Toward a third generation torsion pendulum for the femto newton

Transcription

UNIVERSITÀ DEGLI STUDI DI TRENTO
Facoltà di Scienze Matematiche, Fisiche e Naturali
Corso di Laurea in Fisica
Elaborato Finale
Toward a third generation torsion
pendulum for the femto newton level
testing of free fall in the laboratory
Verso un pendolo di torsione di terza generazione per l’indagine
al livello del femto newton della caduta libera in laboratorio
Laureando: Daniele Nicolodi
Relatori: William Joseph Weber
Stefano Vitale
Anno Accademico: 2006 - 2007
Contents
1 Abstract
1
2 The need for free fall testing
2.1 Gravitational waves . . . . . . . . . . . . . . . . . .
2.1.1 Gravitational wave sources . . . . . . . . .
2.1.2 Gravitational wave detection . . . . . . . .
2.2 The LISA experiment . . . . . . . . . . . . . . . . .
2.2.1 Signal sources for LISA . . . . . . . . . . .
2.2.2 Achievement of LISA sensitivity . . . . . .
2.2.3 Drag free control . . . . . . . . . . . . . . .
2.2.4 Gravitational reference sensor . . . . . . .
2.2.5 LISA pathfinder . . . . . . . . . . . . . . . .
2.3 Ground testing of the GRS . . . . . . . . . . . . . .
2.3.1 The need to investigate force noise sources
2.3.2 Force noise sources arising in the GRS . .
2.3.3 Other disturbance sources . . . . . . . . . .
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Recent improvements
4.1 New GRS prototype integration . . . . . . . . . . . . . . . . . . . . . . .
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Small force measurement with a torsion pendulum
3.1 Torsion pendulum operation principles . . . . . . . . .
3.2 First generation torsion pendulum facility . . . . . . . .
3.2.1 Pendulum position readout . . . . . . . . . . . .
3.2.2 Experimental facility details . . . . . . . . . . . .
3.3 Second generation torsion pendulum facility . . . . . .
3.4 Force noise excess data analysis . . . . . . . . . . . . . .
3.4.1 Spectral estimation . . . . . . . . . . . . . . . . .
3.4.2 Torque estimate . . . . . . . . . . . . . . . . . . .
3.4.3 Readout noise rejection . . . . . . . . . . . . . .
3.4.4 Uncertainties and data reduction . . . . . . . . .
3.4.5 Time domain disturbance subtraction . . . . . .
3.4.6 Example with previous torsion pendulum data
3.5 First generation torsion pendulum achieved results . .
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CONTENTS
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5 Future improvements
5.1 Fused silica torsion fibre . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Interferometric readout . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Possible force sensitivity of the torsion pendulum . . . . . . . . . . . . .
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4.3
4.4
Optical readout improvements . . .
Magnetic shield . . . . . . . . . . . .
Preliminary results . . . . . . . . . .
4.4.1 Cross correlation analysis . .
4.4.2 Initial pendulum suspension
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Chapter 1
Abstract
The realization of free fall is central for the experimental detection of gravitational
waves. The concept that a particle in free-fall under the influence of gravity alone
follows a geodesic in space-time is at the foundation of General Relativity. Within
General Relativity gravity is not a force acting on material particles but instead is
identified with curvature in space-time geometry. Particles in absence of forces travel
in the straightest possible way in curved space-time: this path is called a geodesic.
In the absence of gravity space-time is flat and geodesics are simply straight lines
travelled at constant velocity.
All experiments aimed at directly measuring curvature caused by celestial bodies
like gravitational wave observatories require particles in geodesic motion. In addition
all experiments aimed at probing the limits of General Relativity and the possibility of
alternative theories of gravitation search for violations of geodesic motion. Achieving
high purity geodesic motion is made difficult by non-gravitational forces acting on
masses accelerating them away from the geodesic lines. Gravity is by far the weakest of all fundamental interactions. Achieving the required extremely low level of
non-gravitational acceleration implies the reduction and control of the disturbances
produced by a wide range of physical phenomena. For most of these phenomena this
requires ground-breaking achievements within their relevant fields of science.
Gravitational waves have been predicted by A. Einstein as one of the most fascinating and controversial consequences of the theory of General Relativity. Many excellent
attempts in detecting gravitational waves have been done in the past and continuous
effort is put in this challenging experimental quest. However due to the weakness of
the gravitational interactions gravitational waves have never been directly detected.
Their existence is only confirmed as an indirect result of the observation of the binary
pulsar PSR 1913+16 discovered by R. A. Hulse and J. H. Taylor. The experimental
detection of gravitational waves is not only important as relevant confirmation of the
theory of General Relativity. Detecting gravitational waves emitted by astronomical
sources would be the first fundamental step to create the new field of the gravitational
wave astronomy.
Among the other General Relativity experiments the Laser Interferometer Space
Antenna - LISA - is a joint effort of the European Space Agency - ESA - and of the
1
2
CHAPTER 1. ABSTRACT
National Aeronautics and Space Administration - NASA - to realize the first high
sensitivity gravitational wave observatory in space. It aims to observe gravitational
waves emitted from galactic and cosmological sources in the frequency range from
0.1 mHz to 0.1 Hz. The LISA mission comprises a constellation of three identical
spacecraft located 5 × 106 km apart forming an equilateral triangle. Each spacecraft
hosts a pair of test bodies which are namely in free fall along their geodesic orbits. The
distance between the test bodies in different spacecrafts is measured by means of laser
interferometry ranging technique with picometer precision. The spacecraft mainly
serve to shield the proof masses from most of the external disturbances. Its position
does not directly enter into the measurement. It is nevertheless necessary to keep all
spacecraft moderately accurately centred on their respective proof masses to reduce
spurious local noise forces. This is achieved by a drag free control system consisting of
an inertial sensor and a system of thrusters. Capacitive sensing in three dimensions is
used to measure the displacements of the proof masses relative to the spacecraft. These
position signals are used in a feedback loop to command ion-emitting proportional
thrusters to enable the spacecraft to precisely follow its proof masses. The ensemble
of test mass and position sensor is here called Gravitational Reference Sensor. Big
scientific and technological effort is ongoing to qualify the purity of free fall of the
LISA test masses with the described drag free control loop scheme. The importance
of the development and qualification of the gravitational reference sensor to meet the
LISA scientific goals is the driving force behind the experimental activity discussed in
this thesis.
The LISA working group at the Trento University has the duty of designing and
testing the gravitational reference sensor for LISA. This has been the motivation of
the development and realization of a ground based test bench for small force measurements. The realization of this test bench has the aim to conduct an experimental
campaign focused on characterisation of the disturbances exerted in the mHz and sub
mHz frequency region on the proof masses by the gravitational reference sensors under design. Based on the experience of several other experiments it has been chosen
to base the test bench facility employed for the investigation of the free-fall of LISA
proof masses on a torsion pendulum.
The work described in this thesis consists in the analysis of the existing testing facility and in the implementation and verification of some simple improvements brought
about the torsion pendulum experimental facility. Furthermore some preliminary investigation for radically importing the torsion pendulum performance as a torque
meter has been done. This investigation covered the analysis of the the possibility
of suspending the torsion pendulum with a fused silica fibre to obtain a much lower
mechanical thermal noise and the possibility of implementing an interferometric angular position readout with a sensitivity much higher than the currently available
readouts. An improved torsion pendulum experimental facility would permit to set
more stringent upper limits to the force noise acting on the LISA test mass inside the
gravitational reference sensor. Furthermore it would lower the uncertainties on the
measurement of the investigated effects of single noise sources for LISA.
3
The current torque sensitivity obtained with the torsion pendulum facility used to
investigate noise sources for LISA is roughly 1 fN m Hz −1/2 . Torque is converted into
equivalent force by means of a suitable arm length that for our pendulum geometry
is roughly 10 mm. The current force sensitivity is then roughly 100 fN Hz −1/2 . Force
resolution of 1 fN can now be achieved with an integration time of 10000 s. This
thesis aims toward the development of the torsion pendulum facility to obtain force
resolution better than the current one.
In chapter 2 we introduce the reader to the problem of gravitational wave detection
and to the challenges that this task offers. We will describe the LISA experiment and
the gravitational reference sensor developed at the Trento University. The main sources
of disturbance noise are analysed and some of the reasons for the detailed ground testing are highlighted. In chapter 3 we introduce the use of a torsion pendulum for the
measurement of tiny forces. The two testing facility developed at the Trento University
are be described and the data analysis techniques necessary to obtain the desired information on the force disturbances acting on the test mass is presented. Finally those
techniques are used to demonstrate the sensitivity reached by the torsion pendulum
facility. In chapter 4 we motivate and describe the realised changes to the experimental facility. The achieved results are presented. In chapter 5 we briefly discuss the
improvements we envision for the torsion pendulum testing bench. The challenges
that the realisation of those improvement presents and the possible enhanced torsion
pendulum sensitivity to small forces are reported.
4
CHAPTER 1. ABSTRACT
Chapter 2
The need for free fall testing
The realisation of free fall is central for the experimental detection of gravitational
waves. Free falling test masses describe geodesic curves and realise a space-time
transverse and traceless coordinate frame. The operative possibility of realizing such
reference frame is the basis for experimental detection of gravitational waves by an
interferometric detector. In a transverse and traceless coordinate frame a free particle
remains at rest and the proper time of a clock sitting on the particle coincides with the
coordinate time but the perturbation of the metric tensor due to the gravitational field
produces a change in time of the distance among particles. This change in time can be
detected by the laser interferometer.
Gravitational waves have been predicted by A. Einstein as one of the consequences
of the theory of General Relativity [1–3]. Gravitational waves are ripples of space-time
curvature which are generated by accelerated masses and propagate across the universe at the speed of light. Due to the weakness of the gravitational interactions gravitational waves have never been directly detected. Their existence is only confirmed as
an indirect result of the observation of the binary pulsar PSR 1913+16 discovered by
R. A. Hulse and J. H. Taylor [4, 5].
Detecting gravitational waves is not only important as relevant confirmation of the
theory of General Relativity. Detecting gravitational waves emitted by astronomical
sources would be the first fundamental step to create the new field of the gravitational
wave astronomy.
During the last decades, several experiments have been developed and are currently operating to detect gravitational waves in the frequency range from about 10 or
100 Hz up to the kHz. Within the gravitational waves observatories, the Laser Interferometer Space Antenna - LISA - will be the first space borne, low frequency, gravitational
wave detector, and its aim is to observe gravitational waves emitted from galactic
sources, like binary stars, and cosmological sources, like massive black holes, in the
frequency range from 0.1 mHz to 0.1 Hz [6].
5
CHAPTER 2. THE NEED FOR FREE FALL TESTING
6
2.1
Gravitational waves
Gravitational waves are emitted by any system that interacts by gravity. However
gravitational fields themselves carry energy and momentum and must therefore contribute to their own source. The effect of gravitation on itself is represented by the non
linearity of the Einstein equations and it is an important complication in the theory of
General Relativity. This complication prevents our being able to find general radiative
solutions of the exact Einstein equations. However, driven by the fact that any observable gravitational radiation is likely to be of very low intensity, we can overcome this
difficulty studying only the weak field radiative solutions of Einstein equations. Those
solutions describe waves carrying not enough energy and momentum to affect their
own propagation.
The theory of the General Relativity, in the approximation of linearised Einstein
equations, predicts gravitational waves as ripples on the flat space time. In this framework the the curvature of the space time is defined by the metric tensor gµν which
obeys the Einstein field equations:
Rµν − 21 gµν R = −8πGTµν
(2.1)
where Tµν is the energy-momentum tensor, c is the speed of light, and G the Newton
gravitational constant. In the limit of nearly flat space time the metric tensor can be
written as
gµν = ηµν + hµν
(2.2)
where |hµν | 1 is a perturbation of the Minkowski metric tensor ηµν of the flat space
time. Using the general relativity gauge invariance to chose an harmonic coordinate
frame, substituting equation 2.2 in equation 2.1, and assuming the vacuum condition
Tµν = 0, the Einstein equation can be approximated at the first order in hµν obtaining
the linearised Einstein equations:
2 hµν =
2
1 ∂2
∇ − 2 2
c ∂t
2
hµν = 0
(2.3)
Gravitational waves are solutions of equation 2.3. They propagate at the speed of light
c perturbing the Minkowski metric tensor of the flat space-time.
The freedom in the choice of the coordinate system can be used again to impose
the tensor hµν to be transverse and traceless. In this reference frame the coordinates
are marked by the world lines of free falling masses, hence the coordinate of free
falling test masses are constant. If we consider a monochromatic gravitational wave
propagating along the z axis the solutions of equation 2.3 can then be written as



hµν = 

0 0
0
0 h xx h xy
0 h xy −h xx
0 0
0
0
0
0
0





(2.4)
2.1. GRAVITATIONAL WAVES
7
This example shows that gravitational waves have only two independent polarisations:
h xx (z, t) = h+ e i(ωt−kz)
(2.5)
h xy (z, t) = h× e i(ωt−kz)
(2.6)
The tensors h+ and h× represent the two possible transverse orthogonal polarisations
for waves propagating along the z axis. This is true despite the fact that one goes
into the other through a rotation of only 45o . The perturbation called h+ momentarily
lengthens distances along the x axis, simultaneously shrinking them along the y axis.
The polarisation h× has its principal axes rotated by 45o .
2.1.1 Gravitational wave sources
In analogy to the treatment of electromagnetic radiation, not highly relativistic gravitational wave sources can be decomposed into multipoles.
The monopole moment of the mass distribution is the total mass. The energy
radiated through gravitational waves is quadratic in the amplitude. The energy loss
through gravitational radiation is then a second order effect in the perturbation hµν of
the metric tensor and can be neglected in the weak field approximation, obtaining the
first order conservation of the total mass. This implies that the monopole moment is
constant and that there is no monopole emission of gravitational radiation.
The mass distribution dipole moment is also conserved, because its time derivative
is the total momentum of the source, that is conserved in the same way. Therefore there
is no dipole emission of gravitational radiation either. The dominant gravitational
radiation from a source comes then from the time variation of the quadrupole moment
of the system.
Under the approximation of a vanishing energy-momentum distribution Tµν equation 2.1 can be rewritten in the form
1 ∂2
16πG
2
2
2 hµν = ∇ − 2 2 hµν = − 4 Tµν
(2.7)
c ∂t
c
That is the linearised Einstein equation and it shows that gravitational waves are produced by accelerated masses. It can be shown [3] that, for an observer located at a
distance r from the source, the gravitational wave amplitude hij can be written as
hij =
2G
Q̈ij
rc4
(2.8)
where i and j represent two generic spatial coordinates, and Qij is the second time
derivative of the mass quadrupole moment of the source, defined as
Qij =
Z
Ω
ρ(r ) xi x j − 31 δij r2 dΩ
where ρ(r ) is the local mass density of the source, and δij is the Kröneker delta.
(2.9)
8
It must be pointed out that due to the G/c4 term in equation 2.8 the gravitational
wave amplitudes are extremely feeble: in practice it is impossible to observe gravitational waves produced in a laboratory in a Hertz like experiment. However it is possible to detect gravitational waves emitted by celestial objects: with the high masses
involved in highly accelerated motions, though distance r is high as well, different
kinds of celestial objects are expected to produce gravitational waves with amplitudes
up to h ' 10−21 in the frequency band from 10−5 Hz up to the kHz [6].
An interesting system is the binary pulsar PSR 1913+16 discovered by R. A. Hulse
and J. H. Taylor [4, 5]. It apparently consists of two compact stars orbiting each other
closely. The orbital period is 7 h 45 m 7 s = 25907 s and both stars have masses
approximately equal to 1.4 solar masses. If the system is 1.5 kpc = 1.5 × 1017 m away,
then its radiation will have amplitude h ' 10−20 at the Earth [3].
2.1.2
Gravitational wave detection
Astronomy used to rely totally on visible light as carrier of information from distant objects. Only since 1940 it has been possible to detect electromagnetic radiation
from the sky in other frequency bands. Nowadays the useful forms of electromagnetic
waves span the entire spectrum, from radio waves to γ-rays. The new ways of looking
at the sky have lead to the discovery of phenomena of radically distinct and previously
unexpected kind: large fraction of important new results in the history of astronomy
can be directly traced to innovation in instrumentation that gave qualitatively new or
substantially improved sensitivity to cosmic signals [7]. The development of gravitational wave observatories will open a new window in astronomy, that will permit to
reveal facets of the universe previously unseen with electromagnetic waves.
In the last decades several gravitational wave detectors have been developed and
built in order to detect the gravitational radiation. The historically first generation detectors have been the resonant bars, which are currently in operation with the goal
of detecting gravitational waves at the acoustic frequencies around 1 kHz and with
a sensitive band of order 100 Hz. Modern detectors of this kind are ALLEGRO [8],
AURIGA [9], NIOBE [10], EXPLORER and NAUTILUS [11, 12]. Those detectors aim
to observe gravitational radiation making use of resonant quadrupole antennas, which
are mechanical systems with a natural mode of oscillation. If the frequency of the the
incident wave is close to a natural frequency of free oscillation of the antenna system,
the incident wave serves to excite this free oscillation in what is called resonant scattering of gravitational waves. The narrow sensitivity frequency band of such instruments
is then explained by the difficulty of constructing mechanical systems with wide band
resonances. The sensibility limit of those detectors in approaching the level where they
are able to detect gravitational waves emitted by catastrophic galactic events. However such events are expected to be broadband high frequency sources of gravitational
waves. The narrow sensitivity frequency band is then a sensible limit in the detection
of such sources.
The second generation detectors are the ground based Michelson interferometers.
Those are currently leaving their commissioning phase to enter scientific operation at
2.1. GRAVITATIONAL WAVES
9
full sensitivity, with the aim to detect gravitational waves in the frequency band from
10 or 100 Hz up to few kHz. Detectors of this kind are GEO600 [13], LIGO [14], TAMA
[15] and VIRGO [16, 17]. In those experiments the detection is based on a more direct
measurement of the space-time distortion caused by the gravitational wave. Let us
consider a «plus» polarised gravitational plane wave of amplitude h and frequency ω g
propagating in the z direction. From equation 2.4 h xx = −hyy = h and accordingly to
equation 2.2 the metric element is written as:
ds2 = gµν dx µ dx ν =
= −c2 dt2 + [1 + h(z, t)] dx2 + [1 − h(z, t)] dy2 + dz2
(2.10)
Suppose now to have a Michelson interferometer, whose beam splitter is placed in
the origin of the reference frame, and has its two end mirrors lying at a distance L
respectively along the x and y axis, as in figure 2.1.
y
M
L
h+
M
BS
x
L
Figure 2.1: Michelson interferometer: the beam splitter is placed in the origin of the reference
frame, and the two end mirrors are at distance L respectively along the x and y axis.
Let the optics of the interferometer be freely falling masses. For a light ray ds2 = 0
in all the reference frames, for the light ray traveling from the beam splitter to the end
mirror of the x arm, it is then possible to write the equation:
L = x (t x1 ) = c
Z tx1
du
t x0
(1 + h(u))1/2
(2.11)
where t x0 and t x1 are respectively the times in the transverse traceless reference frame,
when the light ray is emitted and when it reaches the mirror and is reflected back, and
u is an affine parameter. For the reflected ray we can write the equation:
0 = x (t x2 ) = L − c
Z tx2
du
t x1
(1 + h(u))1/2
(2.12)
where t x2 is the time in the transverse traceless reference frame, when the ray reaches
the beam splitter and is made to interfere with the light ray coming from the y interferometer arm. These equations can be rewritten in the form:
L=c
Z tx1
t x0
du
(1 + h(u))
1/2
=c
Z tx2
du
t x1
(1 + h(u))1/2
(2.13)
10
Taking the derivative of equation 2.13 with respect to the observation time t x2 , and
observing that the interferometer arm length L is constant in the transverse traceless
reference frame, so that ∂L/∂t x2 = 0, we obtain:
du
(1 + h(t x2 ))
1/2
=
du
(1 + h(t x1 ))
1/2
dt x1
dt x0
du
=
1/2
dt x2
(1 + h(t x0 )) dt x2
(2.14)
We compute a similar equation for the y arm of the interferometer, using the symmetry
between the component x and y in equation 2.10 and sending h in −h. In a notation
similar to the one used for the x arm we obtain:
du
1 − h(ty2 )
1/2 =
du
1 − h(ty1 )
1/2
dty1
dty0
du
=
1/2
dty2
dty2
1 − h(ty0 )
(2.15)
It is then possible to calculate the derivative of the phase shift δθ = ω t x0 − ty0 of the
two light rays, with respect to the observation time t x2 = ty2 = t, when they are made
to interfere, and the intensity of the interference light is measured:
"
1/2 #
1/2
dty0
1 − h(ty0 )
2πc (1 − h(t x0 ))
2πc dt x0
dδθ
−
=
=
−
(2.16)
1/2
dt
λ dt x2
dty2
λ
(1 + h(t x2 ))1/2
1 + h(ty2 )
where λ = 2πc/ω is the light wavelength. In the weak field approximation the square
root can be expanded at the first order of the binomial series, setting t x0 ' ty0 =
t − 2L/c we obtain a much simpler expression:
dδθ
2πc
2L
'
h(t −
) − h(t)
(2.17)
dt
λ
c
We can then integrate this to obtain the observed phase shift:
Z t
Z t
Z
dδθ
2πc
2L
2πc t
0
0
0
δθ (t) =
=
h
(
t
−
)
−
h
(
t
)
dt
=
h(t0 )dt0
0
λ
c
λ t− 2Lc
0 dt
0
(2.18)
Further simplification is possible in the low frequency limit ω g 2L/c, where the
space-time perturbation h is supposed to be constant during the interaction with the
light beam. Integrating equation 2.17 we obtain:
δθ (t) =
4πL
h(t)
λ
(2.19)
This phase shift produces a modulation in intensity of the recombined light beam
that can be measured with high accuracy. From equation 2.18 it is evident how the
sensitivity of the ground based interferometers is limited mainly by the constrain to a
few kilometres of their length. The sensitivity limiting factors at frequency below 10
Hz are instead the local gravitational and seismic noise, both unavoidable on ground.
