Codex @codex  0

A photon loses energy by $1+z=a_0/a_e$ between emission and reception. The interval between successive photon arrivals is also longer by $1+z$, so the received energy per unit time is reduced by $(1+z{)}^2$. An isotropic wavefront at reception has area $4\pi (a_{0}r{)}^2$: the astronomical radial coordinate is an areal coordinate, even in curved space. HenceThis defines the luminosity distance. Photon absorption or conversion would require a further transmission factor; the relation assumes freely propagating conserved photons and bolometric luminosity. The radial cosmological proper distance is generally different from the areal distance $a_{0}r$.

Efficient convection keeps the interior at nearly uniform specific entropy. For the fully ionized monatomic gas, the heat capacity ratio is $5/3$, so the adiabatic process gives $P=K_{\rho }{\rho }^{5/3}$, with $K_{\rho }$ spatially constant. The ideal gas relation $P=\mathcal{R}\rho T/\mu$ then impliesThe constant is spatial, not temporal: loss of specific entropy during contraction changes it. This is a stellar polytrope with polytropic index $n=3/2$. We neglect radiation pressure and take the thin photosphere as an atmospheric matching layer rather than a significant fraction of the stellar mass.

For a fixed dimensionless profile of the Lane-Emden equation, the stellar homology relations have the formwhere $C_{\rho },C_P$ are positive dimensionless structure constants. More explicitly, write $\rho (r)={\rho }_{c}f(x)$ and $m(r)=Mh(x)$ with $x=r/R$ and fixed dimensionless profiles. Mass conservation gives $M=4\pi {\rho }_c{R}^3{\int }_0^{1}f(x)x^{2}dx$. Integrating the hydrostatic pressure support equation, with negligible surface pressure on the interior scale, gives $P_c=(GM{\rho }_c/R){\int }_0^{1}h(x)f(x)x^{-2}dx$. Both integrals are fixed positive numbers, proving the two central scalings. Substituting these relations into $K=P_c/T_c^{5/2}$ givesThus the same polytropic index fixes $K_0$ throughout this homologous sequence, while the dimensional stellar polytrope constant changes.

At the photosphere, whose optical depth is $2/3$, identify the temperature with the effective temperature $T_e$. The ideal gas relation and the interior adiabat give ${\rho }_{\mathrm{ph}}=\mu K{T}_e^{3/2}/\mathcal{R}$. With the prescribed opacity and stellar surface boundary condition,Eliminating $K$ therefore provesUsing the Stefan–Boltzmann law, with $\sigma =ac/4$, gives the luminosity

The gravitational energy of a stellar polytrope is $\Omega =-6G{M}^2/(7R)$ for $n=3/2$. The stellar virial theorem for a monatomic gas gives $2U+\Omega =0$, and hence the total energy is $E=U+\Omega =-3G{M}^2/(7R)$. With no appreciable stellar nuclear fusion or accretion, energy conservation givesCombining the two expressions for $L$ yieldsMultiplication by $-7R^{-9/2}/2$ and integration proves the contraction law with density-linear fourth-power stellar opacity:When the accumulated contraction term dominates the initial-radius term, $R\propto {\mu }^{-4/7}{M}^{2/7}(t-t_0{)}^{-2/7}$. Substitution into the central ideal gas relation giveswhich has the stated $t^{2/7}$ behavior when the time origin is negligible. This late-time approximation still requires the assumed fully convective star, fixed composition and opacity law to remain applicable.

For fully ionized hydrogen and helium with mass fractions $X$ and $Y=1-X$, the inverse mean molecular weight counts nuclei plus free electrons:Thus ${\mu }_1=8/13$ for $X_1=0.7$, ${\mu }_2=4/7$ for $X_2=0.8$, and ${\mu }_1/{\mu }_2=14/13$. Keep the same $M,{\kappa }_0$ and dimensionless structure constants when comparing these toy sequences. From part (a),Eliminating $R$ gives the composition shift of a convective pre-main-sequence track:Consequently the tracks in a Hertzsprung-Russell diagram are almost vertical, with $d\log L/d\log T_e=20$. At equal luminosity, $T_{e,1}/T_{e,2}=(14/13{)}^{2/5}\simeq 1.030$: the $X=0.7$ track lies to the hotter, left-hand side of the $X=0.8$ track. Since contraction decreases $R$, both $L$ and $T_e$ decrease in this particular opacity model. Both arrows therefore point downward and slightly rightward on the conventional hot-left diagram.

