Pre-Phase A Report - Lisa - Nasa

96 Chapter 4 Measurement Sensitivity
The acceleration noise, ∆an, will depend on the spectral noise density, ãn, andthemeasurement
bandwidth of the dither sensing. The dither frequency must be high enough
that the measurement time does not impinge much on the science observations. With
ameanchargingrateof5×10 −18 C/s the charge limit of 2×10 6 electrons is reached in
some 6.4×10 4 seconds and so any charge measurement, whether it be continuous or intermittent,
must have a response time small compared to this. The integration time, τ
(∼ inverse bandwidth), needed to achieve the required charge sensitivity is
τ = m2 a 2 n C2
4V 2
d Q2 s
−2 ∂Co
. (4.24)
∂k
The integration times given by this equation using a 1 volt dither voltage are completely
negligible compared to the charge build-up time and dither frequencies can be selected
just above the science signal measurement range.
The proposed technique for control of the charge on the proof mass is described in section
3.2.6 .
Momentum transfer. Performing an analysis using poissonian statistics for arrival
times to calculate the fluctuating force due to momentum imparted from cosmic ray
interactions yields the following expression for the spectral density of momentum transfer
in a given direction
SM ≈ 2 p 2 λ (N 2 s 2 / Hz) , (4.25)
where p is the momentum (in the given direction) per particle stopped in the proof
mass, and λ is the number of particles stopped per second. Summing the effects of all
particles (protons and helium), taking into account their directions, yields an acceleration
of ∼ 2×10 −18 ms −2 / √ Hz , which is two orders of magnitude below the desired sensitivity.
4.2.5 Disturbances due to minor bodies and dust
In order to provide a rough estimate of how often spurious signals might be generated by
gravity forces due to minor bodies or dust grains passing by one of the LISA spacecraft, it
is assumed that the disturbances take place with the point of closest approach along one
of the interferometer arms. Only the acceleration of one proof mass is considered. Then
the Fourier component of the proof mass acceleration at angular frequency ω is given by
a(ω) = GM
ωR 2
∞
−∞
z 3 cos y
(z2 + y2 dy , (4.26)
3/2
)
where R is the distance of closest approach, V is the minor body relative velocity, M is
the disturbing mass, and z is defined as z = ωR/V .Since
∞
−∞
z2 cos y
z2 dy = πz e−z
+ y2 (4.27)
3-3-1999 9:33 Corrected version 2.08