Modélisation du syst`eme triple autour du pulsar radio PSR J0337+

Transcription

Master Science de la matière
École Normale Supérieure de Lyon
Université Claude Bernard Lyon I
Stage 2013–2014
Guillaume Voisin
M2 Physique
Modélisation du système triple autour
du pulsar radio PSR J0337+1755
Résumé : Ce stage avait pour but de parvenir à une description précise et stable au cours
du temps des temps d’arrivée au radio-télescope de Nançay des impulsions du pulsar PSR
J0337+1755 récemment découvert par Ransom et al. (2014). Ce pulsar est en effet en orbite relativement proche, inférieure à la taille d’un système planétaire comme le système solaire,
avec deux naines blanches. De ce fait les intéractions gravitationnelles à trois corps doivent être
prises en compte, numériquement, dans la reconstitution des orbites.
Ce système est unique en son genre dans la mesure où il est à ce jour le seul système constitué
de trois objets de masse stellaire comportant un pulsar radio milliseconde. Il est par conséquent
possible de l’étudier en profitant de l’époustouflante précision offerte par cette ”horloge interne”
au système qu’est le pulsar. Néanmoins, pour être efficace, le chronométrage de pulsar doit
s’appuyer sur un modèle réaliste des délais de propagation de la lumière, modèle qui n’existe pas
à ce jour pour un tel système triple.
Mots clefs : pulsar, étoile à neutron, chronométrage, trois corps
Stage encadré par :
Lucas Guillemot
[email protected] / tél. (+33) 2 38 25 52 87
Ismaël Cognard
[email protected] / tél. (+33) 2 38 25 79 08
Laboratoire de Physique et de Chimie de l’Environnement et de l’Espace 3A, Avenue de
la Recherche Scientifique 45071 Orléans cedex 2 France
http://lpce.cnrs-orleans.fr/
August 2, 2014
Remerciements
Je remercie chaleureusement mes tuteurs Ismaël Cognard et Lucas Guillemot pour leur accueil
et leur bienveillance. Je remercie également Jean-Matthias Grießmeier pour toutes les discussions enrichissantes tenues au cours de ces quatre mois à partager le même bureau ainsi que
pour ses conseils, sans oublier les deux autres membres de l’équipe pulsar du LPC2E : Gilles
Theureau et Kuo Liu. Merci aussi à Claire Revillet pour ses conseils et son assitance sur le
plan informatique. Plus généralement je salue l’accueil du LPC2E, et toutes les personnes avec
qui j’ai eu des échanges amicaux.
I would like to thank Anne Archibald for her very appreciated help during my visit at ASTRON.
Also I would like to thank Jorge Piekarewicz1 and Farrukh Fattoyev2 for answering my questions
about neutron stars mechanical properties.
Contents
1 Introduction
1
2 Introduction to the J0337+1755 system
2.1 Previous work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Orders of magnitude and accuracy . . . . . . . . . . . . . . . . . . . . . . . . . .
2
2
4
3 A full three-body timing model
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Computing the orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Retrieving the two-body results with a numerically integrated trajectory
3.2.2 Computation of the three-body orbits . . . . . . . . . . . . . . . . . . .
3.2.3 Numerical intricacies . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 First order systematic delay : the Rømer delay . . . . . . . . . . . . . . . . . .
3.3.1 First order correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 Second order correction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Second order systematic delays . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 The Einstein delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.2 Tidal effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4 Fitting to data : a very delicate optimization problem
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Starting solution and the choice of parameters . . . . . .
4.2.1 Initial solution . . . . . . . . . . . . . . . . . . .
4.2.2 Choice of fitting parameters . . . . . . . . . . . .
4.3 Minimization with the Minuit library . . . . . . . . . . .
4.4 The Markov-Chain-Monte-Carlo (MCMC) approach . . .
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5 Conclusion
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References
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1
Jorge Piekarewicz, Department of Physics; Florida State University Tallahassee, FL 32306-4350
Farrukh Fattoyev, Department of Physics and Astronomy, Texas A&M University-Commerce, Commerce,
Texas 75429-3011, USA ; Institute of Nuclear Physics, Tashkent 100214, Uzbekistan
2
Modélisation du système triple autour du pulsar radio PSR J0337+1755
Guillaume Voisin
Appendix A The gravitational two-body system defined by orbital elements
23
A.1 Relation between orbital elements and state vectors for a single body . . . . . . 23
A.2 The case of two bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
A.3 The mass function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Appendix B Jacobian of the Newtonian three-body differential system
24
Appendix C Likelihood function
24
3
1
Guillaume Voisin
Introduction
The first pulsar was discovered by Jocelyn Bell and Tony Hewish in 1968 (Hewish et al., 1968).
Its periodic radio pulse was soon identified to be that of a highly magnetized neutron star (see
in particular Gold (1968) and Gold (1969)), until then hypothetical objects proposed among
others by Robert Oppenheimer in the 30’s. A neutron star is believed to be the remnant of
a supernova, made of a neutron dominated matter with a density of the order of that of an
atomic nucleus. Nevertheless, it is of macroscopic size with a radius of about ten kilometers,
thus weighting about 1.5 Solar masses (just above the Chandrasekhar limit). The consequence
is that these stars are part of the so-called compact objects, with a compactness parameter
of about 0.5 (1 for a black hole, 10−6 for the Sun), in the vicinity of which the curvature of
space-time cannot be neglected at any rate.
The surroundings of the star are filled with a magnetosphere supported by a gigantic magnetic field that typically ranges from 108 Gauss for millisecond pulsars to 1015 Gauss3 for
magnetars (Earth’s magnetic field is about 0.5 Gauss for comparison). It is widely believed to
be the locus of the electromagnetic emissions of the star as first suggested Gold (1968). The
detailed mechanisms are poorly understood but observation proved that it ranges over a very
large spectrum from GHz in radio, observed with radio-telescopes, to gamma rays, observed
recently with the Fermi-LAT satellite. Besides, this emission must be very localized in the
magnetosphere, such that it emits in a narrow beam that we can see only once at every pulsar
rotation if by chance it crosses our line of sight, just like a cosmic light house. These radiations
are powered by the rotation of the pulsar that together with the magnetic field makes the star
behave like a giant dynamo. The pulsar consequently slows down at a measurable pace, related
to its magnetic field intensity and spin frequency.
Pulsar spin periods range from a few milliseconds to several seconds, a ”normal” pulsar
turning about himself in typically one second. The former are naturally called millisecond
pulsars (MSPs). They form the most interesting family for pulsar timing, which consists in
measuring with a high accuracy the times of arrival of the pulses on Earth. Indeed, their
particularity is to have a lower magnetic field (108 Gauss as said above) and thus a very slow
spin down rate. In other words they are extremely stable clocks which pulse can be predicted
within a few hundred nanoseconds interval (Ransom, 2012). These pulsars had a rich life,
they are sometimes referred to as recycled pulsars. Indeed, contrary to what their rapid spin
frequencies suggest they are old pulsars, but that have lived with a companion, which makes a
big difference ! The commonly accepted scenario is the following : in a binary one of the stars
goes supernova, leaving a neutron star that slows down to become a normal, one-second pulsar
in a hundred Myr while the companion continues its evolution during Gyrs. When the latter
starts ascending the giant branch, it transfers angular momentum and mass to the neutron
star via accretion, thus spinning up again the old neutron star. Accretion also significantly
consumes the magnetic field, and we end up with an old, low magnetic field and so frequencystable, millisecond pulsar.
These MSPs thus provide a very accurate internal probe of the system they belong to. This
is the reason why they are sometimes called ”neutron star laboratories”, because it is one of the
rare astrophysical precision measurements one can achieve. Moreover, the extreme conditions
of gravitation and magnetic fields surrounding these stars make them unique candidates to
test the limits of fundamental theories, and in particular the theory of gravitation. To do this
one needs to take into account all the systematic effects, and in particular the orbital motion
of the binary system, the delays due to deformations of space-time, interstellar dispersion,
3
1 Gauss = 10−4 Tesla.
1
Guillaume Voisin
the motion of the Earth, atmospheric delays etc... Models were developed since the 70’s,
including an increasing number of systematics to an always higher order of approximation,
in particular for the post-Newtonian description of orbital motion (Blandford and Teukolsky
(1976) and Damour and Deruelle (1986) in particular). Nevertheless, MSPs always had only
one companion star, usually a white dwarf, but that recently changed with the discovery of
PSR J0337+1755 (Ransom et al., 2014).