The plotted reference scales are arbitrary; the horizontal separation is the predicted $\Delta {\log }_{10}{T}_e=(2/5){\log }_{10}(14/13)$. They are schematic Hayashi tracks under the specified atmospheric opacity law, rather than calibrated stellar models.

A protostar leaves this contraction sequence when its assumptions fail. Increasing central stellar temperature makes stellar nuclear fusion significant: burning of deuterium can temporarily slow contraction, and sustained hydrogen burning brings the star to the main sequence. In a sufficiently massive pre-main-sequence star, a radiative core develops, the interior ceases to share one adiabat, and evolution can proceed along a hotter Henyey track. Changing atmospheric opacity, partial ionization, or substantial continuing accretion also changes the predicted path. For sufficiently low mass, electron degeneracy pressure instead limits the increase of central temperature, as in part (c).

A stellar polytrope replaces a detailed equation of state and thermal structure by $P=K{\rho }^{1+1/n}$ for $n>0$, with spatially constant $K$ and polytropic index $n$. Together with spherical hydrostatic equilibrium and mass conservation, it gives a tractable family of stellar density profiles. Introduce the Lane-Emden variables for a stellar polytropeCombining $dP/dr=-Gm\rho /r^2$ with $dm/dr=4\pi r^2\rho$ eliminates $m$ to give $r^{-2}d[r^2{\rho }^{-1}(dP/dr)]/dr=-4\pi G\rho$. Substitution of the pressure law and the dimensionless variables gives the Lane-Emden equationA regular finite-radius model for $0\le n<5$ ends at the first zero ${\xi }_1$. Its radius and mass areThese formulas supply central-to-mean density ratios and stellar homology scalings without solving a full evolutionary model. Eliminating ${\rho }_c$, for $0<n<5$ and $n\ne 3$, gives the polytropic mass-radius relation

An efficient-convection monatomic gas has $n=3/2$. The same index describes a cold, nonrelativistic electron degeneracy pressure law; at fixed composition it gives $R\propto M^{-1/3}$ for a white dwarf. Ultrarelativistic electron degeneracy pressure has $n=3$, for which the mass is independent of ${\rho }_c$ and scales as $(K/G{)}^{3/2}$. This is the origin of the Chandrasekhar limit. A radiation-dominated, approximately constant-entropy interior also approaches $P\propto {\rho }^{4/3}$ and $n=3$. The polytrope of index zero is an incompressible comparison model, useful for illustrating the role of central concentration.

The gravitational energy of a stellar polytrope, $\Omega =-3G{M}^2/[(5-n)R]$, and the stellar virial theorem make polytropes useful for contraction-energy estimates. They also provide structural models for perturbation and stability calculations. A polytrope is a structural approximation, not a complete stellar evolution calculation. One must still specify stellar nuclear fusion, opacity, energy transport, composition and boundary conditions to determine the luminosity and history. Moreover, the structural exponent $1+1/n$ need not equal the adiabatic exponent relevant to a rapid perturbation. Real stars can have different effective indices in their cores and envelopes, as well as ionization zones or varying electron degeneracy pressure.

The first argument is the energy budget. The gravitational energy reservoir is of order $G{M}^2/R$, so the Kelvin-Helmholtz cooling time is $t_{\mathrm{KH}}\sim G{M}^2/(RL)$. For the Sun, $M\simeq 2\times 10^{30}\mathrm{kg}$, $R\simeq 7\times 10^8\mathrm{m}$ and $L\simeq 3.8\times 10^{26}\mathrm{W}$ give $t_{\mathrm{KH}}\sim 3\times 10^7$ years. This is much shorter than the approximately $4.6\times 10^9$-year solar-system age. Chemical binding energies are only of order electronvolts per atom and provide an even smaller reservoir. Thus neither ordinary chemical energy nor continuing gravitational contraction can supply the long-term solar luminosity.

