock and look for the change in its momentum
shock
u1
u2
Zone 2
Zone 1
Figure 12.4: Interaction geometry for the diffusive shock acceleration process. The
particle distribution is isotropised at both sides, outside the shock through diffusion
on collective magnetic quasi-particles.
124
as measured from the fluid on either side. (We follow here John Kirk in the Saas
Fee course 24)
When crossing the shock, a particle does not change its velocity, neither in direction
nor in magnitude (the shock is collisionless). However, seen in the different rest
frames of the fluid on either side of the shock, the velocities and the momentum do
differ in the following way (µ is the cosine between p2 and the x-axis):
The y and z component of the relative momentum don’t change, since the particle
moves parallel to the x axis, but the x component changes by a factor m(u1 − u2 ),
which is the difference of momentum due to the velocity change of the plasma:
pz1 = pz2 ;
px1 = px2 − m(u1 − u2 ).
p y1 = p y2 ;
(12.13)
Using the conservation of the square of the momentum one finds
(12.14)
|p1 |2 = p2x1 + p2y1 + p2z1
¯
= |p2 |2 − 2mp2 (u1 − u2 )
(12.15)
¯
¯ 2
¯ ¯
= |p2 | − 2µ1 |p2 |m∆u
(12.16)
¯
¯
where we have neglected the term in (u1 − u2 )2 and µ1 is the cosine of the angle
¯
¯
between p2 and (u1 − u2 ).
¯
¯
¯
In order for the particle to cross the shock front we must have µ1 v1 > −u1 , and so
µ1 has to satisfy the condition 1 > µ1 > − uv11 .
Using |p2 | ' |p1 | one finds
¯
¯
|p1 |2 = |p2 |2
¯
¯
' |p2 |2
¯
!
2µ1
1−
m∆u
|p2 |
!
¯
2µ1
m∆u
1−
|p1 |
¯
(12.17)
(12.18)
And hence,
!
µ1
|p2 | ' |p1 | 1 +
m∆u
(12.19)
|p1 |
¯
¯
¯
Behind the collisionless shock zone the particles are re-isotropised. Our particle can
therefore cross the shock again, it will have a momentum
µ2 ∆u
p̄1 = p2 1 −
,
v2
(12.20)
where µ2 must satisfy the condition −1 < µ2 < − uv22 .
The difference of momentum of the particle after these two crossings, when it is back
in the zone 1 using Eqs 12.19 and 12.20
125
∆p = p̄1 − p1
µ2 ∆u
− p1
= p2 1 −
v2
µ2 ∆u
µ1 ∆u
1−
− p1
= p1 1 +
v1
v2
µ1 µ2
'
−
p1 ∆u
v1
v2
(12.21)
(12.22)
(12.23)
(12.24)
The gain of momentum per particle is therefore
∆p
∆u
'
(µ1 − µ2 ).
p1
v
(12.25)
To calculate the mean gain of momentum one considers a population of isotropised
particles. The probability of crossing the shock region is therefore proportional to
|µv + u|. Hence,
∆p
p1
R −u2 /v2
dµ2 |µ2 v2 + u2 | ∆p
dµ
|µ
v
+
u
|
1
1
1
1
p
−1
−u/v1
= R1
R −u2 /v2
dµ1 |µ1 v1 + u1 | −1
dµ2 |µ2 v2 + u2 |
−u/v1
R1
Performing the integration using eq. 12.25, we find at first order in
∆p
4∆u
, with v ' v1 ' v2 .
=
p1
3v
This is indeed a first order process in
(12.26)
∆u
v
(12.27)
∆u
.
v
The spectrum of the particles can also be calculated under the assumption that
escape from the shock region is independent of the energy and is found to depend
only on the shock properties. This means that strong shocks will systematically lead
to similar particle distributions. We can therefore understand that the difficulties
of the second order Fermi process are largely overcome in this scenario.
12.2.1
Highest energy particles
The spectrum shown in Fig. 12.1 shows evidently that there are particles at energies
up to 1020 eV. This is a macroscopic energy concentrated n a microscopic particle.
This is the energy that corresponds to a powerfully served tennis ball. The mere
existence of particles at these energies raises a number of questions: How are they
accelerated? Where are they accelerated? Can they propagate all the way to us?
First one can set the scene by looking at the Larmor radius:
RL =
v
,
νL
where
126
(12.28)
νL =
eB
γmc
(12.29)
is the Larmor frequency. At 1019 eV the Larmor radius is 1022 cm or 100 kpc for a
typical magnetic field of 10 microGauss.
Particles can only be accelerated in regions where they stay. Therefore, if the confinement is magnetic, and it is difficult to imagine any other confinement vessel at
these energies, the region must be larger than the Larmor radius of the particles at
the highest energies observed. This can only be in the lobes of radio galaxies. Note
that if the field is much larger, the region can be correspondingly smaller. The sizes
are nonetheless of galactic scales.
At very high energies hadrons interact with photons to create π0 particles. This
reaction has a threshold at
γ · = 144.7M eV
(12.30)
and a resonance at 1.2 GeV. This means that there must be a decrease in the flux
of very high energy particles for energies γ ≥' 1GeV . For photon energies of 3 K,
i.e. 10−4 eV the corresponding particle energy is of the order of 1020 eV. This cut-off
is called Griesen-Zatsepin-Kuzmin, or GZK, following the names of the people who
predicted this effect. With the cross section of the π0 creation and the density of the
micro wave background photons one can calculate that the Universe is opaque to
the creation for distances larger than about 50 Mpc. This distance hardly includes
many potential high energy cosmic ray accelerators.
There is a large project now to measure the highest energy cosmic rays, the Auger
project. This works by observing the interaction of the incoming cosmic ray with the
atmosphere. This interaction creates a shower of particles that leave an observable
signature in the form of light (Cherenkov and fluorescence) and particles. The rates
(events per square km and century) imply very large area coverage. The instrument
is located on a high plateau in Argentina (Fig.12.7).
The Auger detector started taking data in 2004. There wre some contradictory
claims on the rate of the highest energy events with one collaboration (AGASA)
claiming rates that indicated that the GZK cut-off was not observed. The data as
of 2006 (Watson, CERN Courrier July 2006) are shown in Fig. 12.6. They indicate
that the Auger instrument is probably seeing less events at the highest energies than
claimed by the AGASA collaboration and therefore more in line with the presence
of a GZK cut-off. Better statistics is, however, needed to make definitive claims.
The main difficulty resides in the measurement of the energy of each event. With a
spectrum as steep as that of the cosmic rays, even a small uncertainty in the event
energies leads to large effects on the spectral slope.
In the November of 2007 the Auger collaboration (Science 318, 5852,938) claimed a
correlation between the highest energy events and the position of AGN as reported
in the Veron catalogue of AGN. This correlation is shown in Fig. ?? from the Science
paper. This correlation is in some sense expected if the AGN are the acceleration
sites, and if the cosmic rays at these energies travel along lines such that they are
127
only weakly deflected by the extragalactic magnetic field. It is possible that the
correlation is rather simply with very large scale structures in the Universe that are
themselves correlated with AGN.
Figure 12.5: The Auger detector in Argentina.
Figure 12.6: The Auger detector in Argentina.
128
Figure 12.7: The 27 highest energy cosmic rays (circles of 3.1 degree) detected by
the Auger collaboration overlaid on a plot of the Veron catalogue of AGN, taking
only the objects at less than 75 Mpc. The blue coloured regions indicate sky areas
of equal coverage, the darker the region, the deeper the exposure.
129
(Second part)
High Energy Astrophysics
(Second part)
130
Introduction
Compact objects are at the origin of deep gravitational wells in which matter moves
with very high, even relativistic, velocities. These velocities were seen in the first
chapter of these lectures to be at the origin of very high temperatures that can exceed
the MeV. Hence compact objects are at the origin of most of the phenomenology that
is observed in the high energy part of the electromagnetic spectrum. Conversely,
compact objects can often be observed and therefore studied best if not only through
high energy observations. Much of what we know on this component of the Galaxy
and the Universe is therefore young and has been obtained since the beginning of
orbital flights.
One example in place is given by the clusters of galaxies. Even though these objects
are very big in the sky, they are also extremely massive and we have seen that most
of the radiation comes not from the galaxies of the clusters but from the intra-cluster
gas that is heated in the deep well of the cluster to temperatures corresponding to
several keV. Without these observations, the understanding one could have of these
clusters was very fragmentary and in some senses misleading.
In this part of the course we will study many forms of compact objects. We will start
with some basics on general relativity and black holes, trying to understand how
much energy can be made available to radiate as matter falls onto the black hole.
We will then have a look at the structure of neutron stars and their magnetospheres
(which in contrast to what we have just established radiate mostly in the radio
part of the spectrum) and then go on to describe the properties of several types of
compact objects, within and without the Galaxy.
Whenever studying the general relativistic properties of compact objects we will
follow the main steps of the derivations without attempting to be complete in the
calculations. We will also not derive any general relativity results, but rather see
how they work when applied to the problems at hand. The books I enjoy most and
therefore used in writing these lectures are Misner, Thorne and Wheeler (Gravitation), Weinberg (Cosmology and Gravitation) and Shapiro and Teukolsky (Black
holes, white dwarfs and neutron stars).
131
Chapter 13
Black holes and accretion efficiency
In this chapter we will use units such that c = G = 1.
13.1
Relativistic hydrostatic equilibrium
Neutron stars have a mass and radius such that matter at their surface is bound with
a binding energy of some 10% mc2 (calculate). Thus these objects are clearly relativistic. The understanding of their structure requires then that they are described
in the frame of general relativity. This is in particular true of the hydrostatic equilibrium which tells us how much mass is in any layer to match the change of the
gravitational potential over the thickness of the layer. We will see here how this
equation is derived. In the following section we will specialise our derivation to the
region outside the object, in the vacuum, and get, as a bonus, the shape of the
metrics in a spherically symmetric gravitational field, the Schwarzschild metrics.
First write the metrics of Minkowski space:
ds2 = −dt2 + dr2 + r2 dΩ2 , dΩ2 = dθ2 + sin2 θdφ2
(13.1)
and generalise it so that the "t" and "r" components are not constant anymore in
the following way
ds2 = −e2Φ dt2 + e2Λ dr2 + r2 dΩ2 .
(13.2)
Φ and Λ are functions of r, but not of t, as we consider static problems. The
angular part of the metrics is left unchanged, as we start by considering a spherically
symmetric case.
Whereas space is described by its metrics, matter is described as a perfect fluid by:
• ρ(r), the density of mass energy in the matter rest system
• n(r), the number density in the matter rest system
132
• p(r), the isotropic pressure in the same system
µ
• uµ (r), the fluid 4-velocity, dx
dτ
• and the stress-energy tensor
T µν = (p + ρ)uµ uν + pg µν .
(13.3)
dr
= 0 = uθ = uφ .
dτ
(13.4)
In a static star we have
ur =
Timelike observers or particles have uµ uµ = −1 we can deduce
gµν uµ uν = gtt ut ut = −e2φ ut ut = −1
(13.5)
and therefore
ut =
dt
= e−Φ .
dτ
(13.6)
The diagonal components of the stress energy tensor (the only non vanishing in a
perfect static fluid) are
T 00 = (p + ρ)e−2φ − pe−2φ
= ρe−2φ
T rr = pg rr = pe−2Λ
p
T θθ = pg θθ = 2
r
p
φφ
φφ
T
= pg = 2 2
r sin θ
(13.7)
(13.8)
(13.9)
(13.10)
(13.11)
The conservation equations are obtained from the Bianchi identities that read
T µν ;ν = 0
(13.12)
where a ";" indicates a covariant derivative, i.e. one in which the change of the
metrics between two adjacent points has been taken into account:
aµ;ν =
∂aµ
+ Γµνδ aδ
∂xν
(13.13)
The Γµνδ are the Christoffel symbols, they are deduced in a straight forward way, if
sometimes algebrically complex, manner from the metric coefficient.
The "0" component of eq.13.12, T 0µ ;µ = 0, can then be derived to obtain
133
(p + ρ)
∂p
∂φ
=−
∂r
∂r
(13.14)
In addition to the conservation equation 13.14 we can also use the Einstein field
equation Gµν = 8πTµν which relates the geometrical properties of space (i.e. Φ and
Λ) to the matter content. The left hand side of the Einstein equation is the Einstein
tensor which describes indeed how the metrics changes locally. It is derived from
the metrics elements. We want to write the "00" component of this equation in
the system of reference in which the fluid is at rest and the local metrics has the
Minkovski form. This system always exists, you may imagine it as the local elevator
from which you have just cut the rope. We will use "ˆ" on the indices to indicate
that the corresponding vector or tensor is expressed in these coordinates.
In the "ˆ" system, we have ur̂ = uφ̂ = uθ̂ = 0 and ut̂ = 1 since the fluid is at rest,
and hence
T 0̂0̂ = (ρ + p)u0̂ u0̂ + pg 0̂0̂
= ρ+p−p=ρ
(13.15)
(13.16)
T îî = (ρ + p)uî uî + pg îî
= p
(13.17)
(13.18)
The "0̂0̂" component of the Einstein tensor in this system is
G0̂0̂ =
1 d r(1 − e−2Λ )
2
r dr
(13.19)
The "0̂0̂" Einstein equation therefore reads in this sytem
1 d r(1 − e−2Λ ) = 8πρ
2
r dr
(13.20)
We introduce the function m(r) through the following definition:
2m(r) = r(1 − e−2Λ ),
(13.21)
from which we see that
2Λ
e
−1
2m(r)
.
= 1−
r
(13.22)
2 dm(r)
= 8πρ,
r2 dr
(13.23)
We now re-formulate eq.13.20:
which can be integrated to yield
134
Z
m(r) =
r
4πρr2 dr,
(13.24)
0
from which one understands that m(r) is the mass within the sphere of radius r.
The "1̂1̂" Einstein equation reads
G1̂1̂ = 8πp
(13.25)
and gives in a similar way
−r−2 + r−2 e−2Λ + 2r−1 e−2Λ
dΦ
= 8πp,
dr
(13.26)
in which we can introduce m(r) instead of Λ to obtain the following expression for
dΦ
:
dr
dΦ
m(r) + 4πr3 p
=
.
dr
r(r − 2m(r))
(13.27)
Finally, we introduce this in 13.14 to obtain the result we were seeking:
dp
(p + ρ)(m + 4πr3 p)
=−
.
dr
r2 (1 − 2m(r) )
(13.28)
r
This is called the Oppenheimer Volkov equation or the Tolman Oppenheimer Volkov
(TOV) equation. It is one of the very few results that were obtained in astrophysics
using general relativity before the discovery of neutron stars in the 1960’s. This
equation gives the pressure gradient that is needed to compensate the change in
gravity as one moves by dr in a spherically symmetric static matter distribution
described by m(r). In other words, this is the general relativistic formulation of the
hydrostatic equation familiar in Newtonian physics. This latter equation reads:
ρm(r)
dp
=
.
dr
r2
(13.29)
The general relativistic form of hydrostatic equilibrium differs in a marked way from
the equivalent Newtonian equation.
Indeed, the right hand side of eq.13.29 is modified in that ρ is replaced by ρ + p,
in other words pressure contributes to the energy density, which is expected as a
pressure is an energy density and energy is a source of gravity (like mass). m(r) is
replaced by m(r)+4πpr3 , pressure contributes to the energy within the radius r; and
1
1
is replaced by r12 [ 2m(r)
]. All three modifications tend to increase the pressure
r2
1−
r
gradient in the relativistic case when compared to the Newtonian approach.
We will make an explicit use of this equation when studying the structure of neutron
stars.
135
13.2
Schwarzschild metrics
We have now all the elements we need to consider the same system, but outside the
boundary of the star, i.e. where ρ = 0. This means that we study the properties of
space outside a spherically symmetric static mass. In this case eq.13.22 becomes
e
2Λ
−1
2M
= 1−
,
r
(13.30)
where M is the total mass of the star. Outside the star the vacuum Einstein equation
must be used (Gµν = 0). This equation gives, after some algebric transformations
e2Φ · e2Λ = 1.
