The Boundaries of Stardom

Section 6 of 6

Reference and Synthesis

Reference Tables

Mass Limits at a Glance

Quantity	Value	Physical Origin
Minimum H-burning mass	$\sim 0.08\,M_\odot$ ( $\sim 80\,M_\text{Jupiter}$ )	Quantum degeneracy halts contraction
Deuterium-burning limit	$\sim 0.013\,M_\odot$ ( $\sim 13\,M_\text{Jupiter}$ )	Lower Coulomb barrier for D+H
Eddington luminosity	$L_\text{Edd} \approx 3.8 \times 10^4 (M/M_\odot)\,L_\odot$	Radiation force = gravity
Maximum stellar mass	$\sim 100$ – $150\,M_\odot$	Radiation pressure and winds dominant near the Eddington regime
Salpeter IMF slope	$dN/dM \propto M^{-2.35}$	Empirical (origin debated)

Symbol Legend

Symbol	Meaning	CGS Units
$\Delta x$	Position uncertainty	cm
$\Delta p$	Momentum uncertainty	$\text{g}\,\text{cm}/\text{s}$
$\hbar$	Reduced Planck constant ( $h/2\pi$ )	$1.055 \times 10^{-27}~\text{erg}\cdot\text{s}$
$\sigma_T$	Thomson cross-section	$6.65 \times 10^{-25}~\text{cm}^2$
$d$	Interparticle spacing ( $\sim n^{-1/3}$ )	cm
$\lambda_\text{dB}$	de Broglie wavelength	cm

Summary: Gravity’s Playground Has Walls

The most important ideas from this reading:

Quantum mechanics sets the minimum stellar mass — below $\sim 0.08\,M_\odot$ , electron degeneracy halts contraction before the core reaches fusion temperatures. Objects below this limit are brown dwarfs: slowly cooling, never truly shining.
The Heisenberg uncertainty principle ( $\Delta x \cdot \Delta p \geq \hbar/2$ ) means confining particles to small spaces gives them momentum — and therefore pressure. This is the origin of degeneracy pressure, which we’ll explore fully in Reading 3.
Radiation pressure sets the maximum stellar mass — above $\sim 100$ – $150\,M_\odot$ , stars approach the Eddington regime, the simple $L \propto M^{3.5}$ scaling breaks down, and strong winds make further growth difficult.
Both limits are built from fundamental constants — the mass range of stars is not accidental but encoded in $\hbar$ , $c$ , $G$ , and $m_p$ . The universe permits stars only in a narrow sweet spot where quantum mechanics allows fusion and radiation allows stability.

Reading the Limit · Reading 1 of 5 — the Gravity Scoreboard opens

This module is one long contest: gravity against a series of opponents, with each round decided by a wall built from fundamental constants. Reading 1 sets the board.

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	$M_\text{max} \sim 100$ – $150\,M_\odot$	$G, c, \kappa$

Status: gravity is constrained — it can only build stars in a narrow window. Too small and electron degeneracy halts contraction before fusion ignites; too large and radiation drives the star apart through winds. The new weapon introduced here — degeneracy pressure, quantum momentum from confinement that works even at $T = 0$ — has only made a brief appearance. It returns in Reading 3 as gravity’s most stubborn opponent.

We’ve mapped the boundaries of the main sequence. But what happens when a star inside that window runs out of fuel? Fusion has been gravity’s defender for billions of years. When hydrogen is exhausted, the star must find a new one — or gravity wins.

Glossary

Brown dwarf: A substellar object below the hydrogen-burning minimum mass ( $\sim 0.08\,M_\odot \approx 80\,M_\text{Jupiter}$ ) in which electron degeneracy halts contraction before the core reaches sustained hydrogen-fusion temperatures. It glows faintly from gravitational (Kelvin-Helmholtz) contraction and brief deuterium burning, then cools and fades.
Heisenberg uncertainty principle: The quantum law $\Delta x \cdot \Delta p \geq \hbar/2$ : a particle’s position and momentum cannot both be sharply defined. Confining a particle to a region of size $\Delta x$ forces a minimum momentum $p \sim \hbar/\Delta x$ — so compression alone gives particles momentum, and therefore pressure, even at zero temperature.
Initial mass function: The distribution of stellar birth masses, $dN/dM$ . The high-mass end follows the Salpeter power law $dN/dM \propto M^{-2.35}$ ; the full IMF flattens below $\sim 0.5\,M_\odot$ (Kroupa/Chabrier forms). It encodes that low-mass stars vastly outnumber high-mass stars.