Indeed, this MSP has two companion white dwarves in rather close orbits. For the historical reasons I just gave no model was developed so far to deal with such a system. It was
consequently impossible to time it to a high level of accuracy for the simple reason that every
systematic was not taken into account. When the LPC2E team in Orléans began recording
data from this system with the Nançay radio-telescope, they quickly faced that problem. That
is the reason why they proposed an internship to deal with that issue. That supposed to make
a new numerical model from scratch. In the next section, I present in more details the system
we are dealing with and why it needs a specific modelization, in section 3 I present the model
itself as I developed it so far, including orbital motion and some post-Newtonian corrections,
then in section 4 I show how I could fit this model to the data at my disposal and the current,
encouraging results that were obtained.
2
Introduction to the J0337+1755 system
2.1
Previous work
The discovery of this system was initially published by Ransom et al. (2014). It is constituted
of a radio pulsar with a spin frequency of 366Hz orbiting two white dwarves in hierarchical
order : the inner white dwarf lies within 1 ls (light second) of the pulsar for an orbital period
of 1.6 days while the outer is within 100 ls for an orbital period of about 327 days (see table
?? for more detailed parameters and error bars). Thus this system is quite compact and the
interactions are rather strong compared to anything that was discovered before : the previously
known three-body systems involved planetary mass objects, such as B1620−26, while here we
have three stellar-mass objects in rather close orbits.
A qualitative knowledge of the parameters of the system can be achieved thanks to the
tools that already exist to deal with pulsars with planets. More specifically the so-called BTX
model implemented in Tempo (Hobbs et al. (2006) and Edwards et al. (2006)) was made to deal
with planets in the approximation that they do not cross interact. It is actually good enough,
replacing planets by white dwarves, to keep track of the phase of the pulsar at plus or minus
half a turn as shown in figure 1.
Nevertheless this is not enough to fold the raw data on a long period of time, that is piling
up consecutive pulse profiles to make an averaged one with higher accuracy, since this demands
to predict the next pulse with a below-microsecond accuracy4 . This is the main goal of this
internship to be able to do that. Besides that, the fact that there are three bodies with nonnegligible interactions allows to solve some degeneracies in the orbital parameters and thus to
extract more information about the masses or the inclination of the orbits than is possible with
an ordinary binary system, as we will see in section 4.2. One may also want to take advantage
of the incomparable regularity of pulsars and the unique configuration of this system to test
fundamental principles of general relativity such as the strong equivalence principle. This is
what we are looking forward.
4
The data is recorded with a binning of 250ns which sets the higher precision limit
2
Guillaume Voisin
Figure 1: Residuals of the BTX model applied to the last-to-date Nançay data, that is the
difference between the time of arrivals (TOAs) predicted by the model and the measured times.
i
Barycenter
Direction of ascending node
Earth
Earth
Figure 2: Sketch of the orbits as fitted in Ransom et al. (2014), not to scale. The neutron star is
the smallest of the bodies but the heaviest so has a smaller amplitude of motion. Together with
the closest (red) white dwarf they form the inner system. To a good approximation this one
can be considered as a body orbiting the outer (green) white dwarf to form the outer system.
To achieve these goals it is necessary to solve the exact three-body equations of motion since
the gravitational coupling is too strong to keep the desired level of accuracy (see below) over
years of data. This is the object of sections 3. Additionally one needs to take into account
3
Guillaume Voisin
systematic effects such as Roemer, Einstein or Shapiro delays (see section 3.3 and 3.4).
During this work, we took contact with Anne Archibald5 who works on the fitting model
used in Ransom et al. (2014) and started a collaboration realized by a visit in late June. We
then agreed that the numerical codes would remain fully independent from each other so as to
be able to confirm results. Indeed, on each side this computation revealed to be delicate and
sometimes tricky because of the very high level of accuracy to achieve.
2.2
Orders of magnitude and accuracy
The input we have is a set of 5019 times of arrival (TOAs) of the pulsar radio pulse received at
the Nançay decimeter telescope6 between August 2013 and July 2014. We corrected them from
all the systematics related to the Solar system and the interstellar medium using the standard
tools provided in Tempo (Hobbs et al. (2006) and Edwards et al. (2006)) : mainly Solar-system
orbital motions and interstellar dispersion. Consequently we will from now on only consider
barycentric time of arrivals at infinite frequency, that is pulses as if they arrived directly at
the Solar-system barycenter, considered a Galilean frame, at infinite energy (so without being
dispersed at all).
These TOAs are recorded with an accuracy of about 3µs. They are given in modified Julian
days7 (MJD), for example the first one is : 56,492.297,192,640,59 MJD. Given that there are
86,400 s in one MJD we soon realize that the accuracy of our TOAs lies in the last digit. From
a numerical point of view, such a number is a float coded on 64 bits (double in C or Fortran)
and we readily understand that numerical round-off will be an issue. The solution will be to use
80-bits-C long doubles for most computation (see section 3.2.3) to avoid loosing the physical
accuracy in numerical round-off. From the point of view of the pulsar itself, it means that its
relative location must be known with an accuracy of a few hundred meters at any time, as
compared to the outer orbit wideness of about 1 AU8 , and this from an estimated distance of
more than 1kpc9 !
In table 2.2, I summarized the relative weight of interactions in several known 3-body systems
in order to get a broader picture of the system. We see that the interactions in the system
J0337 clearly follow a hierarchy that somewhat lies in between the Earth-Moon-Sun and the
Sun-Jupiter-Saturn hierarchies. Note that these can be treated with a very reasonable accuracy
by perturbation methods, but those will intrinsically fail to give the amazing accuracy needed
here over long time periods.
Another reason not to use perturbation methods is that post-Newtonian effects are likely to
appear in this system, such as precession of the periastron, that would make them even harder
to derive. Indeed, contrary to anything in the Solar system, we here deal with compact objects
only. The compactness parameter is given by the ratio of Schwarzschild radius Rs = 2GM/c2
(Misner and Wheeler, 1973) of the object over its radius, and is about 10−4 for a white dwarf to
0.5 for a neutron star. Post-Newtonian effects are likely to appear for two reasons : if the speed
is a reasonable fraction of the speed of light or if the gravitational field is sufficiently large. The
speed of the pulsar is about 10−4 c (a few tens of kilometers per seconds) which is enough to yield
a significant Einstein effect (see section 3.4.1) while the relative strength of the gravitational
field can be determined looking at RS /r where r is the distance between the two bodies. If this
ratio is close to zero then gravity tends to be Newtonian (Will, 2014). Here it is about 10−5
5
Netherlands Institute for Radio Astronomy (ASTRON), Dwingeloo, The Netherlands
Radiotélescope décimétrique de Nançay, Observatoire de Paris, route de Souesmes, 18330 Nançay, France
7
The length of a modified Julian day is by definition always equal to 86400 seconds
8
1AU = 1.496 · 1011 m
9
1kpc ' 3 · 1019 m
6
4
Energy 0 - 1
Energy 0 - 2
Energy 1 - 2
Force 0 - 1
Force 0 - 2
Force 1 - 2
Earth - Moon - Sun
1/10
1
1/300
1
1/100
1/240
Sun - Jupiter - Saturn
1
1/6
1/3000
1
1/12
1/100,000 to 1/10,000
Guillaume Voisin
Pulsar - Inner WD - Outer WD
1
1/230
1/1000
1
1/450
1/3100
Table 1: Summary of the relative weights of gravitational energy and force terms in different
known triple systems. Numbers 0,1,2 refer to the bodies of a given system in the same order
as they are mentioned. WD stands for White Dwarf.
for the inner white dwarf in the field of the neutron star. Hence, both because of speed and
gravity this system is in a weak field regime where a first order post-Newtonian description
should be sufficient (Will, 2014). For comparison, for Earth around Sun, RS /r⊕ ' 10−9 and
v⊕ /c ' 10−4 .
