The Chandrasekhar Limit

Equivalent forms

M_{\text{Ch}} = 1.44\,M_\odot \;(\mu_e = 2)

M_{\text{Ch}} \sim \left(\frac{\hbar c}{G}\right)^{3/2}\frac{1}{m_H^2}

Three constants — ħ, c, G — combine into a mass, and that combination is roughly the mass of a star.

Symbol	Name	SI unit	Dimension	Range
$M_Ch$	Chandrasekhar massoutput Maximum mass a white dwarf can support	kg	M	1e+30 – 5e+30
$μ_e$	Mean molecular weight per electron Nucleon mass units per free electron (2 for C/O)	dimensionless	1	1 – 2.5
$ħ$	Reduced Planck constant Sets the scale of quantum degeneracy pressure	J·s	ML^2T^-1	1.05e-34 – 1.06e-34
$G$	Gravitational constant Strength of gravity	m^3·kg^-1·s^-2	M^-1L^3T^-2	6.6e-11 – 6.7e-11

Unit systems

Symbol	SI	CGS	Imperial
$M_Ch$	kg	g	M_sun
$ħ$	J·s	erg·s	J·s
$G$	m^3·kg^-1·s^-2	cm^3·g^-1·s^-2	ft^3·lb^-1·s^-2

Where it holds

Cold, non-rotating, non-magnetic white dwarf supported by ideal electron degeneracy pressure; corrections from rotation, general relativity, inverse beta-decay, and Coulomb effects shift the limit by a few percent.

Dimensional analysis

[M]

([ML^{2}T^{-1}][LT^{-1}] / [M^{-1}L^{3}T^{-2}])^{3/2} \times [M]^{-2} = ([M^{2}])^{3/2}

...

\to [M]

(ħc/G)^{3/2} has units of mass

^{3}

, divided by

m_H^{2}

gives mass.

\checkmark

Discovery

Subrahmanyan Chandrasekhar · 1930

At 19, sailing from India to Cambridge in 1930, Chandrasekhar combined special relativity with electron degeneracy and found white dwarfs have a maximum mass. Eddington publicly ridiculed the result, but it was right — it earned Chandrasekhar the 1983 Nobel Prize and underpins Type Ia supernovae and neutron-star formation.

Try this

Pile more than about 1.4 Suns into a white dwarf and quantum pressure simply gives up — the star collapses.

Electron degeneracy pressure holds a white dwarf up, but for ultra-relativistic electrons it scales the same way as gravity. Above a critical mass nothing can win, and the star detonates as a Type Ia supernova.

Research status: stable

Real-world applications

Type Ia supernovae as cosmological standard candles
Predicting the white-dwarf / neutron-star / black-hole fate of stars
Nucleosynthesis of iron-peak elements in supernovae
Constraining dark energy via the supernova Hubble diagram

Common misconceptions

Any white dwarf above 1.44 M_☉ instantly exists then collapses — they accrete TOWARD the limit; you don't find stable ones above it
Degeneracy pressure is thermal — it is purely quantum (Pauli exclusion), present even at zero temperature
The limit depends on the star's temperature — to leading order it depends only on ħ, c, G, and composition $\mu _e$

Experimental verification

Observed white-dwarf masses cluster below

\sim 1.4\,\mathrm{M}

_☉; Type Ia supernovae — thought to be white dwarfs reaching M_Ch — are standardizable 'standard candles' that revealed cosmic acceleration (1998 Nobel 2011); mass–radius relations from Sirius B and Gaia match degenerate-star models.

Derivation

Balance gravitational energy E_grav

\approx -GM^{2}/R

against the degenerate electrons' energy.

Non-relativistic electrons give kinetic energy

E_kin \propto N^{5/3}/R^{2}

(so the star finds a finite equilibrium radius).

But as M grows, the Fermi momentum makes electrons ultra-relativistic and

E_kin \propto N^{4/3}/R

— the SAME 1/R dependence as gravity.

Then the total energy is (A

N^{4/3} - B M^{2})/R

, which has no minimum: if gravity wins (large M) it drives

R\to 0

.

Setting the coefficients equal gives the maximum number of nucleons and

M_Ch \approx

(ħ

c/G)^{3/2}/(\mu _e m_H)^{2} \times O(1) \approx 1.44 (2/\mu _e)^{2} M

_☉.

Limiting cases

\mu _e = 2

(carbon/oxygen)⟶

M_Ch \approx 1.44\,\mathrm{M}

_☉The standard value for the C/O white dwarfs that produce Type Ia supernovae.

M < M_Ch⟶ Stable white dwarfNon-relativistic electron degeneracy pressure

(P \propto \rho ^{5/3})

rises faster than gravity and halts collapse.

M \to M_Ch

⟶ Radius

\to 0

, collapseElectrons turn ultra-relativistic

(P \propto \rho ^{4/3})

, pressure and gravity scale identically, and equilibrium is lost.

What if…

What if the white dwarf keeps accreting from a companion?

On reaching M_Ch it ignites runaway carbon fusion — a Type Ia supernova that leaves nothing behind.

What if it can't ignite and just collapses?

Electrons get captured onto protons (inverse beta decay) and it becomes a neutron star, held up by neutron degeneracy.

What if G were larger?

$M_Ch \propto G^{-3/2}$ , so stronger gravity would make the maximum white-dwarf mass smaller.

1

Standard C/O dwarf

Given ·

μ e:: 2

Find · M_Ch

Steps

$\mu _e = 2$ for fully ionized carbon/oxygen
$M_Ch = 1.44 (2/\mu _e)^{2} M$ _☉
$= 1.44\,\mathrm{M}$ _☉ $= 2.86\times 10^{30} kg$

Result ·

M_Ch = 1.44\times (2/2)^{2} = 1.44\,\mathrm{M}

_☉

\approx 2.86\times 10^{30} kg

2

Hydrogen composition

Given ·

μ e:: 1

Find · M_Ch

Steps

Pure hydrogen $has \mu _e = 1$ (one electron per proton)
$M_Ch = 1.44\times 4 = 5.76\,\mathrm{M}$ _☉
Heavier composition (more nucleons per electron) lowers the limit

Result ·

M_Ch = 1.44\times (2/1)^{2} = 5.76\,\mathrm{M}

_☉

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