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:
We must have:
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.
Research Papers Available Online
(Anna Nobili- firstname.lastname@example.org)