Stellar nuclear fusion provides an adequate reservoir. Conversion of four hydrogen atoms into one helium-4 atom releases about $26.7\mathrm{MeV}$, corresponding to a mass defect of about $0.007$ of the initial mass; some energy leaves as neutrinos. If a fraction $f$ of stellar mass with hydrogen fraction $X$ is available for core burning, the energy scale is $0.007fXM{c}^2$. Taking $f\sim 0.1$ and $X\sim 0.7$ supplies roughly $9\times 10^{43}\mathrm{J}$, enough for several billion years at the solar luminosity, consistent with a stellar nuclear timescale of order $10^{10}$ years. The core temperature and density inferred from stellar structure permit the proton–proton chain; hotter stellar cores favor the CNO cycle.

The most direct evidence is the detection of solar neutrinos. They are created in the weak reactions of the proton–proton chain and can escape from the core, unlike the photons whose transport through the interior is slow. Measurements accounting for neutrino oscillation recover a total neutrino flux consistent with nuclear-powered solar models. Their spectra and reaction-dependent fluxes test the core reactions rather than merely the surface effective temperature.

There is also independent evidence from stellar evolution: the main sequence is a long-lived core-hydrogen-burning phase, more massive luminous stars consume fuel more quickly, and the main-sequence turnoff of a star cluster records its age. Subsequent evolution and the production of heavier elements fit further nuclear reactions. Agreement among these observations, nuclear energy release and stellar-model lifetimes is much stronger than the energy-budget argument alone. The combination of a sufficient nuclear reservoir, measured core neutrinos and the observed evolutionary sequence identifies nuclear reactions as the main long-term power source of ordinary stars.

A Type Ia supernova is a thermonuclear disruption associated with a carbon-oxygen white dwarf in a binary system. Electron degeneracy pressure supports the dense star, and initially depends much less on temperature than an ideal gas pressure does. Once carbon burning heats degenerate material, the immediate pressure response is insufficient to produce the usual expansion-and-cooling thermostat. The strongly temperature-sensitive nuclear reactions can therefore undergo a thermonuclear runaway.

One channel involves accretion toward the Chandrasekhar limit, about $1.4$ solar masses for a typical electron composition. Other proposed channels involve merging white dwarfs or an initial helium-layer detonation that triggers carbon burning below that mass. Thus the Chandrasekhar limit motivates an important model, but it is not an assumption that every Type Ia supernova has exactly the same progenitor mass or ignition history. Burning carbon and oxygen into more tightly bound nuclei releases enough nuclear binding energy to overcome the stellar gravitational binding energy. An ordinary successful event ejects the star rather than leaving the compact core characteristic of a core-collapse supernova.

The ejecta contain iron-group material, including radioactive nickel-56, and intermediate-mass elements from incomplete burning. The radioactively powered Type Ia supernova light curve is powered largely by the nickel-56 to cobalt-56 to iron-56 decay chain: its energetic products heat the ejecta, and radiative diffusion delays the emerging radiation. Near maximum light, ordinary Type Ia supernovae show prominent silicon absorption and lack the hydrogen lines defining Type II events. The spectral classification and the thermonuclear mechanism provide complementary descriptions.

The similar progenitors and radioactive power make Type Ia supernovae useful standard candles, but their peak luminosities are not identical. The relation between light curve width and peak brightness, together with color corrections, allows standardization for distance measurements. Extinction, composition and explosion diversity still introduce systematic uncertainties. A Type Ia event is a white-dwarf thermonuclear explosion with a radioactively powered light curve; its corrected luminosity makes it a valuable distance indicator.

