AST-21

Free-fall time

t_ff = √(3π / 32 G ρ). Collapse time of a uniform sphere.

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GravityFree-fall

Governing equation

t_{ff}=\sqrt{3\pi/(32 G\rho)}

where

\rho: Density (kg/m³)
t_{ff}: Free-fall time (yr)

Lecture brief

Historical brief

Hubble expansion, Jeans collapse, Eddington luminosity, Bondi accretion and Stefan–Boltzmann stars are the first astrophysical budgets. The sheets scale a star, a cloud and a horizon. This sheet (AST-21 — Free-fall time) is the form associated with Free-fall. Working symbols: $\rho$ $\rightarrow$ $t_{ff}$ . Independent of radius. Molecular clouds: ~1 Myr. Stellar cores: minutes.

Purpose

Purpose: compute

t_{ff}

from

\rho

in Astrophysics via

t_{ff}=\sqrt{3\pi/(32 G\rho)}

t_ff = √(3π / 32 G ρ). Collapse time of a uniform sphere. Use it when a real astrophysics question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

\rho = 1.000e-16\,\mathrm{kg/m^{3}}

, the governing relation

t_{ff}=\sqrt{3\pi/(32 G\rho)}

yields

t_{ff} = 210500.798\,\mathrm{yr}

. A cloud, a shrinking radius, a clock. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Density \rho1e-16 kg/m³

Outputs

Free-fall time t_{ff}210500.798 yr

Reading speed

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Narration of this film

A cloud, a shrinking radius, a clock.

Independent of radius. Molecular clouds: ~1 Myr. Stellar cores: minutes.

Reading speed

Watch on YouTube