Test bodies are neither point like nor spherical. Therefore any source mass (e.g. the
Earth and the spacecraft body) will interact with their mass moments (higher than the
monopole) giving rise to differential effects between test masses. In the case of the
Earth, because of the 
rotation of
the test masses and the sensors, the effect will have this frequency, just like an EP
violation. On the contrary, when the source is an unbalanced spacecraft mass the effect is
DC because both the source and the test mass spin at the same rate. Let us compute the
quadrupole acceleration due to the Earth (regarded as a point mass) on a test body in the
shape of hollow cylinder with inner radius a, outer radius b, height L,
a fractional difference 
between its principal moments of inertia and the symmetry axis at an
angle 
with respect to the normal to
the orbit plane ( at most a few degrees).
We get:
in the plane of symmetry of the cylinder (the effect along the symmetry axis is from 10 to
100 times smaller). The function 
is about 1 for small .
For two concentric test cylinders of equal mass, the dimensions and 
will be in general different,
thus giving rise to a differential effect in competition with an EP violation signal. We
have checked that in the 3-chamber setup of Fig. 1 which
was used for the finite element numerical stability analysis presented in Section 3 (equal composition bodies in the central chamber,
different composition bodies in the others) the resulting value of the quadrupole
acceleration given by Eq. 55 is for all bodies at
least one order of magnitude smaller than the expected signal 
. Evidently, their
differences in each chamber will also be below the signal. We therefore did not devote any
effort to making the numerical value of the quantity 
equal for the two masses. It is worth noticing
that the values of
(see Table 1 ) are not too small (from a few percents
to 
) in order to avoid additional
instabilities.
As for the interaction of an unbalanced mass of the spacecraft with
the quadrupole moment of a test body, we note that despite a much smaller source mass this
effect is in fact larger because of its dependence on the fourth inverse power of the
distance. However, it is constant because of the spinning of both the spacecraft and the
test masses at the same frequency.
Copyright © 1998 Elsevier Science B.V., Amsterdam. All Rights Reserved.
(Anna Nobili- nobili@dm.unipi.it)
