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For a gas-pressure-supported stellar core of fixed mass and composition, neglect boundary terms for the illustrative whole-core estimate. The virial theorem gives $E=U+\Omega =-U$, and the monatomic gas gives $U=3M_c{k}_B\overline{T}/(2\mu m_u)$. Thus the effective heat capacity along the equilibrium sequence isThis negative heat capacity of a virialized gravitational system means that net energy loss makes a self-gravitating core contract and heat. It is different from the positive local material specific heat capacity. On the main sequence, hydrogen burning supplies most of the radiated energy, so rapid contraction is arrested. Hydrogen depletion increases the mean molecular weight; slow structural readjustment usually raises the central temperature rather than making it fall.

After core hydrogen exhaustion, a massive stellar core contracts and heats more rapidly until a later nuclear fuel ignites. There is no general reversal to cooling simply upon leaving the main sequence. If radiation pressure becomes dominant, the internal energy becomes $U=U_{\mathrm{gas}}+U_{\mathrm{rad}}$ with $3\int PdV=2U_{\mathrm{gas}}+U_{\mathrm{rad}}$. The virial theorem therefore gives $E=-U_{\mathrm{gas}}$; the radiation-dominated limit is marginally bound and its effective adiabatic index approaches $4/3$.

Quasistatic contraction can still maintain virial equilibrium. It fails dynamically only when sufficient support is lost, for example if pair creation reduces the effective adiabatic index below $4/3$, or when an iron core undergoes electron capture and photodisintegration. A fixed-composition gas-dominated contracting core with $T\propto 1/R$ is already a counterexample to any assertion that leaving the main sequence alone reverses the heating or destroys virial equilibrium.

For a regular spherical star, let ${\rho }_c=\rho (0)$ and $P_c=P(0)$. The stellar mass conservation equation and stellar hydrostatic equation areAssuming the smoothness needed for the requested Taylor series, write $M=a_0+a_{1}r+a_2{r}^2+a_3{r}^3+\cdots$. There is no point mass at the centre, so $a_0=0$, and matching the stellar mass conservation equation gives $a_1=a_2=0$, $a_3=4\pi {\rho }_c/3$. Substitution in the stellar hydrostatic equation gives $P^{\prime }(r)=-(4\pi G/3){\rho }_c^{2}r+O(r^3)$, henceThe remainder orders use a smooth spherically symmetric scalar density, for which $\rho (r)={\rho }_c+O(r^2)$. Merely finite continuous density gives the same leading coefficients with $o(r^3)$ and $o(r^2)$ remainders. In particular $P^{\prime }(0)=0$, despite the apparent $r^{-2}$ in the stellar hydrostatic equation.

For a family of spherical profiles with a fixed dimensionless shape, write the gravitational potential energy as $\Omega =-\alpha G{M}^2/R$, where $\alpha >0$ depends on that shape. The virial theorem in b and the sphere volume $V=4\pi R^3/3$ giveFor a uniform density, $\alpha =3/5$; formally integrating $-{\int }_0^{R}GM(r)dM(r)/r$ for $M(r)=Mr/R$ gives $\alpha =1$. Neither the virial theorem nor the initial profile guarantees a specific final shape, so the proportionality requires the stated homologous-profile assumption. Surface pressure must be handled consistently with b before using either coefficient as an equilibrium model.

Assume an isolated, bound, nonrotating Newtonian gas cloud, negligible surface pressure, negligible magnetic and radiation stresses, no persistent bulk flow, and a stationary or time-averaged virialized state. The stellar virial theorem then reduces towhere $\Omega <0$ is the total gravitational potential energy. Define the volume-averaged pressure $\overline{P}=V^{-1}\int PdV$ and average gravitational energy density $u_g=\Omega /V$. HenceEquivalently, for a nonrelativistic monatomic gas, the thermal internal energy $U=(3/2)\int PdV$ obeys $2U+\Omega =0$. This second derivation must not count the thermal motions again as a separate bulk kinetic energy. If the external pressure is $P_s$, the correct result is $\overline{P}-P_s=-u_g/3$; non-negligible bulk motion adds $2T_{\mathrm{bulk}}$ to the virial theorem. The initial sharp-edged $r^{-2}$ profile need not survive virialization; maintaining that profile hydrostatically to its edge would generally require a surface term.

