The effect of nongravitational forces - such as air drag and solar radiation pressure -
acting directly on the outer surface of the spacecraft and not on the suspended masses
inside, is twofold. On one side they shake the spacecraft and produce vibrational noise
whose spectral distribution covers a wide frequency range and depends on the particular
spacecraft. This is not a matter of concern for the GG experiment thanks to the PGB
mechanical suspension which is particularly effective at the 
frequency of the signal (Fig. 3 ). On the other side, nongravitational forces
accelerate the satellite itself. Let us consider air drag, which at 
altitude dominates over solar radiation
pressure (the effect of the Earth's albedo is even smaller). The main component of air
drag is in the along track direction of the satellite with relatively large intensity
variations and a slow change of direction over the orbital period of the satellite around
the Earth. If the spacecraft is not designed to counteract air drag, it will loose
altitude and accelerate in the along track direction, with the result that the suspended
bodies inside (the PGB laboratory as well as the test masses) will be subject to inertial
translational forces. The lifetime of the satellite because of its orbit decay is 
, by far longer than a few
hours integration time needed for the EP experiment (Section
6.2 ) and also much longer than a reasonable mission duration which we foresee of
several months to a year. The price we have to pay in order to avoid the complexity and
cost of a sophisticated drag free system is exactly in the need to deal with these
inertial translational forces. However, the first important fact to learn is that, unlike
the forces which act directly on the surface of the spacecraft, inertial forces on the
suspended bodies inside do not depend - in any way - on the surface properties of
these bodies. Whatever the non gravitational acceleration on the satellite, the inertial
force is simply given by the mass of the suspended body times this acceleration, it is
centred at the centre of mass of the body and directed opposite to the nongravitational
acceleration of the satellite. The balance between this force and the restoring force of
the suspending spring gives the new position of equilibrium. Therefore, two suspended
bodies with ideally equal masses and equal suspensions would be subject to exactly the
same inertial forces: were they originally centred on one another with good enough
precision for the EP experiment, they would continue to be centred on one another with the
same precision in spite of air resistance acting on the spacecraft as well as of its
variations.
This amounts to saying that the effect of the translational inertial
forces on the suspended coaxial cylinders of the GG experiment is intrinsically
common mode. Any difference that may arise, due to either a difference in the masses
or a difference in the suspensions, will give a signal at the spinning frequency,
i.e. the same frequency as the signal. However, they differ in phase as well as in the
important property that while inertial differential forces go to zero with the relative
differences in m and k (i.e. they can in principle be made zero by
reducing 
and 
for the two bodies) the
differential force due to an EP violation would still be there no matter how equal
the masses of the test bodies and their suspensions are. The crucial question is
therefore: How much do we need the inertial forces on the suspended test bodies to equal
each other in order to be able to detect an EP differential acceleration 
(i.e. 
)? Making the signals equal is
usually referred to as Common Mode Rejection (CMR). Given the goal in EP testing,
the required CMR factor does depend on the value of the inertial force in common mode,
hence on the intensity of the air drag acceleration on the spacecraft which we therefore
try to keep as low as possible by making it small and compact. We have 
, the same as the
area-to-mass of LAGEOS, the passive spherical satellite devoted to geodynamical studies.
It is compatible with the 3-chamber setup of Fig. 1 and
a total mass of about 
. Once the orbiting altitude has been chosen the air density depends on the
solar index, which changes with a period of 
. We refer to data on atmospheric density provided by the GTDS (Cappellari et al., 1976), as function of the
solar index and satellite altitude (a minimum and a maximum value are given for each
altitude because of the nonspherical shape of the isodensity surfaces of the Earth
atmosphere). The GTDS tables are based on the original atmospheric model of Harris & Priester (1952); Harris & Priester (1962) which has been
modified to include the dependence on the solar index and the asphericity of the
atmospheric bulge. They are used by NASA and other space centres around the world for
mission analysis and have been shown to be in very good agreement with the more recent and
more sophisticated models (Jacchia, 1971; Roberts, 1971). If the mission is flown at altitude and the solar index is 
(the solar index F in the GTDS tables
goes from a minimum of 65 to a maximum of 275), the average air density is 
. The corresponding
acceleration on the spacecraft is: 
, with 
the orbital velocity of the spacecraft and 
its aerodynamic coefficient. The inertial common
mode acceleration on the test masses is therefore 
with direction opposite to the drag acceleration.
The equilibrium position discussed in Section 3 (where
the centrifugal force due to the spin is balanced by the restoring force of the mechanical
suspensions) will be displaced - in common mode - by an amount 
, each body spinning around
its own axis displaced in the new position. Since the precise location of the laboratory,
as well as the test bodies (in common mode!) is not needed, this displacement of the
position of equilibrium - which is the main effect of the translational inertial forces -
is no problem at all.
