Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2012 / S26 (Stars and galaxies)

For a spherical star in hydrostatic equilibrium, $dP/dr=-Gm(r)\rho /r^2$. Multiply by $4\pi r^{3}dr$ and integrate to its radius $R$:Integration by parts gives $4\pi R^3{P}_s-3\int PdV=\Omega$. Neglecting the external surface pressure $P_s$, an ideal gas with constant heat capacity ratio $\gamma$ has thermal energy density $u=P/(\gamma -1)$. ThereforeThis is the hydrostatic virial theorem. A nonzero confining pressure retains the right-hand side $4\pi R^3{P}_s$; varying composition, radiation contributions or a spatially varying effective exponent require the integral relation, rather than pulling one $\gamma$ outside it.

For a fully ionized, nonrelativistic monatomic ideal gas, $\gamma =5/3$ and the virial theorem becomesSince $U$ increases with its particle-weighted mean temperature, the equilibrated self-gravitating object has negative heat capacity: adding total energy allows expansion, makes $\Omega$ less negative, and reduces $U$. Conversely, radiating energy permits contraction and raises the internal temperature. This refers to a sequence of gravitational equilibria, not to the positive local heat capacity of a parcel at fixed volume.

In an ordinary nondegenerate nuclear-burning star, a rise in nuclear heating tends to expand and cool the burning region, lowering its temperature-sensitive reaction rate. Reduced heating leads to contraction and reheating. This expansion feedback is the nuclear thermostat; it can fail in degenerate matter where pressure responds weakly to temperature. During contraction of a pre-main-sequence star, gravitational binding supplies the radiated luminosity and heats the interior. For a fixed-structure scaling, half the released gravitational energy increases $U$ and half escapes as radiation, until hydrogen burning can support a long-lived main sequence equilibrium.

Write $\Omega =-aG{M}_{\odot }^2/R_{\odot }$ with a structure factor $a$ of order unity. Then the thermal radiation timescale isFor a uniform sphere $a=3/5$, this is about $9.4\times 10^6$ years. The conventional Kelvin-Helmholtz cooling time $G{M}_{\odot }^2/(R_{\odot }{L}_{\odot })$ is about $3.1\times 10^7$ years; a centrally concentrated solar model changes the coefficient. Both estimates are far shorter than the nuclear lifetime.

Ionization and dissociation can store compression energy without the full temperature rise of a monatomic gas, lowering the effective adiabatic response in partially ionized outer layers. Radiation pressure tends towards an adiabatic exponent $4/3$. These effects, changing composition and degeneracy make the effective exponent position dependent. The local dynamical exponent is ${\Gamma }_1=(\partial \log P/\partial \log \rho {)}_s$; it equals $C_P/C_V$ in the simple fixed-composition ideal-gas model, but need not do so for an ionizing mixture.

If an envelope has an effective exponent at or below $4/3$, its pressure response is soft and sufficiently extensive such regions can promote radial pulsation, envelope instability or mass loss. In the ideal homologous global model, $\gamma =4/3$ is marginal and $\gamma <4/3$ is unstable to contraction or expansion. A low value confined to a thin outer layer does not by itself prove collapse of the entire star: global stability depends on the pressure-weighted structure. Its location and mass extent matter.

Displace a small parcel upward, with negligible heat exchange, rapid pressure balance, and the same composition as its surroundings. Define the environmental gradient $\nabla =d\log T/d\log P$. An adiabatic ideal gas parcel obeys $P{\rho }^{-\gamma }=\mathrm{constant}$ and hence $T\propto P^{(\gamma -1)/\gamma }$, giving ${\nabla }_{\mathrm{ad}}=(\gamma -1)/\gamma$.

The upward displacement has $\delta \log P<0$. If $\nabla <{\nabla }_{\mathrm{ad}}$, the parcel's temperature changes more negatively than the environment's. At their common pressure it is cooler, denser and driven back down. Therefore the Schwarzschild criterion for stability isThe reversed inequality produces buoyant growth, and equality is neutrally stable in this approximation. A composition gradient requires the Ledoux criterion; pressure equilibrium alone does not ensure equal mean molecular weight.

