AST-08

Schwarzschild radius

r_s = 2 G M / c². Event-horizon scale of a non-spinning black hole.

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GravitySchwarzschild

Governing equation

r_s=2GM/c^2

$M \in \left[0.1, 1\times 10^{9}\right]$

where

M: Mass (M_\odot)
r_s: Schwarzschild radius (km)

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

r_s=2GM/c^2

$M \in \left[0.1, 1\times 10^{9}\right]$

$M = 0.1$ $r_s = 0.295$

Along the curve

②

Domain of the variable — blue

M_{\min} = 0.1,\quad M_{\max} = 1\times 10^{9}

$M \in \left[0.1, 1\times 10^{9}\right]$

$M = 0.1$ $M = 0.1$

Along the curve

③

Response curve — green

r_s=2GM/c^2

$M \in \left[0.1, 1\times 10^{9}\right]$

$M = 0.1$ $r_s = 0.295$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}r_s}{\mathrm{d}M}

$M \in \left[0.1, 1\times 10^{9}\right]$

$M = 6.29\times 10^{6}$ $r_s^{\prime} = 2.953$

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-08 — Schwarzschild radius) is the form associated with Schwarzschild. Working symbols: $M$ $\rightarrow$ $r_s$ . The Sun's r_s is 3 km. Nothing inside r_s communicates with infinity.

Purpose

Purpose: compute

r_s

from

M

in Astrophysics via

r_s=2GM/c^2

r_s = 2 G M / c². Event-horizon scale of a non-spinning black hole. Use it when a real astrophysics question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

M = 10.000\,\mathrm{M_odot}

, the governing relation

r_s=2GM/c^2

yields

r_s = 29.53\,\mathrm{km}

. A hole, a photon sphere, an r_s ring. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Mass M10 M_\odot

Outputs

Schwarzschild radius r_s29.53 km

Reading speed

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Narration of this film

A hole, a photon sphere, an r_s ring.

The Sun's r_s is 3 km. Nothing inside r_s communicates with infinity.

Reading speed

Watch on YouTube