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.

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