After the Main Sequence

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
$\hbar$	$1.055 \times 10^{-27}~\mathrm{erg\,s}$
$k_B$	$1.38 \times 10^{-16}~\mathrm{erg\,K^{-1}}$
$m_e$	$9.11 \times 10^{-28}~\mathrm{g}$
$m_p$	$1.67 \times 10^{-24}~\mathrm{g}$
$M_\odot$	$2.0 \times 10^{33}~\mathrm{g}$
$R_\odot$	$7.0 \times 10^{10}~\mathrm{cm}$
$T_{\text{eff},\odot}$	$5{,}800~\mathrm{K}$
$1~\mathrm{AU}$	$1.5 \times 10^{13}~\mathrm{cm}$
electron Fermi energy (scaling)	$E_F \sim \hbar^2 n_e^{2/3}/(2 m_e)$ , with $n_e = \rho/(\mu_e m_p)$ , $\mu_e \approx 2$

Useful scalings from the reading: virial $E = U_\text{grav}/2 = -K_\text{th}$ ; Stefan–Boltzmann $L/L_\odot = (R/R_\odot)^2(T_\text{eff}/T_{\text{eff},\odot})^4$ ; $\varepsilon_{3\alpha}\propto T^{40}$ ; $P_\text{deg}\propto\rho^{5/3}$ (no $T$ ); $R_\text{WD}\propto M^{-1/3}$ ; lifetime $t\sim E/L$ .

Conceptual

Problem

⭐⭐ Will the Sun go supernova? A student writes: “At the end of its life the Sun will run out of fuel, collapse, and explode as a supernova, scattering its elements.”

(a) Identify what is wrong for a low-mass star like the Sun.
(b) Give the correct endpoint and the correct way the Sun returns material to space.
(c) One clause of the student’s sentence is essentially right. Which, and why?

Problem

⭐⭐ Read the interior off the HR diagram. You observe a star that is cool ( $T_\text{eff}\approx 3{,}800~\mathrm{K}$ ) yet $\sim 1{,}000\times$ more luminous than the Sun.

(a) State the observable contradiction (what would an ordinary main-sequence star of that temperature look like?).
(b) State the model: what interior energy source and structure resolve it?
(c) State the inference about the star’s radius and evolutionary stage.
(d) Put these five steps in causal order to explain why the envelope expands (rather than just glowing brighter at fixed size): convection turns on; core contraction; the hydrogen shell’s luminosity rises; a high-opacity transport bottleneck steepens the gradient; the envelope swells to a new equilibrium. Then say in one sentence why “convection makes giants big” is an incomplete explanation.

Problem

⭐⭐ Who flashes and who doesn’t? Helium ignites with a flash in some stars and quietly in others.

(a) For a $1\,M_\odot$ star and a $5\,M_\odot$ star, predict which has a helium flash.
(b) State the physical condition — in terms of $E_F$ and $k_B T$ in the helium core at the moment helium ignites — that decides the outcome.
(c) Explain why a degenerate core makes ignition run away but a non-degenerate core does not.

Calculation

Problem

⭐⭐ How big is an AGB star? An AGB star has $L = 5{,}000\,L_\odot$ and $T_\text{eff} = 3{,}000~\mathrm{K}$ .

(a) Use the Stefan–Boltzmann ratio to the Sun to find $R/R_\odot$ .
(b) Convert to AU. If this star sat where the Sun is, which solar-system bodies would it engulf? (Mercury $0.39$ , Venus $0.72$ , Earth $1.0$ AU.)
(c) Sanity check: should an AGB star be larger or smaller than the $\sim 87\,R_\odot$ RGB star from the reading? Is your answer consistent?

Problem

⭐⭐ Is the helium core degenerate? Just before the flash, a helium core has $\rho \sim 1\times10^6~\mathrm{g\,cm^{-3}}$ at $T \sim 1\times10^8~\mathrm{K}$ . Use $E_F \sim \hbar^2 n_e^{2/3}/(2 m_e)$ with $n_e = \rho/(\mu_e m_p)$ , $\mu_e \approx 2$ .

(a) Compute the electron number density $n_e$ , then the Fermi energy $E_F$ (in erg).
(b) Compute the thermal energy $k_B T$ and form the ratio $E_F/k_B T$ .
(c) Is the core degenerate? Does this predict a flash? Tie your answer to “Reading the Limit.”

Problem

⭐⭐ Heavier means smaller. A $0.6\,M_\odot$ white dwarf has radius $R = 0.013\,R_\odot$ . Using $R_\text{WD}\propto M^{-1/3}$ :

(a) Find the radius of a $1.0\,M_\odot$ white dwarf.
(b) By what factor does the mean density differ between the two? (Use $\bar\rho\propto M/R^3$ .)
(c) Interpret: which way does the mass–radius relation push as a white dwarf approaches the Chandrasekhar limit, and why is that ominous?

Synthesis

Problem

⭐⭐⭐ Why the phases last as long as they do. Use $t\sim E/L$ (available energy over luminosity) to reason about post-main-sequence timing.

(a) Rank these by duration, longest to shortest: main sequence, RGB ascent, planetary nebula, white-dwarf cooling.
(b) Explain why white-dwarf cooling is the longest phase even though it has no fusion energy source at all.
(c) Explain why the planetary-nebula phase is the shortest, despite the nebula being the most visually dramatic.

Problem

⭐⭐⭐ The engine and the brake. The virial theorem gives a star “negative heat capacity”: a core that loses energy heats up.

(a) Explain how this single fact drives the entire evolutionary ladder — why each time one fuel is exhausted, the core contraction tends to ignite the next.
(b) Ordinary (non-degenerate) fusion is self-regulating (a thermostat). Explain why, using negative heat capacity: what happens to a non-degenerate core that briefly over-burns?
(c) Explain why degeneracy breaks that thermostat — and why that is exactly what makes the helium flash (and, in Reading 3’s preview, the Type Ia runaway) possible.

Problem

⭐⭐⭐ Two flashes, two fates. The helium flash and a Type Ia supernova share the same underlying logic (a thermonuclear runaway in a degenerate gas) but have opposite outcomes.

(a) State the shared mechanism in one sentence (what makes a degenerate gas prone to runaway?).
(b) The helium flash leaves the star intact; a Type Ia destroys the white dwarf. What physical difference produces the opposite outcome?
(c) Why does the existence of the Chandrasekhar limit (Reading 3) make Type Ia supernovae such reliable “standard candles”? Reason from the fact that they all detonate at nearly the same mass.