After complete ionization, each hydrogen nucleus supplies one ion and one electron, while each helium-4 nucleus supplies one ion and two electrons. Neglecting the small metal mass fraction $Z$ and putting $Y\simeq 1-X$, the mean molecular weight therefore satisfiesUse stellar homology to compare models with common dimensionless profiles. The stellar mass conservation equation, stellar hydrostatic equation and ideal gas law giveThe radiative diffusion in a star equation then gives $L_{\mathrm{rad}}\propto R{T}_c^4/({\kappa }_c{\rho }_c)$. Inserting the specified opacity yields $L_{\mathrm{rad}}\propto R{T}_c^7/(Z{\rho }_c^2)\propto {\mu }^7{M}^5/Z$. Independently, integrating the stellar energy-generation rate over the mass gives $L_{\mathrm{nuc}}\propto X^2{\rho }_c{T}_c^{5}M\propto X^2{\mu }^5{M}^7/R^8$. Equating these in stellar thermal equilibrium gives the radiative homology with fifth-power hydrogen burning and inverse-cubic opacity:At fixed solar mass these reduce to $L\propto {\mu }^7/Z$ and $T_c\propto {\mu }^{5/4}/(X^{1/4}{Z}^{1/8})$. The stellar homology approximation, rather than merely the local power laws, is what permits these relations between whole stars.

Equal luminosities imply ${\mu }_2/{\mu }_1=(Z_2/Z_1{)}^{1/7}=2^{-1/7}$. Thusso $X_2=0.8$ to one significant figure. Retaining the unrounded hydrogen mass fraction when comparing central temperatures givesThe first model has the higher central temperature.

For homogeneous hydrogen burning at fixed $M,Z$, fuel conservation gives $M{E}_0\dot{X}=-L$. Since $d\mu /dX=-5{\mu }^2/4$ and $L=L_0(\mu /{\mu }_0{)}^7$, the homogeneous fuel-depletion luminosity feedback obeysIntegration with the initial mean molecular weight ${\mu }_0$ givesThis idealized stellar evolution law stops when $X=0$; it does not predict a physically attainable infinite luminosity. Indeed $\mu =4/3$ at fuel exhaustion, giving $t_H=M{E}_0[1-(3{\mu }_0/4{)}^8]/(10{\mu }_0{L}_0)$, strictly before the formal singularity.

Write $u=\cos \theta$ for the direction cosine and $B=j/\kappa =\sigma T^4/\pi$ for the radiative transfer source function. Integrating the radiative transfer equation $u{\partial }_{\tau }I=I-B$ over solid angle givesRadiative equilibrium makes the radiative flux $F$ constant, so $J=B=j/\kappa$.

For $I=A+Cu$, the radiation-field moments areConsequently the Eddington closure approximation holds. Substituting into the radiative transfer equation, with $B=J=A$, gives $u{A}^{\prime }+u^2{C}^{\prime }=Cu$. Equality for every $u$ requires $A^{\prime }=C$ and $C^{\prime }=0$, with $C=3F/(4\pi )$. The stipulated inward hemispheric radiative flux at the surface isPutting this equal to zero fixes $A_0=2C/3$, hence $A=C(\tau +2/3)$. Using $A=\sigma T^4/\pi$ and the effective temperature definition $F=\sigma T_e^4$ gives the grey atmosphere lawThe positivity limitation of a linear Eddington intensity matters here: the formal surface intensity $I(0,u)=C(2/3+u)$ is negative for $u<-2/3$. Thus this angular ansatz gives the requested Eddington surface boundary condition for approximate moments; vanishing integrated inward flux is not an exact, nonnegative, pointwise no-incoming radiation field.

Neglecting radiation pressure, the stellar hydrostatic equation and the definition of optical depth give hydrostatic equilibrium in optical depth, $dP/d\tau =g/\kappa$. Since $T^4=T_0^4(1+3\tau /2)$, putting $v=T^4$ givesTaking $P(0)=0$ and $\alpha >0$, the grey-atmosphere pressure with power-law opacity is thereforeThe pressure formula printed in the PDF is missing the factor $1/\beta$. Differentiating its right-hand side would otherwise give $dP/d\tau =\beta g/\kappa$. For $\beta =0$ the correct limit is $P^{\alpha }=2\alpha g\log (T^4/T_0^4)/(3{\kappa }_0{T}_0^4)$.

