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An ultrarelativistic degenerate electron gas has $P=K{\rho }^{4/3}$, so it forms an index-three stellar polytrope. Insert $P_c=c_{3}G{M}^2/R^4$ and ${\rho }_c=3a_{3}M/(4\pi R^3)$ into that equation of state: the radius cancels, leaving the displayed mass. For fully ionized helium, $n_e=\rho /(2m_p)$ and $K=(3/(2\pi ){)}^{1/3}hc/(16m_p^{4/3})$. This is the ideal limiting Chandrasekhar mass, rather than a claim that every ordinary white dwarf has that mass.

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.

For an isotropic degenerate electron gas, pressure is one third of the momentum flux. In the ultrarelativistic limit each occupied mode has speed $c$, so the supplied momentum-space state count givesA fully ionized helium nucleus provides two electrons for approximately $4m_p$ of mass, so $n_e=\rho /(2m_p)$. Eliminating the Fermi momentum $p_0=(3h^3\rho /(16\pi m_p){)}^{1/3}$ therefore givesThe two-spin-state degeneracy is already included in the supplied state count; no further factor 2 is needed.

If electron degeneracy pressure supplies the star's pressure, this is an index-three stellar polytrope. Its central pressure and density obeyThe radius cancels, leaving $M^{2/3}=K(3a_3/(4\pi ){)}^{4/3}/(c_{3}G)$. Thus the Chandrasekhar mass from central polytropic constants isThe dependence on microscopic constants reflects the balance of fermionic pressure and gravity. The absence of radius dependence is special to the limiting $4/3$ pressure exponent. Ordinary partially relativistic white dwarfs can have a range of masses below this Chandrasekhar limit; the formula does not force every white dwarf to have this mass. As a degenerate stellar core grows toward the limiting mass, compression cannot provide an arbitrarily larger supporting pressure relative to gravity. Above it the ideal cold, nonrotating electron-supported configuration cannot remain in equilibrium; nuclear ignition, collapse or other changes in the support mechanism must intervene. Which outcome occurs depends on the core's composition and evolutionary state.

Electrical neutrality gives $n_e\simeq n_p$ at the accuracy of the proposed estimate. With $V\sim R^3$ and $n_p\sim M/(m_p{R}^3)$, the degenerate electron gas result givesThe gravitational potential energy is $E_{\mathrm{grav}}\sim -G{M}^2/R$. Both scale as $R^{-1}$, so in the strictly ultrarelativistic approximationFor sufficiently large mass the bracket is negative, and contraction lowers the energy without bound in this model: electron degeneracy pressure cannot halt collapse. Equating the scales gives the Chandrasekhar massBelow this scale, the strictly ultrarelativistic expression alone has no finite-radius energy minimum; including the transition to nonrelativistic electrons produces the stable white dwarf branch. Composition and the stellar density profile supply the numerical factors omitted by this scaling estimate.