In 1930, on a ship from India to England, the 19-year-old Subrahmanyan Chandrasekhar added special relativity to the theory of white-dwarf sizes and found that there is a largest possible white dwarf. The result earned him a share of the 1983 Nobel Prize in Physics.
💡 Why it works
As a white dwarf gains mass and shrinks, its electrons are squeezed to higher momenta. Once $p_F \gtrsim m_ec$, the fastest electrons move at nearly $c$ instead of $p/m_e$. The pressure, roughly $n_e \times p_F \times v$, then grows as $n_e \times n_e^{1/3} \times c$, so $P \propto \rho^{4/3}$ instead of $\rho^{5/3}$. Compare with gravity: $P \propto (M/R^3)^{4/3} = M^{4/3}/R^4$, while gravity needs $M^2/R^4$. Both go as $1/R^4$, so shrinking no longer helps. Only one mass balances; above it, gravity wins at every radius and the star collapses. Dimensional analysis gives that mass from the constants alone: $$M \sim \left(\frac{\hbar c}{G}\right)^{3/2}\frac{1}{(\mu_em_p)^2} \approx 0.5\,M_\odot \text{ for } \mu_e = 2,$$ and the full calculation multiplies it by about 3.1.
🔑 Key idea
The Chandrasekhar limit is $M_{\text{Ch}} \approx 5.83\,M_\odot/\mu_e^2$, about $1.4\,M_\odot$ for carbon, oxygen or helium ($\mu_e = 2$). No white dwarf, and no stellar core supported by electron degeneracy, can be heavier.
Worked examples
✏️ Worked example: The dimensional estimate
Evaluate $(\hbar c/G)^{3/2}/(\mu_em_p)^2$ for $\mu_e = 2$ ($\hbar c = 3.16 \times 10^{-26}$ J·m).
- $\hbar c/G = 3.16 \times 10^{-26}/6.67 \times 10^{-11} = 4.74 \times 10^{-16}$ kg², and its $3/2$ power is $1.03 \times 10^{-23}$ kg³.
- $(2m_p)^2 = 1.12 \times 10^{-53}$ kg², so the ratio is $9.2 \times 10^{29}$ kg.
Answer: About $0.46\,M_\odot$; times 3.1 gives about $1.4\,M_\odot$.
✏️ Worked example: When do the electrons become relativistic?
At what density does $p_F = m_ec$ for $\mu_e = 2$?
- $n_e = \dfrac{1}{3\pi^2}\left(\dfrac{m_ec}{\hbar}\right)^3 = \dfrac{(2.59 \times 10^{12})^3}{29.6} = 5.9 \times 10^{35}$ m⁻³.
- $\rho = 2m_pn_e$.
Answer: About $2 \times 10^{9}$ kg/m³, a few times the mean density of a typical white dwarf and about its central density. In heavier white dwarfs, the central electrons are strongly relativistic.
✏️ Worked example: An iron core
What is the limit for a core of iron-56 ($Z = 26$)?
- $\mu_e = 56/26 = 2.15$.
- $M_{\text{Ch}} = 5.83/2.15^2$.
Answer: About $1.26\,M_\odot$ in this simple model: iron has fewer electrons per nucleon, so it can support less.
⚠️ Common mistake
Applying the limit to the whole star. A star of 6 solar masses can still end as a white dwarf: it sheds most of its mass in winds and a planetary nebula, and only its core, below the limit, remains. The limit concerns the mass supported by degenerate electrons: a white dwarf, or the core inside a massive star. When a massive star's iron core grows past it, the core collapses (next topic). A white dwarf in a binary that gains matter until it nears the limit can explode as a type Ia supernova.
The 5.83 comes from solving the structure equations exactly for fully relativistic electrons. Detailed models with the electrons' finite mass and other small effects give about $1.4\,M_\odot$ for carbon–oxygen white dwarfs, and no white dwarf heavier than that has been found.