Within the interferometric detectors, the Laser Interferometer Space Antenna - LISA is a joint effort of the European Space Agency - ESA - and of the National Aeronautics
and Space Administration - NASA - to realize the first high sensitivity gravitational
2.2. THE LISA EXPERIMENT
11
wave observatory in space. The operation in space allows both for very long arm
length and for a friendlier environment: less gravity gradient noise and smaller external disturbances. This will open to the possibility to perform observation of gravitational waves emitted in the low frequency region from 0.1 mHz to 0.1 Hz. The
LISA gravitational radiation observations will be complementary to the ground based
experiments, in a frequency range where several gravitational wave signals candidate
sources are known with well measured parameters [6].
2.2
The LISA experiment
The LISA mission comprises a constellation of three identical spacecraft each one hosting a pair of test bodies which are namely in free fall along their geodesic orbits. The
three spacecraft are located 5 × 106 km apart forming an equilateral triangle, whose
centre lies in ecliptic plane, 1 au from the Sun and 20◦ behind the Earth. The plane of
the triangle is inclined at 60◦ with respect to the ecliptic. The independent heliocentric
orbits for the three spacecraft were chosen such that the triangular formation is maintained throughout the year, with the triangle appearing to rotate about the centre of
the formation and about the Sun once per year, allowing to explore a wide region of
the sky.
Figure 2.2: LISA configuration: three spacecraft located 5 × 106 km apart forming an equilateral
triangle, whose centre lies in ecliptic plane, 1 au from the Sun and 20◦ behind the Earth
Drawing not to scale: the LISA triangle is drawn one order of magnitude too large [6].
Each test body is the end mirror of a single arm interferometer. The other mirror is
the corresponding test body in one of the other two spacecraft. The triangular formation basically reconstructs two giant semi-independent Michelson interferometers with
arms rotated by 60◦ and one common arm. Each interferometer measure the difference
of the lengths of its arms: gravitational waves are thus detected as relative variation
of the two optical paths. The common arm gives independent information on the two
12
gravitational wave polarisations and redundancy. The distance between the spacecraft
defines the the interferometer arm length and determines the frequency within LISA
can make observations. It has been carefully chosen to allow for the observation of
most of the interesting sources of gravitational radiation. As the three-spacecraft constellation orbits the Sun in the course of one year, the observed gravitational waves
are Doppler shifted by the orbital motion. For periodic waves with sufficient signalto-noise ratio, this allows the direction of the source to be determined.
Figure 2.3: Annual revolution of LISA configuration around the Sun. One selected two arm
interferometer is highlighted by heavier interconnecting laser beams. The green trajectory of
one individual spacecraft is shown, inclined with respect to the blue Earth orbit.
While LISA can be described as a big Michelson interferometer, the actual implementation in space is very different from a laser interferometer on the ground. It is
more reminiscent of the spacecraft tracking technique. The laser light going out from
the centre spacecraft to the other corners is not directly reflected back. Because of the
small but finite divergence of the laser, very little light intensity would be left over that
way, determining a very high shot noise. Instead, the laser on the distant spacecraft is
instead phase-locked to the incoming light and directed toward the centre spacecraft,
providing a return beam with full intensity.
After being transponded back from the far spacecraft to the censer spacecraft, the
light is superposed with the on-board laser light serving as a local oscillator in a
heterodyne detection. This gives information on the length of one arm modulo the
laser frequency. The other arm is treated in the same way. Information on the length
of the other arm is obtained modulo the same laser frequency. The difference between
these two signals will thus give the difference between the two arm lengths and then
on the gravitational wave signal. The sum will give information on laser frequency
fluctuations. More generally, the combination of the three interferometer arms give
six optical signals, that allow the reconstruction of 18 different signals combinations.
Those signals provide more detailed informations on the laser frequency fluctuations
13
or optical bench motion as well as on the remaining system noise [6, 18].
The goal of the LISA interferometry is to measure distance fluctuations between
freely falling test bodies on different spacecrafts with sub-angstrom precision. Combined with the large separation between the spacecrafts, this will allow LISA to reach
the strain sensitivity goal of S1/2
' 4 × 10−21 Hz−1/2 in the mHz frequency region
h
and detecting gravitational wave strains down to a level of order h = 10−23 with a
signal-to-noise ratio of 5 in one year of observation.
Each LISA spacecraft contains two optical assemblies, each one pointing towards
an identical assembly on each of the other two spacecraft. The interferometric ranging
technique employes a 1 W infrared laser beam with wavelength of 1064 nm, transmitted to the corresponding remote spacecraft via a 30 cm aperture telescope. The
same telescope is used to focus the very weak light intensity of few pW coming from
the distant spacecraft on a sensitive photo-detector, where it is superimposed with a
fraction of the original local light.
At the heart of each assembly is a vacuum enclosure containing a 46 mm polished
platinum-gold cubic proof mass. It serves as optical reference for the light beams and
should nominally follow as close as possible a perfect geodesic orbit. The spacecraft
mainly serve to shield the proof masses from most of the external disturbances, and
its position does not directly enter into the measurement. It is nevertheless necessary
to keep all spacecraft moderately accurately centred on their respective proof masses
to reduce spurious local noise forces.
This is achieved by a Drag Free Control System consisting of an inertial sensor and
a system of thrusters. Capacitive sensing in three dimensions is used to measure the
displacements of the proof masses relative to the spacecraft. These position signals
are used in a feedback loop to command µN ion-emitting proportional thrusters to
enable the spacecraft to precisely follow its proof masses. The thrusters are also used
to control the attitude of the spacecraft relative to the incoming optical wavefronts.
The ensemble of test mass and position sensor is here called Gravitational Reference
Sensor. Big scientific and technological effort is ongoing to qualify the purity of free
fall of the LISA test masses with the described drag free control loop scheme. The
importance of the development and qualification of the gravitational reference sensor
to meet the LISA scientific goals is the driving force behind the experimental activity
discussed in this thesis.
2.2.1 Signal sources for LISA
As previously described, LISA is focused in the frequency region between 0.1 mHz
and 0.1 Hz. It will be complementary to the ground based interferometers, that are
sensitive in the region from 10 or 100 Hz up to few kHz. The interest for a gravitational
wave detector in the mHz frequency range arises from the wide number and variety
of sources which are expected to emit gravitational waves in this region [6].
Several astronomical sources like binary systems in our own galaxy have been
identified to certainly produce gravitational wave signals around the mHz [19, 20].
This represents a big advantage for LISA compared to the ground based detectors:
14
besides ensuring gravitational wave detection, these sources, whose period, mass, position and distance are known from other astronomical observations, can be used as
gravitational wave calibration signals for the LISA instrument [6, 19, 20]. However, more
than a direct observation the gravitational wave signal produced by these sources in
our galaxy, LISA aims to learn about the formation, growth, space density and surroundings of massive black holes and about the capture of small compact objects into
a BH as well. Here, we briefly list the most interesting scientific objectives of the LISA
mission and the expected sources.
Galactic binary systems. LISA is expected to observe gravitational waves and to
provide information about the population of various types of binary systems in our
galaxy, with masses of order of the solar mass: pairs of neutron stars, neutron stars
and black holes binary systems, and possibly BH-BH binaries with masses ranging
from about one solar mass up to few tens solar masses. Additionally it is expected
to observe gravitational wave signals emitted by binary system in our galaxy whose
electromagnetic spectrum is not currently detected. LISA aims to detect these gravitational wave signal with a signal-to-noise ratio of 5, such that the time evolution of the
signal amplitude and the source location in the sky can be resolved [6].
Gravitational wave signals from our galaxy are also expected from close white
dwarf binary systems. Related estimates suggest that there will be such a high number
of white dwarf binaries signals that they would not be resolvable by LISA below few
mHz: they would overwhelm other kind of gravitational wave signals, giving rise to a
confusion limited background.
Unfortunately, the famous binary pulsar PSR 1913+16, discovered by R. A. Hulse
and J. H. Taylor, that would serve as a well known calibration signal, has an orbital
period of 7.68 hours, thus it falls below the LISA frequency band.
Massive black holes in distant galaxies. The most interesting scientific objective of
the LISA mission will be the search and the study of signals from sources which
involve massive black holes. LISA aims to the identification of several MBH in nearby
galaxies, which would confirm their existence. Moreover, one of the LISA goals is to
study the formation, growth and space density of MBH with masses ranging from 106
up to 108 solar masses.
A big scientific objective is also the observation of the MBH-MBH merging, whose
waveform signals would provide extremely sensitive tests of General Relativity in
post-Newtonian conditions. Finally, another type of MBH signal of interest is given
by the gravitational capture of small objects, like stars and stellar-mass black holes
orbiting and eventually falling into a MBH.
Primordial background. A non-thermal cosmological background of gravitational
waves is expected to come from many different sources: density fluctuations produced
by cosmic strings or cosmic textures have been much discussed. There is general
agreement that inflation would amplify early quantum fluctuations into a stochastic
background [21].
This stochastic background radiation would consists of a huge number of incoher-
15
ent waves arriving from all directions and with all frequencies. In order to discriminate it against the instrument noise floor, suitable LISA signal combinations have been
studied, allowing suppression of the interferometer sensitivity to gravitational waves
and subsequent evaluation of the instrument noise level.
2.2.2 Achievement of LISA sensitivity
The LISA gravitational wave detector aims to be sensitive to gravitational waves in
the wide frequency band between 0.1 mHz and 0.1 Hz, with a maximum sensitivity
of 4 × 10−21 Hz−1/2 around few mHz, and slightly relaxing this requirement at both
ends of the band.
The sensitivity of the LISA mission is determined by two competing features: the
output signal of the interferometer to a given gravitational wave strain, and the effect
of various noise sources that fake such gravitational wave signals. There are two main
categories of such sensitivity limiting noise effects: noise sources that fake fluctuations
in the measured lengths of the optical paths, and noise sources that contribute forces
acting on the proof masses, and thus changing their physical distance. The most
prominent types of noises in the first category are the photon shot noise and the
interferometer laser beam pointing instabilities. The drag free environment effectively
shields the proof masses from outside influences but some residual force noise arises
from the interactions with the spacecraft and the gravitational reference sensor. Those
interactions build up the second noise category. Force noise on the proof masses
dominates in the detector noise in the low frequency range, leading to a decrease in
sensitivity toward lower frequencies roughly proportional to the inverse of ω 2 . The
shot noise dominates instead at high frequency, where the decline of the gravitational
wave antenna transfer function causes a decrease in sensitivity roughly proportional
to ω.
As shown in section 2.1.2, a laser beam traveling back and forth between two free
falling proof masses, in a transverse and traceless coordinate frame, along an axis x
normal to the direction z of propagation of a gravitational wave, is subject to a phase
shift δθ whose time derivative, in first approximation, is given by:
πc
2L
dδθs
=
h(t −
) − h(t)
(2.20)
dt
λ
c
where h is the amplitude of the gravitational perturbation, L is the distance between
the particles, λ is the wavelength of the laser, and t is the time at which light is collected
and the frequency shift is measured. The x axis has been used to define the wave
polarisation so that the phase shift is contributed by the «plus» polarisation only.
However the proof masses are subject to certain force noise: they are not a rest in
the transverse and traceless coordinate frame. If the particles moves slowly relative
to light, their relative motion does not affect the transverse traceless construction, but
competes with the gravitational wave signal providing a phase shift:
2L
L
2π
x1 ( t ) + x1 ( t −
) − 2x2 (t − )
(2.21)
δθn (t) =
λ
c
c
16
at first order in v/c and where x1 is the component along the laser beam of coordinate
of the particle sending and collecting the laser beam, and x2 is that of the particle
reflecting the light. At measurement frequencies much lower than c/L this equation
becomes:
4π
δL(t)
(2.22)
δθn (t) '
λ
where δL(t) = x1 (t) − x2 (t) is the optical path difference. The optical path difference
is governed by the equation of motion of the proof masses:
δFn (t) ∂2 δLn (t)
∂2 δL
=
+
(2.23)
∂t2
m
∂t2
where δFn is the force noise and δLn is the path length noise. The observed phase shift
δθ is the sum of signal δθs and noise δθn . In the frequency domain this is:
δθ (ω ) = δθs (ω ) + δθn (ω )
2πL
ωL
ωL
4π δFn (ω )
=
h(ω ) exp −i
sinc
−
−
δL
(
ω
)
n
λ
c
c
λ
mω 2
(2.24)
where sinc( x ) = sin( x )/x. At the purpose of the noise analysis it is customary to
describe the LISA interferometer as a stationary causal linear system, whose frequency
response g(ω ) is:
δθ (ω )
2πL
ωL
ωL
g(ω ) =
=
exp −i
sinc
(2.25)
h(ω )
λ
c
c
The power spectral density of the noise on phase shift δθ is obtained from equation 2.24:
2 SδFn (ω )
4π
Sδθ (ω ) =
+ SδLn (ω )
(2.26)
λ
m2 ω 4
and it is translated into equivalent power spectral density of the noise on the gravitational wave amplitude h dividing by the square module of the frequency response.
In this extremely simplified model of the LISA interferometer, the sensitivity of the
detector in measurement of the amplitude h of a gravitational wave is then:
SδFn (ω )
1
Sδθ (ω )
2ω 2
Sh ( ω ) =
=
+ SδLn (ω )
(2.27)
c
m2 ω 4
sin2 ωL
| g(ω )|2
c
The scientific goals of the LISA experiment drive stringent limits on the noise sources
that contribute to the total noise on the observed signal. The error appointment for
LISA requires total path length noise and total force noise with power spectral density
respectively not exceeding:
"
2 #
3
mHz
1/2
SδL
(ω ) =2 × 10−11 1 +
m Hz−1/2
(2.28)
n
ω/2π
"
2 #
ω/2π
1/2
SδF
(ω ) =3 × 10−15 1 +
m s−2 Hz−1/2
(2.29)
n
3 mHz
Inserting those values into equation 2.27 gives the gravitational wave amplitude noise
power spectral density S1/2
h ( ω ) of figure 2.4.
17
−17
10
−18
S1/2
[1 / Hz1/2]
h
10
−19
10
−20
10
−21
10
−4
10
−3
−2
10
10
−1
10
frequency [Hz]
Figure 2.4: Gravitational wave amplitude noise spectral density predicted with a simple model
of the LISA interferometer described by equation 2.24. The power spectral density of the
optical path and force noise are those of equations 2.28 and 2.29. The blue line represents
the sensitivity calculated with the «LISA Sensitivity Curve Generator» [22]. The smoothing of
the sinc peaks is due to the averaging of the sensitivity over the possible gravitational wave
incidence angles. The dashed line represent the supposed white dwarf noise.
2.2.3 Drag free control
A key point in achieving the extremely high quality of free fall required for the LISA
test masses is keeping the spacecraft as stationary as possible around the proof masses,
in order to minimise the force noise arising from stray coupling between the proof
masses and the spacecraft. This is realized by a drag-free control loop scheme: the
spacecraft is driven by a high precision thrusters array to follow the test masses in
their geodesic motion, according to a position sensor, the Gravitational Reference Sensor,
which measures the spacecraft-test mass relative position.
In the drag-free control loop scheme, the spacecraft is used as a shield for external
environmental disturbances, either from constant forces as well as from fluctuating
disturbances, as for example those related to the solar radiation pressure. However,
both the satellite and the position sensor themselves could produce force disturbances
on the test masses. In order to let the LISA test masses fall as close as possible along
the geodesic orbit defined by the external gravitational field, the gravitational reference
sensor should perform the position measurement with sufficiently high precision but
minimising the residual force disturbances on the test mass.
We can analyse the drag-free control loop scheme in a very simplified configuration
where the spacecraft contains a single test mass. In this configuration, the equations
of motion for the spacecraft and for the single test mass are:
m ẍ (t) + k [ x (t) − X (t)] = f (t)
(2.30)
18
M Ẍ (t) − k [ x (t) − X (t)] = − f (t) + G [ x (t) + xn (t) − X (t)] + F (t)
(2.31)
where x (t) and X (t) are the positions relative to the inertial frame of the proof mass
and of the spacecraft, m and M are the masses of the proof mass and of the spacecraft,
k is the stiffness of the coupling between the spacecraft and the test mass, f (t) is the
total force noise acting on the test mass, beside the elastic coupling to the spacecraft, G
is the open loop gain of the drag free feed back system, F (t) is the force noise acting on
the spacecraft, xn (t) is the noise of the inertial sensor that measures the position of the
test mass relative to the spacecraft. The acceleration of the test mass in the frequency
domain a(s) is determined from those equation via the Laplace transform:
a(s) =
ω 2f b
(
ω02 + s2 + µω02 + ω 2f b
#
"
#)
"
f (s)
F
(
s
)
s2
1 + 2 + ω02 xn (s) +
m
ωfb
Mω 2f b
(2.32)
where ω 2f b = G/M is the square of characteristic frequency of the feedback loop, if
we think to the test mass and its coupling to the spacecraft as an oscillator, ω02 = k/m
is the resonance frequency square of the oscillator, and µ = m/M is a coefficient
introduced to simplify the writing. Since it is expected to be ω 2f b ω02 we can neglect
the terms ω02 in the denominator of the first factor, taking only the terms at the first
order in ω02 /ω 2f b we obtain:
"
#
F (s)
f (s)
2
+ ω0 x n ( s ) +
a(s) '
m
Mω 2f b
(2.33)
We are interested in analysing the contributions to the force noise on the test mass.
We calculate the total force acting on the test mass multiplying by m this expression,
obtaining:
F (s)
δFn (s) ' f (s) + k xn (s) +
= f (s) + k∆x
(2.34)
G
This equation describes two different contributions to force noise acting on the test
mass: position independent stray forces f (ω ) arising either in the position sensor or in
the satellite, acting directly on the test mass, and spring like couplings k, arising from
position dependent forces between the proof mass and the spacecraft, which couple
residual spacecraft-test mass relative motion into force noise. The residual jitter in the
relative motion of the satellite ∆x arises in part from the position sensor noise xn (s)
and in part from forces acting on the satellite that are not perfectly compensated with
a finite drag-free control loop gain.
The residual acceleration noise goal of equation 2.29 doe not only set serious constraints on the residual forces f (ω ). It places constraint on the parasitic test massspacecraft coupling k, on the motion sensor position noise power spectral density
xn (ω ), and on the feed back open loop gain G, that must be maximised due to the
need to limit the impact of external disturbances on the spacecraft itself. In turn this
requires that the GRS needs motion sensing resolution good enough to keep proof
mass and spacecraft sufficiently centred, but must minimise its contributions to the
19
spacecraft-test mass coupling and to the direct force disturbances. Accordingly to
the LISA noise budget apportioning [6, 23] the GRS requirements are for a position
sensing noise spectral density
1/2
Sδx
≤ 1.8 nm Hz−1/2
n
(2.35)
and for an overall stiffness due to the sensing smaller than k s ' 1 × 10−7 N m−1 .
It is worth to note that the spacecraft to test mass coupling arises mainly from the
gravitational interaction between the test mass and the spacecraft, and from electric
forces between the test mass and the gravitational reference sensor.
Beside our simple model of the drag-free control loop, in each LISA spacecraft
there are actually two proof masses, and it is impossible for the spacecraft to follow
the motion of both. It is then necessary for the GRS to provide actuation system, with
sufficient authority to make the proof masses to follow the spacecraft along the axes
but the interferometric one sensible to gravitational waves.
2.2.4 Gravitational reference sensor
A Gravitational Reference Sensor - GRS - based on a capacitive readout and actuation
scheme have been developed to meet the LISA requirements in terms of position noise,
residual couplings and force noise, outlined in the previous section. We describe
here the main features of this capacitive position sensor, describing the geometrical
configuration of the electrodes, the choice of the composing materials, the machining
tolerances and the readout and actuation scheme. The most important sources of
disturbances, envisioned during the design and testing activity, and which could affect
either the test mass acceleration noise performances, as well as the residual couplings
between the GRS and the test mass, will be analysed in section 2.3.2.
The current LISA gravitational reference sensor design is based on the theoretical
and experimental studies performed at Trento University [24–26, 23, 27–30] and takes
advantage of the experience of the space qualified accelerometers developed by the
ONERA [31, 32]. It is however mandatory to recall that the goal of the GRS is to
minimise the disturbances to the proof mass, sacrificing its performances as a position
sensor.
The GRS creates a nearly closed cavity a round the proof mass, and its gold coated
surfaces serve as electrostatic shield. Thus the inertial sensor shields the test mass
from external surface force disturbances. It is then likely that most of the parasitic
forces that disturb the proof mass geodesic motion arise in the GRS itself. This has
important consequences in the design:
- The gaps between the proof masses and the surrounding electrodes are kept as
large as possible, compatible with achieving sufficient position sensitivity. Large
gaps readily suppress disturbances due to uncontrolled potentials and to the possibility of developing pressure gradients in the sensor housing. The forces connected to stray dc potentials on the metallic surfaces, known as «charge patches»
and described in section 2.3.2, decay with the gap width d with a functional form
20
that depends on the specific noise mechanism. Some of these effects may decay
as 1/d3 or even more rapidly [33]. Thus the GRS works with 4.0 mm gaps for
the x interferometric axis, sensitive to the gravitational signal, and 3.5 mm and
2.9 mm respectively for the y and z axes.
- The voltage Vtm used to sense the test mass motion is kept as low as possible. Voltages create stiffness much more rapidly than they increase sensitivity:
stiffness is proportional to h(Vtm )2 i while the position noise spectral density is
inversely proportional to h|Vtm |i.
