Frequency of gravitational waves

Question

Is there a way by means of dimensional analysis we can obtain the order of frequency of gravitational waves emitted by massive bodies?

Answer 1

To first order we could say that the frequency of the gravitational waves (GWs) will be at, or at small integer multiples, of the inverse of the characteristic timescale upon which the gravitational field can change. In turn this depends on the characteristic mass and size of the system - i.e. the density.

Dimensionally speaking, we could equate the acceleration to the gravitational acceleration and define the acceleration as a length scale divided by the square of a timescale. $$ \frac{L}{T^2} = G\frac{M}{L^2}$$ The frequency is $f=1/T$, so $$f = \left(G \frac{M}{L^3} \right)^{1/2}$$ But the density of a system is $\rho \sim M/L^3$, so this is telling us that the characteristic frequency will be $f \sim (G\rho)^{1/2}$.

In practice this will be a largest possible frequency, the frequencies could always be smaller than this. For instance a binary system will emit GWs with a frequency of twice the orbital frequency. If you do a rough analysis you can see that the orbital frequency when the two objects are touching is $\sim (G \rho)^{1/2}$, where $\rho$ is the density of each star, but is obviously lower if the orbital period is longer and the objects are further apart.

Similarly, an asymmetric rotating object will emit GWs at the rotation frequency. A simple calculation for the maximum mass that can be held on the surface by gravity indicates that objects break up if they rotate at frequencies higher than $\sim (G\rho)^{1/2}$, but of course they can rotate at any frequency slower than this.

A final example woud be GWs from a supernova collapse. This takes place on a freefall timescale, which is about $0.5(G \rho)^{-1/2}$, so again leading to GWs with a principle frequency of $\sim (G\rho)^{1/2}$.

In summary then, a dimensional analysis suggests that the maximum frequency of GWs from a particular type of source will be given by $\sim (G \rho)^{1/2}$, where $\rho$ is the density of matter making up the source.