Past exam of the astrophysics course of the University of Oxford / 2021 (A11044W1)

Source: www.whatdotheyknow.com/request/oxford_physics_past_papers_2/response/2340696/attach/9/Physics%20Papers%202020.pdf?cookie_passthrough=1

The cover identifies the 2021 examination; the source bundle's filename uses 2020. This paper occupies bundle pages 21–27. TODO: recover the damaged source plots for question 5(a)–(c). The solutions below distinguish results derivable from the surviving material from measurements that require an intact figure. Units are SI unless specified.

A hydrogen recombination line is a discrete spectral line produced when radiative recombination captures an electron into an excited bound level and a subsequent radiative cascade emits photons. The free-to-bound capture itself produces a continuum; the later bound-to-bound transitions produce the recombination lines. For example, H-alpha is the $n=3\to 2$ transition in the Balmer series.

For the neutral hydrogen absorbing an ionizing stellar continuum, the principal bound-free transition is photoionization from the $1s$ ground state into the continuum. Most neutral hydrogen is in its ground state, and the photoionization cross section is largest near its $13.6\mathrm{eV}$ threshold, falling approximately as ${\nu }^{-3}$ above it. Dominance of the population-weighted opacity is intended here; this does not mean that every excited-state microscopic cross section is smaller at every frequency.

In an ionization-bounded, optically thick nebula, direct radiative recombination into $1s$ emits an ionizing photon that promptly photoionizes another hydrogen atom. One ion disappears and another is created: there is no net removal of an ion from the global budget. Case B recombination therefore uses ${\alpha }_B={\sum }_{n\ge 2}{\alpha }_n$, with local recycling of ground-state recombination photons. This is the on-the-spot approximation, rather than an unconditional property of radiative recombination: if those photons escape, direct capture does produce net neutralization and a Case A treatment is appropriate.

Write $Q={\dot{N}}_{\gamma }$ for the hydrogen-ionizing photon production rate, counting only photons above the hydrogen ionization energy, not all stellar photons. In steady photoionization equilibrium, the external supply balances the volume integral of net Case B recombination:For uniform, almost completely ionized pure hydrogen, $n_e=n_p=n_H$. HenceThe final approximation requires $r_{\mathrm{in}}\ll r_S$. The Strömgren sphere also assumes spherical symmetry, a sharp ionization front, a steady source older than the recombination time, uniform number density and temperature so that ${\alpha }_r$ is constant, negligible dust absorption of ionizing photons, negligible leakage, and no important helium correction or collisional ionization. Clumping would replace $n_H^2$ by a volume average of $n_e{n}_p$.

An additional atomic input is needed: the total Case B recombination coefficient does not specify the number of H-alpha photons per net radiative recombination. Let $y_{\alpha }={\alpha }_{\mathrm{H}\alpha }^{\mathrm{eff}}/{\alpha }_r$, where ${\alpha }_{\mathrm{H}\alpha }^{\mathrm{eff}}$ is the effective recombination coefficient. At the approximately $10^4\mathrm{K}$ temperature implicit in the supplied ${\alpha }_r$, take $y_{\alpha }\simeq 0.45$. The coefficient 2.2 used later in the paper corresponds to $y_{\alpha }=1/2.2\simeq 0.455$, which we use consistently here.

For an unextinguished, fully filled Strömgren sphere with negligible central cavity, the H-alpha luminosity and observed radiative flux satisfySubstitution eliminates $r_S$ and givesConvert the angular diameter to radians: $\theta =56\times 10^{-3}/206265=2.7150\times 10^{-7}$. Also $hc/{\lambda }_{\mathrm{H}\alpha }=3.0268\times 10^{-19}\mathrm{J}$. ThusThe implied hydrogen-ionizing photon production rate is $Q\simeq 1.00\times 10^{39}{\mathrm{s}}^{-1}$. Using $y_{\alpha }=0.45$ instead gives $D\simeq 505\mathrm{pc}$. These values use the printed milliarcseconds and integrated line observed radiative flux. Assuming one H-alpha photon per recombination would give a different, unjustified distance. An appreciable cavity or a nonunit filling factor must also enter the emitting volume.

