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 / S2 / 1

The distance modulus at $100\mathrm{pc}$ is five magnitudes. Thus the two approximate main-sequence endpoints read from the original figure, $m_V\simeq 0,20$, correspond to $M_V\simeq -5,15$. With the given solar reference,Use the figure's approximate endpoint effective temperatures $25,000$ K and $2500$ K. If $V$-band luminosity is used as a rough bolometric proxy, the Stefan–Boltzmann law givesTaking $T_{\odot }\simeq 5800$ K givesThese are schematic estimates, not precision radii. The diagram supplies $V$ magnitudes rather than bolometric luminosities, and hot and cool endpoints need different bolometric corrections. The exact range cannot be recovered without those corrections; the quoted values state the intended approximation explicitly.

Integrating $dP/dr=-G{M}_r\rho /r^2$ in order of magnitude, with $\rho \sim M_{\ast }/R_{\ast }^3$, givesThe ideal gas law independently gives $P_c\sim M_{\ast }{k}_B{T}_c/(\mu m_H{R}_{\ast }^3)$. Equating the two estimates yieldsFor similar composition and an approximately fixed hydrogen-burning central temperature, $R_{\ast }\propto M_{\ast }$. Calibrating this rough relation on the Sun and using part a gives a nominal mass range of about $0.05$ to $5M_{\odot }$.

The endpoint luminosity ratio is $10^8$, whereas the inferred mass ratio is about $10^2$, giving $L\propto M^4$ across that range. The exponent is therefore consistent at this schematic level. The solar normalization and individual endpoint masses are much less reliable: using the luminosity relation itself gives approximately $0.1$ to $10M_{\odot }$. Central temperatures are not truly constant, low-mass stellar structure differs, and the uncorrected $V$ luminosities in the diagram are not bolometric. These limitations prevent treating the rough radius-based masses as a contradictory precision measurement.

If radiation dominates the central pressure, equate $P_c\sim G{M}_{\ast }^2/R_{\ast }^4$ to $a{T}_c^4/3$. This givesAt roughly fixed burning temperature, the corresponding mass-radius scaling is $R_{\ast }\propto M_{\ast }^{1/2}$, rather than the gas-pressure estimate $R_{\ast }\propto M_{\ast }$. Radiation pressure alone does not determine a unique radius at a specified mass: a thermal or nuclear-burning condition is still needed to specify $T_c$. The relation is a homology estimate, not a universal exact law for all massive stars.

The numbered features in the original colour-magnitude diagram have the following interpretation.

Continuous star formation explains the coexistence of newly formed massive main-sequence stars with evolved low-mass remnants; interpreting a single sharp turnoff as the age of a coeval cluster would be inappropriate here.