3
3.1
A full three-body timing model
Introduction
As I tried to show in the previous section we need to compute an exact solution, at least to
desired accuracy, at all times. For a three-body system it is usually well accepted that no
analytical solution exist to date. This is not exact since Karl F. Sundman published such a
solution in 1909 (see Henkel (2001) for an historical approach and Sundman (1913)). Nevertheless this solution is based on slowly converging series and thus is of little interest for actual
computation, as compared to purely numerical approaches. However the numerical way is not
without difficulties as we shall see in section 3.2.3.
Once the motion of the pulsar and its two companion white dwarves is computed we need
to model the times of arrivals of the radio pulses. This involves taking into account intrinsic
spinning parameters of the pulsar as well as systematic delays due to geometry (Roemer delay),
post-Newtonian effects (Shapiro delay, Einstein delay...) or tidal deformations. We will consider
that the Roemer delay is first-order and the others second-order meaning that the former is at
a 100 s level while the latter are at most a few hundreds of microseconds. In the proper frame
of the pulsar one just computes the phase, or number of turns N , with a Taylor expansion to
second order (Blandford and Teukolsky, 1976) :
N (τ ) = N (τ0 ) + f (τ0 )(τ − τ0 ) +
1 df
(τ0 )(τ − τ0 )2
2 dτ
(1)
Where τ is the proper time of the pulsar.
The timing model must then be expressed in the Solar-system-barycenter frame. Now delays
come in :
N (ta ) = N (t0 ) + f (t0 )(ta − ∆t(1) − ∆t(2) − τ0 ) +
1 df
(t0 )(ta − ∆t(1) − ∆t(2) − τ0 )2
2 dt
(2)
Where ta is the time of arrival, ∆t(1) and ∆t(2) are respectively the first and second order
systematic delays and ta − ∆t(1) − ∆t(2) = τre is the retarded proper time of emission of
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Guillaume Voisin
the pulsar. Indeed we cannot know what was the actual time of emission unless we know
the distance between the pulsar and us. Fortunately, although this parameter is often poorly
measured, it is not necessary to analysis. We will thus assume that all our results are retarded by
an unknown constant delay. However the system could undergo a relative radial motion with
respect to the Solar system. Ongoing observations with the VLBA10 telescope will address
that matter (Ransom et al., 2014), but as long as this relative proper motion is constant (no
acceleration), it essentially amounts to a linear drift of the phase which is equivalent to a slight
shift in spin frequency : consequently our model will be just as efficient but with an effective spin
period different from the intrinsic one. τ0 is our retarded proper time of reference at which are
taken all the orbital and spin parameters. Since coordinates systems can be choosen arbitrarily,
I decided for convenience to take τ0 = t0 the first Nançay TOA : 56,492.297,192,640,59 MJD.
3.2
3.2.1
Computing the orbits
Retrieving the two-body results with a numerically integrated trajectory
The aim here was to reproduce the residuals of programs such as Tempo for known binary
pulsars but, instead of using the well known analytical expressions for the Newtonian two-body
problem, I replaced it by a fully numerical approach.
The numerical integrator used was the routine odeint of the scipy library11 . It implements
a predictor-corrector algorithm (Press et al., 1992) using double floats (64 bits).
The equations were cast into a dimensionless form using the typical scales of the problem.
Typically :
 dq1
 dτ = q˙1


 dq2 = q˙2
dτ
−q2
dq˙1
(3)
= −M2 kqq1−q
k
3

1
2k
 dτ

−q1
 k dq˙2
= −M1 kqq2−q
dτ
k3
1
2
Where, if r and t stand for the dimensional position and time, P is the orbital period, a
may be the semi-major axis of one of the bodies if they are similar (which is the case for
similar masses) or an intermediate value otherwise, M is the mass of the Sun, and G is the
gravitational constant, I define :
a3
M P 2 G
τ = t/P
q = r/a
q̇ = ṙP/a
k =
(4)
(5)
(6)
(7)
Thus all variables and interaction terms should be of the order of one. For a predictorcorrector scheme one needs to provide the Jacobian as well but we shall see that in the next
section, for the more general three-body case.
3.2.2
Computation of the three-body orbits
We shall have here exactly the same approach as before, but we will somewhat generalize
the notations. The adimensionned state vector Q gathers all the eighteen components of the
10
11
Very Large Base Array
Scientific Python : http://www.scipy.org/
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J1455+3330 Residuals
15
10
Residuals (µs)
5
0
5
10
15
55800
56000
56200
56400
56600
Barycentric time of arrival (Julian days)
56800
Figure 3: Residuals obtained for the pulsar J1455+3330. The weighted standard deviation is
2.81µs while with tempo we have 3.55µs. The error bars for each TOA, though not represented
here for the sake of clarity, are mostly between 2 and 3 µs and thus are compatible with a null
residual.
problem (9 components of position, and 9 in the tangential velocity space). Schematically :
Q = (q1 , q2 , q3 , q˙1 , q˙2 , q˙3 ).
The differential system becomes :
k
dQ
= F (Q)
dτ
(8)
Where Qi≤9 = F (Qi+9 ) and
F (Q{I 0 ,J 0 ,K 0 } ) = −MJ
QI − QJ
QI − QK
and circular permutations of {I, J, K}.
(9)
3 − MK
kQI − qJ k
kQI − QK k3
I is the position space of the first body and I 0 its associated tangential space, i.e. the
velocity space of the first body. Hence QI must be understood as ”the projection of Q on I”
or equivalently ”the 3-vector (Q1 , Q2 , Q3 ) = q1 ”. As for J and K they refer respectively to the
spaces of the second and third body.
∂Fi
From there one can also derive the Jacobian of the system Jij = ∂Q
, needed by some
j
integration schemes such as the Scipy predictor-corrector, or to perform perturbations of an
initial solution. This Jacobian is worked out in appendix B
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Guillaume Voisin
Figure 4: A three body system with the characteristics given in table 3, as well as numerical
errors on first integrals of motion. L is the component of the angular momentum directed
toward the observer, the impulsion P and the Barycenter numerical errors are represented in
the plane of sky.
3.2.3
Numerical intricacies
Once the mathematical problem is posed one needs to integrate numerically. Naturally, the
easiest way would be to to use again the predictor-corrector integrator provided with Scipy
12
. This proves to work well if one integrates over no more than a few weeks but beyond
that roundoff errors become more important than a few microseconds which our observational
accuracy as previously stated.
The reason for that lies in the numerical accuracy of the variables : Python defines floats
on 64 bits giving a mantissa as long as 16 digits in decimal base. One will easily check that to
keep an accuracy at the microsecond level for Roemer delays, it is necessary to compute the
location of the pulsar with about 100m precision. Given that we are dealing with distances as
large as 1011 m, it means a relative accuracy of 109 and thus numerical round-off errors have
only 6 decimal places to go before reaching our desired level of accuracy. This is easily reached
since the number of evaluations of the right hand side of the differential equation 9 reaches a
few millions.
As a consequence we cannot do otherwise than using another type of variable and a library
that can deal with it. It is the case of the C++ library BOOST 13 which provides numerous
differential integrators as well as multi-type support and a Python interfacing library. The type
used will be the native C++ long double which encodes floats on 80 bits on most processors
: it is usually the longest type supported in hardware, that keeps it fast14 , and it provides a
twenty-digits mantissa that proves sufficient for our purpose. The numerical integrator chosen
follows a Bulirsch-Stoer scheme, since it is famous for its high accuracy (Press et al., 1992).
12
See note page 6.
Boost : http://www.boost.org
14
More bits would have to be handled through a specific software which would be much slower.
13
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3.3
Guillaume Voisin
First order systematic delay : the Rømer delay
All of this is fine but one needs to give himself/herself a frame to perform his/her computations.
Since it is by far preferable to have a Galilean frame it is reasonable to use integrals of motion
for this purpose, since these will remain constant with respect to such a frame. More specifically
we’ll use the invariant plane given by nH = H/ kHk where H is the total angular momemtum of
the system. Moreover, the unit vector n going from the solar system barycenter to the pulsar
system barycenter can be considered as constant in time given the distances at stake (though
this might need to be corrected in case of high proper motion). Then it proves useful to take
the intersection between the invariant plane and the plane of the sky (the line of ascending
×nH
node) : na = knn ×n
Hk
From those vectors, we can draw two others to form two direct basis, n3 = na × nH and
n03 = na ×n , such that eventually we have two frames RH = (na , nH , n3 ) and R = (na , n , n03 ).