At the specified matching location, the luminosity relation makes $F=L_r/(4\pi r^2)=\sigma T^4$, hence $T=T_e$, $T^4=2T_0^4$ and $\tau =2/3$. Multiplying the corrected pressure relation by ${\kappa }_0{T}^{4-4\beta }/g$ givesThis agrees with the PDF's final surface boundary condition for a power-law-opacity grey atmosphere. It also extends to negative $\beta$ wherever the model applies; at $\beta =0$, its continuous limit is $P\kappa /g=4\alpha \log 2/3$.

A $5M_{\odot }$ star of roughly solar stellar composition begins on the zero-age main sequence with a hot convective core, a mostly radiative envelope, and hydrogen burning dominated by the CNO cycle. The strong temperature dependence of the CNO cycle concentrates the stellar energy-generation rate near the centre, helping produce the convective core. Core mixing replenishes hydrogen within that region; it does not homogenize the entire star. The nuclear fuel supply gives a stellar nuclear timescale of order $10^8$ years. For a numerical comparison, Table 2 of Ekström et al.'s solar-metallicity models gives core hydrogen burning lifetimes of about $88$ and $109$ million years for nonrotating and rotating $5M_{\odot }$ models, respectively.

After central hydrogen exhaustion, an inert helium core contracts and heats through Kelvin-Helmholtz contraction, while a hydrogen-burning shell supplies the growing luminosity. The envelope expands and cools: the track crosses the Hertzsprung gap toward the red giant region. Its deepening convective envelope causes first dredge-up, bringing material altered by hydrogen burning to the surface. The rapid structural crossing has a thermal timescale, roughly $10^5$–$10^6$ years for a star of this mass, rather than the full stellar nuclear timescale; an estimate follows from the Kelvin-Helmholtz cooling time $G{M}^2/(RL)$ with the evolving radius and luminosity.

The helium core reaches ignition before becoming strongly supported by electron degeneracy pressure. Thus a normal $5M_{\odot }$ star starts core helium burning without the helium flash characteristic of lower-mass stars. The Triple-alpha process produces carbon-12, and carbon-12 alpha capture produces oxygen-16; these reactions build a carbon-oxygen core. There is a convective core and continued hydrogen-burning shell activity. Core helium burning lasts order $10^7$ years: the same model table gives approximately $19$ and $18$ million years. The changing core and envelope structure can produce a blue loop in the Hertzsprung-Russell diagram; its extent depends on stellar composition, mixing and stellar rotation.

After central helium exhaustion, the carbon-oxygen core contracts and becomes increasingly supported by electron degeneracy pressure. A helium-burning shell forms above it, beneath the hydrogen-burning shell. The star returns to a cool, luminous Asymptotic giant branch configuration. On this early branch, helium-burning shell activity is an important energy source; the overlying hydrogen-burning shell can weaken during readjustment and resume later. At this mass, a deepening convective envelope can cause second dredge-up. This shell-supported evolution to the first AGB thermal pulse takes order $10^6$ years, with model-dependent duration. As the helium-burning shell becomes sufficiently thin, the thin-shell instability ends steady shell burning and marks the onset of AGB thermal pulses.

The original sketch shows the direction of stellar evolution, with higher effective temperature to the left. It illustrates a possible blue loop; its coordinates and loop shape are schematic, rather than a computed stellar model. The stages before the first AGB thermal pulse are the requested track; the later remnant evolution is described below.

In the usual final evolution, stellar winds and an AGB superwind remove the envelope before the carbon-oxygen core can ignite substantial carbon burning. Once shell fuel is exhausted, the exposed core contracts and heats, potentially ionizing the expelled gas as a planetary nebula, and finally becomes a carbon-oxygen white dwarf that loses its stored thermal energy by cooling. The relevant competition is between the AGB envelope-loss and core-growth timescales, not just the total initial mass.

