The Death of Giants

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
$G$	$6.67 \times 10^{-8}~\mathrm{cm^3\,g^{-1}\,s^{-2}}$
$c$	$3.0 \times 10^{10}~\mathrm{cm\,s^{-1}}$
$M_\odot$	$2.0 \times 10^{33}~\mathrm{g}$
$M_\odot c^2$	$1.8 \times 10^{54}~\mathrm{erg}$
$1~\mathrm{yr}$	$3.16 \times 10^{7}~\mathrm{s}$
Chandrasekhar mass	$M_\text{Ch} = 5.83\,Y_e^2\,M_\odot$
Iron-peak $B/A$	$\approx 8.8~\mathrm{MeV/nucleon}$ (the maximum)

Useful scalings from the reading: burning lifetime $\tau\sim qM_\text{fuel}/(L_\gamma+L_\nu)$ ; main-sequence $L\propto M^{3.5}$ so $\tau_\text{MS}\propto M^{-2.5}$ ; collapse energy $\Delta E_\text{grav}\sim GM^2/R_f$ ; SN energy budget $\sim99.7\%$ neutrinos, $\sim0.3\%$ kinetic, $\sim0.003\%$ light.

Conceptual

Problem

⭐⭐ Most fuel, longest life? A student says: “The most massive stars have the most nuclear fuel, so they must live the longest.”

(a) Identify the error using $\tau\sim E/L$ and the mass–luminosity relation $L\propto M^{3.5}$ .
(b) State the correct trend (longest- vs shortest-lived) with a scaling.
(c) The late burning stages shorten even faster than this scaling predicts. Name the two effects (from the reading) that accelerate them.

Problem

⭐⭐ Where did each element come from? For each, name the dominant nucleosynthetic site (and the process):

(a) the oxygen in your lungs,
(b) the iron in your blood,
(c) the gold in a ring,
(d) the carbon in your DNA.

Problem

⭐⭐ Read the binding-energy curve. Use the $B/A$ -versus- $A$ curve (binding energy per nucleon) to explain the iron endpoint.

(a) State the rule that decides whether a fusion reaction releases or absorbs energy, in terms of $B/A$ .
(b) Why does fusing toward iron release energy but fusing past it cost energy?
(c) “Iron is the end of fusion” is precise only with a qualifier. State the precise version.

Calculation

Problem

⭐⭐ How much faster is silicon burning? For a $25\,M_\odot$ star, carbon burning lasts $\sim 600~\mathrm{yr}$ and silicon burning $\sim 1~\mathrm{day}$ .

(a) Compute the ratio $\tau_\text{Si}/\tau_\text{C}$ (dimensionless).
(b) Using $\tau\sim qM_\text{fuel}/(L_\gamma+L_\nu)$ , name the two factors that make $\tau_\text{Si}$ so much smaller, even though there is still silicon fuel.
(c) Sanity check: is your ratio consistent with the reading’s $\tau_\text{Si}/\tau_\text{H}\sim4\times10^{-10}$ ?

Problem

⭐⭐ The ejecta’s share. A supernova ejects $\sim 5\,M_\odot$ of material carrying $\sim 10^{51}~\mathrm{erg}$ of kinetic energy.

(a) Estimate the mean ejecta speed from $\mathrm{KE} = \tfrac12 M v^2$ . Give it in $\mathrm{km\,s^{-1}}$ .
(b) The total explosion energy budget is $\sim 5\times10^{53}~\mathrm{erg}$ . What fraction is the kinetic energy?
(c) Where did the other $\sim99.7\%$ go, and why is it nearly invisible?

Problem

⭐⭐ The wall comes down to meet the core. An iron core has mass $1.3\,M_\odot$ . It starts with $Y_e = 0.50$ , but electron capture during contraction lowers it to $Y_e = 0.42$ . Use $M_\text{Ch} = 5.83\,Y_e^2\,M_\odot$ .

(a) Compute $M_\text{Ch}$ before and after the drop in $Y_e$ .
(b) Is the $1.3\,M_\odot$ core stable before? After?
(c) Explain how this is a “threshold crossing” even though the core’s mass barely changed.

Synthesis

Problem

⭐⭐⭐ Two different walls. Core collapse requires crossing two thresholds: the binding-energy peak and the Chandrasekhar mass.

(a) One is an energy threshold, the other a mechanical (support) threshold. Which is which, and what does crossing each one mean physically?
(b) Why is the first threshold a precondition for the second? (What does reaching the iron peak do to the core’s pressure support?)
(c) Explain why a star needs to cross both — why neither alone causes the collapse.

Problem

⭐⭐⭐ The invisible explosion. Roughly $99.7\%$ of a core-collapse supernova’s energy leaves as neutrinos, which barely interact — yet the star still explodes.

(a) State the paradox sharply: if almost all the energy escapes invisibly, what drives the visible ejection?
(b) Describe the modern resolution (the role of the stalled shock and neutrino heating).
(c) How did the SN 1987A neutrino detection (about 20 neutrinos, $\sim10~\mathrm{s}$ , $\sim3$ hr before the light) confirm this picture?

Problem

⭐⭐⭐ Trace one atom; why Earth is late. Follow a single oxygen atom now in your body back to its origin.

(a) List, in order, what had to happen for that atom to exist and reach you (its synthesis, its dispersal, its incorporation).
(b) Why could no atom heavier than helium have existed in the very first generation of stars?
(c) Explain why a planet around a first-generation (Population III) star almost certainly could not host Earth-like life.