The first one will be useful as a local frame to carry out analytical calculations helping to see
what is going on, while the second one is going to be used to compute numerically the Rømer
delay as seen from the barycenter of the Solar system. Let’s remark that these two are related
by a rotation of angle i = (n , nH ) ∈ [0, π[.
Now let’s compute a general formula for the so-called barycentric time of arrival at infinite
frequency (Blandford and Teukolsky (1976)). This is the time of arrival of the pulsar light in the
Solar-system barycenter, assumed to be a good approximation to a Galilean frame, corrected
from any local and interstellar delays such as the dispersion of the Earth atmosphere or the
dispersion of the intergalactic medium.
ta = tem + ttravel
(10)
And, neglecting for now relativistic effects :
distance(P, b )
(11)
c
P is the pulsar location, b the solar-system barycenter and bP shall be that of the pulsar
kbP −P k
system. To first order in kb
≡ bbPbPP we end up with :
−bP k
ttravel =
ttravel = (b bP + n · P )/c
(12)
Further we shall get rid of the term in b bP above, since it is hard to measure and does not
provide any essential information as far as we neglect the effect of proper motion ( even so we
would only need its derivative). Thus we redefine ta as :
ta = tem + n · P/c
(13)
The only difference being that any function of tem , such as the frequency of the pulsar and
its derivative, will be a retarded function of the ”True” tem : the state of the pulsar is known
at an undetermined epoch in the past .
We now need to parametrize the position of the pulsar P with respect to its barycenter. In
RH spherical coordinates seem appropriate, especially since in the case of the triple system, the
orbits are almost coplanar (see Ransom et al. (2014)). We define :
θ = (nH , P ); φ = (na , P − nH · P ); rP = kP k
(14)
Thus, the Rømer delay is :
∆R (tem ) = n · P/c = rP /c (cos(i) cos(θ) + sin(i) sin(θ) sin(φ))
In R , it simply stands as ∆R (tem ) = y/c where y is the position component along n .
9
(15)
3.3.1
Guillaume Voisin
First order correction
Since the only thing we measure is ta we would need to solve ta − ∆R (ta − ∆R (tem )) = tem
to know the Rømer delay at tem . Fortunately the Rømer delay is about a few seconds while
the characteristic time scale T of the orbits is at least a day. Thus we will use a first order
correction in = ∆R (ta )/T (the time derivative hereafter yields a 1/T in order of magnitude).
This is the approach used by Damour and Deruelle (1986) or Blandford and Teukolsky (1976)
:
d∆R
(16)
∆R (tem ) = ∆R (ta ) − ∆R (ta )
dt ta
In R this gives : ∆R (tem ) = y/c(1 −
3.3.2
y0
).
c
Second order correction
For most applications i.e. for most binary pulsars, a first order correction is sufficient. However,
it is certainly a careful practice to check that against at least the following order. Moreover, we
are using microsecond accurate data here, and it is likely that the second order will eventually
matter for high accuracy applications.
Indeed, if we take T to be the inner orbit period and ∆R to be at most the time needed to
travel across the outer orbit, be 100 seconds, ' 7 · 10−4 and 2 ' 5 · 10−7 . Then the first order
yields at most a 0.1s correction while the second order yields a few microseconds and the next
order would not contribute to more than tens on nanoseconds, well below our observational
accuracy. Let’s remark that in the traditional case of a binary pulsar, it is the same orbit that
generates both the biggest delay and the time scale of orbital motion, which since light travels
much faster than the pulsar end up in an already very accurate formula at first order. For
instance, in the case of the binary millisecond pulsar J1455+3330 we already mentioned, this
correction amounts to at most 160µs while the second order falls below one nanosecond.
The expansion of Rømer delay to second order in ∆R (ta )/T reads :
1
2 00
0 2
0
(17)
∆R (te ) = ∆R (ta ) − ∆R (ta )∆R |ta + ∆R (ta )∆R |ta + ∆R (ta ) ∆R |ta + ◦(2 )
2
In R , ∆00R |ta = y 00 /c. Computationally y 00 is picked on-the-fly from the right-hand-side of
the differential system during the integration of the equations.
3.4
3.4.1
Second order systematic delays
The Einstein delay
As I mentioned in the introduction of this section (3.1), from the pulsar viewpoint, namely in
its proper time, the only variation of its phase occurs quadratically. However, we need to turn
it into our own time to model observations, and that is the so-called Einstein effect.
Actually we shall simplify the problem and rather express equation 1 in the Solar-system
barycenter (SSB) since other programs like Tempo can convert from an Earth-based-observatory
frame with a lot of refinements.
Let’s precise what the proper frame of the pulsar is : it is a frame following the speed of the
barycenter of the pulsar and undergoing every external gravitational fields. By symmetry the
pulsar gravitational field does not play any role, and it is good because we don’t want it : it
would only contribute to a constant deformation of space-time that would not be detectable.
To perform computation, we are going to put ourselves in the first post-Newtonian approximation (Will, 2014). Namely, if one considers a scale given by ∼ v 2 /c2 ∼ U/c2 where U is
10
80
a)
Delay (µs)
60
20
0
20
40
60
b)
First order
72.77
+5.649e4
c)
Delay (µs)
Delay (s)
40
5000
4000
3000
2000
1000
0
1000
2000
3000
40002.30
0.1
0.0
0.1
0.2
0.3
0.4
0.5 2.30
Guillaume Voisin
Roemer delay
80 56492.3
56599.4
56707.7
Time since first toa (MJD)
Second order
37.75
+5.649e4
Figure 5: Second Order Correction of Rømer delay for the fitted parameters given in table 3.
a) is the complete delay to second order, small variations can be distinguished ; b) Zoom on
the first order correction, an envelope with a longer time scale can be seen until the first half
pseudo-period ; c) Zoom on the second order correction to Rømer delay, here again an envelope
can be distinguished plus some specific patterns
the external gravitational potential and v the speed of the pulsar with respect to the SSB, the
line element is given by :
dτ 2 = (1 − 2U )dt2 − (1 + 2U )dr2 /c2 + ◦()
Where t is the proper time of the SSB and U =
From what it is straightforward that :
Gmi
|ri −rp |
+
(18)
Gmo
|ro −rp |
dτ
v2
= 1 − U/c2 − 2 + ◦()
dt
2c
The Einstein delay to the first post-Newtonian order comes out as :
Z
v2
∆E = t − τ = U/c2 + 2 dt
2c
(19)
(20)
That being said, it is clear from this formula that this delay will increase to infinity. That is
normal, times are not passing at the same speed. If the integrand where to remain constant it
would just be a constant deformation of time, as before, and the only effect would be to change
the apparent spin frequency : we get the ”real” spin period, as it is from the pulsar viewpoint.
A much more interesting effect is due to the variations of the integrand with time. It can
be seen just by removing the linear component in equation 20 :
Z
v2
v2
δ∆E = U/c2 + 2 − hU/c2 + 2 idt
(21)
2c
2c
What remain are the variations due to orbital motion. It thus has two pseudo-periods
corresponding to PI and PO , as can be seen on figure 6.a. Hereafter, we shall always implicitly
remove this linear component.
11
Guillaume Voisin
b)
a)
c)c)
Figure 6: a) Einstein delay for the parameters drawn from 8-month data (green dots) of J0337
at Nança, b) Component due to the outer white dwarf during the same time as in a), c) Zoom
on the component due to the coupling between the outer and inner speeds. The pseudo-period
is that of the inner orbit, about 1.6 days.
We can get more insight by decomposing equation 20 in terms of inner, outer and coupled
components. Namely :
U = Ui + Uo
2
2
v 2 = vp/I
+ vI2 + vI · vp/I
(22)
(23)
Where Ui and Uo are respectively the gravitational potential of the inner and the outer companion, vI is the speed of the inner system barycenter and vp/I is the velocity of the pulsar with
respect to that barycenter.
