 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2017/iii/paper-315/4/i/solution

 Top articles  Latest articles New article in topic

Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 315 / 4 / i / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-06

At fixed mass, three broad controls of a super-Earth-sized body's observed radius are:

Bulk composition and material compressibility. A larger iron-core fraction usually makes an exoplanet interior denser and smaller, whereas silicate or water-rich interiors are larger. The planetary mass-radius relation therefore differs among compositions; mass and radius alone retain an exoplanet interior-composition degeneracy.
A gaseous envelope and its retention. Even a modest hydrogen-helium mass fraction can substantially increase radius through the transit-radius contribution of a gaseous envelope. Its mean molecular weight, total atmospheric mass and opacity determine how far the slant-optical-depth surface lies above the condensed interior. Atmospheric escape can strip that envelope, producing a much smaller object without a comparable loss of core mass.
Thermal and irradiation history. Higher interior entropy, youth and stellar heating tend to expand an envelope; Kelvin-Helmholtz contraction reduces its size over time. Stellar high-energy exposure affects the envelope through atmospheric escape. Exoplanet clouds and wavelength-dependent opacity shift the measured transit radius even when the deep interior is unchanged.

Bulk composition, envelope fraction/composition, and thermal/irradiation history are three independent controls. If mass is not fixed, the mass itself is an additional major variable. The label super-Earth does not ensure a rocky composition or an Earth-like atmosphere, and a radius alone does not determine which of these effects dominates.

 Read the full article

 New to topics? Read the docs here!