In any small force gravitational experiment electric charges on the test bodies must be
absolutely avoided since they can easily produce forces far much bigger than the
gravitational signal. The GG spacecraft does not contain free floating masses, and
therefore no electrostatic charges will be able to build up inside it. Potential
differences between the test masses can be avoided by coating them with a thin layer of
the same conductive material. The PGB laboratory can be made of 
, or another highly conductive material, so
as to work as a Faraday cage, shielding the experiment from external electric fields. Any
small residual charge inside the spacecraft will only produce a constant effect on the
output signal and can therefore be neglected. It is worth stressing that this is a very
important advantage of GG as compared to STEP, where the test bodies are suspended by
means of magnetic bearings and the problem of discharging them without producing unwanted
perturbations is a serious one. The problem is even worse if the spacecraft orbit goes, as
in the case of STEP, through the Van Allen belts and is subject to the bombardment of
charged particles in the so-called South Atlantic Anomaly. The orbit of GG is equatorial
and at a low enough altitude to avoid the South Atlantic Anomaly.
Fig. 18. In an equatorial orbit the direction of the Earth's magnetic field is not more than about
away from the perpendicular to the orbit. Therefore its effects in the plane of the orbit will be reduced to about
times those of the full field. These effects are distinguishable from a violation of the equivalence principle because they change sign every half orbit.
Magnetic disturbances can be of two types: interactions of magnetized
or magnetizable materials between themselves and interactions between these materials and
the Earth's magnetic field. All magnetic and especially ferromagnetic materials inside the
spacecraft should be avoided, i.e. magnets, electromagnets and electric motors. All
electric currents should flow in shielded cables. For small controlled displacements we
shall use piezoelectric actuators and active damping will be realized with electrostatic
actuators. Any residual constant magnetization inside the spacecraft, as for the case of
electrostatic charges, would produce only a DC effect (unless there is a relative motion
at the spinning frequency). If the GG satellite is essentially non magnetic, its attitude
is almost unaffected by the Earth's magnetic field 
because on average over one orbit is almost parallel to the spin axis. Along the satellite orbit the
instantaneous orientation of is no
more than about away from the spin
axis of the satellite (Fig. 18 ). Thus, the component
of in the orbit plane can only
produces a signal whose amplitude is reduced by a factor 
. If the interaction is between the proper test
mass magnetization and the Earth's magnetic field, the frequency of this signal is 
, because the shape of the
magnetic field rotates with the Earth. The signal also changes sign every half orbit of
the satellite. When the interaction is between induced magnetization on the test mass and
the Earth's magnetic field, the frequency is 
because it depends on 
. We can argue in the same way for the
component of parallel to the spin
axis. The interaction between magnetized test masses gives an signal only if the
interaction is between an induced magnetized test mass and a test mass with its own
magnetization. A signal at this frequency could also arise if the test masses have their
own magnetization and are subject to deformations (e.g. due to nonuniform thermal
expansion) at the 
frequency.
Let us now estimate the intensity of these perturbations. The most
dangerous ones are those which act at (or close to) the spin/signal frequency. The largest
among them is due to the interaction between the magnetic moment 
(due to ferromagnetic
impurities) of one test body and the magnetization induced on the other (with
susceptibility 
) by
the magnetic field of the Earth. In the worst case hypotheses the resulting perturbing
force which competes with the signal is (in MKS):
with 
(
the permeability of
vacuum), 
, 
the mutual distance and 
the volume of body 1. For it
to be smaller than the signal 
it must be:
Since reasonable values for the susceptibility are 
, we need 
. From experimental data reported in textbooks we find that the
magnetic moment of a cube of magnet of 
size is about 
. Since
this is in fact quite a large impurity, it appears that the inequality (Eq. 59 ) can be satisfied by the test bodies thus ruling
out any need of reducing the magnetic field of the Earth inside the satellite. Another
magnetic perturbation which acts at the spin/signal frequency is due to the interaction of
the magnetic moment of one test body with the magnetic field of the Earth. Worst case
hypotheses give:
hence we need 
,
which can be easily satisfied. A DC magnetic perturbation comes from the interaction of
the magnetic moments of two test bodies with one another:
For it to be smaller than the signal it must be 
, which, from the previous discussion, is a
reasonable requirement for a DC effect. We complete the analysis by estimating the
magnetic perturbations which contain 
and therefore appear at a frequency close to twice the spin/signal frequency.
One of these effects is due to the interaction of the magnetic moments induced on the test
bodies with one another:
which requires 
and
is satisfied because 
.
Another 
effect comes from the
interaction between B and the magnetization induced on a test body by B
itself. The resulting perturbation force is:
and it is smaller than the signal provided that 
, which is surely the case.
In conclusion, as far as magnetic perturbations in the GG experiment
are considered, we have demonstrated that the only one which sets a somewhat demanding
requirement is due to the coupling of the magnetic field of the Earth with the magnetic
moment of either test body due to residual ferromagnetic impurities. For this effect to be
neglected we need the magnetic moment of the test bodies not to exceed a few 
, which appears to be feasible. On the
contrary, in the torsion balance of the Eöt-Wash experiment (Su, 1992; Su et al.,
1994) the magnetic field of the Earth near the balance was reduced by a total factor
of 
(partly with 
-metal shielding, partly with
Helmotz coils). This seems to contradict our previous conclusion, especially if one
considers that they have reached a sensitivity 
while the GG target is 
. Indeed, it is not so and we
are going to show why. The first important fact to bear in mind is that, despite its
higher target sensitivity the GG expected force signal is 
times larger than it is in Eöt-Wash, because of the bigger EP
signal in space and the larger mass of the test bodies. In GG there are two test bodies
of 
each while in
the Eöt-Wash torsion balance there are 4 masses of 
each; the force signals are 
and 
respectively. Note that the force, not the acceleration, is
relevant when dealing with nongravitational perturbations. Secondly, since the Eöt-Wash
experiment is a torsion balance experiment it is sensitive to torques, hence also to the
magnetic torque generated by the interaction of the magnetic field of the Earth with
magnetic moments of the test bodies (due to residual ferromagnetic impurities). Indeed, it
turns out that the magnetic moment of the tray on which the test bodies are positioned
gives an even larger perturbation than the test masses themselves. For this torque to be
smaller than that due to an EP violation it must be:
where 
is the length of the arm. It
must therefore be 
(having used 
as above, although
the value used by Su et al. (1994) is
actually 
). From measurement of
the torsion angle in absence of any shielding or coils the Eöt-Wash group finds that the
residual magnetic moment of the tray (made of Al) is 
(Su, 1992; Su et al., 1994), thus making a reduction of B
by crucial for the success of the
experiment. This is achieved by means of a 3-layer -metal shielding for a factor 
and of Helmotz coils for a factor 28. As for
the Eöt-Wash test masses, the measured value of the residual magnetic moment is 
while the requirement imposed by the
magnetic torque is about 
; with a
factor of reduction of the
magnetic field of the Earth made necessary by the tray, this effect is no problem. The
magnetic dampers, used to kill the swing and wobble modes so that the motor can provide a
smooth rotation, will also benefit of the reduction of the magnetic field. In GG we have
symmetric and concentric masses and the signal is a force, not a torque, thus we have
nothing like the torque (Eq. 62 ); we do not have any
motor or magnetic dampers either.
Copyright © 1998 Elsevier Science B.V., Amsterdam. All Rights Reserved.
(Anna Nobili- nobili@dm.unipi.it)
