The Boundaries of Stardom

Show explicit units, and run a sanity check on every result. A scaling answer is judged by its exponents, not its coefficient. Worked solutions are released after the homework due date.

Useful constants (CGS):

Constant	Value
$G$	$6.67 \times 10^{-8}~\mathrm{cm^3\,g^{-1}\,s^{-2}}$
$c$	$3.0 \times 10^{10}~\mathrm{cm\,s^{-1}}$
$\hbar$	$1.055 \times 10^{-27}~\mathrm{erg\,s}$
$k_B$	$1.38 \times 10^{-16}~\mathrm{erg\,K^{-1}}$
$m_e$	$9.11 \times 10^{-28}~\mathrm{g}$
$m_p$	$1.67 \times 10^{-24}~\mathrm{g}$
$m_e c^2$	$8.2 \times 10^{-7}~\mathrm{erg} = 0.511~\mathrm{MeV}$
$1~\mathrm{keV}$	$1.6 \times 10^{-9}~\mathrm{erg}$
$M_\odot$	$1.99 \times 10^{33}~\mathrm{g}$
$L_\odot$	$3.8 \times 10^{33}~\mathrm{erg\,s^{-1}}$
$\kappa$ (electron scattering)	$0.34~\mathrm{cm^2\,g^{-1}}$
Natural mass scale $M_0 = (\hbar c/G)^{3/2}/m_p^2$	$\approx 1.8\,M_\odot$
Ignition temperature $T_\text{ign}$	$\sim 3 \times 10^6~\mathrm{K}$

Useful scalings from the reading: $T_\text{max} \propto M^{4/3}$ (peak core temperature reached during contraction); $L_\text{Edd} = 4\pi GMc/\kappa$ ; main-sequence $L \propto M^{3.5}$ ; Salpeter IMF $dN/dM \propto M^{-2.35}$ .

Conceptual

Problem

⭐⭐ More mass, hotter peak? A student reasons: “A more massive protostar has more material to heat with the same gravity, so its core reaches a lower peak temperature than a low-mass one. That’s why only low-mass objects can ignite.”

(a) Identify the error, using the result $T_\text{max} \propto M^{4/3}$ .
(b) State the corrected physical picture in one sentence: which way does the peak temperature go with mass, and why?
(c) Explain how this — not the student’s claim — is what creates a minimum mass rather than a maximum one.

Problem

⭐⭐ What really sets the ceiling? A student claims: “The maximum stellar mass is the mass at which gravity finally overwhelms fusion, so the star can’t hold itself up.”

(a) Identify what is physically wrong with attributing the upper limit to a failure of fusion.
(b) Name the force that actually competes with gravity at the top end, and write the balance it comes from.
(c) The lower limit is about fusion failing to ignite, while the upper limit is not about fusion at all. Contrast the two opponents in one sentence each.

Problem

⭐ Reading the boundary as O→M→I. The main sequence ends near $0.08\,M_\odot$ ; below it sit the brown dwarfs.

(a) State the observable that marks the boundary (what do the faintest true stars look like, versus what lies just below?).
(b) State the model — the two competing pieces of physics whose race decides the boundary.
(c) State the inference — what the boundary tells us about nature that a mere catalog of masses would not.

Calculation

Problem

⭐⭐ Does it ignite? Take the floor to be the mass whose peak core temperature just reaches ignition: a $0.08\,M_\odot$ object reaches $T_\text{max} = T_\text{ign} = 3 \times 10^6~\mathrm{K}$ . Using $T_\text{max} \propto M^{4/3}$ :

(a) Find the peak core temperature of a $0.04\,M_\odot$ object.
(b) Does it ignite hydrogen? State what it becomes instead.
(c) Sanity check: without computing, should a $0.12\,M_\odot$ object’s peak be above or below $3 \times 10^6~\mathrm{K}$ ?

Problem

⭐⭐ A shallower slope moves the ceiling. The reading found the maximum mass by setting $L = L_\text{Edd}$ with $L \propto M^{3.5}$ , giving a naive crossover $M_\text{max} \sim 70\,M_\odot$ . Suppose instead the mass–luminosity relation were $L/L_\odot = (M/M_\odot)^{3}$ , with $L_\text{Edd}/L_\odot = 3.8 \times 10^4 (M/M_\odot)$ unchanged.

(a) Solve for the new crossover mass $M_\text{max}$ .
(b) Is the ceiling higher or lower than the $M^{3.5}$ case?
(c) Explain physically: does a steeper $L(M)$ raise or lower the maximum mass, and why?

Problem

⭐⭐ Electrons on the edge of relativity. In a white dwarf the electrons are packed to a spacing $d \sim 1 \times 10^{-10}~\mathrm{cm}$ . Treat each electron as confined to a box of that size, with $E_\text{min} \sim \hbar^2/(2 m_e d^2)$ .

(a) Compute $E_\text{min}$ in erg, then convert to keV.
(b) Compare it to the electron rest energy $m_e c^2 = 511~\mathrm{keV}$ . What fraction is it?
(c) White-dwarf electrons are mildly relativistic. Explain what would happen to this fraction as the white dwarf is made more massive (and therefore denser), and why that matters for a limit you’ll meet in Reading 3.

Synthesis

Problem

⭐⭐⭐ Same constants, two different walls. The minimum stellar mass and the Chandrasekhar mass (Reading 3) are both built from the same natural scale $M_0 = (\hbar c/G)^{3/2}/m_p^2 \approx 1.8\,M_\odot$ , yet one is $\sim 0.08\,M_\odot$ and the other $\sim 1.4\,M_\odot$ .

(a) Why do a quantum floor (minimum mass) and a quantum ceiling (Chandrasekhar) share the same combination of fundamental constants? Name the physics they have in common.
(b) The minimum mass carries an extra factor $(k_B T_\text{ign}/m_e c^2)^{3/4} \sim \text{a few}\times10^{-3}$ ; the Chandrasekhar mass does not. What is physically different about the two limits that gives the floor this extra (small) factor?
(c) In one sentence: both are “gravity versus degeneracy,” but the floor also answers “…and does it get hot enough?” Explain.

Problem

⭐⭐⭐ Why the galaxy is full of red dwarfs. Combine the Salpeter IMF ( $dN/dM \propto M^{-2.35}$ , the birth distribution) with the nuclear-lifetime scaling $\tau \propto M^{-2.5}$ (Module 3).

(a) Of every $10^4$ stars born, roughly how do the numbers born at $\sim 0.3\,M_\odot$ compare to those born at $\sim 30\,M_\odot$ ? (A scaling estimate is enough.)
(b) Now fold in lifetimes: the galaxy is $\sim 10^{10}~\mathrm{yr}$ old. Which of those two populations is almost entirely still shining, and which has mostly died?
(c) Conclude in one sentence why the most common stars in the galaxy today are faint red dwarfs — citing both effects.

Problem

⭐⭐⭐ A universe with weaker gravity. Imagine $G$ were smaller than in our universe. Use the scalings from the reading: the minimum-mass scale $M_\text{min} \propto M_0 \propto G^{-3/2}$ , and $L_\text{Edd} \propto G$ (so the coefficient $3.8\times10^4$ in $L_\text{Edd}/L_\odot$ is $\propto G$ ). Treat the mass–luminosity coefficient as fixed.

(a) Which way does the floor ( $M_\text{min}$ ) move as $G$ decreases?
(b) From the crossover $(M/M_\odot)^{2.5} \propto G$ , show $M_\text{max} \propto G^{0.4}$ , and say which way the ceiling moves.
(c) Does the allowed stellar mass window get wider or narrower in a weaker-gravity universe? State which end moves more.