It is well known that a small force gravitational experiment in space should avoid the presence of nearby moving masses. Therefore, were a refrigerating material carried on board the spacecraft - close to the apparatus - in order to lower the temperature, it should be accurately confined, which is neither easy nor inexpensive. We choose to operate the experiment at room temperature and find that, because of the (relatively) high spin rate of the spacecraft and the intrinsic differential nature of the signal, it is possible to obtain an adequate level of temperature stability by passive thermal isolation only (Section 6.1 ). As for thermal noise, we use bigger test masses than STEP in order to compensate for the fact of operating at higher temperature (Section 6.2 ). A capacitance read out system which exploits the differential nature of the experiment can provide an adequate measurement accuracy (Section 7 ) with no need to resort to a low temperature measurement device.
For the reasons discussed in Section 2.2 a 
altitude, almost equatorial, almost circular
orbit is the best choice for the experiment. However, on this orbit the satellite spends
almost half of its time in the shadow of the Earth and the rest in sunlight, thermal
equilibrium temperatures in the two cases differing by several tens of degree. While
azimuthal temperature variations are inexistent because of the the fast spin, temperature
gradients between the illuminated and the dark side of the satellite when exposed to
radiation can in principle be very large. These gradients can be essentially eliminated
inside a rapidly spinning spacecraft if it is properly insulated. Insulation and vacuum
serve also the purpose of reducing the rate of temperature variation with time.
Temperature stability in time inside the PGB laboratory should be of 
for the required integration time of about
2.6 hours (Section 6.2 ), but possibly longer.
Vacuum is needed also not to reintroduce acoustical noise on the PGB
laboratory and the test masses. The vacuum level that can be achieved by means of a hole
to open space is that corresponding to atmospheric density. At altitude (where atmosphere is mostly
constituted by molecular oxygen) 
, hence 
. In fact the pressure may be about a factor 
bigger because of outgassing. However, care should be taken in
avoiding materials with high levels of outgassing and in setting the size of the holes on
the basis of the outgassing area. Experience with resonant bar antennas for the detection
of gravitational waves shows that gold coated Kapton should be used instead of Mylar
because of its lower outgassing level. Any anisotropy in the internal outgassing (from
some particular spots) rotates with the test bodies and the sensors. It therefore gives an
essentially DC effect.
The heat sources that the satellite is exposed to are the Sun (
at the Earth's distance) and
the Earth itself, in the visible as well as in the infrared range. The Earth emits a
fraction of the sunlight given by its albedo (
on average, although widely variable) i.e. 
. In the infrared, at the surface, we have 
. A source of heat from a
given direction will produce a temperature gradient across the satellite between the side
facing the source and the one away from it, and eventually a gradient across the test
bodies and the suspended springs. A temperature gradient across the test bodies is a
source of radiation pressure, and since the satellite is spinning, the resulting signal is
modulated at the spinning frequency (as measured with respect to the source of heat). The
spinning frequency with respect to the Earth, namely the source of an EP driving signal,
is not the same as the one with respect to the Sun, and they can be distinguished from one
another. However, possible temperature gradients due to the infrared radiation from the
Earth would have exactly the same signature as the EP violation we are testing. We must
therefore make sure that internal temperature gradients be adequately small. Indeed, we
can see that temperature gradients across a rapidly spinning spacecraft can be reduced by
many orders of magnitude by means of an insulating outer shell and the vacuum inside.
A cylindrical shell spinning with period P and exposed to the
solar flux 
(e.g.
perpendicular to the spin axis) will have a temperature gradient 
where 
is the density of the
shell, 
its
absorption coefficient (absorbed to impinging flux), w its thickness and 
its specific heat. For a spin
period 
and 
the typical value of 
is around 
degree for a good insulator (e.g. glass) and 
for a good heat conductor
(e.g. Copper). However, if the spin period of the shell is much smaller than the timescale
of its thermal inertia, it is as if it were subject to an isotropic flux, hence resulting
in a negligible gradient.