There are two different physical interpretations. The supplied logarithmic integral signals free fall in a fixed singular isothermal sphere, with a test particle moving through a frozen background. Write $C={\rho }_0{r}_0^2$. By the shell theorem, the enclosed mass is $M(r)=4\pi Cr$ and the radial equation of motion isThe second equality follows by multiplying by $\dot{r}$ and using the initial rest condition. The free fall trajectory is given implicitly byThus the intended fixed-profile answer isA collapsing material shell gives a different answer. If the entire cloud undergoes pressureless gravitational collapse from rest, the edge follows the fixed enclosed mass $M_0=4\pi {\rho }_0{r}_0^3$, not the instantaneous fixed-profile value $4\pi Cr$. With no shell crossing, $\ddot{r}=-G{M}_0/r^2$, soThis is a concrete counterexample to treating the fixed-profile result as the exact collapse time of the whole nebula. The initial mean density inside $r_0$ is $3{\rho }_0$, consistent with the usual free-fall time of a uniform sphere expressed using that mean density. The logarithmic integral is valid for the explicitly frozen background approximation.

Adopt the stipulated neutral fraction $f_{\mathrm{HI}}=10^{-6}$, independently of the equilibrium estimate in e. For pure hydrogen, $n_e=(1-f_{\mathrm{HI}})n_{\mathrm{HI}}/f_{\mathrm{HI}}\simeq 8.3{\mathrm{m}}^{-3}$. The thermal Sunyaev-Zeldovich effect measures integrated electron pressure:Using the same absorbing line of sight as in d,Thus one such filament is below the stated measurement sensitivity. If the number density inferred transversely is instead used to view the same cylinder end-on, its path is twenty times longer and $y=2.9\times 10^{-10}$, still below the stated sensitivity. Changing the assumed path of the original absorption measurement while keeping its column density fixed does not change its predicted $y$. The quoted neutral-fraction floor is a premise of this estimate, not a universal bound derivable from thermal equilibrium.

True thermal equilibrium requires a thermal ionizing radiation field as well as chemical balance; equality of photoionization and recombination rates alone would require a specified external radiation field. Under the stronger equilibrium assumption, apply the Saha equation to pure hydrogen. The electron spin factor cancels the ground-state electronic degeneracy, givingAt $T=10^4\mathrm{K}$, $k_{B}T=0.862\mathrm{eV}$ and $S=3.38\times 10^{20}{\mathrm{m}}^{-3}$. Charge neutrality gives $n_e=n_p$. Importantly, the known quantity from d is the neutral number density, not the total number density; use Saha neutral fraction from a measured neutral density to obtain $n_p=\sqrt{S{n}_{\mathrm{HI}}}=5.31\times 10^7{\mathrm{m}}^{-3}$. ThereforeThis extraordinarily small neutral fraction is the result of applying thermal equilibrium to dilute intergalactic gas; it is not a realistic inference from photoionization equilibrium with a nonthermal cosmic background. For the axial geometry of d, the equilibrium neutral fraction is smaller by $\sqrt{20}$.

For the unresolved Lyman-alpha line, the lower $1s$ and combined upper $2p$ state degeneracies are $g_1=2$ and $g_2=6$, so the Einstein coefficients giveThe measured equivalent width $W=7.0\times 10^{-12}\mathrm{m}$ then impliesAssume a central transverse line of sight through a uniform filament, so $\ell =d=0.5\mathrm{Mpc}=1.54\times 10^{22}\mathrm{m}$. Nearly all neutral hydrogen is in its ground state at this temperature, henceThe cylinder's length $L$ determines its total volume but does not enter a transverse column density. Geometry must be specified: an axial line of sight has $\ell =L$ and gives $n_{\mathrm{HI}}=4.2\times 10^{-7}{\mathrm{m}}^{-3}$ for the same measured equivalent width. Intermediate chords give intermediate path-dependent estimates. The numbers above treat $W$ as a rest-frame equivalent width; an observed value at redshift $z$ must first be divided by $1+z$.

With the spontaneous emission term suppressed, $j_{\nu }=0$. A uniform absorption coefficient over length $\ell$ gives directlyThe normalized specific intensity deficit is unchanged by conversion between wavelength and frequency, since both numerator and denominator acquire the same Jacobian determinant. The Boltzmann distribution and the Einstein coefficients give $n_2{B}_{21}/(n_1{B}_{12})=e^{-h{\nu }_{12}/(k_B{T}_e)}\ll 1$. Consequently, in the optically thin limit,Here $N_1=n_1\ell$ is the column density, and $\nu \simeq {\nu }_{12}$ has been used in slowly varying factors. This linear relation between equivalent width and column density fails once the central optical depth becomes appreciable.

