GG is an experiment in space to test the Equivalence Principle through a test of the so called Universality of Free Fall, namely that two bodies of different composition orbiting around the Earth should move with the same acceleration, hence along the same orbit. In order to eliminate major, purely classical, differential effects due to non uniformity of the gravity field of the Earth the test bodies are concentric hollow cylinders, as they were in the STEP project since its very beginning.

                      2.1

Figure 2.1 (to scale) Section through the spin axis of the GG test cylinders (10 kg each) and the capacitance plates of the read-out in between. The lower density cylinder (21 cm in height) encloses the higher density one. Inside the inner cylinder is a narrow tube rigidly connected to a laboratory (also of cylindrical shape) called PGB (Pico Gravity Box) enclosing the test bodies and the read-out capacitance plates shown here. The PGB in its turn is mechanically suspended inside the spacecraft (not shown; see Fig. 2.8 for an overall view). For coupling the test bodies there are two "coupling" arms (shown in light blue) located inside the PGB tube but not in contact with it; the inner test cylinder is suspended from the coupling arms at its center by means of two helical springs; the outer one is also suspended from the arms with helical springs, one at the top and one at the bottom of its symmetry axis. The only connection between the coupling arms and the PGB laboratory is via two flat gimbals at the midpoints of each arm. Being pivoted on torsion wires the gimbals allow conical movements of the coupling arms around their midpoints, e.g. in response to a differential force between the test bodies. The piezoelectric actuators shown next to the gimbals are for adjusting the length of the two halves of each coupling arm. The capacitance plates of the read- out are shown in between the test cylinders; they are connected to the PGB tube and have inch- worms for adjusting their distance from the surfaces of the test cylinders. On the PGB tube are shown the mechanical stops which constrain the test bodies to only slight movements. The small capacitance sensors/actuators (with plates of about 2 cm²) are for sensing and damping the slow whirl motions of the test bodies with respect to the PGB (see Sec. 2.1.5 and Chap. 6).

Figure 2.2 One of the 4 helical springs to be used for suspending the GG test bodies; two such springs are needed for each test body (as shown in Fig. 2.1). This spring has been manufactured in CuBe by DG Technology Service, Parma (Italy) according to our design. They have manufactured the spring by electroerosion in 3D from a single piece of CuBe with special equipment, and then have applied Brush-Wellman heat treatment and ultrasound cleaning. The elastic properties are close to the desired ones. Each spring is clamped by the thick rings at its ends. It is well known that most energy losses (which reduce the mechanical quality) occur at the clamping, no matter what clamping means are used - screws, welding, glue…; if clamping takes place far from where deformations occur during the oscillations (such as the thick rings in this case), this will reduce clamping losses significantly. Half turns of this spring are clockwise and the other half counter- clockwise; this is for de-coupling torsional from longitudinal (axial) oscillations.

Figure 2.3 One of the two flat gimbals to be used for coupling the GG test bodies (as shown in Fig. 2.1). The outer ring is clamped to the PGB tube and the inner one to the coupling arm (in Fig. 2.1 the PGB tube is shown in dark green and the coupling arm in light blue). There are 6 wire sectors in between the clamping rings; 3 of them (in alternation) carry electric signals and are insulated at the clamping (on the outer and inner clamping rings); no electric insulation is needed on the thin wires themselves where deformations occur. The manufacturer of the gimbals is the same as for the helical spring; the same heat treatment and cleaning procedure have been applied.

The way each gimbal is mounted is shown schematically if Fig. 2.4, where it is apparent that it allows conical movements of the coupling arm at its midpoint. By commanding the piezoelectric actuators it is possible to change the relative axial position of the centers of mass of the test bodies, allowing axial centering; in addition, it is possible to change the length of the two halves of the arm, which is extremely important for balancing the test bodies under the effect of the residual drag (Sec. 2.1.4, Fig. 2.13). Note that, if inch-worms are used rather than ordinary piezo, it is possible to switch off the electric potential once the desired adjustment has been achieved; in this way they will not disturb the EP measurements nor produce joule heating inside the PGB (small force gravitational experiments should be as passive as possible).

Figure 2.4 Schematic view of one flat gimbal pivoted on torsion wires. The inner ring of the gimbal is clamped on the coupling arm (in light blue) and the outer one on the PGB tube (of which only a section is shown). The length of the two halves of the arm can be changed by means of the piezoelectric actuators (schematized by their end faces). Torsion of the wires allows conical movements of the coupling arm at its midpoint.

Low Noise Suspended Laboratory (PGB). As shown in Fig. 2.1 that the only connection between the (coupled) test bodies and the "outside world" is through the flat gimbals. It is very important for such a connection not to be directly to the spacecraft, but through an intermediate stage, in its turn suspended from the spacecraft. This intermediate stage is a cylindrical laboratory whose mechanical suspension from the spacecraft (also a very weak suspension, as it is possible in space), provides an effective, passive (i.e. at essentially no cost) vibration isolation above its natural frequency of oscillation. This laboratory is shown in Fig. 2.8 under the name of PGB (Pico Gravity Box), from the original name given to a passive noise attenuator of this kind to be used for low gravity experiments, typically in a box of available volume, on board the space station (Nobili et al., 1991, Catastini et al., 1992).

Figure 2.5 One suspension spring of the PGB laboratory

In summary, the weak mechanical suspensions of the PGB - which can be used only thanks to weightlessness- provide an effective, passive means of isolation from the (relatively) high frequency vibrations around the spin/signal modulation frequency; in addition, they provide electric grounding (see Sec. 2.2.4) and make the PGB laboratory essentially thermally de-coupled from the spacecraft (see Sects. 5.4 and 4.4). All of these are very important advantages for the purpose of achieving a high accuracy test of the Equivalence Principle.

