Radiation Transport

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
$\kappa_\text{es}$	$0.34~\mathrm{cm^2\,g^{-1}}$ (electron scattering)
$\rho_{c,\odot}$	$\approx 150~\mathrm{g\,cm^{-3}}$
$\ell_\odot$ (core)	$\approx 0.02~\mathrm{cm}$
$t_{\text{diff},\odot}$	$\approx 2.6\times10^{5}~\mathrm{yr}$
$a$	$7.56\times10^{-15}~\mathrm{erg\,cm^{-3}\,K^{-4}}$
$G$	$6.674\times10^{-8}~\mathrm{cm^3\,g^{-1}\,s^{-2}}$
$c$	$3.0\times10^{10}~\mathrm{cm\,s^{-1}}$
$R_\odot,\ M_\odot,\ L_\odot$	$7.0\times10^{10}~\mathrm{cm}$ , $2.0\times10^{33}~\mathrm{g}$ , $3.8\times10^{33}~\mathrm{erg\,s^{-1}}$
$T_{c,\odot}$	$\approx 1.5\times10^{7}~\mathrm{K}$
$1~\mathrm{yr}$	$3.16\times10^{7}~\mathrm{s}$

Conceptual

Problem

⭐⭐ The travel-time paradox. Photons move at $c$ , yet the Sun’s energy takes $\sim 10^5\,\mathrm{yr}$ to reach the surface.

(a) Why is the delay not because photons move slowly?
(b) What physical quantity actually diffuses slowly?
(c) Name the observable, the model, and the inference that connect “the sunlight at Earth” to “energy released in the core $\sim 10^5\,\mathrm{yr}$ ago.”

Problem

⭐⭐ Why $\sqrt{N}$ , not $N$ . A photon takes $N$ steps of length $\ell$ in random directions, reaching net displacement $d \sim \sqrt{N}\,\ell$ .

(a) Explain physically why the net displacement grows as $\sqrt{N}$ rather than $N$ .
(b) If the steps were all in the same direction, how far would the photon get — and why doesn’t that happen?
(c) Show that this $\sqrt{N}$ behavior is what makes the diffusion time scale as $R^2$ , not $R$ .

Problem

⭐⭐ Two pressures, two temperature laws. Gas pressure scales as $P_\text{gas} \propto \rho T$ ; radiation pressure as $P_\text{rad} = aT^4/3$ .

(a) Why do the two scale so differently with temperature?
(b) In which stars does radiation pressure matter, and why?
(c) A student says, ” $P_\text{rad}$ and $P_\text{gas}$ are both just pressure, so they are interchangeable.” Identify what is wrong and correct it.

Calculation

Problem

⭐⭐ Mean free path in a massive-star core. A massive star’s core is hot but tenuous: $\rho_c \approx 5~\mathrm{g\,cm^{-3}}$ (vs the Sun’s $150$ ), with $\kappa \approx 0.34~\mathrm{cm^2\,g^{-1}}$ .

(a) Compute the photon mean free path $\ell = 1/(\kappa\rho)$ .
(b) Compare it to the Sun’s $\ell \approx 0.02~\mathrm{cm}$ .
(c) What does a longer mean free path imply about how readily energy escapes?

Problem

⭐⭐ Turn up the opacity. Suppose the solar interior’s opacity were $10\times$ higher (a more metal-rich composition), with everything else held fixed.

(a) Using $t_\text{diff} \sim R^2\kappa\rho/c$ , by what factor does the diffusion time change?
(b) Using the diffusion luminosity scaling $L \sim a c\,T_c^4 R^4/(\kappa M)$ , by what factor does the luminosity change?
(c) Explain physically why higher opacity both slows the photon escape and lowers the luminosity.

Problem

⭐⭐ The diffusion luminosity scaling. Use the Reading-the-Math result $L \sim a c\,T_c^4 R^4/(\kappa M)$ .

(a) At fixed $T_c$ , $\kappa$ , and $M$ , a star has twice the Sun’s radius. By what factor is its luminosity larger?
(b) Now also let its core be $1.2\times$ hotter. What is the combined factor?
(c) In a real star $T_c$ and $R$ are not independent (hydrostatic equilibrium links them). Explain why that is exactly why Reading 5 must substitute $T_c$ to get the true mass–luminosity law.

Problem

⭐⭐ Eddington luminosity of a massive star. The Eddington luminosity is $L_\text{Edd} = 4\pi GMc/\kappa$ .

(a) Compute $L_\text{Edd}$ for a $50\,M_\odot$ star with $\kappa = 0.34~\mathrm{cm^2\,g^{-1}}$ , in $\mathrm{erg\,s^{-1}}$ and in $L_\odot$ .
(b) Such a star shines at $L \approx 10^6\,L_\odot$ . What fraction of $L_\text{Edd}$ is that?
(c) Why does operating near the Eddington limit make a star drive strong winds?

Synthesis

Problem

⭐⭐⭐ When does the star convect? Convection sets in where $\nabla_\text{rad} > \nabla_\text{ad} \approx 0.4$ .

(a) Explain in words what $\nabla_\text{rad} > \nabla_\text{ad}$ means physically — the buoyancy argument.
(b) Two very different regions go convective: a cool stellar envelope (high $\kappa$ ) and a massive star’s core (concentrated CNO burning). Explain each using $\nabla_\text{rad}$ .
(c) Why is $\nabla_\text{ad} \approx 0.4$ nearly universal while $\nabla_\text{rad}$ varies enormously?

Problem

⭐⭐⭐ Assemble the mass–luminosity law. Combine Reading 2 and Reading 4 into the headline scaling.

(a) Run the diffusion Reading the Math: approximate $d(aT^4)/dr$ and extract $L \sim a c\,T_c^4 R^4/(\kappa M)$ .
(b) Substitute the hydrostatic core temperature $T_c \sim \mu GMm_p/(k_BR)$ and show the factors of $R$ cancel, leaving $L \propto M^3$ .
(c) Which assumptions in the audit most threaten the exponent 3 (as opposed to merely the coefficient)?

Problem

⭐⭐ The buffered Sun. The Sun’s energy takes $t_\text{diff} \sim 10^5\,\mathrm{yr}$ to diffuse out, against a dynamical time $\sim 1\,\mathrm{hr}$ and a nuclear time $\sim 10^{10}\,\mathrm{yr}$ .

(a) Order the three timescales from shortest to longest.
(b) The Sun’s luminosity is rock-steady on human timescales. Which timescale guarantees that short-term core fluctuations are smoothed before we see them?
(c) If the core’s fusion rate suddenly jumped, roughly how long until the surface luminosity responded — and what does that imply about observing core physics directly?