Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2012 / S2 / 2

Build the star by adding spherical shells. With $m(r)$ the enclosed mass, its gravitational binding energy isFor a nondegenerate extended star, $r(m)<R_{\odot }$ on the interior, soThis is a gravitational binding-energy lower bound for a spherical star. The Kelvin-Helmholtz cooling time is of orderThe displayed energy bound gives $|\Omega |/L_{\odot }>1.6\times 10^7$ years; it supplies a lower bound on that reservoir-to-luminosity ratio. A stellar structure model sets its coefficient. For an ideal monatomic, gas-supported star the virial theorem puts about half the released gravitational energy into heat, leaving half available for radiation; a uniform-density model would give about $9\times 10^6$ years. These contraction estimates are all of order $10^7$ years, much shorter than the nuclear lifetime.

Use the supplied atomic masses consistently. Their mass defect per four hydrogen atoms is $4(1.0078)-4.0026=0.0286$ atomic mass units. The conversion efficiency isThe atomic-mass bookkeeping includes the associated positron-annihilation energy; an exact photon-heating budget also subtracts energy carried away by neutrinos. With the stated ten-percent fuel fraction and constant present solar luminosity, the main sequence lifetime estimate isThe much longer timescale explains why stellar nuclear fusion can power the Sun throughout its long main-sequence history.

A Hertzsprung-Russell diagram plots stellar luminosity upward and effective temperature increasing toward the left. The main sequence runs from hot luminous stars at upper left to cool faint stars at lower right, with the Sun near $T_{\mathrm{eff}}\simeq 5800\mathrm{K}$, $L=L_{\odot }$.

In the evolution of a solar-mass star, core hydrogen fusion initially supplies the luminosity. As core hydrogen is exhausted, hydrogen-shell fusion around a contracting helium core carries the star through the subgiant and red giant phases toward higher luminosity and lower surface temperature. Helium ignition is followed by core helium fusion with a hydrogen-burning shell, placing a solar-mass star in the red clump. The later Asymptotic giant branch has hydrogen- and helium-burning shells around a carbon-oxygen core. Envelope ejection exposes a hot central star of a planetary nebula; residual shell burning ends and the resulting white dwarf radiates stored thermal energy as it cools.