(2.2)

Finally, as mentioned in Sec. 1.4, mechanical suspensions allow electric grounding of the test bodies and eliminate the major effects of electric charging. There are no floating bodies in GG, as it is shown in Fig. 2.8.

Figure 2.6 Transfer function of the GG Spacecraft/ PGB system as seen in the non rotating frame (upper figure) and in the frame rotating at 2 Hz (lower figure) together with the whole system. The noise reduction factor plotted on the vertical axis is the ratio: amplitude of disturbing vibration at the PGB level to amplitude of vibration at the GG spacecraft level. The lower this ratio, the lower the platform noise of the experiment, since the experiment is carried out inside the PGB. We consider the two cases Q=90 (currently measured) and Q=400 (readily achievable). In both figures we plot also the transfer function in the zero-spin case (i.e. PGB suspended inside the spacecraft with the system not spinning) to recover the familiar shape of the transfer function, with noise attenuation above the natural oscillation frequency of the system and the resonance peak; the peak height is low for a finite, low, Q. As viewed in the non rotating system (top), the transfer function is not changed very much by the spin because it depends on the elastic properties of the suspensions, and rotation does not bring in any dramatic changes (deformations are minute). It is apparent that the system is transparent to frequencies lower than the natural frequency of the system, such as the orbital frequency while it provides good attenuation close to the spinning frequency (the modulation frequency of the EP violation signal). When viewed in the rotating frame (bottom) there is a peak with value 1 at the spinning frequency, showing that vibrations at very low frequency w.r.t. the fixed frame, particularly the DC ones, are not attenuated; the observer co-rotating with the system sees these DC perturbations as 2 Hz, and finds that the suspension does not reduce them, namely that it is transparent to 2 Hz effects. Perturbations which are seen at 2 Hz by the non rotating observer (and attenuated), have frequencies 0 and 4 Hz for the rotating one, and in fact he too finds that they are attenuated.

Alternative Weak Coupling Design. Although helical springs and flat gimbals have many advantages and allow to weakly couple the GG test cylinders as it is required for high accuracy EP testing, they are not the only solution. An alternative design to the one shown in Fig. 2.1 and discussed above, is shown in Fig. 2.7, based on thin, curved laminar suspension strips to be manufactured from a CuBe foil in the required curved shape shown in the figure (they should not be bent elastically) so as to avoid release of accumulated stress. The advantage of this design is to be more sensitive to differential forces and stiffer in common mode. The laminar suspensions should also be easier to manufacture and handle.

Figure. 2.7 Alternative design, with respect to the baseline one shown in Fig. 2.1, for a weak coupled suspension of the GG test cylinders; section through the spin/symmetry axis. The two balancing arms are twice as long as in Fig. 2.1, which increases the sensitivity by a factor of 4. The bodies are coupled by thin, curved laminar suspension strips; there are 3 of them at each level, symmetrically placed around the axis (there can be more, but always symmetric). The particular geometry of mounting them ensures that the pivoting points of the two balancing arms are well defined and that those at the center of the rotor coincide well one with the other (at the cross shown). These suspensions are very soft for lateral differential oscillations of the two test masses and at the same time are particularly rigid for unwanted motions, like common mode lateral oscillations and rotational oscillations around the spin axis. They have enlarged ends for clamping (so as not to add dissipation at the clamps), and those in the central part of the rotor are fastened by electrically insulating clamps, so that these suspensions can be used as conducting leads between the PGB tube and the two balancing arms for the driving voltages of the four axial inch-worms, which are used for balancing the two rods and for adjusting the axial position of the two test masses. Once the desired balancing has been achieved the driving voltages of the inch-worms are switched off, leaving them blocked; in this way they will not disturb the EP measurements nor produce joule heating inside the PGB laboratory. The small capacitance sensors/actuators for sensing and damping the whirl motions, and the capacitance plates of the read-out are shown, as in Fig. 2.1.

2.1.2 Implications on Spacecraft, Attitude and Orbit

Need for a Spin Stabilized Spacecraft in Low Earth Orbit. The test bodies, their mechanical coupling and the capacitance read-out are the core of the GG mission. Once the experimental design outlined in the previous section has been conceived, the features of the required spacecraft, its attitude and orbit are also identified. In the first place, the cylindrical symmetry of test bodies and PGB and the request to spin, suggest a spacecraft of cylindrical symmetry too, stabilized by one-axis rotation along the symmetry axis. The nature of the signal (see Fig. 1.1 and Eq. 2.1) requires the spin axis to be as close as possible to the orbit normal; the need to reduce non gravitational perturbations on the spacecraft surface (Sec. 2.2.1) suggest that it should be small and compact (which in addition helps reducing its cost); the need to reduce perturbations on the test bodies from nearby moving masses suggests to use thrusters of high specific impulse (such as FEEP; see Sec. 4.2) in order to reduce the amount of propellant required for drag compensation during the mission.

A section of the GG satellite through its spin/symmetry axis showing how the PGB and the experimental apparatus is accommodated, in a nested arrangement inside it, is shown in Fig. 2.8; a 3-D view is given in Fig. 2.9 (see Chap. 5 for details). The spacecraft is 1 m wide and 1.3 m high. The area of the external (cylindrical) surfaces covered by solar cells is dictated by the power needs of the mission (somewhat more than 100 W); the compactness of the spacecraft (similar to a spinning top in shape) is for maximizing the moment of inertia with respect to the symmetry axis whereby providing passive spin stabilization around it. The current nominal spin rate is 2 Hz (120 rpm), yielding a peripheral acceleration of about 8 g, which is well doable. From the viewpoint of the EP experiment the modulation frequency of the signal, hence the frequency of spin, should be as high as possible: the higher the better, to reduce mechanical noise and low frequency "1/f" electric noise. In addition, a higher spin rate, with very weak suspensions at it is possible to use in space, makes the test bodies closer to being free, ideal rotors (see Sec. 2.1.5), which is advantageous for the EP experiment. However, there are practical constraints which is better not to push to their limits in order to reduce the complexity and cost of the spacecraft. A spin frequency of 5 Hz, with spacecraft dimensions close to the current ones, was the original choice for GG (Nobili et al., 1993; Nobili et al., 1995). The current baseline is safer and poses no problems. Indeed, the first Italian satellite, Sirio, flown more than 20 years ago, was very similar to GG: cylindrical in shape, passively stabilized by one axis rotation, with mass, spin rate and maximum peripheral acceleration very close to those of GG. In any case, a spin frequency of 2 Hz already provides a modulation of the expected EP signal about 10⁴ times higher than ever achieved.

