A newly characterised planet orbiting a Sun-like star has measured mass and measured…

PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Astrophysics & Cosmology · Temperature, heat and the macroscopic behaviour of matter

Stars from the Inside, Ch Planetary systems: formation estimates, transits and the atmospheres of other worlds

Problem

A newly characterised planet orbiting a Sun-like star has measured mass

M_p = 5.0\,M_\oplus

and measured radius

R_p = 2.0\,R_\oplus

, where

M_\oplus = 5.972\times10^{24}\,\mathrm{kg}

and

R_\oplus = 6.371\times10^{6}\,\mathrm{m}

. This radius sits just above the radius valley separating rocky planets from those with substantial volatile envelopes. The mass-radius diagram's composition curves cross a radius of

2.0\,R_\oplus

at masses that imply the following bulk densities for planets of pure composition at that radius: pure iron:

9.0\times10^{3}\,\mathrm{kg\,m^{-3}}

pure silicate rock:

4.6\times10^{3}\,\mathrm{kg\,m^{-3}}

pure water/ice:

1.9\times10^{3}\,\mathrm{kg\,m^{-3}}

a rocky core with a hydrogen-helium envelope making up only

2

per cent of the planet's mass:

1.2\times10^{3}\,\mathrm{kg\,m^{-3}}

. ((a)) Compute the planet's measured bulk density. ((b)) State which of the four compositions above are consistent with the measured density and which are excluded. ((c)) Model the interior as a mixture of rock and water/ice whose volumes add, so that

\dfrac{1}{\rho_p} = \dfrac{f}{\rho_{\rm rock}} + \dfrac{1-f}{\rho_{\rm water}}

, where

f

is the mass fraction in rock and

\rho_{\rm rock}

\rho_{\rm water}

are the pure-composition densities above. Find

f

, and explain why this value is not the only interior structure consistent with the measured mass and radius.

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