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4. Axial centering and Earth tides

We have seen in Section 3 that a position of relative equilibrium exists, naturally provided by the physics of the system, where the spin axes of the test bodies are extremely close to one another ( in this case). It is worth recalling that the direction of this miscentering is fixed in the rotating system, its location depending on the direction of the original unbalance (by construction and mounting). We have also seen that slow (at the natural frequencies) whirl motions around this relative equilibrium can be damped by active electrostatic sensors/actuators rotating with the system. If the sensing errors allow them to detect displacements as small as the miscentering, the tidal perturbation (in the transverse plane), which is linear with the miscentering, is about 20 times smaller than the EP violation signal we are trying to measure. In case of higher sensing errors, the output signal can be analized to remove tidal variations at the natural frequencies.

The centres of mass of the test bodies could as well be a distance  away from one another along the axis itself, thus also giving rise to a tidal force. Were the spin axis exactly perpendicular to the orbit plane, the tidal force would have no component perpendicular to it. This not being exactly the case, such a component will appear, causing a relative displacement of the centres of mass with respect to one another. This tidal perturbation is, as in the case of an EP violation, a differential force directed towards the centre of the Earth slowly changing its direction as the satellite orbits the Earth. However, it can be distinguished from an EP violation signal and indeed used to drive a servo mechanism for reducing the vertical displacement  from its initial value of  obtained by construction down to below the sensitivity of the experiment. This is possible for two reasons: i) Unlike EP violation, these tidal forces have opposite directions on opposite sides of the Earth, depending on which mass is closer to the Earth (Fig. 10 ). This means that the tidal signal differs from an EP violation signal by the orbital frequency of the satellite, a difference which is detectable by measuring the spin rate of the spacecraft. ii) While the tidal force goes to zero with , the EP violation force doesn't, i.e. mass centering does not change an EP violation signal.

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Fig. 10. Simple scheme of Earth tidal forces on two test bodies which rotate around the same axis but are displaced along it. The figure shows how the component of the tidal force towards the Earth changes phase by  every half orbital period of the satellite around the Earth. Only this component does produce a differential displacement of the centres of mass which can be recorded by the spinning capacitors. It is apparent that a differential force due to a violation of the equivalence principle would not change sign every  orbit and would not go to zero with the separation distance .

If the spin axis is an angle  away from the perpendicular to the orbit plane, and the bodies are at a distance  along the spin axis, the tidal acceleration on each test mass has a component perpendicular to the spin axis:

Allowing for an angle , this gives  in order to have a perturbation to the level of the signal. With the capacitance read out system discussed in Section 7 it is no problem at all to detect miscentering down to the required value and use the piezoelectric actuators shown in Fig. 2 for active centering. Once the tidal acceleration signal has become too small to be detected it will also be too small to perturb the EP experiment. Using the tidal signal for axial centering of the test bodies was originally suggested for STEP (Worden et al., 1990). Besides the differential effect of Earth tides on the test masses, one must also consider the common mode tidal perturbation with respect to the centre of mass of the entire system. In particular, for the test bodies located inside the two experimental chambers above and below the centre of mass (Fig. 1 ), the distance  in Eq. 28 cannot be smaller than about , thus resulting in a common mode tidal perturbation of about , also differing from an EP violation signal by the orbital frequency of the spacecraft (Fig. 10 ). An adequate level of common mode rejection is therefore necessary (see Section 5.2 ).

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

Research Papers Available Online

       (Anna Nobili-