Ages & Lifetimes

Section 6 of 6

Reference and Synthesis

Reference Tables

The Three Stellar Timescales

Timescale	Formula	Physical Meaning	Sun Value
Dynamical $\tau_\text{dyn}$	$\sim 1/\sqrt{G\bar{\rho}}$	Free-fall time; how fast gravity rearranges matter	$\sim 50~\text{min}$
Thermal $\tau_\text{KH}$	$\sim GM^2/(RL)$	Time to radiate the binding-energy reservoir	$\sim 30~\text{Myr}$
Nuclear $\tau_\text{nuc}$	$\sim \varepsilon f_M Mc^2/L$	Time to exhaust nuclear fuel	$\sim 10~\text{Gyr}$

Symbol Legend

Symbol	Meaning	CGS Units
$G$	Newton’s gravitational constant	$6.674 \times 10^{-8}~\text{cm}^3\text{g}^{-1}\text{s}^{-2}$
$\bar{\rho}$	Mean density	$\text{g cm}^{-3}$
$M$	Stellar mass	g (or $M_\odot = 2.0 \times 10^{33}~\text{g}$ )
$R$	Stellar radius	cm (or $R_\odot = 7.0 \times 10^{10}~\text{cm}$ )
$L$	Luminosity	erg/s (or $L_\odot = 3.8 \times 10^{33}~\text{erg/s}$ )
$c$	Speed of light	$3.0 \times 10^{10}~\text{cm/s}$
$\varepsilon$	Nuclear conversion efficiency	dimensionless ( $\approx 0.007$ for H→He)
$f_M$	Accessible fuel fraction ( $f_M M$ actually burns)	dimensionless ( $\approx 0.10$ for a Sun-like star)

Summary: Stars Are Clocks

Three timescales govern stellar physics — dynamical ( $\sim$ minutes), thermal ( $\sim$ Myr), and nuclear ( $\sim$ Gyr) — and their hierarchy $\tau_\text{dyn} \ll \tau_\text{KH} \ll \tau_\text{nuc}$ makes stars stable, quasi-static objects.
The Kelvin-Helmholtz controversy showed that observations (geological ages) can demand new physics (nuclear energy). Kelvin’s calculation was correct; his assumptions were incomplete.
Nuclear lifetime scales as $\tau_\text{nuc} \propto M^{-2.5}$ — massive stars burn far faster than they gain extra fuel. A $10\,M_\odot$ star lives $\sim 30~\text{Myr}$ ; a $0.1\,M_\odot$ star outlasts the universe.
The main-sequence turnoff gives cluster ages: the most massive star still on the MS sets the clock.

Where we stand · Reading 1 of 5

The through-line of Module 3: what holds a star up, and what makes it shine?

✓ Settled. Three clocks govern a star, and their separation — $\tau_\text{dyn} \ll \tau_\text{KH} \ll \tau_\text{nuc}$ — is what lets us model it as a static, balanced structure. The longest, $\tau_\text{nuc} \propto M^{-2.5}$ , sets the lifetime.

? Still open. What actually holds the star up — and powers it for $\sim 10~\text{Gyr}$ ? Gravity alone would collapse the Sun in $\sim 50$ minutes, yet it has shone steadily for $4.6~\text{Gyr}$ . Something balances gravity, and something keeps replenishing what it radiates.

→ Next. What force balances gravity? Reading 2 — hydrostatic equilibrium.

Quick retrieval: a cluster still showing blue main-sequence turnoff stars — is it young or old?

Looking Ahead

You now know stars live for $\tau_\text{nuc} \propto M^{-2.5}$ — but why do they last this long? What holds them up against gravity, and what generates the energy that maintains the balance? In Reading 2 the answer is hydrostatic equilibrium — the precise balance between gravity’s inward pull and pressure’s outward push. And in Reading 3, the energy source maintaining that balance turns out to demand all four fundamental forces of nature, including quantum mechanics. Keep one number in your pocket as you go: $\tau_\text{nuc} \propto M^{-2.5}$ . It means every star is born onto a clock that runs out — fast for the massive, almost never for the faint — and what happens when it runs out is where the second half of this module is headed: how stars age, swell, and die.

Glossary

Dynamical timescale: The characteristic time for gravity to rearrange a star if pressure support failed: $\tau_\text{dyn} \sim 1/\sqrt{G\bar{\rho}}$ — about 50 minutes for the Sun. The shortest of the three stellar clocks; it sets how fast a star restores hydrostatic equilibrium after a perturbation.
Kelvin-Helmholtz timescale: The time a star could shine on gravitational contraction alone: $\tau_\text{KH} \sim GM^2/(RL)$ — about 30 Myr for the Sun. It is a quasi-static contraction time, not a free fall; nuclear fusion makes the real lifetime hundreds of times longer.
Main-sequence turnoff: The point on a star cluster’s HR diagram where the most massive stars are just leaving the main sequence. Because nuclear lifetime falls steeply with mass, the turnoff mass is a clock: a high (blue) turnoff means a young cluster, a low (red) turnoff an old one.
Nuclear timescale: The time core hydrogen fusion can sustain a star’s luminosity: $\tau_\text{nuc} \sim \varepsilon f_M M c^2 / L$ — about 10 Gyr for the Sun. It is the longest of the three clocks and sets a star’s true main-sequence lifetime.