The Boundaries of Stardom

Section 3 of 6

The Minimum Stellar Mass

Part 3: The Minimum Stellar Mass

The Physical Argument

For a collapsing gas cloud to become a hydrogen-burning star, its core must reach $T_c \sim 3 \times 10^6~\text{K}$ — the minimum temperature for pp-chain fusion to sustain energy losses. (This is lower than the Sun’s $15~\text{MK}$ because fusion rates have a steep temperature dependence — even a slow trickle of fusion at $3~\text{MK}$ can sustain a very low-luminosity star.)

As the protostar contracts, the virial theorem tells us the core heats up: $T_c \propto M/R$ . But contraction also increases the density, and eventually the electrons become degenerate. Once that happens, the gas behaves differently:

Pressure no longer depends on temperature. Degeneracy pressure is set by density, not $T$ . So adding heat doesn’t increase pressure — the star can’t expand in response to heating.
Contraction halts. Degeneracy pressure balances gravity at a specific radius, regardless of temperature.
The core may never get hot enough. If degeneracy kicks in before the core reaches $\sim 3 \times 10^6~\text{K}$ , the star is stuck — it has a cold, dense, quantum-pressure-supported core that will never achieve sustained fusion.

The Critical Mass

The hydrogen-burning minimum mass (HBMM) depends on when degeneracy sets in relative to the fusion ignition temperature. Detailed calculations give:

Objects below this mass are brown dwarfs — failed stars that glow faintly from residual gravitational contraction energy (Kelvin-Helmholtz) and possibly brief deuterium burning, but never achieve sustained hydrogen fusion.

Why 0.08 Solar Masses? — Reading the Limit

We now have the ingredients to derive the floor, not just assert it. This is the first appearance of the move that runs through all of Module 4 — Reading the Limit.

① Write the balance. A contracting protostar carries two energies per particle, and as it shrinks at fixed mass $M$ they pull the temperature in opposite directions. Virial heating supplies a thermal energy

k_B T \sim \frac{GM m_p}{R} \propto \frac{1}{R},

while the electrons carry a quantum (Fermi) energy set by how tightly they are confined,

E_F \sim \frac{\hbar^2}{m_e}\,n_e^{2/3} \sim \frac{\hbar^2}{m_e}\left(\frac{M}{m_p R^3}\right)^{2/3} \propto \frac{1}{R^2}.

As the protostar contracts, both energies rise — but the quantum energy rises faster ( $R^{-2}$ beats $R^{-1}$ ). So the core temperature climbs, peaks, and then falls as degeneracy takes over. The hottest the core ever gets is the moment the two energies meet, $k_B T \sim E_F$ .

② Solve for the critical scale. Set them equal; the radius drops out of the temperature:

\frac{GM m_p}{R} \sim \frac{\hbar^2}{m_e}\left(\frac{M}{m_p R^3}\right)^{2/3} \;\Longrightarrow\; k_B T_\text{max} \sim \frac{G^2 m_e\, m_p^{8/3}}{\hbar^2}\,M^{4/3}.

This is the key result: the maximum core temperature a star can ever reach scales as $T_\text{max} \propto M^{4/3}$ . Halve the mass and the peak temperature drops by $2^{4/3} \approx 2.5$ . Light enough objects never get hot enough — they reach their peak temperature below the fusion threshold and then cool forever. The minimum mass is the one whose peak just touches the ignition temperature, $T_\text{max} = T_\text{ign}$ .

③ Read off the constants. Setting $T_\text{max} = T_\text{ign}$ and solving for the mass,

M_\text{min} \sim \underbrace{\left(\frac{\hbar c}{G}\right)^{3/2}\frac{1}{m_p^2}}_{\textstyle M_0\,\approx\,1.8\,M_\odot}\;\left(\frac{k_B T_\text{ign}}{m_e c^2}\right)^{3/4}.

