The Chandrasekhar Mass and White-Dwarf Sequences

The Chandrasekhar mass is the mass scale at which an ideal electron-supported white dwarf loses the freedom to increase pressure stiffness by raising its central density. It is not obtained by setting the electron Fermi energy equal to a gravitational energy for one particle. The coefficient comes from a global index-three stellar profile, while composition enters through the number of electrons per baryon. Real sequences end through a competition among electron relativity, nuclear composition, capture reactions, Coulomb physics, rotation, and general relativity.

Required background. The Einstein Equation supplies the relativistic corrections and mass definition used near the endpoint.

Helpful background. Flat-Background Hilbert Stress-Energy from Metric Variation supplies energy and pressure conventions; Relativistic Matter and Compact Objects supplies the EOS setting; Computational Relativity supplies sequence and convergence checks.

Why the ultrarelativistic index is special

Assume cold, fully ionized matter whose mass density is dominated by nuclei and whose pressure is dominated by ideal degenerate electrons. Charge neutrality gives $n_e=\rho/(\mu_e m_u)$, where $\mu_e=A/Z$ for one nuclear species. In the ultrarelativistic limit,

$$p=K_{\rm UR}\rho^{4/3}, \qquad K_{\rm UR}=\frac{(3\pi^2)^{1/3}}{4(\mu_e m_u)^{4/3}}.$$

This is a Newtonian polytrope of index $n_p=3$. The Lane–Emden scalings give

$$M\propto K_{\rm UR}^{3/2}G^{-3/2}, \qquad R\propto \rho_c^{-1/3},$$

so the mass is independent of central density while the radius shrinks as the center is compressed. Solving the dimensionless profile supplies the coefficient,

$$M_{\rm Ch}\simeq\frac{5.83}{\mu_e^2}M_\odot,$$

under the ideal Newtonian assumptions. Chandrasekhar’s original limiting calculation is given in Chandrasekhar 1931, Eqs. (2)–(6), pp. 81–82. His numerical example used a different mass per electron; the commonly quoted composition-dependent coefficient above uses modern constants.

Deriving the coefficient

Combine $dp/dr=-Gm\rho/r^2$ with $dm/dr=4\pi r^2\rho$. For a polytropic EOS $p=K\rho^{1+1/n_p}$, set

$$\rho=\rho_c\theta^{n_p},\qquad r=a\xi,\qquad a^2=\frac{(n_p+1)K}{4\pi G}\rho_c^{1/n_p-1}.$$

Both dimensional equations then reduce to the Lane–Emden problem,

$$\frac{1}{\xi^2}\frac{d}{d\xi} \left(\xi^2\frac{d\theta}{d\xi}\right)=-\theta^{n_p}, \qquad \theta(0)=1,\qquad \theta'(0)=0.$$

Its first zero $\xi_1$ is the surface: $R=a\xi_1$. Integrating the density, or using the equation once, gives

$$M=4\pi a^3\rho_c[-\xi_1^2\theta'(\xi_1)].$$

At $n_p=3$, $a^3\rho_c=(K_{\rm UR}/\pi G)^{3/2}$, so all dependence on $\rho_c$ cancels. The reproducible Python calculation gives $\xi_1\simeq6.896849$ and $-\xi_1^2\theta'(\xi_1)\simeq2.018236$. Restoring $\hbar$ and $c$ in $K_{\rm UR}$ therefore gives

$$M_{\rm Ch} =\frac{\sqrt{3\pi}}{2} [-\xi_1^2\theta'(\xi_1)] \frac{(\hbar c/G)^{3/2}}{(\mu_e m_u)^2} \simeq\frac{5.825}{\mu_e^2}M_\odot.$$

As a dimensional check, $(\hbar c/G)^{3/2}$ has units of mass cubed; dividing by $(\mu_e m_u)^2$ leaves a mass. The rounded $5.83$ coefficient also depends on the dimensionless density profile. For $\mu_e=2$ this ideal model gives $1.456M_\odot$. The linked script records its constants, refines the Lane–Emden grid three times, and checks the same algorithm against the exact index-one solution $\theta=\sin\xi/\xi$. It also reproduces the relativistic polytrope in Mass–Radius Sequences. Run it with Python 3.9 or later; no external packages or tables are needed.

