The Quantum Limit

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
$\hbar$	$1.055 \times 10^{-27}~\mathrm{erg\,s}$
$c$	$3.0 \times 10^{10}~\mathrm{cm\,s^{-1}}$
$G$	$6.674 \times 10^{-8}~\mathrm{cm^3\,g^{-1}\,s^{-2}}$
$k_B$	$1.38 \times 10^{-16}~\mathrm{erg\,K^{-1}}$
$m_e$	$9.11 \times 10^{-28}~\mathrm{g}$
$m_p$	$1.67 \times 10^{-24}~\mathrm{g}$
$m_e c^2$	$0.511~\mathrm{MeV} = 8.2\times10^{-7}~\mathrm{erg}$
$M_\odot$	$2.0 \times 10^{33}~\mathrm{g}$
Natural mass scale $M_0 = (\hbar c/G)^{3/2}/m_p^2$	$\approx 1.8\,M_\odot$
Chandrasekhar mass	$M_\text{Ch} = 5.83\,Y_e^2\,M_\odot$ (so $1.44\,M_\odot$ at $Y_e=0.5$ )

Useful scalings from the reading: $p_F\sim\hbar n_e^{1/3}$ ; non-rel $P_\text{deg}\sim\hbar^2 n_e^{5/3}/m_e\;(\propto\rho^{5/3})$ ; rel $P_\text{deg}\sim\hbar c\,n_e^{4/3}\;(\propto\rho^{4/3})$ ; $P_\text{deg}/P_\text{gas}\sim\hbar^2 n_e^{2/3}/(m_e k_B T)$ ; $P_\text{grav}\sim GM^2/R^4$ ; $n_e=\rho/(\mu_e m_p)$ with $\mu_e\approx2$ .

Conceptual

Problem

⭐⭐ Cooling won’t save it. A student says: “A white dwarf just above the Chandrasekhar mass could be saved from collapse by cooling it down — colder means lower pressure demand, so it would stabilize.”

(a) Identify the two physics errors (one about what degeneracy pressure depends on; one about what the limit depends on).
(b) State correctly what the Chandrasekhar limit is about.
(c) What would change the limit for a real white dwarf — name the one composition parameter.

Problem

⭐⭐ Three cores, three fates. Three stellar cores arrive at the end of fusion with masses $1.0$ , $2.0$ , and $5.0\,M_\odot$ .

(a) Predict the remnant of each and the pressure that (tries to) support it.
(b) For each, name the physical reason that support succeeds or fails.
(c) The boundaries near $1.4$ and $\sim3\,M_\odot$ are different kinds of wall. How does the $1.4\,M_\odot$ wall differ in origin from the $\sim3\,M_\odot$ one?

Problem

⭐⭐ Why relativity is fatal. Non-relativistic degeneracy pressure ( $P\propto\rho^{5/3}$ ) can always win against gravity, but relativistic degeneracy pressure ( $P\propto\rho^{4/3}$ ) cannot.

(a) Gravity’s pressure demand scales as $P_\text{grav}\propto\rho^{4/3}$ (at fixed $M$ ). Explain why a $\rho^{5/3}$ supply always eventually overtakes it as you compress, but a $\rho^{4/3}$ supply does not.
(b) In the relativistic case, what determines whether equilibrium exists, if not the radius?
(c) Why is this a “hard wall” rather than a gradual weakening?

Calculation

Problem

⭐⭐ Build both pressure laws. Starting from $p_F\sim\hbar n_e^{1/3}$ and $P\sim n_e E$ (pressure $\sim$ energy density):

(a) For non-relativistic electrons ( $E\sim p_F^2/2m_e$ ), derive the scaling $P_\text{deg}\propto n_e^{5/3}$ (hence $\propto\rho^{5/3}$ ).
(b) For relativistic electrons ( $E\sim p_F c$ ), derive $P_\text{deg}\propto n_e^{4/3}$ (hence $\propto\rho^{4/3}$ ).
(c) State, from the exponents alone, why the relativistic gas is “softer.”

Problem

⭐⭐ An iron white dwarf. Using $M_\text{Ch}=5.83\,Y_e^2\,M_\odot$ :

(a) Confirm the carbon-oxygen value ( $Y_e=0.5$ ).
(b) Iron-56 has 26 electrons per 56 nucleons, so $Y_e\approx0.464$ . Compute the Chandrasekhar mass of an iron white dwarf.
(c) Is it larger or smaller than the C/O value? Explain physically why fewer electrons per nucleon lowers the limit.

Problem

⭐⭐ Classical or degenerate? Use $P_\text{deg}/P_\text{gas}\sim\hbar^2 n_e^{2/3}/(m_e k_B T)$ to classify two gases.

(a) Solar core: $\rho=150~\mathrm{g\,cm^{-3}}$ (ionized hydrogen, $n_e\approx\rho/m_p$ ), $T=1.5\times10^7~\mathrm{K}$ .
(b) White-dwarf interior: $\rho=10^6~\mathrm{g\,cm^{-3}}$ ( $n_e\approx\rho/2m_p$ ), $T=10^7~\mathrm{K}$ .
(c) Which is degenerate? Note that the two have similar temperatures — so what is doing the work in the ratio?

Synthesis

Problem

⭐⭐⭐ One scale, the floor and the ceiling. Three walls bound a star’s life: the minimum mass $M_\text{min}\sim0.08\,M_\odot$ , the Chandrasekhar mass $M_\text{Ch}\sim1.4\,M_\odot$ , and the maximum stellar mass $M_\text{max}\sim100\,M_\odot$ .

(a) Which of the three are built from the natural scale $M_0=(\hbar c/G)^{3/2}/m_p^2$ , and which is not?
(b) Explain why $M_\text{min}$ and $M_\text{Ch}$ share $M_0$ but $M_\text{max}$ does not — in terms of which forces are in balance.
(c) What is remarkable about the fact that both edges of degenerate stardom (the lightest star, the heaviest white dwarf) are written in the same four constants?

Problem

⭐⭐⭐ What Eddington couldn’t accept. Eddington accepted Chandrasekhar’s mathematics but rejected its conclusion.

(a) State precisely what one is forced to believe exists if the Chandrasekhar limit is real and stellar cores can exceed it.
(b) Explain the logical chain: limit is real $\Rightarrow$ massive cores have no white-dwarf endpoint $\Rightarrow$ what must happen to them?
(c) Why does the existence of a maximum white-dwarf mass essentially force the existence of neutron stars and black holes?

Problem

⭐⭐⭐ A universe of bosons. Imagine electrons were bosons (no Pauli exclusion) instead of fermions.

(a) What would happen to degeneracy pressure, and therefore to white dwarfs?
(b) Would there still be a Chandrasekhar limit? Explain what “limit” even means in this universe.
(c) Pauli exclusion also underlies everyday matter. Name one consequence for atoms or solid matter if electrons were bosons.