The floor is built from $\hbar$ (quantum mechanics), $G$ (gravity), $c$ (relativity), and the particle masses $m_p, m_e$ . The leading combination $M_0 = (\hbar c/G)^{3/2}/m_p^2 \approx 1.8\,M_\odot$ is a natural stellar mass scale — and you will meet it again, almost unchanged, as the Chandrasekhar mass in Reading 3. The dimensionless factor $(k_B T_\text{ign}/m_e c^2)^{3/4}$ — the ignition temperature measured against the electron rest energy — is only a few $\times 10^{-3}$ , which is why the floor sits far below that natural scale, in the brown-dwarf range. Taken literally, this stripped-down scaling lands near $\sim 10^{-2}\,M_\odot$ (we dropped numerical prefactors — the $(3\pi^2)^{2/3}$ in the Fermi energy, the electron fraction $\mu_e$ , the precise ignition criterion); restoring them lifts the result to the observed $\sim 0.08\,M_\odot$ . The scaling delivers the origin and the order of magnitude — which constants, and a small fraction of a solar mass — and detailed models supply the exact coefficient.

Name the assumptions — the audit

Every limit we “read” rests on approximations. For the minimum mass:

Assumption	Why we made it	When it breaks
$k_BT \sim GMm_p/R$ (one virial energy per particle)	turns gravity into a temperature	rotation/magnetic support; core not strictly uniform
Non-relativistic $E_F \sim \hbar^2 n_e^{2/3}/m_e$	electrons are slow at ignition densities	fails for the most massive degenerate cores (Reading 3)
Peak temperature at $k_BT \sim E_F$	marks where contraction stalls	a smooth crossover, not a sharp switch
$T_\text{ign}$ taken as given	hides the fusion microphysics	$T_\text{ign}$ itself depends on composition and the Coulomb barrier

None of these change the scaling $M_\text{min} \propto (\hbar c/G)^{3/2}/m_p^2$ — they live only in the O(1) coefficient.

The minimum stellar mass is not a coincidence of astrophysics — it is built into the laws of physics. At the low-mass end, quantum mechanics stops gravity from finishing the job. At the high-mass end, gravity succeeds in making the star extremely luminous — but that luminosity creates a new opponent: radiation force.

Observable

The faintest main-sequence stars: type ~L0, ~2,000 K, ~1e-4 Lsun

Below this boundary, objects cool and fade over time instead of settling onto a stable hydrogen-burning main sequence.

Model

Virial heating versus electron degeneracy

Virial heating gives the rough core-temperature trend, while quantum mechanics says sufficiently dense electrons become degenerate and generate pressure even without thermal support.

Inference

The star / brown-dwarf boundary near ~0.08 solar masses

This marks the mass where electron degeneracy halts contraction before sustained hydrogen fusion can take over.

Quick check

Brown dwarfs with masses $0.013$ – $0.08\,M_\odot$ briefly burn deuterium ( $^2\text{H} + {^1\text{H}} \rightarrow {^3\text{He}} + \gamma$ ) but not hydrogen. Why is the deuterium-burning threshold lower than the hydrogen-burning threshold?

The More You Know: Enrichment: Brown Dwarfs — The 'Failed Stars'

Brown dwarfs occupy a fascinating middle ground between stars and giant planets:

Property	Brown Dwarf ( $0.05\,M_\odot$ )	Low-mass Star ( $0.1\,M_\odot$ )	Jupiter
Core H fusion?	No	Yes	No
D burning?	Yes (briefly)	Yes (early)	No
Energy source	Gravitational contraction	Nuclear fusion	Gravitational contraction
Luminosity	$\sim 10^{-5}\,L_\odot$ (fading)	$\sim 10^{-3}\,L_\odot$ (stable)	$\sim 10^{-9}\,L_\odot$
$T_\text{eff}$	$\sim 1{,}000~\text{K}$ (cooling)	$\sim 3{,}000~\text{K}$ (stable)	$\sim 130~\text{K}$
Fate	Cools forever	Burns for trillions of yr	Cools forever

Brown dwarfs were predicted theoretically in the 1960s but not discovered until 1995 (Gliese 229B, detected in the infrared). They are extremely common — possibly as numerous as stars — but very difficult to detect because they are so faint. Modern infrared surveys (e.g., the James Webb Space Telescope) have revealed that brown dwarfs bridge the gap between the smallest stars and the largest planets, with some having atmospheres with clouds of iron and silicate droplets.