Indeed, the perturbation due to the air drag can be further reduced by
flying the spacecraft at higher altitude: in going from to 
altitude the EP violation signal decreases only slightly while the drag effect is
significantly reduced due to a much smaller air density. For instance, at altitude during a solar minimum (
in the GTDS tables) the maximum air density
is 
, resulting in
an acceleration on the spacecraft of 
, only slightly exceeding the perturbation due to the solar radiation
pressure: 
(with 
the solar
constant and c the velocity of light). This effect has a phase difference as well
as a small relative difference in frequency with respect to an EP violation signal in the
field of the Earth, as discussed in Section 2.3 . The
effect of Earth's albedo is smaller than the solar radiation effect because of the albedo
coefficient of the Earth (
); it also has
an extremely clear signature because of its following the entering of the satellite in and
out of the Earth's shadow.
The accuracy that can be achieved in the detection of an EP violation differential
force relies crucially on the ability to reject the common mode effect of inertial forces
on the test masses, which in turn is limited by the ability to balance the test bodies.
This because it is much easier to make the masses equal than it is to equal the
suspensions. As we have seen, the largest perturbation to compete with an EP violation
signal is the inertial force resulting from air drag (
), with the same frequency and 
phase difference, while a 
confidence level in the expected signal
requires a sensitivity of 
. If we
show that a total common mode rejection by 
can be achieved, so as to bring the differential effect of the inertial force
due to air drag down to 
then, since a factor 2 for the separation of effects with about phase difference is generally accepted among
experimentalists, we are below the required sensitivity (see Fig. 11 ).
Fig. 11. Qualitative representation, in the orbital plane and for one pair of test masses, of the differential displacements obtained from the synchronous demodulation of the 2-phase
is the differential displacement, directed along the x-axis and constant in amplitude, of the two masses due to an EP violation. The perturbation
due to the initially unbalanced atmospheric drag will be found in the area between the two dotted lines crossing in P: the angle between them is about
, and is due to the fact that the drag has a variable component in the radial direction because of the solar radiation pressure (of amplitude about
times the atmospheric drag and in the Sun-satellite direction). Smaller contributions to the
is the instability due to the internal dissipation of the springs (Section 3.2 ), slowly rotating and increasing: it must be actively damped until Q is as close as possible to D (and P). The circle around point Q represents the error in the measurement due to the thermal noise of the mechanical oscillations in a few days of integration time. The actual values of all these quantities are discussed in the text.
As for the masses, 
cylinders can certainly be made of equal mass to better than 1
part in 
. Making suspensions
springs that are equal to 1 part in and stay equal for the required integration time (see Section 6.2 ) and hopefully longer, is more difficult.
First of all, we must distinguish between springs being equal and springs being stable.
The problem of stable (helicoidal) springs has been widely investigated and techniques
have been developed to reduce the release of accumulated stress. These include annealing
as well as carving from a single piece of material. An important area of application for
stable springs is in the construction of high sensitive gravimeters for accurate
measurement of Earth tides (Melchior, 1978). In
the 1-g environment, in absence of large shocks and temperature variations,
gravimeters with metallic springs have reached an absolute drift value of of 1 part
in 
per day and a thermal stability
of 
(Melchior et al., 1979). These figures come from
data recorded during a 1-yr (1978) measurement campaign at Alice Springs in Australia by
P. Melchior, B. Ducarme and collaborators using the Gravimètre 4084 (Melchior et al., 1979). It has been pointed out
by Dr. Ducarme (Ducarme, 1994) that, although
these figures are instrument dependent, drifts of 1 part in 
per day are common in this field. This is
encouraging, especially taking into account that gravimeters' springs on Earth are subject
to deformations comparable to their length (along the direction of local gravity), while
inside the GG spacecraft the largest acceleration is only 
and the springs are deformed by less than 
. With an absolute drift of 
per day, it would take 100 days before
springs originally equal were to differ by 1 part in .
It is apparent that the capability to reject the common mode effect of
inertial forces on the test masses is crucial for the EP experiment. Let us therefore
consider the problem of manufacturing equal suspension springs for the test masses inside
each chamber. A level 
can be achieved by construction on Earth with computer controlled precision
techniques. Once the springs have been properly built (carved, annealed, etc.) the problem
arises to check their properties, in order to establish to which extent they equal each
other, and to measure the level of flicker noise. It is apparent that hair-like springs
such as these cannot be loaded on Earth with the masses they are supposed to suspend in
space, and a scaled model would not be of great value. It has been suggested by V.B.
Braginsky (Braginsky, 1993) to use appropriate
torsion balances such that motion in the horizontal plane is dominated by the elastic
constant of the springs; by mounting the springs as in Fig.
12 , and using optical measurement techniques it is possible to measure how equal are
the elastic constants, the amount of their variation with time and the level of flicker
noise.