Hence we expect that those components will show the same periodicity as the sub-systems
they belong to. As we are only interested in variations it is clear that eccentricity will play a
major role, since a circular orbit would offer neither variation of speed nor of potential. One
can estimate these variations to the first non-null order in eccentricity e, assuming the motion
follows momentarily its osculating orbits :
1
Ûi eI cos(2ωI t) + ◦(e2I )
2
1
=
Ûo eO cos(2ωo t) + ◦(e2O )
2
= v̂I2 e2O cos(2ωI t) + ◦(e2o )
2
= v̂p/I
e2I cos(2ωo t) + ◦(e2I )
δUi =
(24)
δUo
(25)
δvI2
2
δvp/I
(26)
(27)
Where the hat means ”value at periastron” and the ω = 2π/P are the instantaneous pulsations
of the orbits. Let’s put numbers (see table 3 or Ransom et al. (2014)). The inner white
dwarf (WD) being the lightest with a small eccentricity ( 10−4 ), it scarcely contributes a few
microseconds. The comparatively more eccentric and larger orbit of the outer WD makes it
12
Guillaume Voisin
contributes hundreds of microseconds (see figure 6.b). Looking at velocities gives similar results
except that the outer motion contributes only at a microsecond level.
The coupling term, however, shows a particular behaviour which is by definition specific to a
three-body system. Indeed, the full amplitude of the term plays a role as the pulsar goes forward
and backward along the outer orbit. Moreover we expect properties of both sub-systems : a
2
fast pseudo-period Pi (and not PI /2 as the vp/I
term), an envelope of pseudo-period PO /2 due
to the outer eccentricity and a drift of pseudo-period PO because of the misalignment of the
two speeds at the end of each inner period, as we can see in figure 6.c.
Not only the Einstein correction makes our model more accurate, but it also separates
otherwise highly correlated parameters. More specifically, we saw in section 3.3 that the Rømer
delay is mostly proportional to a sin i which makes the inclination angle hardly possible to work
out independently from the semi-major axis. With the Einstein delay, we can pull apart these
two parameters since it does not depend on i while it heavily depends on a. Moreover, we
may expect that this will help to determine masses more accurately : masses are somewhat
correlated with period and semi-major axes by the mass function (see section 4.2 and appendix
A.3) and here these parameters come in differently, with a pseudo period often divided by two.
The Einstein correction is thus a major effect, with characteristics specific to a three-body
system, that needs to be tackled to achieve a high accuracy model for this system.
3.4.2
Tidal effects
Tidal effects can in principle severely affect both the orbital parameters of the system and
the intrinsic parameters of the neutron star. Nevertheless, we shall give orders of magnitude
to demonstrate that such effects are negligible here. To do so we will focus on the strongest
interaction, between the inner companion and the neutron star, and consider negligible the
effect of the outer companion.
We shall first consider how it might change the spin parameters. Indeed, if the star flattens
perpendicularly to its spinning axis ~z, its radius orthogonal to ~z will become larger and the star
will spin down as a consequence of conservation of angular momentum. Here we will assume
that the orbital axis of the companion is aligned with the spin axis of the pulsar, since this is the
case for which the effect is maximized and we do not actually know there relative orientations.
Conservation of angular momentum of the neutron star in this configuration just reads :
LNS = ωNS INS
(28)
R
Where ωNS is the spin pulsation and INS = NS (z 2 − r2 /3dV ) is the associated moment of
inertia. In case of a perturbation of order we would have :
LNS = ωNS INS + δωNS INS + ωNS δINS + ◦()
(29)
Where first order terms must cancel, yielding :
δωNS = −ωNS
δINS
INS
(30)
Now we need to estimate
the δINS involved by the companion. If the star was Newtonian, we
R
would have INS = NS ρ(~r)(x2 + y 2 )dV , with ρ the local density, and the deformation induced
by the external gravitational field of the white dwarf would be computed using Hooke’s law,
as it can be found in standard textbooks such as Beutler. Here we need to be more careful for
two reasons : the neutron star matter is in the strong field regime and its equation of state
is highly uncertain. In the case of a weak gravitational perturbation (the distance from the
13
Guillaume Voisin
source is much greater than its Shwartzschild radius) given by the quadrupolar tidal field Eij ,
we may however compute the quadrupole moment of the star Qij through the so-called tidal
polarizability λ (Hinderer (2008), Thorne (1998), Fattoyev et al. (2013)) :
δQij = −λEij
The quadrupole moment of the star δQij tends to δQij
(31)
−→
Newtonian
δρ(~r)(xi yj − ~r2 /3)dV in
the Newtonian limit, while the external quadrupolar tidal field Eij tends to the Hessian of
2V
the external gravitational potential V : Eij → ∂x∂i ∂x
j . In our case we shall use these limits.
Assuming that the external tidal field is mainly along the axis x between the two bodies, and
the neutron star remains axisymmetric around z we can relate the variation of the moment of
inertia to the quadrupole moment as:
δINS = δIzz = 3Qxx + 3δIxx
(32)
The tidal polarizability needs much more cumbersome computations : one needs to integrate
the Tollman-Oppenheimer-Volkoff system along with a given equation of state and that cannot
usually be achieved analytically. Therefore, we will restrain ourselves to the results obtained
by Hinderer (2008) and Fattoyev et al. (2013) for a range of polytropic equations of state,
considered as realistic, and take the largest. For a star of radius R = 10km this is about
λ ' 3 · 1029 m2 s2 · kg−3 .
Let’s now put everything together. It is clear that for a circular orbit the external field would
remain constant and the effect would result in a constant shift irrelevant to observation. For
a small eccentricity e however, the distance varies by a small amount ed0 from the distance of
closest approach d0 , such that the quadrupolar field Exx changes by an amount δExx ' 3eExx
during one orbital period. During this period δIxx remains constant and so cancels in equation
32. Thus the shift in spin frequency is equal to :
λExx
ωNS
(33)
δωNS ' 9e
INS
Finally we put some conservative numbers in this equation. ωNS = 325Hz, e = 10−3 ,
INS = 2π
MNS R2 ' 1.2 · 1038 kgm2 (this is for a homogeneous density, but in a neutron star most
15
G
' 9.9 · 10−7 N/m where
of the mass lies in the core, so this is conservative) and Exx = MWD
d30
we took MWD = 0.2M , MNS = 1.5M and d0 = 1 light-second. It yields :
δωNS ' 7 · 10−15 Hz
(34)
From what we easily conclude that this effect cannot be observed.
Another effect would be the torque of the white dwarf on the neutron star tidal deformation.
This one could be more important since it would be cumulative over time due to dissipation of
the spinning energy (while the previous one was only periodic over an orbital period). However
Thorne (1998) reminds us that the neutron matter has a very low viscosity, actually neutron
stars are believed to be partly made of a superfluid, and therefore the deformation is well
aligned with the field at any time, thus yielding no significant torque.
4
4.1
Fitting to data : a very delicate optimization problem
Introduction
Our purpose is to fit to data from Nançay the model described by equation 2. The data are
TOAs ti with Gaussian uncertainties characterized by their standard deviations σi . If we take a
14
Guillaume Voisin
short enough period we can also infer how many turns the pulsar did in between two TOAs so as
to work out a set Ni of turn numbers. Thus we may just fit N (ti , {θk }) to Ni , where {θk } are the
parameters. We need to work out the likelihood of the Ni s from that of the ti s. If one neglects
the frequency derivative term as well as delays in equation 2 this is fairly straightforward : one
will end up with a Gaussian of standard deviation σi0 = f σi and an amplitude corrected by 1/f .
The density of likelihood for the whole set of TOAs thus reads :
Y
i
p(Ni )dNi ∝
Y1
(N (ti , {θk }) − Ni ))2
exp
dNi
f
2σi0
i
(35)
The point is that this approximation makes things very simple while we miss very little :
as I point out in appendix C, this should not make any significant difference unless we had
decades of data.
We now have to maximize the likelihood function so as to find the most probable set of
parameters that describe the system. In practice we will rather minimize the absolute value
of the logarithm, since it is much easier to handle numerically. Given the complexity of the
problem we must find an appropriate set of parameters to make optimization easier as well as
an appropriate starting solution. Then we need to do the actual minimization with a library
called Minuit (James and Winkler, 2004), developed at CERN for 40 years to solve multivariate
minimization problems. Finally, we shall find the uncertainties of the parameters we found using
a Markov-Chain-Monte-Carlo program called emcee (Goodman and Weare, 2010) that returns
a sample of the probability distribution of each parameter.
