Problems for Chapter 10

Builds on: Sections 10.1 to 10.7
Reading time: 45 min
Exercises: 0
Olympiad problems: 2

The past problems in the earlier sections of this chapter each used one or two ideas at a time. The two in this section join them up. The first, from IOAA 2016, asks what kind of body could have made the first gravitational waves ever detected, using only their frequency just before the merger. It needs the sizes of main-sequence stars, white dwarfs and neutron stars from Section 10.1, the innermost stable orbit of a black hole from Section 10.2 and the frequency of the waves from Section 10.7. The second, also from 2016, uses the bending of light from Section 10.6 to ask how the Sun's gravity could serve as a giant lens for a telescope placed far out in the Solar System.

10.8.1From textbook to olympiad

Part (f) of the first problem below asks which kind of body, orbited by a small mass, could make waves of 143 Hz just before the merger. The method of Section S2 takes us from the chapter to the solution in these steps.

We start by drawing it. A small mass circles a large one, closer and closer, until it touches the surface or, for a black hole, reaches the innermost stable orbit. The waves are fastest at that last orbit.
We name the physics. By (10.17), $f_{GW} = (1 / π) \sqrt{G M / r^{3}}$ , so the highest frequency that a body can make is set by its mass over its radius cubed, which is proportional to its mean density.
We work backwards from the target. For each kind of body we need the largest possible value of $M / r^{3}$ , and then we compare the highest frequency with 143 Hz.
We use every given, and the earlier parts are hints. The problem gives the law $R \propto M^{α}$ for main-sequence stars, the radii of white dwarfs and neutron stars and the innermost orbit at $3 R_{s}$ . Stars reach at most 2.5 mHz, white dwarfs at most 0.36 Hz, and neutron stars at least 2000 Hz. Only a black hole fits, with a mass of $4400 / 143 = 31$ solar masses.
We check the answer. LIGO found two black holes of 36 and 29 solar masses, 65 in all. Our model of a small mass round a large one is rough for two nearly equal masses, but it still picks out black holes and gives a mass within a factor of about two of the total.

Olympiad problemsPast IOAA problems, restated2 problems

2016What made the first gravitational waves?50 points

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 $R / R_{⊙} = (M / M_{⊙})^{α}$ , with $α = 0.8$ above one solar mass and $α = 1.0$ 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 $M_{BH}$ in terms of $M_{BH} / M_{⊙}$ .

(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.

I solved it without the solution

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

2016The Sun as a telescope20 points

Adapted from IOAA 2016 (Bhubaneswar), theory problem T10. We have restated it in our own words and kept the original data. The solution was last checked on 23 October 2026.

A ray that passes a massive body at the distance $r$ from its centre is bent by $θ_{b} = 2 R_{s} / r$ , where $R_{s}$ is the Schwarzschild radius of the body (the original writes $R_{sch}$ ). Take the bending to happen at one point. Rays from a distant source that pass the body at the same $r$ meet on the axis at a distance $f_{r}$ from its centre, where an observer would see the source greatly amplified.

(a) Find the shortest distance $f_{min}$ from the centre of the Sun, in astronomical units, at which rays can be focused. (6 points)

(b) A small circular detector of radius $a$ is placed at $f_{min}$ , centred on the axis and facing the Sun. Only rays that pass the Sun in a thin ring of width $h$ , much less than $R_{⊙}$ , reach it. The amplification is the ratio of the light falling on the detector with the Sun to the light without it. Find the amplification in terms of $R_{⊙}$ and $a$ . (8 points)

(c) For a round cloud of mass, such as the dark matter of a cluster of galaxies, let a ray with impact parameter $r$ be bent only by the mass $M (r)$ inside $r$ . How must $M (r)$ grow with $r$ for the cloud to act as a perfect lens, bringing all parallel rays to one focus? (6 points)

I solved it without the solution

Answer(a) $f_{min} = R_{⊙}^{2} c^{2} / (4 G M_{⊙}) = 548$ au. (b) The amplification is $R_{⊙} / a$ , which for a detector of radius 1 m is $6.96 \times 10^{8}$ , or 22.1 magnitudes. (c) $M (r) \propto r^{2}$ .