Fig. 12. Testing of suspension springs. By means of a torsion pendulum one can make ground tests of the longitudinal (a) and of the transversal (b) characteristics of the very thin and weak springs that will be used in orbit at zero g. One can test one spring at a time or, as shown in this figure, measure the differences between two of them.
We therefore assume a common mode rejection, by construction, of 
and plan to achieve the required level
of during a calibration phase at
the beginning of the mission in which any signal detected by the read out system is reduced down to the expected
signal. In order to make this possible a coupled suspension of the test masses is proposed
(see Fig. 2 ). The masses are attached by means of soft
springs to the ends of two rods, and each rod is pivoted at its middle point on elastic
gimbals so that it can change orientation by small amounts in all directions. In this way
the frequency of oscillations in common mode depends on the elastic constant k of
the suspension springs, while the frequency of oscillations in differential mode (i.e. of
one mass with respect to the other) depends on 
, where 
is the elastic constant of the torsion wires of the
gimbals and l is the length of each of the four halves of the two rods connecting
the gimbals to the springs. Thus, by adjusting the length of the rods it is possible to
balance the effect of forces which are inherently common mode but nonetheless give a
differential effect because of small differences in the elastic constant of the
suspensions. Were the masses suspended independently (i.e. springs attached directly to
PGB laboratory without gimbals) the only way to balance the effect of perturbations in
common mode would be by in flight changes of the elastic constant of the springs itself,
which would be very hard to achieve and control.
Fig. 13. The system of piezoelectric actuators placed in the two balancing rods. The
and
signs represent the intrinsic polarization of the actuators, i.e. how each one of them must be oriented when mounted. Control voltages are applied to the actuators (when they are applied with the opposite polarity they should not exceed a certain value, which however is relatively high, so as not to risk to depolarize the piezoelectrics): the sum
determines the relative axial position of the barycentres of the test masses and is used for axial centering (Section 4 ). The voltage difference
can be used to change the lengths of the four halves of the rods so as to balance out the effect of transverse inertial forces (in particular the along-track component of the air drag).
Fig. 13 shows an enlargement of the
elastic gimbals and the piezoelectric actuators, with the polarization and the applied
constant voltages whose sum and difference allow us to adjust the axial position of the
barycentres (Section 4 ) and to displace the centres of
mass of the four halves of the two rods. In this way it is possible to compensate for any
remaining differences in the suspensions that would otherwise produce differential motions
of the test bodies under the effect of inertial and tidal forces, the driving signal for
these adjustments being the (demodulated) signal itself. To the achieved level of balance, no inertial force - no
matter how variable - will produce any relative displacement of the test masses. The
system is conceptually similar to an ordinary balance on the ground. Considering that
sensitive balances on the ground, at 1 g, can detect changes of weight of -
(Quinn, 1993), and given the extremely
good properties of piezoceramics for fine adjustments, a similar common mode rejection
factor can be achieved with this system at almost zero g. Once no further reduction
is possible the phase and frequency of the signal must be analysed in order to establish
whether it is due to an EP violation. How can one be sure that an EP violation signal is
not eliminated together with the perturbing effects? This would only be possible for a
competing effect with the same frequency and phase as the signal in the case that it were
also constant in time. If the effects of drag and EP violation were parallel to each other
one could, for one particular value of the drag, balance the sum of the two; but drag is
variable, and therefore balancing would not hold. Furthermore, the two effects are in fact
about apart. Once the largest
common mode effect has been balanced it means that - to this level - the suspensions
respond the same, therefore also balancing all other common mode effects. Later checks of
the observed (if any) signal
should be performed to make sure that there has been no long term variation of the
suspensions which may require to repeat the initial adjustment procedure.
Other experiments which require a good rejection of common mode
effects are worth considering in order to compare the different levels of CMR. Let us
first consider STEP, which is the closest experiment to GG and plans to reach a CMR factor
of 
(Worden et al., 1990; Barlier et al., 1991; Blaser et al., 1993; Blaser et al., 1994; Blaser et al., 1996). There is a major difference
between STEP and GG. The STEP configuration is unfavourable because a displacement signal
about 10 orders of magnitude smaller than the length of the symmetry axes of the test
masses is expected to take place just along the common direction of these axes, which
obviously means that any transversal perturbation in common mode can generate the same
type of signal if the two axes are not sufficiently well aligned. The CMR goal of STEP is
limited precisely by the capability to align the axes of the two accelerometers on one
another. This is not so in GG, where the signal acts in the plane perpendicular to the
spin/symmetry axes of the cylinders and common mode forces can be rejected very
effectively like in sensitive balances on the ground. Instead, STEP relies on drag-free
technology to partially compensate for the drag, which necessarily means relying on the
technology of mechanically tuned Helium thrusters. More advanced thrusters under
development in Europe have been proposed for STEP (Blaser
et al., 1994), namely FEEP, which have the advantages of very high specific impulse,
very fine electrical (rather than mechanical) tuning and negligible mass of propellant
(liquid Cs). However, since a large quantity of He propellant must be carried on board of
STEP anyway (in order to make the experiment cryogenic) and the boiled off He must in any
case be eliminated in a carefully controlled manner in order not to disturb the
experiment, it seems reasonable to use the boiled off He as propellant for the drag
compensation system, as originally proposed by Worden and Everitt. The use of FEEP
proposed by Blaser et al. (1994) for STEP was
instrumental to the competition within ESA and was soon proved not viable (Blaser et al., 1996).
