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

2. Experiment setup and orbit choice

2.1. The spacecraft, the experimental apparatus and the signal

In the present nondrag-free version of the mission the GG experiment is carried by a small, cylindrical, spin-axis stabilized spacecraft of about 60 cm base diameter, 70 cm height and 600 kg mass. The symmetry axis of the cylinder is, by construction, the axis of maximum moment of inertia so as to stabilize the rotation around it. The fact of not needing any active attitude control reduces the complexity of the mission and the experiment (see Section 2.2 ). The spacecraft is very compact (with an area-to-mass ratio ) in order to make the effect of non-gravitational forces, such as air drag and solar radiation pressure, as small as possible. The orbit is almost circular, almost equatorial at  altitude and the spin axis of the satellite is almost perpendicular to the orbit plane. This maximizes the signal and makes it unnecessary to perform any attitude manoeuvres after the initial setup. The satellite is therefore very close to a truly passive one, which is extremely desirable when carrying out a small force experiment. The outer surface of the spacecraft is available for solar cells so as to generate the required power.

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Fig. 1. Schematic section through the spin axis of the spacecraft showing the spacecraft, the suspended laboratory and the three experimental chambers containing two test masses each. The spacecraft is a cylinder of  height and  base diameter. The figure is to scale and shows that the experiment can be performed in a very small and compact satellite (the area-to-mass ratio is . The central chamber contains two masses made of the same (dense) material for a null test. Each of the other two chambers contains two test masses of different materials, including low density ones. The numbers 1-8 are referred to in Section 3 .

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Fig. 2. Section through the spin axis of the spacecraft showing (not to scale) the spacecraft, the PGB laboratory and (for simplicity) only one experimental chamber. The PGB laboratory and the test masses are suspended with springs and their equilibrium positions can be stabilized by means of electrostatic active dampers (see Fig. 19 for a top view). The suspensions of the test masses also employ "elastic" gimbals (i.e. gimbals pivoted with torsion wires) on two movable rods for the balancing of inertial forces discussed in Section 5.2 . The axial position of each half of these rods can be finely adjusted by means of piezoelectric actuators (see also Fig. 13 ). The capacitive plates of the read out system, between the test masses, are attached to inch-worms for adjusting their distance from the surfaces of the test masses.

Inside the spacecraft it is possible to accommodate three experimental chambers, each one carrying a couple of test masses for an EP violation experiment. Fig. 1 (to scale) shows schematically how actual test masses of  each can be accommodated inside such a satellite. However, it is easier to understand the experiment in the case of a single experimental chamber, as shown in Fig. 2 . Vibrational noise of the spacecraft around the spin/signal frequency is reduced by suspending the test masses inside a low noise laboratory (also of cylindrical shape and maximum moment of inertia with respect to the symmetry axis) which we call PGB (Pico Gravity Box). Inside PGB a very low noise level is attained by suspending it to the spacecraft with appropriate springs of low elastic constant  and low mechanical quality factor . Thanks to weightlessness a mechanical suspension can drastically reduce the vibrational noise of the spacecraft above a low threshold frequency as shown in Fig. 3 .

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Fig. 3. Noise reduction factor (i.e. amplitude of disturbing vibration at the suspended mass over amplitude of vibration at the suspension point) as function of frequency for a suspended laboratory of  and mechanical suspensions with stiffness  and quality factor of order 1.  and all the rest of the system (spacecraft plus test masses).