- No mechanical contacts are allowed to the test mass. This require that an alternative solution must be found to manage the charge accumulating on the proof
mass, other than the µm size wires used in accelerometers, as they create both
stiffness and noise in the form of brownian forces.
- No dc voltages are allowed on the test mass and electrode surfaces, because these
couples to other sources of voltage noise or to stray charges, producing forces
noise. The electrostatic actuation needed by drag-free control loops have to be
applied via ac voltage carriers Vac that produce forces proportional to the square
of the potential.
- The mass value of the proof mass is chosen to be the largest possible, because
all forces except gravitational ones produce accelerations inversely proportional
to the mass. Many of the force disturbances are proportional to the test mass
surface, however for a cubic test mass of side length L and uniform density, the
surfaces scales as L2 , while the mass scales with the volume and then with L3 :
it is always advantageous to increase the mass. There are however engineering
limits to both mass and size of the proof mass.
- The achievement of high mass value in limited geometrical dimensions for the
proof masses requires a high density material. The core of the GRS is a cubic
test mass of side length 46 mm and weight 2 kg, made of a monophasic goldplatinum alloy, obtained by a rapid quenching technique. The alloy composition
and production procedure has been chosen in order to obtain a high density
material and to minimise the magnetic susceptibility down to 3 × 10−6 and the
residual magnetic moment below 0.02 µA m2 to minimise the magnetic force
noise [29, 34].
The position sensor is based on a capacitive readout and actuation scheme and it
is schematized in figure 2.5. The test mass is surrounded by an array of six pairs of
electrodes, each one entering in a capacitive-inductive resonant readout circuit. The
geometrical configuration provides the information of all six d.o.f. of the test mass
by linear combination of the six readout channels, and similarly permits electrostatic
actuation on the same d.o.f. The readout scheme is schematized in in figure 2.6. It
is based on a capacitive-inductive resonant transducer with capacitive and inductive
components chosen to be resonating at the frequency of ω0 = 2π × 100 kHz. The test
21
Figure 2.5: Scheme of the current capacitive sensor electrode configuration design. Sketch of
the test mass surrounded by the sensing and injection electrodes. Sketch of the electrode configuration respectively on the x y and z electrode housing faces. Holes for the interferometer
laser light beam, in x and y faces, and for the caging mechanism, in z faces, needed for holding
the test mass during the launch phase are visible. Reproduced from [30].
mass is polarised at ω0 by injecting a current through a set of six injection electrodes,
2+2 on the z faces plus 1+1 on the y faces, on which a ω0 voltage bias Vac is applied.
Figure 2.6: Scheme of the capacitive resonant bridge readout and actuation circuitry. For
simplicity only one channel is shown. A 100 kHz voltage bias Vac is applied through the
«injection electrodes» such that the test mass is biased to a rms voltage Vtm ' 0.6 V. The
100 kHz signal is also the reference for the phase sensitive detector at the output of the preamplifier. The test mass motion modulates the gap which the test mass forms with the sensing
electrodes C1 and C2 and modifies the resulting capacitances causing a current imbalance of
the bridge through the two inductances L1 and L2 . The signal is detected by the pre-amplifier
and read by the phase sensitive detector. The actuation force signals Vact are applied directly
to the electrodes through the transformers. Reproduced from [30].
Test mass motion changes the gap between the test mass and the opposing electrodes, modulating the difference of the two capacitances C1 and C2 formed by the test
mass and the electrode pair facing the test mass from opposite sides. These electrodes
make part of the same resonant bridge. The change in capacitance induces then a difference of the current flowing through the two inductance arm L1 and L2 of the bridge.
This is hence read as the current flowing through the final transformer by an amplifier
which is then extracted by a phase sensitive detector locked to injection signal. The
employment of the phase sensitive detector guarantee the rejection of electrical noise
22
at frequencies different from the injection frequency and the effective measurement of
the signal, despite the tiny variations in the voltage across the secondary coil of the
transformer.
Each side of the test mass is faced by a pair a electrodes. Those are connected
to form two inductive-capacitive bridges read by two independent readout channels
for each coordinate axis. The sum of the currents produced by the two channels
provides the translational displacement of the test mass with respect to the centre
of the electrode housing. The difference provides the test mass rotation. Voltages
Vact can applied to the electrodes at audio frequency across the transformer inputs:
this produces forces Fact ∝ h(Vact )2 i on the test mass and thus provides authority of
actuation.
The electrode configuration shown in figure 2.5, as well as the gap sensing readout
scheme, have been chosen among other analysed configurations in order to meet all
position sensing and stiffness requirements for the different d.o.f. of the test mass.
The almost symmetrical electrode configuration, in which no strongly preferential
axis is present, provides roughly equal sensitivities and stiffness along the different
translational d.o.f. and of the test mass. Furthermore, this configuration allows similar actuation authorities on all axes, and allow to limit cross-talking effects between
d.o.f. [23, 35].
Many of the force disturbances acting on the test mass originate from temperature
gradients across the sensor housing. High thermal conductivity of the GRS is then
desiderable in order to limit those gradients. The materials used for the realisation of
the GRS have then to fulfil those requirement: high thermal conductivity, mechanical
reliability to the stresses suffered during the launch phase, and requiring simple and
common machining procedures.
The first sensor prototypes has been realised as high thermal conductance composite structures of Molybdenum, for the electrical conducting parts, and Shapal, a high
thermal conductivity Al ceramic, for the electrical insulators. They were the base for
the previous experimental investigations [24–26, 23, 27–30].
Sapphire is instead used for the electrical insulators in the current flight model
baseline design. The main concern toward the employment of Shapal however comes
from recent investigations. It has been pointed out degradation of Shapal dielectric
properties under the exposure to UV light. Since UV light will be employed for active
proof mass charge control in both LISA and LTP, as described in section 2.2.5, this is
a major problem. Sapphire is preferred also for its more reliable dielectric properties,
especially the dielectric losses much smaller than Shapal, for the known resistance to
radiation, for its better machining properties, and for the higher level of purity and
cleanness.
Those materials were chosen because their thermal expansion and high thermal
conductivity properties are matched and the machining properties and feasibility are
already well investigated, and because they allow also to meet the mechanical tolerances issues, permitting a final tolerance of the whole assembly of order ' 10 µm
given the machining and assembling procedure [23, 36].
23
Guard-ring surfaces are introduced between the «sensing electrodes» in order to
avoid cross-talks between d.o.fėither for readout, stiffness and actuation or noisy
forces. Guard-ring surfaces have been added also between the injection and sensing electrodes in order to avoid a direct coupling of the sensing bias electrical field.
All sensing and injection electrode surfaces as well as the guard-rings and the test
mass are be gold coated, to allow electrostatic homogeneity and avoid the formation
of stray dielectric layers. Parts of exposed insulators, needed to electrically isolate the
electrodes from the sensor housing, are as small as possible and recessed such that
they do not face the test mass.
2.2.5 LISA pathfinder
Achieving the purity of free-fall requested for LISA is a challenging objective: the
level of isolation from stray disturbances requested has never been reached in dragfree flight experiments [37], and the best performance ever demonstrated in laboratory
is more than two orders of magnitude higher than LISA requirements [38]. Aiming
to achieve this demanding scientific and technological goal, with the purpose of risk
reduction for the LISA mission, two preliminary steps are planned: ground testing
activity, and a preliminary space experiment.
The ground testing has been proposed and is currently performed at the Trento
University by means of torsion pendulum experiments. It has the goal of validating
the functionality of the GRS as a position sensor, and to demonstrate the purity of
free-fall allowed by the GRS within two orders of magnitude from LISA.
The preliminary space experiment is the LISA Testflight Package - LTP - which is
currently in preparation and will fly aboard of the LISA Pathfinder space mission that
is scheduled to be launched in 2010 [39, 40]. LTP aims to demonstrate free-fall quality
within one order of magnitude from LISA, such that the residual acceleration noise of
the test masses is proven to be below
"
1/2
SδF
(ω ) ≤ 3 1 +
ω/2π
3 mHz
2 #
× 10−14 m s−2 Hz−1/2
(2.36)
in the frequency range between 1 mHz and 30 mHz, which is also relaxed by one
order of magnitude with respect to the LISA goal. We remark that the achievement of
free fall at this level would still be sufficient to allow the observation of gravitational
waves.
The basic idea of the LTP experiment is to shrink a 5 × 106 km LISA interferometer
arm down to a 30 cm arm-length interferometer, into a single spacecraft, and perform
an ideal geodesic deviation experiment on a pair of test masses, by measuring with
high precision laser interferometry their differential motion. The spacecraft serves
mainly for shielding the proof masses from external disturbances. It is equipped with
a set of precision µN thrusters, needed for compensating external forces, in a dragfree scheme similar to the one envisioned for LISA. In the baseline configuration, the
spacecraft follows one of the two test masses in its geodesic motion, accordingly to
24
the informations on the spacecraft-test mass relative position supplied by the GRS.
The second test mass is instead forced to follow the first, by means of the electrostatic
actuation, provided by the gravitational reference sensor. LTP will be equipped with
the most crucial aspects of the LISA technology [23, 34, 41, 42]:
- The spacecraft control system, based on Field Emission Propulsion thrusters [43].
LTP implements a flight formation of three orbiting bodies, namely the spacecraft
and the pair of proof masses. This formation is kept with sub-micron precision
by the drag-free control system. Position informations, supplied by the GRS
and by the interferometer, are used to control the electrostatic actuation and the
spacecraft thrusters, by 18 d.o.f dynamical control laws, without losing track of
the main objective of the mission: the proof masses must follow geodesics.
- The Gravitational Reference Sensor, which serves as a reference for the spacecraft
control system, and we already lengthly described.
- The Low Frequency Interferometer, which tracks the proof mass relative motion.
Precision interferometric tracking is routinely done at ground, at frequencies
higher than 100 Hz. The LISA and LTP requirements of picometer resolution,
over a large dynamic range of about 1 mm, low frequency stability, and space
operation, lead to the development of a new domain of instrumentation [44–46].
- The apparatus called Caging Mechanism, which holds the test masses in place
during the launch phase, and releases them for the scientific measurement phase.
The proof masses are quite heavy and can not be let to shake freely in the GRS
during the launch: they must be blocked with forces of a few kN. The challenge is
in release the proof masses with kinetic energy sufficiently small to be mastered
by forces of order of few µN, provided by the electrostatic actuation of the GRS.
- The apparatus called Charge Management System, which controls and remove the
net charge accumulated on the test mass due to the exposure to the cosmic ray
radiation [47]. Charge on the proof masses produces forces whose intensity is not
compatible with the quality of free fall requested. Also electrostatic potentials,
commonly used to manage charges, produce unwanted disturbances. The charge
management is then actuated just by means of UV light induced photoelectric
effect.
LTP represents an important verification of functionality and performances of each of
these subsystems and generally of the LISA drag-free control loop scheme.
The LTP experiment will not just place upper limits on the overall acceleration
noise on the proof masses, demonstrating the quality of free fall at unprecedented
level. It will extensively investigate each single possible source of acceleration disturbances, which has been envisioned during the design and ground testing activity.
This is with the objective to confirm the physical model of the stray forces acting on
the proof masses. This model will serve to determinate the transfer function between
individual environmental disturbances and acceleration noise.
2.3. GROUND TESTING OF THE GRS
25
During the scientific operation, disturbance sources are monitored and recorded.
Their expected acceleration noise contribution can be calculated, by using the measured transfer function, and subtracted from the recorded scientific data. If this procedure is applied correctly, the noise power spectral density of the post processed
data decreases, allowing for an increased sensitivity. In this way, taking into account the uncertainty on the model parameters, LTP may be able to put an upper limit on the unmodelled proof masses relative acceleration disturbances at δa '
7 × 10−15 m s−2 Hz−1/2 at 1 mHz, with some slightly improvements up to about 3 mHz
[42]. Above this frequency the laser readout noise is expected to limit the sensitivity,
while below 1 mHz noise degradation is expected to show a inverse square frequency
dependence, limited by the noise due to the electrical actuation.
It is worth to note that, born with the purpose of testing the feasibility of LISA,
the LISA pathfinder mission grown the relevance of an important and autonomous
experiment. The LTP scientific requirements pushed for the development of many new
innovative techniques in the field of precision metrology that will be demonstrated with
the LISA pathfinder mission. The proof masses on LTP will define the best ever local
Lorentz frame. The availability of such a frame will also make the LISA pathfinder
spacecraft the most perfect inertial orbiting laboratory available for fundamental physics
experiments. The technology of LISA pathfinder will open new ground for an entire
new generation of missions in general relativity and in fundamental physics.
2.3
Ground testing of the GRS
The gravitational reference sensors for LISA has to fulfil very stringent requirements:
it has to realise a high sensitivity position sensor, with displacement sensitivity, along
1/2
the interferometer gravitational wave sensitive axis, better than Sδx
' 1.8 nm Hz−1/2
down to 0.1 mHz. And it has to do so without exerting any parasitic force on the proof
1/2
mass, namely with related acceleration noise below Sδa
' 3 fm s−2 Hz−1/2 at 1 mHz.
The design of the GRS on the balance of those requisites has been carried out at
the Trento University in conjunction with the development and construction of a test
bench facility based on a torsion pendulum. This facility has been used for ground
testing activity, that has been crucial in the development of the GRS design that is now
the baseline for the model that will aboard of LISA pathfinder.
Accordingly to its ambitious initial goals, the ground testing has demonstrated
the possibility to achieve free fall better than two orders of magnitude from the LISA
acceleration noise goal. The torsion pendulum facility has been successfully used to
1/2
measure acceleration noise with sensitivity better than Sδa
' 300 fm s−2 Hz−1/2 in
the mHz region. Additionally it has been used to characterise individual sources of
disturbances by means of precision force measurements and to check for unpredicted
disturbance down to the better experimental sensitivity reached up to now.
It is useful to give here an overview of the force disturbances acting on the proof
masses. They are the object of the experimental investigation, but at the same time
the predominant ones constitute a limit in the sensitivity of the torsion pendulum, for
26
the investigation of smaller forces. Force noise sources and their physical models are
presented in section 2.3.2 and section 2.3.3.
2.3.1 The need to investigate force noise sources
The need for an on ground investigation of the envisioned disturbance sources arises
from the high degree of unpredictability of their properties. Investigation in a LISA
like environment, characterising their effects in a real GRS prototype, is necessary for
most of those sources.
For noise sources related to the read-out and actuation scheme, for instance back
action noise or sensing bias stiffness, the goal is the verification of the electrostatic
model of the sensor and of the readout circuitry noise model used to predict them.
Preliminary estimates of noisy disturbances related to stray dc voltages, are based on
experiments which are only partially representative of the LISA case [48]. Initial estimates and measurements of the force noise related to the coupling between proof
mass charging and stray dc voltages predict a related acceleration noise contribution
which could even overwhelm the LISA goal. For disturbance sources like the noisy related to the temperature gradient fluctuations, the existing models are just qualitative
in their predictions, and contain parameters whose values are not clearly predictable.
On ground measurement and characterisation would highlight deviations from the
model estimates and would serve to fix the model parameters. Moreover ground testing is necessary to possibly highlight any other unmodelled or unpredicted source of
disturbance.
Ground testing has also the goal of the development and verification of most of
the measurement techniques that will be used in LISA to characterise the experiment,
in order to achieve its full sensitivity and its scientific goal. Furthermore, it provides
real experimental data that closely resembles the ones that will obtained in flight.
Those are very useful both in the verification of the data analysis procedures, that
will be used to analyse the scientific data obtained by the mission, and in verification
and tuning of the spacecraft control system procedures, that will be embedded in the
onboard computer.
The current experimental effort has the goal to push the sensitivity limit of the
testing facility closer to the LISA requirements. It is with the aim of improving our
understanding of the origin of the force disturbances that act on the proof mass in
the GRS, and for highlighting any possible unmodelled disturbances that produce
accelerations below the current experimental sensitivity, but still important for the
achievement of LISA goals. We recall that the physical modelling of the force noise
acting on proof masses is important for the application of the disturbance subtraction
procedure described in section 2.2.5. Validation of this model will be one the final
objectives of LISA pathfinder mission.
27
2.3.2 Force noise sources arising in the GRS
Readout circuitry related noise
The capacitive readout scheme itself is responsible for the first unavoidable sources
of disturbance on the LISA test masses [24, 25]. The circuitry back action noise is of
difficult estimation for the readout scheme employed for the GRS capacitive sensor.
However it is expected to contribute a white acceleration noise with spectral density
nominally negligible. However an experimental investigation is needed to verify that
unmodelled effects are not present at significant levels.
The capacitive readout scheme contributes also from the point of view of the
sensor-test mass spring like couplings: the electric potential Vac used to polarise the
test mass for the position sensing induces a translational force gradient. The electrostatic coupling0 k e along the x axis can can be computed as:
∂2 Ci
h(Vm )2 i
ke = −
∑
4
∂x2
i
(2.37)
where Vm ' 0.6 V is the test mass rms voltage potential due to the 100 kHz sensing
bias envisioned for the GRS operation, and the sum extends over all sensor conducting
surfaces i facing the test mass, forming a capacitance Ci with respect to the test mass
itself. This coupling can be roughly estimated with a simple model where the test
mass and its surrounding electrodes are considered to for a bank of plane capacitors.
Neglecting the electrodes geometry details and the border effects, the capacitance for
each side i of the test mass is:
ε 0 L2
Ci =
(2.38)
di
where L = 46 mm is the side length of the cubic test mass, di is the distance between
each side of the test mass and its facing electrodes, and ε 0 is the permittivity of free
space. In this simplified model only the x faces of the test mass contribute to the
electrostatic coupling along the same axis, equation 2.37 reduces then to:
2
h(Vm )2 i ∂2 C1 ∂2 C2
h(Vm )2 i ∂2 Cx
2 ε0 L
ke = −
+
=
−
=
−h(
V
i
)
m
4
∂x2
∂x2
2
∂x2
x3
(2.39)
where x = 4 mm is the distance between the test mass and the x electrodes. This stiffness evaluates to k e ' −10−7 N m−1 and represents the main contribution to spring
like coupling between the gravitational reference sensor and the test mass and represents about half of the total electrostatic stiffness. This contribution needs to be
investigated to demonstrate the reliability of the electrostatic model of the sensor.
Stray electrostatics disturbances
Stray electrostatic in the GRS represents a threatening force noise source. Three main
phenomena contribute to produce electrostatic fluctuations that give rise to disturbances that can not be neglected:
28
- Charge is accumulated on the LISA proof masses because of the exposure to
cosmic rays. The accumulation of charge on the test mass due to the arrival of
cosmic radiation is a poissonian process. The net charging rate is estimated to
be λ ' 60 fundamental charges per second, while the total charge q on the test
mass is a stochastic variable with spectral density
ep
2λe
(2.40)
Sq1/2 (ω ) =
ω
where λe ' 260 is the charge arrival in elementary charges per second [49].
The accumulated charge q causes Coulomb forces due to its interaction with
the conducting sensor inner surfaces, and Lorentz forces for its motion in the
interplanetary magnetic field. These forces play a relevant role among the other
disturbances.
- Stray constant voltages δV are produced on sensor surfaces by the variation of
the surface potential due to different work functions of different metallic domains exposing different crystalline facets. For the gold coated surfaces of the
sensor, those domain have size of order a few µm, and their effect averages statistically, to produce voltages of order 1 mV [50]. Patch fields with larger coherence
lengths are caused by surface contamination or outgassing of the sensor materials [33]. Those can produce dc biases up to about 100 mV and with comparable
rms values [26].
- Voltage thermal noise δU originates in lossy dielectric layers and because of
electrons hopping among work function minima on the conducting surfaces.
An estimation of the force noise induces by these electrostatic disturbances can be
obtained in a simple model where we consider the test mass surrounded by i independent surfaces. The force acting on the test mass along the x axis can be computed
as:
1
∂Ci
Fx = ∑
(2.41)
(δVi + δUi − Vm )2
2 i ∂x
where Ci is the capacitance of the i surface with respect to the test mass, δVi and δUi
are respectively the averaged voltage fluctuations noise and dc patch fields for the
i surface, and Vm is the test mass electric potential due to the accumulated charge,
calculated as Vm = q/CT where CT is the test mass total capacitance to the grounded
sensor.
Without entering in the details of each single contribution, we can point out that
each term in equation 2.41 produces two distinct effects: force gradients, that contribute to the overall test mass to sensor stiffens, similarly to the case of the sensing
bias, and time dependent forces, that give rise to acceleration noise with spectral density related to the spectral density of the originating electric field. The most important
contribution to the stiffens is given by the test mass accumulated charge:
2
1
q
∂2 Ci
kq = −
(2.42)
∑
2 CT
∂x2
i
29
This contributions sets a stringent requirement to the maximum electrostatic charge
accumulated on the test masses. Modifications of overall stiffness above the 1% of
the total can spoil the stability of the control laws used for the drag-free control loop.
This sets the maximum charge allowed to the level of ' 106 elementary charges [51].
This upper limit to the test mass charge makes negligible the disturbances caused
by Lorentz forces due to the the interplanetary magnetic field, and the acceleration
noise due to Coulomb forces arising with the induced mirror charges in surrounding
electrodes. The need to constrain the level of net charge makes necessary to develop
a method to remove the excess charge accumulated by the test mass as described in
section 2.2.5. To avoid other electrostatic noise sources to overrun the LISA acceleration noise budget, all fluctuating voltages in the GRS can not have spectral density
exceeding 10 µV Hz−1/2 [52].