4.2
4.2.1
Starting solution and the choice of parameters
Initial solution
Since we still are in a weak regime of gravity, it is possible to use a zeroth-order approximation
of the orbits, in other words to assume there are no interactions in between the companions.
Such a model is already implemented in the Tempo software under the name of BTX, as
previously stated. It was initially designed to take into account the effect of planets turning
around pulsars. Though, planets are much lighter than white dwarfs and the BTX model will
neither give a reliable solution on a long time period nor an accurate one, but it shall be a good
initial guess. Moreover, it proves accurate enough to keep track of the number of turns about
itself the pulsar did, and thus provides the N (ti ) needed for the fit.
The BTX model does only describe a superposition of two-body motions, involving the
neutron star and each of its companions. Such a motion is usually described by so-called
orbital elements. In the case of a bounded, elliptic orbit these sum up as : the semi-major axis
a, the eccentricity of the orbit e, the orbital period P , the time of passage to the periastron tp ,
the longitude of the periastron with respect to the line of ascending nodes ω and the angular
inclination with respect to the plane of the sky i. The line of ascending nodes is characterized
by the intersection of the plane of the ellipse with the plane of the sky. The angle ω is taken
with respect to the point of the ellipse where the pulsar is moving away from Earth. These
definitions can be found in any celestial mechanics textbook such as Beutler. In the case
of the BTX, Tempo will output two sets of orbital elements, one for the inner system and
another for the outer system, as well as the spin characteristics of the pulsar f and f 0 . Masses
can be determined by taking the root of the so-called mass function (Lorimer and Kramer)
f (m, M ) = 0 when one of the masses m or M is known.
f (m, M ) = m3 −
4π 2 a3
(m + M )2 = 0
GP 2
15
(36)
Guillaume Voisin
In the BTX, the mass mp of the pulsar is arbitrarily set to 1.35M , from which one gets
the mass of the inner companion using f (mi , mp ) and the mass of the outer companion with
f (mo , mi + mp ). More details on the mass function are given in appendix A.
4.2.2
Choice of fitting parameters
Since we are using an initial solution defined in terms of orbital elements, it is quite natural to
remain with them. Indeed there is a bi-univoque relation between a set of six orbital elements
and the position and speed of a body at a given time (Beutler). In the general case they describe
the so-called osculating orbit to the actual motion, namely its tangential Keplerian orbit. But
the best reason to keep this particular set of parameters is that it avoids degeneracies. Indeed, a
naive approach to the problem suggests to fit on a total of six parameters, position and velocity,
for the motion of each body, two intrinsic parameters of the pulsar that are its spin period and
derivative, and three masses. Eventually we count 23 unknown parameters ! Fortunately, the
laws of mechanics can help us to lower it down. If we assume the three bodies constitute
an isolated gravitational system, then it has 10 constants of motion among which six we can
arbitrarily choose : the center of mass which we set to zero and the impulsion which can also
be set to zero as a beginning (proper motion is seldom detectable). Eventually we remain with
17 unknowns.
From our previous discussion of the BTX model, we have 12 orbital elements, the pulsar
spin period and its derivative. We still need two more parameters reflecting the interactions
between the companions that were neglected so far. These can be the masses of the white
dwarfs since the mass function is not valid anymore. Indeed, the mass function arises from
Kepler’s third law which is only valid for the two-body motion. However this could be quite a
hazardous choice of parameters since they come in very symmetric ways in the equations and
would likely yield high covariances. Hence I suggest to rewrite equation 38 as follow :
4π 2 a3
GP 2
m3
=µ
(m + M )2
(37)
In the two-body case, µ = 1 and we get back to equation 38. In the three-body case, however, we
obtain two new parameters µpi and µIo 15 . We retrieve masses as before by solving a generalized
mass function :
4π 2 a3
(m + M )2 = 0
(38)
fµ (m, M ) = m3 − µ
GP 2
This parametrization allows to have a direct idea of how the system departs from a zerothorder, double two-body system such as the BTX while reasonably avoiding high covariances.
Table 2 summarizes the set of parameters presented in this section.
4.3
Minimization with the Minuit library
Minuit is the name of a library that contains several minimizers. The most famous is called
Migrad, and is mainly the one I used.
Migrad is a variable metric minimizer (DAVIDON, 1991). The so-called metric is actually
nothing more than the Hessian H of the the function f of variables x to minimize. It roughly
follows the scheme below :
• Compute the local Hessian.
15
I will use index i to point out the inner companion but index I for the inner system, pulsar and inner
companion.
16
Names of intrinsics
Pulsar spin frequency
Pulsar spin frequency derivative
Pulsar mass
Mass coefficient
Mass coefficient
Intrinsic
f
f0
mp
µip
µIo
Inner
eI
ap
Ωp
TI
PI
iI
Outer
eO
ao
Ωo
TO
PO
iO
Guillaume Voisin
Names of orbitals
Eccentricity
Semi-major axis
Longitude of periastron
Periastron epoch
Orbital period
Inclination of the orbit
Table 2: Summary of the parameters of the system. Capital indices are used when the
parameter applies to the whole (inner or outer) system, while lower-case indexes apply when it
is specific to one of the bodies (in this case the parameter of the other body is a combination
of the given orbital elements).
• Perform a Newton-Raphson step from the current position xc to evalutate the position
xmin of the local minimum : xmin = xc + H −1 gc where gc is the current gradient. This
step is exact for a purely quadratic function.
• Evaluate the difference between f (xmin ) and its quadratic approximation. If inferior to
an arbitrary limit, it stops, otherwise the metric is modified and a new step is attempted.
There are different ways to update the metric, Minuit uses the scheme proposed by
Fletcher (1970), that uses the local gradients and avoids an expensive computation of the
full Hessian.
As we can see, the accurate computation of derivatives is a critical point in Migrad. Since it
proceeds with finite differences, numerical accuracy may be, and was, an issue. Another point
is due to the high number of parameters. It implies a high number of local minima, or more
likely there might be a local minimum for a given set of parameters that disappears because
of covariances as soon as one allows to fit on a larger set of parameters. Alternatively, fitting
all the parameters at the same time can fail simply because the parameter space is too big and
the minimizer gets stuck with an unphysical solution. For example it may start from position
where two variables are very covariant, and thus hardly sees a minimum, even though in the
real solution these parameters are not that much correlated. In this case, a solution is to start
by fixing one of the parameters and to proceed in two successive fits.
To be honest, there is a fair portion of try and miss in this kind of search. However, it is
very helpful to develop tools to diagnose why a fit may have failed. I mostly developed two
such tools. One is a tracker, that picks and records every step that Migrad does during a fit. It
proved particularly useful to understand numerical accuracy issues. For example if numerical
steps in the likelihood function occur for a variation of eccentricity of order 10−14 and that
Migrad tries to compute a gradient by doing two such steps, it might yield a catastrophe, even
though we are well below our desired physical accuracy.
The other tool that proved helpful was the use of fake pulsars. Here is the principle : take
data as well as some plausible set of parameters and, instead of fitting parameters to data, fit
TOAs to parameters. Then just change the parameters by some random very little amount that
gives a posterior probability similar to the one we have with the actual data. Once you have
it, just apply the same fitting strategy to both fake and real pulsar. The former will tell you
with certainty whether you got closer to the solution or not (you can have a better probability
but got further away from the real solution), and thus allow you to reject or keep the solution
of the latter.
On figure 7 I show the current status of the fit. The weighted standard deviation of these
17
100
Guillaume Voisin
Time residuals
80
Residuals (µs)
60
40
20
0
20
40
60
56450
56500
56550
56600
56650
Times of arrival (MJD)
56700
56750
Figure 7: Best timing residuals obtained so far including Rømer and Einstein delays. It
includes 3063 TOAs from Nançay spaning over 200 days.
residuals is 23µs. The parameters corresponding to this fit are given in table 3. One of the
current limitations for this fit is that the time span does not cover the full outer orbit, thus
making it difficult to resolve precisely. This might be provided by sharing data with the Ransom
et al. (2014) team, in the framework of the collaboration with Anne Archibald.
