AST-11

Escape velocity

v_esc = √(2 G M / r). The speed to reach infinity from r.

Reading speed

GravityEscape velocity

Governing equation

v_{\mathrm{esc}}=\sqrt{2GM/r}

$r \in \left[0.01, 100\right]$

where

M: Mass (M_\odot)
r: Radius (R_\odot)
v_{esc}: Escape speed (km/s)

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.

00:0 / 00:08

①

Governing law — red

v_{\mathrm{esc}}=\sqrt{2GM/r}

$r \in \left[0.01, 100\right]$

$r = 0.01$ $v_{esc} = 6176.8$

Along the curve

②

Domain of the variable — blue

r_{\min} = 0.01,\quad r_{\max} = 100

$r \in \left[0.01, 100\right]$

$r = 0.01$ $r = 0.01$

Along the curve

③

Response curve — green

v_{\mathrm{esc}}=\sqrt{2GM/r}

$r \in \left[0.01, 100\right]$

$r = 0.01$ $v_{esc} = 6176.8$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}v_{esc}}{\mathrm{d}r}

$r \in \left[0.01, 100\right]$

$r = 0.639$ $v_{esc}^{\prime} = -4474.9$

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-11 — Escape velocity) is the form associated with Escape velocity. Working symbols: $M$ , $r$ $\rightarrow$ $v_{esc}$ . √2 times circular speed. At r_s it equals c — the black-hole edge.

Purpose

Purpose: compute

v_{esc}

from

M

r

in Astrophysics via

v_{\mathrm{esc}}=\sqrt{2GM/r}

v_esc = √(2 G M / r). The speed to reach infinity from r. 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 = 1.000\,\mathrm{M_odot}

r = 1.000\,\mathrm{R_odot}

, the governing relation

v_{\mathrm{esc}}=\sqrt{2GM/r}

yields

v_{esc} = 617.68\,\mathrm{km/s}

. A world, a departing probe, a v_esc. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Mass M1 M_\odotRadius r1 R_\odot

Outputs

Escape speed v_{esc}617.68 km/s

Reading speed

Watch on YouTube

Free library

Full library

Free PDF / open book

YouTube channels

00:0 / 00:08

Narration of this film

A world, a departing probe, a v_esc.

√2 times circular speed. At r_s it equals c — the black-hole edge.

Reading speed

Watch on YouTube