From Eq. (1.2), which gives the intensity of the expected signal acceleration for an EP violation expressed by the adimensional Eötvös parameter h , it is apparent that the lower is the orbiting altitude the stronger is the signal. The altitude h of the satellite orbit is chosen having in mind two competing needs. On one side there is the need for an orbit altitude as low as possible, as this increases the strength of the signal; however Eq. (1.2) shows that the dependence on h of the acceleration signal is very slow for low Earth satellites (RÅ >>h). On the other hand, the altitude should be high in order to reduce the relevant disturbing effect from the residual atmosphere, at least as long as this becomes comparable to the effect of solar radiation pressure, which cannot be avoided anyway. The value of h is much more relevant for air drag acceleration than it is for the strength of the signal, because of its linear dependence on air density. In the current baseline we have h @ 520 km. It is worth noticing that in the case of GG the only concern about a higher altitude is for a slightly weaker signal; there is no special concern on charged particle effects (as it is the case in STEP) because there are no relevant electrostatic effects (Sec. 2.2.4).

Why an Equatorial Orbit. Due to the flattening of the Earth any satellite orbit whose inclination I over the equator is different from exactly 0 or p /2 is subject to the regression of its line of nodes. The dominant effect on the longitude W of the ascending node of the satellite orbit is due to the zonal harmonic of the gravity field of the Earth (which expresses its polar flattening) with a rate of change:

(2.3)

where w _orb is the orbital angular velocity of the satellite at altitude h around the Earth. The formula is valid for small values of the orbital eccentricity e (e² terms neglected) and non-zero inclination (for symmetry reasons, there is no such effect at I = 0); there is either regression or advance of the line of nodes depending on whether I is smaller or bigger than 90° ; there is no motion of the nodes for an exactly polar orbit (I = p /2). For any inclination 0 < I < p /2, assuming the spin axis unit vector of the satellite to be initially parallel to the orbit normal (and neglecting at this point all perturbations on (see below) the two vectors will be an angle 2I away from one another after a time i.e. after half period of the regression of the nodes (see Fig.2.10). For low orbit altitude and inclinations of 5 to 20 degrees half the period of the nodes is from 25 days to about 1 month, to be compared with a mission duration of 7 months, which means that the higher the orbit inclination the more important would be - in order not to loose in signal strength- to perform attitude maneuvers so as to keep the spin axis close to the orbit normal. No such maneuvers are needed at inclination close to zero, and this is why an equatorial orbit is preferable. An orbit very close to a polar one would also be suitable from this point of view; however, it would offer no major advantages from the viewpoint of launch while it is certainly less advantageous for ground operation from the Italian equatorial station of Malindi. Although GG is a mission far from heavy on data rate (see Chap. 7) its operation if close to equatorial orbit is definitely easier for the Italian ground station.

Figure 2.10 The figure shows the relative position of three unit vectors: the spin axis of the Earth , the spin axis of the GG satellite (both fixed in time) and the normal to the orbit plane of GG for a given orbital inclination I . The orbit normal is drawn in two positions: at initial time, as assuming that it coincides with and after 1/2 period of the nodes, as . It is apparent that the originally aligned axes become an angle 2I apart. The higher is the inclination (0<I <p /2), the larger their angular separation will be.

The price to pay for this choice is to deal with severe thermal variations due to the satellite coming in and out of the shadow of the Earth every orbit. And to be without sun power for a significant fraction of the orbital period. The only orbit which would solve (yet not totally) this problem is sun-synchronous, which however cannot (by definition) have zero motion of the node line (since its orbital plane is meant to follow the Sun) and is therefore incompatible with the GG need for a spin axis close to the orbit normal during the entire mission and no attitude maneuvers. However, we have found (see Sects. 4.4 and 4.5) that the GG thermal problems can be solved, thanks to the fast spin of the spacecraft and good vacuum inside it, using, in addition, passive insulation and a spacecraft structure in carbon fiber composite (with very low thermal expansion coefficient). The required thermal stability can be met by purely passive means.

A circular orbit is preferable, but a small eccentricity can be accepted in order not to put stringent requirements on the performance of the launcher. Residual eccentricity from the launcher can be as low as 0.01, a value which is acceptable for GG. An orbit inclination of no more than 0.2° is within the current launcher performances; as for having the spin axis close to the orbit normal, active maneuvers can be performed (before spin up to the nominal rate) reaching a deviation <1° . Apart from the regression of the line of nodes, and its consequences on the strength of the EP signal, perturbations on the GG orbit do not have any relevant consequences over the 7 months nominal duration of the mission. The pericenter will precess; the orbit will circularize and spiral in due to air drag; orbital effects of radiation pressure in semimajor axis will average out every orbit (Milani et al. 1987, Chap. 4). All these non gravitational effects are significantly reduced by FEEP drag compensation; in any case, none of them is of any concern for of the experiment and the mission. No precise satellite tracking is required; tracking with an ordinary accuracy of several km along track is sufficient for the purposes of the EP experiment.