Thermal gradients related disturbances
Dominant factors in the force noise budget are forces generated by fluctuations of thermal gradients across the drag free sensor. It is possible to identify three main force
noise sources due to thermal gradients: the radiometer effect, the thermal radiation
pressure, and asymmetric outgassing. The three physical processes are analysed here
in a very simplified model, where the GRS is supposed at a temperature T and the
sensors inner walls undergo small temperature fluctuation δT T. Given the high
thermal conductivity of the constituting materials, the temperature is considered uniform on the sensor faces and, since the thermal conductance between the sensor and
the test mass is negligible, the test mass is assumed to be stable at the temperature T.
Radiometer effect. Radiometer effect arises from the momentum transfered to the
test mass by the gas molecules in the sensor housing. Molecules coming from higher
temperature zones transfer more momentum than molecules coming from lower temperature zones, producing a force on the test mass.
The pressure between two parallel plates at different but constant temperatures T1
and T2 , containing a gas in molecular regime, can be evaluated as the time derivative of the flux of momentum orthogonal to the plates, trough an imaginary surface
separating the plates. Accordingly to the gas kinetic theory, molecules of mass m at
temperature T have average velocity v and average momentum p given by:
r
p
3k B T
p = 3mk B T
(2.43)
v=
m
where k B is the Boltzmann constant. The number n of molecules crossing, in each
direction, the imaginary boundary between the surfaces, per unit time, per unit area,
is then computed as:
r
Z 2π Z π/2
n
N
1 N 3k B T
=
v cos θ dφ dθ =
(2.44)
dt ds
4V
m
φ =0 θ =0 V
The momentum of the molecules crossing the boundary in each direction is defined
by the temperature of the surface they are leaving. At the equilibrium, the number of
30
molecules crossing the boundary in the two directions must be equal. The pressure
between the parallel plates is then:
r
p
√ dF
d dp x
1 N 3k B T p
P p
=
=
3mk B T1 − 3mk B T2 = √
T1 − T2
(2.45)
ds
dt ds
6V
m
2 T
where we introduced P as the average pressure inside the inertial sensor, when it
is at the equilibrium temperature T. Supposing the test mass at the temperature T,
and opposite the sensor inner walls respectively at temperatures T1 = T + δT and
T2 = T − δT, we calculate the net force acting on the test mass, as linear function of
∆T = 2δT, performing a first order series expansion, obtaining:
√
√ √
AP
AP √
T + δT − T − T − δT + T '
∆T
Fre = √
2T
2 T
(2.46)
Thermal radiation pressure. The effect of thermal radiation pressure can be evaluated from the momentum transfer of thermal photons emitted by each radiating
surface inside the gravitational reference sensor.
We a simple model of infinite parallel plates, where all the surface have the same
emissivity, and the test mass is isothermal at the temperature, each surfaces at temperature T. Additionally, we assume that all the radiation is absorbed, after a sufficient
number of reflections on the surfaces. Each surface emits thermal radiation as described by the black body equation E = σT 4 . This determines a momentum transfer,
per unit area, that can be computed as:
dp
h̄ω
E
σT 4
=
= =
ds
c
c
c
(2.47)
The pressure between two parallel plates at uniform temperature T1 and T2 is then:
dF
d dp x
2σ 4
=
=
T1 − T24
ds
dt ds
3c
(2.48)
Supposing again the test mass at the temperature T, and opposite the sensor inner
walls respectively at temperatures T1 = T + δT and T2 = T − δT, it is possible to
calculate the net force on the test mass, as linear function of ∆T = 2δT, performing a
first order series expansion, obtaining:
Frp =
2σ 8σAT 3
( T + δT )4 − ( T − δT )4 '
∆T
3c
3c
(2.49)
Asymmetric outgassing. Outgassing of molecules absorbed onto the drag free sensor walls increases the residual pressure surrounding the test mass. Any asymmetry
in the in the molecular outflow produced by temperature gradients gives differential
pressure on different faces on the test mass producing a net force on it. The flow of
gas from surfaces can be described by mean of the temperature activation law:
Q = Q0 e−T0 /T
(2.50)
31
where Q0 is a flow pre factor, and T0 the activation temperature of the molecular
species under consideration. The gas flow will produce an overpressure on the test
mass P = Q/C, where C is the conductance between the outgassing surface and the
test mass. Assuming identical outgassing rate for all the surfaces, any temperature
gradient ∆T between two opposite walls of the drag free sensor at temperatures T1
and T2 would induce an asymmetric rate of outgassing. In turn it will generate a
differential pressure on the test mass that will produce a force that can be computes
as:
Q0 −T0 /T1
e
F = ∆PA = A
− e−T0 /T2
(2.51)
C
This can that can be expressed as linear function of the temperature difference ∆T
with a first order series expansion, obtaining:
Q0 T0 −T0 /T
APQ
Q0 −T0 /(T +δT )
e
− e−T0 /(T −δT ) ' A
e
∆T =
∆T
(2.52)
Fog = A
2
C
CT
CT 2
Total thermal gradient noise. All the thermal processes that contribute to the force
noise are driven by the same temperature gradient and are thus coherent noise source.
They must be added linearly to calculate the total effect. The total force acting on the
test mass induced by variations of the thermal gradients is then given by:
A P 8σT 4
T0 Q
Fth =
+
+
∆T
(2.53)
T 4
3c
TC
In the assumption to know the power spectral density ST (ω ) of the temperature fluctuations of the sensor housing, we can calculate the thermal noise power spectral
density:
2
A P 8σT 4
T0 Q
SδFth (ω ) =
+
+
S∆T (ω )
(2.54)
T 4
3c
TC
Some of the parameters that enter in this formula are well known: the area A of each
side of the test mass is 462 mm2 , the temperature inside the spacecraft during scientific
operation is T ' 300 K, the pressure inside the vacuum enclosure that contains the
GRS will be kept to P ' 10−5 Pa [6]. However the precise determination of the outgassing parameters is difficult because those depend strongly on the thermodynamic
history of the surfaces and on the chemical species adsorbed during the manufacturing
and handling.
From semi qualitative estimations for similar surfaces in similar conditions, it is
obtained that the outgassing activation temperature is T0 ' 104 K, and the average
outgassing rate is of order Q = 10−9 Pa m3 s−1 . The conductance C between the walls
of the housing and the test mass can be roughly estimated as the conductance of a
short tube with square section A and length d = 4 mm [53]:
r
2 2 8RT 1
C= L
' 0.12 m3 s−1
(2.55)
3
π M 2dL
1/2
The spacecraft is designed with the goal of S∆T
= 10−5 K Hz−1/2 for the thermal
stability inside the vacuum enclosure containing the test mass and the gravitational
32
reference sensor [6]. Substituting those numbers in equation 2.54 we obtain a total
1/2
thermal induced force noise with spectral density SδF
' 5 × 10−16 N Hz−1/2 .
th
We emphasise how the determination of the thermal gradient related disturbances
is not based on first principles but much more on phenomenological considerations.
Recent measurements of the thermal induced force noise conduced with the torsion
pendulum testing facility confirmed the qualitative model [54] but further testing is
necessary to fully understand the possible effects. In particular the data so far available
has been obtained with a setup that did not permit the direct measurement of forces
but only their determination from torque measurements via a suitable conversion arm
length inferred from the physical model.
2.3.3
Other disturbance sources
Residual gas damping
Residual gas produces brownian noise on the test masses, exerting a viscous drag,
modelled with a force proportional to the test mass velocity: F̄ = − βv̄. This produces
damping of the proof mass motion in a time τ that can be estimated to be:
m
m
=
τ=
β
PA
s
kB T
mg
(2.56)
where m is the proof mass, P is the residual gas pressure, A is the test mast side area,
k B is the Boltzmann constant, T is the gas temperature, and m g the average molecular
weight of the gas. With the envisioned pressure of operation of order P ' 10−5 Pa the
related acceleration noise is currently estimated to be [55]:
r
1/2
Sδa
=
4k B T
' 0.6 fm s−2 Hz1/2
mτ
(2.57)
Cross-talks
The coupling of residual satellite motion along the non-measurement translational
and rotational degrees of freedom into acceleration along the critical x axis is an important and complicated noise source [34]. The current estimates apportion a related
frequency independent acceleration noise of order 0.8 fm s−2 Hz−1/2 . However also in
this case an experimental investigation is needed to verify that unmodelled effects are
not present at significant levels.
Magnetic noise
The remnant magnetic moment µ̄ and magnetic susceptibility χ of the test mass couples to the fluctuations of magnetic field and magnetic gradient fields to produce force
noise. In the limit of weakly magnetic materials, the component of the force acting on
the mass along the LISA interferometer axis x can be, in first approximation, expressed
33
as [56]:
f x (t) = µ̄ ·
∂ B̄ χV
∂ B̄
+
B̄ ·
∂x
µ0
∂x
(2.58)
where B̄ is the magnetic field, and V is the test mass volume, and µ0 is the magnetic
permeability of free space. Similar relations hold for f y (t) and f z (t). Here we describe
the test mass magnetic properties by its permanent, remnant magnetic dipole moment
µ and its magnetisation induced through the small magnetic susceptibility χ by the
externally applied magnetic field B̄. Fluctuations of the magnetic field are expected to
be dominated by the interplanetary magnetic field, while fluctuations of the magnetic
field gradient are expected to be produced by sources on the satellite itself.
With a remnant ferromagnetic moment µ below 0.02 µA m2 and a susceptibility χ
below 3 × 10−6 for the gold-platinum proof mass, this effect is estimated to contribute
to the overall acceleration noise at the level of 0.4 fm s−2 Hz−1/2 [23, 34].
34
Chapter 3
Small force measurement with a
torsion pendulum
The possibility to realise the high isolation from force disturbances, required for the
LISA test masses to achieve geodetic motion, needs experimental investigation. This
is the motivation of the development and realisation of a ground based test bench for
small force measurements. The realisation of this test bench has the aim to conduct
an experimental campaign focused on characterisation of the disturbances exerted in
the mHz and sub mHz frequency region on the proof masses by the gravitational
reference sensors under design.
Several experiments have operated torsion pendulums at the thermal noise limit,
obtaining torque sensitivities of order few 10−15 N m Hz−1/2 in the mHz frequency
region [38, 57–59]. On the base of such successes the test bench facility envisioned for
the investigation of the free-fall of LISA proof masses consists of a torsion pendulum.
The experimental setup is rather simple: a representative copy of the LISA test
mass is suspended by a thin torsion fibre and hangs inside a prototype of the gravitational reference sensor. Proper choice of the pendulum geometry and materials
ensures that the suspended test mass is nearly free along the torsional degree of freedom. The natural alignment of the suspension fibre along the vertical orthogonalises
the rotational mode of the pendulum, with respect to the terrestrial gravitational field.
The test mass rotational d.o.f is thus «free» and allows for high precision measurement
of differential forces induced on the suspended test mass along the horizontal plane.
The high force isolation along the pendulum torsional d.o.f. allows also significant
validation of the techniques that will be employed in important measurements during
the LTP test flight and in the LISA mission. The goal of such measurements will be the
characterisation of the gravitational reference sensor performances during operation
and the identification of acceleration disturbances. Through their physical model the
effect of those disturbances can then be correlated and eventually subtracted from the
scientific data as we described in section 2.2.5. The results of those measurements are
crucial to allow the experiments to reach their full sensitivity [34, 60].
35
36
3.1
CHAPTER 3. SMALL FORCE MEASUREMENT WITH A TORSION PENDULUM
Torsion pendulum operation principles
The basic idea of the torsion pendulum test bench is that any torque N (ω ) acting at a
frequency ω on the suspended test mass can be detected as deflection of the pendulum
angular rotation φ(ω ) through the transfer function of the pendulum:
N (ω ) =
φ(ω )
H (ω )
(3.1)
The pendulum equation of motion for the torsional d.o.f is φ̈I + Γφ + iδΓφ = N
where I is the pendulum inertial moment, Γ is the fibre torsional elastic constant, and
δ is the loss angle of the spring constant. The loss angle δ is assumed to be frequency
independent for damping due to intrinsic fibre dissipations [38, 61]. Translating this
equation in the frequency domain, it is easy to compute the torsion pendulum transfer
function, obtaining:
−1
H (ω ) = Γ 1 − (ω/ω0 )2 + i/Q
(3.2)
where ω0 = Γ/I is the pendulum resonance square, and Q = 1/δ defines the pendulum mechanical quality factor. The measured torque N (ω ) can be converted into a
force F (ω ) by means of a suitable conversion arm length:
F (ω ) =
N (ω )
Rφ
(3.3)
where Rφ depends on the source itself and is of order half of the test mass width. In
order to obtain a torsion pendulum sensible to very weak forces, the torsional elastic
constant must be minimised. This indeed produces a pendulum with a oscillation
frequency in the few mHz frequency region.
The sensitivity of the pendulum as a meter of the torque exerted on the test mass
is limited by the true pendulum torque noise S N (ω ) and by the angular read out noise
Sφro (ω ). The torque measured by means of the torsion pendulum has power spectral
density that is given by:
Sφ ( ω )
S Nm (ω ) = S N (ω ) + ro
(3.4)
| H (ω )|2
This sets upper limits on the low frequency stray forces δF exerted by the gravitational
sensor on the suspended test mass. This upper limit to the force noise power spectral
density SδF (ω ) is computed as:
SδF (ω ) =
S N (ω )
R2φ
(3.5)
still making use of a suitable conversion arm length Rφ depending on the specific class
of the noise source [62–64].
The limit set by equation 3.4 to the torsion pendulum sensitivity can be overcome
when two independent readout of the angular position are available. In this case it is
possible to apply the cross correlation technique explained in section 3.4.3 to effectively
3.1. TORSION PENDULUM OPERATION PRINCIPLES
37
reject the correlated noise of the independent readouts. In this way the ultimate torque
sensitivity is limited by the the pendulum torque noise S N (ω ). The individual readout
noise levels will however still determine the uncertainty on the estimated value of the
pendulum torque noise.
The maximum torque sensitivity of the pendulum is reached when it is limited by
the mechanical thermal noise, that can be determined via the fluctuation-dissipation
theorem [61]:
Γ
(3.6)
S Nth (ω ) = 4k B T
ωQ
where k B is the Boltzmann constant, and T is the temperature. The intrinsic thermal
noise can be made as small as possible enhancing the mechanical quality factor Q
of the pendulum. This enhancement is realized reducing the energy dissipations.
Therefore the pendulum is operated in high vacuum, and the test mass is suspended
from fibres with intrinsically low mechanical losses.
The torque sensitivity can also be maximised minimising the torsion constant Γ of
the torsional fibre. In the case of round fibres the torsion constant is computed as:
Γ=E
πr4
2L
(3.7)
where E is the elastic modulus of the fibre material, L and r are the fibre length and
radius, and m is the total mass of the pendulum. While thinner fibres lower the spring
constant it must be thick enough to sustain the weight m of the pendulum. The radius
r of the fibre must be so that:
r
mg
r≥
(3.8)
Yπ
where Y is the high yield strength of the fibre material.
Equation 3.4 states also that for a finite sensitivity of the torsion pendulum angular
position readout Sφro (ω ) a better force sensitivity is achieved with a torsion fibre with
low torsional elastic constant. A small elastic constant minimises the modulus of
the pendulum transfer function and maximises the angular deflection produced by a
certain torque. The bigger angular deflection helps to overcome the readout noise.
The high torque sensitivity torsion pendulum bench can be used also to investigate
single effects generated by the interaction of the sensor with the suspended test mass
[29, 63]. Measurements of individual noise sources are performed by modulating
the disturbance source itself. Searching for a signal coherent with the modulation of
the source of the disturbance in the measured pendulum angular position it is then
possible to distinguish tiny disturbances from the pendulum intrinsic noise. This allow
to resolve differential forces down to
r
2 SδF (ωm )
δF =
(3.9)
T
where ωm is the modulation frequency, and T the observation time. An example of this
kind of investigation is the measurement of thermal gradients induced disturbances,
38
where it is the temperature of the gravitational reference sensor surfaces to be modulated. This technique is also applicable to disturbances whose source is not directly
under control. For those it is possible to modulate the quantity to which they couple.
For example we saw in section 2.3.2 that the stray dc electric potential fluctuations
on the inner surfaces couple to the test mass potential to produce force disturbances.
Those are indeed measured modulating the test mass potential through the injection
electrodes and searching for coherent deflection of the test mass angular position.
Similar measurements permit also to identify the stray interactions of the torsion
pendulum with the surrounding environment. Investigation of those disturbances is
important to distinguish them from disturbances attributed to interactions of the proof
mass with the sensor under test. For example the effects of external magnetic fields
are investigated modulating an oscillating magnetic field B̄(t) by forcing an oscillating
current to flow through a coil placed in the proximity of the torsion pendulum. In first
approximation this would produce a torque
Nm (t) = [µ̄ × B̄(t)] · n̄φ
(3.10)
where µ̄ is the time independent magnetic moment of the test mass and n̄φ is the
torsion pendulum axis. In turn this will produce a coherent deflection of the pendulum
inertial member.
With this procedure it is indeed possible to infer the transfer function that translates the different disturbance sources into external torques applied to the pendulum.
With the knowledge of those transfer functions is then possible to subtract the known
disturbances from the acquired angular position time series as will be detailed in section 3.4.5. This is very useful for removing the disturbances that are not related to
effects arising in the GRS and that will not effect the test mass in flight. Examples of
well understood disturbances that are in this way routinely subtracted are the magnetic forces and those due to the tilt of the whole experimental facility.
The external torque noise excess S Ne (ω ) that we aim to measure is then represented by the fraction of measured external torque that can not be attributed to known
disturbance sources:
S Ne (ω ) = S Ñ (ω ) − S Nth (ω )
(3.11)
where S Nth (ω ) is the torsion pendulum mechanical thermal noise, and S Ñ (ω ) is the
power spectral density of true external torque acting on the torsion pendulum. It can
be computed as:
Ñ (t) = N (t) − ∑
n
Z +∞
−∞
xn (t) hn (t)
(3.12)
where xn (t) are the measured known disturbance sources and hn (t) are the transfer
function of each disturbance. In this relation we are assuming that the noise on the
measure of each disturbance is negligible and that each disturbance can be converted
into applied torque with a simple frequency independent transfer function.
As already pointed out a similar this procedure will be also used for the analysis
of LISA and LTP data. The work with the torsion pendulum is then important not
3.2. FIRST GENERATION TORSION PENDULUM FACILITY
39
only for the investigation of the disturbance sources but also for the development of
suitable measurement techniques to be applied in flight.
3.2
First generation torsion pendulum facility
The LISA working team at the Trento University has been the first to extensively test
the quality of free-fall that can be achieved in a ground based experiment. The main
goal of the experimental investigation is to characterise of the gravitational reference
sensor developed for LISA and to prepare for the LTP space experiment. However the
achieved sensitivity to small forces can be applied to other kind of experiments.
The first generation torsion pendulum employed for investigation of free-fall uses a
simple design where a test mass hangs from a torsion fibre attached to its centre. It
takes then the nickname of «single mass» facility.
Suspending the 2 kg test mass envisioned for LISA and that will fly onboard of
LTP would require with a very thick torsion fibre. Such a thick torsion fibre would
determine a very high torsional elastic constant and thus a very low force sensitivity.
Therefore a lightweight replica of the LISA proof masses is used to construct the inertial member of the torsion pendulum [62]. This allows to employ fibres with a smaller
radius and thus torsionally softer, that permit to achieve a better force sensitivity.
The most dangerous effects for the sensor under development are likely the ones
originating from surface interactions. Changing the geometric dimensions of the the
test mass from those of the LISA proof masses, would therefore spoil the representativeness of the test. It has then been chosen to use an hollow test mass whose surface
properties are made as similar as possible to the one envisioned for the LISA proof
masses. This pendulum configuration maximises the sensitivity to surface effects, but
it is rather insensitive to force disturbances originating in the bulk properties of the test
mass, such as magnetic and gravitational forces. Anyhow those are in principle independent from the position sensor itself, and are therefore investigated independently,
with dedicated test benches and analysis [29, 56, 65].
The proper torsion pendulum is composed by a test mass suspended to a thin
torsion fibre via a supporting shaft. The test mass is a gold coated 46 mm aluminium
hollow cube, realised welding together six 2.5 mm thick aluminium plates, for a overall
weight of about 75 g. The machining tolerances guarantee a parallelism of opposite
faces within 10 µm. The test mass is electrically insulated from the rest of the inertial
member by means of a fused quartz ring. Shape and composition of the insulator
are chosen in order to minimise the stray capacitance of the test mass to ground and
to provide a very high electrical resistance. The combination of these two features
ensures a long discharge time for the test mass, that has been measured to be higher
than 106 s. This is to obtain conditions similar to the operative ones of the gravitational
reference sensor.
The supporting shaft is also made of gold coated aluminium. It carries a gold
coated glass mirror, used for the independent optical readout of the pendulum angular
position. The mirror lays on a cross shaped support carrying four adjustment screws,
40
used to minimise the gravitational quadruple moment of the inertial member of the
pendulum, to avoid coupling to gravitational disturbances. The cross shaped support
serves also as safety device to prevent damages that can be caused by the test mass
hitting the inner walls of the GRS. When the test mass moves to much off centre of the
sensor, the cross shaped support hits carefully positioned stoppers on the upper part
of gravitational reference sensor. The shaft and the gold coated mirror are grounded
through the conducting torsion fibre.
For the torsion fibre, the best compromise between small torsional spring constant,
high mechanical quality factor, and high yield strength has been found in tungsten,
which has been largely employed in similar experiments [38, 57–59]. We use a commercial bare Tungsten wire, which has circular section, diameter of 25 µm, and length
of about 1 m. The fibre is attached to the supporting shaft by means of high electrical
conductivity glue. Due to the pendulum weight the fibre is loaded to roughly 60%
of its yield strength and experiences a total elongation of nearly 10 mm. For the low
frequency torsion pendulum measurements it is important the characteristic helicity
of the tungsten fibres. It produces a steady unwinding of the fibre during time, that
decays exponentially with time, and that is strongly temperature dependent. Due to
the relaxation of the inner structure of the fibre, every time the fibre is unloaded, by
sitting the pendulum on the supporting structure, the drift rate for the next suspension
jumps back to the a value similar to the initial.