Now that we have a candidate solution, we want to get the uncertainties on it. This is the
purpose of the next section.
4.4
The Markov-Chain-Monte-Carlo (MCMC) approach
The MCMC algorithms are not really minimizers but rather statistical samplers. Indeed they
output a sample of the probability distribution of each parameter. On the one hand they can
be used to fit a model to data, since in this case we are looking for the most probable model,
but they often reveal themselves to be pretty slow at it and thus should only be used when
everything else fails. On the other hand, the statistical samples can be used to get realistic
error bars on each parameter, which is not possible with minimizers like Minuit that implicitly
assume that the model is quadratic (this is usually true locally but it can severely bias the
global probability distribution). As a consequence we will use MCMC to get the error bars of
our model.
To be a bit more specific, let’s introduce the posterior probability density of having a given
set of parameters Θ = {θi } for a given set of data D, p(Θ|D), the likelihood of D given Θ,
p(D|Θ), and the prior probability p(Θ). We then have the simple bayesian relation :
p(D|Θ) × p(Θ) = p(Θ|D) × p(D)
(39)
The prior is usually flat, there is no reason to privilege a set of parameters over another,
and will rather be used to restrict the search over an arbitrary parameter-space volume. The
likelihood is known from section 4.1. Hence we know the posterior probability density up
to a constant factor p(D) which all the Metropolis algorithm needs (see Diaconis (2009) or
Foreman-Mackey et al. (2013)).
18
Guillaume Voisin
6.9
6.93
6.94
6.95
6.96
7
3.5
2
3.5 5
3
3.5 0
3
3.5 5
40
3.5
−4
3e.5O (x
3.5103.5)
25 30 35 40
Indeed the algorithm starts with an arbitrary probability p(Θ0 |Θ) to go from a set to another.
Then it successively tries to jump to new parameter sets at each time adjusting p(Θ0 |Θ) until
it becomes stationary. When it is so, the fundamental theorem of Markov chains (see Diaconis
(2009) for a rigorous mathematical treatment) tells us that the chain has converged and that
from now on any sample of Θs drawn from the successive trials of the chain will follow a
distribution equal to p(Θ|D) (To be mathematically exact the sample must of course be infinite).
This algorithm is very robust : the only assumption is the simple connectedness of the
parameter space, the likelihood and a fortiori the posterior probability do not even need to be
analytical. Moreover, it will theoretically explore completely the allowed volume and find every
local minimum. In practice, this would require a real random generator and an infinite time
ahead, which is why it is preferable to already have a good solution to start with. Moreover,
although random evaluations are slow, it is easily parallelized by running several chains at
the same time (a hundred in our case). One can even use parallel tempering, which consists
in running chains with broader likelihood (actually it enlarges the standard deviations of the
Gaussian, just like a temperature coefficient would do, hence the name of the algorithm), that
will spend less time in local minima and go through parameter space faster, before eventually
sharing this information with other, more accurate, chains (Earl and Deem, 2005).
eI (x10−2 )
eO (x10−4 )
Figure 8: Statistical distribution of the inner and outer eccentricities as well as their correlation
plot. We can see that this last plot is roughly isotropic, and so that the statistical correlation is
low, which is the case for all outer parameters with respect to inner parameters, as one might
expect. The blue lines show where the fitted value with Minuit is.
In practice I used a program called emcee implemented by Foreman-Mackey et al. (2013)
featuring the affine invariant scheme16 of Goodman and Weare (2010).
A run of this program thus gives all the statistical information one might need about each
parameter, including correlations with other parameters, as illustrated in figure 8. The final
results for the first 3063 TOAs from Nançay are given in table 3 with there error bars estimated
at a 90% confidence level 17 .S
16
Affine invariant scheme : such a scheme sees linear combinations of parameters as a same class of equivalence.
For instance, it yields the same results whatever a length was scaled in micrometers or in light years. It is thus
very robust in that matter.
17
For example, x = 1.230+6.7
−4.5 means that, if the real value xr is above x it has 90% of being between 1.23
and 1.297, and respectively if it is below.
19
f (s−1 )
f 0 (s−2 )
Mp (M )
µip
µIo
365.95332+1.4
−2.2
−14
−4.443+0.73
−1.1 · 10
+6.1
1.4325−4.7
1.0000+1.4
−1.9
1.00000+2.6
−2.9
eI
ap (ls)
Ωp (rad)
TI (MJD)
PI (days)
iI (rad)
−4
6.9569952+2.7
−3.2 · 10
1.2178+2.5
−3.4
1.6447+6.7
−8.4
55917.5+2.0
−7.2
1.6294+4.9
−5.7
1.5483+9.6
−7.2
Guillaume Voisin
−2
eO
3.53150+1.3
−1.2 · 10
aI (ls)
74.677+5.1
−5.0
ΩI (rad)
1.6708+4.1
−3.5
+3.3
TO (MJD) 56317.21−2.6
+2.0
PO (days) 327.262.0
iO (rad)
1.5709+5.5
−7.9
Table 3: This table shows the best fitted parameters for the 3063 first TOAs from the Nançay
decimetric telescope, for the system J0337+1755, with their error bars. The errors are given
for the last digit at a 90% confidence level. (ls stands for light-second)
5
Conclusion
The principal aim of this work was to improve the model used to analyse the raw data from the
Nançy radio-telescope. A look at the timing residuals of the initial model (figure 1) compared
to the residuals given by the model developed during this internship (figure 7) suggests that
this goal was somewhat achieved. Indeed the residuals were decreased by almost two orders of
magnitude and are now stable over long periods. To achieve this, a model was realized from
scratch with comparison to the existing binary models when possible. Post-Newtonian effects
such as the Einstein correction were taken into account and proved to matter at a high level.
From a more technical point of view, numerical round-off errors were systematically taken
into account and tested against. It involves a set of tools : the main frame is written in Python
with an underlying layer of Cython 18 for all the computations where accuracy and speed are
critical, iMinuit and emcee libraries are in charge of the minimization operations, and the
whole interoperates with a C++ code featuring the library Boost for numerical integration of
the differential equations of motion. From a user point of view, it presents itself as a fully
documented Python class that can be forked from a Subversion19 repository hosted at LPC2E.
Though this model and code is not yet complete. A better accuracy can undoubtedly be
achieved and must be if one wants to consider tests of the strong equivalence principle. It will
need to include more post-Newtonian effects, in particular in the equations of motion which
are currently merely Newtonian. Indeed, orders of magnitude clearly suggest that periastron
advance is to be considered in this system. For the sake of completeness, one should also work
out the Shapiro delay20 even though this one is not likely to matter at a very significant level
in this system, as deduced both from orders of magnitude and according to the more advanced
work of Anne Archibald. Moreover, a greater number of times of arrival covering a larger period
of time will be needed to perfectly resolve all the parameters. This point might eventually be
solved in the frame of the collaboration with Anne Archibald, by sharing data, and might also
be critical to achieve the most accurate possible test of the strong equivalence principle.
In the next weeks, Ismaël Cognard and Lucas Guillemot will try to use the model developed
during this internship to improve the accuracy of the times of arrival recorded at Nançay,
with the hope of decreasing the uncertainty by an order of magnitude, down to a few hundred
nanoseconds.
18
Cython : C-extension to Python that allows to use C-static-type definitions and optimisations.
http://www.cython.org/
19
Subversion is a centralized version control system. https://subversion.apache.org/
20
Shapiro delay : delay added by the gravitational field of a companion when the light passes close to it on
its way to a telescope.
20
Guillaume Voisin
References
Scipy. URL http://www.scipy.org/.
Gerhard Beutler. Methods of Celestial Mechanics: Volume I: Physical, Mathematical, and
Numerical Principles. Springer, 1st softcover edition without CD-ROM of original hardcover
edition. edition edition. ISBN 9783540407492.
Roger Blandford and Saul A. Teukolsky. Arrival-time analysis for a pulsar in a binary system.
The Astrophysical Journal, 1976.
Thibault Damour and Nathalie Deruelle. General relativistic celestial mechanics of binary
systems. II. the post-newtonian timing formula. 44:263–292, 1986.
William DAVIDON. VARIABLE METRIC METHOD FOR MINIMIZATION. 1991.