Passive Compensation of Differential Rotation. With this choice of the orbit (almost equatorial, almost circular with the spin axis close to the orbit normal) there is an effect induced by eclipses which gives rise to an angular phase lag between the spacecraft outer shell and the payload suspended inside it, and it must therefore be taken care of (Lund, 1995).

The spin rate w _os of the outer shell will change if the moment of inertia of the shell changes while the spin angular momentum remains constant. This indeed happens every orbit due to temperature variations of the outer shell as the GG satellite gets in and out of the Earth shadow, which is not the case for the PGB laboratory since it is very well insulated and thermally de-coupled from the spacecraft (see Sects. 4.4 and 5.4). Hence, a differential rotation rate of the outer shell with respect to the PGB is to be expected. The relative change of the angular velocity of the outer shell w _os with respect to its nominal spin angular velocity w _s must equal the relative change of its moment of inertia J_z with respect to the symmetry/spin axis (due to the conservation of spin angular momentum), and it must be:

(2.4)

where a _os is the thermal expansion coefficient of the outer shell and D T its temperature variation due to the eclipse. Because of the low torsion constant of the PGB springs the natural period for torsion oscillations of the spacecraft/PGB systems is longer than the eclipse duration (about 1/3 of the orbit period); as a consequence, a differential rotation rate will accumulate over the eclipse passage, yielding a corresponding phase lag between the two bodies. We shall have:

(2.5)

for the variation of the spin angular velocity in 1 eclipse passage of duration t _eclipse, and a corresponding phase lag:

(2.6)

The GG thermal model gives D T @ 30 degrees for the temperature variation of the spacecraft in 1 eclipse passage; if the spacecraft is made of carbon fiber composite it is reasonable to have a thermal expansion coefficient a _{os @} 0.9× 10^-6/K; thus:

(2.7)

This result has been confirmed by detailed theoretical and numerical analysis; it is clearly unacceptable and calls for a solution. A way to solving it comes from noticing that such a large phase lag is due to the high spin rate of the spacecraft (together with the fact that the phase difference grows quadratically in time) while the absolute change in the moment of inertia is indeed very small. This means that it can be balanced by a small compensation mass. Moreover, compensation can be passive (Marchal, 1996); the idea is to have a mass which expands and contracts in anti-phase with respect to the outer shell of the spacecraft so as to keep the total moment of inertia (of the outer shell plus the compensation mass) essentially constant. Since the compensation mass should be small compared to the mass of the outer shell, clearly its expansion coefficient must be larger than the expansion coefficient of the outer shell (if the masses have about the same distance from the spin axis). Expansion is required in the radial direction (normal to the spin axis) because this is how the outer shell expands and contracts. Hence, there must be bars located radially from the cylindrical surface towards the spin axis, while the bulk of the compensation mass should be as close as possible to the outer shell in order to give the highest contribution to the total moment of inertia (and thus to its variations as well). Also, materials with high expansion coefficients tend to have low density. From all this we are led to the design shown in Fig. 2.8: a number of bar shaped supports with large thermal expansion coefficient a _bar are located radially from the outer shell surface (in thermal contact with it) towards the spin axis; from the other ends, getting back towards the surface of the spacecraft, there are thermal insulating low expansion bars at whose ends the compensation masses are attached. As the spacecraft outer shell expands outward the compensation masses are displaced inward, and viceversa. The compensation masses are far away from the spin axis thus giving a large contribution to the moment of inertia. Care should be taken in maintaining symmetry in azimuth as well as top/down. Also, thermal insulation of the top radial bars and insulating masses is necessary not to bring heat inside. Three possible mass compensation designs are shown in Fig. 5.7. For the total moment of inertia to remain constant the ratio of the thermal expansion coefficients and the ratio of the moments of inertia (with respect to the spin axis) must satisfy the equation:

(2.8)

and, with the design outlined above:

(2.9)

with m_cm the total compensation mass (with bars of negligible mass), m_os and r the mass and radius of the outer shell, r_cm the radial distance of the compensation mass from the spin axis, l_bar the length of the high thermal expansion bar. Using materials with a _bar a factor 50 to 100 times larger than the expansion coefficient of the outer shell in carbon fiber it is found that a compensation mass not larger than 5 kg is sufficient. Note that a small compensation mass, and a small mass for the bar, i.e. a small mass for the entire system means that heat can be transferred (hence expansion/contraction achieved to compensate for those of the spacecraft) in a very short time.

How rapidly must this passive mass compensation system work in order to keep the phase lag between the outer shell and the PGB within a required phase difference (namely, 1/10 of the of angle available by the mechanical stops)? Assuming the temperature change to be linear in time we get a quadratic growth of the phase lag angle with acceleration:

(2.10)

giving about 240 sec for the response of the compensation system. Materials with thermal expansion coefficient as large as 10^-4/K to be used for manufacturing the bars tend to have low thermal conductivity; they can have a thermally conductive inner bar so as to increase the surface of thermal contact with the expanding/contracting outer shell of the spacecraft thus reducing the response time. In any case there is sufficient time available. Instead, it is possible to use bars is special Al alloy which have somewhat smaller thermal expansion coefficient but high thermal conductivity and need no inner bar. The gravitational perturbation on the test bodies of all the moving mass (from the outer shell as well as the compensation mass) has been estimated and found not to be a problem (Sec. 2.2.5).

In the conservative assumption that 10% of (2.10) is not compensated passively, for the residual angular acceleration between the spacecraft and the PGB (3.4× 10^-8 rad/sec²) the response time is 770 sec; it can be sensed and corrected. It is sensed by placing a small mirror on the PGB tube and a photo-detector on the spacecraft (it adds no wire to the PGB); the angular resolution (À 0.01 rad) is certainly not a demanding one. The photo-detector can then drive the FEEP thrusters to correct for the phase lag by spinning the outer shell up or down as necessary. The torque to be provided is of 10 m N× m, well within the capabilities of the thruster authority of the FEEP. Indeed, the FEEP could correct for the entire effect (no passive mass compensation), since this requires 100 m N× m, but it is preferable to limit the FEEP to fine adjustments only in order to reduce power consumption.