The obtained inertial member has an overall weight of about 100 g and moment of
inertia I = 4.3 × 10−5 kg m2 . The pendulum typical oscillation period is found to be
about 560 s without any electrical field applied to the electrodes. Purposely applied
or parasitic electrical fields can increase this number by introducing a corresponding
negative stiffness. The quality factor of the oscillator is Q = 2900 ± 500 with an energy
decay time of order 2.3 × 105 s.
Some different gravitational reference sensor prototypes has been integrated in the
torsion pendulum testing facility during the development phase. We report here on the
last scientific run with an engineering model replica of the GRS that will fly onboard
of LTP. This prototype is very similar to the final gravitational reference sensor design
for the LTP experiment. The difference from the flight model relevant for the force
noise characterisation is that it employes Shapal instead of Sapphire for the eclectic
insulators. Also the UV light optical fibre seats will be in a slightly different position
in flight, influencing the operation of the charge management system.
3.2.1 Pendulum position readout
The pendulum swing and twist modes are monitored by two independent readouts:
the gravitational reference sensor under investigations, and an optical readout based
on a commercial two axis autocollimator «Möller-Wedel ELCOMAT vario 300/D40».
The GRS can provide informations on all six degrees of freedom of the test mass
with about the same sensitivity. However the design is optimised to give best performance and less force disturbances along what would be in LISA the interferometric
axis sensible to the gravitational wave signals. This is referred in the text as the x axis.
41
The GRS prototype is then integrated in the torsion pendulum facility so that the x
axis provides the torsion pendulum angular position φ, important for the measurement of the torque acting on the pendulum, and then the determination of th force
noise excess.
In the last scientific run of the torsion pendulum testing facility, the readout of the
capacitive sensor signals was performed with a bread board version of the Front End
Electronics - FEE - designed for LTP and constructed by the Swiss Federal Institute of
Technology in Zurich - ETHZ. The FEE implements the read out scheme illustrated in
section 2.2.4 based on a inductive-capacitive resonant bridge and the employment of
a phase sensitive detector. It provides as well the injections voltage needed to polarise
the test mass and the actuation voltages that can be applied to the electrodes to exert
forces and torques on the test mass.
The torsion pendulum angular position sensitivity achieved by the capacitive sensor is limited by the intrinsic thermal noise of the resonant bridge circuitry. With the
test mass polarisation voltage Vac ' 0.6 V and the capacitive and inductive components of the bridge chosen to have resonance at about 100 kHz, the electronic thermal
noise is about 100 µV Hz−1/2 . This translates into translational sensitivity of roughly
2 nm Hz−1/2 and a rotational sensitivity of roughly 200 nrad Hz−1/2 .
The independent optical readout monitors the torsion pendulum twist mode φ and
the swing mode along the x axis η. The commercial autocollimator has a bit resolution
of 25 nrad and a full scale range of about 5.5 mrad. However its noise power spectral
density has never been demonstrated in operative conditions below 100 nrad Hz−1/2
in our measurement frequency range. The independent optical readout is also used
for calibration of the capacitive readout.
3.2.2 Experimental facility details
Detailed descriptions of this apparatus have been already given [29, 62, 63]. We recall
here the most important details.
The design requirements for the torsion pendulum experimental setup are long
time mechanical and thermal stability, magnetic cleanliness, and high vacuum operation. The main structure of facility is thus composed of stainless steel vacuum components: a cylindrical vacuum chamber of about 50 litres capacity and 40 cm radius
accommodates the gravitational reference prototypes surrounding the test mass, a 80
cm long and 6 cm thick vacuum tube is mounted on the top of the vacuum vessel and
encloses the pendulum torsion fibre.
The whole apparatus is placed on a concrete slab that is not rigidly connected to
the rest of the laboratory floor. This gives some degree of insulation from seismic
noise produced by human activity in the surrounding of the experiment. The whole
apparatus sits on three legs whose height can be adjusted to align the experiment on
the horizontal plane. A description of the main aspect of the experimental setup is
given in the following few pages.
Gravitational reference sensor integration. The GRS is mounted on a five d.o.f. man-
42
ual micro-manipulator which provides the two relative translations on the horizontal
plane and all three rotations. It sits on four Macor legs that provide electrical and
thermal isolation from the vacuum vessel itself.
Alignment system. The torsion pendulum hangs from a two d.o.f micro-manipulator
mounted on top of the vacuum tube which provides vertical and rotational alignment.
With the micro-manipulator that supports the GRS this constitutes a positioning system providing a total of six d.o.f. plus one for redundancy. It allows to centre the
sensor about the suspended test mass within few µm in translation and few tens of
µrad in rotation. Additionally the sensor prototype sits on a motorised rotational
stage which has a resolution better than 1µrad and is mounted on the five d.o.f. manual micro-manipulator. The motor allows coherent modulation of the sensor-test mass
relative rotation angle, needed for the characterisation of the overall spring-like couplings, and is also used for fine alignment of the sensor-test mass relative rotation
angle, needed for the electrostatic characterisation of the position sensor itself.
Isolation from seismic disturbances. The 25 µm thick and 1 m long torsion fibre
hangs from an upper pendulum stage consisting in a shorter and thicker Tungsten
fibre. To the 100 µm thick and 15 cm long fibre is attached an aluminium disk. The disk
is surrounded by toroidal rare earth magnets whose magnetic field gradient provides
eddy current viscous damping of the pendulum horizontal motion due to swing or
violin modes. If the pendulum swing losses were produced only by the fibre internal
friction with typical mechanical loss angle φ ' 10−4 − 10−3 the decay time would
be of order of few hours [61]. In this way the pendulum swing mode decay time is
reduced down to about 70 s without affecting the torsional mode performances by the
cylindrical symmetry of the design. Additionally it acts as a pre-hanger that ensures
that the torsion fibre always hangs vertical. From the torsional point of view, the upper
pendulum is about a factor 2500 more stiff than the lower torsion fibre itself and thus
does not influence the overall torsional elastic constant of the pendulum.
Vacuum. The residual pressure inside the vacuum vessel is kept to about 3 × 10−8 mbar
by means of a turbo-molecular pump directly mounted onto the vacuum vessel itself.
The backing pump is much more vibrationally dangerous. It is than isolated from
the experiment being mounted on a different laboratory floor platform. Additionally
the vacuum tube that connects the backing pump to the vacuum chamber is passed
through a vibration damping system realised with a sand box and a big lead weight.
Temperature control. The entire apparatus is enclosed inside a thermally isolated
room made out of 4 cm thick foam panels connected by a wooden frame. The vacuum
chamber sits on a thick metallic plate raised from the laboratory floor by mean of
three legs. That offers thermal inertia to the experiment and mechanical stability.
The three legs offer a very high impedance thermal link with the laboratory floor.
The torsion fibre tube is covered with an additional layer of thermal shielding. The
temperature inside the room is actively controlled by means of a closed loop control
by means of a water bath that stabilises the circulating air. A thermal stability more
than one order of magnitude better than daily laboratory temperature fluctuations is
43
achieved: the resulting long term stability inside the thermal room is about 50 mKpp,
whereas inside the vacuum the temperature daily fluctuations are reduced down to
about 10 mKpp. The system is put through low temperature bake-out cycles where
the vacuum chamber internals reach about 340 K. This serves mainly to drastically
reduce the long term unwinding of the fibre from about 1 mrad per hour to less than
1 µrad per hour.
Environmental disturbances monitoring. Several environmental variables that could
affect the pendulum performances are continuously monitored. The temperature of
the experiment is measured by several PtAu100 thermometers placed in key points of
the apparatus. Those are read by s digital multimeter obtaining a measurement resolution about 2 mK Hz−1/2 . The magnetic field is monitored by a three-axis flux-gate
magnetometer with 10 nT resolution placed in the proximity of the pendulum. The
capacitive sensor itself is used for monitoring the low frequency tilt of the apparatus
through the equilibrium position of the simple pendulum motion along the two d.o.f.
η and θ.
Charge management system. In section 2.3.2 it has been shown how charge on the
test mass couples with stray electric fields to produce force disturbances. During the
ground testing the charging rate of the test mass is not so hough as in space environment conditions, because of the much reduced intensity of the cosmic radiation.
However an important source of negative charge in the testing facility is the ion gauge
used to measure the pressure inside the vacuum chamber. The pendulum test mass is
electrically insulated and it is then necessary to provide a method to remove the charge
that accumulates on it. The same discharge mechanism briefly described in discharge
system as described in section 2.2.5 is employed in the ground testing facility. A simplified version of the flight model of such system, composed by UV lamps, optical
fibres and vacuum feed-throughs has been provided by Imperial College of London
[47]. Its integration in the torsion pendulum facility has also the purpose of testing
the functionality of the charge control scheme under development for LISA and LTP.
Electrostatic actuation. The front end electronic bread board provided by ETHZ provides also the electrostatic actuation circuitry. Following the LISA actuation scheme
[34] it can apply both audio frequency and dc voltages directly to the sensing electrodes to exert forces on the test mass. During space operation of the GRS the actuation is used mainly for forcing the proof masses to follow the spacecraft on the non
interferometric axes. In the ground testing it is used to control the dynamics of the
torsion pendulum. The actuation circuitry scheme has been designed to minimise its
impact on the GRS performance. It should not not contribute with excess noise to the
sensor displacement sensitivity, and it should not exert any disturbance on the d.o.f.
that are not actuated. This is one of the important test performed with the torsion
pendulum facility.
The actuation voltages produced by the FEE are controlled by a 16 bit PCI-DAC.
The dc voltages are mainly used for the characterisation of the position sensor electrostatics. The audio frequency voltages are often used to apply torques an forces along
all the test mass degree of freedom. The possibility of actuation is exploited in the
44
implementation of PID control of the pendulum torsional and translational mode. The
PID control has a torque resolution of about 2 aN m and a dynamical range of about
±2 nN m and allows a controllable range of angle of order several degrees.
This torsion pendulum has dynamics with very long characteristic times. The possibility of actuation on the test mass via the feed back control loop permits to reduce
drastically the time needed to bring the torsion pendulum in equilibrium positions.
This is important for the measurement phase of the experiment and fundamental during the first phase of the setup.
Data Acquisition. The pendulum position measured by the capacitive and optical
readouts, as well as the monitored possible sources of environmental noise, are continuously recorded by a dedicated data acquisition. It is based based on two integrated
workstation system and has been developed as a set of «Labview» routines that also
perform data visualisation and storage. The system implements also the PID control
on the pendulum angular and translational positions by controlling the actuation voltages in a 10 Hz quasi real-time control loop. The PID can be driven either by the
pendulum position read by the capacitive sensor or by the autocollimator.
The capacitive sensor and the magnetometer data are acquired by 16 bit PCI-ADC
boards installed on the workstations. The temperature are read by digital multimeter,
and then acquired through a GPIB interface. The autocollimator data, the motorised
stage encoder, and the pressure data are read via workstation serial ports. The sensing
and actuation voltages produced by the FEE and applied to the sensing and injection
electrodes are commanded via an analog interface by means of two dedicated 16 bit
PCI-DAC boards.
The high level of integration and automation of the data acquisition and control
system guarantees a very high duty cycle for this ground testing facility. This is very
important considered the very long measurement time necessary to acquire the data
necessary to cover the frequency range subject of the tests performed with the torsion
pendulum facility.
3.3
Second generation torsion pendulum facility
The «single mass» torsion pendulum has proven to be very effective in testing and
characterising the gravitational reference sensor design and the noise sources that
may arise inside it. However it is intrinsically limited to the measurement of only
a rotational d.o.f. of the test mass. This only permits to measure torques applied to
the test mass, that can be translated into forces by mean of a suitable conversion arm
length. Disturbances acting symmetrically in the respect of the torsional d.o.f of the
pendulum can not be detected, and the effect of disturbances on test mass translational
motion is only inferred through a physical model. It is a rather obvious concern that
there could be uncertainties with this physical model and with the determination of
the conversion arm length, and that some of the disturbances may have no effects on
the rotational d.o.f. for example those acting orthogonally in the centre of one face or
those that are symmetrical with respect to a plane containing the fibre.
3.3. SECOND GENERATION TORSION PENDULUM FACILITY
45
The original idea of a torsion pendulum test bench where the test mass from a
cross shaped inertial member [66] has then been recently reconsidered: a torsion pendulum where the test mass is suspended off centre with respect of the fibre axis using
a relatively long arm has been developed. It makes possible to directly measure forces
instead of just torques. This is the second generation torsion pendulum for the investigation of free-fall and takes the nickname of «four mass» facility.
Being able to directly measure forces acting on the test mass allows a more reliable
and model independent estimation of the contribution of the different sources to the
total force noise, and an evaluation of other possible sources of force noise that do
not translate into torques relative to the test mass axis. With the test mass suspended
off-centre with respect to the fibre axis, the translational forces acting in the direction
orthogonal to the arm that connects the test with the fibre translates into torques that
the torsion pendulum can measure.
Suspending the test mass off centre with respect to the torsion fibre requires at
least an extra mass on the opposite side of the suspension point to balance the pendulum. However the quadruple moment of such a system is proportional to the square
of distance between the masses. This configuration would have a large gravitational
quadruple moment that would couple the pendulum to the gravitational noise of the
laboratory. To reduce the gravitational quadruple it has been then chosen a symmetrical configuration with four identical test masses installed at the ends of the arms of a
cross-shaped central support suspended by its centre. Similar balancing problems are
not present in a single mass pendulum even with complex shape: in such geometry
the arm length is to short to produce moments of noticeable effects.
There is very limited possibility in manufacturing pendulum test masses lighter
than the one used in the the first generation torsion pendulum facility. The inertial
member of the second generation torsion pendulum is then almost exactly four times
heavier than the single mass pendulum. From equation 3.8 we see that the fibre load
limit is proportional to the square of the fibre radius and then that the fibre necessary
to support a four times heavier pendulum is twice as thick. From equation 3.7 we
see also that the torsional spring constant is proportional to the fourth power of the
radius. Such a fibre would then have a 16 times larger stiffness. Supposing to operate
the pendulum with a fibre that gives the same mechanical quality factor Q as the one
obtained for the single mass pendulum the four mass torsion pendulum will suffer
of a mechanical thermal noise with power spectral density 16 times bigger than the
single mass torsion pendulum as for equation 3.6.
However the increase of the mechanical thermal noise is compensated by the increased force sensitivity due to the longer arm length. The typical effective arm length
estimated for the single mass pendulum depends on the under investigation and on
how it is modelled range and ranges from about 1 to 2 cm [62–64]. The 10 cm arm
length of the four mass pendulum then gives roughly an overall factor 2 better force
sensitivity depending on the effect under investigation.
The four mass torsion pendulum facility is entering just now the scientific phase.
Since it shares the same structural design of the single mass torsion pendulum all the
46
improvement activity described in this thesis have been thought and implemented on
the single mass pendulum. They can be easily introduces to the four mass pendulum
as well.
3.4
Force noise excess data analysis
In this section we report on the data analysis techniques employed to extract the maximum amount of informations from the data obtain from the torsion pendulum gravitational reference sensor testing. The data analysis procedures have been implemented
as «matlab» routines.
3.4.1 Spectral estimation
Our data analysis procedure for power spectrum estimation is standardised on the
well known Welch averaged periodogram method [67]. The Cross Spectral Density CSD - between two processes x (t) and y(t) is defined as:
Sxy (ω ) = 2
Z +∞
−∞
h x (t) y(t + δt)i e−iωδt dδt
(3.13)
where the angle brackets indicate the ensemble average of the quantity between them,
and the factor 2 is introduced to comply with the convention of using mono lateral
quantities. The Power Spectral Density - PSD - of a process x (t) is then defined as a
special case of equation 3.13:
Sx (ω ) = Sxx (ω ) = 2
Z +∞
−∞
h x (t) x (t + δt)i e−iωδt dδt
(3.14)
Time domain data strings x [m] and y[m] are obtained by sampling and digitisation
of the signals x (t) and y(t) after the application of proper anti aliasing filters. Those
data string are then properly detrended: a straight line x [m] = a + b m is fit to the
data and subtracted from them. From the data string x [m] we then compute the
periodogram x̃ (k ) by discrete Fourier transform:
N
x̃ (k ) =
∑
2π
w[m] x [m] e−i N km
(3.15)
m =1
where N is the number of points in the data string, and the time function w[m] is a
windowing function, necessary to remove the artefacts introduced by the unavoidable
truncation of the data string. We assume that this function describes a normalised
window:
1 N
w [ m ]2 = 1
(3.16)
N m∑
=1
The periodogram ỹ(k ) is similarly computed from the y[m] data string. The cross
spectral density is then estimated from the periodograms x̃ (k ) and ỹ(k ) as:
2πk
Sxy ω =
' Pxy (k) = 2∆T x̃ (k) ỹ(k)∗
(3.17)
NT
3.4. FORCE NOISE EXCESS DATA ANALYSIS
47
where ∆T is the inverse of the sampling frequency. For x (t) and y(t) stationary normal
processes the relative uncertainty on the estimation of the cross spectral density Sxy (ω )
accordingly to this procedure is one. This holds in good approximation also for other
signals slightly deviating from the assumption of stationary normal processes. To
reduce the uncertainty the data string is divided into Nw segments of the same length.
The described operations are performed on each of them, and the estimated cross
spectra Snxy obtained from each segment are averaged:
Sxy (ω ) =
1
Nw
Nw
∑ Snxy (ω )
(3.18)
n =1
In this way a reduction of the uncertainty by a factor square root of Nw is obtained.
The minimum frequency to which is possible to compute the cross spectral density
and the frequency resolution of the obtained spectrum are set by the time length of
the data string:
2π
(3.19)
∆ω = ωmin =
T
Dividing the data string into shorter segments information at the low frequencies is
then lost. In experiments where the interest is on the very low frequency range the
desired uncertainty on the estimation of the spectral densities must be balanced with
the affordable measurement duration.
The multiplication by the windowing function in equation 3.15 reduces the statistical weight in the computation of the data at the beginning and end of each segment.
To obtain maximum information from the digitised data string it is then possible to
use the head and tail of each data segment more than once in the estimation of the
cross spectral density. This is done dividing the data string in overlapping segments.
The amount of overlapping that maximises the information retrieval from the measured data depends on the chosen windowing function and is typically between the
30% and 50% of the data segment length.
Different windowing functions have been investigated: data strings with known
power spectral density have been synthesised and processed as described, using with
different windowing functions and different overlapping fractions of the data segments. The Blackman-Harris 3rd order function with overlapping of the 50% of the
segment length has been found to give the best results for the torsion pendulum data
[68]. The Blackman-Harris 3rd order function is defined as:
w[n] = q
e[n]
N
2
∑m
=1 e [ m ]
(3.20)
35875
48829
2πn
14128
4πn
1168
6πn
e [ n + 1] =
−
cos
+
cos
−
cos
100000 100000
N
100000
N
100000
N
3.4.2 Torque estimate
Suppose to have compute the power spectral densities Sφgrs (ω ) and Sφac (ω ) respectively for the correctly sampled and digitised angular position of the pendulum, mea-
48
sured by the capacitive sensor φgrs (t) and by the optical readout φac (t), accordingly to
the procedure of the previous section.
In principle it is then possible to compute the power spectral density of the external
torque S N (ω ) acting on the torsion pendulum simply dividing the power spectral
density of the pendulum angular position by the square modulus of the pendulum
transfer function H (ω ) of equation 3.2:
S N (ω ) =
Sφ ( ω )
H (ω )
(3.21)
However a problem arises in the application of this procedure to our torsion pendulum
data. The problem is directly related to the physical system with which we are dealing.
The torsion pendulum described by equation 3.2 is a damped harmonic oscillator
with characteristic time τ = Q/ω0 . The pendulum approaches its equilibrium condition throw an exponential decay from its initial conditions. Our pendulum has period
T ' 560 s and mechanical quality factor Q ' 2900 which determine a characteristic
time τ ' 260000 s ' 3 days. The average measurement time is of the same order.
This implies that the pendulum motion can be influenced by the initial conditions
for the whole measurement. This can manifest in the angular position time series as a
weakly damped oscillation at the pendulum resonance frequency ω0 . Computing the
power spectral density of the pendulum angular position φ(t) we can then obtain a
pronounced peak in correspondence of the resonance frequency, whose intrinsic width
is 1/τ, due to the effect of the damping.
When the angular position power spectral density is divided by the transfer function, to obtain the external torque power spectral density, this peak is not completely
cancelled out by the complementary resonance peak in the transfer function, because
it is much wider. What is it obtained is a torque power spectral density that presents
artefacts in an narrow interval around the pendulum resonance frequency. The problem is emphasised when the measurement duration is not enough for the resolution
of the resonance peak: in this condition the peak is artificially broadened and the
frequency region subject to distortion becomes larger. This happens especially when
the time series is divided into shorter segments that are analysed separately and then
averaged, to obtain smaller uncertainties on the power spectral density, as described
in section 3.4.1.
To avoid those artefacts in the torque power spectrums, the external torque is computed in the time domain instead than in the frequency domain: the angular motion
of the pendulum φ(t) is converted into an instantaneous applied torque N (t) as
N (t) = I φ̈(t) + βφ̇(t) + Γφ(t)
(3.22)
where the derivatives φ̈(t) and φ̇(t) are estimated from a sliding second order fit
to five adjoining data points. This is in practice obtained by multiplying the data
points around φ[m] by the coefficients k [m + n] found by expliciting minimising the
χ2 of a second order polynomial fit to five points. With this processing the initial
conditions influence only the first point of the computed torque time series. The
49
power spectral density can be then computed rejecting this point. We emphasise that
the calculated torque N (t) is really the external torque acting on the pendulum and
not that generated by the torsional fibre or by the damping mechanism.
