AST-20

Hill sphere

r_H = a (m / 3 M)^{1/3}. The satellite's gravitational territory.

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GravityHill

Governing equation

r_H=a(m/3M)^{1/3}

$a \in \left[0.1, 40\right]$

where

a: Orbit radius (AU)
m: Satellite mass (M_\oplus)
M: Primary mass (M_\odot)
r_H: Hill radius (AU)

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_H=a(m/3M)^{1/3}

$a \in \left[0.1, 40\right]$

$a = 0.1$ $r_H = 0.001$

Along the curve

②

Domain of the variable — blue

a_{\min} = 0.1,\quad a_{\max} = 40

$a \in \left[0.1, 40\right]$

$a = 0.1$ $a = 0.1$

Along the curve

③

Response curve — green

r_H=a(m/3M)^{1/3}

$a \in \left[0.1, 40\right]$

$a = 0.1$ $r_H = 0.001$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}r_H}{\mathrm{d}a}

$a \in \left[0.1, 40\right]$

$a = 0.351$ $r_H^{\prime} = 0.01$

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-20 — Hill sphere) is the form associated with Hill. Working symbols: $a$ , $m$ , $M$ $\rightarrow$ $r_H$ . Moons and Trojan clouds live inside r_H. Beyond, the star (or planet) wins.

Purpose

Purpose: compute

r_H

from

a

m

M

in Astrophysics via

r_H=a(m/3M)^{1/3}

r_H = a (m / 3 M)^{1/3}. The satellite's gravitational territory. 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_oplus}

M = 1.000\,\mathrm{M_odot}

, the governing relation

r_H=a(m/3M)^{1/3}

yields

r_H = 0.0100\,\mathrm{AU}

. A planet, a surrounding sphere of control. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Orbit radius a1 AUSatellite mass m1 M_\oplusPrimary mass M1 M_\odot

Outputs

Hill radius r_H0.0100 AU

Reading speed

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Narration of this film

A planet, a surrounding sphere of control.

Moons and Trojan clouds live inside r_H. Beyond, the star (or planet) wins.

Reading speed

Watch on YouTube