Pre-Phase A Report - Lisa - Nasa

110 Chapter 4 Measurement Sensitivity
The Doppler modulation. The translational motion of the detector relative to the
source leads to a phase modulation of the measured gravitational wave signal. This
modulation can easily be calculated with the so-called barycentric transform between
time of arrival at the Solar System and time at the detector [114]. In the former system,
which can be considered to be a convenient inertial frame, the signal is not modulated
and therefore of fixed frequency. Let sd and sb be the signals at the detector and at the
barycenter, respectively; then by definition
sd(td) =sb (tb[td,θ,φ]) , (4.65)
where (θ, φ) is the angular position of the source (see Figure 4.9). The relation between
the two time variables tb,td is given by
tb[td,θ,φ]=td + n(θ, φ) d(td)
c
, (4.66)
with n being a unit vector pointing towards the source and d a vector connecting LISA
and the sun:
⎛
⎞
cos φ sin θ
n = ⎝ sin φ sin θ ⎠
cos θ
⎛
d = R ⎝
⎞
⎠ . (4.67)
cos 2πt
T
sin 2πt
T
0
Therefore the relation between the two signals sd and sb as functions of time is
R sin θ
2πt
sd(td) =sb td + cos − φ
c T
. (4.68)
So if the signal in the inertial frame is pure sinusodial of frequency f GW, in the detector
response it appears as
sd(td) = sin(2πfGWtb)
= sin
c
2πt
cos − φ ,
T
Φ(t)
(4.69)
2πf GWtd + 2πf GWR sin θ
including a phase modulation Φ(t) with a modulation index m of :
m = 2πfGWR sin θ
c
fGW
≈ π sin θ . (4.70)
1mHz
The LISA response to a gravitational wave. A gravitational wave which is purely
sinusoidal in the barycentric frame causes a detector response given by :
⎛
0
⎜
H = T ⎜ 0
⎝ 0
0
h+
h×
0
h×
−h+
⎞
0
0 ⎟
0 ⎠
0 0 0 0
Tt exp {i [2πfGWt +Φ(t)]} , (4.71)
3-3-1999 9:33 Corrected version 2.08