Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2018 / S26 / 2 / c

For fully ionized hydrogen with a small metal fraction, there is approximately one proton and one electron per hydrogen mass, so ${\mu }_H\simeq 1/2$. The Kramers coefficient is proportional to $Z(1+X)\simeq 2Z$. Denote this star's luminosity by $L_H$; the supplied proportionality law alone does not fix its absolute normalization.

Relative to the helium-dominated comparison star, the calculation in part ii givesIf one uses a roughly solar luminosity as an illustrative calibration for this roughly solar-mass, solar-central-temperature reference, the helium star would be of order a hundred solar luminosities. That calibration is only approximate because a mostly hydrogen composition with $\mu \simeq 1/2$ is not the actual solar composition.

Fully ionized helium supplies three particles per four hydrogen masses, so ${\mu }_{\mathrm{He}}\simeq 4/3$. The same small $Z$, but negligible hydrogen fraction, gives ${\kappa }_{0,\mathrm{He}}\propto Z$, half the hydrogen-star coefficient. The relative luminosity isConsequentlyThe very strong molecular-weight dependence outweighs the reduction in mass. Treating the two opacities as identical would miss a factor two, and using neutral-gas molecular weights would be inconsistent with the stated hot ionized stars.