Gruppo di Meccanica Spaziale, Dipartimento di Matematica, Università di Pisa, Via Buonarroti 2, I-56127 Pisa, Italia
Thermal noise due to dissipation by the electrostatic dampers has been computed and taken into account in Sec. 2.3.5 of the GG Pre Phase A Report and in Sec. III C of the Pisa Preprint ``Proposed New Test of the Equivalence Principle in Space'', 1995, available in Internet and quoted in the same Report. In accordance with those calculations we show here that losses in the dampers themselves, when operated to provide the forces necessary to stabilize the whirl motions due to losses in the mechanical suspensions, are by far negligible compared to the latter. This result is summarized in the statement no. 5 of the Note by S.H. Crandall and A.M.N. (January 23, 1997).
Let us assume for the moment that the only losses in the system are the mechanical ones, hence expressed by the quality factor Q of the suspensions when deformed at the spin frequency (statement no. 1 of the Note by S.H. Crandall and A.M.N.). If so, the stabilizing force to be provided by the electrostatic dampers in order to stabilize the whirl motion of a GG test body is (statement no. 3 of the Note by S.H. Crandall and A.M.N.):
with 
the passive force of the suspension springs (of stiffness k)
with 
the displacement from the equilibrium position we require not to be
exceeded, 
the natural frequency of oscillation of the test body (and the frequency
of whirl too), 
the period of the natural oscillation, 
the mass of the test body.
The active dampers are therefore required to provide a force (slightly larger than):
with 
the vacuum dielectric constant, 
the effective surface of the capacitor's
plate, 
the gap between the plates, 
the voltage of the capacitor. Thus:
The thermal noise 
of the electrostatic damper is
where 
is the Boltzmann constant and: 
, 
, 
, 
, 
are
the operating temperature, the equivalent resistance, the operating frequency, the
electric quality factor (10 is a typical value) and the capacity of the electrostatic
damper. The perturbing force due to the thermal noise of the damper is derived from (3)
using (5):
The electrostatic dampers are fixed on the rotating body spinning at angular velocity 
, so
they must apply the stabilizing force at the frequency 
, with 
the whirl
frequency. The dissipation cycle for the electrostatic damper is the spin period 
,
hence:
From the perturbing force (7) due to losses in the damper itself the corresponding
quality factor 
is derived (not to be confused with the electrical quality factor) which
is a measure of the ratio between the total energy stored in the damper and the energy
dissipated by the damper itself:
In the worst case assumption that all 8 dampers of each test body dissipate according
to (8) (i.e. as if each damper were to provide the entire stabilizing force at the same
time), the total quality factor 
of the whole system of dampers would be:
yielding:
The numerical coefficient in (10) is not much different for the PGB. It is apparent that even if we had a quality factor Q=1 for the mechanical suspensions the quality factor of the electrostatic dampers, accounting for their thermal losses, would be much higher, meaning that dissipation in the dampers is totally negligible. Better than that, we have Q=16,000 for the suspensions of the test masses and even Q=90 for the suspensions of the PGB.
A calculation of losses in the dampers due to shot noise (see Sec. III C of the Pisa Preprint mentioned above) shows that they are only a few times larger than the thermal losses given by (10), hence still negligible compared to the mechanical losses. All other parts of the GG system are rigid and do not dissipate. In addition, and very importantly, GG has no bearings, which is where most of the dissipation (mostly viscous dissipation) takes place.
Clearly, the assumption made in writing (1) is correct. Namely, the only relevant
losses in GG are the mechanical ones, taking place in the suspension springs when deformed
at the spin frequency of 
. This is the statement no. 1 of the Note
by S.H. Crandall and A.M.N. This being the case, there is no doubt that high Q
values can be achieved by means of well clamped suspensions of high mechanical quality.
Indeed, special effort should be devoted to lower the quality factor of these suspensions
down to values close to, or smaller than, 1. This is opposite to what we do. For instance,
the value Q=90 of the PGB suspensions is due to they being made of 3 insulated
electric wires. Instead, we can easily manufacture a spring made of 3 separated wires
insulated at the clamping only, and this would give a higher Q. However, since the
perturbing effects of the forces which stabilize the PGB are common mode, Q=90 is
acceptable.
(Anna Nobili- nobili@dm.unipi.it)