Use the Einstein coefficients convention in which $B_{ij}$ multiplies spectral radiation energy density $u_{\nu }$, with $u_{\nu }=c^{-1}\int I_{\nu }d\Omega$. A normalized spectral line profile obeys $\int \phi (\nu )d\nu =1$. The spontaneous emission probability per upper-state atom per unit time is $A_{21}$; the stimulated emission and upward absorption rates per atom are respectively $B_{21}\int u_{\nu }\phi (\nu )d\nu$ and $B_{12}\int u_{\nu }\phi (\nu )d\nu$.

For isotropic spontaneous emission, $n_2{A}_{21}$ transitions occur per unit volume and time, each supplying photon energy $h{\nu }_{12}$. Dividing among $4\pi$ steradians givesFor a small incident beam, its contribution to $u_{\nu }$ is $I_{\nu }d\Omega /c$. Absorption removes and stimulated emission adds photons in the incident direction. Matching the net loss to ${\alpha }_{\nu }{I}_{\nu }d\Omega$ therefore givesReplacing ${\nu }_{12}$ by $\nu$ across the narrow spectral line profile gives the printed expressions. The negative sign is essential: an inverted population can amplify the incident specific intensity. Thermal equilibrium and the Planck function imply $g_1{B}_{12}=g_2{B}_{21}$ and $A_{21}/B_{21}=8\pi h{\nu }_{12}^3/c^3$, where $g_i$ is the state degeneracy. A convention using mean intensity instead changes both $B$ coefficients by $4\pi /c$.

The specific intensity $I_{\nu }$ is energy transported per unit time, projected area, solid angle and frequency. The path coordinate $s$ increases along the ray. The emission coefficient $j_{\nu }$ is emitted energy per unit time, volume, solid angle and frequency, whereas the absorption coefficient ${\alpha }_{\nu }$ has units of inverse length. Thus $j_{\nu }ds$ adds specific intensity and ${\alpha }_{\nu }{I}_{\nu }ds$ removes it. Stimulated emission can be included as negative absorption.

Define the optical depth between two points by ${\tau }_{\nu }(s_1,s_2)={\int }_{s_1}^{s_2}{\alpha }_{\nu }(u)du$. Multiplying the radiative transfer equation by the integrating factor $\exp [{\tau }_{\nu }(s_0,s)]$ and integrating givesEach local contribution is attenuated only by the material between its emission point and the observer. Where ${\alpha }_{\nu }\ne 0$, the source function is $S_{\nu }=j_{\nu }/{\alpha }_{\nu }$ and the same formal solution of the radiative transfer equation is $I_{\nu }(\tau )=I_{\nu }(0)e^{-\tau }+{\int }_0^{\tau }{S}_{\nu }(t)e^{-(\tau -t)}dt$. A constant source function gives $I_{\nu }=I_{\nu }(0)e^{-\tau }+S_{\nu }(1-e^{-\tau })$.

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 $\Sigma$.

A ring of radius $s$ and width $ds$ has mass $dM=2\pi \Sigma sds$. By axial symmetry its transverse gravitational force components cancel. At signed height $z$, its vertical force on a star of mass $m$ isIntegrating the rings gives, for $z\ne 0$,ThusFor a positive distance above the plane, this is $\mathbf{F}=-2\pi Gm\Sigma [1-z/\sqrt{z^2+R_d^2}]\hat{\mathbf{z}}$, directed toward the disk. With $R_d=16\mathrm{kpc}$, the inferred $\Sigma \simeq 178.3M_{\odot }\mathrm{p}{\mathrm{c}}^{-2}$ sets its normalization.

The far-field expansion gives $F_z\simeq -GMm\mathrm{sgn}(z)/z^2$, as required for a point-like total mass. Close to either face, $F_z\to \mp 2\pi Gm\Sigma$, the infinite-sheet limit. The ideal zero-thickness model has a jump in gravitational field at $z=0$; the formula is understood through its one-sided limits. Its symmetric mean is zero on the plane, and a disk of finite thickness resolves the discontinuity.

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.

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.

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.

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.

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.

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.

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: 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.