Past exam of the mathematics course of the University of Cambridge 2015 iii Paper 59 4 e Solution Created 2026-10-03 Updated 2026-10-06
The circumstellar habitable zone is the range of orbital distances where a terrestrial planet with a specified atmospheric inventory can retain liquid water at its surface. It is a conditional climate criterion, not a guarantee of life or a requirement for every possible subsurface habitat.
Four influential factors are:
- The incident stellar energy and spectrum, including orbital distance and long-term stellar evolution.
- Atmospheric pressure and composition, greenhouse effect, Bond albedo and clouds, which set the relation between absorbed light and surface temperature.
- The water and volatile inventory, together with planetary mass and the ability to retain or replenish an atmosphere.
- Internal and surface evolution, including plate tectonics, outgassing and the carbonate-silicate cycle, which can regulate climate over geological time.
For a Sun-like present-day star and an Earth-like planet, a useful conservative range is approximately –. The inner limit depends on the adopted moist or runaway greenhouse condition, and the outer limit on the maximum greenhouse outer habitable-zone limit. More restrictive water-loss choices put the inner edge near . Empirical optimistic limits based on past Venus and Mars are about –. These are model conventions rather than exact universal boundaries.
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.
Plate tectonics 2026-10-06
A mobile-lid mode of planetary evolution in which coherent pieces of the lithosphere move, form at spreading regions and can recycle through subduction. Slab pull, ridge push and mantle flow compete with deformation resistance. It affects heat transport, volcanism, earthquakes and the carbonate-silicate cycle. It is not guaranteed solely by an exoplanet's mass.