(13.31)
One therefore immediately knows that
e2Φ = 1 −
2M
r
(13.32)
and that the metrics is written
2M
1
dr2 + r2 dΩ2 .
ds = − 1 −
dt2 +
r
1 − 2M
r
2
(13.33)
This is the so-called Schwarzshild metrics, it describes how a mass M curves the
space outside the mass. It is singular at the origin and at r = 2M . Whereas the
singularity at the origin is a real one in the sense that the curvature is infinite there,
the singularity at r = 2M is only due to the coordinate system. Space is regular
there, curvature is finite, but the description of the metrics in the coordinate system
chosen here is singular, in some sense in the same way as the Earth longitude and
latitude coordinates system is singular at the poles, without the Earth being in any
way "special" there.
The proper time of an observer is given by the distance between 2 points on a
)dt2 , or
trajectory with r, θ, φ constant: dτ 2 = −ds2 = (1 − 2M
r
1/2
2M
dτ = 1 −
dt.
r
(13.34)
Black holes are objects for which the mass is within the horizon. These objects are
causally disconnected from the outside world by the presence of the horizon.
13.3
Particle motion around Schwarzschild black holes
We want now to study the motion of massive test particles in the Schwarzschild
metrics in order to assess how much energy may be gained in the accretion process.
136
Figure 13.1: The mass and radius of a number of objects in the Universe, together
with the size of the horizon for all masses. From P. Doherty
In other words, we study the orbits of particles around a spherically symmetric nonrotating mass. We thus leave out the vast majority of the black hole related physics
which might have been the subject of a set of lectures on their own.
We start by constructing an orthonormal system
−1/2
2M
et̂ =
1−
et ,
r
1/2
2M
er̂ =
1−
er ,
r
1
eθ̂ =
e,
r θ
1
eφ̂ =
e .
r sin(θ) φ
(13.35)
(13.36)
(13.37)
(13.38)
This is a realisation of the coordinate system in which matter is at rest and which
is locally flat that was introduced in section 13.1. It is in this system that we have
137
a familiar understanding of the physical quantities. The equations of motion are
derived from the Lagrangian:
2L = gαβ ẋα ẋβ ,
(13.39)
with
ẋα =
dxα
,
dλ
(13.40)
where λ is a parametrisation of the path xα (λ). Note the parallel with the non
relativistic case in which a force free movement is described by the Lagrangian
L = 1/2 · mv 2 .
The Lagrangian with the Schwarzschild metrics explicitely introduced leads to
−1
2M
2M 2
(13.41)
ṫ + 1 −
ṙ + r2 θ̇2 + r2 sin2 (θ)φ̇2 .
2L = − 1 −
r
r
The equations of movement are the Lagrange equations:
d
dλ
∂L
∂ ẋα
−
∂L
= 0.
∂xα
(13.42)
For the t equation this gives:
∂L
2M
ṫ
=− 1−
r
∂ ṫ
(13.43)
for the first term and 0 for the second. The equations of movement therefore read:
d
dλ
2M
1−
ṫ = 0
r
d 2
(r θ̇) = r2 sin(θ) cos(θ)φ̇2
dλ
d 2 2
(r sin (θ)φ̇) = 0
dλ
(13.44)
(13.45)
(13.46)
For the last equation we use
gαβ pα pβ = −m2 ,
(13.47)
where pα is the 4-impulsion. We can choose the parametrisation λ = τ /m, where τ
is the proper time and m the mass of the particle. We therefore have
138
dt
dt
= m
= pt
dλ
dτ
dr
dr
ṙ =
= m
= pr
dλ
dτ
dφ
dφ
φ̇ =
= m
= pφ ,
dλ
dτ
ṫ =
(13.48)
(13.49)
(13.50)
where we have used
pα =
∂L
.
∂ ẋα
(13.51)
Therefore,
2M
∂L
= − 1−
pt =
ṫ
r
∂ ṫ
−1 2M
2M
t
tt
p = g pt = − 1 −
− 1−
ṫ = ṫ
r
r
(13.52)
(13.53)
In a spherical symmetrical case, as we have with the Schwarzschild metric, we may
choose any plane through the centre in an arbitrary manner to discuss the orbits.
We therefore use in the following sin(θ) = 1 and write for the φ component of 13.44:
r2 φ̇ =: l = constant.
(13.54)
Similarly for the t component of the 13.44 we read:
2M
1−
r
ṫ =: E = constant.
(13.55)
Using Eq. 13.52 we finally deduce that E = −pt .
l and E are the constants of movement associated with the symmetries in φ and t
of the problem in discussion.
The meaning of E can be seen from the following: consider the t component of the
4-impulsion p~ in the local orthonormal system of the observer and use eq. 13.35:
Elocal
−1/2
−1/2
−1/2
2M
2M
2M
~et = − 1 −
pt = 1 −
· E.
:= −~p~et̂ = −~p 1 −
r
r
r
(13.56)
For r tending towards infinity E tends towards Elocal and E is called the energy at
infinity. Both are related by a redshift factor (1 − 2M
)1/2 .
r
139
In order to see the meaning of l, we consider the tangential velocity in the local
orthonormal system of the observer v φ̂ :
v φ̂ =
p~~eφ̂
p~~eφ /r
pφ /r
gφφ pφ /r
dφ̂
dφ̂ dt
pφ̂
=
=
=
= rφ̇/Elocal . (13.57)
=
= t̂ =
dt dt̂
Elocal
Elocal
Elocal
Elocal
p
dt̂
In other words: rφ̇ = Elocal v φ̂ . Using 13.54 we finally have:
l = r2 φ̇ = r · Elocal · v φ̂ ,
(13.58)
which we can compare with the Newtonian equivalent of l = r · m · Ω. We see that
l is the angular momentum.
13.4
Orbits in Schwarzschild geometry
Let’s introduce:
e = E,
E
m
`
`e =
m
(13.59)
so we can write the last component of the equations of motion 13.47:
gαβ pα pβ = −m2
(13.60)
gαβ ẋα ẋβ = −m2
| {z }
(13.61)
and therefore with eq. 13.48:
2L
therefore:
L=−
m2
2
(13.62)
then with the explicit form of the Lagrangian:
2
m =
2M
1−
r
2M
ṫ − 1 −
r
2
−1
ṙ2 − r2 θ˙2 − r2 φ̇2
(13.63)
where we have set: θ = π2 . Some algebra leads to:
2M
1−
r
2 2
2M
ṫ
ṙ2
2M r2 θ̇2
2M r2 φ̇2
= 1−
−
− 1−
− 1−
2
r
m2 m2
r
m2
r
|m
{z }
|
{z
}
|
{z
}
0
e2
E
`e2 /r2
(13.64)
140
see 13.56 and 13.58.
Considering λ =
dr
dτ
2
τ̄
m
dr
,
dλ
and ṙ =
we can re-arrange the terms of eq. 13.64 to obtain
!
e2 e2
`
2M
2M
2M
`
e2 −
e2 − 1 −
=− 1−
+E
1−
=E
1+ 2
2
r
r
r
r
r
(13.65)
where τ is the proper time.
From 13.54 and 13.55 we find:
`e
dφ
= 2
dτ
r
e
E
dt
=
dτ
1 − 2M
r
13.4.1
(13.66)
(13.67)
Radial geodesics
In radial geodesics Φ and θ are constants and `e vanishes. In this case Eq. 13.65
becomes
or
dr
dτ
dr
dτ
2
2M
2
e
=E − 1−
r
(13.68)
1/2
2M
2
e − 1−
=− E
.
r
(13.69)
1/2
2M
2
e
=− E −1+
r
(13.70)
dr
dτ
In order to proceed further, we must look at three cases:
dr
= 0 for r < ∞
dτ
e = 1 → dr = 0 for r → ∞
E
dτ
e > 1 → v∞ = − dr > 0 for r → ∞
E
dτ
e<1
E
→
(13.71)
(13.72)
(13.73)
We further look at the first case, that of a particle at rest at some finite distance R
dr
and falling towards the black hole. We introduce R, R so that dτ
|r=R = 0. Therefore
e2 + 1 =
−E
141
2M
R
(13.74)
and
dr
=−
dτ
2M
2M
−
r
R
1/2
.
(13.75)
This may be integrated in the following way:
−q
dr
2M
r
−
= dτ
(13.76)
2M
R
Z r2
dr
−1
q
τ = √
1
2M r1
− R1
r
"
#
r
r2
Z r2
√
R
dr
R
√
− rR − r2 +
= −
2M
2 r1
rR − r2
r1
" r
1/2
# r2
R3
r
r2
2r
=
2
−
− sin−1 1 −
8M
R R2
R
(13.77)
(13.78)
(13.79)
r1
Choosing τ = 0 at r = R, the fall begins at the origin of τ and to obtain τ for all r
less than R we select the boundaries of the integration so that r1 = R and r2 = r.
This gives the following function τ (r):
1/2 " #
1/2
r
r2
R3
2r
2
−
− sin−1 1 −
τ =
8M
R R2
R
3 1/2
R
+
sin−1 (−1)
| {z }
8M
−π/2
3 1/2 " 1/2
#
R
r
r2
2r
=
−
2
− cos−1 1 −
.
8M
R R2
R
(13.80)
(13.81)
We can look at the value of τ for r = 2M . In other words calculate the proper time
of the observer falling from R to the horizon:
τ (r = 2M ) =
R3
8M
1/2 " 1/2
#
2M
4M 2
4M
2
− 2
− cos−1 1 −
< ∞ (13.82)
R
R
R
This shows explicitly that the proper time when reaching the horizon is finite. In
other words an observer falling onto a black hole will reach the horizon in a finite
proper time. Note that the time to reach the singularity at the center of the black
hole (at r = 0) is also finite.
Let us now consider the coordinate time t elapsed during the same fall from R to
2M .
142
As before we use τ (r = R) =
dr
dτ
(r = R) = 0.
e
dt
E
=
,
dτ
1 − 2M
r
(13.83)
which we may use to calculate the coordinate time with
dt dr
dt
=
·
dτ
dr dτ
−1
e
dt
E
dr
⇒
=
·
dr
dτ
1 − 2M
r
−1/2
e
E
2M
2
e
= −
· E −1+
r
1 − 2M
r
(13.84)
(13.85)
(13.86)
and then
Z
e
Edr
t=−
1−
2M
r
e2 − 1 +
E
2M
r
1/2
(13.87)
for which the solution is the cycloid
(
t = 2M ln
R
2M
R
2M
)
1/2
1/2 −1
+ tan( η2 )
R
R
· η+
+ 2M
−1
(η + sin η) ,
1/2
2M
4M
−1
− tan( η2 )
(13.88)
where η is the cycloid parameter defined by
R
(1 + cos η) .
2
r =
(13.89)
Note that with this parametrisation, the solution of the τ (r) equation can be expressed as
τ =
R3
8M
1/2
(η + sin η) .
(13.90)
For r = 2M we obtain
4M
−1
R
1/2
R
η
tan
=
−1
.
2
2M
cos η =
143
(13.91)
(13.92)
We see that the first term of Eq. 13.88 is singular and t tends towards infinity. The
behavior of the function t(r) is given in figure 13.2. This shows that indeed t tends
towards infinity when r approaches 2M . It takes an infinite coordinate time for the
particle to fall towards the horizon of the black hole. The coordinate time is the one
an observer at large distances would measure. This means then that while it takes
a finite particle proper time to fall onto the horizon of a black hole, an observer at
large distances would see the process to take an infinite time.
f, Schwarzchild
coordinatetime
7, propertime
u
5
1
0
1
5
2
TimelM
0
2
5
3
0
Figure l2.l Fall from rest toward a Schwarzschildblack hole as described( a by a comovinl
)
observer(proper time r) and (ô) bV a distant observer(Schwalzschildcoordinatetime /). In the onr
:
description,the point r 0 is attained,and quickly [seeEq. (12.4.23)].In the other descriptior,r : (
is neverreachedand evenr :2M is attainedonly asymptoticallytF.q.(12.4.24)1.
[From Graoitationb,
Charles W. Misner,.Iftp S. Thorne, and John Archibald Wheeter, 'W. H. Freeman and Compani
Copyright o t973.1
i
Figure 13.2: Radial fall into a black hole as observed in proper time of the falling
body or in coordinate time at a large distance
13.4.2
Non radial orbits
Radial orbits are interesting for the properties we have just discussed. Physically,
however, non radial orbits are much more relevant. We will discuss them now with
thai aim of calculating the amount of energy we can hope to gain from matter falling
into a black hole. This amount of energy is what we can expect to have radiated
from the vicinity of the black hole.
Let’s write Eq. (13.65):
dr
dτ
2
e 2 − V 2 (r)
=E
(13.93)
where we keep the specific angular momentum in V (r):
"
V (r) =
2M
1−
r
144
`e2
1+ 2
r
!# 12
(13.94)
e is shown in Figure 13.3. The shape of the function
The function V 2 (r), for a given `,
can be described as follows:
1. At r = 2M → V 2 (r) = 0;
2.
dV 2 dr r=2M
>0
The extrema of the function are found for
∂V 2
= 0
∂r
"
!# 2M
`e2
∂
1−
1+ 2
=
∂r
r
r
r=rc
! 2
`e
`e2
2M
2M
=
1
+
+
1
−
(−2)
rc2
rc2
rc
rc3
!
M `e2 `e2 2M `e2
1
.
= 2 M+ 2 − +
rc
rc
rc
rc2
(13.95)
(13.96)
(13.97)
(13.98)
The extrema are therefore located at r = rc with
0 = M rc2 + 3M `e2 − `e2 rc
so we can find:
rc =
`e2 ±
p
`e4 − 12M 2 `e2
.
2M
(13.99)
(13.100)
Two extrema exist for `e4 − 12M 2 `e2 > 0, i.e.
`e2 > 12M 2
√
`e > 2 3M
√
while no extremum exists for `e < 2 3M .
3. V 2 (r) → 1 for r → ∞.
√
4. The figure 13.3 gives the shape of the potential for `e > 2 3M in both the
relativistic and the Newtonian case. In the Newtonian equivalent discussion,
the orbits are not bound for E > 0 and are bound and elliptical for E < 0.
The relativistic case is different from the classical case in that:
1. There are bound orbits similar
√ to the Newtonian case (but note that the
periastron precesses) for `e > 2 3M .
√
2. For `e < 2 3M all orbits fall inside the hole.
2
e > Vmax
3. All orbits with E
fall inside the hole, they are called capture orbits.
2
4. For `e = 4M we have Vmax
= 1 (to be proved) so all orbits with `e < 4M coming
from infinity are capture orbits.
145
V2
1
0
2
r/M
√
Figure 13.3: Left: The relativistic "potential" V2 (r) for `e > 2 3M . Right: The
Newtonian potential for a non zero angular momentum. It differs in important ways
from the relativistic analogous.
A more quantitative picture is given in Fig. 13.4.
Circular orbits are given by the minimum of the potential. This means that r is
constant. This minimum of the potential lies at (13.100):
p
e2 + `e4 − 12M 2 `e2
`
rc+ =
,
(13.101)
2M
where we have taken the extremum at the larger distance, the one inside being
unstable, as can be seen from the figure. The location of this extremum, and hence
the radius of the circular orbits, depends on `e in the following way
rc+ → ∞ for `e → ∞ and decreases with `e reaching:
rc = 6M
(= 3Rs )
(13.102)
√
√
for `e = 2 3M . Since for `e < 2 3M there is no extremum, this radius correponds
to the last possible stable circular orbit. It is therefore also the inner boundary of
an accretion disk around a Schwarzschild black hole.
The binding energy of a particle per unit mass on the last stable orbit is:
ebinding := m − E = 1 − E,
e
E
m
(13.103)
where E is the particle energy “at infinity” in rc , i.e. the minimum value of the
potential at the last stable circular orbit.
dr
e 2 = V 2 (r), and considering
For a circular orbit dτ
= 0 so using Eq. (13.93) we have E
the last stable orbit we find:
146
√
e 2 = V 2 (rc , `e = 2 3M )
E
12M 2
2M
1+
=
1−
6M
36M 2
2 4
=
·
3 3
8
=
9
(13.104)
(13.105)
(13.106)
(13.107)
so finally:
r
ebinding = 1 −
E
8
= 0.0572
9
(13.108)
This means that when a particle is on the last stable orbit around a black hole it
Figure 13.4: A quantitative plot of V 2 (r) from page 662 of Gravitation by Misner,
Thorne and Wheeler
147
has lost an energy equivalent to about 6% of the mass it had at infinity. This is,
for example, the energy available for the luminosity produced by an Active Galactic
Nucleus.
Note that there exist circular orbits at the maxima of V 2 (r) (closer in than the one
we have just considered). These orbits are, however, not stable.