This equation approximates the damping as viscous while really it is dominated
by structural dissipation within the fibre. While the damping model used in equation 3.22 is not physical, it is not possible to write a time domain equation with a
complex torsional elastic constant, necessary to model the structural dissipations as in
the transfer function of equation 3.2. The effect of this approximation is negligible.
The torque data string obtained from the above procedure has strong high frequency content. This can be naively understood recalling that the time domain derivation operation is a multiplication by iω in the frequency domain, and thus the second
derivative is a multiplication by −ω 2 in the frequency domain. What is obtained is
then a convolution of signals that produces high frequency components. To achieve
clean spectra and a legible time series it is possible to process the torque data string
with a digital low-pass filter. Anyway this influences only the high frequency part of
the power spectral density, that is usually not interesting for our purposes. The filter
is then not applied in the common data analysis procedure.
3.4.3 Readout noise rejection
We can take full advantage of the possibility of measuring the pendulum angular
position with two independent readouts. Imagine to form the average NΣ (t) and the
semi difference N∆ (t)
Ngrs (t) + Nac (t)
2
Ngrs (t) − Nac (t)
N∆ (t) =
2
NΣ (t) =
(3.23)
(3.24)
of the torques Ngrs (t) and Nac (t) calculated from the angular position measured by the
capacitive readout φgrs (t) and from the optical readout φac (t). For ideally uncorrelated
readout noise the power spectral densities of these signals would be:
S NΣ (ω ) =
S N∆ (ω ) =
Snφgrs (ω ) + Snφac (ω )
4 | H (ω )|2
Snφgrs (ω ) + Snφac (ω )
4 | H (ω )|2
+ S N (ω )
(3.25)
(3.26)
where Snφgrs (ω ) and Snφac (ω ) are respectively the power spectral densities of the uncorrelated angular noise of the capacitive and the optical readout, H (ω ) is the pendulum
transfer function as in equation 3.2, and S N (ω ) is the power spectral density of the
correlated part of torque time series. It is intuitive to associate this correlated part
of torque time series to real external torque noise acting on the pendulum inertial
member. In the assumption that the noise of the capacitive readout and of the optical
readout are uncorrelated we obtained complete rejection of the measurement noise in
the computation of the external torque power spectral density.
50
By mean of the properties of the cross spectral density it is possible to prove that
the torque power spectral density S N (ω ) can be also estimated as:
S N (ω ) = S NΣ (ω ) − S N∆ (ω ) = Re{S Nac Ngrs (ω )}
(3.27)
The power spectral density of the true torque noise excess acting on the test mass is
then equivalent to the correlated noise between the two signals S NΣ (ω ) − S N∆ (ω ) and
can be calculated as the real part of the cross spectral density Re{S Nac Ngrs (ω )} between
the torque noise signals computed from the angular measurement of the capacitive
sensor and of the optical readout Ngrs (t) and Nac (t). This is the most powerful tool in
our data analysis procedure.
We can exemplify the analysis of the torsion pendulum data with two opposite
situation. If the angular position data string recorded from the two readouts is strongly
dominated by the real motion of the torsion pendulum the computed torque data
strings would be dominated by true external torque acting on the test mass. Those
data strings would almost equal and thus strongly correlated. If the angular position
data string recorded from the two instruments are instead dominated by readout noise,
the computed torque data strings would be completely uncorrelated. The estimated
cross spectral density would then be on average zero and its uncertainty would be
determined by the individual power spectral densities. As measure of the level of
correlation of two processes x (t) and y(t) we define their cross coherence r xy (ω ) as:
Re{Sxy (ω )}
r xy (ω ) = q
Sxx (ω )Syy (ω )
(3.28)
This is a function that equals zero when the two processes are completely uncorrelated
and one when they are completely correlated.
3.4.4
Uncertainties and data reduction
The relative uncertainty on the cross spectral density estimated from a single data segment is assumed to be similar to that for uncorrelated stationary normal processes.
We are working at the sensitivity limit of the readout devices. The measured angular
position φ(t) is thus a deterministic signal superimposed with a large fraction of random noise. In this condition we can approximate very well the signal with a stationary
normal processes and estimate the relative uncertainty on the estimation of the power
spectral density to be of order one.
Being able to determine the power spectral density of the true torque noise excess
as in equation 3.27 with relative uncertainty equal to one is not of physical significance.
We saw how the uncertainty on the estimation of the cross spectral density can be
reduced dividing the data string into shorter segments and averaging the cross spectral density computed from each segment. However this possibility is limited by the
necessary measurement time. The frequency range of interest for the torsion pendulum measurements is the same where LISA aims to be sensible to gravitational waves.
The requirement of sensitivity down to 0.1 mHz and sets the minimum measurement
51
time to 10000 s for equation 3.19. We saw however how the first three data points at
the lowest frequencies in the power spectral density computed using the BlackmanHarris 3rd order windowing function are not accurate. Having to reject those points
the minimum measurement duration is increased to 30000 s. One thus needs more
than 8 hours for obtaining a single data segment and even then with one hundred
percent uncertainty.
To reduce those uncertainties to the detriment of the spectral frequency resolution
is possible to perform the binning of the spectral data. In this procedure the value
of nearby points in the spectrum are averaged and the error bars on each point are
computed as the standard deviation. To obtain more legible spectra we often chose to
use a logarithmic density for the binning obtaining an uniform distribution of points
in the logarithmic plots.
After the binning is usually then possible to infer a model for the shape of the
spectrum. This model is usually a polynomial law and can be easily fit to the spectral
data via a least square fit - LSF - procedure. The least square fit of the power spectrum
of the single data series however requires itself an estimation of the uncertainty of
each point of the spectrum. We know that the relative uncertainty on each point is one
but taking the value of each point as its uncertainty results in the fact that the points
closer to zero are weighted more than the one far from zero in the fitting procedure.
To solve this problem we make use of the fact that the uncertainty on the estimated
cross correlation Sxy (ω ) of two uncorrelated stationary normal processes x (t) and y(t)
obtained accordingly to the procedure of section 3.4.1 is given by:
q
(3.29)
δSxy (ω ) = Sxx (ω )Syy (ω )
where Sxx (ω ) and Syy (ω ) are the autocorrelations of the single processes. This relation is in good approximation still valid slightly deviating from the assumption of
normal stationary processes and for processes are not completely uncorrelated. We
then perform the least square fitting of the cross spectral density in two steps. First we
estimate the power spectral density of the single processes, then we use this estimation
to compute the uncertainty on each point of the cross spectral density with the relation
of equation 3.29 to be used in the least square fitting procedure.
The estimation of the power spectral density of the single processes however suffers
from the same problems of the estimation of the cross spectral density. Those are
solved again performing a least square fit of the power spectral density to a inferred
model. We are again dealing with the fact that is not accurate to assign the value of
each point as its uncertainty. This time there is no escape from the problem. What we
do is an iterative fitting. In this procedure to each point is initially assigned a nominal
uncertainty and the least square fitting is performed once, then the uncertainty of each
point is recalculated as the value of the fitted curve, and the fitting is repeated with
the new determined uncertainties. The procedure is repeated until convergence is
reached. We observe that the convergence of the fit to a representative curve and the
convergence itself are not guaranteed by the simple procedure. They depend much on
the choice of an accurate model and on the initially estimated uncertainties.
52
It is important to note that the determination of the cross spectral density model
and of its parameters is not only a way to obtain more accurate estimations. It permits
to better understand the nature and origins of the noise excess. We will better see this
in the following example.
3.4.5
Time domain disturbance subtraction
In equation 3.11 we show how the torque excess that we want to investigate is given
by the true torque noise acting on the pendulum minus the known disturbances. In
equation 3.27 we found how to exploit the possibility of measuring the angular position of the pendulum with two independent readouts to obtain an estimation of the
power spectral density of the true external torque acting on the pendulum despite of
the readout noise.
To estimate the torque noise excess we then need to subtract the known disturbances from our data strings. In equation 3.11 we supposed that each disturbance
source xn (t) acts on the torsion pendulum with a torque that can be computed trough
its transfer function hn (t). The transfer functions hn (t) can be measured with the modulation technique briefly presented at the beginning of the chapter. The disturbance
sources xn are measured and recorded during the measurements with the torsion pendulum. Neglecting any noise on the measurements of the disturbance sources it is
then possible estimate the power spectral density of the disturbance sources as for
section 3.4.1 and perform the subtraction prescribed by equation 3.11.
It can be argued that the assumption of simple frequency independent transfer
function for the disturbance sources is too simplistic and that bringing the computation of the torque exert by the disturbances in the frequency domain it can be possible
to accommodate for more complex transfer functions. However it can be seen how
calculating the torque disturbances through the power spectral density of the disturbance sources becomes easily quite complex. Taking into account the cross correlation
between different disturbance sources and between those and the measured torque
results in complex relations between spectral densities. Another limit of this procedure would be that it does not permit direct access to torque time series cleaned from
the known disturbances. If the goal of the measurement is just the determination of
the torque noise excess this is not a problem. But it is not optimal if the torque time
series are used to investigate some other phenomena that can be hidden by the effect
of the known disturbances. We standardise then on performing the know disturbance
subtraction in the time domain.
Transfer functions between disturbance sources and apparent torque exert on the
torsion pendulum can be simply treated in the time domain as simple frequency independents constants. For instance one would expect the effect of magnetic field to
be just the one described by equation 3.10. The time domain subtraction of the known
disturbances reduces then to:
Ñ (t) = N (t) − ∑ k n xn (t)
(3.30)
n
where N (t) is the torque computed from the measured pendulum angular position
53
and k n is the constant that relates the measured disturbance source xn (t) to the corresponding exert torque. In this very simplified model of known disturbances is also
possible to compute the relations k n minimising the torque noise excess power spectral
density.
3.4.6 Example with previous torsion pendulum data
This example of the data analysis procedure is based on data obtained from an experimental run with the «old single mass» pendulum. It gives us the possibility to show
the torque noise excess sensitivity level reached by the torsion pendulum facility during the last scientific operation. The analysed data comes from from experimental run
#2775 but completely similar outcomes are obtained for other runs. Results of similar
analysis have already been reported [69].
optical
readout
φac (t)
disturbance
subtraction
torsion
pendulum
xn (t)
φgrs (t)
PSD
time
torque
iterative
LSF
hS Nac (ω )i
Nac (t)
hδS N (ω )i
S N (ω )
disturbance
monitor
disturbance
subtraction
capacitive
readout
S Nac (ω )
time
torque
CSD
×
LSF
Ngrs (t)
hS N (ω )i
S Ngrs (ω )
PSD
iterative
LSF
hS Ngrs (ω )i
Figure 3.1: Diagram representation of the data reduction procedure employed to estimate
the true torque noise excess exert on the torsion pendulum inertial member by unmodelled
disturbances.
The purpose of this measurement is the determination of the true torque noise
excess acting on the test mass. To extract this information from the experimental data
the data analysis procedures explained in the previous sections will be all used. We
summarise the data reduction procedure with the diagram of figure 3.1.
Run #2775 consists of a 200000 s measurement while the torsion pendulum has
been let free to oscillate. In figure 3.2 and figure 3.3 we plot the angular position measured by the capacitive sensor and by the autocollimator. In the first plot it is possible
to observe the relative drift of the angular position measured by the two readouts,
due to mechanical distortions of the apparatus and less likely to gain changes of the
capacitive sensor readout electronics. The second figure highlights a calibration error
between the optical readout and the capacitive one, noticeable because the calibration
54
of the readouts has been performed long after the end of the run. It is consequence
of the relative drift of the readout outputs. Effects of sensor drift can effect noise estimate and uncertainties at the lowest frequencies, but both those effects can do not
have influence on the estimation of the torque noise excess. In figure 3.3 we also show
the coherence of the measured pendulum oscillation amplitude with the average oscillation amplitude expected from the thermal noise. The oscillation
amplitude due to
√
mechanical thermal noise can be statistically computed as 2 times the mean rms of
the mechanical thermal noise:
r
2k B T
φth =
' 1.2 µrad
(3.31)
Γ
where k B is the Boltzmann constant, T is the temperature, and Γ is the torsional elastic
constant of the pendulum.
The capacitive sensor sensitivity is limited by the electrical thermal noise in the
readout electronics. The electronics thermal noise can be measured suppressing the
signal from the capacitive sensor by setting to zero the ac voltage that polarises the test
mass. To evaluate the performance of the capacitive sensor during the measurement
we use data from run #2797. During this run the pendulum was still free to oscillate
but the ac voltage that polarises the test mass was set to zero. The spectral density of
the readout noise of the capacitive sensor is plotted in figure 3.4. An average noise
spectral density Sφ1/2
= 210 ± 2 nrad Hz −1/2 is obtained, roughly equivalent to the
grs
expected 100 µV Hz−1/2 electronic thermal white noise
Once obtained the necessary information on the readout noise floor we can go
back to analyse the scientific run #2775. First understanding of the system behaviour
is obtained observing the spectral density of the angular position of the torsion pendulum. In figure 3.5 we plot the spectral density of the angular position of the torsion
pendulum measured by the capacitive sensor. The resonance frequency oscillation at
roughly 1.75 mHz is clearly visible. We observe also how at high frequencies the spectral density is coherent with the noise power spectral density observed during the «no
bias» run. The sensitivity of the torsion pendulum operated with a single readout is
at high frequency indeed limited by the readout noise.
Direct informations on the torque noise excess are available observing the measured torque spectral density. We compute the torque time series with the procedure
described in section 3.4.2. In figure 3.6 we plot the spectral density of the external
torque acting on the torsion pendulum calculated from the pendulum angular position measured by the capacitive sensor. The expected pendulum torque noise floor
is given by equation 3.4 where the pendulum torque noise is the mechanical thermal
noise only:
Sφ ( ω )
S Nm = S Nth + ro
(3.32)
| H (ω )|2
The expected torque noise is is dominated by the readout noise converted into torque
via the pendulum transfer function at high frequency, and by the mechanical thermal
noise at low frequency. Excess torque noise is clearly observed at the low frequencies
by comparing the torque noise floor with the measured torque spectral density.
55
We investigate the possible origins of this excess checking for correlation between
the measured external torque and the known disturbances sources. Correlations with
tilt of the experimental facility, with temperature measured in different points of the
experiment, and with magnetic fields measured nearby the torsion pendulum, turn
out to be negligible. Only extra coupling to motion of the test mass in the y direction
as measured by the capacitive sensor is observed. We plot torque and y translation
cross coherence in figure 3.7.
The force disturbance produced in the testing facility by this coupling would be unacceptable for the performances requested to the gravitational reference sensor. However we should note that the translational motion of the test mass is a few order of
magnitude larger in the torsion pendulum testing than any test mass-spacecraft relative motion relevant for LISA or LTP operations. The large translational motion is due
to the unavoidable tilt of the experimental facility at these frequencies. The small coupling converts then into a large torque that would not translate into a corresponding
force disturbance on the LISA or LTP proof masses.
We get rid of this important contribution to the torque disturbances with the known
disturbance subtraction procedure described in section 3.4.5. We employ a simple
model where the y motion motion measured by the capacitive sensor translates into
torque by multiplication of a constant factor. We determined this factor by minimising
the the residual torque noise remaining after the disturbance subtraction procedure.
It has been found to be ∂N/∂y = −9.8 ± 2.2 nN. The nature of this stray coupling
between the GRS and the test mass has been investigated on the base of the available
information [69, 70]. It has been found to be related to the presence of observed
defects in the electrode gold coating that exposed the potentially charged underneath.
A roughly measurement of the effect produces an estimation of this coupling fully
coherent with the value obtained by spectral density minimisation.
To verify the effectiveness of the known disturbance subtraction we plot in figure 3.8 the power spectral density of the external torque acting on the torsion pendulum, calculated from the pendulum angular position measured by the capacitive
sensor, after the subtraction of the coupling with the translational motion. Significant
reduction of the external torque spectral density at the low frequencies is observed.
However comparing the torque power spectral density with the expected pendulum
torque noise limit as in figure 3.6 we still observe significant excess.
To obtain more precise informations we apply now the full blown data analysis
procedure described in the previous sections. In figure 3.9 we plot the cross spectral
density of torque, calculated from the angular position measured by the capacitive
sensor and by the optical readout, before the known disturbance subtraction. In section
3.4.3 we demonstrated how this gives an estimation of the true torque acting on the
pendulum with the readout noise rejected. The torque noise excess computed with
this method must then be compared with a torque noise floor represented by the
mechanical thermal noise only. Comparing the estimation external torque with the
mechanical thermal noise we observe torque noise excess at low frequency, compatible
with the excess identified in the torque spectral density obtained from the angular
56
position measured by the capacitive readout. Additionally we observe previously
unseen torque noise excess at high frequency.
We then proceed with the known disturbance subtraction. In figure 3.10 we plot
the cross spectral density of torque, calculated from the angular position measured by
the capacitive sensor and by the optical readout, after the subtraction of the coupling
with the y translation. The subtraction shows that the coupling to the apparatus tilt
was responsible for more than half of the excess torque noise below roughly 5 mHz
with negligible effect at other frequencies. The remaining excess is compatible in the
limit of the uncertainties with the thermal noise in the frequency region between 1
and 4 mHz. The uncertainty the thermal noise reported in the plot is due to the
uncertainty on the value of mechanical quality factor Q. The mechanical quality factor
is determined exciting torsion pendulum oscillations well over its thermal level and
measuring the damping rate of the oscillations.
To verify the effectiveness of the known disturbance subtraction we also plot in
figure 3.11 torque and y translation cross coherence after the subtraction of the identified coupling with the y translation. We observe that some degree of correlation is still
observable at frequencies between 2 and 3 mHz. The nature of this residual coupling
is still not clear. Unfortunately the pendulum has been unmounted, for starting the
upgrade of the experimental facility, right after this measurement, and before the analysis of the data that lead to the identification of the correlation. Therefore dedicated
measurements were not performed to investigate this phenomena. Some hypothesis
can be formulated. It is possible that the coupling between y translation and applied
external torque can not be approximated with the very simple frequency independent
transfer function we used. It is also possible that there is a disturbance that couples
to both rotation and translation of the pendulum and it is then detected in the cross
correlation of those.
In order to gain better knowledge and to quantify the observed torque noise excess we perform a fit to the torque cross spectral density with a polynomial law in
the frequency range from 0.2 to 30 mHz. Testing different power laws with linear
combinations of positive and negative powers of the frequency we obtained that the
polynomial giving best results is:
S N (ω ) =
b
a
+
+ c (ω/2π )4
2
(ω/2π )
ω/2π
(3.33)
where we use the 1/ω 2 term to represent the torque noise excess at the low frequencies
and the ω 2 term to represent the noise excess at the high frequency. The 1/ω term is
instead to represent the thermal noise. It would be also possible to subtract the thermal
noise from the torque noise excess before fitting the data. We decided to fit also the
thermal noise term to obtain a more stringent test for the model of the excess torque
noise. The least square fitting of the polynomial law to the cross spectral density gives
57
the following results:
a = (2.7 ± 0.7) × 10−36 N2 m2 Hz
b = (5.0 ± 0.7) × 10−33 N2 m2
c = (1.8 ± 0.7) × 10−21 N2 m2 Hz−5
(3.34)
We observe that the coefficient of the 1/ω term is in good agreement with the expected
mechanical thermal noise. For the thermal noise contribution only this coefficient
would be:
Γ
' (4.6 ± 0.7) × 10−33 N2 m2
(3.35)
bth = 4k B T
2πQ
Figure 3.10 shows that the measured torque noise given by the one σ band of the fitted
model is compatible with the expected mechanical thermal noise in the frequency
region around 2-4 mHz. From the fit parameters it is obtained that at 3 mHz the
measured external torque power spectral density is 1.1 ± 0.3 the expected thermal
noise.
Below the mHz region the excess is up to two times the expected background. The
exact origin of this noise is not yet completely understood but there are few evidences
that it is a property of the pendulum and that it is not connected with test massgravitational reference sensor interaction. Possible explanation of this excess is the
progressive unwinding of the torsional fibre [69]. The noise excess at high frequency
is unexpected and would be worrisome if it were traceable to a force noise arising
in the inertial sensor. However many indications exists to make us suppose that this
excess is not attributed to any physical torque acting on the test mass [69, 70].
The ω 4 dependence in the power spectrum of the high frequency noise suggests
that it can be converted into rotational white noise. It can be thus interpreted as
rotation of the entire apparatus relative to the local frame of inertia. The rotational
white noise necessary to explain the high frequency excess can be computed from the
fit parameter:
SN
c
Sφ ( ω ) = 2 4 = 2
= 25 ± 4 nrad Hz−1/2
(3.36)
I ω
I 16π 4
Such rotation can be explained by either a rotation of the whole apparatus due to
seismic noise or by mechanical distortions of the apparatus itself.
It is worth to note that this excess is detected only with the aid of the cross correlation analysis. This region of the spectra is indeed dominated by the read out noises
and what we are observing is a residual correlation of the readout noises. We make
this statement more clear plotting in figure 3.12 the cross coherence of the torque computed from the angular position time series measured by the capacitive and optical
readouts. The cross coherence is still 0.10 ± 0.05 between 3 and 30 mHz and it is non
negligible up to about 80 mHz. However it is not possible at this point to exclude that
this correlation is due to cross talk between the two readouts.