The Brackett series consists of transitions ending at $n=4$, at infrared wavelengths. They suffer much less interstellar extinction than H-alpha, so their corrected observed radiative fluxes can give a more reliable version of the same inference. At fixed measured angular size and number density, the formula has $D\propto F$: uncorrected interstellar extinction therefore makes this particular distance estimate too small. Infrared lines are weaker and need their own effective recombination coefficients; their advantage is reduced sensitivity to dust.

Let $q_{\mathrm{in}}$ be the photoionization rate per neutral hydrogen atom just outside the empty cavity. With an isotropic central source and no attenuation inside that cavity,The cross section is zero below threshold; this permits the lower integration limit zero. Pure hydrogen with hydrogen ionization fraction $\zeta$ has $n_{\mathrm{HI}}=n_H(1-\zeta )$ and $n_e=n_p=n_H\zeta$. Local photoionization equilibrium is thereforeDefine the dimensionless quantity ${\Theta }_{\mathrm{phys}}={\alpha }_r{n}_H/q_{\mathrm{in}}$. Solving the resulting quadratic equation, ${\Theta }_{\mathrm{phys}}{\zeta }_{\mathrm{in}}^2+{\zeta }_{\mathrm{in}}-1=0$, gives the physical rootThe rationalized form avoids numerical cancellation for small ${\Theta }_{\mathrm{phys}}$. Strong illumination gives ${\zeta }_{\mathrm{in}}\simeq 1-{\Theta }_{\mathrm{phys}}$; weak illumination gives ${\zeta }_{\mathrm{in}}\simeq {\Theta }_{\mathrm{phys}}^{-1/2}$.

Eliminating $Q$ with the H-alpha luminosity, $Q=2.2(4\pi D^2{F}_{\mathrm{H}\alpha }){\lambda }_{\mathrm{H}\alpha }/(hc)$, yieldsThe printed definition of $\Theta$ is inconsistent with the defined radii. The source calls $r_{\ast }$ a physical stellar radius, but its factor $(2\pi r_{\ast }D{)}^2$ gives $\Theta$ units of length to the fourth power, rather than a dimensionless value. It also omits the physically necessary cavity radius $r_{\mathrm{in}}$. For a concrete counterexample, double $r_{\mathrm{in}}$ while holding the source, $D$, and $n_H$ fixed: the incident photoionization rate falls by four, so ${\Theta }_{\mathrm{phys}}$ rises by four; the printed expression does not change. The root has the requested form with the corrected ${\Theta }_{\mathrm{phys}}$ above. There is no supplied relation allowing $r_{\mathrm{in}}$ to be replaced by $r_{\ast }$.

Retain the consistent ${\Theta }_{\mathrm{phys}}$ of d. Across a sufficiently thin layer, the neutral hydrogen column density is $N_{\mathrm{HI}}=n_H(1-{\zeta }_{\mathrm{in}})\Delta r+O(\Delta r^2)$. The direct stellar continuum acquires optical depth ${\tau }_{\nu }=N_{\mathrm{HI}}{\sigma }_{\nu }$. Define the photon-weighted second momentExpanding the attenuating exponential in the photoionization rate givesThe second cross section moment appears because the radiation both undergoes absorption and is weighted by the photoionization cross section in computing the rate.

In a local plane-parallel approximation that suppresses geometrical dilution, setThen local photoionization equilibrium becomes ${\Theta }_{\mathrm{phys}}{\zeta }^2=\Gamma (1-\zeta )$, soThis gives the requested algebraic dependence, with the corrected definition of $\Theta$. Besides $\Delta r\ll r_{\mathrm{in}}$, linear attenuation requires $N_{\mathrm{HI}}{\sigma }_{\nu }\ll 1$ over the frequencies contributing appreciably to the rate. In particular $\Gamma$ must remain positive.