13.5
Kerr black holes
“ If only it were not so damnably difficult to find exact solutions” wrote Einstein to
Born in a 1936 letter about the search for solutions to the Einstein equations.
The “no hair theorem” indicates that only mass M, angular momentum J and charge
Q can influence the geometry outside a mass distribution at large distances. There
is no astrophysical reason that would lead us to think that the charge is meaningful.
We therefore do not consider it in the following. The resulting metrics is called the
Kerr metrics, it reads:
M r sin2 (θ)
Σ 2
M ra2 sin2 (θ)
2M r
2
2
2
2
dt −4a
dtdφ+ dr +Σdθ + r + a + 2
sin2 (θ)dφ2
ds = − 1 −
Σ
Σ
∆
Σ
(13.109)
2
with
J
,
M
∆ := r2 − 2M r + a2
a :=
(13.110)
(13.111)
and
Σ := r2 + a2 cos2 (θ)
(13.112)
In a unit system in which c = G = 1, you can show that the unit of a is that of a
length (or equivalently of a mass or time).
The metric coefficients are independent of t and φ. Setting a = 0 leads to the
Schwarzschild metrics, as it should.
The metrics is singular at Σ = 0, which is a real singularity and at ∆ = 0, which is
not. This latter singularity defines the horizon:
r± = M ±
√
M 2 − a2 .
(13.113)
Physically this means that the singularity is at the larger of the two, i.e. r+ . The
horizon exists only for a < M . Since one expects that there exists no naked singularity, one deduces that a = M is a limit that cannot be exceeded. a = M black
holes are called maximally rotating.
Let’s assume a timelike particle i.e. one for which u · u < 0, where u is the 4-velocity
(you and I belong to this category).
148
u · u = gtt ut ut + 2gtφ ut uφ + gφφ uφ uφ .
(13.114)
gtt ut ut + 2gtφ ut ut Ω + gφφ ut ut Ω2 ,
(13.115)
which we can write as
where we have introduced
dφ dτ
dφ
=
·
dt
dτ dt
Ω=
(13.116)
the angular velocity.
The condition u · u <0 then reads
gtt + 2gtφ Ω + gφφ Ω2 < 0.
(13.117)
Looking at the Kerr metric you will find that gφφ >0. Expression 13.117 therefore
defines a parabola for Ω which tends towards infinity for very small and large Ω.
The inequality can only be fullfilled for a range Ωmin < Ω < Ωmax , where Ωmin and
Ωmax are the zeros of 13.117.
In other words:
−gtφ ±
Ωmin,max =
q
2
gtφ
− gtt gφφ
gφφ
.
(13.118)
You will therefore find non rotating particles only for Ωmin = 0 (or less), which
r
= 1 which is satified for r0
occurs for gtt = 0, or looking at equation (13.109) 2M
Σ
r0 = M +
p
M 2 − a2 cos2 (θ).
(13.119)
For r < r0 (but greater than r+ ), there is no solution with Ω = 0, this means that
there is no static solution with φ constant. The region between r+ and r0 is called
the ergosphere for reasons that will become clear later.
The Figure 13.5 gives the shape of the region.
For a = 0, it is clear that r0 = 2M , i.e. r0 is at the horizon and there is no
ergosphere.
The trajectories of the massive particles (to which we limit our discussion of the
Kerr black holes) can be derived, as in the case of Schwarzschild, starting from the
Lagrangian:
2L = gαβ ẋα ẋβ
(13.120)
α
with ẋα = dx
.
dλ
Let’s consider now the case θ = π2 , for the Kerr metrics we obtain:
149
2L = gtt ṫ2 + gtφ ṫφ̇ + grr ṙ2 + gφφ φ̇2
(13.121)
2
2
2M a
2M 2 4aM
r
ṫ −
φ̇2 (13.122)
= − 1−
ṫφ̇ + ṙ2 + r2 + a2 +
r
r
∆
r
The metrics does not include explicitly t and φ. The Lagrange equations 13.42
therefore lead to the following conservation laws:
∂L
:= −E
∂ ṫ
∂L
pφ =
:= `
∂ φ̇
pt =
(13.123)
(13.124)
and the third law of conservation is given for the rest mass:
|~p2 | = −m2 where 2L = −m2
(13.125)
d ∂L
∂L
− α =0
α
dλ ∂ ẋ
∂x
(13.126)
The equations of motion:
give for t and φ:
∂L
=
− 1 − 2M
ṫ − 2aM
φ̇
= −E
r
r
∂ ṫ
∂L
2M a2
2
2
= − 2aM
φ̇ = `
ṫ
+
r
+
a
+
r
r
∂ φ̇
(13.127)
(13.128)
Figure 13.5: In a Kerr black hole there are two event horizons, the outer and the
inner. The region of space in-between the two horizons is the ergosphere. Anything
inside the ergosphere will be dragged by the black hole and rotate with it but it
can still escape. However, anything inside the inner event horizon can never escape
(image credit: Kwong-Sang Cheng, Hoi-Fung Chau and Kai-Ming Lee).
150
which can be solved for ṫ and φ̇, giving the following results:
(r3 + a2 r + 2M a2 )E − 2aM `
r∆
(r − 2M )` + 2aM E
φ̇ =
r∆
ṫ =
(13.129)
(13.130)
We can obtain an equation similar to the Eq.(13.66,13.67), using the relation 2L =
−m2 and Eq.(13.130).
We find:
r3 ṙ2 = R(E, `, r) =:
(13.131)
2 3
2
2
2
2
= E (r + a + 2M a ) − 4aM E` − (r − 2M )` − m r∆, (13.132)
which defines R(E, `, r). Note that we have not an expression of the form (E 2 −
V 2 (r)) anymore.
Circular orbits are obtained as in the previous case and the extreme points of the
effective potential are given by:
∂R
=0
∂r
R = 0,
(13.133)
which gives:
√
2
E
r
−
2M
r
±
a
Mr
e=
= E
√
1
m
r(r2 − 3M r ± 2a M r) 2
√
√
2
`
M
r(r
∓
2a
M r + a2 )
e
= `= ±
√
1
m
r(r2 − 3M r ± 2a M r) 2
(13.134)
(13.135)
The sign on the top means co-rotation while the one on the bottom refers to a
counter-rotation.
We find circular orbits for all r values for which the denominator is not vanishing.
The zeros of the denominator of Eq.(13.135) are:
a
2
r0 = 2M 1 + cos
arccos ∓
3
M
(13.136)
This is equal to 3M for a = 0 (note that this orbit is not the last stable orbit). In
some cases the orbits correspond to the pseudo-potential maxima.
e < 1) by looking at the numerator of 13.135:
We find circular bound bound orbits (E
e = 1 for the marginally bound orbit
Eq.(13.135), with the additional condition E
(rmb ), gives:
√
1
(13.137)
r > rmb = 2M ∓ a + 2 M (M ∓ a) 2
and finally:
rmb (a = 0) = 4M
151
(13.138)
All these cases do not necessarily give stable orbits. To ask the stability we have to
consider the following additional condition:
∂ 2R
≤0
∂r2
(13.139)
which we have not considered yet together with the Eq.(13.132):
∂ 2R
= 6E 2 r − m2 6r + 4M m2 ≤ 0
∂r2
E2
2M
−1 ≤
2
m
3 r
e2 ≥ 2 M
1−E
3 r
(13.140)
(13.141)
(13.142)
e for circular orbits, where we have included condition 13.142
Eqs.(13.135) give us E(r)
at the limit of equality:
1
rms = M [3 + z2 ∓ [(3 − z1 )(3 + z1 + 2z2 )] 2 ]
1 a 13 a 13
a2 3 1+
+ 1−
z1 = 1 + 1 − 2
M
M
M
2
21
a
z2 =
3 2 + z12
M
(13.143)
(13.144)
(13.145)
For a = 0 we found again rms = 6M , the radius of the last stable orbit using the
Schwarzschild metrics.
For a = M , a black hole in maximal rotation we find from 13.145:
z1 = 1,
z2 = 2.
(13.146)
We have for the radius of this orbit from 13.145:
1
rms = M 3 + 2 ∓ [2.8] 2 = M (5 ∓ 4)
(13.147)
This means that the last radius of a stable circular orbit lies at rms = M for a direct
orbit particle and at rms = 9M for a retrograde orbit one. This is much closer to
the central singularity than we had found for the Schwarzschild black holes.
In order to calculate the binding energy associated to this last stable orbit we have
to use the condition in Eqs.(13.135,13.142) removing, in this case, r.
For the marginally stable orbit (= in Eq.(13.142)) and extracting Ma , we obtain:
√
e 2 ) 12 − 2E
e
4 2(1 − E
a
=∓
√
e2)
M
3 3(1 − E
e2 =
For a = 0 we find E
case.
8
9
(13.148)
(which you may want to verify) as in the Schwarzschild
152
For a = M we have:
e=
E
q
1
3
direct orbit
e=
E
q
25
27
retrograde orbit
(13.149)
So the binding energy is:
e = 42.3%
1−E
(13.150)
for an orbit around a black hole in maximal rotation.
This energy is much more impressive than what we calculated in the case of a
Schwarzschild black hole.
13.5.1
Relativistically broadened emission lines
The Doppler broadening of a line is
∆v
∆λ
=
.
λ
c
(13.151)
So clearly, when lines are very broad, this indicates relativistic velocities. Such
velocities in a gravitational context can only be reached very deep in the potential
well of a compact object. This means that the line profile will not only include
the Doppler effect and the aberration due to the velocities, but also effects due to
general relativity.
This subject became important few years ago with the discovery in ASCA data of a
very broad Fe line in the AGN MCG 6-3015 (fig. 13.6). The line is at a rest energy
of 6.4 keV, it is therefore a line emitted by cold Fe. The line width is 2.5 keV or so,
clearly indicating relativistic velocities. The blue "horn" of the line is considerably
brighter than the red "horn". This is due to Doppler boosting of the photons in
the line of sight. The profile fitted is that expected from a relativistic accretion disc
around a massive black hole. The line profile can be calculated if, the velocity profile
of the emitting matter is known, as well as the emission profile and the alignment
of the emititng material with respect to the black hole, so that the trajectories of
the photons in the curved space around the black hole can all be calculated.
These studies have become an industry in the X-ray spectral analysis of compact
objects. There are, however, many effects that must be kept in mind when considering these results. The line is almost as wide as the line energy (∆λ ' λ). This
means that the continuum must be known very well before subtraction in order to
measure the profile of the line. This is often not easy at all and requires data at
energies higher than those observable with X-ray telescopes. The line may also be a
complex due to the superposition of many different ionisation levels of Fe. Further,
the emission law of the disk (or whatever structure) must be known, as well as the
geometry of the emission region with respect to the black hole.
153
Since the geometry of the space around the black hole depends on its spin, and
since the properties of the accretion flow also depend on the black hole spin, it can
be expected that the line profiles will carry the signature of the black hole spin.
Indeed this is one way in which one may hope to measure the spin of a black hole.
Simulations show that this is a realistic possibility with the next generations of X-ray
instruments (XEUS) (see fig, 13.7).
13.6
Energy gain from a Kerr black hole
The end mass of a black hole of mass M in which a particle of mass m falls from
infinity (at rest there) is m+M. Let’s call the falling object A and imagine that it
explodes close to the black hole in 2 pieces that we’ll call B and C. Imagine further
that that B is accreted while C returns to infinity. The change of mass of the black
hole will be
∆M = EA − EC ,
(13.152)
as the part EC of the energy of the first object A has returned to infinity. In the
inertial system in which the explosion is taking place, we have
p~A = p~B + p~C
(13.153)
from which we read that ∆M = EB . Interestingly, there exists orbits with negative
E in the vicinity of Kerr black holes. If you solve Eq. 13.132 for E you will find
Figure 13.6: The X-ray spectrum (continuum subtracted of the Seyfert galaxy MCG
6-30-15 observed by ASCA (Tanaka et al. Nature......).
154
1/2
E=
2aM l + (l2 r2 ∆ + m2 r∆ + r3 ṙ2 )
r3 + a2 r + 2M a2
,
(13.154)
which has negative solutions for l < 0 (retrograde orbits) and
4a2 M 2 l2 > l2 r2 ∆ + m2 r∆ + r3 ṙ2 .
(13.155)
The region in which such orbits can exist (which does not extend to infinity) is the
same as that described as the ergosphere above.
It is therefore sufficient to organise the explosion A to B + C such that EB < 0 to
extract energy from the black hole. Indeed in this case ∆M = EB < 0 and EC > EA .
You might imagine a civilisation living around a black hole and that organises its
garbage managemet as suggested by MTW in the way described by Fig. 13.8.
13.7
Black hole radiation
Black holes are not quite as black as one would think.
In a very strong electro-magnetic field pairs of particles-anti-particles are created
when the electric potential energy over a Compton length λC is sufficient:
Figure 13.7: Simulated line profile assuming a Kerr or a Schwarzschild geometry for
a 10 ks XEUS observation of the Seyfert galaxy MCG 6-30-15. The differences in
the profiles are clearly seen. (Reeves et al Xeus documentation).
155
eEλC > 2mc2 ,
(13.156)
where m is the mass of the corresponding particle and E the electrical field. The same
is expected to be true if the tidal force, and energy, is sufficient (remember that tidal
forces are the only "true" forces in a gravitational field, as the field can always be
eliminated through an adequate coordinate transformation). One therefore expects
pair creation for particles of mass m for
GM m 2
· λC > 2mc2 .
r3
(13.157)
G2 M 2
c2 r 3
'
,
GM
c4
(13.158)
This can be solved for λC :
λ2C '
where we have used the fact that we are close to the gravitational radius of a black
. We therefore see that particles with λC ' GM
hole of mass M, and therefore r ' GM
c2
c2
Figure 13.8: Misner Thorne and Wheeler view of a civilisation extracting its energy
from garbage dump onto a Kerr black hole. The dump at ejection point must
only be organised such that the orbits of the garbage falling into the black hole are
charatcerised by negative energies. The containers will then reach the large distances
with energies larger than they had when dropping.
156
are created close to the horizon of a black hole. In a fraction of the cases one of
the particles created will fall into the black hole and the other will escape towards
infinity. Thus the black hole will be seen to "radiate". Hawking has shown that the
radiation field at infinity is that of a black body of temperature T:
< N >=
1
hν
)
exp ( kT
(13.159)
±1
and
h̄
T =
' 10−7 K
8πkM
M
M
.
(13.160)
The luminosity of the black hole will be given by the L ∝ surface · T 4 law. With the
surface proportional to the square of the horizon size and Eq. 13.160, we have
L ∝ M2 ·
1
∝ M −2 .
M4
(13.161)
The time it takes for the black hole to emit an energy equal to its mass, the black
hole evaporation time is
tev =
E
M
∝ −2 ∝ M 3 .
L
M
(13.162)
Numerically this gives
10
tev ' 10 years ·
M
1015 g
3
.
(13.163)
We can therefore deduce that black holes formed at the origin of the Universe and
which have a mass ' 1015 g evaporate now with a luminosity L
20 ergs
L ' 10
s
1015 g
M
(13.164)
emitting radiation of characteristic energy
hν ' kT ' 100 M eV ·
1015 g
M
.
(13.165)
The total energy emitted is of the order of M c2 ' 1036 ergs (much less than the
energy radiated by gamma ray bursts).
157
Chapter 14
Neutron Stars
The existence of neutron stars was predicted by Baade and Zwicky in 1934 (Phys.
Rev.45, 138, 1934; Fig. 14.1), shortly after the discovery of the neutron by Chadwick
in 1932. The paper already suggests that neutron stars could be the product of
stellar explosions in supernovae and that they could be at the origin of cosmic rays.
Neutron stars were then almost completely forgotten for more than 30 years, with
the exception of some work e.g. by Oppenheimer, Volkoff in the 30’s and Harrison
and Wheeler in the 60’s. The situation changed drastically with the discovery of
pulsars in 1967.
It is interesting that the authors did not know that most of the energy of a supernova
is emitted as neutrinos (not even postulated by then) and therefore they overestimate
the energy available to produce cosmic rays, they also underestimate the rate of
supernovae. They thus still get a roughly correct energy flux in cosmic rays.
The structure of neutron stars can be calculated to some extend. The major ingredients needed are the hydrostatic equilibrium equation and the equation of state.
While the first may be deduced without problems from general relativity, the second
requires knowledge about the equation of state of matter above nuclear density, a
topic which is not solved with certainty now.
