Is gravitational wave frequency always equal to double the orbital frequency?

Question

If the binary does not evolve into merger stage (i.e.: it is still a steady binary), does the gravitational wave frequency have to be 2*orbital frequency? Could the frequency be, for instance, 2/3*orbital frequency and steady, rather than a chirp? And if not, could the frequency 2/3*orbital frequency be the result of interactions with other waves of this binary? The frequency range is about several milihertz, low frequency.

Answer 1

A binary will always emitted a spectrum of gravitational wave frequencies. As long as the binary is in the adiabatic regime (meaning that changes to the orbit due to the emission of gravitational waves happen on a timescale much longer than the orbital period) this spectrum is determined by the frequencies of the orbit.

When the binary is circular and non-precessing, this spectrum consists purely of integer multiples of the orbital frequency. Usually (although there are extreme counter examples) the mode at double the orbital frequency dominates the spectrum, often being strong then all other modes combined. The other modes tend be suppressed exponentially, but generically all integer multiples of the orbital frequency will be present.

General binaries however are not circular, but have some eccentricity and feature precessing spins. Such a system is characterized the by four fundamental frequencies, the orbital frequency $\Omega_\phi$, the frequency of the radial oscillation $\Omega_r$, and the precession frequencies of the two spins $\Omega_\theta$ and $\Omega_\psi$. The gravitational wave spectrum of such a binary consists of integer combinations $m \Omega_\phi+ n\Omega_r + k_1\Omega_\theta+k_2\Omega_\psi$, and generically all such combinations are present, but a few will dominate.

In the weak field regime all these frequencies are approximately equal, and the spectrum again consists (to first approximation) of only integer multiples of the orbital frequency. However, being in the weak field, is not the only way of being in the adiabatic regime. The evolution rate of a binary is proportional to the mass ratio. Consequently, binaries with small mass-ratios can be in the slowly evolving while being in the strong field regime, where the fundamental frequencies can be very different.

To achieve the milliHertz frequencies request while in the strong field regime the total mass of the binary needs to be of the order a million solar masses. So we could get an (extremely) small mass-ratio slowly evolving binary by considering a solar mass object, orbiting a million solar mass supermassive black hole (e.g. the one in the center of our own galaxy).

Now can we get a situation where there GW modes with 2/3s of the orbital frequency. The answer is yes. One situation is as follows. Consider a binary with eccentricity $e= 0.1$, and semi-latus rectum $p$ roughly 10.8 times $GM/c^2$. In this case, the radial frequency $\Omega_r$ is roughly $2\Omega_\phi/3$. The spectrum of this binary contains a mode with frequency $\omega = 2\Omega_\phi - 2 \Omega_r = 2\Omega_\phi/3$. However such a mode will be fairly weak with a strain amplitude of only about 0.5% of the dominant mode at twice the orbital frequency.

Answer 2

The gravitational waves (GWs) from a perfectly stable, circular binary system are completely dominated by GWs at twice the orbital frequency.

If the binary orbit is eccentric then the GWs have a frequency spectrum; the waves are a combination of discrete frequencies at integer multiples of the orbital frequency (e.g. Wen 2003). At low eccentricity, most of the power is still at twice the orbital frequency, but for eccentricities greater than about 0.3, the peak frequency moves to higher and higher multiples.

Despite mmeents objections, I believe what I've written is (approximately) true for the circumstances posed in the question - i.e. a non-evolving binary, where the GW energy losses are small, the binary components are thus widely separated and can be treated as point-like, and the rate of periastron precession is small compared with the orbital frequency (note that if the eccentricity were really high, $e \sim 1$, then the latter may not be true). Closer binaries may have significant higher order multipole emission and modifications to their frequency spectrum caused by the decay of the orbit, precession of their periastron, their spin etc.