Past exam of the mathematics course of the University of Cambridge 2015 iii Paper 59 4 g Solution Created 2026-10-03 Updated 2026-10-06
For typical well-mixed structures, the dominant processes are:
- Gas giant interiors: efficient convection through most of the deep fluid envelope; radiative transfer releases the heat near the photosphere.
- Rocky interiors: slow solid-state mantle convection over geological time, with heat conduction dominant across the rigid lithosphere. A liquid core can also convect; being solid does not prevent creep-driven heat transport in the mantle.
- Weakly irradiated giant atmospheres at –: usually convection in the planetary troposphere, becoming radiative near and above the tropopause. The transition pressure and cloud or compositional effects vary between planets.
- Strongly irradiated hot Jupiter atmospheres at –: usually a stable radiative region; the deep radiative-convective boundary can lie at substantially larger pressure. Atmospheric winds also redistribute energy horizontally.
Representative temperatures must specify the level: Earth has about at the surface (about effective emission temperature); Jupiter has about near one bar (about effective temperature); hot Jupiters commonly have photospheric temperatures of order –; and the Sun's photosphere is about . Upper layers, nightsides, deep interiors and the solar corona have different temperatures. These are characteristic values, not constant temperatures throughout each atmosphere.
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 2026 iii Paper 332 1 d Solution Created 2026-09-24 Updated 2026-09-25
Treat the warm ice shell as a very viscous layer and estimate its Rayleigh numberThe temperature-dependent ice viscosity should be evaluated carefully, because deformation is concentrated near the warm base; a useful first estimate uses a representative basal or depth-averaged viscosity. The onset time is obtained by inserting and into this expression and finding when first exceeds the critical value for the shell's mechanical boundary conditions, usually of order .
Above onset, mantle convection within the ice transports heat more efficiently than thermal conduction, increases the basal heat loss for a given thickness, and generally limits further thickening. For otherwise similar bodies, a larger planetary radius normally gives larger gravitational acceleration and hence a larger Rayleigh number. Convection is therefore more likely and begins in a thinner or younger shell, although changes in pressure-dependent melting temperature, shell thickness, and viscosity can alter that comparison.
Planetary ice shell Created 2026-09-24 Updated 2026-09-24
A planetary ice shell is a solid water-rich layer overlying an ocean or deeper planetary interior. Its thickness can be controlled by thermal conduction, basal heat flux, surface radiation, and solid-state mantle convection.