Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 55 / 3 / iv / Solution

For polytropic index three, the Lane-Emden mass formula cancels the central mass density:Substituting the preceding constant $K$ givesFor fixed composition this is $M\propto \sqrt{1-\beta }/{\beta }^2$, as required. Squaring and rearranging yields the Eddington quartic relation, $(1-\beta )/{\beta }^4\propto {\mu }^4{M}^2$. The larger masses in this model have smaller gas fractions and relatively more radiation support.