AST-16

Virial mass

M = 5 R σ² / G for a uniform self-gravitating sphere (η ≈ 5).

Reading speed

GravityVirial

Governing equation

M=5R\sigma^2/G

$\sigma \in \left[0.5, 400\right]$

where

R: Radius (pc)
\sigma: Dispersion (km/s)
M: Virial 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=5R\sigma^2/G

$\sigma \in \left[0.5, 400\right]$

$\sigma = 0.5$ $M = 2906.3$

Along the curve

②

Domain of the variable — blue

\sigma_{\min} = 0.5,\quad \sigma_{\max} = 400

$\sigma \in \left[0.5, 400\right]$

$\sigma = 0.5$ $\sigma = 0.5$

Along the curve

③

Response curve — green

M=5R\sigma^2/G

$\sigma \in \left[0.5, 400\right]$

$\sigma = 0.5$ $M = 2906.3$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}M}{\mathrm{d}\sigma}

$\sigma \in \left[0.5, 400\right]$

$\sigma = 3.013$ $M^{\prime} = 70042.9$

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-16 — Virial mass) is the form associated with Virial. Working symbols: $R$ , $\sigma$ $\rightarrow$ $M$ . 2K + W = 0. Galaxy clusters and globulars are weighed this way.

Purpose

Purpose: compute

M

from

R

\sigma

in Astrophysics via

M=5R\sigma^2/G

M = 5 R σ² / G for a uniform self-gravitating sphere (η ≈ 5). Use it when a real astrophysics question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

R = 10.000\,\mathrm{pc}

\sigma = 5.000\,\mathrm{km/s}

, the governing relation

M=5R\sigma^2/G

yields

M = 290626.89\,\mathrm{M_odot}

. A swarm of stars, a velocity dispersion, a mass. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Radius R10 pcDispersion \sigma5 km/s

Outputs

Virial mass M290626.89 M_\odot

Reading speed

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Narration of this film

A swarm of stars, a velocity dispersion, a mass.

2K + W = 0. Galaxy clusters and globulars are weighed this way.

Reading speed

Watch on YouTube