2 Gravitational waves from binaries

The gravitational wave luminosity of a binary in the nth harmonic is given by (Peters & Matthews 1963)

$$L (n, e) = \frac{32}{5} \, \frac{G^4}{c^5} \frac{\,M^2 \,m^2 \,(M+m)}{a^{5}} g(n, e).$$ (1)

Here M and m are the masses of the components, a is their orbital separation and e is the eccentricity of the orbit. The function g(n, e) is the Fourier decomposition of the GW signal.

The measurable signal for gravitational wave detectors is the amplitude of the wave - h₊ and $h_{\times}$ for the two polarisations. These can be computed from the GW flux at the Earth (Press & Thorne 1972)

$$\frac{L_{\rm GW}}{4 \pi d^2} = F = \frac{c^3}{16 \pi G} \langle \dot{h}_+^2 + \dot{h}_{\times}^2 \rangle\cdot$$ (2)

Assuming the waves to be sinusoidal and defining the so called strain amplitude as $h = ({1 \over 2} [h_{\rm +,\max}^2 + h_{\rm\times,\max}^2])^{1/2}$ one obtains

 

$$h (n, e) = \left[ \frac{16 \pi \, G}{c^3 \, \omega_{\rm g}^2} \fr... ...\rm 1\, {\rm hr}} \right)^{-2/3} \left( \frac{d}{\rm 1~{\rm kpc}} \right)^{-1},$$ (3)

where $\mathcal{M} = (M \, m)^{3/5} / (M + m)^{1/5}$ is the so called chirp mass and $\omega_{\rm g} = \pi n/P_{\rm orb}$ is the angular frequency of the emitted wave. In Fig. 1 we plot the values of $\sqrt {g(n, e)}/n$ for different eccentricities. High eccentricity binaries emit most of their energy at higher frequencies than their orbital frequency, reflecting the fact that the radiation is more effective near periastron of the orbit. Thus, eccentric compact binaries may be detectable sources of GW signals at frequencies higher than their orbital frequency (cf. Barone et al. 1988; Hils 1991).

Figure 1: Scale factor of the GW strain amplitude $\sqrt {g(n, e)}/n$ for the different harmonics (Eq. (3)) for e = 0, 0.2, 0.5 and 0.7.

 

 

Table 1: Current birth rates ($\nu$) and merger rates ( $\nu _{\rm merg}$) per year for Galactic disk binaries containing two compact objects and their total number ($\char93$) in the Galactic disk, as calculated with the SeBa population synthesis code (see text).

Type	$\nu$	$\nu _{\rm merg}$	$\char93$
[2.5ex](wd, wd)	2.5 $\times$ 10-2	1.1 $\times$ 10-2	1.1 $\times$ 108
[wd, wd)	3.3 $\times$ 10-3	-	4.2 $\times$ 107
(ns, wd)	2.4 $\times$ 10-4	1.4 $\times$ 10-4	2.2 $\times$ 106
(ns, ns)	5.7 $\times$ 10-5	2.4 $\times$ 10-5	7.5 $\times$ 105
(bh, wd)	8.2 $\times$ 10-5	1.9 $\times$ 10-6	1.4 $\times$ 106
(bh, ns)	2.6 $\times$ 10-5	2.9 $\times$ 10-6	4.7 $\times$ 105
(bh, bh)	1.6 $\times$ 10-4	-	2.8 $\times$ 106