A red dwarf star has surface temperature and radius

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Condensed Matter, Materials & Soft Matter · Origins of quantum theory

Thermal Physics, Ch Photons and phonons: blackbody radiation and the heat capacity of solids

Problem

A red dwarf star has surface temperature

3200\,\mathrm{K}

and radius

0.35\,R_\odot

, with

R_\odot = 6.96\times10^{8}\,\mathrm{m}

. A rocky planet orbits it at

0.12\,\mathrm{AU}

, where

1\,\mathrm{AU} = 1.496\times10^{11}\,\mathrm{m}

, and has albedo

A = 0.15

. Treat the star as a black radiator, the planet as a bare rotating ball of uniform temperature with no atmosphere, absorbing the fraction

1-A

of the starlight and radiating like a black surface, and ignore any internal heat source. You may use

\int_0^\infty x^2/(e^x-1)\,\mathrm{d}x = 2.404

. ((a)) Find the radiation flux

S

at the planet's orbit. ((b)) Find the planet's surface temperature. ((c)) A sealed lava tube of volume

1.0\,\mathrm{m^3}

sits just below the surface at the planet's temperature. Find the number of photons it holds, and, using the effusion result of Chapter 2 with every photon moving at

c

, the rate at which photons would stream out through a

4.0\,\mathrm{cm^2}

opening. ((d)) A lump of iron (

\theta_D = 470\,\mathrm{K}

) from the planet's surface is at

50\,\mathrm{K}

on the night side. Using the low-temperature

T^3

law, estimate its lattice heat capacity per mole, ignoring the electronic contribution.

Related theory

Quantum Mechanics

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