The Stellar Blueprint

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.

Toy main-sequence scalings (anchored at the Sun): $L \propto M^3/\kappa$ · $T_c \propto M/R$ · $R \propto M^{3/7}$ · $\tau_\text{nuc} \propto M/L \propto M^{-2}$ . Solar anchors: $L_\odot = 3.8\times10^{33}~\mathrm{erg\,s^{-1}}$ , $R_\odot = 7.0\times10^{10}~\mathrm{cm}$ , $T_{c,\odot} = 15~\mathrm{MK}$ , $\tau_\odot \approx 10~\mathrm{Gyr}$ . Constants: $G = 6.674\times10^{-8}$ , $M_\odot = 2.0\times10^{33}~\mathrm{g}$ , $m_p = 1.67\times10^{-24}~\mathrm{g}$ , $k_B = 1.38\times10^{-16}~\mathrm{erg\,K^{-1}}$ , $\mu = 0.6$ .

Conceptual

Problem

⭐⭐ Why the radius drops out. In the toy derivation, the radius cancelled and left $L \propto M^3/\kappa$ .

(a) Does that mean a star’s radius is irrelevant to its structure?
(b) Explain what “the radius drops out of the scaling” actually means.
(c) Why does this cancellation make mass the dominant control parameter of the main sequence?

Problem

⭐⭐ Which exponent is robust? Across the assumption audit, the exponent in $L \propto M^3$ stays near 3 while the coefficient swings by orders of magnitude.

(a) Why is the exponent of a scaling more trustworthy than its coefficient?
(b) Give one audited assumption that shifts the exponent and one that shifts only the coefficient.
(c) Connect this to Kelvin’s lesson from Reading 1.

Problem

⭐⭐ Reading the HR diagram. The main sequence is a narrow band; massive stars sit at the hot, luminous (upper-left) end.

(a) What is the observable that defines the main sequence?
(b) What model turns it into a mass sequence?
(c) What is the inference — why is it a tight band rather than a scattered cloud of points?

Calculation

Problem

⭐⭐ Anchor and predict: a $5\,M_\odot$ star. Use the toy scalings anchored at the Sun.

(a) Predict its luminosity in $L_\odot$ ( $L \propto M^3$ ).
(b) Predict its radius in $R_\odot$ ( $R \propto M^{3/7}$ ).
(c) Predict its nuclear lifetime ( $\tau \propto M^{-2}$ , with $\tau_\odot \approx 10~\mathrm{Gyr}$ ).
(d) Compare your luminosity to the empirical $L \propto M^{3.5}$ — which is larger, and why?

Problem

⭐⭐ Two ends of the main sequence. Compare a $0.5\,M_\odot$ red dwarf (star A) with an $8\,M_\odot$ B star (star B).

(a) Using $L \propto M^3$ , find the luminosity ratio $L_B/L_A$ .
(b) Using $\tau \propto M^{-2}$ , find the lifetime ratio $\tau_A/\tau_B$ .
(c) Using $T_c \propto M^{4/7}$ , by what factor is star B’s core hotter?

Problem

⭐⭐ The mass-radius relation, tested. Use the toy result $R \propto M^{3/7}$ .

(a) Predict the radius of a $10\,M_\odot$ star in $R_\odot$ .
(b) The observed radius is $\approx 3.7\,R_\odot$ . By what percentage does the toy model fall short?
(c) The toy mass-radius relation is shallower than observed. Which simplifying assumptions are responsible?

Problem

⭐⭐ Lifetime of a $25\,M_\odot$ star. Anchor at the Sun ( $\tau_\odot \approx 10~\mathrm{Gyr}$ ).

(a) Estimate $\tau_\text{nuc}$ with the toy scaling $\tau \propto M^{-2}$ .
(b) Estimate it with the empirical $\tau \propto M^{-2.5}$ .
(c) The observed value is $\approx 6~\mathrm{Myr}$ . Which estimate is closer, and why is $M^{-2.5}$ the better exponent?

Synthesis

Problem

⭐⭐⭐ The coefficient catastrophe. Pushed through with its full prefactor, the one-zone model overestimates the Sun’s absolute luminosity by a factor of $\sim 10^3$ .

(a) Which single assumption is most responsible for that enormous coefficient error?
(b) Why does the luminosity’s $T^4$ dependence amplify that error so severely?
(c) Why is the $M^3$ exponent still trustworthy even though the coefficient is $\sim 1000\times$ off?

Problem

⭐⭐⭐ Stack the audit. For each real-star observation below, name which audited assumption fails and what it does to the toy result.

(a) The most massive stars follow $L \propto M$ , not $M^3$ .
(b) Over most of the main sequence the empirical law is $L \propto M^{3.5}$ , not $M^3$ .
(c) Stars below $\sim 0.35\,M_\odot$ are fully convective rather than radiative.

Problem

⭐⭐⭐ Build the main sequence from scratch. Reconstruct the whole blueprint in your own words.

(a) Walk the chain $\rho \sim M/R^3 \rightarrow P_c \sim GM^2/R^4 \rightarrow T_c \sim M/R \rightarrow L \propto M^3/\kappa \rightarrow R \propto M^{3/7} \rightarrow \tau_\text{nuc} \propto M^{-2}$ , naming the physical principle, the new assumption, and the inference at each arrow.
(b) In one paragraph, explain why the main sequence is a mass sequence.
(c) The main sequence has edges — a minimum and a maximum stellar mass. State physically what sets each edge.