No such phase lags as those discussed above will take place between the PGB and the test bodies because of the very good thermal insulation (see Sec. 4.4); spring coupling will take care of eliminating residual small phase lags (e.g. left out by the unlocking procedure) within 2 weeks (see Sec. 2.1.6).

Effects on the Spin Axis. We have analyzed the effects of the regression of the nodes on the relative angle between the orbit normal and the spin axis, assuming the latter fixed in space. Although the approximation is correct, the spin axis is not exactly fixed in space. This is because a body with different moments of inertia (the GG satellite) whose spin axis is not exactly normal to the orbit plane will be forced to precess about the orbit normal by the monopole gravitational attraction of the central body (the Earth). The effect is similar to the well known luni-solar precession (with 26,000 yr precession period) of the spin axis of the Earth about the normal to the ecliptic produced by the monopole of the Moon and the Sun on a planet Earth which has a non zero quadrupole moment.

For an axisymmetric body having moments of inertia J_z with respect to the symmetry axis and J_x with respect to any axis in the transversal plane (J_z>J_x), whose spin axis - assumed to be coincident with the symmetry axis at this point- is at an angle q with respect to the orbit normal, it can be shown that the spin axis will precess about the orbit normal with a precession period P_pr given by (e.g. Afonso et al., 1989):

   (2.11)

This formula has been obtained after averaging over the fast variables, both the true anomaly and the longitude of the node. Clearly, the more the body differs from a sphere i.e. the larger is its fractional difference in the principal moments of inertia (J_z-J_x)/J_z, the more relevant this effect will be (i.e. faster precession). Once all the bodies have been unlocked, since they are weakly coupled and have different values for the fractional difference in the principal moments of inertia (J_z-J_x)/J_z they will precess at different rates, slower the test masses than the outer shell and the PGB (the test masses are manufactured with lower quadrupole moments). The fastest precession rate is that of the outer spacecraft, whose precession period amounts to P_{pr @} 6 yr. (and it is the shortest of all). This means that the assumption made for the longitude of the nodes to be a fast variable too was indeed correct. Note that the precession angular velocities are different for the different bodies due to their different quadrupole moments, but they have all the same sign (in fact opposite to the sign of the spin angular velocity, which is obviously the same). The precession rate of the PGB is slightly slower than that of the outer shell, while those of the test bodies are at least a factor of ten slower. Let us consider the worst possible case: that of one body precessing at the fastest rate (i.e. the spacecraft outer shell) and another body with zero precession rate. In a given time interval D t the precession angle a _pr covered by the vertex of the spin axis of the spacecraft in its precessional motion in the sky is a _pr = (2p /P_pr)× D t. However, the spin axis is sweeping a precession cone of semi-aperture q , and therefore the angular displacement with respect to its original position (also the position of any axis not affected, or much less affected by this torque) is F =a _prq (all angles in radians). Taking the entire 7 months nominal duration of the mission as D t, we get 9× 10^-3 rad (0.6° ) for the total angular displacement. This value, which is an upper limit, is also much smaller (about 1/10) of the maximum physical room allowed by the mechanical stops for relative angular movements (in all degrees of freedom) for the suspended bodies.

However, in the above reasoning we have ignored the mechanical coupling between the two bodies. Instead, there are coupling springs which will be elastically deformed in response to the external torque until an equilibrium position is reached. The effect is like the so called "gyroscopic effect" for rotors on Earth - in the case that the coupled rotors be suspended by their center of mass- , noticing that on the ground precession would be around the diurnal angular velocity vector of the Earth, hence yielding a much stronger effect. The problem arises for the GG prototype experiment in the laboratory and it has been carefully investigated: the resulting angular displacement at equilibrium between the spin axes is very small even on Earth (10^-5 rad or smaller depending on the arrangement; Comandi, 1998). It will be certainly smaller in space. Even more important, the GG test bodies in space are suspended from the center of mass (or symmetrically with respect to it): as a consequence, gyroscopic effects - in addition to causing very small angular displacements- will not affect the centers of mass of the test cylinders whose relative displacement is the only physical quantity relevant for the EP experiment.

The outer shell of the GG satellite is subject also to a number of non gravitational perturbations that can change its spin axis while not affecting the PGB laboratory and test masses suspended inside. As a result, there will be tilts which, although not affecting the centers of mass, may be unacceptable for the EP experiment. If so one should resort to active control of the spin, that is, GG should rely on compensation (to some extent) of non gravitational effects also in angle. From the analysis reported below we conclude that no active control is needed on the direction of the spin axis.

The main non gravitational torques are due to solar radiation pressure and air drag. Radiation pressure on the GG top and bottom covers is usually asymmetrical. The Sun is not always at zero declination over the equator of the Earth, the orbit of GG is not exactly equatorial, the spin axis is not exactly normal to it. So one cover is illuminated while the other is in the dark. The reflected part of this asymmetrical radiation gives a force along the spin axis, hence no tilt. Instead, the effect due the absorbed fraction has the same direction as the surface-to-Sun direction, and this has a component perpendicular to the spin axis which, in the case of asymmetrical illumination, results in a torque that can tilt the spin axis of the outer shell with respect to that of the PGB inside. The radiation pressure force on a GG cover can be estimated to be:

(2.12)

where F _sun is the solar flux, c the speed of light, r the radius of the GG spacecraft, d the declination of the Sun over the equator of the Earth and a _ab the absorption coefficient of the surface. The corresponding torque is:

(2.13)

where h is the height of the GG outer shell. In the unfavorable, simplifying assumptions that d is always at its maximum, that no shadowing occurs, and with a large value for a _ab (0< a _ab<1) the maximum value of N_rp is only of few 10^-7 N× m, to be compared with the enormous spin energy of the spacecraft, E_spin= J_{zw s}^{2 @} 5× 10³ N× m (a factor at least 10¹⁰ bigger). As far as air drag, is concerned, since the GG spacecraft is highly symmetric only fluctuations in the residual (at the orbiting altitude) atmospheric density can give rise to a tilting torque. Furthermore, only a fraction of such fluctuations will produce a tilting torque always in the same direction. In the very conservative assumption that fluctuations producing a tilting torque always in the same direction amount to 0.01 of the average air density, the resulting torque would still be of the same order as in the torque due to solar radiation estimated above; hence, also negligible compared to the spin energy. For completeness we have estimated also the effect of a magnetic torque. Any electric charge that were to materialize inside the spacecraft (e.g. interaction with cosmic rays) will move to the external surface of its outermost conducting structure because all parts are connected via conducting suspensions and there are no free floating masses. Since GG is spinning this charge will create a current loop whose magnetic moment (parallel to the spin vector of the spacecraft) will interact with the dipole magnetic field of the Earth giving rise to a torque As a result, the spin axis will precess around . With an upper limit for the electric charge obtained from charging on LAGEOS (an upper limit, because LAGEOS orbits in the middle of the Van Allen belts) this magnetic torque turns out to be totally negligible.

Having analyzed all non gravitational perturbing torques (that we are aware of) which can tilt the spin axis of the GG spacecraft outer shell with respect to the PGB laboratory inside, we conclude that their effects are negligible by far and that no active control of the direction of the spin axis in space is needed. Such direction is absolutely stable. The simple physical reason behind this fact is that the kinetic energy of spin once at the nominal rate of 2 Hz is so high compared to all perturbing torques that they would need a very long time to even slightly displace the spin axis; with the addition of the restoring coupling of the springs, these torques are of no concern at all. From the viewpoint of satellite operation, this result is a very important simplification; in addition, a reduced complexity of the spacecraft is clearly beneficial for the experiment. By comparison, the STEP attitude needs to be actively controlled; moreover, it must be drag free also in angle to very high accuracy.

Besides the precessions caused by external torques (of both gravitational and non gravitational origin) there is a precession, known as Eulerian or free precession, and also as Eulerian or free nutation, which takes place in the absence of any external torque. For an axis-symmetric body with the symmetry axis being the axis of maximum moment of inertia, there is free precession whenever the body is put into rotation about an axis at non zero angle with respect to the symmetry/principal axis. The spin axis precesses around the symmetry/principal axis of the body (moving inside the body itself) at a frequency smaller than the spin frequency and given by n _pr=n _spin× (J_z - J_x)/J_z. A well known example is the polar motion or Chandler wobble of the spin axis of the Earth by which the north pole moves on the surface of the Earth along a circle-like curve of about 6 m radius (0.2 arcsec) every @ 420 days. In the case of spinning spacecraft free precessions must be damped in order to stabilize their attitude. In the case of GG the presence of low stiffness mechanical suspensions coupling the outer spacecraft to the PGB can reduce the amplitude of the free precession cones (see Fig. 2.6). The free precession movement causes forced oscillations (and deformations) of the PGB springs in the transversal plane, just like any other vibration perturbation; this happens at the free precession frequency given above, which is higher than the threshold frequency of the system for the attenuation of vibration noise, thus reducing the amplitudes of these oscillations and closing the precession cone.

Innovative Drag Compensation. So far the GG spacecraft is standard. However, it is an innovative spacecraft in that it performs an accurate drag-free control at the orbital frequency. The request for drag compensation comes from the experiment, because the effect of air drag is by far larger - by many orders of magnitude; see Sec. 2.2.1- than the expected signal. The residual atmosphere at the satellite orbiting altitude (as well as radiation from the Sun, the Earth’s albedo and the infrared Earth radiation) act on the outer surface of the GG spacecraft but not on the test bodies suspended inside. Since gyroscopic effects are negligible (because of the very high energy of spin the GG spinning bodies are essentially unaffected by tilts and torques), these non-gravitational effects appear as inertial accelerations on the test bodies to which they are transferred via the flat gimbals on the coupling arms (as shown in Fig. 2.1), ideally - i.e. in case of perfect balancing of the test bodies; see Sec. 2.1.4- equal and opposite to the acceleration acquired by the spacecraft. Most of the effect (along-track) is at the satellite orbiting frequency around the Earth; smaller, low frequency variations are also to be expected. In the ideal case of perfect balancing of the test bodies air drag effects would cause no differential displacement of the bodies (with respect to one another), hence not competing with the EP differential signal. In reality balancing is not perfect, and it is a good strategy that the burden of dealing with air drag effects be shared between the experiment core, by accurate balancing of the test bodies, and the spacecraft itself - by making it drag-free, i.e. capable of compensating for air drag.

In GG drag compensation is performed with FEEP thrusters, whose advantages for high precision missions in Fundamental Physics devoted to the detection of very small forces, are numerous: high specific impulse, negligible amount of propellant (few grams for a few months mission duration), no moving parts, fine electric tuning and consequent high level of proportionality. The motion of the PGB with respect to the spacecraft is monitored by small capacitance sensors (shown in Fig. 2.8) which drive the FEEP thrusters (spinning with the system) for compensation. The actual configuration of FEEP thrusters on GG for drag compensation (6 in total) is given in Sec. 5.5 and Fig. 5.15. The drag-free control is based on a notch filter, and requires the thrusters to fire close to the spin frequency; it has been designed and implemented in a finite element numerical simulation of the full GG system (see Sec. 6.15). The corresponding vibration noise (close to the signal modulation frequency) is very effectively reduced by the PGB suspensions according to the transfer function of Fig. 2.6, and therefore does not reach the test bodies. The FEEP thrusters and their control electronics are discussed in Sec. 4.2. The current GG requirements on drag compensation are given in Sec. 2.1.1

2.1.3 The Read-Out System

(2.14)

(2.15)

The GG requirements for the mechanical balancing of the capacitance bridge are given in Sec. 2.2.1 and can be easily met by means of inch-worms mounted a shown in Fig. 2.1. They will not disturb the experiment because they can be switched off after having been positioned for the required balancing has been achieved. Any disturbances from parasitic capacitances depend on the geometry of the system and therefore act as DC effects, while the signal is detected at the spin frequency.