The source's $\Gamma$ neglects $-2\Delta r/r_{\mathrm{in}}$, which is also first order in layer thickness. A consistent spherical first-order answer uses ${\Gamma }_{\mathrm{sph}}=\Gamma -2\Delta r/r_{\mathrm{in}}$ in the same root. Neglecting dilution relative to absorption additionally requires $n_H(1-{\zeta }_{\mathrm{in}})⟨{\sigma }^2⟩/⟨\sigma ⟩\gg 2/r_{\mathrm{in}}$. As a concrete limiting test, with negligible absorption the source's $\Gamma =1$ predicts no change, while inverse-square dilution still decreases $q$ and thus $\zeta$.

Take the initially expanding branch ${\dot{r}}_i=H_i{r}_i$; the printed squared velocity alone also permits an initially contracting branch, which would not reach a future maximum radius. Multiply the equation of motion by $\dot{r}$ and integrate. The conserved outer-shell specific orbital energy isThe initial mass follows from the background critical density:ConsequentlyIf $E$ means the energy of a test shell of mass $m$, $E=mϵ$. If it means the total energy of the entire uniform, homologously moving perturbation, direct integration instead gives $K=3M{\dot{r}}^2/10$, $U=-3G{M}^2/(5r)$ andThe common convention of writing the first integral as $E$ is therefore an energy per unit shell mass, not the total energy of the sphere.

At spherical-collapse turnaround, $\dot{r}=0$, so $ϵ=-GM/r_m$. HenceAny positive initial density contrast binds this pressureless perturbation in the stated Einstein-de Sitter universe.

For $r=A(1-\cos \theta )$ and $t+T=B(\theta -\sin \theta )$, differentiation givesComparison with $-GM/[A^2(1-\cos \theta {)}^2]$ proves $A^3=GM{B}^2$. Substitution into the conserved specific orbital energy gives $ϵ=-GM/(2A)$. ThereforeThese constants exactly reproduce both the initial radius and expanding velocity. The initial parameter obeysIn an Einstein-de Sitter universe whose background big bang is $t=0$, $t_i=2/(3H_i)$. Thus the exact required offset isExpanding ${\theta }_i-\sin {\theta }_i=\frac{4}{3}{\delta }_i^{3/2}-\frac{8}{5}{\delta }_i^{5/2}+O({\delta }_i^{7/2})$ givesOne cannot set $T=0$ exactly at finite ${\delta }_i$ while retaining the stated initial Hubble velocity and background time origin. For example ${\delta }_i=0.01$, $t_i=1$ gives $T=-0.00199148$. Shifting to $t^{\prime }=t+T$ removes the offset from the local orbit, but the background then scales as $a\propto (t^{\prime }-T{)}^{2/3}$, not $t^{\prime 2/3}$. Setting $T\simeq 0$ is justified in the early-initial-time limit for subsequent evolution; the following standard growing-mode formulae use that approximation.

First retain the exact initial normalization. The scale factor obeys $a/a_i=(t/t_i{)}^{2/3}$, so the cosmological redshift isAlso $\rho =3M/(4\pi r^3)$ and ${\rho }_m=1/(6\pi G{t}^2)$. Using $A^3=GM{B}^2$ givesThese equations exactly accommodate the finite-time initial conditions in b.

Now neglect $T$ and retain leading order in ${\delta }_i\ll 1$. Since $t_i/B=4{\delta }_i^{3/2}/[3(1+{\delta }_i)]$, the cosmological redshift relation reduces toDefine ${\delta }_0=\frac{3}{5}{\delta }_i(1+z_i)$. ThenThe second equality is exact for the ideal common-bang growing-mode spherical-collapse model; it is approximate for the paper's finite-time initial Hubble-velocity data.

