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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 3 10 Mo . to 8 10 Mo . 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 [21Jump To The Next Citation Point]. 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 [20Jump To The Next Citation Point]. Here we briefly mention the main points and the final results.

A source fixed in the Solar System Barycentric reference frame in the direction (hB,fB) is considered. But as the LISA constellation moves along its heliocentric orbit, the apparent direction (hL,fL) of the source in the LISA reference frame (x ,y ,z ) L L L changes with time. The LISA reference frame (x ,y ,z ) L L L has been defined in [20Jump To The Next Citation Point] as follows: The origin lies at the center of the LISA triangle and the plane of LISA coincides with the (xL,yL) plane with spacecraft 2 lying on the xL axis. Figure (9View Image) displays this apparent motion for a source lying in the ecliptic plane, that is with hB = 90o and fB = 0o. 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.

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Figure 9: Apparent position of the source in the sky as seen from LISA frame for o o (hB = 90 ,fB = 0 ). The track of the source for a period of one year is shown on the unit sphere 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 A 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 A, E, T 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 v+,× belonging to two non-zero eigenvalues. The two SNRs are labelled as SNR+ and SNR ×, corresponding to the two orthogonal (thus statistically independent) eigen-vectors v+,×. 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
2 2 2 SNR network = SNR + + SNR × . (79)
The corresponding observable is called the network observable. The third eigenvalue is zero and the corresponding eigenvector orthogonal to v+ and v× gives zero signal.

The eigenvectors and the SNRs are functions of the apparent source direction parameters (hL,fL) 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 10View Image.

It shows the sensitivity curves of the following observables:

  1. The Michelson combination X (faint solid curve).
  2. The observable obtained by taking the maximum sensitivity among X, Y, and Z for each direction, where Y and Z are the Michelson observables corresponding to the remaining two pairs of arms of LISA [1]. This maximum is denoted by max[X, Y, Z] (dash-dotted curve) and is operationally given by switching the combinations X, Y, Z so that the best sensitivity is achieved.
  3. The eigen-combination v + which has the best sensitivity among all data combinations (dashed curve).
  4. 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 max[X, Y,Z] does not do much better than X. This is because for the source direction chosen h = 90o B, X is reasonably well oriented and switching to Y and Z combinations does not improve the sensitivity significantly. However, the network and v+ observables show significant improvement in sensitivity over both X and max[X, Y,Z]. This is the typical behavior and the sensitivity curves (except X) 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 max[X, Y, Z] rather than X. This comparison of sensitivities is shown in Figure 11View Image, where the network and the eigen-combinations v+,× are compared with max[X, Y,Z].

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Figure 10: Sensitivity curves for the observables: Michelson, max[X, Y,Z], v +, and network for the source direction (o hB = 90, o fB = 0).
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Figure 11: Ratios of the sensitivities of the observables network, v+,× with max[X, Y,Z] for the source direction hB = 90o, fB = 0o.
Defining
SNRa(f ) ka(f ) = ----------------, (80) SNRmax[X,Y,Z](f)
where the subscript a stands for network or +, ×, and SNRmax[X,Y,Z] is the SNR of the observable max[X, Y,Z], 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 v+ combination shows improvement of 12% at low frequencies rising to over 50% at about 20 mHz.


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