Figure 2.11 Schematic drawing of the two capacitance sensor of the bridge of the GG read out system for detecting relative displacements of the inner and outer test body with respect to one another. Each capacitor is formed by two surfaces, one for each of the two grounded bodies, and one plate, to which a sinusoidal voltage is applied. The other two capacitors of the bridge are fixed capacitors. Any differential displacement of the test masses with respect to the plates causes a loss of balance of the bridge and therefore an output signal.

2.1.4   Balancing of the Test Bodies and Common Mode Rejection

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 acceleration transferred to the coupled test bodies is simply opposite to that acquired by the spacecraft because of drag; and if the test bodies are perfectly balanced, there is no differential displacement between the two. In such an ideal case, drag would not affect the expected signal of an EP violation at all since this is differential. The mathematical dynamical model equivalent to the GG coupled test bodies is shown in Fig. 2.13, from which it is apparent that the system is in all similar to an ordinary beam balance, save for the fact that it is a vertical rather than a horizontal balance. Although not perfectly balanced so as to be totally unaffected by drag, the system can be balanced in order to reject common mode forces (such as the inertial forces resulting from drag) to very high accuracy, leaving only a much smaller differential residual disturbance to compete with the signal. Piezoelectric actuators are well suited to accurately balance the system, as shown in Fig. 2.14. The required electric connections are through the flat gimbals (as discussed in Sec. 2.1.1) and, by using inch-worms, all tensions can be switched off after balancing.

Figure 2.13 The GG system of coupled test bodies (shown in Fig. 2.1 and discussed in Sec. 2.1.1) is equivalent to the mathematical model shown on the right, which is in essence a vertical balance. F represents any force acting on the system in common mode (i.e. the same on both bodies); being the beam of the balance vertical, it can balance forces in the horizontal plane (i.e. the orbital plane). F represents here the main effect of drag as the GG satellite orbits around the Earth (orbit frequency). It is apparent that if the arm lengths and masses were exactly equal F would cause no relative displacement of the centers of mass of the bodies one with respect to the other. An EP violation signal would be also in the horizontal plane, but at about 90° from F , i.e. about normal to the plane of the figure.

Figure 2.14 An enlarged view of the system of piezoelectric actuators placed on the two balancing arms of the GG test bodies (see Fig. 2.1 for an overall view). 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 V₁₊ V₂ determines the relative axial position of the centers of mass of the test bodies and is used for axial centering. The voltage difference V_1- V₂ can be used to change the lengths of the four halves of the arms so as to balance out the effect of transverse inertial forces (in particular the along-track component of the air drag).

(2.16)

Perturbations such as air drag and solar radiation pressure acting on the external surface of the spacecraft produce a non gravitational acceleration of its center of mass. In the reference frame of the spacecraft the bodies will therefore be subject to inertial forces in the opposite direction, which will move the masses to new displaced positions of equilibrium where the perturbation is balanced by the restoring force of the spring. This is shown very clearly in Fig. 6.4 obtained from the numerical simulation (Fig. 6.4 shows also the whirl, that we introduce below). It is worth noticing that, because of the supercritical state of rotation, the displaced body will always spin around its own axis, which means that no centrifugal force due to the spin will result because of this displacement. The only centrifugal force due to the spin come from the off-centering given by Eq. (2.16) and is balanced by the restoring force of the suspension springs.

No matter how good is the mechanical quality of the suspensions and how accurately they are clamped, as the system rotates they will undergo deformations and, in this process, will dissipate energy. The only energy that can be dissipated is the spin energy of the rotor, which means that the spin angular velocity must decrease. As a consequence, the spin angular momentum also decreases, and since the total one must be conserved, a motion of the two bodies one around the other (in the same direction as the rotational motion) will develop. Which is what is observed, and is referred to in the literature on Rotordynamics as whirling motion. Friction between rotating parts of the system (i.e. friction inside the suspension springs) is the physical cause of the whirl motion, and it is referred to as rotating damping. Except in the case of rotating machines with very viscous bearings, the rotor whirls at essentially its natural frequency (with respect to a fixed frame of reference). In GG there is no motor, there are no bearings, no fluids, no oils, no greases; only carefully clamped suspensions of high mechanical quality (particularly for the test bodies) which undergo only minute deformations. Hence, the GG bodies whirl at their natural frequencies of oscillation.

Can the whirl motion be damped and the rotating system be stabilized? For more details than those given in the remaining of this Section see: Bramanti et al., 1996; Nobili et al., 1996; Nobili et al., 1997b; Nobili et al., 1997c.

In the GG space experiment where there are no non-rotating parts an equivalent non-rotating damping must be provided by an active control system of sensors and actuators fixed with the rotating bodies. Before any such device can be designed, it is obviously necessary to establish the magnitude of the forces which destabilize the system and which will therefore need to be counteracted actively. The actual implementation of the active control forces for the real case, with realistic errors in all components of the control system, can only be faced once the magnitude of the destabilizing forces has been firmly established. In turn, this requires to firmly establish the amount of energy losses in the GG rotating system.

