Coalescing binary features measurable with a spherical gravitational wave detector

UCLEAR PHYSICS

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Nuclear Physics B (Proc. Suppl.) 48 (1996) 113-115

Coalescing binary features measurable with a spherical gravitational wave detector E. Coccia and V. Fafone Dipartimento di Fisica - Universit& di Roma "Tor Vergata" and INFN Via della Ricerca Scientifica, 1 - 00133 Roma (Italy) We discuss the interaction of the gravitational waveform emitted by a coalescing binary during its inspiral phase with the two lowest-order quadrupole modes of a resonant spherical mass. We find that a large spherical detector can determine the chirp mess, the orbit inclination, the distance and the direction of the source.

Coalescing compact binaries consisting of either neutron stars or black holes are one of the most important potential sources for groundbased gravitational wave (gw) detectors. These sources have received a lot of attention because of the existence of PSR 1913+16 [1] and the clean analytic behaviour of the source. In particular, the gravitational waveforms of increasing amplitude and frequency, called "chirp", expected for the inspiral phase, can be well modeled. Today the main interest in this type of source is connected with the new large interferometer detectors [2,3], since resonant bar antennas [4]-[6] are considered of insufficient sensitivity to detect chirps. We have reexamined this matter, stimulated by the potentialities of the proposed resonant detectors of spherical shape [7]-[12]. Let us consider the equations governing the gravitational radiation emitted by a binary system in the Newtonian regime, assuming orbits of zero eccentricity. The waveform is thus sinusoidal with a frequency and amplitude increasing in time. In finite time the system will decay to coalescence, so the system state can be parametrized by the time to coalescence ~-. Using this parametrization, the frequency of the gw, twice the orbital frequency, is given by [13]:

termines the overall frequency acceleration of the chirp signal and is called the chirp mass. From the quadrupole formula for masses in circular orbit, the waveform amplitudes of the emitted gravitational radiation can be calculated. In the proper reference frame of the wave, the h+ and h× oscillations are 90 ° out of phase respect to each other and their amplitude are:

1 (5Gs'~ 1/4 h0+(7-) = ~ r \ - ~ ]

M5c/47--1/4(1 + cos2 t.)(3)

1 ( 5 G 5 ~ 1/4 5 / 4 - 1 / 4 _ h0×(7-) = r \-~iq-] Mc 7cos~

where ~ is the angle of inclination of the orbit to the line of sight. Equation 1 for wg can be integrated to yield an expression for the amount of phase until coalescen ce:

ca

McSlSr s/s

Mc ==-.(mlm2)3/5(rr~l + m2) -1/5

(2)

and ml and m2 are the masses of the two compact objects. Mc is the only parameter that de0920-5632/96/$15.00© 1996 ElsevierScienceB.V. All fights reserved. PII: S0920-5632(96)00222-8

(5)

Using this phase the gravitational waveform of the generic polarisation state can be readily approximated by h(7-) = ho(~) sin[q~(7-) + qO~b]

where

(4)

(6)

where ~ b is an arbitrary initial phase different by 90 ° for the 2 polarisation amplitudes. At some point (small value of 7- and large a)g) the real physical system will deviate from the waveform given above under the assumption of ideal point-like masses, due to the influence of tidal, mass-exchange and post-Newtonian effects.

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E. Coccia, V. Fafone /Nuclear Physics B (Proc. Suppl.) 48 (1996) 113-115

According to Dewey [14] we set a limit that is reasonably independent of the source model by considering this equation inaccurate when there are less then five cycles remaining in the waveform. It is easy to show, using eq. (1) and (5), that ws~v = 27r(1525Hz)

~

.

(7)

The detector analysis will then require that wse~ for the source be larger than the resonant frequencies of the detector. Following General Relativity, gw's interact with the vibrational modes of a solid body that show a nonzero quadrupole mass-moment. Spherical gw detectors appear very promising because the lowest order, five-fold degenerate, quadrupole mode (whose angular dependence can be described in terms of the five spherical harmonics Yt,~(0, ¢) with 1 = 2 and m = - 2 , . . . + 2) presents an isotropic cross section, much larger than that of a cylindrical antenna made of the same material and with the same resonant frequency, and allows one to determine the gw intrinsic amplitudes of the two polarisation states and the two angles of the source direction [8]. The Fourier components of the gw amplitudes h+(t) and h× (t) at the fundamental quadrupole frequency and the two angles defining the source direction can be obtained as proper combinations of at least five transducer outputs. Practical and elegant configurations have been discussed in ref. [9,10,15], where explicit formulas can be found. A recent result is that a resonant sphere can be used as a multifrequency detector: for the cross section of quadrupole modes of a spherical detector it has been found that [11,12]:

~(~)

= f

GMv~ I',~ ~ (~ _ w~) + r ~ / 4

(8)

where M is the sphere's mass, w,~ is the n-th 5fold degenerate resonance frequency of the sphere, and F,, is the linewidth of the mode. Finally F~ is a dimensioneless coefficient which is characteristic of each mode. It has been found numerically for the first-order quadrupole mode F1 = 2.98, and for the second-order quadrupole mode F2 = 1.14. It is remarkable that the second-order quadrupole mode cross section is only a factor 2.61 lower than

that of the first-order quadrupole mode. This means that this detector can be used to advantage at two frequencies. We find that the multimode and multifrequeney nature of a spherical detector can be used to determine the main features of a coalescing binary: i) Chirp mass. Suppose that during its sweep, the chirp frequency reaches the value wl (first sphere quadrupole frequency) at time rl and the value w2 (second sphere quadrupole frequency) at time 72. Measurement of the time delay 7 1 - r 2 between excitations of the first and second quadrupole modes on the same detector will then determine the frequency acceleration of the chirp signal and allow the chirp mass to be measured automatically. An explicit formula can be easily found by subtracting eqs. (1), calculated for wg(71) = wl and wg(72) = ~2:

