Pre-Phase A Report - Lisa - Nasa

108 Chapter 4 Measurement Sensitivity
Figure 4.8 Annual revolution of LISA configuration around the Sun, describing
a cone with 60 ◦ half opening angle. One selected 2-arm interferometer is
highlighted by heavier interconnecting laser beams. The “tumbling” motion
of a single LISA interferometer allows the determination of the position of the
source as well as of the polarisation of the wave. The (heavier) trajectory of
one individual spacecraft is shown, inclined with respect to the (fainter) Earth
orbit.
modulation and Doppler modulation will spread a sharply peaked monochromatic signal
into a set of sidebands separated from the carrier at integer multiples of the fundamental
frequency (1 year) −1 . It is easy to see that the effects of the beam-pattern modulation and
Doppler modulation are of roughly the same size. Consider a monochromatic signal with
frequency f0. In Fourier space, the effect of the beam-pattern modulation is to spread
the measured power over (roughly) a range f0 ± 2/T ,whereT is one year. (The factor
of 2 arises because the beam pattern is quadrupolar.) The effect of the periodic Doppler
shift coming from the detector’s center-of-mass motion is to spread the power over a
range f0(1 ± v/c), where v/c ∼ 10 −4 . These two effects are therefore of roughly equal size
at f0 ∼ 10 −3 Hz, which is near the center of the LISA band; beam-pattern modulation is
more significant at lower frequencies and Doppler modulation is more significant at higher
frequencies.
In the following chapters we review the way the source position, polarisation and intrinsic
amplitude are encoded in the LISA datastream. After reviewing some standard methods of
parameter estimation, we then present results on how accurately these physical parameters
can be determined for two LISA sources of particular interest: stellar-mass binaries and
merging MBH binaries. We refer the reader to [112, 113] for more details.
The beam-pattern modulation. The beam-pattern modulation can be calculated by
transforming the metric-tensors
h× := h×
⎛
⎝
0 1 0
1 0 0
0 0 0
⎞
⎛
⎠ and h+ := h+ ⎝
1 0 0
0 −1 0
0 0 0
⎞
⎠ (4.60)
3-3-1999 9:33 Corrected version 2.08