Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 347 / 3 / b / Solution

For a dusty shell, direct ultraviolet absorption, infrared trapping, and incomplete ultraviolet absorption give the radiation force

F_{rad} = \frac{L}{c} (1 + τ_{IR} - e^{- τ_{UV}}) .

Balancing it against gravity gives the general critical luminosity of a dusty shell

L_{crit} = \frac{G c M ( < r ) M _{s} ( r )}{r ^{2} [ 1 + τ _{IR} - e ^{- τ_{UV}} ]} .

For a singular isothermal sphere,

M (< r) = \frac{2 σ ^{2} r}{G}, M_{s} = f_{gas} M (< r) = \frac{2 f _{gas} σ ^{2} r}{G},

so

F_{grav} = \frac{GM M _{s}}{r ^{2}} = \frac{4 f _{gas} σ ^{4}}{G}

and

τ_{i} = \frac{κ _{i} M _{s}}{4 π r ^{2}} = \frac{κ _{i} f _{gas} σ ^{2}}{2 π G r} .

In the infrared-thick limit

F_{rad} ≃ τ_{IR} L / c

; in the single-scattering limit it is

L / c

; and in the ultraviolet-thin limit it is

τ_{UV} L / c

. Therefore

L_{crit, IR} = \frac{8 π c σ ^{2} r}{κ _{IR}}, L_{crit, s.s.} = \frac{4 c f _{gas} σ ^{4}}{G}, L_{crit, UV} = \frac{8 π c σ ^{2} r}{κ _{UV}} .

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