====== Stellar timescales ======

A star may remain in a state of **hydrostatic equilibrium** for most of its lifetime, but different physical processes act over different characteristic timescales. Once equilibrium is disturbed—by exhaustion of fuel, loss of pressure support, or radiative imbalance—the **rate of change** is governed by these intrinsic timescales.

Three important timescales determine the pace of stellar evolution:

  * **Thermal (Kelvin–Helmholtz) timescale** — governs how long a star can radiate away its thermal energy if nuclear fusion ceases.  
  * **Dynamical timescale** — measures how fast a star would collapse under gravity if pressure vanished.  
  * **Diffusion timescale** — quantifies how long it takes photons to random-walk from the stellar core to the surface.

===== Thermal (Kelvin–Helmholtz) timescale =====

The **thermal timescale** \( \tau_{\mathrm{th}} \) (also called the *Kelvin–Helmholtz timescale*) is the time required for a star to radiate away its total internal (thermal) energy \(E_k\) at its luminosity \(L\):

$$
\tau_{\mathrm{th}} \approx \frac{\sum E_k}{L}.
$$

From the **virial theorem**, the total kinetic energy is half the magnitude of the gravitational potential energy:

$$
\sum E_p = -2 \sum E_k = -\frac{G M^2}{R},
$$

where \(M\) is the stellar mass, \(R\) its radius, and \(G\) the gravitational constant. Substituting gives

$$
\tau_{\mathrm{th}} \approx \frac{G M^2}{R L}.
$$

If nuclear reactions in the Sun stopped today, it would radiate away its stored thermal energy in roughly **30 million years (30 Myr)**.  
However, we know the Sun’s age exceeds **4.5 billion years (4.5 Gyr)**—older than Earth itself—implying that **nuclear fusion** provides a continuous source of internal energy far exceeding the Kelvin–Helmholtz limit.

---

===== Dynamical timescale =====

The **dynamical timescale** \( \tau_{\mathrm{dyn}} \) is the time a star would take to collapse under its own gravity if pressure were suddenly removed.

Consider a small test mass \(m\) on the stellar surface. Its inward acceleration due to gravity is

$$
a = \frac{F}{m} = \frac{G M}{R^2}.
$$

If we (unrealistically) assume \(a\) remains constant during the fall, then the free-fall time for a distance \(R\) follows from \(s = a t^2 / 2\):

$$
\tau_{\mathrm{dyn}} = \sqrt{\frac{R}{a}} = \sqrt{\frac{R^3}{G M}}.
$$

Expressing the mean density as \( \rho = M / R^3 \), we obtain the elegant form

$$
\tau_{\mathrm{dyn}} = (G \rho)^{-1/2}.
$$

For the Sun (\(\rho_\odot \approx 1.4\times10^3\ \mathrm{kg\,m^{-3}}\)),  
\(\tau_{\mathrm{dyn}} \approx 50\ \text{minutes}\), reducing to about **20 minutes** when accounting for internal density variations.

> **Exercise:**  
> A white dwarf has \(M \approx M_\odot\) but \(R \approx 0.01 R_\odot\).  
> Estimate its density and corresponding \(\tau_{\mathrm{dyn}}\).  
> What about a neutron star with nuclear density \( \rho \sim 10^9\ \mathrm{kg\,m^{-3}} \)?  
> How short can \(\tau_{\mathrm{dyn}}\) become?

---

===== Diffusion timescale =====

The **diffusion timescale** \( \tau_{\mathrm{diff}} \) measures how long a photon takes to travel from the stellar core to the surface.  
Because photons are constantly scattered or absorbed and re-emitted (e.g. by [[en:Thomson scattering]]), their path is a **random walk** rather than a straight line.

Let the step size be the **mean free path** \(l\). In a one-dimensional random walk of \(N\) steps, the root-mean-square (rms) displacement is

$$
x_{\mathrm{rms}} = \sqrt{N}\, l.
$$

To traverse a distance \(x\), the number of steps required is

$$
N \approx \left(\frac{x}{l}\right)^2.
$$

In three dimensions, replacing \(x\) by the stellar radius \(R\):

$$
N \approx \left(\frac{R}{l}\right)^2.
$$

Since each step takes a time \(l/c\) (where \(c\) is the speed of light), the total diffusion time is

$$
\tau_{\mathrm{diff}} = \frac{N l}{c} = \frac{R^2}{c l}.
$$

To compute \(l\), we use the **Thomson cross-section** for electron scattering:

$$
\sigma_T = \frac{8\pi}{3} r_e^2,
$$

where the **classical electron radius**

$$
r_e = \frac{1}{4\pi \epsilon_0} \frac{e^2}{m_e c^2}.
$$

For a fully ionized hydrogen plasma, the electron number density equals the proton number density \(n_e = n_p = \frac{M / R^3}{m_p}\), giving

$$
l = \frac{1}{\sigma_T n_e} = \frac{m_p R^3}{\sigma_T M}.
$$

Substituting this into the earlier expression yields

$$
\tau_{\mathrm{diff}} = \frac{\sigma_T}{c m_p} \frac{M}{R}.
$$

For solar parameters, this gives approximately **10,000 years**, and accounting for absorption–reemission processes, the actual photon escape time is closer to **20,000 years**.

---

The Sun’s **luminosity** can be roughly estimated from its internal energy and photon diffusion time:

$$
L_\odot \approx \frac{a T^4 (4\pi R_\odot^3 / 3)}{\tau_{\mathrm{diff}}} \approx 10^{27}\ \mathrm{W},
$$

or about **1 ronnawatt (RW)**—an octillion watts.

Interestingly, photons have about a thousand times less energy than typical particles in the solar plasma.  
Thus, if the Sun were composed entirely of photons, it would take roughly **a thousand times longer** to lose all its energy—comparable to the **thermal timescale**.

---

===== Insights =====
  * The thermal timescale (\(\tau_{\mathrm{th}}\)) determines how fast a star cools without fusion.
  * The dynamical timescale (\(\tau_{\mathrm{dyn}}\)) sets the limit for structural collapse or oscillations.
  * The diffusion timescale (\(\tau_{\mathrm{diff}}\)) explains the long delay between core energy generation and surface radiation.
  * For the Sun, these timescales are hierarchically ordered:
    $$
    \tau_{\mathrm{dyn}} \ll \tau_{\mathrm{diff}} \ll \tau_{\mathrm{th}} \ll \tau_{\mathrm{nuc}}
    $$
showing why stars appear so stable over billions of years.

===== Inquiries =====
  - Derive the ratio of \(\tau_{\mathrm{dyn}}\) to \(\tau_{\mathrm{th}}\) for the Sun and interpret its meaning.  
  - What physical processes dominate in objects where \(\tau_{\mathrm{dyn}} \approx \tau_{\mathrm{th}}\)?  
  - How would the diffusion timescale change for a red giant compared to a main-sequence star?  
  - Why is the Kelvin–Helmholtz timescale much shorter than the nuclear timescale?  
  - Explain how the virial theorem connects gravitational and thermal energies in estimating \(\tau_{\mathrm{th}}\).