Ages & Lifetimes

Section 2 of 6

Three Timescales That Govern a Star

Part 2: Three Timescales That Govern a Star

Stars are governed by three fundamental timescales, each tied to a different physical process. Their hierarchy — which is shortest, which longest — tells you what a star is doing.

Generated white-background log-log plot of stellar timescale versus stellar mass in solar units, with three labeled curves for dynamical, Kelvin-Helmholtz, and nuclear timescales, a marked Sun point, a dashed age-of-the-universe line, and shaded caveat bands at low and high mass. — Figure 2The hierarchy tau_dyn << tau_KH << tau_nuc survives across the main sequence. Massive stars still adjust quickly, but their thermal and nuclear clocks shrink drastically. Caveat bands remind us where simple scaling laws bend.ASTR 201 (generated)

Timescale	Question it answers
$\tau_\text{dyn}$	How fast would gravity rearrange the star if pressure support failed?
$\tau_\text{KH}$	How long could the star shine if gravity were its only long-term energy source?
$\tau_\text{nuc}$	How long can nuclear fusion sustain the star’s luminosity?

The Dynamical Timescale: How Fast Could a Star Collapse?

Imagine you could suddenly “turn off” all the internal pressure holding a star up. How long would it take to collapse under its own gravity? This is the dynamical timescale $\tau_\text{dyn}$ . Dimensionally, the only timescale you can build from $G$ and the mean density $\bar{\rho}$ is:

The equation is not predicting what a healthy star is doing now — it predicts how fast gravity would win if pressure support suddenly failed. (Exact free-fall calculations add numerical factors of order unity; here we keep the scaling and meaning.)

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.

Worked Example 1The Sun's Dynamical Timescale

Problem

The Sun’s mean density is $\bar{\rho}_\odot \approx 1.4~\text{g cm}^{-3}$ (about 1.4× water). Estimate its dynamical timescale.

StepEvaluate inside the square root

$G\bar{\rho}_\odot \approx (6.674 \times 10^{-8})(1.4) \approx 9.3 \times 10^{-8}~\text{s}^{-2} \Rightarrow \sqrt{G\bar{\rho}_\odot} \approx 3.1 \times 10^{-4}~\text{s}^{-1}$

StepInvert

$\tau_\text{dyn,\odot} \sim \frac{1}{3.1 \times 10^{-4}~\text{s}^{-1}} \approx 3.2 \times 10^3~\text{s} \approx 50~\text{min}$

Dimensional check

$[G\bar{\rho}] = (\text{cm}^3\text{g}^{-1}\text{s}^{-2})(\text{g cm}^{-3}) = \text{s}^{-2}$ ; the inverse square root gives seconds ✓.

Result

If pressure support were removed, the Sun would collapse in about an hour. Since it clearly is not collapsing, something holds it up — pressure. Understanding that balance is the next reading.

Problem

A white dwarf has roughly the Sun’s mass but the Earth’s radius ( $R \approx 0.009\,R_\odot$ ). Estimate its mean density and dynamical timescale. How do they compare to the Sun’s?

The Thermal (Kelvin-Helmholtz) Timescale: How Long Could Gravity Alone Power a Star?

Before anyone knew about nuclear reactions, the best guess for the Sun’s energy source was gravitational contraction — the Sun shines by slowly shrinking, converting gravitational potential energy into heat and light. Using the same grammar, $\tau \sim (\text{energy reservoir})/(\text{energy-loss rate})$ , the reservoir is the star’s gravitational binding energy: $|E| \sim GM^2/R$ (a scaling statement; a virial treatment changes only the prefactor). If there is no nuclear source, the luminosity must leak away that bound energy, $L = -dE/dt$ (the minus sign matters: a star losing energy has $dE/dt$ negative while the outward $L$ is positive). Together these give the Kelvin-Helmholtz timescale:

Here $M$ is the mass, $R$ the radius, $L$ the luminosity. The hidden assumption: this is the lifetime if gravitational contraction is the only long-term energy source.

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.

You can also see it from a contraction-rate argument: with $E \sim -GM^2/R$ and roughly constant mass, $dE/dt \sim (GM^2/R^2)\,dR/dt$ ; for a contracting star $dR/dt$ is negative, so $L = -dE/dt \sim (GM^2/R^2)|dR/dt|$ , giving a contraction time $R/|dR/dt| \sim GM^2/(RL) = \tau_\text{KH}$ . The Kelvin-Helmholtz timescale is not a free-fall time — it is the time for a star to contract quasi-statically as it radiates energy away.

Worked Example 2The Sun's Kelvin-Helmholtz Timescale

Problem

Estimate $\tau_\text{KH}$ for the Sun ( $M_\odot = 2.0 \times 10^{33}~\text{g}$ , $R_\odot = 7.0 \times 10^{10}~\text{cm}$ , $L_\odot = 3.8 \times 10^{33}~\text{erg/s}$ ).

StepPlug into GM²/RL

$\tau_\text{KH} \sim \frac{(6.674 \times 10^{-8})(2.0 \times 10^{33})^2}{(7.0 \times 10^{10})(3.8 \times 10^{33})} = \frac{2.67 \times 10^{59}}{2.66 \times 10^{44}}~\text{s} \approx 1.0 \times 10^{15}~\text{s}$

Dimensional check

$\dfrac{(\text{cm}^3\text{g}^{-1}\text{s}^{-2})(\text{g}^2)}{(\text{cm})(\text{erg s}^{-1})} = \dfrac{\text{erg}}{\text{erg s}^{-1}} = \text{s}$ ✓.

Result

$\tau_\text{KH} \approx 30~\text{Myr}$ . This was Lord Kelvin’s answer in the 1860s — he concluded the Sun could not be much older than 20–30 million years. Then the geologists objected.

Multiple choice

If a star has the same mass and radius as the Sun but twice the luminosity, is its Kelvin–Helmholtz timescale longer or shorter?

LongerShorter