The structure of neutron stars must be calculated in a general relativity frame. In
order to see this you can consider the binding energy of a particle at the surface of
a neutron star. You will find that for a roughly solar mass neutron star of some 10
km radius, this binding energy is of the order of 10% of the rest mass of the particle,
which shows that relativistic effects cannot be neglected.
14.1
Relativistic hydrostatic equilibrium
In order to capture the essence of the relativistic hydrodynamic equilibrium we
consider a perfect fluid as already described in Section 13.1. with
ρ(r) the mass-energy density in the local rest system and
p(r), the isotropic pressure in the same system,
n(r), the number density of the particles in the system and
uµ (r), the 4-velocity of the fluid.
158
Figure 14.1: The original Phys. rev. article of Baade and Zwicky (Phys. Rev.45,
138, 1934).
The mass-energy tensor of this fluid is:
T µν = (p + ρ)uµ uν + pg µν .
(14.1)
The equations of conservations
T;νµν = 0,
(14.2)
and the Einstein equation
Gµν = 8πGTµν
lead to the relativistic hydrostatic equation as we had seen in a previous chapter:
dP
G(P + ρ)(m(r) + 4πr3 P )
=−
,
dr
r2 (1 − 2G m(r)
)
r
where P is the pressure and m(r) the mass within the radius r.
159
(14.3)
14.2
Equation of state
We consider only the simplest case of a mixture of electrons, protons and neutrons
in beta equilibrium and at T = 0. The particles we consider have all half integer
spin and obey the Fermi statistics
The equations needed for the number density, the mass-energy density and the
pressure (the elements of the equation of state for zero temperature) are given by
the ideal zero temperature Fermi gas:
Z pF
8π
p2 dp
n=
3
2π~ 0
Z pF p
8π
ρ=
p2 + m2 p2 dp
2π~3 0
Z pF
1 8π
p2
p
P =
p2 dp.
2
2
3 2π~3 0
p +m
(14.4)
(14.5)
(14.6)
where pF is the Fermi momentum, ρ is the mass energy density, n the number
density and P the pressure (remember that the pressure is given by the exchange of
momentum per unit time at the wall (∝ p · v, v = p/m)).
The particles we consider are electrons, protons, neutrons and neutrinos. The reactions between them are the β reactions:
n → p + e− + ν
e− + p → n + ν
(14.7)
(14.8)
In order to calculate the relative numbers of the different kind of particles we consider
the conservation laws for charge and baryons. Charge conservation and neutrality
imply ne = np and baryon conservation implies nn + np = constant.
Consider = (n, S, Yi ) = energy density in the gas. The first law of thermodynamics
dQ = T dS = d + P dV + µdN , the first term gives the free energy, the second the
mechanical work and the third the chemical energy.
Considering this equation per particle and solving for the energy /n, we obtain
1
= −P d
+ T dS − Σµi dYi ,
d
n
n
with
160
(14.9)
∂
P =−
∂
∂
n
,
1
n
n
,
∂S ∂ n
µi =
.
∂Yi
T =
(14.10)
(14.11)
(14.12)
Looking for the structure of a neutron star with no exchange of energy with the
external world, the mechanical work is zero, so is the entropy change. The only
variation in the state of the gas is that of the relative abundances Yi of the particles.
One therefore obtains
Σµi dYi = 0.
(14.13)
µi is the chemical potential of the i-th sort and Yi := nni is the relative density of the
i-th particle sort. This condition expresses that changes in the relative concentration
of the particle sorts may not change the energy of the system as there is no energy
source available.
Charge conservation imposes dYe = dYp and baryon number conservation demands
dYp = −dYn . Equation 14.13 therefore becomes
µe dYe + µn dYn + µp dYp + µν dYν = 0.
(14.14)
The last term is set to 0 because the ν leave freely the system and therefore do not
contribute. We finally have µn = µp + µe .
In an ideal Fermi gas q
at T=0, the chemical potentials are given by the Fermi energy
of the species: µi = p2F,i + m2i . This is a straight function of the density of the
particles, as is seen from the first equation of 14.4. The rest is algebra and will not
be pursued further here (see e.g. Shapiro and Teukolski section 2.5). It is clear,
however that we have enough elements to deduce the composition of the material.
This is given in Fig. ??, which gives the ratio of proton to neutron. It is important
to note that the neutrons are stable in this configuration because the Fermi energy
of the electrons is larger than the mass difference between neutron and protons. The
neutrons cannot, therefore, decay into a propton an electron and an anti-neutrino,
there is simply not enough energy available.
14.2.1
The Harrison-Wheeler Equation of State
At low densities and pressure the most bound nucleus is 56 Fe. The lowest energy
state of matter that has been given time to settle will therefore be a collection of
56
Fe nuclei. When the density increases, however, the electrons become degenerate
and therefore energetic. In these conditions the equilibrium mixture of particles
is not any more 56 Fe nuclei and electrons, but ever richer neutron elements and
161
Figure 14.2: The composition of a cold mixture of neutrons, protons and electrons
as a function of the density.
electrons. At densities above about 4 1011 g cm−3 the matter is so neutron rich that
some neutrons drip out of the nuclei. The mixture is then one of nuclei of mass A and
charge Z, free neutrons and relativistic electrons. In order to study the composition
of this mixture consider its energy density
= nN M (A, Z) + e ne + n (nn ),
(14.15)
where ni is the density of a type of particle and M(A,Z) is the mass (energy) of
the (A,Z) nucleus. The function M(A,Z) is given by a model of the nuclear forces.
Harrison and Wheeler used a then known semi empirical mass formula (of Green)
based on the liquid drop model of the nucleus. One then has to minimize the
energy density of the mixture while satisfying the conservation laws. There results
an equation for P (ρ), the equation of state of a (very simplified) neutron star.
Since then there have been a number of equation of state that have been developed,
based on more sophisticated models of the nucleus and treatments of the particle
interactions.
Using the Harrison Wheeler equation of state and the relativistic hydrostatic equation one can deduce the total mass of the configuration as a function of the central
density. This is shown in Fig. 14.4 for the white dwarfs (where the pressure is given
by a degenarate electron gas) and the neutron stars. The stable configurations are
those for which the function’s slope is positive and the limit of stability is found at
the maxima. The maximum mass of a white dwarf is seen to be, as expected, at
1.4 solar masses, the Chandrasekhar mass, while the maximum mass of a Harrison
Wheeler neutron star is seen to be at 0.7 solar masses with a radius at the maximum
mass of 9.6 km and a central density of ρc = 5 · 1015 g cm−3 . Clearly the maximum
mass of a given configuration is set by the equation of state. Different, more sophisticated equations of state will therefore lead to different maximum masses, a point
to which we will return.
Realistic equations of state consider more aspects of the interactions between neutrons, electrons and protons than just the beta reactions and the perfect fluid approximations used by Harrison and Wheeler. Baryon interactions may, as an example,
162
Figure 14.3: The Harrison Wheeler equation of state (from Shapiro and Teukolsky,
1983).
give rise to a population of π mesons that should be included in the model as a π
condensate in the central regions. The baryons also have an internal structure and
may not be considered as point like particles at the very high densities met in the
central regions of the neutron star. Taking these effects and many others has led
to a number of equation of states that in turn and with the relativistic hydrostatic
equilibrium gives a number of mass vs density relations (see Fig.??). These lead to
different maximum masses for neutron stars, as the maximum in the M (ρ) relations
is at different masses.
It should be noted that the density in the inner regions of neutron stars is larger
than the density of normal nuclear matter. This implies that we have very few tools
that can be used to measure the properties of the equation of state in the laboratory.
There remains a large uncertainty in the inner properties of neutron stars, and hence
on their possible maximum mass.
One way of progressing on this issue is to measure the radius and mass of neutron
stars. While the mass can be measured for some neutron stars in binary systems, the
radius is much more elusive. It has, however, become recently possible to measure
the X-ray spectrum of bursting neutron stars with XMM-Newton. The lines a shited
towards the red by the gravitational redshift at the surface of the neutron stars, thus
163
1.8
1.6
1.4
1.2
1,O
M
Mo o.B
0.6
HW
Stable
white dwarfs
HW,OV
0.4
o.2
0
105
OV
107
10e
10rI
t ol3
Stable
neutronstars
1015
1017
,|
ote
P" (gcm-31
Figure9.1 Gravitationalmassversuscentraldensityfor the HW (1958)and OV (1939)equationsol
state.The stablewhite dwarf and neutron star branchesof the HW curve are designatedby a âeny
solid line.
Figure 14.4: Total mass vs central density for white dwarfs and Harrison Wheeler
neutron stars. (from Shapiro and Teukolsky, 1983).
giving a measure of M/R for the star. Fig. 14.6 gives one such measurement for the
EXO0748-676 from which one deduces z ' 0.35, which for a reasonable mass of 1.41.8 solar masses gives a radius of the order of 9-10 km (Cottam, Parels and Mendez,
Nature 420,51, 2002).
14.2.2
structure of Neutron stars
Together with the hydrostatic equilibrium equation given above this gives a first
glimpse into the possible configurations of a neutron star. This is given in Figure 14.7 for two different equations of state, that take different physical processes
into account. It is already apparent in this figure that the gravitational redshift at
the surface will be different.
The composition of the neutron stars is made of 56 Fe nuclei at the surface (an Iron
atmosphere). As one progresses towards the interior, the nuclei become more and
more neutron rich, as the Fermi energy of the electrons increases until the density
reaches the neutron drip point at 41011 g/cm3 , where free neutrons begin to appear.
The composition is then made of neutrons, electrons and protons. Further inside
more exotic forms of matter may appear.
It is also worth mentionning that while the collapse is ongoing, the chemical potential
of the neutrinos is not zero, as the neutrinos are trapped within the collapsing star.
The formalism described must therefore take this into account for these very early
phases.
164
3.0
2.O
M
Mo
1.0
0
1014
1016
1ots
Pc (gcm-g)
Figure9.2 Gravitational mass vs. central density for various equations of state. The letters labeling
thedifferent curvesare identified in Table 8.2 with the exception of.m, which denotesa Reid equation
of statemodified by charged-pioncondensation.The ascendingportions of the curyes representstable
neutronstars.[After Baym and Pethick (1979).Reproducedwith permission,from the Anntnl Raniew
Vol. 17. @ 1979by Annual ReviewsInc.J
ol Astronomyand Astrophysrcs,
Figure 14.5: Total mass vs central density for a number of equation of states that
are more realistic than free particles under beta equilibrium (from Shapiro and
Teukolsky, 1983).
14.2.3
The maximum mass of a neutron star
The question of the maximum mass a neutron star can have is of prime importance
when one wants to assess the existence of black holes in our Galaxy. Since this
maximum mass depends on the details of the equation of state and this is poorly
known, there is some level of uncertainty in this maximum mass.
One treatment of this question using as much knowledge as possible is given by the
following assumptions:
• The hydrostatic equilibrium is given by 13.28. This means that general relativity is applicable.
• Matter is microscopically stable, i.e.
dP
dρ
≥ 0.
• ρ≥0
• The equation of state is known up to some density.
• One may consider a further condition which is dP
≤ c2 which states that the
dρ
speed of sound must be less than the velocity of light. One should remark,
however, that the speed at which information are transported in a medium is
not the phase velocity and therefore that it is not clear that this condition is
a causality condition.
With these considerations one estimates that the maximum mass of a neutron star
is around 3 solar masses and certainly below 6.
165
D(Oo748-676
phases
Early-burst
Ëo
_
-
"k
0.006
ïa
T
E
o
(f,
iî
pu
0.004
a
+t
c
:J
E o.oo2
o.oo8
o
c
'i^
i
J' f
E
t)
g
g
f
0.004
0.003
(f)
c{rE
=*Î
x oèr
olJz
D
x
EgES
= = = =
B:3
O O
= oo
z
J
=
0 6
=
0.002
o
v
o.oo1
20
Wavelength
6)
Figure 14.6: XMM-Newton spectrum of a burst of EXO0748-676 (from Cottam,
Parels and Mendez, Nature 420,51, 2002).
It is highly interesting to compare these considerations with the observed masses
of neutron stars (Figure 14.8).
It is very striking that all these masses are extremely close to the Chandrasekhar
mass.
166
Figure 14.7: The structure of a neutron star (from Shapiro and Teukolsky, 1983).
167
Figure 14.8: Neutron star masses from observations of radio pulsar systems
(Thorsett & Chakrabarty 1999, ApJ 512, 288). All error bars indicate central 68%
confidence limits, except upper limits are one-sided 95% confidence limits. Five
double neutron star systems are shown at the top of the diagram. In two cases,
the average neutron star mass in a system is known with much better accuracy
than the individual masses; these average masses are indicated with open circles.
Eight neutron-star–white-dwarf binaries are shown in the center of the diagram, and
one neutron-star–main-sequence-star binary is shown at bottom. Vertical lines are
drawn at m = 1.35 ± 0.04M .
168
Chapter 15
Pulsars
Neutron stars in isolation have first been observed as pulsars which have been discovered completely unexpectedly in 1967 when Jocelyn Bell, who was looking for
scintillation in the flux of radio sources, noted the appearance of regular pulses in
the light curve she was looking at. See Figure 15.1 for an example of pulsar light
curve. This discovery gave a Nobel prize for Hewish, the thesis advisor of J. Bell.
15.1
Basic observational facts
Pulsar periods range from ms to few s. For a long time the Crab pulsar with a period
P = 33 ms was the fastest known. In the 1980’s, however, pulsars with P ∼ few ms
where discovered. We will come back to these objects called millisecond pulsars in a
later chapter. We will also not discuss X-ray pulsars in this chapter. We limit here
the discussion to the "classical" radio pulsars, which we write simply as pulsars, the
other categories will be discussed later.
The pulse shape of pulsars taken as an average of some 1000 pulses is very stable,
allowing for very accurate timing measurements (Figure 15.2). Thanks to pulsars
astronomy is thus back in the business of precise time measurement.
The periods of pulsars increase slowly with time: Ṗ ∼ 10−12 − 10−13 (Figure 15.3).
Figure 15.1: Individual pulses from the 0.714 s pulsar PSR 0329+54 at 410 MHz
(from Manchester and Taylor 1977).
169
Figure 15.2: A sequence of 100 pulses from PSR 1133+16 at 600 MHz. An average
of 500 pulses is shown at the top (Cordes 1979, SSRv 24, 567).
For example ṖCrab = 4.22×10−13 (note that Ṗ is a unitless number). The ms pulsars
have much smaller Ṗ (Ṗ ∼ 10−19 ).
In some pulsars the period abrubtly decreases (the spin increases) at few year intervals. Those events are called glitches (Figure 15.4). They are caused by a reorganisation of the structure of the neutron star (starquake) in which the moment
of inertia changes leading to the observed rotation rate change to keep the angular
momentum constant.
Pulsars are broadly distributed around the plane of the Galaxy (Figure 15.5). There
are some 500 of them that are known. The width of the distribution with respect
to the plane of the Galaxy indicates that the pulsars have non negligible proper
velocities.
The short periods of pulsars indicate that the objects must be compact (second light
in size at most). This leaves white dwarfs, neutron stars or black holes a possible
candidates. A simple arguments allows to limit the possible candidates to neutron
stars:
The maximum rotation velocity Ω that an object can have while gravitationally
bound is
170
Figure 15.3: P versus Ṗ diagram for pulsars. Diagonal lines indicate constant
ages according to the formula t = P/2Ṗ (Lyne and Graham-Smith 1990, Pulsar
Astronomy).
Ω2 R '
GM
,
R2
(15.1)
which states that rotational kinetic energy at the surface of the object cannot exceed
the binding energy at the surface. This may be expressed as:
Ω'
p
Gρ,
(15.2)
or
2π
Pmin = 2π/Ω ' √
Gρ
(15.3)
which is of the order of 1 s for ρ ' 108 g cm−3 , the density of a white dwarf. The
existence of periods well below 1 s among pulsars thus clearly excludes rotating white
dwarfs as a possible explanation of the phenomenon. A similar argument can be
171
Figure 15.4: Pulse period of PSR 0833-45 from 1968 to 1980. The period increases
over the time except for short glitches at intervals of few years (Downs 1981, ApJ
249, 687).
Figure 15.5: Distributions of 558 pulsars in Galactic coordinates (Taylor et al. 1993,
ApJS 88, 529).
made in the case of vibration. Another explanation that can readily be ruled out is
based on orbital periods of black holes and/or neutron stars. Whereas such periods
might be envisaged, the derivatives would be such that the periods decrease with
time, as the orbit shrinks by the emission of gravitational radiation.
