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Past exam of the mathematics course of the University of Cambridge 2015 iii Paper 59 2 b Solution Created 2026-10-03 Updated 2026-10-06
Use a dry ideal gas of fixed composition with specific gas constant and constant specific heat capacity at constant pressure . For a fixed-mass parcel, constant and implyAlong a hydrostatic adiabat, ; hence the dry adiabatic lapse rate isFor an actual pressure-balanced parcel rising in an ambient atmosphere, instead gives . The usual lapse-rate expression is exact for a hydrostatic adiabatic column and is the local first-order result at the launch point where , as needed in a linear stability test. Treating an already much hotter parcel as an exact copy of the ambient hydrostatic column would be an extra approximation.
After a small upward displacement from temperature equilibrium, its temperature excess isAt equal pressure, warmer gas is less dense and continues to rise. Thus the Schwarzschild criterion in altitude form isThe supplied non-strict inequality includes the marginal case; strict growth requires the strict inequality. A downward displacement gives the same stability conclusion. Efficient convection normally adjusts an initially superadiabatic gradient to a nearly adiabatic one.
Deep envelopes of gas giants and ice giants commonly transport intrinsic heat by convection, as do the planetary tropospheres of many weakly irradiated atmospheres. Earth's dry troposphere provides another approximate example, with moisture changing the lapse rate. Strongly irradiated hot Jupiters can still have deep convective interiors, while their upper radiative regions need not be convective. Composition gradients can modify the homogeneous-gas criterion and inhibit overturning even in an interior.
Past exam of the mathematics course of the University of Cambridge 2015 iii Paper 59 3 b Solution Created 2026-10-03 Updated 2026-10-06
The mass-radius curve of solar-composition substellar objects reflects the transition from weak compression to pressure ionization and electron degeneracy pressure, followed by sustained hydrogen burning. A schematic joining representative object classes is:
Schematic mass-radius sequence from ice giants through gas giants and brown dwarfs to low-mass stars
. The illustration is not an age-specific numerical evolutionary model. In particular, ice giants contain much more heavy material than a solar-composition giant, so a single uniform-composition equation of state does not describe the entire joined curve.
- At the low-mass, weak-compression end, a fixed-density or fixed-composition approximation gives . Ice giants such as Neptune and Uranus have substantial water/rock-rich interiors and modest H/He envelopes; changing envelope fraction changes the radius markedly. Their heat comes from retained formation energy, contraction and radiogenic heating of heavy material. Fluid interiors generally convect, while composition stratification can impede mixing; outer radiative layers release the heat.
- Ordinary gas giants reach radii of order over a broad range around Jovian masses: an effective polytrope explains the approximate segment. Increased mass compresses material enough to offset the added volume. Cooling and Kelvin-Helmholtz contraction, with additional differentiation energy such as helium settling in Saturn, supply the intrinsic luminosity. Their deep envelopes are usually convective, with radiative photospheres.
- More massive brown dwarfs become increasingly supported by electron degeneracy pressure. The cold nonrelativistic limit gives , but finite entropy and Coulomb effects flatten actual giant/brown-dwarf curves and their radii depend on age. They cool and contract; temporary deuterium fusion occurs above a composition-dependent deuterium-burning mass near . This threshold does not cause a sharp structural kink or permanent stellar luminosity. Interiors are largely convective and surface emission is radiative.
- Near the hydrogen-burning minimum mass, roughly – or – for near-solar composition, sustained fusion prevents indefinite cooling into a degenerate object. The low-mass main-sequence star branch turns upward, with approximately over the illustrative interval. Its entropy is not constant across masses, so this branch is compatible with an approximately internal profile. Hydrogen fusion through the proton–proton chain provides energy; the lowest-mass main-sequence stars are fully convective, capped by radiative atmospheres.
Planet/brown-dwarf naming conventions and deuterium burning do not define a universal discontinuity in the mass-radius relation. Composition, age and irradiation move the curves; the hydrogen-burning transition changes the long-term energy source more fundamentally.
Uranus 2026-10-06
An ice giant with substantial heavy material beneath its H/He envelope. Its low intrinsic luminosity makes efficient, fully mixed cooling an imperfect universal assumption.