The Chandrasekhar Mass

The Chandrasekhar Mass

When a star exhausts its nuclear fuel, it may leave behind a white dwarf — a dense remnant supported not by fusion but by electron degeneracy pressure (a quantum mechanical effect from the Pauli exclusion principle).

Maximum Mass

There is a fundamental upper limit on white dwarf mass, the Chandrasekhar mass:

$M_{\rm Ch} = \frac{5.87}{\mu_e^2}\ M_\odot$

where $\mu_e$ is the mean molecular weight per electron:

• $\mu_e = 2$ for carbon/oxygen white dwarfs → $M_{\rm Ch} \approx 1.47\ M_\odot$
• $\mu_e = 1$ for hydrogen → $M_{\rm Ch} = 5.87\ M_\odot$

Above this mass, degeneracy pressure cannot halt gravitational collapse and the star implodes — triggering a Type Ia supernova.

White Dwarf Mass–Radius Relation

Unlike normal stars, white dwarfs shrink as mass increases (non-relativistic approximation):

$R_{\rm WD} \approx R_0 \left(\frac{M}{M_{\rm Ch}}\right)^{-1/3}$

where $R_0 \approx 0.01\ R_\odot$ is a characteristic radius for typical white dwarfs.

Gravitational Binding Energy

The gravitational binding energy of a uniform sphere:

$E_{\rm bind} = -\frac{3GM^2}{5R}$

For a 0.6 $M_\odot$ white dwarf, this is a few $\times 10^{42}$ J.

Your Task

Implement three functions. Use $G = 6.674 \times 10^{-11}$ m³ kg⁻¹ s⁻², defined inside each function.

• chandrasekhar_mass_solar(mu_e) — returns $M_{\rm Ch}$ in solar masses
• wd_radius_solar(M_solar, mu_e) — returns white dwarf radius in solar radii
• gravitational_binding_energy_J(M_kg, R_m) — returns $E_{\rm bind}$ in joules (negative)