The factor $3/5$ has a physical origin. Linear mass conservation gives ${\dot{\delta }}_i=0$ when the initial local expansion equals the background Hubble flow. The general linear dust solution is ${\delta }_L=C_g(t/t_i{)}^{2/3}+C_d(t/t_i{)}^{-1}$. Imposing ${\delta }_L(t_i)={\delta }_i$ and ${\dot{\delta }}_L(t_i)=0$ yieldsThus ${\delta }_0$ is the present extrapolation of the growing component, after neglecting the decaying transient. It is not the exact present value of the full linear solution for finite $t_i$.

In the growing-mode spherical-collapse model, use $\theta -\sin \theta ={\theta }^3/6-{\theta }^5/120+\cdots$ and $1-\cos \theta ={\theta }^2/2-{\theta }^4/24+\cdots$. Their ratio in c gives $\delta =3{\theta }^2/20+O({\theta }^4)$. Meanwhile $t\propto \theta -\sin \theta$, and the linear growing density contrast scales as $t^{2/3}$. Matching at small $\theta$ therefore fixesThis is a linearly extrapolated growing-mode amplitude at any epoch, including epochs where the actual density contrast is nonlinear. It carries the initial-condition qualification in c.

Numerical comparison isThe last column measures the fractional underestimate of the actual density contrast. Linear theory is accurate at $\delta \ll 1$, but substantially underpredicts it by the time $\delta$ approaches unity. At $\theta =\pi$, the shell reaches spherical-collapse turnaround; $1+\delta =9{\pi }^2/16\simeq 5.55$.

At $\theta =2\pi$, $r=0$ and the ideal pressureless density diverges at finite time $t_c=2\pi B$. Yet the extrapolated linear value remains finite:This is the linear spherical-collapse threshold, not the true density contrast at collapse. Actual nonspherical collisionless systems undergo shell crossing and virialization rather than becoming uniform point singularities. The ideal virial theorem and conservation of energy estimate a final radius $r_m/2$ if the structural form is preserved.

Work in the source rest frame and neglect the source's small cosmological redshift. Over an emission interval $\Delta t$, a ballistic relativistic jet feature travels $\beta c\Delta t$, advancing $\beta c\Delta t\cos \theta$ toward the observer and $\beta c\Delta t\sin \theta$ across the sky. The second photon has a shorter journey, so its arrival is delayed by onlyDividing the projected displacement by this arrival interval givesApparent superluminal motion is a combination of projection and light-travel delay. The feature's local speed remains $\beta c<c$; its inferred transverse displacement divided by the observer's arrival interval is not a local signal velocity.

For fixed $\beta$, differentiate the apparent speed with respect to angle:The maximum lies at $\cos \theta =\beta$ and has value ${\beta }_{\mathrm{app},\max }=\beta /\sqrt{1-{\beta }^2}=\gamma \beta$. Hence strict apparent superluminal motion requiresThe threshold is an infimum: equality yields exactly ${\beta }_{\mathrm{app}}=1$ at $\theta =\pi /4=45^{\circ }$, rather than a strictly superluminal value. For a chosen larger $\beta$, the optimal inclination is $\theta =\arccos \beta$, and the full superluminal interval satisfies $\sin \theta +\cos \theta >1/\beta$.

For intrinsically symmetric oppositely directed relativistic jets, their proper motions areThe ratio removes $D$ and yieldsTaking the product instead givesCausality requires $\beta \le 1$, so the limiting distance and inclination areHere and in e, “maximum” denotes the limiting $\beta \to 1$ bound; massive ejecta obey strict inequalities. The conversion is $1\mathrm{mas}\mathrm{da}{\mathrm{y}}^{-1}=5.6113\times 10^{-14}\mathrm{rad}{\mathrm{s}}^{-1}$, henceBoth bounds occur together in the light-speed limit. At any specified smaller distance, ${\beta }^2=x^2+(1-x^2)(D/D_{\max }{)}^2$ determines the intrinsic speed.

