   Nobili et al. / Proposed noncryogenic, nondrag-free test of ...

# 8. Coupling to higher mass moments of the test bodies

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.   