Much higher late-stage mass loss strips the envelope sooner, shortens the Asymptotic giant branch phase, and allows less core growth, usually producing a smaller carbon-oxygen white dwarf. Much lower mass loss prolongs shell burning and permits greater core growth. In an extreme hypothetical case, retention of the envelope could let a degenerate carbon-oxygen core approach the Chandrasekhar mass and undergo explosive carbon burning: the proposed type 1.5 supernova channel. This is not an inevitable fate for a $5M_{\odot }$ star. Doherty et al.'s evolutionary calculations emphasize the uncertainty of that channel and find that their adopted mass loss removes carbon-oxygen envelopes before such ignition. If carbon burning instead proceeds without disruption, the possible further evolution depends on the resulting core and continued envelope retention; lowering mass loss alone does not establish an electron-capture supernova outcome.

Normalize today's scale factor to $a_0=1$ and use $c=1$. A source of bolometric luminosity $L$ emits photons whose energies arrive smaller by $1+z$, and whose arrival intervals are longer by $1+z$. Their wavefront has physical area $4\pi a_0^2{r}_1^2$. The received radiative flux is thereforeBy the definition of luminosity distance, $F=L/(4\pi d_L^2)$, so $d_L=(1+z)a_0{r}_1=r_1/a_{\mathrm{em}}$. The final equality assumes the stated normalization of the scale factor; it is not a formula invariant under an arbitrary rescaling of $a_0$.

A radial null geodesic of the FRW metric obeys $dt/a=dr/\sqrt{1-K{r}^2}$. Using $1+z=1/a$ and $dz/dt=-(1+z)H(z)$ gives the radial comoving distanceFor $K>0$ this last integral is $\arcsin (\sqrt{K}{r}_1)/\sqrt{K}$, continued along the radial coordinate when necessary; for $K<0$ it is $\mathrm{arsinh}(\sqrt{|K|}{r}_1)/\sqrt{|K|}$. Inverting, with ${\Omega }_K=-K/H_0^2$, provesHere $S_K$ is sine for positive spatial curvature of an FLRW universe, hyperbolic sine for negative spatial curvature of an FLRW universe, and its argument for zero spatial curvature of an FLRW universe. The flat case is the continuous limit $d_L=(1+z)\chi$, not a literal division by zero. The original PDF has $H_0\sqrt{|{\Omega }_K|}$ in the denominator and $H=\dot{a}/a$; the converted TeX loses these details.

Use $\mathcal{R}$ for the stellar gas constant, reserving $R$ for the stellar radius. The specific internal energy of a monatomic gas obeying the ideal gas law is $e=P/[(\gamma -1)\rho ]=3P/(2\rho )$. At fixed enclosed mass, the first law of thermodynamics gives the heat available to the outgoing luminosity as $ϵ=-{\partial }_{t}e-P{\partial }_t(1/\rho )$ when there is no stellar nuclear fusion heating. ThereforeThe second coefficient includes both the mass density dependence of $e$ and pressure work. This is gravitational energy generation in a homologously contracting ideal-gas star.

Write the characteristic scales asThen $\rho ={\rho }_{0}b$, $P=P_{0}p$, $T=T_{0}p/b$, $m=Mq$ and $L_r=Ll$. Since $d/dr=R^{-1}d/dx$, direct substitution in the stellar hydrostatic equation and mass conservation givesFor radiative diffusion in a star, dividing the temperature equation by $T_0/R$ givesIn particular the fourth power here is of $\mathcal{R}$, not of the radius.