The data available for this verification are not many. We can use the very short
run #2797 already used to evaluate the capacitive sensor noise floor. immediately after
the torque noise run. In this run the autocollimator was still observing an oscillating
58
pendulum and this is reflected into the spectral density. Anyhow we are interested in
the high frequency region where we expect the spectral density to be dominated by
the readout noise and not by real motion if the pendulum. The low frequency part of
the power spectral density can be simply neglected. We plot in figure 3.13 the cross
spectral density of the torsion pendulum angular position measured by the capacitive
and optical readout during the «no bias» run. We then compute the average of the
cross spectral density for frequencies higher that 5 mHz. The cross spectral density
highlight a residual correlation of the two readouts compatible with a white angular
noise of 15 nrad Hz−1/2 with a one σ interval from −8 to 38 nrad Hz−1/2 . This is
consistent with no correlation but also with the correlation necessary to explain the
torque noise excess. The possibility that the observed torque noise excess at high
frequency is due to cross talk between the two readouts can not be rejected.
The detected torque noise excess sets an upper limit to the equivalent acceleration
noise acting on the LISA proof masses. Though the detected excess is supposed not to
be all due to the gravitational reference sensor itself and is not then relevant to LISA,
we still include it when estimating the upper limit to the disturbances acting on the
test mass. We convert the detected torque noise excess into an equivalent acceleration
noise that the corresponding forces would exert on the 1.96 kg LISA proof masses by
using a pessimistic estimation of the conversion arm length. Simple assumptions on
the effects contributing force noise on the test mass bring to a conversion arm length
between 1-2 cm. We use the more pessimistic assumption is Rφ = 10.75 cm as if the
excess torque noise would be determined by front end electronic electrostatic back
action [63, 69].
We plot in figure 3.14 the estimated acceleration noise upper limit computed from
the parameters of the fit to the torque noise excess of figure 3.10 computed as:
SδF (ω ) =
SδN (ω ) − S Nth (ω )
R2φ M2
(3.37)
where M is the mass of LISA proof mass. The plotted upper limit also includes the
fit uncertainty and the uncertainty in the evaluation of thermal noise coming from the
error in mechanical quality factor Q measurement. Between 2 and 4 mHz we obtain
a torque noise excess smaller than 1.1 fN m Hz−1/2 that sets an upper limit to the
acceleration disturbances on the LISA proof not higher than 5.5 × 10−14 m s−2 Hz−1/2 .
This level of isolation from force disturbances would be sufficient for the observation
of gravitational waves in LISA.
It should be anyhow remarked that the pendulum geometry does not detect disturbances applied uniformly and normal across the x face or tangentially to the y and
z faces or disturbances acting normally onto the centre of the x face or in a few other
symmetric positions. It also lack sensitivity to volume forces and all effects due to coupling among different d.o.f. that would take place for a fully free-falling proof mass
are absent in the ground testing. Additionally the environment on board LISA will
be substantially different from the laboratory one. In space the proof mass charging
rate will be higher due to cosmic rays but the level of temperature and magnetic field
fluctuations will be lower.
3.5. FIRST GENERATION TORSION PENDULUM ACHIEVED RESULTS
3.5
59
First generation torsion pendulum achieved results
The torsion pendulum testing facility has long been in operation at Trento University
to investigate the performances of the gravitational reference sensor prototypes there
developed. We show in the data analysis the last experimental campaign of the torsion pendulum obtained a differential force sensitivity in the mHz range better than
100 fN Hz−1/2 . Unmodelled disturbance that can clearly attributed to the test massgravitational reference sensor interaction has not been detected [69]. This is within
two orders of magnitude from LISA required force isolation and less than a factor ten
from the LTP goal. It represent the first step toward the demonstration of the success
of the LISA mission.
Important verifications performed with the single mass torsion pendulum facility
other than the identification of the force noise upper limit include:
- Measurement of the dc bias and verification of the possibility of their compensation at a level better than 1 mV [63].
- Verification of the electrostatic model of the capacitive sensor and measurement
of the induced test mass-gravitational reference sensor coupling [68].
- Exclusion of stray test mass-gravitational reference sensor coupling at the 1%
level of the overall stiffness [68].
- Verification of the thermal gradients effects and radiometric effect accordingly to
the model within 25% uncertainty. [54].
- Verification of the possibility of bipolar discharge of the test mass with UV light
accordingly to the system designed for LISA and LTP [68].
- Verification of the possibility of operating charge measurement and continuous
discharge of the test mass without measurable increased force noise [68].
60
−20
−30
φ [µrad]
−40
−50
−60
−70
−80
0
50
100
150
200
time [s] × 103
Figure 3.2: Torsion pendulum angular position time series, measured by the capacitive and
optical readouts during run #2775. The optical readout reading has been artificially shifted of
10 nrad to make the curves distinguishable. The blue an red lines are obtained filtering out
the pendulum oscillation at the resonance frequency. It is not possible to distinguish the single
pendulum oscillation. The reading of the two instruments is slightly drifting.
−71
−72
φ [µrad]
−73
−74
−75
−76
−77
0
0.5
1
1.5
2
time [s] × 103
Figure 3.3: First 2000 s of the time series of figure 3.2. The blue and red lines are obtained
with two poles phase preserving low pass filtering at 0.1 Hz. The noise is superimposed to the
clearly visible pendulum equilibrium oscillation of period T ' 571 s. The shift between the
two curves is due to errors in the calibration of the readouts. The dashed lines represent the
averaged peak to peak amplitude of the oscillation due to mechanical thermal noise computed
as two times equation 3.31.
61
3
2
10
1/2
Sφ [nrad / Hz
1/2
]
10
1
10
−4
10
−3
10
−2
10
frequency [Hz]
−1
10
0
10
Figure 3.4: Spectral density of the torsion pendulum angular position noise measured by the
capacitive sensor during the «no bias» run #2797 subsequent to scientific run #2775. The red
line is a straight line fitted to the data. White noise of 210 ± 2 nrad Hz−1/2 is obtained.
6
10
5
1/2
Sφ [nrad / Hz
1/2
]
10
4
10
3
10
2
10
1
10
−4
10
−3
10
−2
10
frequency [Hz]
−1
10
0
10
Figure 3.5: Spectral density of the torsion pendulum angular position noise measured by the
capacitive readout during the scientific run #2775. The red line is a straight line fitted to the
data for frequencies higher than 10 mHz. A white noise floor compatible with the «no bias»
run of figure 3.4 is obtained.
62
4
10
3
S1/2
[fN m / Hz1/2]
N
10
2
10
1
10
0
10
−1
10
−4
−3
10
−2
10
−1
10
10
frequency [Hz]
Figure 3.6: Spectral density of external torque computed from the pendulum angular position
time series measured by the capacitive sensor. The gray line is obtained with a single data
segment. The blue line is obtained by averaging 18 data segments with 50% overlapping. The
red line is the sensitivity limit computed as for equation 3.4. The dashed lines are the single
contributions from thermal mechanical noise and from read out noise. External torque noise
excess is clearly present at frequencies lower than 5 mHz.
0.2
cross coherence
0
−0.2
−0.4
−0.6
−0.8
−1
−4
10
−3
10
−2
10
frequency [Hz]
−1
10
0
10
Figure 3.7: Cross coherence between external torque computed from the torsion pendulum
angular position and y translation measured by the capacitive readout. The red markers are
obtained by averaging of nearby data points with logarithmic density of ten points per frequency decade. The uncertainty on each point is the standard deviation. High degree of
correlation is shown at frequencies lower than 30 mHz.
63
4
10
3
S1/2
[fN m / Hz1/2]
N
10
2
10
1
10
0
10
−1
10
−4
10
−3
−2
10
10
−1
10
frequency [Hz]
Figure 3.8: Spectral density of external torque computed from the pendulum angular position
time series measured by the capacitive sensor after subtraction of the coupling with the y
translation. The gray line is obtained with a single data segment. The blue line is obtained
by averaging 18 data segments with 50% overlapping. The red line is the sensitivity limit
computed as for equation 3.4. The dashed lines are the single contributions from thermal
mechanical noise and from read out noise. External torque noise excess is still however clearly
present at frequencies lower than 5 mHz.
3
10
2
S1/2
[fN m / Hz1/2]
N
10
1
10
0
10
−1
10
−4
10
−3
−2
10
10
−1
10
frequency [Hz]
Figure 3.9: Cross spectral density of torque calculated from the angular position measured by
the capacitive sensor and by the optical readout. The gray line is obtained with a single data
segment. The red markers are obtained by averaging of nearby bins with logarithmic density
of ten points per frequency decade. The uncertainty on each point is the standard deviation.
The blue contour region is a fit to the data of the polynomial model of equation 3.33 plus or
minus one sigma. The black line is the expected mechanical thermal noise.
64
3
10
2
S1/2
[fN m / Hz1/2]
N
10
1
10
0
10
−1
10
−4
−3
10
−2
10
−1
10
10
frequency [Hz]
Figure 3.10: Same as the plot of figure 3.9 but after the time domain subtraction of the identified
coupling with the y translation. The dotted lines represent the uncertainty on the estimation
of the mechanical thermal noise. The dashed line represents the fit to the torque noise excess
before the subtraction of the coupling. A reduction of the external torque noise between a
factor 0.2 and 2 is observed. The remaining excess is compatible with the thermal noise in the
frequency region between 1 and 4 mHz.
0.6
cross coherence
0.4
0.2
0
−0.2
−0.4
−0.6
−4
10
−3
10
−2
10
frequency [Hz]
−1
10
0
10
Figure 3.11: Same as the plot of figure 3.7 but after the subtraction of the identified coupling
with the y translation. Some degree of correlation is still observable at frequencies between 2
and 3 mHz.
65
1
cross coherence
0.8
0.6
0.4
0.2
0
−0.2
−4
10
−3
10
−2
10
frequency [Hz]
−1
10
0
10
Figure 3.12: Cross coherence between external torques computed from the torsion pendulum
angular position measured by the capacitive and optical readouts. The red markers are obtained by averaging of nearby data points with logarithmic density of ten points per frequency
decade. The uncertainty on each point is the standard deviation. Non negligible degree of
correlation is still present up to 30 mHz.
4
10
3
2
10
1/2
Sφφ [nrad / Hz
1/2
]
10
1
10
0
10
−4
10
−3
10
−2
10
frequency [Hz]
−1
10
0
10
Figure 3.13: Cross spectral density of the torsion pendulum angular position measured by capacitive and optical readouts during the «no bias» run #2797. The red markers are obtained by
averaging of nearby data points with logarithmic density of five points per frequency decade.
The blue line is average of the data points for frequencies higher than 10 mHz. The blue
dashed line is the one sigma limit from the obtained average.
66
−10
10
−11
S1/2
[m s−2 / Hz1/2]
δF
10
−12
10
−13
10
−14
10
−15
10
−4
10
−3
−2
10
10
−1
10
frequency [Hz]
Figure 3.14: Upper limit to acceleration disturbances acting on the free-fall test mass obtained
from the torsion pendulum ground testing. The red line is the limit obtained from the best fit
of the experimental data. The gray contour region is the one sigma limit to this estimation.
The solid and dashed blue lines represent the acceleration noise limit set respectively for LISA
and LTP.
Chapter 4
Recent improvements to the single
mass torsion pendulum facility
In this chapter we briefly describe the improvement introduced during the setup of
torsion pendulum facility for the new scientific run. We think that those improvements
will set the best sensitivity that we can reach with the single mass torsion pendulum
without introducing more substantial changes.
4.1
New GRS prototype integration
The current experimental run with the torsion pendulum facility aims at testing a new
gravitational reference sensor prototype. This new prototype is as close as possible to
the final design of the GRS that will fly onboard of LTP. Noticeable changes from the
one previously employed are the replacement of Shapal insulators with the Sapphire
ones and the modification of the UV light fibres seats position. The change in the insulators material should not produce any change in th behaviour of the GRS during the
ground testing. However experimental confirmation of the new sensor electrostatic
characteristics will be the first goal of the new scientific run of the torsion pendulum.
The new positioning of the UV light fibres instead will modify the efficiency of the
charge management system but this can be corrected by redefining the parameters
that control the system. Those parameters will be used to control the discharge mechanism during the LTP experiment. The ground testing for the determination of those
parameters is thus crucial for the mission.
We also radically redesigned the mounting of the GRS in the vacuum chamber.
In section 3.2.2 we described how the sensor prototype were mounted on a manual
five d.o.f. micro-manipulator. The micro-manipulator was previously mounted on a
big side flange of the vacuum chamber, sustaining the full weight of the GRS and of
the mechanical interfaces needed for the assembly, included a motorised rotational
stage. The weight of the assembly was well over the maximum load suggested by
the manufacturer of the micro-manipulator. The high load, united with the very long
arm length determined by the GRS position, were exerting big torques on the micro67
68
CHAPTER 4. RECENT IMPROVEMENTS
manipulator suspensions. We were not confident of the mechanical stability of the
system.
The new mounting substitutes the micro-manipulator with a simpler two d.o.f.
motorised translational stage, that permits the test mass-capacitive sensor alignment
along the x and y axes. Over the translational stage is mounted the motorised rotational stage that permits to obtain control over three d.o.f. The alignment along the z
axis is permitted by the micro-manipulator that supports the suspension point of the
pendulum. We renounced to have fine control over test mass-capacitive sensor alignment in the η and θ angles. Rough alignment in those d.o.f will be obtained by tilting
the whole experiment, by acting on the adjustable feets of the metallic plate where the
facility is mounted. Redesigning the GRS mounting in the vacuum chamber, we suppose to have removed a possible cause of mechanical instability, that can contribute
to test mass-capacitive sensor relative motion. The motorisation of the translational
stage will be also extremely useful in characterisation of tilt-twist couplings as the
ones observed in run #2775 and analysed in section 3.4.6.
The new sensor prototype has been equipped with a home-brew readout circuitry
which follows the scheme envisioned for LTP and LISA, employing high stability
transformers, commercial low noise differential preamplifiers, and phase sensitive detectors. The actuation voltages are generated by home-brew circuitry based on commercial oscillators and self made electronics. It follows the principles of the actuation
scheme envisioned for LTP and LISA, but in a simpler design, that permits dc voltage
compensation on all the electrodes, but ac actuation in only one rotational and one
translational degree of freedom, along the same test mass axis. The readout circuitry
is roughly limited by intrinsic thermal noise originating in the transformers losses that
is evaluated to be roughly 90 nV Hz −1/2 .
Characterisation of the new sensor performances with the home-brew readout circuitry has been performed with good results. We plot in figure 4.1 the spectral density
of the translational position of the test mass along the x axis, measured by the capacitive sensor, while the test mass polarising voltage was set to zero obtaining. It
is indeed an estimation of the capacitive readout noise floor. We obtained a readout
limited by the electric thermal noise, which is equivalent to a translational sensitivity
1/2
Sδx
= 0.95 nm Hz−1/2 fully compliant with the requirement of 1.8 nm Hz−1/2 . Using the nominal conversion arm length of 10.75 mm this is equivalent to a rotational
sensitivity Sφ1/2 = 88.4 nrad Hz−1/2 .
Figure 4.2 shows the time series of the translational position of the test mass along
the x axis, measured by the capacitive sensor, with the polarisation voltage set to the
nominal value to obtain test mass polarisation at 0.6 V. The test mass was sitting on its
support inside the gravitational reference sensor and was thus nominally steady. The
observed motion is probably due to temperature driven mechanical distortions of the
test mass support. In figure 4.3 we plot the spectral density of the same measurement.
In the plot we observe that the capacitive readout noise floor at high frequency is
compatible with the before measured electric thermal noise, while at low frequency it
increases. The increase is not due to readout noise but to real motion of the test mass.
4.2. OPTICAL READOUT IMPROVEMENTS
4.2
69
Optical readout improvements
The mounting of the commercial autocollimator used for the optical readout has been
largely improved in the new setup of the experimental facility. In the old configuration
it was looking at the mirror connected to the shaft of the pendulum inertial member
thought a view port located on the side of the vacuum chamber. The autocollimator
was positioned on a metallic shelf bolted to the side of the vacuum chamber. The
system so composed was not enough mechanically stable to permit the autocollimator
to operate at the best sensitivity possible.
It is very difficult to realise a support that hangs from the side of the vacuum
chamber that is enough reliable on the point of view of the mechanical stability at the
low frequency. This is mainly due to the distortions that the load of the suspended
object and the temperature fluctuations produce in the materials. It has then been
decided to mount the autocollimator on the top of the vacuum chamber. To do so we
made the autocollimator to look at the mirror mounted on the inertial member of the
pendulum through a view port placed on the top flange of the vacuum chamber. We
realised a custom autocollimator support to replace the general purpose one used in
the past. The new support can be roughly described as stainless steel hollow cylinder
with about 2 cm thick walls. The hole on the upper side of the cylinder precisely
accommodates the metallic tube that contains the objective of the autocollimator. The
hole on the bottom side of the cylinder precisely accommodate the view port making
the whole structure to sit on the metalling view port frame. The mechanical tolerances
are so to reduce to the minimum the possibility of movement. The alignment of the
autocollimator to the test mass mirror is obtained only by the careful design of the
apparatus. The light beam of the autocollimator is directed on the inertial member
mirror making it bounce on a second mirror. This second mirror is realized with a
gold coated burnished stainless steel surface. It is constructed in a single piece and
it is directly bolted on the optical bench on the bottom of the vacuum chamber. A
similar «mock up» mirror has been also manufactured to be used to simulate a steady
test mass during the tests in the setup of the experimental facility. The autocollimator
is surrounded by a metallic shield that should protect it from air turbulence caused by
the temperature stabilisation system when the thermally stabilised room will be setup.
Despite the careful design of the components of the experimental facility and of
their connections, the whole apparatus can not be seen as a rigid body. Vibrations
and thermal distortion cause respectively hight frequency and low frequency relative
motion of the different component of the experimental apparatus. The autocollimator
support and the gravitational reference sensor are not excluded. The relative motion
of those components sums to the readout angular position noise.
The possibility of testing the autocollimator performance with the nominally fixed
mock up mirror to substitute the test mass gave us the possibility to extensively investigate the vibration sources in our experimental facility. Precautions to vibrationally
insulate the torsion pendulum facility from nearby source of mechanical noise has
been in place since the beginning. However important sources of mechanical vibrations has been identified in the vacuum pumps and in the water circulatory used to
70
stabilise the temperature. The autocollimator readout noise has been measured in different condition in a long preliminary experimental investigation of the mechanical
noise sources. Finally a suitable configuration has been found.
The optical readout noise floor, obtained after the isolation of the vibrational noise
source and after the setup of the new autocollimator mounting, is presented in figure 4.4. Here we plot the spectral density of the angular position of the torsion pendulum test mass, while it was sitting on its support inside of the gravitational reference
sensor, and thus nominally steady. Read out noise floor of less than 6 nrad Hz−1/2
is obtained at high frequency. Below roughly 20 mHz the readout noise increases inversely proportional to the frequency. Detailed analysis of the autocollimator noise
floor with the old mounting are not available. Anyhow from the analysis of the data
of the old facility setup we infer an improvement of a factor up to roughly ten at
the frequencies above 10 mHz and of a factor about two to three below. In the plot of
figure 4.4 we report also spectral density of the angular position of the pendulum measured by the autocollimator during run #2775 analysed in section 3.4.6. In this run the
autocollimator noise is superimposed with the real pendulum motion but we know
that at the high frequencies the readout noise strongly dominates. The comparison
between the two spectral densities is a good estimation of the realised improvements.
Furthermore the measurement that lead to the estimation of the spectral density of
figure 4.4 was performed without any thermal stabilisation of the facility. We hope in
a further reduction of the optical readout noise at the lowest frequencies with a more
stable temperature.
4.3
Magnetic shield
In the integration of the new gravitational reference sensor prototype in the experimental facility we reserved some of the limited space available in the vacuum chamber
for a magnetic shield. In the data analysis of section 3.4.6 we explain how the magnetic disturbances to the torsion pendulum were not relevant in the previous setup.
However from separate measurements of the magnetic moment of the test mass and
of the magnetic field fluctuation in the area of the laboratory where we operate the
torsion pendulum we know that the magnetic noise contribution is just below the limit
reached with the analysis of run #2775. We are planning more radical changes to the
torsion pendulum facility as we will describe in chapter 5. Those will hopefully reduce
both the mechanical thermal noise of the pendulum and the readout noise making the
magnetic noise to play an important role in the measured torque noise excess. We decided then to introduce the magnetic shield to test the effectiveness that we can achieve
with passive magnetic shielding in the reduction of thee magnetic field fluctuations.
The realised magnetic shield consist of two concentric layers of high magnetic
permeability material industrially available with the name of «µmetal». Each layer is
obtained rolling in a circular shape up to three bands of 1 mm thick foils and cutting
out the minimum possible holes to let the autocollimator light beam to go thought and
to accommodate cables and UV light fibres. The layers are supported by a very small
4.4. PRELIMINARY RESULTS
71
magnetic susceptibility aluminium structure assembled with bras screws. The two
layers surround the gravitational reference sensor in the horizontal plane at a distance
of about 10 and 18 cm from the centre for a vertical extension of about 18 and 23 cm
respectively.
The effectiveness of the magnetic shield has been tested producing a magnetic
field forcing a current to flow thought a coil placed nearby the experimental facility
and measuring the magnetic field inside the vacuum chamber with and without the
magnetic shield in place. The same flux-gate magnetometer used to monitor the magnetic field during the torsion pendulum operation has been used. It has been placed in
a position representative of the position of the test mass during the torsion pendulum
operation. The magnetic field was square wave modulated at 0.1 mHz and the magnetic field measured by the magnetometer was demodulated obtaining informations
on all the odd harmonics up to 1 Hz. The magnetic field fluctuations suppression
factor has been computed at different frequencies as the ratio of the magnetic field
intensity measured with and without the shield in place.
