6. Concluding remarks

We have built a rotating differential accelerometer, at room temperature, with fast spinning test cylinders (10 kg each) suspended like in a vertical beam balance so as to be weakly coupled in the horizontal plane. In spite of the need to sustain its weight, the coupled system is very sensitive to differential forces acting between the test cylinders in the horizontal plane; in addition, the read-out is made of capacitance bridges which read the relative displacements of the test cylinders directly (instead of deriving them as the difference of their individual displacements). This makes the accelerometer well suited for detecting tiny differential effects; by comparison, the proposed µSCOPE accelerometer (also at room temperature and based on capacitance sensors) is not inherently differential because each test cylinder has an independent suspension and sensing system (although both cylinders are controlled with respect to the same silica frame) (MICROSCOPE Website: http://www.cnes.fr/activites/programmes/microsatellite/1sommaire_microsatellite.htm and http://www.onera.fr/dmph-en/accelerometre; Touboul et al., 2001, Fig. 1). The quality factor of the system has been measured at the spin frequency, as well as at the low frequency of differential oscillations (when at zero spin rate). The results are consistent with those obtained in previous measurements for losses in the mechanical suspensions alone (Nobili et al., 1999, 2000). Unstable whirl motions which are predicted because of such losses have been found to grow very slowly, according to the Q values, and therefore very small forces are needed to stabilize them (see Nobili et al., 1999 for the relevance of this issue). Gyroscopic effects have been measured and shown to be in agreement with their theoretical prediction. Finally, it is found that the stability of the present prototype is such that, at 2.5 Hz spin rate and 3.5 s period of whirl, the 10 kg mass test cylinders remain within 1.5 µm from each other for 1 h.

These results are relevant for the space variant of this instrument, proposed for the GG space mission, in several respects. Losses in the system and whirl motions are in agreement with predictions, giving us confidence in the theoretical analysis and numerical simulations of the GG dynamical system carried out so far ("GALILEO GALILEI" (GG), Phase A Report, 1998, Chapter 6). The relevant quality factor, as measured with the accelerometer in full operation, is only a factor four smaller than the quality factor required in the GG error budget for its target sensitivity in EP testing of 10¹⁷: 4900 instead of the 20,000 value required ("GALILEO GALILEI" (GG), Phase A Report, 1998, Section 2.2.7). (Note that we have measured Q=19,000 for a low stiffness CuBe suspension, suitable for use in space, when set in horizontal oscillation at 5 Hz (Nobili et al., 1999, 2000). The read-out (mechanical parts and electronics), data acquisition and data analysis (including the need for accurate coordinate transformation from the rotating to the non-rotating frame of reference) are of direct relevance to the space instrument and its operation. The stability observed in the relative position of the test cylinders can be compared with the GG requirement as follows. The spin rate is almost the same (the nominal spin rate of GG is 2 Hz), but the test cylinders in space can be coupled much more weakly than on the ground, thanks to the absence of weight. We have 3.5 s whirl period in our recent measurement runs and expect to be able to reach 540 s in space (as in the GG mission baseline at Phase A study level ("GALILEO GALILEI" (GG), Phase A Report, 1998), the relative displacement of the test cylinders in response to differential forces being proportional to the square of the differential period (and inversely proportional to the stiffness of the suspensions). An EP violation signal would have a well defined signature (frequency and phase), in both the ground and the space experiment, so the relevant sensitivity of the instrument has to be assessed for this target signal. In space (Fig. 1) the signal is a relative displacement vector of fixed length pointing to the Earth and therefore changing direction with the orbital period of the spacecraft. On the ground it is a fixed displacement in the North-South direction if the source mass under consideration is the Earth; it is a displacement vector whose length and direction change with the daily (and also annual) motion of the Sun if the Sun is the source mass. In all cases, the rotation of the instrument provides higher frequency modulation of the displacement vector. For GG to reach its target sensitivity, the relative displacement of the test cylinders in the satellite-to-Earth direction, modulated at the high frequency of spin and then transformed into a constant signal in the non-rotating reference frame, should not exceed r_GG=6.2·10¹¹ cm ("GALILEO GALILEI" (GG), Phase A Report, 1998, Section 2.1.1). Bench tests have demonstrated that the sensitivity of our read-out electronics is of 5·10¹⁰ cm in 1 s of integration time ("GALILEO GALILEI" (GG), Phase A Report, 1998, Section 2.1.3), allowing us to detect the target displacement r_GG of the space experiment in about 100 s. So, the observed 1.5 µm separation between the centers of mass of the test cylinders is due to the ground perturbations mentioned at the end of the previous section, while the read-out electronics could detect much smaller displacements. The ground prototype, whose measurements of the relative displacements of the test cylinders are reported here, is stiffer than the one proposed for flight by a factor =24,000, and consequently it is 24,000 times less sensitive to differential displacements. In order to demonstrate the feasibility of the space experiment to that level of sensitivity it should have detected relative displacements between the centers of mass of the test cylinders of ·r_GG=1.5·10² µm, while so far we have achieved only 1.5 µm. In order to gain this factor of 100, so as to perform a better demonstration, we need to reduce the effects of the ground perturbations by the same amount. The significance of the ground demonstration improves by reducing the stiffness of the accelerometer (hence the scaling factor ), together with a corresponding reduction of the effects of the ground perturbations. An improved version of the prototype currently under construction is designed to reach a scaling factor =2400 and a stability in the relative displacements of the test cylinders of 1.5·10³ µm. By comparison with the target of the GG space experiment in testing the equivalence principle: _GG=a/a=10¹⁷ (a=840 cm s², a=8.4·10¹⁵ cm s²) this corresponds to a full scale test at the level _prototype=²_GG=5.8·10¹¹, because a=²_diff·r_GG, the differential natural frequency _diff being proportional to the coupling stiffness of the suspensions.

