AST-04

Jeans mass

M_J = (π/6) c_s³ / √(G³ ρ). The mass that just collapses against pressure.

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GravityJeans

Governing equation

M_J=\dfrac{\pi}{6}\dfrac{c_s^3}{\sqrt{G^3\rho}}

$\rho \in \left[0.001, 1000\right]$

where

c_s: Sound speed (m/s)
\rho: Density (10⁻¹⁸ kg/m³)
M_J: Jeans mass (M_\odot)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

M_J=\dfrac{\pi}{6}\dfrac{c_s^3}{\sqrt{G^3\rho}}

$\rho \in \left[0.001, 1000\right]$

$\rho = 0.001$ $M_J = 122.2$

Along the curve

②

Domain of the variable — blue

\rho_{\min} = 0.001,\quad \rho_{\max} = 1000

$\rho \in \left[0.001, 1000\right]$

$\rho = 0.001$ $\rho = 0.001$

Along the curve

③

Response curve — green

M_J=\dfrac{\pi}{6}\dfrac{c_s^3}{\sqrt{G^3\rho}}

$\rho \in \left[0.001, 1000\right]$

$\rho = 0.001$ $M_J = 122.2$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}M_J}{\mathrm{d}\rho}

$\rho \in \left[0.001, 1000\right]$

$\rho = 6.29$ $M_J^{\prime} = -9.626$

Along the curve

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-04 — Jeans mass) is the form associated with Jeans. Working symbols: $c_s$ , $\rho$ $\rightarrow$ $M_J$ . λ_J = c_s √(π / G ρ). Clouds heavier than M_J fragment into stars.

Purpose

Purpose: compute

M_J

from

c_s

\rho

in Astrophysics via

M_J=\dfrac{\pi}{6}\dfrac{c_s^3}{\sqrt{G^3\rho}}

M_J = (π/6) c_s³ / √(G³ ρ). The mass that just collapses against pressure. Use it when a real astrophysics question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

c_s = 200.000\,\mathrm{m/s}

\rho = 1.000\,\mathrm{10⁻¹⁸ kg/m^{3}}

, the governing relation

M_J=\dfrac{\pi}{6}\dfrac{c_s^3}{\sqrt{G^3\rho}}

yields

M_J = 3.863\,\mathrm{M_odot}

. A cloud, a critical sphere, a collapse. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Sound speed c_s200 m/sDensity \rho1 10⁻¹⁸ kg/m³

Outputs

Jeans mass M_J3.863 M_\odot

Reading speed

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Narration of this film

A cloud, a critical sphere, a collapse.

λ_J = c_s √(π / G ρ). Clouds heavier than M_J fragment into stars.

Reading speed

Watch on YouTube