Persi Diaconis. The markov chain monte carlo revolution. Bulletin of the American Mathematical Society, 2009.
David J. Earl and Michael W. Deem. Parallel tempering: Theory, applications, and new
perspectives. 7(23):3910, 2005. ISSN 1463-9076, 1463-9084. doi: 10.1039/b509983h. URL
http://arxiv.org/abs/physics/0508111.
R. T. Edwards, G. B. Hobbs, and R. N. Manchester. TEMPO2, a new pulsar timing package
- II. The timing model and precision estimates. Monthly Notices of the Royal Astronomical
Society, 372:1549–1574, November 2006. doi: 10.1111/j.1365-2966.2006.10870.x.
F. J. Fattoyev, J. Carvajal, W. G. Newton, and Bao-An Li. Constraining the high-density
behavior of nuclear symmetry energy with the tidal polarizability of neutron stars. Physical
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015806. URL http://arxiv.org/abs/1210.3402. arXiv: 1210.3402.
R. Fletcher. A new approach to variable metric algorithms. The Computer Journal, 13(3):
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http://comjnl.oxfordjournals.org/content/13/3/317.
D. Foreman-Mackey, D. W. Hogg, D. Lang, and J. Goodman. emcee: The MCMC Hammer.
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1086/670067.
T. Gold. Rotating neutron stars as origin of pulsating radio sources. Nature, 218(5143):731–&,
1968. ISSN 0028-0836. doi: 10.1038/218731a0. WOS:A1968B164900007.
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January 1969. doi: 10.1038/221025a0. URL http://www.nature.com/nature/journal/
v221/n5175/abs/221025a0.html.
Jonathan M. Goodman and Jonathan Weare. Ensemble samplers with affine invariance. Communications in Applied Mathematics and Computational Science, 2010.
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5(2):161–184, 2001. ISSN 1281-2463. URL https://eudml.org/doc/103658.
21
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A. Hewish, S. J. Bell, J. D. H. Pilkington, P. F. Scott, and R. A. Collins. Observation of a
rapidly pulsating radio source. Nature, 217(5130):709–713, February 1968. ISSN 0028-0836.
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//arxiv.org/abs/0711.2420. arXiv: 0711.2420.
G. B. Hobbs, R. T. Edwards, and R. N. Manchester. TEMPO2, a new pulsar-timing package I. An overview. Monthly Notices of the Royal Astronomical Society, 369:655–672, June 2006.
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Fred James and Matthias Winkler. Minuit user’s guide. CERN, Geneva, 2004. URL http:
//www.ftp.uni-erlangen.de/pub/mirrors/gentoo/distfiles/mnusersguide.pdf.
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livingreviews.org/Articles/lrr-2014-4/.
22
A
A.1
Guillaume Voisin
The gravitational two-body system defined by orbital
elements
Relation between orbital elements and state vectors for a single
body
There is a bi-univoque relation between between state vectors (position and velocity) and orbital
elements. In the frame we may call R0 made of the semi-major axis, the semi-minor axis and a
third vector orthogonal to the plane of the ellipse (say the angular momentum ~h this relation
stands as follow :
~r = a(1 − e cos E) (cos(v), sin(v), 0)0
√
2π
a
d~r
2
=
− sin E, 1 − e cos E, 0
dt
(1 − e cos E) P
0
(40)
(41)
Where E is the eccentric anomaly and v the true anomaly which can be defined using the
following relations :
t − tp
(42)
E − e sin E = 2π
P
Which is known as Kepler’s equation. It cannot be solved analytically so we use a fixed-point
method to solve it to desired accuracy (see for example Beutler).
Then one needs two more relations which merely account for the geometrical relation between
E and v :
√
1 − e2 sin E
(43)
cos v =
1 − e cos E
cos E − e
sin v =
(44)
1 − e cos E
Let’s remark that we did not use two parameters : i and ω. These allow to orientate the
frame in space. In this case, one will need to perform two rotations on 40 : one around ~h of
angle ω such that the first component be along the line of ascending node n~a , and a second
around n~a of angle i such that the third component be along the line of sight n~ .
A.2
The case of two bodies
In the case of two bodies one only needs to add one parameter, which is the mass of one of the
bodies, the second one being computed using Kepler’s third law (see below).
The reason why one does not need to add as many parameters as there are new initial
conditions to the equations of motion is that six integrals of motion can be set arbitrarily, thus
providing six relations between the initial conditions. Those integrals are the three components
of the center of mass and of the impulsion of the system. Remark that this assumes an isolated
system.
A.3
The mass function
The mass function f (m, M ) is simply the expression of Kepler’s third law, which stands that
the mass m of a body involved into a Keplerian two-body orbit is related to the period P, the
semi-major axis a, the tilt angle i and the mass M of the other object by finding the root of f
:
4π 2 (a sin i)3
f (m, M ) = (m sin i)3 −
(m + M )2 = 0
(45)
GP 2
23
Guillaume Voisin
Where G is the gravitational constant.
This is a third order polynom in m which can be efficiently solved by standard root finding
algorithms, such as those implemented in Scipy (sci).
We clearly see here that the single-body motion is equivalent to the two-body case with one
infinite mass.
B
Jacobian of the Newtonian three-body differential system
Following the notations of section 3.2.2, it is possible to map the subsets I, I 0 , J, J 0 , K, K 0 to
the component indices of Q :
I
I0
J
J0
K
K0
→
→
→
→
→
→
[1 : 3]
[10 : 12]
[4 : 6]
[13 : 15]
[6 : 9]
[16 : 18]
From that we can compute the Jacobian as Jij =











∂FL
=
∂Q0M
∂FI0
∂QI
∂FI0
∂QJ
∂FI0
∂QK
∂Fi
∂Qj
(46)
(47)
(48)
(49)
(50)
(51)
:
δLM where L, M ∈ {I, J, K}
= −MJ JacI (fg (QI , QJ )) − MK JacI (fg (QI , QK ))

= −MJ JacJ (fg (QI , QJ ))

 and circular permutations of {I, J, K}
= −MK JacK (fg (QI , QK ))
Here I used the gravitational force function fg (q1 , q2 ) =
respect to the 3-vector q1 , reads :
q1 −q2
kq1 −q2 k3
(52)
the Jacobian of which, with
∂fg i
δij
3 (q1 − q2 )i (q1 − q2 )j
=
3 −
∂q1 i
2
kq1 − q2 k
kq1 − q2 k5
(53)
And :
JacI (fg (QI , QJ )) = −JacJ (fg (QI , QJ )) = −JacI (fg (QJ , QI )) = JacJ (fg (QJ , QI ))
C
(54)
Likelihood function
Given a Gaussian likelihood density for a toa t, the likelihood for the associated turn number
N is given by straightforward probabilities :
Z
Z
∀∆t,
p(t)dt =
p0 (N )dN
(55)
∆t
∆N (∆t)
dN 0
dt
⇔ p(t)dt = p (N ) (56)
dt 24
Guillaume Voisin
Hence one has to compute the derivative of the phase with respect to time. Though it is
clear that delays will contribute to second order since their characteristic period of variation
is the orbital period (let’s take the shortest PI ) and they are bounded to a hundred seconds
at most (in the case of the Rømer delay). Their derivatives will consequently be bounded and
small compared to the main contribution :
dN
= f + f 0 (t − t0 ) + (delay terms)
dt
(57)
In computations, we try to minimize the logarithm of the likelihood which to first order in
− t0 ) ' 10−11 (t − t0 ) reads :
f0
(t
f
f0
log(p(N )) ' log(p(t)) − log(f ) − (t − t0 )
f
(58)
The last term is the only one new compared to equation 35. It can become important since
it is proportional to time. Also we must remind ourselves that the previous formula must be
summed up for all toas. So assuming we can observe 5000 TOAs a year for n years how long
will it be before it changes the log of the total likelihood by, say, 0.001 (which in practice not
significant for such a number of TOAs) ?
5000n
f0
× 365n > 0.001 ⇒ n > 7
f
(59)
Compared to a purely Gaussian likelihood this implies a shift of about 10−7 turn per turn
or 1ns per TOA, well below the experimental accuracy.
25

Modélisation du syst`eme triple autour du pulsar radio PSR J0337+

Transcription

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