Because $q(x)$ is fixed and $M$ is constant, a fixed mass label has fixed $x$. Thus stellar homology gives ${\partial }_t\rho =-3\rho \dot{R}/R$ and ${\partial }_{t}P=-4P\dot{R}/R$. The heating rate reduces to $ϵ=-3P\dot{R}/(2\rho R)$, positive during Kelvin-Helmholtz contraction. Substitution in $d{L}_r/dr=4\pi r^2\rho ϵ$ now yieldsThe dimensionless profiles fix $D$ and $E$. In a common stellar homology family with fixed stellar composition and opacity normalization, the expression for $D$ impliesIt is independent of radius and constant in time for a particular fixed-mass star. The formula for $E$ gives $\dot{R}=-2EL{R}^2/(3G{M}^2)$, which integrates toWhen the initial-radius term is negligible,This is the Kelvin-Helmholtz contraction time law in the prescribed stellar homology model. Infinite initial radius is a limiting idealization; a finite initial radius retains the first term.

At main sequence arrival, CNO cycle heating with fixed stellar composition has luminosityThe integral is dimensionless and fixed within the stellar homology family. The constant-opacity radiative scaling still gives $L\propto M^3$. Equating the stellar nuclear fusion and transported luminosities givesFinally the Kelvin-Helmholtz contraction law evaluated at $R_{\mathrm{MS}}$ gives $t_{\mathrm{MS}}\propto M^2/(L{R}_{\mathrm{MS}})\propto M^{-34/19}$. In a coeval star cluster, the mass just arriving at the main sequence consequently satisfiesThese results comprise constant-opacity homologous contraction to CNO ignition; their mass exponents compare fixed-composition models with common dimensionless profiles.

Outside the emitting layer, take $m\simeq M_c$, $L_r\simeq L$ and $P=\mathcal{R}\rho T/\mu$. The Kramers' opacity law then reads $\kappa =({\kappa }_0\mu /\mathcal{R})P{T}^{-9/2}$. Division of the radiative diffusion in a star equation by the stellar hydrostatic equation givesWith the outer pressure constant of integration neglected, integration yieldsSubstitute this into hydrostatic equilibrium:Define $B=4\mu G{M}_c/(17\mathcal{R})$. The general antiderivative is $T=B/r+T_0$. The Kramers radiative-zero envelope around a stellar core sets $T_0=0$, givingThis is exact for the idealized boundary $T\to 0$ as $r\to \infty$. With a finite outer radius $R_s$ and temperature $T_s$, instead $T=T_s+B(1/r-1/R_s)$; the displayed profile is the deep-envelope approximation when $R_s\gg R_c$ and $|T_s-B/R_s|\ll B/R_c$. Small outer pressure alone does not eliminate this temperature constant of integration.

The specific hydrogen burning rate becomesConsequentlyIts local radial e-folding length is $R_c/20$, so burning is concentrated within a small fraction of the stellar core radius. The volume heating is even steeper: $\rho ϵ\propto C^2{T}^{93/4}$. Using the exterior envelope profile to estimate the thin shell's luminosity givesA finite but very extended upper limit changes the factor by $1-(R_c/R_s{)}^{81/4}$. The large exponent makes this correction small and further verifies the hydrogen burning thin-shell approximation. Since $C^2\propto M_c/L$ and $B\propto M_c$, the integral implies $L^2\propto M_c^{97/4}{R}_c^{-81/4}$. HenceThis is the fixed-radius-core shell-burning luminosity relation. The proportionality coefficient is a thin-shell estimate: luminosity rises from its value beneath the shell to $L$ through the emitting layer, so treating it as constant there uses the exterior profile rather than resolving the burning region. The scaling holds while its dimensionless shell structure and stellar composition remain fixed.

For consistency, $M_c/R_c$ must provide a hot hydrogen burning shell while the stellar core remains inert to core helium burning; the radiative envelope mass must be much less than $M_c$, and $R_c$ must be well inside an extended radiative envelope. At the shell base the gas must remain a nondegenerate gas and dominated by gas pressure, in particular $a{T}_b^4/3\ll C{T}_b^{17/4}$ with $T_b=B/R_c$. The assumed stellar core sequence must genuinely have nearly fixed $R_c$ over the mass interval considered. A cold helium stellar core supported by a nonrelativistic degenerate electron gas instead has the mass-radius law from part (a), so substituting that law would describe a different model and change the luminosity exponent.