Black holes do not have a structure that could create periods and accretion is a
highly varying phenomenology. One is therefore lead very naturally to the rotation
of neutron stars to explain pulsars.
172
15.1.1
Distances to pulsars
The high time accuracy available on the pulsar measurements allows us to obtain
their distances using the properties of radio wave propagation in the interstellar
medium. Consider the dispersion relation of radio waves in a plasma:
ω 2 = ωp2 + k 2 c2 ,
(15.4)
where k is the wave number (amplitude of the wave vector) and ωp the plasma
frequency
ωp2 :=
4πne e2
.
me
(15.5)
Radio waves will propagate through the medium for ω > ωp . The group velocity of
the waves is given by
1/2
ωp2
ωp2
dω(k)
k · c2
vg =
=
=c 1− 2
'c 1− 2
dk
ω
ω
2ω
(15.6)
for ω >> ωp , where we have used eq. 15.4 (k 2 c2 = ω 2 − ωp2 ).
The time of arrival of a wave pulse around the frequency ω is given by
D
Z
ta (ω) =
0
1
dl
'
vg
c
Z
D
0
ωp2
dl 1 + 2
2ω
(15.7)
where D is the distance to the object.
Writing explicitely the plasma frequency from eq. 15.7 gives this time as a function
of the integral of the electron density along the path:
2πe2
D
+
ta (ω) =
c
mcω 2
Z
L
ne dl.
(15.8)
0
RL
One calls the integral 0 ne dl the dispersion measure, often written DM . This
arrival time is a function of the wave frequency. Deriving the arrival time one
obtains
dt
4πe2
=−
DM
dω
mcω 3
(15.9)
which is an observable quantity. Thus the dispersion measure is known. Assuming,
or knowing from some other source the density of the interstellar medium (in average
< ne >' 0.03 cm−3 ) thus provides the distance to the object.
173
15.2
Magnetic Dipole model
One can gain an excellent understanding of the energy balance in a pulsar by considering the pulsar as a magnetic dipole in which the magnetic axis is not aligned
with the rotation axis (the same is true on the Earth).
→
−
Let’s consider a neutron star with a magnetic field B which is oblique with respect
to its rotation axis.
The magnetic moment is:
|m| =
Bp R3
,
2
(15.10)
where R is the radius of the neutron star and Bp the dipolar field. Since the field
and the rotation axis are not aligned, the magnetic moment varies with time and a
variable magnetic dipole radiates (in a way similar to the variable electric moment
that also radiates as we have seen at the very beginning of these lectures).
The energy loss of the variable magnetic dipole is
Ė = −
2
|m̈|2 ,
3c3
(15.11)
with m :
1
m = Bp R3 · ek cos α + e⊥ sin α cos Ωt + e0 ⊥ sin α sin Ωt ,
2
(15.12)
and where ek and e⊥ are the unit vectors respectively parallel and perpendicular to
the rotation axis.
We find when deriving 15.12 twice and inserting in 15.11:
|Ė| =
Bp2 R6 Ω4 sin2 α
6c3
(15.13)
(calculations are left to the reader).
The radiated energy has to have some origin... . The most readily available source
stems is the slowing down of the spin of the pulsar. The kinetic energy of the rotation of the star and its first derivative are given by:
Erot = 21 IΩ2
E˙rot = IΩΩ̇,
where Ω is the angular rotation and I the momentum of inertia.
174
(15.14)
Neutron Stars
L27
t4
l3
t2
o
!a ll
èo
I
t0
Graveyard
9
8
-3
-2
-l
0
log F (s)
Figure 15.6: B versus P diagram for pulsars. This diagram is equivalent to Fig. 15.3
when using the B(P, Ṗ ) relation.
Combining 15.13 and 15.14, the magnetic field can be expressed as:
IΩΩ̇6c3
B = 6 4 2
R Ω sin α
2
(15.15)
which can be estimated for P = 33 ms, the rotation period of the Crab pulsar, and
Ṗ = 4.22 · 10−13 its derivative.
10km and mass 1033 g,
R 2For a neutron star45of radius
the moment of inertia (I = r dm) is I = 1.4 · 10 g cm2 .
Expression 15.15 gives the magnetic field of a pulsar as a function of its period and
period derivative. It can therefore be used to express the diagram 15.3 not as Ṗ vs
P , but as B vs P. This equivalent diagram is shown in Fig. 15.6.
With sin α ∼ 1 the field we find (B = 5.2 · 1012 gauss) which is remarquably close
to the one found when observing cyclotron emission lines in X-ray pulsars. This
implies that a very plausible scenario for the source of energy of pulsars is indeed
their rotational energy. This is naturally also well in line with the observation that
the pulsar periods increase with time, showing that they indeed loose rotational
energy.
This model of a pulsar is shown in Fig. 15.7, which shows in addition the expectations
one has as to the polarisation of the outcoming radiation. The observed radiation
comes indeed from relativistic electrons that are accelerated and follow the magnetic
field lines. The acceleration is therefore in the plane of the field line and induces a
linear polarisation that will evolve through the pulse, as the observer sees electrons
following different fields lines (remember that the cone of radiation of the electrons
is narrow and forward showing).
We can use these ideas further and calculate the age of the pulsars using Eqs.(15.13
and 15.14).
175
2t7
Pulsars:their origin and evolution
FIELDLINES
LINEOF SIGHT
OF EMISSION
Pofh of
LINEOF
SIGHT
POLARIZATION
ANGLE
Time*
Rofotionr+Àlu
nulgllull
Axis
Time*
Fig. 6. The polar cap model of Radhakrishnanand Cooke (1969).The observationalinspiratior
for this model was the sweepof the position angle of linear polarization within the pulse windov
(schematicallyshown by the frgure on the left). The length and orientation of the arrolrysrepresen
the polarizedflux and position angle.This led to the'light house model'for pulsarsshown oI
the right: the observedradiationis believedto originatevery closeto the magneticpoles,and beamet
into a narro,wcone.The fîgure at the top left shows the geometryof the projectedfield lines anc
the locusof line of sight
Figure 15.7: A rotating neutron star with a misaligned magnetic dipole functions
(and therefore radiates) like a time variable magnetic moment.
Let’s introduce:
Ω
6Ic3
T := = 2 6 2
,
Ω̇ 0 B R sin αΩ20
(15.16)
where the index "0" indicates the present time.
Eqs 15.13 and 15.14 can be combined to express Ω/Ω̇:
B 2 R6 Ω2 sin2 α
Ω̇
=
,
Ω
6Ic3
(15.17)
in which we introduce T in the following way:
Ω̇
B 2 R6 sin2 α Ω2 2
1 Ω2
=
Ω
=
−
.
0
Ω
6Ic3
Ω20
T Ω20
(15.18)
Separating the variables leads to
dΩ
dt 1
=−
,
3
Ω
T Ω20
(15.19)
1
t 1
− Ω−2 = −
+ c0 .
2
T Ω20
(15.20)
which can be integrated to give
176
Let Ω = Ωi be the initial angular velocity at t = 0 to find the integration constant
to find finally for Ω(t) :
− 12
2Ω2i t
.
(15.21)
Ω = Ωi 1 + 2
Ω0 T
This can be inverted to give the age of the pulsar at a given angular velocity Ω0 :
T
t=
2
Ω20
1− 2
Ωi
.
(15.22)
Note that after a significant slowing down, Ω0 << Ωi :
t'
T
.
2
(15.23)
Finally we find for the values of the Crab pulsar we have used :
Tpulsar = 2486 years and therefore an age t of ' 1263 years
which is very close to the real age (2000 − 1050) ' 950 years deduced from the date
of the explosion of the supernova at the origin of the Crab nebula and pulsar.
This result confirms very convincingly the origin of the pulsar in the Crab as the
supernova observed by the Chinese in 1054. It also shows the level of intuition that
Baade and Zwicky had had in 1934.
Although one has understood with the oblique rotating magnetic dipole the loss of
energy of the pulsars, one is still very far from the understanding of the observed
radiation. This requires a considerably more involved consideration on the magnetosphere of the pulsars.
15.3
The aligned rotator and the pulsar magnetosphere
A first approach to the magnetosphere is given by the the aligned rotator model
in which we now consider the case of a rapidly rotating magnetised neutron star in
which the magnetic field and the rotation axis are parallel. At first sight such an
object should not radiate, as the magnetic dipole is not variable in time, both axis
being parallel.
Consider a dipole magnetic field:
3
R
B = Bp
,
r
177
(15.24)
where R is the size of the neutron star and r is the distance to the star. The field
can only have the dipole shape for
c
,
(15.25)
Ω
because at larger distances the rotating magnetic field lines that are attached each
to a point on the surface of the star would move faster than the speed of light. Rc is
called the light cylinder radius. It is located at the distance at which the equatorial
co-rotation velocity equals the velocity of light. At distances large compared to the
2
light cylinder, the magnetic field will be given by the Poynting flux S = cB
that
4π
charaterises the radiation of the star. Close to the light cylinder both approaches
to the magnetic field must match. The energy loss Ė is then given by the Poynting
flux integrated over the sphere:
r Rc =:
|Ė| = 4πRc2 · S
c 2
= 4πRc2 ·
B (Rc )
4π
c 2
6
2R
=
· cBp 6 Ω6
Ω
c
Bp2 R6 Ω4
=
.
c3
(15.26)
(15.27)
(15.28)
(15.29)
This corresponds to the expression we had found in the oblique rotator approach
as Eq. 15.13 without the sin2 (α) term. The link between the slowing down of the
neutron star and its radiation is therefore identical to what we deduced in the
preceding section.
We can now look for the properties of the magnetosphere of the neutron star using
the dipole magnetic field approximation close to the neutron star. The vector form
of the field is
B = Bp R
3
sin θ
cos θ
e +
e .
r3 r
2r3 θ
(15.30)
Inside the star we expect the medium to be highly ionised. The conduction will
therefore be very high. In the infinite conductivity limit (which we
use) the Lorentz
v
force must vanish. Were it not so, the currents j = σ E + c × B would be infinite.
This means that inside the star we have
E+
v
× B = 0.
c
(15.31)
Using the star rotation in Eq. 15.31 we have for the fields inside the star (superscript
"in"):
E in +
Ω×r
× B in = 0.
c
178
(15.32)
In absence of currents at the surface of the star the magnetic field B must be continous at the surface r = R. Using Eq. 15.30, the field is therefore
B
in
sin θ
= Bp cos θer +
e .
2 θ
(15.33)
With Eq. 15.31, we know the electric field inside the star:
E
in
RΩBp
sin θ
=
c
sin θ
e − cos θeθ
2 r
(15.34)
The component of the field parallel to the surface is continous (not necessarily the
component perpendicular to the surface, as there could be a charge at r = R). The
electrical field outside the surface is therefore given by
RΩBp
sin θ cos θ
c
∂ RΩBp
2
= −
sin θ
∂θ
2c
Eθout = −
(15.35)
(15.36)
This is a quadrupolar electric field, its magnitude is
E'
RΩBp
2 · 108
'
B12
c
P
volt cm−1 ,
(15.37)
for a magnetic field of about 1012 G and a period P (in s) of 1 s.
The electric force acting on the elementary charges are much larger than the gravitational binding energy at the surface of the star:
eE ∼ e
RΩBp
GM mp
∼ 109 Fg = 109
.
c
R2
(15.38)
This means that there will be a region where charges will be dissociated in a very
conducting plasma for which E · B = 0. Only thus can the Lorentz force vanish (the
charges cannot be accelerated along the magnetic field lines). This shows that the
vacuum in the vicinity of a rapidly rotating neutron star is unstable.
One can use these fields to get a zeroth approximation to the energy of particles
that can be expected around a pulsar by considering the acceleration of particles in
the field:
γ̇mc = eE =
eRΩBp
,
c
(15.39)
or
γ̇ =
eRΩBp
,
mc2
179
(15.40)
where γmc is the impuls of the particle. For particles accelerated in the vicinity of
the star and traveling close to te speed of light we have
eR2 ΩBp ∼ 11
R
γ ' γ̇∆t ' γ̇ '
= 10 .
c
mc3
(15.41)
This gives for electrons a maximum possible energy of
γmc2 ∼
= 1011 · 500keV = 5 · 1016 eV.
(15.42)
But note that this is a very unrealistic estimate. However, it encourages to look at
the neutron star environement as a source of relativistic particles. These relativistic
particles can then be feeding the supernova remnants that surround some neutron
stars (the plerions, like the Crab nebula) and makes it plausible that synchrotron
radiation is observed. The relativistic particles are also expected to form one component of the cosmic rays, giving right to the speculation of Baade and Zwicky in
1934.
15.3.1
Maximum particle energy
A somewhat more sophisticated approach to the energy of particles that can escape
from a neutron star and thus be observable outside the light cylinder is as follows.
Consider dipolar field for which the field lines are characterised by
sin2 θ
= const.
r
(15.43)
The open field lines are those going through the cap of the neutron star. The last
one being given by the dipole line that extends just to the light cylinder, i.e.
sin2 θ
1
=
.
r
Rc
(15.44)
This field line cuts the surface of the star at θp given by
1
sin2 θp
=
,
R?
Rc
Rc = c/Ω,
(15.45)
R? being the radius of the star. This defines the so-called polar cap (see Fig. 15.8).
We can estimate the potential Φ that corresponds to the electric field, knowing that
the field is the gradient of the potential:
E = −∇φ.
(15.46)
For the quadrupolar electrical field that we found in Eq. 15.36, the potential is
180
φ (r, θ) = −
1 B0 ΩR?5 (3 cos2 θ) − 1
·
.
6 c
r3
(15.47)
The maximum potential that we can use for a particle that escapes the magnetosphere is the potential difference between the pole and θp . We therefore insert the
angle that delimits the polar cap that we found in 15.45 in 15.47 with 15.44 for the
potential at θp using cos2 θ = 1 − sin2 θ = 1 − R? /Rc
Φ(R? , θp ) = −
R?
1 B0 ΩR?2
· (2 − 3 ).
6 c
Rc
(15.48)
1 B0 ΩR?2
· 2,
6 c
(15.49)
At θ = 0, the potential is
Φ(R? , 0) = −
so tat the available potential difference is
Φ(R? , θp ) − Φ(R? , 0) =
1 B0 Ω2 R?3
.
2 c2
(15.50)
For a period os 1ms and a field of 10 1012 G, this gives a maximum available potential
difference of
∆φ ≤ 6 · 1019
volts.
(15.51)
Which gives also the maximal energy that a charge can gain when crossing the
potential difference in eV. Note that this is a very simplified model of a region that
is bound to be very complex and that the potential difference calculated here is
certainly a gross overestimate of the energy that can be gained in the vicinity of a
very powerful pulsar. This energy is, however, still less than the maximum energy
observed in the cosmic rays, which indicates, that pulsars cannot be responsible
directly for these extreme particles.
These results form the basis on which one builds models for the emission of pulsars.
Indeed the oblique dipole model that we considered in section 15.2 shows convincingly what the origin of the energy is, it does not indicate at all, however, how radio
emission is created. Taken at face value, it would indicate a radiation of frequency
comparable to that of the pulsar.
It is thought that the radio emission is produced by relativistic electrons traveling
along the curved open magnetic field lines that emanate from the polar cap. These
electrons are accelerated, the field lines are not straight, and therefore radiate according to the Larmor formula. For a trajectory with a radius of curvature ρ and
energy γ, the frequency of the emitted radiation is
ν∼
3 3c
γ .
4π ρ
181
(15.52)
Realistic models rely on configurations in which the charge distribution σ related to
the quadrupole electric field
σ=−
BΩR
cos2 θ
4πc
(15.53)
accelerates in a region in which E · B 6= 0 charges. This is region may be found
either in the polar regions (polar gap model) or in regions in the magnetosphere.
They are most probably not stable, the accelerations happening in sparks. The
polar cap interpretation also explains naturally that the radiation comes in a cone
and is therefore pulsed seen from the observer far away. It explains the polarisation
structure of the pulses, as hown in Fig. 15.9.
One can also see from the formalism discussed here that the acceleration source, the
potential difference, depends on the period and the magnetic field in the following
way:
∆φ ∼
B
.
P2
(15.54)
Thus at some point the field will have decayed and the period increased in ways such
that the sparks will be develop any more, and the pulsar will cease to radiate radio
waves. This is the death line in Fig 15.6. Neutron stars beyond the death line are in
the so called graveyard and are most difficult to detect. Indeed only their thermal
radiation is observable in the X-rays. Although very weak and difficult to measure,
this radiation is of prime importance as, linked with the ages of pulsars, it provides
information on the cooling of the neutron stars, hence on their heat content and
therefore on their internal structure.