The local acceleration of gravity, because of the need for a stiff suspension in the vertical direction, forces a few asymmetries in the design of the ground accelerometer which are not there in the instrument designed for space (as it is apparent by comparing Fig. 2 and Fig. 6) and reduce the advantages of the instrument for EP testing on the ground. Nevertheless, rotation (especially if at high rate)and the corresponding frequency modulation of the signalis extremely important, as the successful experiments by the "Eöt-Wash" group have demonstrated, in EP testing (Adelberger et al., 1990; Su et al., 1994; Baeßler et al., 1999) as well as in the measurement of the universal constant of gravity (Gundlach and Merkovitz, 2000) and in testing the inverse square law at sub-mm distances (Hoyle et al., 2001). Our accelerometer shows that fast rotation can be achieved, that it can be achieved with large test masses (which is very important to reduce thermal noise), that it is compatible with small force gravitation measurements andmost importantlythat is suitable for use in space. The dynamics of the system is understood, it can be theoretically anticipated and checked by the measurements. Losses measured with the full system in operation (and with mechanical suspensions of quite a complex shape; see Fig. 4), yield a quality factor only four times smaller than the value that is required for the GG space experiment to reach its target. As for the fact that the prototype can only check for violation in the field of the Sun and not of the Earth (because of the gyroscopic effects discussed in Section 2), it is worth stressing that also the best "Eöt-Wash" results have been obtained in the field of the Sun (Baeßler et al., 1999), in spite of the slightly weaker signal and the need for long term measurements in this case. The reason is the difficultywhen searching for an effect in a fixed directionto model the spurious effects of local mass anomalies (the small ones nearby and the very large ones far away) which obviously do not rotate with the instrument. A difficulty which is totally eliminated in space where the whole spacecraft co-rotates with the test masses.

In summary, we can convincingly argue that theoretical understanding, numerical modeling and experimental measurements performed so far put on solid grounds the novel idea of a high accuracy space test of the equivalence principle (to one part in 10¹⁷) with fast rotating weakly coupled test cylinders as proposed for the GG small mission. It has been shown (Nobili et al., 2001) that fast rotation and large mass of the test bodies are pivotal in making it possible to aim at such a high accuracy test in space with an experiment at room temperature. Among the proposed space experiments, GG is the only one in which the accelerometer devoted to EP testing and the one used for zero check (i.e., with test bodies made of the same material) are both centered at the center of mass of the spacecraft, so as to reduce common mode tidal effects and improve the reliability of the zero check. It is also the only one for which a full scale prototype of the accelerometer has been built and can be operated and tested on the ground.