6.2 Optimization of SNR for binaries with known direction but with unknown orientation of the
orbital plane
Binaries will be important sources for LISA and therefore the analysis of such sources is of major
importance. One such class is of massive or super-massive binaries whose individual masses could range
from
to
and which could be up to a few Gpc away. Another class of interest are known
binaries within our own galaxy whose individual masses are of the order of a solar mass but are just at a
distance of a few kpc or less. Here the focus will be on this latter class of binaries. It is assumed that the
direction of the source is known, which is so for known binaries in our galaxy. However, even for such
binaries, the inclination angle of the plane of the orbit of the binary is either poorly estimated or unknown.
The optimization problem is now posed differently: The SNR is optimized after averaging over the
polarizations of the binary signals, so the results obtained are optimal on the average, that is, the
source is tracked with an observable which is optimal on the average [21
]. For computing the
average, a uniform distribution for the direction of the orbital angular momentum of the binary is
assumed.
When the binary masses are of the order of a solar mass and the signal typically has a frequency of a few
mHz, the GW frequency of the binary may be taken to be constant over the period of observation, which is
typically taken to be of the order of an year. A complete calculation of the signal matrix and the
optimization procedure of SNR is given in [20
]. Here we briefly mention the main points and the final
results.
A source fixed in the Solar System Barycentric reference frame in the direction
is considered.
But as the LISA constellation moves along its heliocentric orbit, the apparent direction
of the
source in the LISA reference frame
changes with time. The LISA reference frame
has been defined in [20
] as follows: The origin lies at the center of the LISA triangle and the plane of LISA
coincides with the
plane with spacecraft 2 lying on the
axis. Figure (9) displays this
apparent motion for a source lying in the ecliptic plane, that is with
and
. The source
in the LISA reference frame describes a figure of 8. Optimizing the SNR amounts to tracking
the source with an optimal observable as the source apparently moves in the LISA reference
frame.
Since an average has been taken over the orientation of the orbital plane of the binary or equivalently
over the polarizations, the signal matrix
is now of rank 2 instead of rank 1 as compared with
the application in the previous Section 6.1. The mutually orthogonal data combinations
,
,
are convenient in carrying out the computations because in this case as well, they
simultaneously diagonalize the signal and the noise covariance matrix. The optimization problem now
reduces to an eigenvalue problem with the eigenvalues being the squares of the SNRs. There
are two eigen-vectors which are labelled as
belonging to two non-zero eigenvalues. The
two SNRs are labelled as
and
, corresponding to the two orthogonal (thus
statistically independent) eigen-vectors
. As was done in the previous Section 6.1 F the two
SNRs can be squared and added to yield a network SNR, which is defined through the equation
The corresponding observable is called the network observable. The third eigenvalue is zero and the
corresponding eigenvector orthogonal to
and
gives zero signal.
The eigenvectors and the SNRs are functions of the apparent source direction parameters
in
the LISA reference frame, which in turn are functions of time. The eigenvectors optimally track the
source as it moves in the LISA reference frame. Assuming an observation period of an year, the
SNRs are integrated over this period of time. The sensitivities are computed according to the
procedure described in the previous Section 6.1. The results of these findings are displayed in
Figure 10.
It shows the sensitivity curves of the following observables:
- The Michelson combination
(faint solid curve).
- The observable obtained by taking the maximum sensitivity among
,
, and
for each
direction, where
and
are the Michelson observables corresponding to the remaining
two pairs of arms of LISA [1]. This maximum is denoted by
(dash-dotted curve)
and is operationally given by switching the combinations
,
,
so that the best
sensitivity is achieved.
- The eigen-combination
which has the best sensitivity among all data combinations (dashed
curve).
- The network observable (solid curve).
It is observed that the sensitivity over the band-width of LISA increases as one goes from
Observable 1 to 4. Also it is seen that the
does not do much better than
. This
is because for the source direction chosen
,
is reasonably well oriented and
switching to
and
combinations does not improve the sensitivity significantly. However, the
network and
observables show significant improvement in sensitivity over both
and
. This is the typical behavior and the sensitivity curves (except
) do not show much
variations for other source directions and the plots are similar. Also it may be fair to compare the
optimal sensitivities with
rather than
. This comparison of sensitivities is
shown in Figure 11, where the network and the eigen-combinations
are compared with
.
Defining
where the subscript
stands for network or
,
, and
is the SNR of the observable
, the ratios of sensitivities are plotted over the LISA band-width. The improvement in
sensitivity for the network observable is about 34% at low frequencies and rises to nearly 90% at about
20 mHz, while at the same time the
combination shows improvement of 12% at low frequencies rising
to over 50% at about 20 mHz.