Let the inner surface of the spacecraft cylindrical body be covered with an insulating shell; the temperature behaviour across the shell will be given by:
where 
is the
satellite initial temperature (when injected into its orbit), 
is the temperature of
external environment and d is the penetration depth. Since the spacecraft spins
fast we can assume that is the same all over its surface. We have:
where , and 
are the density, specific
heat and thermal conductivity of the insulating shell. Taking for these quantities typical
values for Kapton or Mylar (as given by DUPONT or found by Immergut, 1984) we get 
. If the insulating shell has a thickness
of 
, it is apparent from Eq. 29 that gradients of temperature in the plane
perpendicular to the symmetry axis are negligible. As for the top and bottom sides of the
cylindrical spacecraft, they can be coated with reflecting material (mirrors) in order to
reduce solar heating, since it would be differential. This technique is widely used, e.g.,
in geosynchronous satellites. In addition, it is worth recalling a very important
peculiarity of the GG experiment, namely that the spacecraft is connected to the
experimental laboratory only through the hair-like suspension springs of the PGB
laboratory (inside which the requirement on thermal stability must be achieved) and that
there is vacuum inside the spacecraft. Hence, heat is transferred essentially by radiation
(see below). Temperature gradients along the symmetry axis modulated at the orbital
frequency would give an effect at the spin/signal frequency. Let us consider the effect of
axial gradients on the arms. A differential expansion of the arms in the coupled
suspension of the test masses would destroy the balancing and therefore reduce the
capability to maintain the required CMR factor of 
discussed in Section 5.2 . We
therefore need:
If 
is the expansion coefficient of
the arms and z the axial coordinate:
Since materials with 
are common
and can be used for manufacturing the arms, it follows that axial temperature gradients
over the arm's length (a few cm) must be smaller than , which is not a stringent requirement. Were it needed, one could
make the arms of a material with 
,
thus allowing for gradients of 
.
The average temperature of the spacecraft changes because of it going
in and out of the Earth shadow, i.e. the relevant frequency is the orbital one. In order
to keep the temperature stable with time we must reduce the incoming heat. Since the
cylindrical surface of the spacecraft must be covered with solar cells (in order to
generate the necessary electric power) the external surface will absorb most of the heat
and reach, more or less, equilibrium with the solar radiation. Hence, we need - on the
internal side of the spacecraft - an insulating shell in order to reduce the amount of
heat that will reach the interior of the satellite. The timescale 
of thermal inertia provided by an insulating
shell of thickness w is:
with , and the density, specific heat
and thermal conductivity of the insulating material. Then, the time variation of the
temperature at the internal surface of the insulating shell is:
( and as defined above). Let us
consider one half satellite orbit in sunlight. At the end of half period in sunlight the
temperature of the internal surface of the insulating shell will be:
where 
is the
equilibrium temperature in sunlight. Using 
and the properties of materials like Kapton and Mylar we have 
, while 
. The corresponding temperature variation (after
half period in sunlight) is:
Similarly, the temperature variation after half period in the shadow of the Earth
(darkness) is:
where 
is the
equilibrium temperature of the satellite when in the dark. Let us consider a large
difference of 
between equilibrium
temperatures in sunlight and in darkness (when the satellite is exposed only to the
infrared radiation form the Earth), and a satellite initial temperature halfway between
the two. Then, the temperature variation on the internal side of the insulating shell
after half orbit period will be (in absolute value) 
Obviously the two temperature variations (increase
when in sunlight and decrease when in darkness) will not cancel out. Let us take half of
each. This means that, after one full orbital period of the satellite around the Earth the
temperature on the internal side of the insulating shell is changed by:
This is too much for our requirements. We now exploit the important fact of having good
vacuum inside the spacecraft, because it ensures that there is only radiative transfer of
heat (apart for the thin hair-like suspension springs of the PGB laboratory). Indeed, the
mean free path of gas molecules at room temperature and 
is about 
,
while we have about 
(having
allowed a factor for outgassing).
We can also coat the external surface of the PGB with Kapton. Then, the amount of energy
transferred from the insulating shell to the Kapton coated external surface of the PGB
after 1 orbital period is:
where 
is the Stefan-Boltzmann
constant, 
is the
emissivity of kapton and A the area of the surface involved (
). The corresponding temperature variation
of the PGB laboratory (made of Copper) is:
where 
is the
specific heat of Copper and 
is the mass of the PGB laboratory. We get 
. After 5 days, i.e. about 76 orbits of the
satellite we shall have:
Multilayers of insulating material are usually used in cryogenic experiments (Haselden, 1971), because, if properly separated in
order to reduce heat conduction, they are known to provide a reduction of the transferred
power proportional to the number of layers employed. For instance, this technique is
successfully used to reduce the amount of input power on the very large cryostats which
enclose resonant bars (
) for low
temperature detection of gravitational waves (the EXPLORER antenna at CERN, in Geneva, and
the NAUTILUS antenna in Frascati, Rome). Our problem is easier because of both the smaller
size of the device and the very tiny connection between the spacecraft and PGB. In any
case, from Eq. 41 it follows that a number of about 30
Kapton layers (such as those commercially available from DUPONT, of 
thickness each) can provide
the required level of thermal stability.
