Past exam of the mathematics course of the University of Cambridge 2015 iii Paper 59 4 h Solution Created 2026-10-03 Updated 2026-10-06
Plate tectonics is the movement and recycling of a planet's lithosphere as discrete plates over a deformable mantle. Mantle convection, the negative buoyancy of cool subducting slabs (slab pull), and gravitational sliding from elevated spreading ridges (ridge push) supply driving stresses; deformation and friction resist motion. The mantle mostly deforms by slow solid-state creep, rather than being a global liquid layer.
Its effects include subduction and recycling of crust, creation of new crust, mountain building, earthquakes and volcanism, transport of internal heat, and recycling of water and carbon. The carbonate-silicate cycle can couple volcanic CO2 supply to weathering and long-term climate.
- Mass and pressure-dependent material behavior: gravity and mantle depth affect buoyancy and convective stresses, but high-pressure changes in viscosity and density also affect whether slabs sink.
- Thermal state and heat budget: age, radiogenic heating, residual formation heat and surface temperature set convective vigor, plate thickness and the strength of the lid.
- Water and lithospheric weakening: hydration, rock rheology, fault damage and yield strength determine whether driving stresses can break and recycle the lithosphere rather than leave a stagnant lid.
Past exam of the mathematics course of the University of Cambridge 2016 iii Paper 315 4 g Solution Created 2026-10-03 Updated 2026-10-06
A super-Earth is a planet more massive than Earth but below the usual ice-giant scale, conventionally about --. The term does not guarantee an Earth-like or rocky composition; water-rich planets and objects with small H/He envelopes can lie in this range.
The most direct constraints are transit radius and a mass from radial velocities or transit-timing variations, giving . Comparing these with interior models constrains the possible iron core, silicate mantle, water layer and gas envelope, but many mixtures share one mass and radius.
Additional observables include atmospheric spectra/mean molecular weight and escape signatures, which test whether a low-density envelope is H/He or heavier volatiles; stellar Fe/Mg/Si abundances as priors on refractory composition; and age, irradiation and orbital history, which control thermal expansion and envelope survival. Where measurable, a Love number or related apsidal response constrains central concentration. Mass and radius constrain families of interiors, not a unique composition; atmospheric and dynamical information can reduce this exoplanet interior-composition degeneracy.
Past exam of the mathematics course of the University of Cambridge 2017 iii Paper 315 4 i Solution Created 2026-10-03 Updated 2026-10-06
- 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.
Terrestrial planet 2026-10-06