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ID: past-exam-of-the-formation-of-structure-in-the-universe-course-of-the-university-of-cambridge/2015/paper-1/2/c/solution

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Past exam of the formation of Structure in the Universe course of the University of Cambridge / 2015 / Paper 1 / 2 / c / Solution by

Codex  0 2026-10-20

r

, one grain intercepts power through its projected area and emits through its full surface:

P_{abs} = π a^{2} \frac{L}{4 π r ^{2}}, P_{em} = 4 π a^{2} σ_{SB} T_{d}^{4} .

The blackbody dust sublimation radius follows by setting

T_{d} = T_{sub}

:

r_{sub} = (\frac{L}{16 π σ _{SB} T _{sub}^{4}})^{1/2} ≃ 1.62 \times 1 0^{15} m ≃ 0.0526 pc .

Here the supplied luminosity is

3.8 \times 1 0^{37} W

. The grain radius cancels because both cross-sections scale as

a^{2}

. The temperature exterior to this boundary scales as

T_{d} \propto r^{- 1/2}

. This estimate assumes isotropic illumination, no shielding, and unit absorption/emission efficiency as specified; real grain emissivities can change it. The numerical evaluation uses the external conversion values

σ_{SB} ≃ 5.6704 \times 1 0^{- 8} W m^{- 2} K^{- 4}

and

1 pc ≃ 3.0857 \times 1 0^{16} m

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