The Doppler factors for the two discrete blobs are ${\mathcal{D}}_{\pm }=[\gamma (1\mp \beta \cos \theta ){]}^{-1}$. Their ratio is $(1+x)/(1-x)=77/33=7/3$. Lorentz-invariant specific intensity means $I_{\nu }/{\nu }^3$ is unchanged; with $\nu =\mathcal{D}{\nu }^{\prime }$ and the same rest-frame power-law spectrum,The brightness ratio at the common observed frequency therefore givesThe stated identical intrinsic specific intensities must refer to comparable intrinsic states. If the blobs fade with their proper ages, equal observer times sample different ages and need that correction. A steady continuous jet has a different boosting exponent, $2+\alpha$; the discrete ejecta and the specified specific intensity transformation justify $3+\alpha$ here.

For optically thin synchrotron radiation from an isotropic electron distribution $dN/dE\propto E^{-p}$, the characteristic emitted frequency scales as $B{E}^2$ and the total single-particle power as $B^2{E}^2$. Thus $j_{\nu }\propto E^{-p}{E}^2/(d\nu /dE)\propto {\nu }^{-(p-1)/2}$; integration over the full single-particle spectrum preserves this exponent. HenceThe test-particle strong nonrelativistic diffusive shock acceleration prediction is $p=(r+2)/(r-1)=2$ for compression ratio $r=4$, giving $\alpha =0.5$. The inferred spectrum is steeper. If interpreted literally within that uncooled formula, $p=2.50$ would correspond to $r\simeq 3$, but cooling, escape, and other acceleration conditions can also steepen it.

The observed Galactic cosmic ray differential energy spectrum below the knee is approximately $E^{-2.7}$, steeper still; its formal synchrotron radiation equivalent would be $\alpha \simeq 0.85$. Thus $p\simeq 2.50$ lies between the simplest source prediction and the propagated cosmic ray slope. The comparison is qualitative: most measured cosmic rays are nuclei, whereas the radio slope here probes electrons, and propagation and radiative losses alter the spectra differently.

Apply the same symmetric-ejection and causality argument to the launch proper motions, when the proposed ejecta are fastest. One obtainsandThe fitted angular decelerations describe subsequent evolution; they do not relax the bound already imposed at launch. They are observer-time phenomenological coefficients, not directly identical physical decelerations in the source frame.

The stellar parallax distance $2.96\pm 0.33\mathrm{kpc}$ is compatible with the constant-velocity bound $3.435\mathrm{kpc}$, but its central value exceeds the decelerating-launch bound by $0.717\mathrm{kpc}$, or about $2.17$ quoted standard deviations if the uncertainty is treated as symmetric Gaussian. Even its one-standard-deviation lower endpoint, $2.63\mathrm{kpc}$, exceeds $2.243\mathrm{kpc}$. At $D=2.96\mathrm{kpc}$ the respective inferred intrinsic speeds are $\beta =0.885$ and $\beta =1.300$. The latter is a concrete violation of causality under the symmetric model assumptions.

Thus the supplied distance favours the constant-velocity fit. This is a consistency comparison, not a full statistical model selection: uncertainties in the fitted proper motions, their correlations, intrinsic asymmetry and the distance likelihood would be required for that.

The Einstein-de Sitter universe has $a\propto t^{2/3}$ and $H=\dot{a}/a\propto a^{-3/2}$. Normalizing $a_0=1$ therefore gives $H(z)=H_0(1+z{)}^{3/2}$. Differentiating $1+z=1/a$ givesIn the local comoving rest frame, a photon crosses proper path length $d\ell =c|dt|$. Galaxies with proper number density $n(z)$ and geometric cross section ${\sigma }_G=\pi r_G^2$ contribute the mean number of intersections $d\tau =n(z){\sigma }_{G}d\ell$. The infinitesimal probability of an intersection is the same to first order, givingThe radius is a physical radius, and $n(z)$ is a proper number density evaluated at epoch $z$, rather than a density already multiplied by $dz$. For a finite path with overlaps, the integrated mean count $\tau$ need not itself be the probability of at least one intersection.