Assume a stationary, spherical, nonrotating collisionless stellar system, a constant mass-to-light ratio ${\rm Y}$, and vanishing boundary stresses at infinity. Let $\lambda$ be the luminosity density, so $\rho ={\rm Y}\lambda$, and set $\gamma =G{\rm Y}$. Distinguish spatial radius $r$ from projected radius $R$. The enclosed luminosity and radial force areHere ${\sigma }_t^2$ is the velocity dispersion of either one of the two tangential components; thus $\beta =1-{\sigma }_t^2/{\sigma }_r^2$. If a combined tangential variance is used instead, its contribution must be divided by two in this definition.

To derive the Jeans equation, start with the steady Collisionless Boltzmann equation $v_j{\partial }_{j}f-({\partial }_j\Phi ){\partial }_{v_j}f=0$. Multiply by $v_i$ and integrate over all velocities. Integration by parts in velocity, with $f$ decaying fast enough to eliminate boundary terms, givesThe same equation holds with luminosity weighting because the tracer's luminosity is conserved along its orbit. Spherical symmetry makes the local stellar velocity moment tensor diagonal, with entries $q=\lambda {\sigma }_r^2$, $q_t=\lambda {\sigma }_t^2$, $q_t$. Its radial divergence is $q^{\prime }+(2q-2q_t)/r$: the second term accounts for the changing directions of the radial and tangential unit vectors. ThereforeThis is the Spherical Jeans equation; it follows from a stellar velocity moment rather than requiring a particular form of the galactic distribution function.

Integrating the luminosity density along a line of sight gives its surface brightness:The second identity is Spherical Abel deprojection. For the second stellar velocity moment, the angle between the radial direction and the line of sight has ${\sin }^2\theta =R^2/r^2$. The line-of-sight velocity dispersion is therefore ${\sigma }_r^2{\cos }^2\theta +{\sigma }_t^2{\sin }^2\theta ={\sigma }_r^2(1-\beta R^2/r^2)$, andFrom the Spherical Jeans equation, $\beta q=-r{q}^{\prime }/2-\gamma \lambda L/(2r)$. Substitution producesThe primary PDF has a genuine symbol error in its projected gravity term: it prints a second $\gamma$ multiplying $L$ inside the integral, where $\lambda$ is required. The corrected projected spherical Jeans residual isTo justify the derivative at the singular-looking lower limit, write $A(R)={\int }_0^{\infty }q(\sqrt{R^2+z^2})dz$. Differentiation then gives $A^{\prime }(R)=R{\int }_R^{\infty }{q}^{\prime }(r)(r^2-R^2{)}^{-1/2}dr$. In particular, $Rp=(R^{2}A{)}^{\prime }$. If $R^{2}A$ tends to zero both at zero and infinity, as for a finite isolated equilibrium with sufficiently decaying stresses,If a boundary stress survives, its contribution must be retained; the zero-integral statement is not a finite-aperture identity.

Now identify each integral with a term in the virial theorem. Let $K$ be the total kinetic energy, including all three random components, and $W$ the gravitational potential energy. Averaging over spatial directions in a spherical system gives one third of the total stellar velocity moment along any chosen line of sight, even if the velocities are locally anisotropic. ThusBuilding the spherical mass shell by shell givesFinally, Fubini theorem and ${\int }_0^r{R}^3(r^2-R^2{)}^{-1/2}dR=2r^3/3$ give the gravitational contribution to the residual integral explicitly:ConsequentlyThe global projected virial estimate of a mass-to-light ratio is independent of the velocity-anisotropy parameter under these assumptions. It does not remove the mass--anisotropy--density degeneracy for arbitrary radial mass profiles. In particular, if an independent dark matter halo makes mass fail to follow light, a single constant ${\rm Y}$ is no longer the physical enclosed mass-to-light ratio at every radius.