In the Sun's cool outer layers, partial ionization lowers the adiabatic temperature gradient, while large opacity makes radiative transport inefficient. The required radiative gradientcan exceed ${\nabla }_{\mathrm{ad}}$, so convection carries part of the outward luminosity. The Sun therefore has an outer convective envelope, above its primarily radiative interior. Here $a$ is the radiation constant, $\kappa$ opacity per unit mass and $m_r,L_r$ the enclosed mass and luminosity. Near the visible surface, radiative losses are again important and the flow need not be perfectly adiabatic.

Near a red giant's compact core, intense nuclear burning in a small region can produce a large $L_r/m_r$ and a steep required radiative gradient. Where this exceeds the adiabatic gradient, convection transports the released energy and mixes fuel. Core helium burning is a clear example: the strongly temperature-sensitive Triple-alpha process commonly produces a convective burning core; shell flashes can also make convective regions.^[ref]

The qualification matters: an inert degenerate helium core and a steady hydrogen-burning shell on the first-ascent red giant branch are not universally convective; conduction or radiation may dominate there. Red giants also commonly have deep convective envelopes. The physical condition is an excessive required radiative gradient, not proximity to the centre alone.

An absorption line depends on the population of its lower atomic level, ionization state, transition strength, pressure and broadening, not just elemental abundance. Boltzmann distribution changes excitation populations with temperature, while the Saha equation changes ionization with temperature and electron pressure. Thus similar overall composition can produce very different spectra. Molecules survive preferentially in cool atmospheres.

The Harvard spectral classification orders the main stellar classes by decreasing effective temperature: O, B, A, F, G, K, M, from blue to red. Numerical subclasses run approximately from 0, the hotter end of a class, to 9, its cooler end. An A0 star, near $10^4\mathrm{K}$, has very strong Balmer series lines; hotter stars ionize much of the hydrogen and cooler stars underpopulate its $n=2$ level. A G2 star, near $5800\mathrm{K}$, has weaker Balmer absorption, many neutral and ionized metal lines, prominent Ca II H and K lines and the CH G band. An M5 star, near $3000\mathrm{K}$, has strong TiO molecular bands, neutral-metal and alkali features, and weak photospheric Balmer absorption.^[ref]

For giant and dwarf stars of the same colour, compare gravity-sensitive line widths and ratios. The giant's larger radius gives lower surface gravity and atmospheric pressure, typically narrower pressure-broadened lines; high-pressure dwarf atmospheres have stronger broad alkali wings and different molecular and ionization ratios. These diagnostics assign a luminosity class, such as III for a giant and V for a dwarf. A line ratio or calibrated gravity indicator is stronger evidence than colour alone.

A $1M_{\odot }$ star spends roughly $10^{10}$ years on the main sequence, deriving most of its luminosity from the proton–proton chain. After central hydrogen depletion, shell hydrogen burning around a contracting helium core expands it into a red giant. The low-mass helium core becomes electron-degenerate before the helium flash. Subsequent core helium burning by the Triple-alpha process, with alpha capture producing oxygen, lasts of order $10^8$ years.

After central helium exhaustion, hydrogen and helium shells burn around a carbon-oxygen core on the Asymptotic giant branch. Strong mass loss removes the envelope, potentially making a planetary nebula, and the remnant becomes a carbon-oxygen white dwarf. It cools using stored thermal energy and does not ignite sustained carbon burning. Detailed durations depend on composition and mass loss; the nuclear endpoint of the isolated solar-mass model is a white dwarf.

A $5M_{\odot }$ star has a much greater luminosity and exhausts central hydrogen in roughly $10^8$ to $2\times 10^8$ years, mainly through the CNO cycle. It crosses the Hertzsprung gap, develops shell hydrogen burning and ignites core helium without the strongly degenerate helium flash of the solar-mass case. Core helium burning lasts roughly $10^7$ years, with possible blue-loop evolution depending on the model.