Conservation of the comoving galaxy population gives $n(z)=n_0{a}^{-3}=n_0(1+z{)}^3$. Holding the proper galaxy radius fixed, integrate a:With $r_G=0.015\mathrm{Mpc}$, $n_0=0.01\mathrm{Mp}{\mathrm{c}}^{-3}$, and $c/H_0=4282.75\mathrm{Mpc}$, set $C=c\pi r_G^2{n}_0/H_0=0.030273$. ConsequentlyFor random uncorrelated intersections, the zero-intersection probability is $e^{-\tau }$ from the Poisson distribution, so the fraction of the sky covered by foreground galaxies out to $z=1$ isIgnoring overlaps gives $f\simeq \tau \simeq 3.7\%$. A literal shell of zero thickness exactly at $z=1$ has zero coverage; the integrated foreground interpretation is needed. Clustering changes the exact overlap probability, although the mean optical depth above still follows from the mean number density.

Finally,At this optical depth, the Poisson sky coverage is $1-e^{-1}\simeq 63\%$, not $100\%$. The large inferred cosmological redshift extrapolates conserved galaxy counts and fixed radii far beyond their plausible physical range; it is the answer within the prescribed toy model.

Use the dimensionally correct flat Friedmann-Lemaître-Robertson-Walker metricThe source's $-cd{t}^2$ is missing a factor $c$, and its angular term must mean the unit-sphere line element $d{\Omega }^2$, not the solid angle element $d\Omega =\sin \vartheta d\vartheta d\phi$. For example, $cd{t}^2$ has dimensions length times time, while $d{s}^2$ has dimensions length squared. The corrected convention above removes both ambiguities.

A radial light ray has $d{s}^2=0$ and $|d\chi |=c|dt|/a$. ThusA proper shell volume at emission is $d{V}_{\mathrm{proper}}=a^3{\chi }^{2}d\chi d\Omega$. Multiplying by $n(z)=n_0{a}^{-3}$ cancels the expansion factors, equivalently using the comoving volume element directly. HenceThis counts every galaxy in the shell per solid angle, rather than intersections weighted by individual galaxy areas. Near $z=0$ it reduces to $n_0(c/H_0{)}^3{z}^2$, as expected for a Euclidean volume shell.

TODO: identify the actual plotted branches from an intact source figure. The descriptions below explain how to do so.

In an old globular cluster's colour-magnitude diagram, the expected principal sequences are the main sequence below the main-sequence turnoff, the subgiant branch leading to the red giant branch, the horizontal branch above the main-sequence turnoff and extending across colour, and an Asymptotic giant branch above the horizontal branch, generally near and somewhat blueward of the red giant branch. A sufficiently deep diagram can also show a faint blue white dwarf cooling sequence; blue stragglers may lie above and blueward of the old main-sequence turnoff.

The two panels use the colour index $B-V$, increasing toward cooler, redder stars. On a conventional apparent-astronomical magnitude vertical axis, brighter stars are higher. These coordinate conventions locate the branches just described.

The supplied PDF does not permit a complete identification of the plotted branches. Its upper image data are corrupt and render as a grey rectangle, covering the main-sequence turnoffs and evolved populations; only lower fragments and $B-V$ axes survive. Thus the list above identifies the relevant stellar evolution phases and how to recognize them on an intact plot, rather than claiming that every listed sequence is visibly present in this damaged source. In particular, neither a plotted white dwarf sequence nor individual blue stragglers can be confirmed from it.

TODO: measure the relative-distance offset from an intact source figure or its numerical photometry.

For corresponding populations whose intrinsic absolute magnitudes are calibrated to match, the distance modulus in the same filter iswhere $A$ is interstellar extinction in that filter. Subtracting the two equations yieldsHere $\Delta m=m_2-m_1$, $\Delta M=M_2-M_1$, and $\Delta A=A_2-A_1$. For a matched horizontal branch after a metallicity correction, or matched unevolved main sequences after correcting colour and composition, $\Delta M\simeq 0$. Neglecting differential interstellar extinction then gives $D_2/D_1\simeq 10^{\Delta m/5}$. A one-astronomical magnitude shift would mean a factor $1.585$ in distance, illustrating the calibration rather than measuring these panels.

