Inspiralling Compact Binaries - I
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In a series of posts I will go through basics of inspiralling compact binaries. In this post, I will go over the study of the study of binary system, and how the orbital motion affects the emission of Gravitational Waves (GWs).
Circular Orbits
The systems that we are interested considyestem of two compact starsm such as neutron stars or black holes. We treat them as point-like particles with masses $m_{1}$ and $m_{2}$ and position vectors $r_{1}$ and $r_{2}$. In the center of mass frame (CoM) this problem reduces to a one-body problem with a reduced mass $\mu = (m_{1} m_{2})/(m_{1} + m_{2}) = (m_1 m_2)/m$ and $r = r_{2} - r_{1}$ is the relative coordinate. The equation of motion is
\[\ddot{r} = -\left(\frac{G m}{r^3}\right) r\]and the orbital frequency $\omega$ is related to the orbital radius R by Kepler’s law
\[\omega^2 = \frac{G m}{R^3}.\]In the Netwonian approximation, the Gravitational Waveform amplitudes from inspiralling circular binary can be witten in terms of the orbital parameters
\[h_{+} = \frac{4}{r} \left( \frac{G M_c}{c^2}\right)^{5/3} \left( \frac{\pi f_{gw}}{c}\right)^{2/3} \left( \frac{1 + \cos^2 \theta}{2}\right) \cos(2 \pi f_{gw} t_{ret} + 2 \phi) \\ h_{\times} = \frac{4}{r} \left( \frac{G M_c}{c^2}\right)^{5/3} \left( \frac{\pi f_{gw}}{c}\right)^{2/3} \cos \theta \ \sin(2 \pi f_{gw} t_{ret} + 2 \phi) \\\]where $f_{gw} = \omega_{gw}/(2 \pi)$ and $\omega_{gw} = 2 \omega$. The Chirp Mass $M_c = \mu^{3/5} m^{2/5} = (m_1 m_2)^{3/5}/(m_1 + m_2)^{1/5}$.
The total power radiated by the gravitational waves is given by
\[P = \frac{32}{5} \frac{c^5}{G} \left( \frac{G M_c \omega_{gw}}{2 c^3}\right)^{10/3}\]We can re-write the power radiated in terms of Post Newtonian (PN) parameter $x = (G m \omega/c^3)^{2/3}$. This parameter is dimensionless and of order $v^2/c^2$. Also by introducing symmetric mass ratio $\nu = \mu/m$, the power radiated becomes
\[P = \frac{32}{5} \frac{c^5}{G} \nu^2 x^5\]Newtonian order Energy spectrum
The GW amplitude stated above is only for binary systes that are fixed. However, the emission of GWs reduce the total orbital energy. The total energy is the sum of kinetic plus potential energy
\[E_{orbit} = - \frac{G m_1 m_2}{2 R}.\]Therefore, the loss of energy suggests R must decrease in time so that $E_{orbit}$ becomes more and more negative. Also as $R$ decreases, $\omega$ increase which also increases the power radiated in GWs. This finally leads to the coalescence of the binary on a sufficientlu long time-scale as long as
\[\dot{\omega} << \omega^2.\]This is the quasi-circular motion regime.
Equating $P$ to $-dE_{orbit}/dt$ we get
\[\dot{\omega}_{gw} = \frac{12}{2} 2^{1/3} \left( \frac{G M_c}{c^3} \right)^{5/3} \omega_{gw}^{11/3}\]or in terms of $f_{gw}$
\[\dot{f}_{gw} = \frac{96}{5} \pi^{8/3} \left( \frac{G M_c}{c^3} \right)^{5/3} f_{gw}^{11/3}\]The fourier transform of $h_{+}(t)$ and $h_{\times} (t)$ can be computed first to calculate the energy spectrum $dE/df$ using
\[\frac{dE}{df} = \frac{\pi c^3}{2 G} f^2 r^2 \int d\Omega (|h_{+}(f)|^2 + |h_{\times}(f)|^2)\]After performing the angular integratuon we get the spectrum in the inspiral phase in Newtonian approximation.
\[\frac{dE}{df} = \frac{\pi^{2/3}}{3 G} (G M_c)^{5/3} f^{-1/3}\]