Pulsars are at the origin of a very wide domain of astrophysics, often because of the
very accurate nature of their clocks. Millisecond pulsars have Ṗ ' 10−18−20 . The
main result that this precision has lead to is probably the discovery of the binary
pulsar 1913+16 by Hulse and Taylor. The changes of the orbital period of this pulsar
shows very convincingly that the system is loosing energy through the emission of
gravitational waves in excellent agreement with the predictions of general relativity
(see below).
182
"
e
Op
!p
nf
dl
ie l
ine
s
Closed field lines
Rc
Figure 15.8: Geometry of the magnetic field and polar cap.
183
2t7
Pulsars:their origin and evolution
FIELDLINES
LINEOF SIGHT
OF EMISSION
Pofh of
LINEOF
SIGHT
POLARIZATION
ANGLE
Time*
Time*
Rofotionr+Àlu
nulgllull
Axis
Fig. 6. The polar cap model of Radhakrishnanand Cooke (1969).The observationalinspiratior
for this model was the sweepof the position angle of linear polarization within the pulse windov
(schematicallyshown by the frgure on the left). The length and orientation of the arrolrysrepresen
the polarizedflux and position angle.This led to the'light house model'for pulsarsshown oI
the right: the observedradiationis believedto originatevery closeto the magneticpoles,and beamet
into a narro,wcone.The fîgure at the top left shows the geometryof the projectedfield lines anc
the locusof line of sight
Figure 15.9: Schematic of the oblique rotator model for pulsars in which the emission
happens in the polar cap region, from Shapiro and Teukolski.
184
Chapter 16
Gravitational Radiation
Pulsar timing with its very high precision can be used to measure, also with great
precision, the orbits of the pulsars in binary systems. This is at the origin of the
indirect discovery of gravitational radiation. The discovery is indirect in that the
gravitational waves have, up to now, never been directly measured. The reaction,
however, of the binary system to the emission of gravitational waves, has been
measured on the so-called Hulse-Taylor pulsar, PSR 1913+16. This has earned
Hulse and Taylor a Nobel price.
Gravitational radiation is a quadrupolar process, i.e. contrary to electro-magnetic
radiation that is emitted when a charge dipole varies in time (but not when a
monopole varies), a time variable quadrupole (Ijk ) is necessary to emit gravitational
waves. (graviton are a massless spin 2 particles). Einstein general relativity allows
to calculate the emission of gravitational waves with the following result (see e.g.
Misner, Thorne and Wheeler for a discussion).
LGW ≡
1 G ... ...
dE
= 5 h I jk I jk i.
dt
5c
(16.1)
The quadrupole moment of a mass distribution is given by
Ijk =
X
A
1
A A
A 2
mA xj xk − δjk x
3
(16.2)
The quadrupole of a system of 2 point masses in a binary orbit can be calculated.
Note, however, that since gravitation is a non linear theory, it is not straight forward
to use the quadrupole formula in the case of two very dense objects, as a complete
discussion of the system would require that the gravitational field generated by
each object be smoothly merged into the binary system, rather than treating the
two masses as point masses. This notwithstanding, let us describe a binary system
consisting of M1 and M2 in circular orbits around the center of mass. In this case
M1 a1 = M2 a2 = µa,
where
185
(16.3)
µ=
M1 M2
M1 + M2
(16.4)
is the reduced mass of the system. The "xx" component of the quadrupole moment
of the circular binary system in the x-y plane is
M1 a21 + M2 a22 cos2 φ + cst
1 2
=
µa cos 2φ + cst
2
Ixx =
(16.5)
(16.6)
and similarly
1
Iyy = − µa2 cos 2φ
2
(16.7)
1
Ixy = Iyx = µa2 sin 2φ.
2
(16.8)
and
There is no z component. Let φ = Ωt, Ω the angular velocity, and derive with
respect to time 3 times and feed into the quadropule formula to find
1 G ... ...
h I jk I kj i
(16.9)
5 c5
2
1 2
1G
6
·
(2Ω)
·
µa
=
hsin2 2Ωt + sin2 2Ωt + 2 cos2 2Ωti (16.10)
|
{z
}
5 c5
2
LGW =
2
3
32 G (GM )
µa
5 c 5 a9
32G4 M 3 µ2
=
.
5c5 a5
=
2 2
(16.11)
(16.12)
We now study the effect that this energy loss has on the orbit parameters of our
binary system. For this consider the orbital period P = 2π
which is an observable.
Ω
The third Kepler law relates the period and a in the following way
(2π)2
GM
= 3 ,
Ω =
2
P
a
2
(16.13)
from which we have
Ṗ
3 ȧ
=
.
P
2a
186
(16.14)
The energy of the binary system is
E=−
1 GµM
,
2 a
(16.15)
which decreases as a result of the gravitational radiation losses.
−Ė = LGW =
1 GµM
ȧ
ȧ = −E · .
2
2 a
a
(16.16)
This leads to
ȧ
Ė
=+ .
a
E
(16.17)
and therefore to an orbital period change:
Ṗ
3 ȧ
3 Ė
3 32 G4 M 3 µ2 (−2a)
96 G3 M 2 µ
=
=+
= ·
·
=
−
.
P
2a
2E
2 5 c 5 a5
GµM
5 c 5 a4
(16.18)
The corresponding calculation for an elliptical orbit of eccentricity e leads to
96 G3 M 2 µ
Ṗ
=−
· f (e)
P
5 c 5 a4
(16.19)
−7/2
73 2 37 4
f (e) := 1 + e + e
1 − e2
.
24
96
(16.20)
with
The binary pulsar PSR 1913+16 was discovered by Hulse and Taylor in 1974 using
the Arecibo radio telescope. The binary nature was clear from the irregularities of
the pulse arrival times (Fig. 16.1), although only one of the two masses making the
binary is a pulsar. The other is not observed.
The main parameters of the pulsar and orbit are:
Ppulsar = 0.059029997929883(7) s
Ṗ = 8.62629(8) · 10−18
e = 0.617127
Porbit = 27906.98163(2) s.
3
(m2 sin i)
In a Newtonian system, only the mass function f = (M
2 of a binary of this
1 +M2 )
nature can be measured. In a relativistic system, however, more observables can be
6πGM2
measured, prima amongst them the rate of change of the periastron ω̇ = a1 (1−e
2 )P c2 .
In the case of the binary pulsar this is found to be 4.22660(4) degrees per year
(see fig. 16.2 to see how the periastron advance modifies the pulse arrival time as a
function of phase). This may be compared to the 43 arcsec/year observed in the
187
Figure 16.1: The pulse arrival time for the binary pulsar PSR 1913+16 (Taylor
Nobel lecture).
solar system for the orbit of Mercury. Gravitational redshift and transverse Doppler
effect can also be measured. This leads to an over constrained system that can
therefore be tested for consistency.
Figure 16.2: The observed changes in the pulse arrival change over a 10 year period.
The changes in the pattern are due to a large variation in the periastron position.
(from ???)
The relativistic effects therefore provide a measurement of both masses of the system:
The pulsar mass is 1.386 (3) solar masses
The companion mass is 1.442 (3) solar masses.
With both masses and the eccentricity known, it is then possible to calculate the
rate of change of the period (from eq. 17.9 and to compare it with observations. This
is shown in Fig, 16.3.
188
Figure 16.3: The rate of change of the orbital period of PSR 1913+16. The dots
refer to the observations, while the curve represents the theoretical prediction from
the quadrupole formula (Will 2006).
The overconstraint of the system maybe be shown on a 2-D diagramme with the
mass of each member of the binary on each axis as in Fig. 16.4.
189
Figure 16.4: The masses of the pulsar and its companion shown with the constraints
on Ṗ , ω̇ and γ̇ from Will 2006.
190
Chapter 17
X-ray Binaries
The X-ray sky is characterised by different source populations.
There is one population of sources that are rather weak (whatever this is supposed
to mean) and isotropic on the celestial sphere. This population is made of Active
Galactic Nuclei (AGN) and it is (most probably) at the origin of the so-called diffuse
X-ray background. Indeed, the more sensitive the instruments for point sources, the
more the “diffuse” background is resolved in individual weak sources.
Another population of weak sources has emerged in the last decade, these are the
coronae of “normal” stars. Paradoxally, often cool stars, when in strong rotation,
have a very active corona that can be at the origin of a substantial X-ray flux that
has been revealed in particular by ROSAT.
There is a population of extended sources, the supernovae remnants.
Another population of extended sources is given by the clusters of galaxies. These
contain large quantities of hot gas (107 K), where large means some ten times the
quantity of matter contained by the galaxies.
Finally, there is a population of bright (up to some 1038 ergss−1 sources, clearly
associated with our Galaxy. These are the sources that were first discovered at Xray energies, they are much more luminous than the other galactic sources and show
very peculiar properties, in particular a variability by many orders of magnitudes
on many different timescales, from less than a second to years and longer. These
are accretion powered binary systems. The so-called X-ray binaries.
The first of these sources to be discovered was Sco-X1 during a ballon flight by
Giacconi et al. in 1962. This brought Giacconi a Nobel prize in 2002. The detection
was completely unexpected, as any extrapolation from the then known X-ray flux
of the Sun to the distances at which most stars showed that would by far not be
observable.
A first survey of the sky for X-ray sources was performed by the UHURU satellite
launched by NASA on the 12.12.1970.
The optical counterpart of Sco-X1 was discovered (“identified”) in 1966. In 1967,
Shklovski proposed a model of the X-ray emission based on the transfer of mass
from the companion to the compact object. This is the paradigm in which we are
191
COMPACT OBJECT
white dwarf
no white dwarf
Cataclysmic variable
X-ray binary
COMPANION MASS
M > M
M < M
HMXRB
LMXRB
COMPACT OBJECT MASS
COMPACT OBJECT MASS
M < 3M
Pulsar X
(with NS)
M > 3M
Black hole
candidate
M < 3M
M > 3M
Burster
(with NS)
Black hole
candidate
Figure 17.1: Schematic view of the many types of X-ray binaries.
going to discuss these binaries in the following subsections.
The population of X-ray sources in our Galaxy (excluding now stars and supernovae
remants) is very varied, many subcategories must be distinguished (Figure 17.1).
When the compact object is a white dwarf one speaks of cataclysmic variables, also
called novae or dwarf novae. In these systems matter is pulled from a (Roche lobe
filling) companion onto the white dwarf. The white dwarf increases in mass and
eventually becomes more massive than the Chandrasekhar mass. When this mass is
reached, the dwarf explodes in a type I supernova. We will not discuss these objects
any further here.
The second distinction is based on the mass of the companion. When the mass is
large, the companion is an O or B star, the X-ray luminosity of the system is less
than that in the optical domain, the system is called a high mass X-ray binary,
abbreviated by HMXRB. When, however, the companion mass is low, less than the
192
mass of the Sun, the X-ray luminosity is larger by about one order of magnitude than
the optical luminosity and the system is called a low mass X-ray binary, abbreviated
as LMXRB.
The distributions of both types of systems are very different. HMXRB are young
systems (high mass stars live for a short time only). They are naturally found in
regions of star formation and associated with the disk of our Galaxy. The LMXRB
are older systems, low mass stars live long, and are therefore less concentrated on the
disk (they show a broader distribution in latitudes), but more concentrated towards
the central regions of the Galaxy (Figures 17.2 and 17.3).
Figure 17.2: Distribution of LMXRB (open circles) and HMXRB (filled circles) in
the Galaxy. A sample of 86 LMXRB and 52 HMXRB is shown. Note the significant
concentration of HMXRB towards the Galactic Plane and the clustering of LMXRB
in the Galactic Bulge (Grimm et al. 2003, ChJAA 3S, 257).
Figure 17.3: Distribution of Galactic HMXRB (solid lines) and LMXRB (thick green
lines) against Galactic latitude (left panel) and longitude (right panel). The arrows
in the right panel mark the positions of the tangential points of spiral arms. Note
that on the right panel the number of LMXRB is divided by 3 (Grimm et al. 2003,
ChJAA 3S, 257).
193
The magnetic fields of both types of systems are very different. The B field of
HMXRB is high, of the order of 1012 G, whereas that of the LMXRB is much smaller,
often of the order of 1011 G or less. This has for consequence that the X-ray properties of both types of systems are also very different. The accretion system is also
different. The HMXRB accrete mass that stems from the companion stellar wind
while the matter falling from the companion of a LMXRB comes from the surface
of the companion that fills its Roche lobe.
These differences imply a very different phenomenology. The HMXRB accrete matter that is linked to the magnetic field lines from far away from the compact object.
The material is channeled by the magnetic field onto the magnetic poles of the neutron star. These systems appear like X-ray pulsars. Note that the energy source in
the X-ray pulsars is accretion onto a neutron star and not rotational energy from
the neutron star as was discussed in the case of the radio pulsars above. Indeed the
periods of the X-ray pulsars are observed to increase or to decrease.
A disk forms in LMXRB and the material is accreted in a more distributed way onto
the neutron star that does not appear like an X-ray pulsar. The material is spread
on to the complete star and every few hours it is ignited in a nuclear reaction that
is observed as an X-ray burst. These systems are called “bursters”.
A further distinction on the type of binary one is dealing from is related to the mass
of the compact object. When this mass exceeds the maximum mass for a neutron
star, the system contains a black hole “candidate”.
In the following we will discuss HMXRB, LMXRB and black hole candidates in 3
subsections.
17.1
High Mass X-ray Binaries (HMXRB)
In many of these systems, or for some periods of time in their lives, the period
decreases, i.e. the angular velocity and the angular momentum increase.
The Figure 17.4 shows some typical light curves and some light curve derivatives
as obtained from the BATSE instrument (Bildsten et al. 1997, ApJS 113, 367).
It must be remarked that the shape of the light curves is very different compared
to that of the radio pulsars. This and the different sign (at times) of the period
derivative clearly indicates that the origin of the energy radiated is different. X-ray
pulsars are driven by the accretion of matter in the potential well and not by the
slowing down of the rotation of the neutron star.
The Figure 17.5 shows how such a system might look like.
The changes of the angular velocity of the neutron stars evidenced by the changes
in their period comes from a torque. This is caused by the accretion of angular
momentum that comes together with the accretion of matter. The process is driven
by the magnetic field.
The radius at which the magnetic field is dominating the geometry of the accretion
flow is called the Alfven radius rA . This radius is given by the equality of the B field
194
Figure 17.4: Long-term frequency histories of three LMXRB (Her X-1, 4U 1626-67,
and GX 1+4) and three HMXRB (Cen X-3, OAO 1657-415, Vela X-1) observed with
BATSE in the period 1993 April 23-1995 February 11 (MJD 48370-49760, thick line
in the figures. Bildsten et al. 1997, ApJS 113, 367).
energy density on one side and that of the accretion flow:
B 2 (rA )
1
= ρ(rA )v 2 (rA ),
2π
2
(17.1)
where B(r) is the magnetic field, ρ(r) is the accretion flow density as a function of
the distance to the neutron star and v(r) is its velocity. Consider a dipole field:
195
Figure 17.5: Left panel: schematic dipole magnetosphere around a neutron star
that is accreting material. Infalling gas is excluded from the toroidal region whose
cross section is shaded (Davidson & Ostriker 1973, ApJ 179, 585). Right panel:
enlargement of the base of an accretion funnel, near a magnetic pole of the neutron
star (Davidson & Ostriker 1973, ApJ 179, 585).
B(r) = B0 (
Rstar 3
)
r
(17.2)
with B(rstar ) of the order of 1012 G and a free fall accretion for which the velocity
is the escape velocity:
v(r) = vf f (r) =
q
2GM
r
(17.3)
ρ(r) = ρf f (r) =
Ṁ
,
4πr2 vf f (r)
and Ṁ is the mass accretion rate. After some algebra one obtains
7/2
rA =
6
B0 Rstar
√
Ṁ GM
(17.4)
−2/7 4/7 12/7
rA ' 3.2 · 10 M˙17
B0,12 Rstar,6 ( MM )−1/7 cm.
8
Note that in realistic cases the accretion is unlikely to be really free fall, but a
significant amount of angular momentum is likely to be present and the radius to
consider is less than the Alfven radius. We will use r0 ' 1/2rA in the following. In
a Keplerian accretion disk, the accreting flux has a specific angular momentum
e =
`(r)
p
GM r0 ,
196
(17.5)
which is tranferred to the star, thus creating a change in the angular velocity.
d
e 0 ).