A firm estimate of the losses in the GG mechanical suspensions requires them to be measured experimentally, by setting the springs in oscillation under realistic operating conditions (oscillation frequency, vacuum, temperature, clamping); note that there is no need to perform this measurement with the system rotating (Crandall and Nobili, 1997; from S.H. Crandall on the subject of damping in Rotordynamics see also Crandall, 1970; Nelson and Crandall, 1992; Crandall, 1995).

(2.17)

(2.18)

Figure 2.17 One helical spring to be used for suspending the GG test bodies (see Figs. 2.1 and 2.2) clamped and ready for measuring its quality factor at a frequency of a few Hz. A small mass is attached to the free end of the spring in order to obtain the desired oscillation frequency. Note that the measurement is done for horizontal oscillations for the result not to be affected by local gravity

Figure 2.18 One helical spring manufactured for suspending the PGB laboratory clamped and in preparation for the measurement of its quality factor in horizontal oscillations at frequencies of a few Hz (the small black sheet at the free end of the spring is for the optical measuring system).

(2.19)

(2.20)

, (2.21)

which means, an average destabilizing force (along-track) of magnitude

(2.22)

Since is the centrifugal force, equal and opposite to the elastic force of the spring , the destabilizing force which generates the unstable whirl motion depicted in Fig. 2.15 turns out to be

(2.23)

The following question becomes relevant at this point: how much energy (per unit time) is gained by the (destabilizing) whirling motion as fraction of the energy lost by the spinning rotor? Let us consider the rotor in the two body model of Fig. 2.15. The spin energy of the rotor is:

(2.24)

(2.25)

, (2.26)

From the conservation of angular momentum it is possible to relate to . The total spin angular momentum of the rotor is:

(2.27)

The angular momentum of the whirl motion is:

(2.28)

Since the total angular momentum has to be conserved, it must be:

(2.29)

(2.30)

Using Eq. (2.30) for, we get from Eq. (2.26):

(2.31)

(2.32)

(2.32)

(2.33)

(2.34)

(2.35)

This force will prevent the whirl motion from growing, i.e. it will maintain the system in a steady condition. To actually damp a whirling motion one has to apply a slightly larger force: for instance, a force twice as large will damp the whirling motion at the same rate at which it would grow in the absence of .

Were the capacitance sensors perfect, this would be enough to stabilize the GG bodies, as it can be also checked with numerical simulations. However, the sensitivity of the sensors in realistic flight conditions has to be taken into account because, being the growth rate of the whirl instabilities very slow, the relative displacement during one spin period is so small to be dominated by the noise of the sensors. As a result, the recovered velocity vector is also very noisy, and much larger than its actual value in the fixed frame. If the actuators were programmed to damp the velocity vector recovered in this way, the resulting active force would be driven by the noise, i.e. it would be much larger than the force which destabilizes the system. Such a controlled system would be dominated by the active control forces rather than by the very low stiffness passive mechanical suspensions. This is in contrast with the basic physical design of the GG experimental test of the universality of free fall whereby the test bodies must be very weakly coupled so as to be sensitive to tiny differential forces.

2.1.6      Locking/Unlocking

The rôle of the coupling springs in damping out small phase lags between the PGB and the GG test bodies inside it, is very important and should not be overlooked. This is unlike the phase lags between the outer spacecraft and the PGB laboratory which are induced by eclipses every orbit; as discussed in Sec. 2.1.2, these phase lags are reduced almost entirely by a passive mass compensation system, and the small remainder by FEEP control.

Figure 2.20 Top view (across the spin/symmetry axis) of one set of 4 inch-worm actuators for fine unlocking of the suspended GG bodies; this figure refers to unlocking the PGB from the outer spacecraft. The inner tube less than 10 cm in radius (belonging to the PGB) encloses one of the PGB helical suspension spring. As viewed in Fig. 2.8, this system is located at the top (or bottom) of the PGB suspensions. Since Fig. 2.8 is a section through the spin/symmetry axis, the capacitance active dampers and the inch-worms cannot be shown together (each damper is 45° from the two nearest inch-worms); Fig. 2.8 shows only the capacitance dampers. Each suspended cylinder needs 2 sets like the one shown here placed at its two axial ends.

2.1.7    Calibration Procedure

Figure 2.21 Qualitative representation, in the orbital plane, of the differential displacements obtained from the synchronous demodulation of the 2-phase 2 Hz signal. The X axis is fixed in the Earth-to-satellite direction; in this non spinning frame the OP vector represents the expected signal, namely a differential displacement, directed along the X axis and constant in amplitude, of the two masses due to a violation of the Equivalence Principle violation. The perturbation PD due to an unbalanced atmospheric drag will be found in the area between the two dotted lines crossing in P: The angle between them is about 0.8 rad, due to the fact that the drag has a variable component in the radial direction because of solar radiation pressure (of amplitude about 0.4 times the atmospheric drag and in the Sun-satellite direction). Smaller contributions to the PD 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 (as discussed in Sec. 2.1.4) the point D is displaced up or down inside this area, and brought close to P. In doing so, also the radial component of the drag is automatically balanced, as pointed out in Sec. 2.1.4. The low frequency variations of the drag (not shown) will oscillate inside the same area. The vector DQ shows here the whirl instability before it is damped by the whirl control whose period in this (non spinning) frame is the natural frequency of oscillation. The circle around point Q represents the error in the measurement due to thermal noise of the mechanical oscillations built up during the integration time.

Large relative displacements of the test bodies will be detected at first. This means that either the capacitance bridge is not balanced or the test masses are not balanced (i.e. disturbances in common modes are not sufficiently rejected), or the axial misalignment between the centers of mass is too large (see the discussion on tidal disturbances, Sec. 2.2.2), or all of them together. The calibration phase consists in applying changes with the piezo actuators (to the capacitance plates, the arms length of the test bodies and the axial positions of their centers of mass) one by one, detecting the corresponding effect and from it deciding about the next change. Fig. 2.21 describes graphically how the signal is demodulated and the balancing is carried out.