Gravitational radiation is an essential result of Einstein's theory of General Relativity. It is an inevitable consequence of time delay of the gravitational fields experienced by moving masses.

While the exact waveform of gravitational radiation emitted by a general mass distribution is derived by using general relativity's formalism, the total power emitted by a system of two point masses can be easily obtained as will be presented on this page.

Problem set up Edit

We consider two point masses moving in a circular orbit about their common center of mass. Strictly speaking, the orbit cannot be circular, since the gravitational radiation carries away orbital energy, thus decreasing the orbital separation between the components. However, as will be shown, at large orbital separations (compared to the masses' Schwarzschild radii), the orbital dissipation timescale is much longer than the orbital period.

Derivation - order of magnitude Edit

Why quadrupole? Edit

This derivation employs the useful analogy between gravitational radiation and electromagnetic radiation.

The Larmor formula is a famous result from the theory of EM radiation, describing the power emitted by an oscillating electric dipole. This is the lowest order radiation term emitted by an accelerating charge distribution. $ P = \frac{2}{3} \frac{(\ddot{d})^2}{c^3} $

Where $ d $ is the dipole moment, and $ c $ is the speed of light.

For an isolated system of gravitational charges, namely, masses, the second time derivative of the dipole moment is always zero. This result originates from the equivalence of inertial and gravitational mass (the equivalence principle). The dipole moment of a distribution of masses represents the center of mass which is non-accelerating in an isolated system (i.e, no external forces).

Therefore, for gravitational radiation, the lowest order radiation term is the quadrupole radiation.

Derivation Edit

The extension of Larmor's formula to the quadrupole term is simply:

$ P \propto (\partial^3 {Q} / \partial t^3)^2 $

Where $ Q $ is the quadrupole moment of the charge distribution. Dimensional analysis in the context of general relativity yields:

$ P \sim \frac{G}{c^5} (\partial^3 {Q} / \partial t^3)^2 $

The quadrupole moment, calculated with respect to the center of mass is simply the moment of inertia of the system.

For now, we will assume that both masses are of the same order, $ M $.

$ Q \sim M a^2 $

Where $ a $ is the orbital separation. The time scale for changes in the quadrupole moment is $ \omega $, the orbital frequency.

In the Newtonian limit,

$ \omega = \sqrt{GM_T/a^3} $

Where $ M_T $ is the total mass, and so:

$ P \approx \frac{GM^2 a^4 \omega^6}{c^5} \approx \frac{G^4 M^5}{c^5 a^5} $

Dissipation timescale Edit

The energy of the system scales as

$ E \sim -GM^2/a $

And so the dissipation timescale is:

$ \tau \sim \frac{E}{\dot{E}} \approx \frac{c^5 a^4}{G^3M^3} $

Now, in terms of the period ($ \sim 1/\omega $), we can write:

$ \tau \sim \omega^{-1} (\frac{c^2 a}{GM})^{5/2} $

And since $ R_{Sch} \sim \frac{GM}{c^2} $, we get:

$ \tau \sim \omega^{-1} (\frac{a}{R_{Sch}})^{5/2} $

This shows that the orbital dissipation timescale is indeed longer than the orbital timescale, as long as the components are far from contact (the most compact "point mass" is at most as compact as a black hole, so we compare $ a $ to the Schwarzschild radius).

Exact dependence on masses Edit

Carrying out a more careful calculation, keeping the geometrical factors, one gets:

$ P = \frac{32}{5} \frac{G^4}{c^5} \frac{(M_1 M_2)^2 (M_1+M_2)}{a^5} $

This is derived by writing the quadrupole moment with the origin at the center of mass (since we are interested in the second time derivative, any constant terms would not contribute).

Masses 1 and 2 are positioned at $ r_1 = a \frac{M_2}{M_T} , r_2 = a \frac{M_1}{M_T} $.

$ Q = M_1 r_1^2 + M_2 r_2^2 = \mu a^2 $

Where $ \mu = \frac{M_1 M_2}{M_T} $ is the reduced mass.

Differentiating with respect to time, we get:

$ P \sim \frac{G}{c^5} (\partial^3 {Q} / \partial t^3)^2 = \frac{G}{c^5} \mu^2 a^4 \omega^6 $

As before, with the Keplerian $ \omega $:

$ P \sim \frac{G^4 M_T^3 \mu^2}{c^5 a^5} $

A useful definition is of the Chirp Mass, ($ M_{ch} $):

$ M_{ch} \equiv (\mu^3 M_T^2)^{1/5} $

(Note the different exponents).

The chrip mass can be also related to two observable quantities measured in gravitational waves detectors (such as LIGO) - $ f $ and $ \dot{f} $ - the signal's frequency, and its time derivative. Through the chirp mass, the masses of the components can be constrained.

Let us consider the total energy emitted in gravitational waves during a merger of two black holes with a large mass ratio. Most of the energy is emitted when the distance between the two black holes is of the order of the Schwartzschild radius of the bigger one, so $ P \propto \left(\mu/M_T\right)^2 $. The orbital period scales as $ \tau \propto M_T $, so the radiated energy per orbit is $ \varepsilon_o \approx \mu^2/M_T $. This would be the total energy emitted in the case of a radial plunge (merger with zero angular momentum). In the case of an inspiral, all the binding energy has to be radiated, so the total emitted energy is $ \varepsilon_i \approx \mu $, and this happens over the course of $ M_T/\mu $ orbits.