Formula to calculate the frequency of gravitational waves emitted by two bodies inspiral?

Question

Is there a small straightforward equation which can give the approximate frequency of the gravitational waves emitted by the inspiral of two bodies?

Answer 1

To leading order, the angular frequency of radial motion in a binary inspiral obeys the equation $$ \frac{d \omega}{dt} = \frac{96}{5} \left( \frac{G \mathcal{M}}{c^3} \right)^{5/3} \omega^{11/3} $$ where $\mathcal{M}$ is the so-called chirp mass, defined by $$ \mathcal{M} = \left( \frac{m_1 m_2}{(m_1 + m_2)^{1/3}} \right)^{3/5}. $$

The frequency of the gravitational wave signal will be twice the orbital frequency. That is, the observed frequency of the signal is $$ \nu = 2 \frac{\omega}{2\pi} = \frac{\omega}{\pi}. $$ You can use these equations to estimate the chirp mass of the system $\mathcal{M}$ using $\nu$ and $\dot{\nu}$, or vice versa. Given a sensible initial condition, you can solve for $\omega(t)$ or $\nu(t)$ analytically and find the evolution of the signal frequency.

Note that the differential equation for $\omega$ implies that $\omega$ increases monotonically ($\dot{\omega} > 0$). This equation also predicts that $\omega$ will diverge in finite time, which corresponds to the collision event and the end of orbital motion.