It subsequently develops a carbon-oxygen core and double-shell burning on the Asymptotic giant branch, with thermal pulses, winds and envelope loss. For ordinary isolated evolution at this mass, the core does not reach sustained carbon ignition and the final remnant is a carbon-oxygen white dwarf. The broad boundary between white-dwarf and core-collapse outcomes depends on composition, overshooting, rotation and mass loss; the quoted mass is comfortably in the usual intermediate-mass regime.

A $15M_{\odot }$ star burns core hydrogen chiefly by the CNO cycle for about $10^7$ years. It commonly evolves towards a red supergiant, though winds and mixing alter its track. Core helium burning lasts of order $10^6$ years. Continued contraction then permits carbon, neon, oxygen and silicon burning, producing successively heavier nuclei up to an iron-group core. Neutrino cooling makes the advanced phases much shorter: carbon burning can last thousands of years, while the final burning stages fall to years or days, with substantial model dependence.

An iron-group core cannot gain energy by ordinary fusion into heavier nuclei. Electron captures, photodisintegration and loss of pressure support lead to stellar core collapse. A core-collapse supernova can eject the envelope and leave a neutron star; strong fallback or a failed explosion can instead produce a black hole. The massive-star endpoint involves core collapse, rather than the quiet carbon-oxygen white-dwarf cooling of the two lower-mass examples.

For a young open cluster, the Hertzsprung-Russell diagram contains a populated main sequence reaching hot, luminous, high-mass stars, with a high-mass main-sequence turnoff determined by its age. Cool low-mass members may still lie on tracks of pre-main-sequence stars above the lower main sequence. Depending on how young the cluster is, a few evolved blue or red supergiants may already be present; an extremely young cluster need not have any. It lacks the dominant old low-mass red-giant and horizontal-branch populations of an old globular cluster.

An old globular cluster has lost its original massive main-sequence members. Its main-sequence turnoff lies near a low stellar mass, typically around $0.8M_{\odot }$ for an age of order $10^{10}$ years. The diagram shows a lower main sequence, a subgiant branch, a prominent red giant branch, a horizontal branch of core-helium-burning stars, an Asymptotic giant branch and a faint white dwarf cooling sequence. The preceding schematic labels these features. A few blue stragglers may lie above the old turnoff because interaction or merger rejuvenated them; they do not imply a young age for the bulk population.

Assume homologous, gas-pressure-supported main sequence stars with nearly constant composition and mean molecular weight $\mu$, radiative energy transport and approximately constant opacity $\kappa$. Hydrostatic and mass scalings giveThe radiative diffusion equation gives the luminosity scalingTherefore, under these explicit assumptions,The radius cancels. Radiation-pressure dominance in the most massive stars changes the relation towards the Eddington luminosity scaling $L\propto M$, so the cubic law is an idealized gas-pressure homology result rather than an exact universal massive-star law.

The trigonometric parallax gives $d=1/0.02=50\mathrm{pc}$. Neglecting extinction, the distance modulus and given bolometric correction giveUsing the specified solar bolometric magnitude,The mass estimate uses the cubic scaling normalized to solar values. A lifetime estimate additionally needs the fraction of fuel accessible to core burning. Take a core-accessible fraction $f_c=0.1$ and initial hydrogen mass fraction $X=0.7$. The supplied fusion energy givesThe alternative common solar-lifetime normalization $t\simeq 10^{10}(M/M_{\odot })/(L/L_{\odot })$ years gives $1.69\times 10^9$ years. Thus this simplified turnoff argument gives an age of order $10^9$ years, with its fuel fraction and mass-luminosity assumptions explicitly stated. It does not use a modern fitted Hyades age or a full stellar isochrone. Unresolved binaries, extinction and deviations from the homology law affect the estimate.

Face-on, the Milky Way is a barred spiral galaxy: a central bulge and bar of scale a few kiloparsecs lie inside a disk of stellar radius roughly $15\mathrm{kpc}$, with spiral arms containing gas, dust, H II regions and young stars. Recent star formation is concentrated in molecular clouds and arm complexes. The Sun lies in the disk about $8\mathrm{kpc}$ from the centre in the source's convention. Old stars are also present throughout the disk, while the bulge, thick disk and extended stellar halo contain important old populations. Globular clusters occupy a roughly spheroidal halo extending well beyond the bright disk.