A defensible numerical ratio cannot be extracted from the surviving figure. The corresponding sequences and most of the vertical scales are hidden by the corrupted image. Different bottom-axis labels or limiting astronomical magnitudes do not establish a common-population vertical offset: exposure depth and intrinsic population differences can also change them. An intact figure or numerical photometry is needed to measure $\Delta m$, then correct $\Delta M$ and $\Delta A$. The equation above is the quantitative inference supported by the available source; inventing an offset would not solve the missing-data problem.

TODO: assign the physical differences to the actual two panels once the missing evolved-star sequences are recovered.

The main intrinsic parameters for old globular clusters are age, metallicity, helium abundance and the distribution of envelope masses after red giant branch mass loss. Distance and foreground interstellar extinction alter the appearance and must first be removed before assigning intrinsic differences.

Age controls the main-sequence turnoff: a more massive star has a shorter main-sequence lifetime, approximately fuel divided by luminosity. An older coeval population therefore has a smaller turnoff mass and a fainter, generally redder main-sequence turnoff. The separation in astronomical magnitude between the main-sequence turnoff and horizontal branch is useful because a common distance modulus cancels.

At comparable age, larger metallicity increases stellar-envelope opacity and atmospheric line blanketing. This generally makes the red giant branch and main sequence redder; metal-poor populations tend to have hotter, bluer red giant branches. Metallicity also affects hydrogen burning through the CNO cycle, so a quantitative age inference needs composition-dependent stellar models.

The horizontal branch is especially sensitive to envelope mass: at similar helium-core mass, a thinner hydrogen envelope gives a higher effective temperature and hence a bluer location. Red giant branch mass loss, age and helium abundance all influence that envelope and produce the familiar “second parameter” complexity. Enhanced helium also changes mean molecular weight, luminosity and evolutionary lifetime, and can shift main sequences and horizontal branches.

Interstellar extinction dims the diagram, while wavelength-dependent interstellar extinction reddens $B-V$; distance primarily supplies a vertical distance modulus shift. A systematic bend or different horizontal branch colour distribution cannot be explained by distance alone. Because the source hides the evolved branches, it is not possible to assign “older”, “more metal rich”, or “bluer horizontal branch” to a particular panel reliably. The preceding mechanisms give the warranted physical interpretation without inventing the lost observational differences.

On the main sequence, central hydrogen burning converts four protons into helium, positrons, neutrinos and released energy. The net reaction is $4p\to {}^4\mathrm{He}+2e^{+}+2{\nu }_e$, with positron annihilation contributing to the total roughly $26.7\mathrm{MeV}$ release before subtracting escaping neutrino energy. In the low-mass main-sequence stars of an old globular cluster, the proton–proton chain generally dominates. The CNO cycle makes the same net conversion and becomes important in hotter burning regions.

After central hydrogen exhaustion, a subgiant contracts its helium core and establishes a hydrogen-burning shell. On the red giant branch, that shell supplies most of the luminosity, commonly through the temperature-sensitive CNO cycle, while the inert helium core grows and becomes supported mainly by electron degeneracy pressure. The envelope expands and its surface cools. The helium core is not yet powered by helium nuclear fusion.

Helium ignition in a sufficiently low-mass degenerate core produces the helium flash, removing the strong electron degeneracy pressure dominance. A horizontal-branch star subsequently burns helium in its core through the Triple-alpha process, $3{}^4\mathrm{He}\to {}^{12}\mathrm{C}+\gamma$, and through ${}^{12}\mathrm{C}(\alpha ,\gamma ){}^{16}\mathrm{O}$. A surrounding hydrogen-burning shell can continue simultaneously.

