The Quantum Limit

Section 6 of 6

Reference and Synthesis

Reference Tables

Degeneracy Pressure at a Glance

Quantity	Formula	Notes
Non-relativistic degeneracy pressure	$P \sim \hbar^2 n_e^{5/3}/m_e$	Scales as $\rho^{5/3}$ ; $T$ -independent
Relativistic degeneracy pressure	$P \sim \hbar c\,n_e^{4/3}$	Scales as $\rho^{4/3}$ ; weaker than non-rel
Chandrasekhar mass	$M_\text{Ch} = 1.44\,M_\odot$	From $\hbar$ , $c$ , $G$ , $m_p$ only
WD mass-radius relation	$R \propto M^{-1/3}$	Reversed: more mass = smaller
Electron Fermi energy at $M_\text{Ch}$	$E_F \sim m_e c^2 = 0.511~\text{MeV}$	Transition to relativistic

Three Kinds of Pressure

Pressure type	Source	$T$ dependence	Scaling with $\rho$
Thermal (ideal gas)	Random thermal motions	$P \propto T$	$P \propto \rho$
Radiation	Photon momentum	$P \propto T^4$	Via $T(\rho)$
Electron degeneracy	Pauli exclusion (QM)	None ( $T = 0$ OK)	$P \propto \rho^{5/3}$ or $\rho^{4/3}$

Symbol Legend

Symbol	Meaning	CGS Units
$p_F$	Fermi momentum	$\text{g}\,\text{cm}/\text{s}$
$E_F$	Fermi energy	erg
$n_e$	Electron number density	$\text{cm}^{-3}$
$P_\text{deg}$	Degeneracy pressure	$\text{dyn}/\text{cm}^2$
$M_\text{Ch}$	Chandrasekhar mass	$1.44\,M_\odot = 2.86 \times 10^{33}~\text{g}$
$Y_e$	Electron fraction (electrons per baryon)	$0.5$ for C/O

Summary: Quantum Mechanics vs. Gravity — The Final Score

The most important ideas from this reading:

The Pauli exclusion principle — fermions cannot share quantum states, so compressing them forces them into higher-momentum states, generating degeneracy pressure that operates at zero temperature.
Degeneracy pressure does not depend on temperature — fundamentally different from thermal pressure. This is why white dwarfs can exist without an energy source.
Relativistic effects impose the Chandrasekhar limit — when electrons approach $c$ , the pressure scaling softens from $\rho^{5/3}$ to $\rho^{4/3}$ . Above $M_\text{Ch} \approx 1.4\,M_\odot$ , no equilibrium exists.
The Chandrasekhar mass is built from fundamental constants — $M_\text{Ch} \sim (\hbar c/G)^{3/2}/m_p^2$ . It divides gentle stellar death (white dwarfs) from catastrophic collapse (neutron stars, black holes).

Reading the Limit · Reading 3 of 5 — the Gravity Scoreboard at the crest

This reading is the high point of the module’s contest. Gravity’s most stubborn opponent — electron degeneracy pressure, which holds the line even at $T=0$ — finally meets a wall it cannot cross.

Round	Gravity’s opponent	The wall	Decided by
Floor	quantum degeneracy (before ignition)	$M_\text{min} \sim 0.08\,M_\odot$	$\hbar, G, c, m_p, m_e$
Ceiling	radiation force	$M_\text{max} \sim 100$ – $150\,M_\odot$	$G, c, \kappa$
Dead-star ceiling	electron degeneracy (relativistic)	$M_\text{Ch} \approx 1.4\,M_\odot$	$\hbar, c, G, m_p$

Status: conditional stalemate. Below $1.4\,M_\odot$ degeneracy wins and a white dwarf stands forever; at $1.4\,M_\odot$ the electrons turn relativistic, the pressure can no longer outpace gravity, and gravity wins. The decisive number $M_\text{Ch} = (\hbar c/G)^{3/2}/m_p^2$ is the same natural mass scale $M_0$ that fixed the minimum stellar mass in Reading 1 — the floor and this ceiling are written in one combination of constants.

For stellar cores above $1.4\,M_\odot$ there is no white-dwarf solution. Massive stars do not quietly fade: they build iron cores, collapse catastrophically, and explode as supernovae, scattering the elements they have forged. Reading 4 follows high-mass evolution to its violent conclusion.

Glossary

Chandrasekhar mass: The maximum mass of a stable white dwarf, $M_\text{Ch} \sim (\hbar c/G)^{3/2}/m_p^2 \approx 1.44\,M_\odot$ (for $Y_e = 0.5$ ). It is set purely by fundamental constants: above it, relativistic electron degeneracy pressure can no longer balance gravity at any radius, so the core collapses to a neutron star or black hole.
Degeneracy pressure: The pressure of a degenerate fermion gas, arising from the Pauli exclusion principle rather than from heat: compression forces fermions into high-momentum states, and those fast particles push back even at $T = 0$ . For non-relativistic electrons $P_\text{deg} \sim \hbar^2 n_e^{5/3}/m_e$ ( $\propto \rho^{5/3}$ ); it is what supports white dwarfs.
Fermi energy: The kinetic energy of the highest-occupied state in a degenerate gas, $E_F = p_F^2/(2m_e)$ (non-relativistic). In a typical white dwarf $E_F \sim 0.1$ – $0.2~\text{MeV} \ll m_e c^2$ , so the electrons are non-relativistic; when compression pushes $E_F$ toward $m_e c^2 = 0.511~\text{MeV}$ the gas turns relativistic.
Fermi momentum: The momentum $p_F$ of the highest-occupied quantum state in a degenerate fermion gas. Set by density through confinement, $p_F \sim \hbar n_e^{1/3}$ , it rises as the gas is compressed — and when $p_F \sim m_e c$ the electrons turn relativistic, the route to the Chandrasekhar limit.
Fermion: A particle with half-integer spin ( $1/2, 3/2, \ldots$ ) — electrons, protons, neutrons. Fermions obey the Pauli exclusion principle, so no two identical ones can share a quantum state; this is what makes them resist compression. (Integer-spin particles are bosons, which can pile into the same state.)
Pauli exclusion principle: The quantum rule that no two identical fermions can occupy the same quantum state at once. In dense matter it forces electrons to fill momentum states from the bottom up, so compression pushes them into ever-higher momenta — the origin of degeneracy pressure.