The limits of accretion

Adapted from IOAA 2022 (Kutaisi), theory problem 10. We have restated it in our own words and kept the original data. In part (f) the official solution keeps the angular speed of the orbit fixed while the separation changes, which gives a change in separation four times too small. Using Kepler's third law, as we do, gives the same conclusion with the right size. The solution was last checked on 23 October 2026.

A compact object of mass $M$ draws in hydrogen evenly from all sides. The infalling gas heats up and radiates, and the radiation pushes on it with the force $F={\sigma }_{\text{T}}I/c$ on each electron, where $I$ is the intensity of the light and

is the Thomson cross-section. Although the force acts on the electrons, it acts on whole atoms.

(a) Find the Eddington luminosity $L_{\text{E}}$, the greatest luminosity the accreting gas can have. (3 points)

(b) An object accretes gas with a luminosity equal to the Sun's. What is its smallest possible mass, in solar masses? (4 points)

(c) The gas starts far away and its energy of falling is turned into radiation. Find the accretion luminosity in terms of $M$, the accretion rate $\dot{M}$ and the radius $R$ of the object. (2 points)

(d) Show that the greatest steady rate of accretion does not depend on the mass of the object. (2 points)

(e) Find this rate, in solar masses a year, for an object of radius $1.2\times {10}^{10}$ m. (2 points)

(f) The object, of mass $M_1$, is now in a binary with a star of mass $M_2$ at a distance $a$, and gains gas from it. Find the condition on the masses for the separation to grow. Ignore the spin of the stars. (7 points)

Answer(a) $L_{\text{E}}=4\pi GM{m}_{\text{H}}c/{\sigma }_{\text{T}}$. (b) $3.04\times {10}^{-5}$ solar masses. (c) $L_{\text{acc}}=GM\dot{M}/R$. (d) ${\dot{M}}_{max}=4\pi m_{\text{H}}cR/{\sigma }_{\text{T}}$. (e) 0.018 solar masses a year. (f) The separation grows when the object gaining mass is the heavier one, as in (8.8).