The Quantum Limit

Section 5 of 6

The White-Dwarf Mass-Radius Relation

Part 5: The Mass-Radius Relation for White Dwarfs

Plot of white dwarf radius in Earth radii versus mass in solar masses, showing non-relativistic power law R proportional to M^(-1/3) as dashed line and full relativistic curve dropping to zero radius at the Chandrasekhar limit of 1.44 solar masses. Sirius B and a typical white dwarf are marked. — Figure 1White dwarf mass-radius relation. More massive white dwarfs are smaller, the counter-intuitive result of R proportional to M^-1/3 from degeneracy pressure. The full relativistic curve plunges to R = 0 at the Chandrasekhar limit (1.44 solar masses), where electron degeneracy can no longer support the star.ASTR 201 (generated)

How Size Changes with Mass

For non-relativistic white dwarfs (well below the Chandrasekhar limit), balance non-relativistic degeneracy pressure against gravity:

\frac{\hbar^2}{m_e}\left(\frac{M}{m_p R^3}\right)^{5/3} \sim \frac{GM^2}{R^4} \;\Longrightarrow\; R \propto \frac{\hbar^2}{G\,m_e\,m_p^{5/3}}\,M^{-1/3}.

More massive white dwarfs are smaller — the opposite of main-sequence stars ( $R \propto M^{0.8}$ ) and ordinary objects. It is a direct consequence of degeneracy: more mass means stronger gravity, tighter electron confinement, higher density, and a smaller star.

The non-relativistic relation $R \propto M^{-1/3}$ predicts a gentle shrinking with mass. Anchoring to a $0.6\,M_\odot$ white dwarf at $R \approx R_\oplus$ :

$M/M_\odot$	$R/R_\oplus$ (non-rel., $\propto M^{-1/3}$ )
0.2	1.44
0.6	1.00
1.0	0.84
1.2	0.79
1.4	0.75

Read the table carefully: the simple $R \propto M^{-1/3}$ formula has no upper limit built in — taken literally it still returns a finite radius ( $0.75\,R_\oplus$ ) at $1.4\,M_\odot$ . The real white dwarf does something more dramatic. As $M \rightarrow M_\text{Ch}$ the electrons turn relativistic, the pressure softens from $\rho^{5/3}$ toward $\rho^{4/3}$ , and the true radius drops below these non-relativistic values, plunging toward zero at the Chandrasekhar mass. The gentle decline in the table is the low-mass behavior; the relativistic correction — absent from this simple formula — is exactly what turns the soft trend into the hard wall.

So the simple non-relativistic model stops admitting a stable solution at $M_\text{Ch}$ . If the core mass exceeds $1.4\,M_\odot$ , electron degeneracy fails entirely, and something new must take over: neutron degeneracy (or gravity wins completely).

Quick check

If more massive stars make more massive cores, why are more massive white dwarfs smaller instead of larger?