Frequency independent magnetic field attenuation in the frequency region between
0.1 mHz and 1 Hz has been measured. The attenuation factor is 45 for the y component
and 75 for the x component. The attenuation along z has not been measured because
magnetic field fluctuations along z are not important for the operation of the torsion
pendulum. The anisotropy of the attenuation can be due to the holes cut in the shields
or to a systematic error. The flux-gate magnetometer is indeed composed by three coils
of diameter roughly 10 mm that serve to measure the three different components of the
magnetic field. The magnetometer was oriented in such a way that the coil measuring
the x component of the magnetic field was centred on the nominal position of the test
mass while the coil measuring the y component was shifted by roughly 30 mm. Due
to the open geometry of the two layer shield the small offset in the position of the
different coils can explain the anisotropy.
Further testing has not been thought necessary at this stage of the experiment.
More accurate measurement of magnetic field attenuation will be possible one the
torsion pendulum will be operative. The magnetic moment of the test mass currently
employed has been measured during the last scientific run of the torsion pendulum
obtaining µ x ' 90 × 10−9 A m2 and µy ' 110 × 10−9 A m2 for the two components
in the horizontal plane. Modulating a known magnetic field and looking for coherent
deflection of the torsion pendulum angular position it is then possible to measure the
attenuation produced by the magnetic shield.
4.4
Preliminary results
The described improvements have been successfully integrated into the existing single
mass torsion pendulum facility. However due to many technical problems encountered during the setup phase of the experiment we are not able to provide full analysis
of the improvements presenting the performance of the torsion pendulum. The debugging of the final pendulum configuration is still in progress. The analysis of some
72
preliminary data is however enough to obtain some interesting informations to complement the torsion pendulum performance analysis of the previous chapter.
4.4.1
Cross correlation analysis
The first measurement done with the new experimental facility had the goal of checking for the high frequency correlation of optical and capacitive readouts observed in
the data analysis presented in section 3.4.6. We checked that the nature of the correlation is cross talk between the two readouts. It is not necessary to have a suspended and
properly working torsion pendulum to obtain this experimental confirm. The presence
of cross talk between the read out has been investigated in two distinct «white» experimental run where we had the two readout looking at physically unlinked references.
The optical readout was looking at a fixed mirror mounted in the vacuum chamber
to perform its performance testing. The capacitive readouts was looking at the test
mass sitting on its support inside the gravitational reference sensor while the sensor
was tested on bench. Run #1601 is a 100000 s measurement where the sensors where
operating as usual. Run #1613 is a 28800 s measurement where the configuration was
similar but the ac voltage that polarises the test mass set to zero.
For those runs there is no point in converting the torsion pendulum angular position in external torque acting on the torsion pendulum, as there should not be any
such torque being the pendulum nominally fixed. We then compute the cross spectral density of the torsion pendulum angular position measured by the capacitive and
optical readouts.
In figure 4.5 we plot the results for run #1601. Averaging the data points between
5 mHz and 0.1 Hz we obtain that the cross spectral density highlight a residual correlation of the two readouts compatible with a white angular noise of −0.5 nrad Hz−1/2
with a one σ interval from −3.9 to 2.9 nrad Hz−1/2 . In figure 4.6 we plot instead the
results for «no bias» run #1613. Averaging the data points between 5 mHz and 0.1 Hz
we obtain that the cross spectral density highlight a residual correlation of the two
readouts compatible with a white angular noise of 5.6 nrad Hz−1/2 with a one σ interval from 4.0 to 6.9 nrad Hz−1/2 . We do not know how to interpret the rather small
uncertainty obtained in this case if not with a curious statistical fluctuation.
The obtained results are definitely not compatible with the correlation necessary
to explain the torque noise excess observed in the previous experimental setup and
discussed in section 3.4.6. However this result does not bring new informations o the
nature of the correlation between optical and capacitive readout observed in the data
of run #2775. The experimental setup is different and the capacitive readout has been
replaced with a different one. It is anyhow interesting to note that already in those
very preliminary measurement we would be able to resolve a residual correlation
between the two readouts of order of few nrad Hz−1/2 . This would permit to lower
the high frequency torque noise excess detected in the run #2775 data up to a factor
one hundred in absence of a true physical correlation.
73
4.4.2 Initial pendulum suspension
We also managed to suspend the pendulum and control its dynamics in a first short
preliminary test run. However alignment problems prevented having a free pendulum without a large displacement of the test mass from the capacitive sensor centre.
Figure 4.7 shows very preliminary data from a 12 hours with the pendulum oscillating
freely. Analysis for torque noise excess will be possible only after reassembly of the
pendulum.
We plot in figure 4.8 the spectral density of the pendulum angular position measured by the optical readout. We compare it with the spectral density of the pendulum angular position measured by the optical readout in run #2775 analysed in
section 3.4.6. We plot in figure 4.9 the external torque acting on the pendulum computed in the time domain from the angular position measured by the optical readout
during run #1174. We compare it with the spectral density of the external torque acting
on the torsion pendulum during run #2775 computed in a similar way. The apparent
torque acting on the torsion pendulum in the last run is up to a factor ten higher.
74
1
S1/2
[nm / Hz1/2]
x
10
0
10
−1
10
−4
−3
10
10
−2
10
frequency [Hz]
−1
10
0
10
Figure 4.1: Spectral density of the translational position of the test mass along the x axis
measured by the capacitive sensor during a 100000 s long «no bias» measurement. The blue
line is obtained averaging five data segment 20000 s long. The red line is obtained averaging
the data points below 0.1 Hz. The dashed line is the translational sensitivity requirement for
the gravitational reference sensor.
1
0.8
x [nm]
0.6
0.4
0.2
0
0
20
40
60
80
100
3
time [s] × 10
Figure 4.2: Test mass position along the x axis measured by the capacitive sensor during a
100000 s long measurement
75
1
S1/2
[nm / Hz1/2]
x
10
0
10
−1
10
−4
10
−3
10
−2
10
frequency [Hz]
−1
10
0
10
Figure 4.3: Spectral density of the translational position of the test mass along the x axis measured by the capacitive sensor during a 100000 s long measurement. The blue line is obtained
averaging nine data segment 20000 s long. The dashed line is the translational sensitivity
requirement for the gravitational reference sensor.
4
10
3
S1/2
[nrad / Hz1/2]
φ
10
2
10
1
10
0
10
−1
10
−4
10
−3
10
−2
10
frequency [Hz]
−1
10
0
10
Figure 4.4: Spectral density of the a 100000 s long angular position measurement of the nominally steady torsion pendulum. The blue line is obtained averaging 9 data segments 20000 s
long. The red line is a polynomial fit to the data showing the increasing of the readout noise
inversely proportional to the frequency. The red scattered line is instead the autocollimator
readout spectral density during run #2775 analysed in section 3.4.6.
76
3
10
2
S1/2
[nrad / Hz1/2]
φφ
10
1
10
0
10
−1
10
−4
10
−3
10
−2
10
frequency [Hz]
−1
10
0
10
Figure 4.5: Cross spectral density of the torsion pendulum angular position measured by
capacitive and optical readouts during the «white» run #1601. The red markers are obtained by averaging of nearby data points with logarithmic density of ten points per frequency decade. The data points between 5 mHz and 0.1 Hz are averaged obtaining residual correlation between the two readouts compatible with angular position white noise level
(−3.0 : 2.9) nrad Hz−1/2 at one sigma. The blue dashed line is the plus one sigma limit. The
plot is of tricky interpretation because the many negative points obtained can not be represented on the logarithmic scale.
77
3
10
2
S1/2
[nrad / Hz1/2]
φφ
10
1
10
0
10
−1
10
−4
−3
10
−2
10
10
frequency [Hz]
−1
0
10
10
Figure 4.6: Cross spectral density of the torsion pendulum angular position measured by
capacitive and optical readouts during the «white» run #1613. The red markers are obtained by averaging of nearby data points with logarithmic density of ten points per frequency decade. The data points between 5 mHz and 0.1 Hz are averaged obtaining residual correlation between the two readouts compatible with angular position white noise level
(4.0 : 6.9) nrad Hz−1/2 at one sigma. The blue line is average of the data points while the blue
dashed lines are the one sigma limit. The plot is of tricky interpretation because the many
negative points obtained can not be represented on the logarithmic scale.
2100
2050
2000
φ [µrad]
1950
1900
1850
1800
1750
1700
1650
0
5
10
15
20
25
30
35
40
3
time [s] × 10
Figure 4.7: Torsion pendulum angular position time series, measured by the capacitive and
optical readouts during the preliminary run with the pendulum suspended. The red line is
obtained filtering out the pendulum oscillation at the resonance frequency. The oscillation
amplitude at the resonance frequency is much bigger than in ideal operational conditions.
78
8
10
6
4
10
φ
S1/2 [nrad / Hz1/2]
10
2
10
0
10
−4
10
−3
−2
10
−1
10
frequency [Hz]
10
0
10
Figure 4.8: Spectral density of the torsion pendulum angular position measured by the optical
readout during preliminary run #1174. The blue line is obtained averaging three data segments. For comparison the spectral density of the pendulum angular position measured by
the optical readout during run #2775 is shown with the red line.
−11
10
−12
S1/2
[N m / Hz1/2]
N
10
−13
10
−14
10
−15
10
−4
10
−3
−2
10
10
−1
10
frequency [Hz]
Figure 4.9: Spectral density of the external torque acting on the torsion pendulum computed
from the angular position measured by the optical readout during preliminary run #1174.
For comparison the spectral density of the external torque acting on the torsion pendulum
computed from the angular position measured by the optical readout during run #2775 is
shown with the red line.
Chapter 5
Future improvements to the single
mass torsion pendulum facility
In this chapter we briefly present the improvements we want to introduce to the single
mass torsion pendulum in the near future. Those changes will determine much better
sensitivity to small forces and will realize the third generation torsion pendulum facility.
In section 3.4.6 we demonstrate that in principle we are able to detect any torque
noise excess above the mechanical thermal noise. However the experimental sensitivity is limited by the uncertainties on the estimation of those excess. To increase the
sensitivity to torque noise excess we can reduce the uncertainties by reducing the readout noises or we can make the excess we want to detect virtually bigger by reducing
the mechanical thermal noise. We plan to act on both sides by employing a torsional
fibre with much higher mechanical quality factor and developing an interferometric
angular position readout.
5.1
Fused silica torsion fibre
In equation 3.6 we see that the thermal noise is determined by two controlled parameters of the experiment: the operation temperature T and the torsion pendulum
mechanical quality factor Q. In principle it would be possible to lower the thermal
noise by decreasing the operation temperature of the experiment. However many
of the disturbances arising in the gravitational reference sensor and that we want to
investigate with the torsion pendulum are directly or indirectly dependant on the temperature. The torsion pendulum experimental facility and thus the GRS are actually
stabilised at the temperature of 293 K. This temperature is roughly the operational
temperature envisioned for the GRS in scientific operation onboard of LISA and LPF
missions [6]. Changing the temperature by a factor that would introduce an interesting
decrease of the mechanical thermal noise would completely spoil the representativity
of our experiments. Furthermore the torsion pendulum is a complex and cumbersome
experimental apparatus. Bringing it to cryogenic temperatures would be an effort as
challenging as the design of the GRS itself.
79
CHAPTER 5. FUTURE IMPROVEMENTS
80
To lower the mechanical thermal noise is then necessary to employ a torsion fibre with higher mechanical quality factor. Incidentally the search for materials with
low mechanical losses and with elastic modulus suitable for the realisation of torsion
pendulums has been carried out by the ground interferometric gravitational wave
detectors community. In those experiments the fibres are used to suspend the interferometer optics. Also in this application the mechanical thermal noise is a limiting
factor for the detectors performance. They found that fused silica permits fabrication of
fibres suitable for the production of reliable suspensions for the interferometer mirrors
and probably the same material can be used as a torsion fibre in a pendulum for the
measurement of small forces.
The currently employed tungsten fibres has elastic modulus E ' 150 GPa and
yield strength Y = 3.5 GPa. Fused silica offers very similar mechanical characteristics
with elastic modulus E ' 31 GPa and a yield strength Y ' 3 GPa. However while
the mechanical quality factor Q obtained with tungsten fibres is of order 3 × 103 the
quality factor demonstrated with fused silica fibres is up to 3 × 106 .
Techniques for the production of fused silica fibres of diameter of order 100 µm
are fully developed to produce the fibres used in the large interferometers. However
those fibres offer a too high torsional elastic constant to produce a torsion pendulum
sensible to small forces. Recalling equation 3.8 we compute that the fused silica fibre
radius necessary to sustain the weight of the single mass torsion pendulum is:
r
rfs ≥ 1.7
mg
' 35 µm
Yπ
(5.1)
where we introduced a security factor 1.7 to load the fibre to roughly the 60% of its
yield strength. A 1 m long fused silica fibre with this radius would have a torsional
elastic constant given by equation 3.7
Γfs = E
πr4
' 4.6 nN m rad− 1
2L
(5.2)
that is comparable with the torsional elastic constant of the currently employed tungsten fibre Γth ' 5.2 nN m rad− 1. The slightly lower torsional elastic constant would
produce wider pendulum oscillations for torque disturbances of the same strength and
would therefore reduce the influence of readout noise on the overall performance of
the pendulum. But the main improvement would be in a largely reduced mechanical
thermal noise. For the similarity in the torsional elastic constants from equation 3.6
we compute that the ratio between the power spectral density of the thermal noise
with the tungsten fibre is approximately the ratio between the Q factors. We obtain
that the same torsion pendulum we are operating now but with the fused silica fibre
would obtain a reduction of a factor about 350 in the power spectral density of the
mechanical thermal noise. This would lead to an improvement of a factor about 20 in
torque sensitivity.
However the possibility of producing fibres of the diameter suitable for the employment in our torsion pendulum has not yet been demonstrated. Some preliminary
5.2. INTERFEROMETRIC READOUT
81
experiments have been performed to realise fibres with the diameters of order 30 to
40 µm but many difficulties have been encountered.
If it would be possible to obtaining the thin fibres required for the pendulum there
would be anyhow another problem in their employment in the facility. Silica contrary
to tungsten is an insulator. Charge would accumulate on the torsion fibre and on
the inertial of the pendulum previously grounded through the conducting torsional
fibre. The electric interactions between the charge accumulated on the fibre and the
surrounding are not expected to produce noticeable torque disturbances due to the
very small arm length of the fibre radius. However in the common torsion pendulum
configuration the accumulation of charge on the inertial member would be a disaster for the measurement of small forces. The electrostatic forces between the charged
inertial member and the surrounding would dominate on any other effect that one
wants to measure. However we have already demonstrated the possibility of effectively managing the charge on the test mass with the aid of UV light without introducing measurable disturbances. The only parts of the inertial member that would
not be under control by the discharging system are the aluminium shaft and the optical read out mirror. Anyhow on the base of the charging ratio measured for the test
mass we are confident that the electrostatic forces that can arise from the charging of
the shaft would be very small. Charges accumulates on the test mass with a ratio of
roughly a fundamental charge per hour making necessary to discharge the test mass
only every few months. The shaft at the contrary of the test mass is not shielded by
the gravitational reference sensor from charged cosmic radiation. However we expect
it to have a similar charging ratio.
We are looking forward to be able to verify those suppositions in the torsion pendulum test bench. We are currently investigating the possibility of produce fibres
suitable for the employment in the single mass pendulum facility in collaboration of
other research groups.
5.2
Interferometric readout
In section 3.4.3 we demonstrate how having two independent readouts of the torsion
pendulum angular position and employing the cross correlation technique we can reject the readout noise. However the noise on the readout contributes to the uncertainty
of our estimations of the external torque acting on the torsion pendulum. To improve
the experimental resolution it is then necessary to decrease the readout noise.
In principle it would be possible to drastically increase the performance of the
gravitational reference sensor as position readout. However this can be done only
increasing the test mass-capacitive sensor stiffness. The increased stiffness would be
not compatible with the small force measurement we want to perform and with the
requirements set by the goal of achieving the purity of free-fall envisioned for LISA.
We would like then to improve the performance of the optical readout. We decided
to look for the possibility of implementing an interferometric torsion pendulum angular position readout. We have been mainly guided in this choice by the achievements
82
of the group designing the interferometer that will be used for test mass ranging in
the LTP experiments. To fulfill the requirements for the interferometer that will fly
onboard of LPF they demonstrated an interferometric setup able to provide an angular sensitivity of better than 2 nrad Hz1/2 in the frequency region between 3 and
30 mHz slightly worsening at the lower frequencies [71, 72]. The baseline for this
interferometer is a heterodyne Mach-Zender design where wave-front sensing with
quadrant photo-diodes is used for the measurement of angles [73]. The LTP interferometer requirement are very similar to ours but the emphasis in the design is on the
measurement of the relative distance of the test masses more than on the measurement
of angles.
Another design that is being investigated is a quadrature phase interferometer
realized with a single light beam. [74]. This interferometer has been designed with
the precise aim of measuring the angular position of a torsion pendulum and has been
demonstrated able of a resolution of 0.68 nrad Hz−1/2 in the frequency region between
10 Hz and 10 kHz with a 0.02 rad range [74]. The interesting features of this design
are the independence of the sensitivity on the thermal drifts and the wide range of
measurement at full accuracy not present in other interferometer designs. We are then
looking into the possibility of making this design compatible with an heterodyne measurement to be able to push its very high performance level into the lower frequencies
region needed by our application. In an heterodyne configuration the light beams going through the two arms of the interferometer are frequency shifted of a frequency
∆ω of order some kHz by means of acusto-optic modulators. When they are made to
interfere the recombined beam measured light intensity has then a component at the
frequency ∆ω. Any difference in the path length of the two beams translates into a
phase shift of the component at this frequency. In a classical interferometric setup the
phase shift between two beams due to an introduced path length difference is instead
measured as a variation of the intensity of the recombined beam. At the very low
frequencies the measurement of intensity variations is limited by electronic noise. In
the heterodyne schema instead what is measured is the phase of a signal with constant
frequency. This permits to use phase locked detectors obtaining high rejection of the
electronic noise.
The introduction of an interferometric readout in the torsion pendulum facility
would also have the interesting advantage of making our testing configuration more
representative of the LTP experimental configuration. The possibility of obtain interferometric data from the torsion pendulum is valuable in the optic of validating the
measurement and data analysis techniques that will be employed in LTP and LISA.
5.3
Possible force sensitivity of the torsion pendulum
We can provide an outlook of the force sensitivity that the torsion pendulum facility
can obtain operating with the fused silica torsion fibre and with the interferometric
readout. We provide an estimation on torsion pendulum force sensitivity by convert-
5.3. POSSIBLE FORCE SENSITIVITY OF THE TORSION PENDULUM
83
ing the torque noise floor into equivalent force by means of a suitable arm length:
SF (ω ) =
S N (ω )
R2φ
(5.3)
Where Rφ is the conversion arm length. For the pendulum geometry a conservative
estimation of the force arm length is 10 mm. We compute the torque noise floor as the
sum of torsion pendulum mechanical thermal noise and readout noise:
S N (ω ) = S Nth (ω ) +
Sφ ( ω )
| H (ω )|2
(5.4)
where H (ω ) is the torsion pendulum transfer function and S Nth is the mechanical
thermal noise computed as in equation 3.6.
Figure 5.1 shows a comparison of torsion pendulum force sensitivity projections
for the new experimental setup and for a possible torsion pendulum employing fused
silica torsion fibre and equipped with an interferometric readout. The force sensitivity
projection for the new experimental setup is obtained by introducing in equation 5.4
the demonstrated readout noises and for the capacitive sensor as for the fit of figure 4.1
converted into torque by mean of the suitable conversion arm length and for the autocollimator as for the fit of figure 4.4. We suppose that the fused silica torsion fibre has
the same torsional elastic constant of the tungsten one but mechanical quality factor
Q = 106 . Furthermore we suppose to have an interferometric readout with an angular
sensitivity of 10 nrad Hz− 1/2 in the frequency range between 0.1 mHz and 1 Hz.
Figure 5.2 is similar to figure 5.1 but introduces the magnetic disturbances. Those
disturbances has been computed as:
SFm (ω ) =
µy 2
S Nm (ω )
µx 2
=
S
(
ω
)
+
SBy (ω )
Bx
k x Rφ
k y Rφ
R2φ
(5.5)
where µ x and µy are the test mass magnetic moment components known from the
last single mass pendulum experimental run, k x and k y are the attenuation factors of
the magnetic shield for the two components as in section 4.3, and SBx (ω ) and SBy (ω )
are the power spectral densities of magnetic field fluctuations in the two components
representative of the laboratory conditions. We observe how the introduced magnetic
shields are already useful at the very low frequency and how are indispensable for
a possible future improved pendulum. For completeness of information we should
note that in run #2775 analysed in section 3.4.6 magnetic shield were installed. They
provided an attenuation of the magnetic field fluctuations of roughly a factor 5. This is
the reason why any correlation with the magnetic field was found during the analysis.
84
5
10
4
S1/2
[fN / Hz1/2]
F
10
3
10
2
10
1
10
0
10
−4
10
−3
−2
10
10
−1
10
frequency [Hz]
Figure 5.1: Force sensitivity projections. The black line is obtained using in the model the
capacitive readout noise demonstrated in the new experimental setup of the pendulum. The
blue line is obtained using in the model the optical readout noise demonstrated in the new
experimental setup of the pendulum. The red line is obtained introducing in the model the
parameters for a fused silica fibre and an interferometric readout as detailed in the text. The
dotted lines represent the thermal noise contribution in the two cases.
5
10
4
S1/2
[fN / Hz1/2]
F
10
3
10
2
10
1
10
0
10
−4
10
−3
−2
10
10
−1
10
frequency [Hz]
Figure 5.2: Same as the plot of figure 5.1 but with the magnetic force noise spectral density
superimposed.
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Toward a third generation torsion pendulum for the femto newton

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