Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 63 / 1 / c

Keep the fixed stellar homology profile, so that ${\rho }_c=C_{\rho }M/R^3$ and $P_c=C_{P}G{M}^2/R^4$. Solving the partially degenerate equation of state for the central temperature giveswhere $A=C_P/C_{\rho }$ and $B=C_{\rho }^{2/3}$ are positive constants. The ion mean molecular weight is distinct from the total-particle mean molecular weight used in part (a).

Set $y=1/R$. The expression in parentheses is a concave quadratic in $y$. Its maximum occurs atSubstitution gives the contraction-limited central temperatureAt this point electron degeneracy pressure supplies half the required central pressure. Further contraction increasingly goes into the density-dependent degeneracy term rather than heating the ions. The formula applies along the positive-temperature part of the idealized homologous sequence; a negative value beyond it is not a physical temperature.

For sufficiently small mass, this maximum never reaches the temperature needed for sustained hydrogen burning. Such an object stays below the hydrogen-burning minimum mass and becomes a cooling brown dwarf, rather than an ordinary main sequence star. Some such objects burn their initial deuterium, but that finite reservoir does not provide lasting hydrogen burning.