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

Radiative transport supplies the temperature gradient used below. A purely radiative equilibrium model also needs stellar convective stability; the transport assumption alone is not a general proof of stability.

The total pressure is the sum of ideal gas and radiation pressure contributions. The gas fraction and its complement will be obtained from the structure equations, rather than prescribed independently at each radius.

Chemical homogeneity makes the mean molecular weight spatially constant. This is what allows the pressure-density coefficient derived below to be a single polytropic constant.

The chosen opacity law makes $\kappa L_r/m_r=\alpha L/M$ spatially constant. This closes the Eddington standard model relation between its radiation-pressure and total-pressure gradients. The four roman-numbered proofs use the whole model, not only this opacity assumption.

Divide the radiation-pressure gradient by the total-pressure gradient. Radiative diffusion in a star and hydrostatic equilibrium giveThis last quantity is independent of radius under the specified opacity assumption. Integration gives $P_r=fP+C$. With the standard idealized zero-pressure outer boundary, where both components vanish, $C=0$. The stellar gas-pressure fraction is thereforeconstant throughout the model. This is the Eddington standard model idealization. A finite photospheric pressure offset would need its boundary treatment; the differential relation alone does not set the integration constant to zero.

Both gas and radiation pressures are nonnegative, so $0\le \beta \le 1$. Rearranging the constant-fraction result givesThis is the Eddington luminosity for the effective opacity constant of this model. Positive gas support gives a strict inequality, with equality only in the radiation-only limiting case. Exceeding the limit would require a negative gas-pressure fraction, which is incompatible with the assumed equation of state.

Put $\mathcal{R}=k_B/m_u$, with $m_u$ the atomic mass constant and $a_r$ the radiation energy-density constant. The ideal gas and radiation pressure contributions obey $P_g=\rho \mathcal{R}T/\mu$ and $P_r=a_r{T}^4/3$. With the constant stellar gas-pressure fraction, eliminate $T=\mu \beta P/(\mathcal{R}\rho )$ to getHenceSince $\mu$ and $\beta$ are spatially constant, so is $K$. Comparing with $P=K{\rho }^{1+1/n}$ gives polytropic index $n=3$. This structural stellar polytrope relation does not assert that every perturbed fluid parcel has stellar adiabatic exponent $4/3$.

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.