AST-10

Circular orbit speed

v = √(G M / r). Centripetal balance of gravity.

Reading speed

GravityCircular orbit

Governing equation

v=\sqrt{GM/r}

$r \in \left[0.01, 40\right]$

where

M: Mass (M_\odot)
r: Radius (AU)
v: Orbital 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.

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①

Governing law — red

v=\sqrt{GM/r}

$r \in \left[0.01, 40\right]$

$r = 0.01$ $v = 297.9$

Along the curve

②

Domain of the variable — blue

r_{\min} = 0.01,\quad r_{\max} = 40

$r \in \left[0.01, 40\right]$

$r = 0.01$ $r = 0.01$

Along the curve

③

Response curve — green

v=\sqrt{GM/r}

$r \in \left[0.01, 40\right]$

$r = 0.01$ $v = 297.9$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}v}{\mathrm{d}r}

$r \in \left[0.01, 40\right]$

$r = 0.262$ $v^{\prime} = -509.5$

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-10 — Circular orbit speed) is the form associated with Circular orbit. Working symbols: $M$ , $r$ $\rightarrow$ $v$ . Low Earth orbit is ~7.8 km/s. Galactic rotation curves that stay flat need dark matter.

Purpose

Purpose: compute

v

from

M

r

in Astrophysics via

v=\sqrt{GM/r}

v = √(G M / r). Centripetal balance of gravity. 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{AU}

, the governing relation

v=\sqrt{GM/r}

yields

v = 29.785\,\mathrm{km/s}

. A planet, a circle, a speed tag. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Mass M1 M_\odotRadius r1 AU

Outputs

Orbital speed v29.785 km/s

Reading speed

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Narration of this film

A planet, a circle, a speed tag.

Low Earth orbit is ~7.8 km/s. Galactic rotation curves that stay flat need dark matter.

Reading speed

Watch on YouTube