M, = 211/5 \ 5 - - ~ ]

G \

72

71

]

(9)

ii) Orbit inclination. The spherical detector is able to measure the amplitudes of the two polarisation states. The orbit inclination ~ can be determined by the proportion of the two polarisation state amplitudes (see eqs. (3) and (4)): ho× /to+

- -

=

2 cos 1 + cos 2

(10)

This measurement can be done separately at each of the two frequencies. iii) Source distance. Having determined the chirp mass M~ and the orbit inclination ~, the measurement of the intrinsic gw amplitudes tells us the source distance r. Moreover, as already mentioned, the spherical detector is able to determine the source direction (with the antipodal ambiguity) . Let us now evaluate the signal-to-noise ratio ( S N R ) of spherical detectors to the chirp signal. A convenient way to write the energy AEn deposited in the n-th vibrational mode of the detector is through its gw energy absorbtion cross section a(w) and incident gw flux ~(w); the sensitivity of the readout system can be defined by the rms values of the detector energy fluctuations at the various modes: ,~ tzr('~) Jt e f f"

E. Coccia, V. Fafone /Nuclear Physics B (Proc. Suppl.) 48 (1996) I13-115

The S N R can be calculated analytically following the treatment presented by Clark and Eardley and Dewey [14]. Approximating the waveform by a truncated Taylor expansion and performing further algebraic manipulations, the S N R for chirp detection of a spherical detector equipped with identical, properly located, transducers is:

SNR~

---- ( F,, Mv~ ~ 1 5~r21/3 i-q-(n) l -g r

k G

"" ~ e f f

5/3

M5/3 - "- C

(11)

6

/

,-1/3 L.~?$

The results of our analyses are reported in fig. 1. We assume quantum limited detectors {bT(n)

250

200 ...............

i ................

i .................

i

~, 150

bl

loo ....

50

¢- ........

.... .

i

0.5

- ---~ y ~-,.c:

i

,

i

i .................

...................

i

,

:

i

m/Mo

i

...................

i

1.5

i

i

J

i

2

Figure 1. Distance at which binaries of the indicated mass (m = ml = m 2 ) give S N R = 1 at the resonance frequency w2. When observed at the frequancy ~1 the same sources gives a S N R about 6. Dotted lines refer to A15056 spheres, continuous lines to CuA1 spheres. The considered diameters are: 4m (a), 3.5m (b) and 3m (c).

Time delays between excitation of the two modes are typically of the order of some hundredths of seconds. Since wl and w2 are usually known with a relative precision better than 10-6, the main relative error in the estimation of the

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chirp mass is due to the timing (see eq. (9)). It has been shown [16,17], that a timing accuracy of the order of one millisecond is attainable in a resonant-mass gw detector even at low SNR. In our opinion the results reported here indicate that large resonant spheres could be very interesting instruments for studying coalescing binaries. The interval of masses covered by these detectors ranges from that of neutron starneutron star binaries to black hole-black hole binaries for very large spherical detectors. We thank Guido Pizzella for his encouragement and valuable advice. We thank Valeria Ferrari and Alberto Lobo for useful discussions. REFERENCES 1. J.H. Taylor Rev. of Modern Physics 66, 711 (1994). 2. A. Abromovici et al., Science 26, 325 (1992). 3. C. Bradaschia et al., Nucl. Instr. Meth. A289, 518 (1990). 4. P. Astone et al. Phys. Rev. D 47, 362 (1993). 5. W.W. Johnson et al. in Gravitational Wave Experiments, edited by E. Coccia, G. Pizzella and F. Ronga (World Scientific, Singapore, 1995). 6. D.G. Blair et al. Phys. Rev. Lett. 74, 1908 (1995). 7. R. Forward, Gen. Rel. and Gray. 2, 149 (1971). 8. R.V. Wagoner and H.J. Paik in Experimental Gravitation, Proceedings of the Int. Symposium held at Pavia 1976 (Acc. Naz. dei Lincei, Roma, 1977). 9. W.W. Johnson and S. M. Merkowitz, Phys. Rev. Lett. 70, 2367 (1993). 10. C.Z. Zhou and P.F. Michelson, Phys. Rev. D 51, 2517 (1995). 11. E.Coccia, J.A. Lobo and J.A. Ortega, Phys. Rev. D 52, 3735 (1995). 12. J.A. Lobo, Phys. Rev. D 52, 591 (1995). 13. R.V. Wagoner and C. Will, Astrophys. J. 210, 764 (1976). 14. D. Dewey, Phys. Rev. D 36, 1577 (1987). 15. J.A. Lobo and M. Serrano, in preparation. 16. M. Cerdonio et al. Phys. Rev. Lett. 71, 4107 (1993). 17. S.Frasca and M.A. Papa, Intern. Journ. of Mod. Phys. D, 4, 1 (1995).