As far as internal power sources are concerned, we recall that they will be rotating together with the entire system, so all resulting effects will be DC. We shall also take care to use low dissipation components. In addition, only the preamplifiers will be positioned on the supporting rod inside the hollow cylinder test bodies; all the remaining power sources will be placed on the internal surface of the spacecraft shell. The required number of wires can be accommodated either as independent helicoidal springs or by grouping them on a plastic support without any serious problem of degrading the reduction of vibrational noise since a low Q is required for the PGB suspensions. Most of the wires will go to the inch-worms, to the active dampers, to the locking-unlocking mechanisms and to the preamplifiers of the capacitance read out while only 4 wires go through the gimbals to the piezoceramics on the balancing arms (see Fig. 2 ). For the passage through the multilayer insulation on the outer side of the PGB laboratory we can use, if needed, a technique known as thermal sink, whereby the wire does not go straight inside but rather makes many turns so as to largely reduce the temperature gradient that it will bring in. This technique is currently employed in the EXPLORER and NAUTILUS cryogenic resonant bar antennas mentioned above.
LISA (Laser Interferometer Space Antenna) (Bender et al., 1994), a proposed space mission
for the detection of gravity waves, is another example of a noncryogenic space experiment
where a very good thermal stability is needed, to be achieved passively. For LISA the
requirement is of 
around 
. LISA has the advantage,
over GG, to be subject to a much smaller variation of the solar flux due to its
heliocentric orbit. However, GG has the advantage, over LISA, of spinning fast (while LISA
is space stabilized). In the ground torsion balance Eöt-Wash experiment (Su et al., 1994) (also noncryogenic) they use two
active temperature stabilization systems based on several temperature sensors in order to
achieve a stability of 
across the
experimental chamber. We recall that their balance rotates very slowly, with a period
of 
, while the spin period of GG is
only 
.
Residual gas particles inside the spacecraft accelerated by a
temperature gradient between two sides of a test body would result in an acceleration
necessarily different for the two bodies: 
with 
the density of the test mass. This is the radiometer effect that the STEP
experiment is concerned about. The reason why this is so is that in STEP the signal is
along the symmetry axis of the test cylinders; for a given residual gas pressure any
temperature gradient between the two bases of each cylinder will result in an acceleration
along its symmetry axis, and it would inevitably be a differential acceleration; if
temperature variations are modulated at orbital frequency this would mimic an EP
violation. In STEP this problem is solved by having an extremely low residual gas pressure
(as low as 
, made possible by very
low temperature) and a requirement for temperature gradients across each test body not to
exceed 
. In the GG design this
effect is of no concern because the signal is normal to the symmetry axis of each
cylinder, hence if its inner and outer surface are at different temperatures, the net
force by thermally accelerated residual gas would be zero for symmetry reasons. Only if
temperature dos not have azimuthal symmetry there can be a radiometer effect; which is not
the case because of fast spin.
The effect of viscous drag due to the residual gas on the centre of mass of the sensing bodies is well below the signal even under conservative assumptions. The effect is estimated according to Milani et al. (1987), Ch. 6, taking into account that the thermal velocity of the gas molecules is much higher than the velocity of the test bodies.
We must not forget that solar cells will necessarily cover the outer
surface of the spinning spacecraft whose temperature gradient can be as large as of a degree. It is only after
an appropriate insulating shell that this gradient is significantly reduced. Therefore we
must expect that the outermost layer of the spacecraft (essentially the solar cells) will
be subject to nonuniform thermal expansion at the spinning frequency. However, since the
part of the spacecraft involved in this oscillation is only the external one (because
temperature gradients are negligible inside) the result is a forced term acting on the
suspension at frequency 
, which will
be reduced according to the transfer function of Fig. 3 ,
and the effect can be neglected. We must also expect that the outermost layer of the
spacecraft will be subject to nonuniform thermal expansion at the orbit frequency,
resulting in a common mode oscillation of the test masses at that frequency similar to the
effect produced by the along track component of air drag. With a typical expansion
coefficient 
and a temperature
gradient which, thanks to the rapid spin, is not larger than about 
, the oscillation amplitude of
the outer shell will be of a few 
.
A common mode rejection of 
is
enough to make this effect smaller than the signal (Section
5.2 ). As for the corresponding gravitational effects, they are found to be
negligible. The expansion/contraction of the outer shell of the spacecraft (due to it
entering and exiting the shadow of the Earth), the resulting changes in moment of inertia
and spin rate, and how to compensate for them are discussed in GALILEO GALILEI (1996) (Ch. 2.1).
We now investigate the effects of thermal expansion of the test
bodies. Let us first consider the simpler case in which the test bodies expand uniformly.
They will have expansion coefficients of the order of 
, because their composition is selected on the basis of EP
violation considerations and obviously not with the purpose of minimizing their thermal
expansion. Furthermore, their expansion coefficients will not be the same, and indeed we
shall assume that they differ by its entire value, namely that there is a difference in
their expansions by too. However,
as long as they expand uniformly, there is no relative displacement of their
centres of mass and therefore no differential effect on the read out capacitors.
