What made the first gravitational waves?

Adapted from IOAA 2016 (Bhubaneswar), theory problem T11. We have restated it in our own words and kept the original data. The original asks for the period to be read from a graph of the signal, and we give the reading of about 0.007 s that the official solution uses. The book prints marks for only three parts, and they disagree with its solution, so we give only the total of 50. In (c) we state the white dwarf law and the upper mass of 1.44 solar masses, which the official solution assumes without the problem giving them. The official solution writes that law upside down and puts the wrong power of the mass ratio under the square root, though its 0.359 Hz follows the correct law. It also writes 166.67 Hz where it means 142.86 Hz, without changing its answers. The solution was last checked on 23 October 2026.

The first gravitational waves seen, in September 2015, had a period of about 0.007 s just before the merger. Model the source as a small mass $m$ on a circular orbit round a much larger mass $M$. The orbit shrinks until the small mass reaches the surface of the large body, or, for a black hole, the innermost stable orbit at three Schwarzschild radii. The frequency of the waves is twice the orbital frequency, and Kepler's laws hold until the merger.

(a) Find the frequency $f_0$ of the waves just before the merger.

(b) For main-sequence stars the radius $R$ and mass $M$ are related by $R/R_{\odot }=(M/M_{\odot }{)}^{\alpha }$, with $\alpha =0.8$ above one solar mass and $\alpha =1.0$ from 0.08 to 1 solar mass. Find the highest frequency of waves from a small mass orbiting a main-sequence star.

(c) Find the highest frequency from a small mass orbiting a white dwarf. Take white dwarfs to have radii of about 3000 to 6000 km, with $R\propto M^{-1/3}$, a radius of 6000 km at one solar mass, and masses up to 1.44 solar masses.

(d) Neutron stars have masses of 1 to 3 solar masses and radii of 10 to 15 km. Find the range of frequencies from a small mass orbiting close to the surface of a neutron star.

(e) Write the frequency from a small mass orbiting a black hole of mass $M_{\text{BH}}$ in terms of $M_{\text{BH}}/M_{\odot }$.

(f) From $f_0$ alone, decide whether the central body is a main-sequence star, a white dwarf, a neutron star or a black hole, and estimate its mass.

Answer(a) 143 Hz. (b) 2.5 mHz. (c) 0.36 Hz. (d) From 2000 Hz to 6350 Hz. (e) $4400\text{ Hz}\times M_{\odot }/M_{\text{BH}}$. (f) A black hole of about 31 solar masses.