Pre-Phase A Report - Lisa - Nasa

94 Chapter 4 Measurement Sensitivity
The charge build up will be the result of the random arrival of cosmic rays, with each
‘hit’ depositing a variable amount of charge according to the stochastic nature of the
interaction process. Following an argument similar to that already used for Lorentz force
noise it turns out that the acceleration noise arising from this process can be approximated
by
˙ne f −1 (m s −2 / √ Hz ) . (4.17)
ãn =10 −29.9 ne
In order to keep this acceleration noise below its budget allocation the amount of accumulated
charge must controlled such that
ne ≤ 2×10 13
f
f 1+
3×10−3
2
electrons/protons. (4.18)
This limit is much less severe than that from the Lorentz force noise. It has been based
on a 1 µm ‘asymmetry’ assumption. Whether this is a reasonable figure to use remains to
be confirmed.
Another source of acceleration noise which will occur if the proof mass becomes charged is
through displacement noise modifying the electrostatic force. There are two components
(arising from the second term in equation 4.15) due to changes in the total capacitance
and in the asymmetry factor, which in turn affects the capacitance gradients. Adding
these two in quadrature gives an acceleration noise (for the displacement type sensor
geometry)
ãn = Q2
mC 2
∂C
∂k
1 1
+
4δk2 C2 2 ∂C
˜kn (m s
∂k
−2 / √ Hz ) , (4.19)
where ˜ kn is the spatial displacement noise spectral density and all other symbols retain
their earlier meanings. Putting in the numericalvalues for the LISA sensor design and
using ˜ kn =10 −9 m/ √ Hz gives ãn =1.8×10 −33 n 2 e ms−2 / √ Hz .
The final noise source which will be discussed in this section arises from the interaction
between any free charges on the proof mass and the applied control voltages, Vi, andany
associated voltage noise, Vni. Using the third term in equation 4.15 this noise component
is given by
a 2 n = Q2
m2C 2
n
∂Ci
Vni
∂k
i=1
2
+ Q2
m2C 4
n
Vi
i=1
2
2
∂Ci
k
∂k
2 n (m 2 s −4 / √ Hz ) . (4.20)
The electrostatic forces which come about through the proof mass becoming charged
bring with them associated effective spring constants. There are three effects, two of
which involve the applied potentials and one which does not, which have been considered.
The two involving the applied potentials are given by terms of the form |K| ≈ QVcm
and |K| ≈ 2QVcm
C
4ɛA
C g3 ∂2Ci ∂k2 ,whereVcm and Vdm are the voltages applied in common mode and
differential mode to electrodes on opposite sides of the proof mass. The second of these
two spring constant terms is the most significant for the current sensor design. The spring
constant term arising out of pure electrostatic interaction with the surrounding conducting
surfaces is of the form (for displacement type sensor geometries) |K| ≈ Q2
2C
4ɛA
g 3 .Thisterm
3-3-1999 9:33 Corrected version 2.08