Ages & Lifetimes

Useful constants: $G = 6.674 \times 10^{-8}~\mathrm{cm^3\,g^{-1}\,s^{-2}}$ ; $M_\odot = 2.0 \times 10^{33}~\mathrm{g}$ ; $R_\odot = 7.0 \times 10^{10}~\mathrm{cm}$ ; $L_\odot = 3.8 \times 10^{33}~\mathrm{erg/s}$ ; $c = 3.0 \times 10^{10}~\mathrm{cm/s}$ ; $\tau_{\text{nuc},\odot} \approx 10~\mathrm{Gyr}$ ; age of the universe $\approx 13.8~\mathrm{Gyr}$ .

Show explicit units, and run a sanity check on every result. Worked solutions are released after the homework due date.

Conceptual

Problem

⭐⭐ Fuel is not lifetime. A student says, “A more massive main-sequence star should live longer because it has more hydrogen fuel.”

(a) Explain why this statement is incomplete.
(b) In $\tau_\text{nuc} \sim \text{fuel}/\text{burn rate}$ , which term grows faster with stellar mass along the main sequence?
(c) Use $\tau_\text{nuc} \propto M/L$ to explain why high-mass stars die young.

Problem

⭐ Which clock answers which question? Match each question to the most relevant timescale, and justify your choice in one sentence:

(a) How quickly would the Sun respond if pressure support suddenly failed?
(b) How long could the Sun shine if gravity were its only energy source?
(c) How long can core hydrogen fusion sustain the Sun’s present luminosity?

Problem

⭐⭐ Turnoff color as an age clue. Cluster A still has blue main-sequence turnoff stars; Cluster B’s turnoff is much redder and cooler.

(a) Which cluster is older?
(b) What is the direct observable in this comparison?
(c) What model connects that observable to the cluster’s age?

Problem

⭐⭐⭐ The Kelvin-Helmholtz controversy. In the 1860s Lord Kelvin computed $\tau_\text{KH} \approx 30~\text{Myr}$ for the Sun and concluded the geologists — who needed hundreds of millions of years — must be wrong.

(a) Was Kelvin’s physics or arithmetic wrong? Defend your answer.
(b) What single assumption made his model incomplete?
(c) What later discovery resolved the controversy, and by roughly what factor was Kelvin’s age too short?
(d) State the general lesson about the difference between a correct calculation and a complete model.

Calculation

Problem

⭐ The Sun’s dynamical timescale. The Sun’s mean density is $\bar{\rho}_\odot \approx 1.4~\mathrm{g/cm^3}$ .

(a) Use $\tau_\text{dyn} \sim 1/\sqrt{G\bar{\rho}}$ to estimate the Sun’s dynamical timescale in seconds.
(b) Convert to minutes.
(c) Explain in one sentence what this timescale physically means.

Problem

⭐⭐ Kelvin-Helmholtz timescale for the Sun.

(a) Use $\tau_\text{KH} \sim GM_\odot^2/(R_\odot L_\odot)$ to estimate the Sun’s Kelvin-Helmholtz timescale in seconds.
(b) Convert to Myr.
(c) If the Sun had the same mass and radius but twice the luminosity, by what factor would $\tau_\text{KH}$ change?

Problem

⭐⭐ Nuclear lifetime, from the Sun to a heavier star.

(a) For a Sun-like star, use $\tau_\text{nuc} \sim \varepsilon f_M M c^2/L$ with $\varepsilon = 0.007$ and $f_M = 0.10$ to estimate the Sun’s nuclear lifetime.
(b) Use solar normalization to estimate the main-sequence lifetime of a star with $M = 2\,M_\odot$ and $L = 10\,L_\odot$ .
(c) Compare your answer in (b) to the Sun’s lifetime.

Synthesis

Problem

⭐⭐ Why M dwarfs live so long. An idealized fully convective M dwarf has $M/M_\odot = 0.2$ , $L/L_\odot \approx (0.2)^{3.5} \approx 0.0036$ , and (because full convection mixes more fuel) $f_M/f_{M,\odot} \approx 10$ .

(a) Estimate its nuclear lifetime in units of the Sun’s.
(b) Convert that result to years.
(c) Compare to the age of the universe, and explain what it implies about the faintest red dwarfs.

Problem

⭐⭐ Turnoff as a clock. A young cluster still contains a B2 main-sequence star with $M \approx 10\,M_\odot$ .

(a) Use $\tau_\text{nuc} \propto M^{-2.5}$ to estimate the star’s main-sequence lifetime, and therefore the cluster’s age scale.
(b) State the observable, the model, and the inference in this dating method.
(c) Explain why the star can remain close to hydrostatic equilibrium while still evolving over millions of years.

Problem

⭐⭐⭐ Reading the age of a globular cluster. A globular cluster’s main-sequence turnoff sits at $M \approx 0.9\,M_\odot$ .

(a) Use $\tau_\text{nuc} \propto M^{-2.5}$ (anchored to the Sun) to estimate the cluster’s age.
(b) Compare your answer to the Sun’s $\sim 10~\text{Gyr}$ lifetime and to the $13.8~\text{Gyr}$ age of the universe. Is the cluster older or younger than the Sun’s lifetime, and what does that say about when it formed?
(c) Name two assumptions hidden in the $\tau_\text{nuc} \propto M^{-2.5}$ scaling, and state which direction each could bias your age estimate.