After central helium exhaustion, the Asymptotic giant branch has an inert carbon-oxygen core with a helium-burning shell and an outer hydrogen-burning shell. Alternation between quiescent burning and helium-shell thermal pulses drives mixing and substantial mass loss. Subsequent post-asymptotic giant branch evolution exposes a hot core that may ionize its expelled envelope as a planetary nebula. Finally a white dwarf cools mainly by loss of stored thermal energy, rather than sustained central nuclear fusion.

For ordinary single stars in these old globular clusters, no current core-collapse supernovae are expected, and no new black holes are expected from the normal plotted population. Their main-sequence turnoff masses are typically around $0.8M_{\odot }$, far below the roughly $8M_{\odot }$ initial-mass scale for advanced burning and core collapse. More massive original stars exhausted their much shorter main-sequence lifetimes long ago. The precise transition mass depends on composition and mass loss, but this large separation makes the conclusion robust.

The low-mass stars now leaving the main sequence evolve through the red giant branch, helium ignition, horizontal branch and Asymptotic giant branch. They lose their envelopes and leave mainly carbon-oxygen white dwarfs, typically around $0.5$–$0.6M_{\odot }$ for current old-cluster progenitors, supported by electron degeneracy pressure. A visible planetary nebula is possible if the exposed core heats before the ejecta disperse; it is not guaranteed for every low-mass remnant. Lower-mass main-sequence stars repeat this evolution only much later, and the very lowest-mass isolated stars can ultimately leave helium-rich degenerate remnants without igniting helium.

The clusters may already contain neutron stars and black holes made by their original massive stars; these remnants need not appear on an optical colour-magnitude diagram. Binary interactions can instead produce stripped helium white dwarfs, mergers or a possible Type Ia supernova; such exceptions are distinct from normal single-star core collapse. No such special progenitor can be established from the damaged plots. In particular, a luminous red giant is not by itself evidence for a massive core-collapse supernova progenitor.

For a spherical mass distribution, the shell theorem says the circular gravitational acceleration is $GM(<R)/R^2$. Balancing it against the centripetal acceleration $V_c^2/R$ givesAt $R=16\mathrm{kpc}=4r_0$, the galaxy rotation curve gives $V_c=200(1-e^{-4})=196.337\mathrm{km}{\mathrm{s}}^{-1}$. With $R=4.93708\times 10^{20}\mathrm{m}$,This is the total dynamically inferred enclosed mass, including any dark matter; the galaxy rotation curve alone does not separate its components.

Put $V_0=200\mathrm{km}{\mathrm{s}}^{-1}$. A Taylor expansion near the centre givesTherefore the circular angular frequency isThus the central motion approaches solid-body rotation. Numerically ${\Omega }_0=1.6204\times 10^{-15}{\mathrm{s}}^{-1}$, and the central orbital period tends toThe finite limit of $\Omega$ accompanies $V_c\to 0$ at $r\to 0$; it does not imply a finite central linear speed.

At large radius $V_c\to V_0$, so the spherical enclosed mass is $M(<r)\simeq V_0^{2}r/G$. A spherical mass density satisfies $dM/dr=4\pi r^2\rho (r)$, and henceEquivalently, a local power law $\rho \propto r^{-p}$ with enclosed mass dominated by that extended profile has $M\propto r^{3-p}$ and $V_c^2\propto r^{2-p}$. A flat galaxy rotation curve requires $p=2$.

Differentiating the exact mass profile verifies the limit and the finite central density:As $x\to 0$, the bracket is $3x^2+O(x^3)$, so $\rho (0)=3V_0^2/(4\pi G{r}_0^2)$. As $x\to \infty$, it tends to one, recovering the $r^{-2}$ tail. Extending the flat galaxy rotation curve to infinity makes total mass diverge linearly; a real finite halo must eventually depart from this idealization.

Let the uniform thin galactic disk have outer radius $R_d$ and surface mass density $\Sigma =M/(\pi R_d^2)$, where $M$ is the mass calculated in a. If the redistribution retains the stated outer radius, set $R_d=16\mathrm{kpc}$; otherwise $M$ alone does not determine both $R_d$ and $\Sigma$.