Passive mechanical suspensions in space for noise reduction in all six degrees of freedom have been the subject of recent extensive work (Nobili et al., 1991) in particular suspensions with low quality factor (which is very easy to obtain, e.g. with PTFE coating) in order to eliminate the resonance peaks (Catastini et al., 1992). The latter work gives an analytical model for longitudinal waves in a thin bar and shows that, with low Q, resonance peaks can be abated while maintaining a very good level of noise attenuation. Such a bar is not effective in the case of transverse waves. However, a mechanical suspension capable to respond with comparable stiffness in all directions (e.g. a helicoidal spring with a length comparable to its diameter) is suitable to reduce noise in all directions. If, in addition, it has a low Q value (for all types of deformations) it will also damp the resonance peaks. In the case of helicoidal springs (other shapes can be suitable too) one can play with the number of turns, their diameter, the way springs are fastened at their ends, the cross section of the wire and the total length of the spring in order to get (for a given spring material) the same, low, longitudinal and transversal elastic constant. If care is taken in using suspensions which have elastic and damping constants of the same order in all directions the analytical model used by Catastini et al. (1992) is a good indication of what should be expected ( Catastini et al. (1992) investigate also the problem of rotational noise showing that it is easier to deal with than the translational one). Laboratory work performed within the VIRGO project has shown that vibrational noise attenuation and damping can be extremely effective even in the more difficult 1-g environment. In order to appreciate the effectiveness and simplicity of a passive noise attenuator in space it is enough to notice that - except during the initial launch phase - the largest acceleration on GG, which is due to friction with the residual atmosphere, is smaller than the local gravitational acceleration on the Earth by a factor , which means that one can suspend  in space inside the GG spacecraft using the same (hair like) springs that one would use for suspending 0.1 milligram in a ground laboratory. Just to give an idea, an elastic constant of  (both transversal and longitudinal) is obtained with helicoidal springs a few cm long made of a few tens of turns each one of cm size and made with a wire of about  diameter. If the spring is coated with PTFE - in order to provide a low quality factor - a transfer function for vibrational noise like the one given in Fig. 3 can be obtained. This analysis has been extended to including the rotation of the spacecraft (Catastini et al., 1996).

It is important to note that the suspension springs of the PGB laboratory, besides ensuring a very low level of platform noise for the experiment, serve also other important purposes. The first is that, with no free floating masses no electrostatic charges will be able to build up anywhere inside the spacecraft. The second is to allow transferring the electric power generated by the solar cells to the experimental apparatus inside. 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. Once at the level of the PGB laboratory further transfer can take place through the rods and the gimbals (see Fig. 2 ).

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Fig. 4. Section across the spin axes of two test bodies with their centres of mass displaced by a distance  due to an equivalence principle violation  in the field of the Earth. For  (and with mechanical properties as given in Section 3.2 ) we get . The centres of mass of the two bodies rotate independently around  and  respectively. The direction of the displacement  changes with respect to inertial space as the satellite orbits around the Earth in .

An EP violation in the field of the Earth results in a differential force between the test masses in the Earth-to-satellite direction which displaces their centres of mass to a new equilibrium where the EP violation force and the restoring force of the suspensions balance each other. At  altitude the differential acceleration of an EP violation at the level  is . Each test body has mass  and is suspended (like the PGB laboratory) by means of two springs with transversal and longitudinal stiffness  (Fig. 2 ). Two rods pivoted on elastic gimbals for each pair of test masses couple the two bodies to one another. The equivalent transverse elastic constant, as derived from the computed natural frequencies (Section 3.2 ) is , hence the relative displacement caused by an EP violation with  is  (Fig. 4 ). Since the displaced equilibrium position is fixed in the Earth-to-satellite direction while the capacitance sensors are spinning, they will modulate this signal at their spin frequency, namely the spin frequency of the spacecraft. In this way the signal is displaced to a higher frequency (by several orders of magnitude) whereby reducing the effect of  noise. We choose for the spin frequency the value of  because it is large enough as compared to the threshold frequency of the noise attenuator to guarantee very good noise reduction, and yet reasonable for a spin-axis stabilized, small and compact satellite (note, for instance, that the european meteorology satellites METEOSAT, whose cylindrical body is more than 3 times bigger in height and diameter, spin at about ). The capacitance bridge is adequately balanced so that common mode displacements at low frequencies, most importantly those at the orbital frequency of the spacecraft (e.g. due to air drag) will give signals always smaller than the differential signal expected from an EP violation (see Section 7 ). These common mode effects at low frequencies must also be adequately rejected (see Section 5.2 ) so that their residual differential effects will not compete with the expected signal. The fact that the entire system is spinning is extremely advantageous because it makes all effects caused by coupling to spacecraft mass anomalies and test masses inhomogeneities to appear as DC effects while the signal of interest is modulated at . The only moving mass on board will be a very limited amount of ordinary propellant which is needed only for the initial orbital and attitude adjustments before unlocking the test masses, and for redundancy. If the propellant is kept in a narrow toroidal tank close to the outer surface of the spacecraft, its motion will be dominated by the centrifugal force, thus ruling out a relative motion at the spinning frequency and therefore any interference with the signal.

