Nobili et al. / Proposed noncryogenic, nondrag-free test of ...

5. Inertial forces

5.1. The common mode effect

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.

5.2. Rejection of the common mode effect

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 ).

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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  signal. The x-axis is in the Earth-to-satellite direction and the vector  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  vector come from the Earth albedo, the Earth infrared radiation and, by a smaller amount, from a possible small eccentricity of the orbit. By finely adjusting the lengths of the suspension arms the point D is displaced up or down inside this area, and this balancing of the drag should be continued until D is as close as possible to P. In doing so, also the radial component is automatically balanced. The resonant variations of the drag (not shown) will oscillate inside the same area. The vector  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.

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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.

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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.

5.3. Low frequency effects

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.

Research Papers Available Online

       (Anna Nobili-