Gravity gradiometers, which have reached a CMR level of (Park,
1990), may appear as possessing similarities with GG. As a matter of fact this is not
the case because they use separate and uncoupled accelerometers. To the contrary, in GG
the test masses are not separate but mechanically coupled: even though they are concentric
and very accurately aligned, the dynamics of their motion is similar to that of balances
and torsion balances (see Fig. 12 b where the wire is
used only for suspending and as an axis of rotation, without torsion) because they are
suspended on the opposite ends of two arms pivoted at their centres, as in balances and
torsion balances. The capability to reject common mode effects depends, in GG, on two
facts: (a) the stability of springs (both in time and with temperature); (b) the
characteristics of the piezoelectric actuators to be used for in flight adjustments of the
length of the arms. None of these crucial issues is of any concern for researchers on
gravity gradiometers or on STEP; hence there is no reason why the CMR factor of the GG
apparatus should have any relation to the CMR factor of these apparata. To the contrary,
the issue of springs stability is typically of great interest for scientists who build and
use room temperature gravimeters for accurate measurements of Earth tides, since spring
gravimeters use similar (metallic) springs and face very similar stability problems.
Although the largest effect of air drag is at the orbital frequency of the satellite,
other low frequency variations (in the range from the orbital frequency to near the
threshold frequency of the PGB laboratory) which are too low to be damped by the
mechanical suspension, cannot be ruled out. They act on the test bodies as inertial forces
of which only the differential effect does matter as far as detecting an EP violation is
concerned; if a total CMR factor of
is achieved, their common mode effect is also reduced by this factor. Furthermore, while
the inertial force resulting from the main along-track component of the drag is seen by
the capacitance sensors at exactly the same frequency as an EP violation signal (the
direction of the acceleration due to air drag changes at the orbital frequency just like
the satellite-to-Earth direction of an EP violation), a higher frequency variation of the
drag is seen by the sensors at a frequency differing from their spinning frequency with
respect to the centre of the Earth (i.e. the frequency of an EP violation) by an amount
which is given by the frequency of that particular air drag variation (Fig. 11 ). Therefore, besides being reduced by common mode
rejection these effects can be distinguished by measuring the rotation rate of the
spacecraft with ordinary star trackers or Earth elevation sensors. After the demodulation
of the signal at (Fig. 11 ) they appear as very regular oscillations (due to
the high Q; see Section 6.2 )) at about with respect to a constant signal, and are
therefore easily distinguishable. By taking the average value of two measurements at a
time interval of half their period we can determine the centre of the oscillations with an
error 
, which is
certainly better than we need.
The GG coupled test bodies have low natural frequencies of oscillation
both in common mode and in differential mode (see Section
3.2 ). There will be air drag disturbances at these frequencies due to air density
variations ("air granularities") over distance scales of about a thousand
km. The corresponding density is smaller than average atmospheric density, typically by at
least a factor of 10. For these disturbances to resonate with the natural frequencies of
the system, they must act at a frequency whose distance from the resonant frequency 
is within the width of the
gaussian, namely 
.
With 
for the test
bodies of the GG experiment (see Section 6.2 ) there
is no way that air granularities over a thousand km can act on the spacecraft so precisely
close to the natural frequencies of the test bodies.
We conclude this section on inertial forces by saying that this
version of GG is designed in such way that EP testing is possible without eliminating the
common mode effect of air drag. Its differential effect on the test masses is transformed
by the mechanical suspension in a difference of inertial forces that can therefore be
adequately reduced, partly by construction in the ground laboratory, partly by small and
fine in flight adjustments at the beginning of the mission. It might be necessary to
repeat the calibration, but only a few times during the 
duration of the mission. Such adjustments of differential effects
are much smaller and less demanding than it would be to counteract the effect of air drag
itself on the spacecraft. Indeed, beside being required only once (or a few times), they
are extremely fine for them to be realized with piezoceramics, whose precision and
reliability as actuators are well known, thus avoiding any thruster firing.
Copyright © 1998 Elsevier Science B.V., Amsterdam. All Rights Reserved.
(Anna Nobili- nobili@dm.unipi.it)