(IΩstar ) ' Ṁ `(r
dt
(17.6)
d
dI
dΩstar
dI
I
Ṗ
Ṗ
Ṗ
(IΩstar ) = Ωstar +I
=
Ṁ Ωstar −IΩstar '
Ṁ Ωstar −IΩstar =: `estar Ṁ −IΩstar ,
dt
dt
dt
dM
P
M
P
P
(17.7)
which defines `estar . Together with 17.6, 17.7 gives:
Ṁ e
Ṗ
e 0 )).
'
(`star − `(r
P
IΩstar
(17.8)
This explains naturally how the sign of the period derivative can change, when the
specific angular momentum of the accreted matter is larger than that of the star,
the latter will be accelerated, while if the specific angular momentum is smaller the
star will be slowed down. In the case when the neutron star angular momentum can
be neglected in 17.8, with 17.5 expressed explicitely as a function of r0 , and with
(17.4) that gives us r0 as a function of Ṁ (and hence L) we obtain:
Ṗ ∝ −
P
Ṁ L−1/7 ∝ −(P L3/7 )2 ,
IΩstar
(17.9)
where we have used that Ṁ ∝ L. (the L dependencies imply an Eddington accretion
that will be explicited later). The Figure 17.6 shows how the relation (17.9) is
satisfied in a number of observed cases.
17.2
Low Mass X-Ray Binaries (LMXRB)
The phenomenology of LMXB is very rich. In particular the timing analysis of the
emission is providing insight into the accretion flow and its interaction with the
surface of the star. The spectral information is also important to know how the
matter distributes itself onto the surface.
We will here limit the discussion to the most important fact that is related to the
fate of the material falling onto the surface of neutron stars. We will give then some
remarks on a presently very important category of LMXRB, the micro-quasars.
17.2.1
Bursters
LMXRB are very often (not always, in particular not when the compact object is a
black hole) observed to emit more or less regularly every few hours important bursts
of X-ray emission, see Fig. 17.7 and 17.8. The very rough ratio of the time averaged
persistent to time averaged burst fluxes is
197
Figure 17.6: The theoretical relation between the spin-up rate, −Ṗ , and the quantity
P L3/7 , superposed on the data (pre-1979) for 9 pulsating X-ray sources. Shown is
the effect of varying the neutron star mass, assuming a stellar magnetic moment
µ30 = 0.48. Each curve is labeled with the corresponding value of M/M . The
shaded area indicates the region where 0.5 ≤ M/M ≤ 1.9 (Ghosh & Lamb 1979,
ApJ 234, 296).
Figure 17.7: The first X-ray burst detected with the ANS satellite showing a very
rapid followed by a slower (10s of seconds) decay in the flux.
198
Figure 17.8: The EXOSAT light curve for the X-ray burster 1636-536 (Breedon et
al. 1986)
R
dtL
R persistent ' 100.
dtLbursts
(17.10)
This is interpreted as stemming from the ratio of gravitational binding energy to
that of nuclear energy liberated when the accreted material, in the form of H and
He, is transformed into Fe, the equilibrium nucleus. This nuclear reaction is verly
likely happening in an explosive way, as the electron degeneracy in the surface layer
of a neutron star is very high. The conditions are indeed close to those met in white
dwarfs. Degenerate electrons imply that there is no thermal inertia in the system.
We had seen when discussing neutron stars that the binding energy of particles at
their surface is of some 10% of the particle rest mass. On the other side, the nuclear
energy available in fusion reactions is of few permille of the rest mass, leading to a
ratio close to the observed ratio of persistent to burst energies.
Spectra of X-ray bursts are well approximated by black body emission with temperatures of few 107 K. The properties of the black body emission can therefore be
used to gain some deep insight on the properties of neutron stars.
The observed flux from a spherical black body is given by:
2
4πRstar
fν =
πBν (T ),
(17.11)
4πD2
where D is the distance to the source and Bν is the black body emissivity. The
luminosity of the source is given by:
2
L = 4πRstar
σT 4 ,
199
(17.12)
Figure 17.9: Variation of the blackbody temperature, blackbody radius and luminosity for a composite of 4 bursts observed with EXOSAT from 1636-536, as obtained
from time-resolved blackbody fitting of burst spectra (Turner & Breedon 1984, MNRAS 208, 29).
where σ is the Stefan-Boltzmann constant. The flux which is measured at the Earth
L
is 4πD
2 . Most of the sources are observed in the direction of the centre of the Galaxy.
Thus, in average, the distance to the sources is the distance to the Galactic centre.
This distance is known, it is of the order of 8 kpc. The temperature is given by the
spectral shape, it is measured by fitting the observed emission. The flux is observed
and the distance known. It is therefore possible to deduce the radius of the neutron
star. It is found (see Figure 17.9) that the radii that are deduced in this way are
indeed of the order of 10 km, as expected from the structure of the neutron stars.
Further information may be obtained from the observation of absorption features in
the spectra. These provide an estimate of the redshift and therefore give another
combination of mass and radius (Waki et al 1984).
The phenomenology of bursters is not limited to bursts. In several LMXRB, including pulsars, dips in the light curves are observed. These dips are highly dynamical,
with variations on short time scales. They are thought to occur in systems that are
seen close to the plane of the disk. When matter flows onto the accretion disk, the
pressure at the point of impact increases thus leading to a thickening of the disk at
200
Figure 17.10: Light curve of the binary 1254-69 showing regular dips and a type
1 burst (Courvoisier et al 1986). The top panel shows the hardness ratio observed
during this observation.
this place. This thickening comes once per revolution across the line of sight and
causes a decrease in the source flux. This is still a subject of many investigations,
not all the observed features fitting with this simple model.
The phenomenology of LMXRB is also marked by the presence of dipping sources
like 1254-69, Fig. ??. In these sources the flux is regularly seen to decrease in a
very irregular way. This is interpreted as beeing due to absorption along the line of
sight caused by material that accumulates at the place where the incoming stream of
matter from the companion, through the L1 point impacts on the accretion disk. the
periodicity is therefore that of the orbit of the binary system. This interpretation is
re-inforced by the observation of a spectral hardening at the tim of the dip, which
implies an increased column density of matter.
17.3
Millisecond pulsars
There are few pulsars that do not fit the general discussion we had in section 15.2.
These have very fast periods, a few milliseconds, and very small period derivatives,
which, as we have seen, indicates weak magnetic fields (108 G). Young objects are
indeed expected to have short periods, they are, however, expected to have large
magnetic fields and therefore to rapidly slow down. They move towards the right
and the bottom in the Fig. 17.11 during their "lives" as normal pulsars. They eventually cross the "death line" and enter the pulsar graveyard, where their periods and
magnetic fields a too weak to sustain the processes that lead to radio emission. Millisecond pulsars are found in the lower left area in Figure 17.11, where the evolution
of single pulsars does not lead.
It is thought that these millisecond pulsars are “recycled” in the following sense.
201
Pulsars are indeed born with fast rotation and large magnetic fields and are thus
relatively quickly slowed down. In this process the magnetic field slowly decays.
The pulsar emission mechanism (which we have not discussed) needs large electrical
and magnetic field to function. Below a critical magnetic field, the electric field
which is induced is not sufficient anymore to sustain the radio emission, the pulsars
therefore disappear from the radio sky. This is the so called death line (graveyard
line in Figure 17.11).
Figure 17.11: Diagram of period-derivative period (P − Ṗ ) for the known population
of pulsars (Kramer 2004, IAUS 218, 13). Lines are drawn for constant magnetic field
and constant characteristic age. Young pulsars should be located in the upper left
area. When pulsars age, they move in the central part of the P − Ṗ diagram where
they spend most of their lifetime. The lack of pulsar in the lower right area is due
to the "death" of pulsars when their slowdown has reached a critical state. This
state seems to depend on a combination of P and Ṗ which can be represented in
the P − Ṗ diagram as a pulsar death line (graveyard in the figure).
In binary systems, however, the companion will eventually evolve to the stage of red
giant and fill its Roche lobe. The binary system then becomes a LMXRB with a
neutron star that radiates through accretion (and nuclear reactions). As the neutron
star accretes material it will also accrete angular momentum, as it has been discussed
in the HMXRB section. As a consequence, the neutron star rotation will accelerate.
As we have seen the limit between acceleration and slowing down of the spin of
a star is when the co-rotation of the magnetic field at rA is equal to the Kepler
rotation at this radius. The co-rotation radius is given by the equality of the star
angular velocity with the Kepler angular velocity at Rc (see ??):
202
rc = (
1
GM 1/3
) ∝ (M P 2 ) 3
2
Ω
(17.13)
Equating this with the alfven radius given in 17.4 gives the equilibrium period Peq
(including the the numerical factors)
6/7
−5/7
Peq = 1.9 msec · B9 M
(
Ṁ
18/7
)−3/7 R6 .
˙
MEddington
(17.14)
Clearly the radius cannot be less than the radius of the neutron star, which leads to
maximum periods of the order of 1-2 msec. The minimum period is a constraint for
the equation of state, as the radius cannot be such that the Kepler period at this
radius is less than the observed period.
In summary the evolution of millisecond pulsars can be seen on Fig. 17.12. In a
binary system the pulsar evolves as in an isolated system to the graveyard, but
when the companinion evolves to the point when it fills its Roche lobe, then angulr
momentum is tranferred to the neutron stars, which period decreases until it crosses
again the "death line". This scenario is very nicely confirmed by observations.
Indeed, one has found in LMXRB signatures of fast pulsars. This was first detected
in the X-ray emission of bursters and then also in sources like IGR J00291+5934,
a transient INTEGRAL source detected in December 2004 (Eckert et al.) and in
which the period was found to decrease while the outburst lasted (see fig. 17.13).
Figure 17.12: The evolution of a pulsar in a binary system (from www, cornell)
17.3.1
The eclipsing pulsar
The model we have just described implies that msec pulsars are to be found in binary
systems. However, it is apparent from Fig. 17.11that there are such objects that are
203
Figure 17.13: The evolution of the phase (signalling the change of frequency of the
pulsar during the observation) for the pulsar IGR J00291+5934 during the outburst
of December 2004 (from Burderi et al. Ap.J. 657, 961, 2007)
single. For those that are found in clusters of stars, the density of stars is such that
collisions between stars are relatively frequent. When a binary system is involved
in such a collision, the gravitational energy exchange can lead to the disruption of
the binary and therefore explain the presence of isolated msec pulsars.
A further possible scenario to explain the presence of isolated msec pulsars is suggested by the eclipsing pulsar PSR1957+20. In this system eclipses are observed,
as shown in fig. 17.14 , indicating that the pulsar is in a binary system. The orbital
period is 9.2 hours and the eclipse lasts a surprisingly long 8% of the period. The
mass function of the system can be used to give an estimate of the mass of the
companion:
(mc sin i)3
−6
2 = 5.2 · 10 M
(mp + mc )
(17.15)
from which it is evident that the companion is a low mass star (remember that in an
eclipsing system sin i ' 1 and the mass of the pulsar is most likely around 1.4 M ).
Indeed with these parameters Eq. 17.15 gives mc = 0.02M .
In a binary system the size of the Roche lobe RL is given by
RL
0.49 q 2/3
'
,
a
0.6 q 2/3 + ln (1 + q 1/3 )
204
(17.16)
Figure 17.14: Period of the pulsar PSR1957+20 as a function of the phase. The
curve indicates the binary nature of the pulsar (the period increases as the pulsar
recesses from the observer while traveling along its orbit and shortens as it nears
again). The eclipse appears as the absence of a pulse between phases .2 and .3.
From Fruchter et al. Nature 333, 237, 1988.
where a = 1.7 1011 cm is the semi major axis and q = 0.014 is the mass ratio. The
size of the Roche lobe is thus found to be RL ' 2.81010 cm.
The size of the eclipse can be estimated as
Recl ∼ πa ·
∆tecl
∼ 4 · 1010 cm
P
(17.17)
shorter than the Roche lobe. In a binary the Roche lobe is the largest size a gravitationally bound system can have, any material outside this radius is not bound
to one of the two stars, but to the whole system. It is therefore not possible that
the eclipse in PSR1957+20 is caused by a gravitationally bound object. Instead it
is proposed that the companion is being evaporated by the energy radiated by the
pulsar and that the eclipse is caused by the "cometary" tail of the material escaping
from the companion (Phinney et al. Nature 1988). This interpretation is firmed
up by the observation that the trailing end of the eclipse is characterized by long
lags of arrival of the pulses (see fig. 17.15). These lags indicate the presence of a
dense plasma that slows the radio waves (see the discussion of the measurement of
distances to pulsars in chapter....).
We can estimate whether it is energetically possible to evaporate a substantial fraction of the companion with energy radiated by the pulsar by comparing the rotational energy of the pulsar
1
−2
erg
Erot = IΩ2 ' 2 · 1052 Pms
2
(17.18)
which is the energy that is available for radiation with the binding energy of the
205
Figure 17.15: Residuals of the period of the pulsar PSR1957+20 as a function of
the phase around the eclipse. The delay observed after the eclipse when taking
the orbit into account indicates that the pulses arrive later than expected from the
binary parameters. This is due to the fact that the radio ewaves are slowed down in
the plasma that emanates from the companion in a form of "cometary tail". From
Fruchter et al. Nature 333, 237, 1988.
companion, which we estimate at first simply as a uniform sphere
Ebin
GM 2
' 1048
=
R
M
M
2 R
R
−1
.
(17.19)
Provided that the efficiency of the energy transfer η is such that
ηErot > Ebin ,
(17.20)
enough energy will be available. The efficiency η has a geometrical bound (provided
that the pulsar energy is isotropically radiated) given by
η<
Rc
2a
2
.
(17.21)
Assuming that the companion fills its Roche lobe, we thus have η < 7 10−3 . Thus
showing that it is energetically possible to evaporate the companion. The time that
this process might take can be estimated by considering the energy loss by the pulsar
Lp = E˙rot which we calculated in Chapter xxxx
Lp = Ėrot
6
2 RN
S 2
=
B
3 c3
206
2π
P
4
(17.22)
with the energy that must be acquired by material escaping the companion at velocity v:
1
Ṁ v 2 = η Ėrot .
2
(17.23)
We parametrize the efficiency as
2
Rc
,
(17.24)
η=f·
2a
with f < 1. The velocity of the escaping material has to be of the order of the
escape velocity
v ∼ vesc =
2GMc
Rc
1/2
.
(17.25)
Combining eqs 17.22, 17.23, 17.24 and 17.25, one obtains an expression for the evaporation time τevap :
τevap :=
3c3 2GMc2 a2 P 4
Mc
·
,
=
6
2RN
Rc3 8π 4 f B 2
Ṁc
S
(17.26)
which has to be compared to the age of the pulsar τpulsar , given by its slowing down
rate
τpulsar :=
=
P
2Ṗ
3c3
P2
·
I
6
B 2 16π 2 RN
S
(17.27)
(17.28)
(You can verify that
τevap
1 Ebin
=
.)
τpulsar
η Erot
(17.29)
For a small mass star like the companion, the structure is that of a degenerate gas
for which the mass radius relation is
R
= 0.013 (1 + X)5/3
R
M
M
−1/3
,
(17.30)
where X is the Hydrogen mass abundance. Inserting this in eq. 17.26, one sees that
the evaporation time is proportional to Mc3 :
τevap ∼
M2
M2
∼
∼ M 3.
R3
M −1
207
(17.31)
When all numerical parameters are included one obtains
τevap
= 0.01
τpulsar
0.1
f
a 2
2R
P
1.55ms
2 Mc
M
2 Rc
R
−1
.
(17.32)
With the properties of the companion of PSR1957+20 (Mc = 0.02M , Rc = 0.166R , a =
2.5R ) one obtains
τevap
= 4 · 10−3 .
τpulsar
(17.33)
This shows that it is indeed quite possible, even with a modest efficiency, that
the pulsar evaporates completely its companion in a time short compared to its
age. Since the age of the pulsar is of the order of 109 years, one also sees that
the companion is evaporated in a time of the order 107 years. It is therefore quite
possible that isolated msec pulsars have indeed undergone an evolution in which the
pulsar has been accelerated in a phase in which it acquired angular momentum from
the binary system in which it lived during a LMXRB period, before it evaporated
the companion.