But test bodies will not be perfect, and in correspondence of
inhomogeneities in their mass distribution there might be a different response to
temperature variations. However, the direction of non uniform expansion being fixed with
the test body it is also fixed with respect to the sensors, which means that its effect is
DC and does not compete with an EP violation signal. Furthermore, it is also small because
of it being proportional to inhomogeneities of the test bodies. With a not too stringent
requirement on the mass density such as 
, and in the conservative assumption that mass inhomogeneities do
necessarily imply a nonuniform thermal expansion of the same level (i.e. with a
proportionality factor of 1), we get a relative motion of the centres of mass by
about 
, which means (for linear
dimensions of about 
) differential
displacements in the direction of the mass inhomogeneity by 
, which is perfectly satisfactory for a DC
signal.
We must also consider the fact that a uniform (but different)
expansion of the test bodies will change the distances a and b of the
capacitors from the outer and inner surfaces - respectively - of the test bodies (Section 7 ). We shall see in Section
7 that in order to be sensitive to an EP violation of 
the read out capacitance plates must be centered
between the test masses to within 
Å which, in relative terms means 
.
Active balancing to this level is done with inch-worm actuators (see Section 7 ) and can be repeated if necessary. A difference
in the expansion coefficients by
and a thermal stability of allow
us to maintain this level of balancing. However, were it necessary, there is a possibility
to reduce the differential displacement of the surfaces of the test bodies with respect to
the capacitance plates in between. We can build the frame which supports the capacitors
(see Fig. 2 ) using an appropriate alloy whose expansion
coefficient must be such to compensate for the differential expansion of the test bodies.
Once the material choice for the EP experiment has been made and the test bodies have been
built their radii and thermal expansion coefficients can be accurately measured, thus
uniquely determining the required value for the expansion coefficient of the frame. In
this way it is possible to reduce the differential expansion by about a factor of 10, thus
reducing the relative displacement.
Finally, temperature variations will affect the stiffness of the
suspension springs and therefore change the value of the transversal elongation 
in response to the inertial force caused by
air drag acting on the spacecraft:
since is the required level of
CMR. With a maximum temperature change of and 
as obtained in gravimeter springs (see Section 3.2 )
the effect is 10 times smaller than it is needed, thus allowing for a less good thermal
stability of the springs.
All the above analysis of the perturbing effects to be expected in performing the experiment at room temperature is based on simple physical principles; a more detailed thermal analysis has been conducted in GALILEO GALILEI (1996) taking advantage of space industry experience on the subject.
Test masses will have their own mechanical thermal noise, resulting in a perturbing
thermal acceleration on the test mass 
that must be smaller than the signal acceleration 
. At room temperature 
we have:
where 
is the
Boltzmann constant, 
is the mass of each test body, 
their natural frequency for differential oscillations, Q the
mechanical quality factor and 
the integration time.
If we take the 
value, as used in the numerical simulation of Section 3.2 , we get 
, which means that an integration time of
about 
is necessary
in order to reach a signal-to-noise ratio of 2. However, we have now manufactured
helicoidal springs similar to the ones to be used in the space experiment to suspend the
test masses GG (GALILEO GALILEI, 1996). By
setting the spring in horizontal oscillation (for the oscillations not to be affected by
local gravity) with vacuum, temperature and clamping similar to those expected in the
space experiment, and at the same frequency at which it will spin - hence undergo
deformations - in space, we could measure the mechanical quality factor getting a value
of 
. Since losses due to the
electrostatic dampers are much smaller (Section 3.2 )
and all other parts are rigid with no expected dissipation, this result gives a realistic
value for the total mechanical losses (further improvement may be possible). Using 
in Eq. 45 we get 
and an integration time of about 2.6 hours for a
signal-to-noise ratio of 2 as before.
Note that in Eq. 44 , Eq. 45 we have the natural frequency of oscillation of one
test body with respect to the other instead of the (higher) natural frequency of
oscillations in common mode (
). This is correct because, since the two frequencies are not close (they
differ by about a factor of 2) and Q is high, the bandwidth of noise is so small
that there is no significant contribution from the thermal noise in common mode to the
thermal noise in differential motion where the effect of an EP violation would appear. It
is also worth noticing that the perturbing effect of thermal noise is proportional
to 
, which explains
why - working at room temperature - it is important to have test bodies of relatively
large mass. In comparison with STEP we work at 100 times higher temperature and therefore
use test bodies about 100 times more massive. As for the thermal noise perturbation due to
the PGB laboratory, i.e. the common mode thermal noise of the platform, it is given by
formulas similar to Eq. 44 , Eq. 45 with the natural frequency, mass and quality factor
of the PGB laboratory (higher mass, lower Q). Being a small common mode effect it
is easily rejected.
Copyright © 1998 Elsevier Science B.V., Amsterdam. All Rights Reserved.
(Anna Nobili- nobili@dm.unipi.it)
