AST-09

Kepler's third law

T² = 4π² a³ / (G M). Period of a circular (or mean) orbit.

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GravityKepler III

Governing equation

T^2=4\pi^2 a^3/(GM)

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

where

a: Semi-major axis (AU)
M: Central mass (M_\odot)
T: Period (yr)

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

T^2=4\pi^2 a^3/(GM)

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

$a = 0.01$ $T = 0.001$

Along the curve

②

Domain of the variable — blue

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

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

$a = 0.01$ $a = 0.01$

Along the curve

③

Response curve — green

T^2=4\pi^2 a^3/(GM)

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

$a = 0.01$ $T = 0.001$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}T}{\mathrm{d}a}

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

$a = 0.262$ $T^{\prime} = 0.729$

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-09 — Kepler's third law) is the form associated with Kepler III. Working symbols: $a$ , $M$ $\rightarrow$ $T$ . For two bodies replace M by M₁+M₂. Solar-system form: T²(yr) = a³(AU) if M = M_⊙.

Purpose

Purpose: compute

T

from

a

M

in Astrophysics via

T^2=4\pi^2 a^3/(GM)

T² = 4π² a³ / (G M). Period of a circular (or mean) orbit. Use it when a real astrophysics question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

a = 1.000\,\mathrm{AU}

M = 1.000\,\mathrm{M_odot}

, the governing relation

T^2=4\pi^2 a^3/(GM)

yields

T = 1.000\,\mathrm{yr}

. An ellipse, a ticking period. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Semi-major axis a1 AUCentral mass M1 M_\odot

Outputs

Period T1.000 yr

Reading speed

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Narration of this film

An ellipse, a ticking period.

For two bodies replace M by M₁+M₂. Solar-system form: T²(yr) = a³(AU) if M = M_⊙.

Reading speed

Watch on YouTube