Hydrostatic Equilibrium

Hydrostatic Equilibrium

A star is not collapsing — it is in a delicate balance between gravity pulling inward and pressure pushing outward. This balance is called hydrostatic equilibrium.

The Equation

At every point inside a star, the pressure gradient must exactly counteract gravity:

$\frac{dP}{dr} = -\rho g = -\frac{G M(r) \rho}{r^2}$

where $M(r)$ is the mass enclosed within radius $r$.

Central Pressure Estimate

Using the virial theorem approximation, the central pressure of a star scales as:

$P_c \approx \frac{G M^2}{4\pi R^4}$

For the Sun, this gives $P_c \approx 9 \times 10^{13}$ Pa — about a billion atmospheres.

Free-Fall (Dynamical) Timescale

If pressure support were suddenly removed, a star would collapse under its own gravity on the free-fall timescale:

$t_{\rm ff} = \sqrt{\frac{3\pi}{32 G \rho}}$

where $\rho$ is the mean density. For the Sun, $t_{\rm ff} \approx 1769$ s — about 30 minutes.

Mean Density

$\bar{\rho} = \frac{M}{\frac{4}{3}\pi R^3}$

The Sun's mean density is about 1410 kg/m³ — slightly denser than water.

Your Task

Implement three functions. Use $G = 6.674 \times 10^{-11}$ m³ kg⁻¹ s⁻², defined inside each function.

• central_pressure_Pa(M_kg, R_m) — virial estimate of central pressure in Pa
• freefall_timescale_s(rho_kg_m3) — free-fall collapse timescale in seconds
• mean_density_kg_m3(M_kg, R_m) — mean density of a spherical body in kg/m³