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.
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.
(Anna Nobili- nobili@dm.unipi.it)