In the GG setup, if the spin angular velocity vector  is at an angle  with respect to the orbital angular velocity  of the satellite around the Earth () the intensity of the differential displacement between the test masses as seen by the rotating sensors is of the form:

where  is the relative displacement of the suspended test masses in the satellite-to-Earth direction caused by an EP violation with  and  is the phase of the EP violation signal, which is known. The factor  comes into play in case the angle  is not zero ( is the phase angle of the sensors with respect to the satellite-to-Earth direction). If  (i.e. the spin axis is exactly perpendicular to orbit plane) , whereas for any  this factor does reduce the intensity of the EP violation effect and introduces a dependence also on the orbital period of the satellite. This is why the spin axis of the satellite should be perpendicular to the orbit plane. It also leads to choosing an equatorial orbit for the satellite.

2.2. The orbit and the attitude

Because of the flattening of the Earth, the ascending node of a satellite orbit not exactly equatorial would regress along the equator, i.e. the normal to the orbit plane would describe a cone around the normal to the equator. The spin axis of the satellite, if not exactly normal to the orbit would in turn precess around the normal to the orbit because of the effect of the Earth's monopole on a body - the satellite - with different principal moments of inertia. Thus, even if the spin axis and the normal to the orbit were originally aligned, they would no longer be so after a few tens of days. Attitude manoeuvres would then be necessary to realign the spin axis to the orbit normal in order to have a factor  in Eq. 2 , hence to maximize the effect of an EP violation. This may require to activate the locking-unlocking device (Section 10 ), which would complicate the mission. Instead, if the satellite is originally injected in an orbit close to equatorial with the spin axis close to the normal to it, the spin axis and the orbit normal will stay close to one another (by the same amount) and attitude manoeuvres will not be required. In addition to that, the equatorial orbit has - if low enough - the advantage of avoiding the perturbing effects of the radiation from the Van Allen belts in the so-called South Atlantic Anomaly. An altitude of  is suitable for this purpose. We therefore assume an equatorial, low eccentricity orbit at  altitude and allow for an angle of a few degrees between the spin axis and the normal to the orbit plane as well as the inclination of the orbit on the equator, which are rather relaxed constraints for orbit injection. In this configuration no active control is needed, neither of the attitude nor of the orbit.

The satellite should be equipped with ordinary star trackers or Earth elevation sensors in order to monitor its spin rate and its instantaneous orientation. Although a predetermined spinning frequency is not needed, a knowledge of the actual spin rate or, more precisely, of the angle  at all times, is required in the process of data analysis for removing small perturbations close to the signal frequency (Section 5.3 ), for checking purposes and to provide the electrostatic damper with a reference signal synchronized to the spin (Section 3.1 ). For communication with the Earth several choices are possible (see GALILEO GALILEI, 1996); a despun antenna should be avoided because moving parts would disturb the experiment. Since the orbit is low and equatorial the satellite will be in view of the ground station only for a fraction of its orbital period. There is no special need for continuous tracking; the experimental data can be stored on board and down loaded once per orbit. The required bit rate is low.

2.3. EP violation signal driven by the Sun and other sources

While moving around the Earth the test masses will also orbit, together with the planet, around the Sun. Therefore, the equivalence between inertial and gravitational mass can also be tested by comparing the gravitational attraction of the Sun with the centrifugal force due to the orbital motion around it. In this case the acceleration of an EP violation is , with  the mass of the Sun acting at its distance from the satellite (in practice the Earth-Sun distance , namely ), and  the Eötvös parameter expressing the violation of the equivalence between inertial and gravitational mass for the test masses in the field of the Sun. Since , while  it is apparent that our experimental apparatus cannot detect an EP violation due to the Sun to the same accuracy as for the Earth. We shall have . For instance, if the experiment is limited to , an EP violation due to the Sun can be tested to , which would be better than achieved on Earth so far. The signal on the sensors will have a frequency which differs from that of an EP violation signal in the field of the Earth by the orbital frequency of the satellite. It will also be modulated by the annual motion of the Earth around the Sun. The two frequencies, from the Sun and from the Earth, can therefore be distinguished.

Similarly, one can analyze the data searching for possible violations of the equivalence principle driven by other sources such as the galaxy. Naturally, the sensitivity that can be achieved will depend on the intensity of the driving signal in each case, which however for the Sun and other sources farther away will be the same as it is on the ground. Indeed, all efforts towards more sensitive ground apparata for testing the equivalence principle should be strongly encouraged because their contribution is unique at short range and very valuable over distances much bigger than the radius of the Earth.

Copyright © 1998 Elsevier Science B.V., Amsterdam. All Rights Reserved.

Research Papers Available Online

       (Anna Nobili-