The finite-density white-dwarf sequence

At lower central density the electrons are nonrelativistic and the effective polytropic index is $3/2$. Increasing mass raises the central Fermi momentum and reduces the radius. A complete zero-temperature sequence integrates the hydrostatic equations with the full electron pressure integral rather than switching abruptly between two power laws.

The sequence needs a surface prescription, composition, and a distinction between mass density and total energy density. In the Newtonian regime the radius scales approximately as $R\propto M^{-1/3}$ far below the limiting mass. Near the endpoint the radius falls rapidly while the mass approaches the ideal scale. The limiting configuration itself is not a finite-radius star: it is an asymptotic feature of the ideal sequence.

Hamada and Salpeter include Coulomb and composition effects in representative zero-temperature models Hamada and Salpeter 1961, §§I–III and Tables 1–4. Their results show that two stars with the same electron ratio can still differ because nuclear charge affects lattice and Coulomb energy.

What sets a real endpoint

Several effects reduce or reorganize the ideal picture:

• inverse beta decay becomes favorable when the electron chemical potential reaches a composition-dependent capture threshold;
• Coulomb and lattice terms lower the pressure relative to the ideal gas;
• general relativity strengthens gravity and can create a radial instability before the formal Newtonian limiting state;
• pycnonuclear reactions can change the nuclei and hence the EOS;
• rotation adds support but also introduces mass shedding and secular instability; and
• strong magnetic quantization matters only in a carefully specified field and density regime and cannot be represented by an isotropic EOS at will.

A general-relativistic, composition-aware treatment and its stability boundaries are developed in Rotondo et al. 2011, §§III–IV, Figs. 3–10, and Tables II–III. Numerical values from any sequence must name the nuclear species, temperature approximation, Coulomb prescription, capture thresholds, gravity model, and rotation state.

Checks and common misreadings

The nonrelativistic low-mass and ultrarelativistic $n=3$ limits are independent checks on a sequence. Refining the central-density grid and ODE tolerance must stabilize the apparent endpoint. Restoring dimensions should reproduce $M\propto G^{-3/2}m_u^{-2}$ and the $\mu_e^{-2}$ composition dependence.

The phrase “Chandrasekhar limit” should not hide the model. A quoted value near $1.4M_\odot$ is not a universal constant for hot, rotating, magnetized, or accreting white dwarfs. Nor does exceeding an ideal nonrotating value determine the outcome: nuclear burning, collapse, mass loss, and rotation depend on the evolutionary channel.

Check the composition scaling

Compare the ideal limiting mass for carbon or oxygen, with $\mu_e=2$, to that for iron, with $\mu_e=56/26$. What fraction of the carbon/oxygen value does the iron value have?

Solution

Because $M_{\rm Ch}\propto\mu_e^{-2}$,

$$\frac{M_{\rm Ch}^{\rm Fe}}{M_{\rm Ch}^{\rm CO}} =\left(\frac{2}{56/26}\right)^2 =\left(\frac{13}{14}\right)^2 \simeq0.862.$$

The ideal iron value is about $86\%$ of the carbon/oxygen value. This scaling does not include the different electron-capture thresholds or Coulomb terms, which must be treated separately in a real endpoint calculation.

References

• Chandrasekhar, Subrahmanyan. “The Maximum Mass of Ideal White Dwarfs.” Astrophysical Journal 74 (1931): 81–82. DOI.
• Hamada, T., and Edwin E. Salpeter. “Models for Zero-Temperature Stars.” Astrophysical Journal 134 (1961): 683–698. DOI.
• Rotondo, Michael, Jorge A. Rueda, Remo Ruffini, and She-Sheng Xue. “Relativistic Feynman–Metropolis–Teller Theory for White Dwarfs in General Relativity.” Physical Review D 84 (2011): 084007. DOI.