Edge-on, the bright thin galactic disk forms a narrow band, with a central bulge protruding above and below it and a thinner gas-and-dust lane along the midplane. Young stars and H II regions occupy a vertical scale of a few tenths of a kiloparsec or less; a broader old thick disk has scale of order a kiloparsec. These are approximate scale heights, not sharp boundaries. The old stellar halo and globular clusters form a much larger, low-surface-brightness spheroidal component, extending to many tens of kiloparsecs or more. The dark matter halo is inferred dynamically rather than directly visible in such a sketch. The preceding figure is a structural diagram, not an external photograph or a reconstruction of a particular survey.

Massive young stars emit ionizing ultraviolet radiation that creates H II regions. Recombination and radiative cascades produce H-alpha emission, including the $n=3\to 2$ transition. Shocks or other ionizing sources can also produce it, but disk star-forming gas is the natural interpretation for an extended spiral-galaxy rotation measurement.

The systemic redshift and inclination of an intrinsically circular thin disk areThe rotation-induced half-width is $(689.7-688.5)/2=0.6\mathrm{nm}$. Divide by the systemic observed wavelength to remove the common redshift factor before applying the small-velocity Doppler shift:The circular-orbit and thin-disk assumptions neglect intrinsic thickness, warps and noncircular gas motion.

Use the low-redshift Hubble law, neglecting peculiar velocity and cosmological distance corrections:The angular radius is $15$ arcseconds, half the stated major-axis diameter. A spherical dynamical estimate givesIt estimates mass inside the observed disk radius, not the entire unobserved halo. Disk geometry changes the exact coefficient.

Using the same approximate distance for the distance modulus,For an illustrative stellar mass-to-light ratio of $1$--$5$ in solar $V$-band units, $M_{\ast }\sim 1.5\times 10^{10}$--$7.4\times 10^{10}{M}_{\odot }$, below the dynamical estimate. This suggests substantial nonluminous matter under the adopted assumptions. Gas, stellar-population uncertainty, inclination, extinction and geometry also affect the comparison; luminosity alone cannot uniquely fix a stellar mass. The single-distance approximation at $z\simeq 0.05$ is the intended introductory estimate.

Use a Tully-Fisher relation $L\propto v_{\max }^4$. A simple justification assumes comparable mean surface brightness, so $L\propto R^2$, and comparable total mass-to-light ratio, so $M\propto L$. The virial scaling $v^2\sim GM/R$ then gives $v^4\propto M^2/R^2\propto L$. Calibrating to the previous galaxy yieldsThis is a rough luminosity estimate: the empirical slope depends on passband and population, and the assumed constant surface brightness and mass-to-light ratio are approximations rather than exact properties of every spiral galaxy.

For a spherical mass distribution with constant circular speed $v_c$,For $v_c=220\mathrm{km}{\mathrm{s}}^{-1}$, this is about $0.0140(r/8\mathrm{kpc}{)}^{-2}{M}_{\odot }\mathrm{p}{\mathrm{c}}^{-3}$. It is the spherical model's total density; it is singular if extrapolated to the very centre and is intended only over the flat-rotation region.

For a sharp outer truncation at $r_{\ast }$, the exterior mass is fixed at $M_{\ast }=v_c^2{r}_{\ast }/G$. Integrating the gravitational work from $r$ to the boundary and then to infinity givesThusThe exterior contribution cannot be dropped: at $r=r_{\ast }$ it correctly gives the point-mass escape speed $\sqrt{2}{v}_c$.

If the fastest measured radial speed is assumed to approximate both the total speed and the local escape speed of a bound star, then at $r=8\mathrm{kpc}$,Without those extra assumptions, the measurement gives only a bound: a bound star's total speed is at least its radial component, and the sample maximum need not reach escape, so $r_{\ast }\gtrsim 19.9\mathrm{kpc}$ within this model. An unbound interloper would invalidate even that bound. This is not a determination of the real Milky Way halo edge.