The physics of accretion

Adapted from IOAA 2024 (Rio de Janeiro), theory problem T9. We have restated it in our own words and kept the original data. In part (f) the official solution rounds the temperature to 4.8 × 10⁴ K and gives 1.3 × 10⁹ solar masses. Without that rounding the mass is 1.2 × 10⁹, a difference of only 1% that falls either side of 1.25. The solution was last checked on 23 October 2026.

A thin, steady disc of gas orbits a compact object of mass $M$ and radius $R$, from an outer radius $R_{max}$ to an inner radius $R_{min}$, with $R_{min}/R_{max}$ much less than 1. Ignore the gravity of the disc itself.

(a) An element of gas of mass $\mathrm{\Delta }m$ moves on a nearly circular orbit and spirals slowly inwards, its lost energy turned into heat by friction. Find the energy per unit mass, $\mathrm{\Delta }E/\mathrm{\Delta }m$, that it releases between $R_{max}$ and a radius $r$ much less than $R_{max}$. (6 points)

(b) Mass enters the disc at the rate $\dot{M}$, and all the energy released becomes radiation. Find the total luminosity of the disc. (5 points)

(c) Find the luminosity from the ring between $r$ and $r+\mathrm{\Delta }r$, per unit width of the ring. (8 points)

(d) The energy released in each ring is radiated from its surface as a blackbody. Find the temperature of the ring. (10 points)

(e) The central object is a black hole of 3 solar masses fed at ${10}^{-9}$ solar masses a year, with $R_{min}=3R_{\text{s}}$. Find the luminosity of the disc and the peak wavelength of its inner part. Ignore the gravitational redshift. (3 points)

(f) Another black hole is fed at one solar mass a year, and the inner part of its disc peaks at $6\times {10}^{-8}$ m. Estimate its mass. (3 points)

Answer(a) $GM/(2r)$. (b) $GM\dot{M}/(2R_{min})$. (c) $GM\dot{M}/(2r^2)$. (d) $T=(GM\dot{M}/(8\pi \sigma r^3){)}^{1/4}$. (e) $4.7\times {10}^{29}$ W, with a peak at $5.2\times {10}^{-10}$ m. (f) About $1.2\times {10}^9$ solar masses, a quasar.