17.4
Micro-quasars
A number of LMXRB have shown in the past years that they emit jets in a way very
similar to the quasars that will be discussed later in these lectures. These systems
have been called micro-quasars. They are of a high level of interest, as they exhibit
in a matter of hours a phenomenology that is observed to take place in years in
quasars.
17.5
Transient sources
208
Chapter 18
Black Hole Candidates
We have seen in chapter XXXXX that neutron stars have a maximum mass beyond
which degenerate neutrons cannot resist the gravitational attraction. This mass is
of about 3 solar masses, the exact value depends on the poorly known equation of
state at densities beyond the nuclear density. Beyond this mass, the structure of
the object must be that of a black hole.
There are systems in which the mass of the compact object can be seen to be larger
than the mass limit for neutron stars. These systems are binaries (Cowley, AARA
1992) in our Galaxy, but also the massive black hole in the center of our Galaxy or
the supermassive black holes that power the phenomenology of AGN.
We discuss in this chapter first the dynamical evidence for compact objects of masses
beyond 3 solar masses and then some direct evidence for black holes.
18.0.1
Dynamical evidence
X-ray binaries
The first system in which it was suspected that the mass of the compact object
is larger than 3 solar masses are X-ray binaries. In these systems it is possible to
measure the radial velocity of the companion object and to deduce the mass function
f (M ) =
(MX sin i)3
,
(MX + MC )2
(18.1)
where we note with MX the mass of the compact X-ray source and MC the mass of
the companion. sin i is the term that gives the inclination of the orbital plane on
the sky.
Cygnus X-1 is a bright X-ray source that was discovered shortly after the discovery
of X-ray sources outside the solar system. It was then associated with a B supergiant
star of 9th magnitude (HDE 226868) which was found to be in a binary of period
5.6 d. Analysis of the radial velocity gives a mass function f (M ) = 0.25 ± 0.01 MM .
The absence of an eclipse implies that i is less than about 33 degree with a most
209
probable value of 65deg. With these parameters, the masses of both stars have most
likely values of MC = 33 ± 9M and MX = 16 ± 5M . Independent analysis yields
a lower mass limit of 9 solar masses for the compact object. Uncertainties in the
analysis stem from the fact that the properties of the companion may be influenced
by the binary nature of the system and the presence of the compact object, and also
from the unknown inclination. Nonetheless this remains a very strong black hole
candidate.
There are other black hole candidates in HMXRB. It is, however, important to look
also at LMXRB. Indeed inspection of eq. 18.1 shows that the mass function is a
lower limit for the mass of the compact object. sin i is always less than or equal 1
and the companion star has a mass larger than zero.
LMC X-3 is a LMXRB in the large Magellanic cloud. The optical identification of
the companion leads to a 17 mag star in a binary of period 1.7 d and a velocity
amplitude of 235 km/s, leading to a mass function f (M ) = 2.3M . Optical obserMC
' 0.6 − 0.8 leading to estimates of
vations indicate that the mass ratio must be M
X
MX ' 4 − 7M . The rotation of the companion allows us to gain some idea on the
inclination of the system which is found to be i = 50 − 70 deg, leading to a lower
mass of 6M for the compact object.
It is also clear that the unseen object, even it were only 2.3M , would be brighter
than the observed limits in the optical, leading to the conclusion that it cannot be
a "normal" star.
An even more extreme system is A0620-00 (= Nova Mon 1975, 1917). This is a
transient source. By this one designates sources that are observed at some epochs
and not at others, i.e. sources that vary by amplitudes such that for period of time
their flux is less than the sensitivity limit of the instruments used to perform the
observations.
The binary period of this system is 7.8 h with orbital velocities of 470 km/s. This
leads to a mass function f (m) = 3.18 ± 0.16M , already larger than the maximum
mass of a neutron star. A reliable estimate for the mass of the companion (0.7 M )
and a limit of 50 degrees for the inclination lead to a lower mass limit for the compact
object of 7.3 M .
Anoter transient system, GS1124-68, Nova Muscae 1991 has been found to have a
period of 10.5 h, a mass function f (M ) = 3.1 ± 0.5M . This system has become
famous for the observation by the SIGMA telescope of a bright 511 keV line for few
hours during the outburst. This has not since then been observed.
The central black hole of the Milky Way
It has been known for a long time that there is a mass concentration in the center of
the Milky Way. This knowledge resulted from the study of the motion of stars in the
central regions of the Galaxy. At first the measurements were of a statistical nature,
the angular resolution did not allow to identify individual stars and to measure their
radial velocity and/or proper motion. The result of these measurements was the
estimate of the total mass within the radius sampled by the observations. The result
210
Figure 18.1: The annihilation line observed by SIGMA in 1991, IKI team.
is shown in fig. 18.2. The flattening of the curve as the enclosed radius decreases
was a clear sign that there was a central compact mass in the Galaxy rather than a
mere compact stellar cluster.
As the angular resolution of the instruments improved, first with speckle techniques,
then with the use of adaptive optics it became possible to measure the positions of
individual stars in the central second of arc of the Galaxy. These measurements
could then be repeated and individual proper motions could be measured. Not only
could the velocity on the plane of the sky, but with still improving resolution, the
acceleration of the individual stars was measured, until, finally, the orbital parameters of the stars were available. Figs 18.3 and 18.4 gives the status of this work as
of 2003.
Once the orbital parameters are determined one can deduce the mass around which
the stars orbit. This is found to be 3.3 ± 0.7 · 106 M . Clearly the mass has to
be confined within the orbit, i.e. in a size of some 0.01". There is no known
configuration of matter that would be stable at the implied density in excess of
211
Figure 18.2: Mass enclosed within the radius r in the center of the Galaxy. (schoedel
et al ApJ 2003)
1017 M /pc3 (Schoedel at al 2003). The conclusion is therefore that there is a black
hole of 3.3 ± 0.7 · 106 M at the center of the Milky way.
18.0.2
Intrinsic signatures?
It is very difficult to "prove" that an object is a black hole based on its emission
signature. The main reason for this is that a neutron star is only slightly larger
than a black hole and that the last stable orbit around a Schwarzschild black hole is
3RS . Since in a black hole system we observe matter basically further than the last
stable orbit, we observe matter in the same region in both neutron star and black
hole systems. In fact proving the existence of a black hole in these terms means
proving the absence of a surface to the object.
On the other side, proving that an object is a neutron star is a much easier task
and is readily made as soon as an X-ray burst is observed. It thus happened in the
course of the years that some objects that were believed to be black holes showed a
burst and were immediately re-classified as neutron stars.
This difficulty is nicely expressed in Fig. 18.5 in which one shows the hard X-ray
luminosity and the soft X-ray luminosity of a number of objects. It was for a long
time believed that neutron stars live in the so called X-ray burster box, but a number
of them have been shown to lie on the right side of the box, leaving only the top
part of the diagramme for black hole systems. This shows that accreting neutron
stars can have bright hard emission tails. These hard tails are thought to originate
212
Figure 18.3: The orbits of individual stars in the center of the Galaxy. The stars
orbit a mass within the orbit. The deduced mass is 3.3 ± 0.7 · 106 M (schoedel et
al ApJ 2003)
from thermal Comptonisation of soft photons by electrons with temperatures above
100 keV.
Black hole binaries can have very complex spectral energy distributions that vary
widely over time. Fig. 18.6 shows the spectrum energy distribution of the black hole
syetem GRO J1655-40 measured at several different epochs. First one should note
that dividing the states in "high" and "low" as had been used for a long time is
insufficient, as the source may be bright in the soft band and weak in the hard band
or the contrary, hence the more complex denominations of the states (LHS, VHS,
etc). It is thought that these different spectral states can be caused by the interplay
between an accretion disk and a hot corona that surrounds it. In essence the soft
emission is made by the accretion disk while the hard emission is due to a hot corona
213
Figure 18.4: details of the orbit of the "S2" star (schoedel et al ApJ 2003). Note
that since the observations at the base of this figure have been obtained, the star
S2 has completed a complete revolution on its orbit.
that Comptonises a fraction of the soft photons from the disk. The corona seems to
be evaporating the inner parts of the disk at times. Note that even if this scenario
does not embed the whole truth, whatever the model of the source it will have to
include elements that have similar phenomenological components and behaviors.
An interesting argument is brought by Tanaka and Lewin (in X-ray binaries, eds
Lewinvan paradijs and vanden Heuvel 1995) who argue that the inner radius of
accretion disks can be measured from the spectral shape of the X-ray emission
(remember that the inner temperature is a function of the distance to the inner
radius of the disk) and who performed this analysis for a number of black hole
candidates obtaining inner radii of 18-40 km and for a number of neutron stars
obtaining inner radii of 4-10 km. The lower inner radii of the neutron star system
can then be interpreted as showing that the central object is indeed less massive
than the black holes for which the inner radius has to be at about 3RS .
Paradoxically, the luminosity of the neutron stars can also be expected to be larger
than that of the black holes. This results from the fact that the kinetic energy of
the accreting material on the last orbit of the accretion disk is radiated when this
material hits the (slowly rotating) surface of the neutron star while it is advected into
the black hole in the last free fall towards the horizon. This leads to the existence of
214
Figure 18.5: Hard and soft X-ray luminosities from a number of X-ray binaries from
di Salvo et al. 2001 ApJ. 554, 49.
Figure 18.6: The spectral states of the black hole candidate GRO J1655-40 from
Done et al. 2007. The right panel gives the different configurations of disk and
corona that are thought to produce the mix of components resulting in each of the
ovearll spectra
215
Figure 18.7: Quasi Periodic Oscillations (QPOs) observed in 2 black hole candidates
with RXTE from Shaposhnikov and Titarchuk 2007 ApJ. 663, 445.
a bright boundary layer in neutron star systems that is not observed in black holes.
High frequency variability with timescales of the order of few RS /(0.5c) ' O(1) msec
is expected when an irregular mass distribution reaches the last stable orbits around
a black hole. This variability is expected to have a quasi periodic behavior rather
than a strictly periodic behavior, as the "clumps" will soon disappear beyond the
last stable orbit. This gives rise to a large body of literature. Fig.18.7 gives an
example of a study of QPOs in two black hole candidates.
Finally and to be complete one should also note that intrinsic signatures for the
existence of black holes is given by the phenomenology of AGN where variability on
short time scales is observed as well very high luminosities, both pointing towards
the presence of a black hole at the centre of the object.
The study of matter close to the horizon of black holes is of very high importance,
not only to demonstrate the presence of black holes, but more generally as the only
available tool to test general relativity in strong fields. Until the detection and
measurement of gravitational waves, X-ray astronomy gives us the only possibility
to probe the metrics in deep gravitational fields. Variability and spectral shapes are
then the doors to tests of the Einstein theory of gravitation.
216
Chapter 19
Gamma ray bursts
19.1
Basic facts
Gamma Ray Bursts (GRBs) are short bursts of gamma rays lasting from a small
fraction of a second to 100 seconds or so that are observed from random directions
of the sky at a frequency of roughly 1 per day.
Figure with different bursts
Distribution of duration
19.2
Short history
GRBs were discovered completely serrendipitously by US military satellites (VELA)
carrying X-ray, gamma ray and neutron detectors in order to monitor eventual
atmosphere nuclear test explosions (from the Soviet Union). While no special nuclear
activity was observed from the Earth, the first GRBs were observed in 1967. It
soon became appparent that origin of these GRBs was "cosmic", a result that was
published in 1973, when the information was declassified.
The question of the origin of these bursts was then on the table. Efforts were made to
localise them as best as possible. For this purpose, a series of gamma ray detectors
on a variety of spacecrafts in the solar system were used to measure the burst
arrival time at each spacecraft and then to use triangulation techniques to locate
the origin of the bursts. This is the so-called interplanetary network (IPN) that has
been functioning since 1976. In 1991 NASA launched the Compton Gamma Ray
Observatory (CGRO) that included one instrument (BATSE) capable of roughly
localising the GRBs. Over 2700 GRBs were measured.
Most astrophysicists would have expected that GRBs are of Galactic origin. Indeed,
if you deduce from the observed fluence (the time integrated gamma ray flux) and
assume the distance to the source to be of the order of several kpc, you would
conclude that the energy at the origin of the bursts is of the order of 1053 ergs, the
binding energy of a neutron star. This provides a "natural" framework in which
217
Figure 19.1: Sky distribution of the BATSE (triggered and non-triggered) GRBs.
3906 bursts are shown on the figure, no deviation from isotropy can be found (from
Stern et al. 2001, ApJ. 563, 80).
the ories of gamma ray bursts could be developed. This predicted that the burst
distribution would have some signature of the Galaxy. Contrary to this expectation,
however, the GRB distribution measured by BATSE was found to be completely
isotropic, with no measurable deviation from isotropy (Fig. 19.1).
This implies that if the bursts are indeed of galactic origin, they must be distributed
in such a way that the offset of the position of the Sun with respect to the center
of the Galaxy is unnoticeable, thus placing the origin of the GRBs in the far halo.
This brings difficulties of its own, because at these distances the brightest bursts
would have such properties that those taking place in the nearby galaxies ought to
be observed.
The story got to be more difficult when the bursts fluence or peak flux distribution
was measured. Fig. 19.2 shows the peak flux distribution (the number of bursts
brighter than the flux P). This distribution is expected to have a slope of -3/2 for
a homogeneous distribustion (see section 19.3). The deviations from the -3/2 slope
indicate that there is a derth of weak bursts. This is as if we were at the center of
a distribution in which the density of GRBs was fading away with distance.
The explanation was given in 1997 when observations with the Beppo-SAX satellite
could locate the position of a GRB with a precision that was sufficient to point an
X-ray telescope also on board Beppo-SAX and then optical telescopes to the location
of the burst some time after. A source was then found that faded with time, clearly
the counterpart of the source. The optical spectrum of this "afterglow" could be
measured and absorption lines were found and their redshift could be measured
and found to be 0.695. This immediately implied that the distance to GRBs is
cosmological and their luminosities a significant fraction of the energy equivalent of
the mass of the Sun.
218
Figure 19.2: The number of GRBs which peak flux is brighter tan flux P. A homogeneous distribution of bursts in space would imply a slope of 3/2 (see below). From
????
19.3
Homogeneous distribution of events
In a homogeneous distribution of sources of luminosity L0 and of density n, e can
calculate the observed flux distribution for a (any) observer. There are ∆N = 4π∆rn
sources in a shell of width ∆r at a distance r from the observer. The sources in the
L0
shell are observed to have a flux s = 4πr
2 . They are observed in a flux interval
∆s = −
Using r =
q
L0
,
4πs
dL
−2L0
∆r =
∆r.
dr
4πr3
(19.1)
one finds
L0
∆N = 4πr ∆rn = 4π
n
4πs
2
4π(
q
L0 3
)
4πs
2L0
∆s.
(19.2)
The number of sources brighter than a given flux s is given by
Z
N (s) =
∆N
ds ∝
∆s
Z
s−5/2 ∝ s−3/2 ,
which explains the slope -3/2 in the log-log plot of fig.19.2.
19.4
Interpretation
energy budget
219
(19.3)
high gamma factor
low hadron numbers
models
19.5
Afterglows
Light curves as measures of the jet opening angle
relation of GRBs with supernovae
19.6
GRBs as cosmological probes
220
Figure 20.1: The B magnitude of 3C 273 as recorded by photographical plates
between 1887 and 1980. (Angione and Smith AJ 90, 2474, 1985)
Chapter 20
Active Galactic Nuclei
20.1
Basic observed properties
-Luminosity, redshift, history of QSO discovery
-emission from radio to gamma rays
- variability—-> compactness, black holes
Basic parameters: Black hole masses and accretion rates out of the luminosity
221
Figure 20.2: The optical spectrum of 3C 273. The lines are identified as the Balmer
seies redshifted by 0.158. This established QSOs as very luminous objects at cosmological distances.
Figure 20.3: Type I and type II Seyfert galaxy optical spectra. Type I have broad
permitted lines, while those of type II galaxies are narrow (from Keel, university of
Alabama)
20.2
Zoology
-Seyfert galaxies, link with QSOs, broad and narrow lines
-radio galaxies
-BL Lacs
222
Figure 20.4: total emission and polarised flux of the Seyfert II galaxy NGC 1068. In
polarised light, the permitted emission lines are seen to be broad (From Antonucci
and Miller, ApJ. 297, 621, 1985).
20.3
The emission components
20.4
statistics and evolution
20.5
Link with host galaxies
223