Gravitational waves from coalescing binaries and Doppler experiments

Doppler tracking of interplanetary spacecraft provides the only method presently available for broadband searches of low frequency gravitational waves $(\sim {10}^{-5}–1\mathrm{Hz}).\mathrm{}$ The instruments have a peak sensitivity around the reciprocal of the round-trip light time T $(\sim {10}^3–{10}^4\hspace{0.5em}\sec )$ of the radio link connecting Earth to the space probe and therefore are particularly suitable to search for coalescing binaries containing massive black holes in galactic nuclei. A number of Doppler experiments—the most recent involving the probes ULYSSES, GALILEO, and the Mars Observer—have been carried out so far; moreover, in 2001–2004 the CASSINI spacecraft will perform three 40-day data acquisition runs with an expected sensitivity about 20 times better than that achieved so far. The central aims of this paper are (i) to explore, as a function of the relevant instrumental and astrophysical parameters, the Doppler output produced by inspiral signals—sinusoids of increasing frequency and amplitude (the so-called chirp), (ii) to identify the most important parameter regions where to concentrate intense and dedicated data analysis, and (iii) to analyze the all-sky and all-frequency sensitivity of the CASSINI experiments, with particular emphasis on possible astrophysical targets, such as our galactic center and the Virgo cluster. We consider first an ideal situation in which the spectrum of the noise is white and there are no cutoffs in the instrumental band; we can define an idealsignal-to-noise ratio (SNR) which depends in a simple way on the fundamental parameters of the source—chirp mass $\mathcal{M}$ and luminosity distance—and the experiment—round-trip light time and noise spectral level. For any real experiment we define the sensitivity function ${\rm Y}$ as the degradation of the SNR with respect to its ideal value due to a colored spectrum, the experiment finite duration $T_1,$ the accessible frequency band ${(f}_b{,f}_e)$ of the signal, and the source’s location in the sky. We show that the actual value of ${\rm Y}$ crucially depends on the overlap of the band ${(f}_b{,f}_e)$ with the instrument response: the sensitivity is best when $f_b\lesssim 1/T$ and $f_e$ coincides with the frequency corresponding to the beginning of the merging phase. Furthermore, for any $f_b$ and $T_1,$ there is an optimal value of the chirp mass—the critical chirp mass ${\mathcal{M}}_c\propto f_b^{-8/5}{T}_1^{-3/5}$—that produces the largest sensitivity function; lower values of $\mathcal{M}$ correspond to a smaller bandwidth and lower SNR. Also the optimal source’s location in the sky strongly depends on ${(f}_b{,f}_e).\mathrm{}$ We show that the largest distance at which a source is detectable with CASSINI experiments is $\sim 600\hspace{0.5em}\mathrm{Mpc}$ and is attained for massive black holes of comparable masses