AST-29

Escape speed

Sun: 618 km/s.

Reading speed

Astroescape

Governing equation

v_e=\sqrt{2GM/R}

$M \in \left[0.1, 100\right]$

where

M: M (M_\odot)
R: R (R_\odot)
v_e: ve (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.

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①

Governing law — red

v_e=\sqrt{2GM/R}

$M \in \left[0.1, 100\right]$

$M = 0.1$ $v_e = 195.3$

Along the curve

②

Domain of the variable — blue

M_{\min} = 0.1,\quad M_{\max} = 100

$M \in \left[0.1, 100\right]$

$M = 0.1$ $M = 0.1$

Along the curve

③

Response curve — green

v_e=\sqrt{2GM/R}

$M \in \left[0.1, 100\right]$

$M = 0.1$ $v_e = 195.3$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}v_e}{\mathrm{d}M}

$M \in \left[0.1, 100\right]$

$M = 0.728$ $v_e^{\prime} = 417.1$

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-29 — Escape speed) is the form associated with escape. Working symbols: $M$ , $R$ $\rightarrow$ $v_e$ . Sun: 618 km/s.

Purpose

Purpose: compute

v_e

from

M

R

in Astrophysics via

v_e=\sqrt{2GM/R}

Sun: 618 km/s. 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_e=\sqrt{2GM/R}

yields

v_e = 617.70\,\mathrm{km/s}

. Sun: 618 km/s. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

M M1 M_\odotR R1 R_\odot

Outputs

ve v_e617.70 km/s

Reading speed

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Narration of this film

Sun: 618 km/s.

Reading speed

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