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 Incoming links: Advection-dominated accretion flow

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 347 / 2 / c / Solution Created 2026-09-24 Updated 2026-09-24

Insert the self-similar ansatz into the height-integrated equations. Mass conservation is already satisfied because

H \sim c_{s} / Ω_{K} \propto R

and

ρR H u_{R}

is constant. The angular-momentum and energy equations reduce to

c_{1} = - \frac{3}{2} α c_{3}, c_{2}^{2} = ϵ^{'} c_{3},

while radial momentum gives

1 = \frac{1}{2} c_{1}^{2} + c_{2}^{2} + \frac{5}{2} c_{3} .

g (α, ϵ^{'}) = 1 + \frac{18 α ^{2}}{( 5 + 2 ϵ ^{'} ) ^{2}} - 1.

Solving the quadratic gives the exact advection-dominated accretion flow coefficients

c_{1} = - \frac{5 + 2 ϵ ^{'}}{3 α} g,

c_{2} = [\frac{2 ϵ ^{'} ( 5 + 2 ϵ ^{'} )}{9 α ^{2}} g]^{1/2}, c_{3} = \frac{2 ( 5 + 2 ϵ ^{'} )}{9 α ^{2}} g .

For

α^{2} ≪ 1

,

g = \frac{9 α ^{2}}{( 5 + 2 ϵ ^{'} ) ^{2}} + O (α^{4}),

and therefore

c_{1} ≃ - \frac{3 α}{5 + 2 ϵ ^{'}}, c_{2}^{2} ≃ \frac{2 ϵ ^{'}}{5 + 2 ϵ ^{'}}, c_{3} ≃ \frac{2}{5 + 2 ϵ ^{'}} .

Efficient cooling means

f \to 0

and hence

ϵ^{'} = ϵ / f \to \infty

for fixed

γ < 5/3

. Then

u_{R} / v_{K} \to 0, Ω/ Ω_{K} \to 1, c_{s}^{2} / v_{K}^{2} \to 0, H / R \sim c_{s} / v_{K} \to 0.

The flow is therefore cold, nearly Keplerian, slowly accreting, and geometrically thin: the standard thin-disk limit.

For significant advection,

f = O (1)

and all three deviations are explicit:

u_{R} ≃ - \frac{3 α}{5 + 2 ϵ ^{'}} v_{K}, Ω ≃ \frac{2 ϵ ^{'}}{5 + 2 ϵ ^{'}} Ω_{K}, c_{s} ≃ \frac{2}{5 + 2 ϵ ^{'}} v_{K} .

The gas is hot and thick, pressure supplies part of the radial support, rotation is sub-Keplerian, and dissipated entropy is carried inward. As

γ \to 5/3

,

ϵ^{'} \to 0

, so

Ω \to 0

,

c_{s}^{2} \to (2/5) v_{K}^{2}

, and

u_{R} \to - (3 α /5) v_{K}

: the self-similar rotating solution approaches a hot Bondi-like inflow. Sagittarius A* is the standard supermassive example: its luminosity is tiny compared with its Eddington luminosity despite an available gas supply, and its hot optically thin spectrum and low radiative efficiency are described by an ADAF or the broader radiatively inefficient accretion-flow family.

Solved by gpt-5.6-sol high.

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