Pre-Phase A Report - Lisa - Nasa

4.2 Noises and error sources 97
and
∞
−∞
z4 cos y π
2 dy =
2 z(1 + z) e−z , (4.28)
(z 2 + y 2 )
it is expected that the effective value of a(ω) will be approximately proportional to the
rms of the above two expressions:
a(ω) ≈ πz GM
ωR2
5 z z2
+ +
8 4 8 e−z = πGM
5 z z2
+ +
RV 8 4 8 e−z . (4.29)
We take the signal of interest to be the second derivative of the difference in length of two
of the interferometer arms. For frequencies higher than the corner frequency fc for the
LISA antenna of about 3 mHz, the expected acceleration noise level ãn for LISA is given
roughly by
ãn =ãc
ω
ωc
2
, (4.30)
where ãc =6×10 −15 ms −2 / √ Hz and ωc =2πfc. Below fc, an is equal to ac down to
at least 1×10 −4 Hz, and then increases again at some lower frequency. Thus, for fixed
R, sincea(ω) is a monotone decreasing function of ω, the ratio of a(ω) toãn will be a
maximum somewhere below fc . For simplicity, it is assumed below that ãn =ãc down to
zero frequency, although this won’t be the case in reality.
The square of the signal-to-noise ratio S/N for detecting the disturbance is given by
(S/N) 2 2 a(ω)
=2
a2 df . (4.31)
n
In terms of the dimensionless variable z = ωR/V this becomes
(S/N) 2 = V
2 a(z)
πR a2 dz . (4.32)
n
The integral divides naturally into two parts:
where
I1 =
I2 =
zc
0
∞
z4 c
z4 zc
5 z
+
8 4
5 z
+
8 4
(S/N) 2 = π(GM)2
R 3 Va 2 c
(I1 + I2) , (4.33)
z2
+ e
8
−2z dz = 13
32 −
13 3
+
32 16 zc + 1
16 z2
c e−2zc , and
z2
+ e
8
−2z dz =
with zc = ωcR/V .HereE1 is the usual exponential integral.
5
24 zc − 1
12 z2 5
c +
24 z3
c e−2zc − 7
12 z4 c E1(2zc)
(4.34)
,
Corrected version 2.08 3-3-1999 9:33