Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2016 / S26 / 1

Dust condenses in cool stellar outflows and ejecta and is processed and grown in interstellar clouds. It absorbs and scatters starlight, usually more efficiently at blue wavelengths, making a background star fainter and redder. RR Lyrae variables are old horizontal-branch pulsators with approximately calibrated absolute luminosities, so their recognizable light curves and standard-candle relation provide distances after metallicity and extinction corrections. They are not exactly identical-luminosity objects in every population.

The color excess is $E(B-V)=0.3-0.2=0.10$ mag, hence $A_V=0.31$ mag. The observed modulus $V-M_V$ is 15.4, while the extinction-corrected modulus is $\boxed{{\mu }_0=15.09}$. Thus $\boxed{d=10^{({\mu }_0+5)/5}\simeq 1.04\times 10^4\mathrm{pc}}$. Distinguishing the apparent and true moduli avoids treating extinction as extra distance.

Hot bright stars lie toward the upper left: smaller $B-V$ means bluer/hotter, and smaller numerical $V$ means brighter. The vertical axis is inverted in the original diagram. The plot shows apparent magnitudes, but cluster members at a common distance permit relative luminosity comparisons.

Cool faint stars lie toward the lower right. Redder color and larger magnitude identify that corner, subject to common extinction and the atmospheric color-temperature relation. A color-magnitude diagram is not a geometrical map of where the stars are located within the cluster.

An isochrone is the locus of stars of a common age and composition over a range of initial masses. A real cluster diagram approximates it, with scatter from binaries, measurement errors, composition spread and differential reddening. Region A is the hydrogen-core-burning main sequence. Region B contains evolved subgiant/red-giant stars, typically with hydrogen-shell burning around an increasingly helium-rich core. Region C is the horizontal branch, occupied by core-helium-burning stars; RR Lyrae lie in its instability strip.

Point D is the main-sequence turnoff: core hydrogen exhaustion drives structural readjustment toward shell burning and expansion. Point E is the tip of the red-giant branch, where a low-mass star ignites helium in its degenerate core, the helium flash, and subsequently moves toward horizontal-branch conditions. The two labeled events are different evolutionary transitions, not merely changes in apparent color from dust.

At D, $M_V=20.3-15.09=5.21$. Ignoring differential bolometric corrections, $L/L_{\odot }\simeq 10^{-0.4(5.21-4.74)}\simeq 0.65$, and $L\propto M^4$ gives $\boxed{M_D\simeq 0.90M_{\odot }}$. If accessible fuel fraction is similar, lifetime scales as $M/L\propto M^{-3}$. Normalizing to a solar main-sequence lifetime about 10 Gyr gives an age near $\boxed{14\mathrm{Gyr}}$.

This is much older than the Sun's current approximately 4.6 Gyr and close to the cosmological age scale. The approximate estimate is not evidence for a cluster older than the Universe: bolometric corrections, abundance, accessible hydrogen fraction and the crude mass-luminosity law limit precision. The stated 26.7 MeV conversion and an assumed core fuel fraction can supply the order-of-magnitude solar lifetime; they do not by themselves specify the burned fraction or composition precisely.