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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

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

ρ∈[1×10−20,1000]\rho \in \left[1\times 10^{-20}, 1000\right]

where

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

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AST-21 · star
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①

Governing law — red

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

ρ∈[1×10−20,1000]\rho \in \left[1\times 10^{-20}, 1000\right]

ρ=1×10−20\rho = 1\times 10^{-20}tff=2.11×107t_{ff} = 2.11\times 10^{7}

②

Domain of the variable — blue

ρmin⁡=1×10−20,ρmax⁡=1000\rho_{\min} = 1\times 10^{-20},\quad \rho_{\max} = 1000

ρ∈[1×10−20,1000]\rho \in \left[1\times 10^{-20}, 1000\right]

ρ=1×10−20\rho = 1\times 10^{-20}ρ=1×10−20\rho = 1\times 10^{-20}

③

Response curve — green

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

ρ∈[1×10−20,1000]\rho \in \left[1\times 10^{-20}, 1000\right]

ρ=1×10−20\rho = 1\times 10^{-20}tff=2.11×107t_{ff} = 2.11\times 10^{7}

④

Rate of change — amber

dtffdρ\frac{\mathrm{d}t_{ff}}{\mathrm{d}\rho}

ρ∈[1×10−20,1000]\rho \in \left[1\times 10^{-20}, 1000\right]

ρ=6.289\rho = 6.289tff′=−1.67×106t_{ff}^{\prime} = -1.67\times 10^{6}

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: ρ\rho →\rightarrow tfft_{ff}. Independent of radius. Molecular clouds: ~1 Myr. Stellar cores: minutes.

Purpose

Purpose: compute tfft_{ff} from ρ\rho in Astrophysics via tff=3π/(32Gρ)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 ρ=1.000e−16 kg/m3\rho = 1.000e-16\,\mathrm{kg/m^{3}}, the governing relation tff=3π/(32Gρ)t_{ff}=\sqrt{3\pi/(32 G\rho)} yields tff=210500.798 yrt_{ff} = 210500.798\,\mathrm{yr}. A cloud, a shrinking radius, a clock. Move a slider: the numbers are this situation, not a canned story.

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Inputs

Outputs

  • Free-fall time t_{ff}210500.798 yr
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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