AST-24

Tidal acceleration

a_tid = 2 G M R / d³. Differential gravity across a body of radius R.

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GravityTides

Governing equation

a_{\mathrm{tid}}=2GMR/d^3

$d \in \left[1000, 2\times 10^{6}\right]$

where

M: Perturber mass (M_\oplus)
R: Target radius (km)
d: Separation (km)
a_{tid}: Tidal acceleration (µm/s²)

The figure, part by part

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①

Governing law — red

a_{\mathrm{tid}}=2GMR/d^3

$d \in \left[1000, 2\times 10^{6}\right]$

$d = 1000$ $a_{tid} = 1.38\times 10^{9}$

Along the curve

②

Domain of the variable — blue

d_{\min} = 1000,\quad d_{\max} = 2\times 10^{6}

$d \in \left[1000, 2\times 10^{6}\right]$

$d = 1000$ $d = 1000$

Along the curve

③

Response curve — green

a_{\mathrm{tid}}=2GMR/d^3

$d \in \left[1000, 2\times 10^{6}\right]$

$d = 1000$ $a_{tid} = 1.38\times 10^{9}$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}a_{tid}}{\mathrm{d}d}

$d \in \left[1000, 2\times 10^{6}\right]$

$d = 13572.3$ $a_{tid}^{\prime} = -55066.2$

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-24 — Tidal acceleration) is the form associated with Tides. Working symbols: $M$ , $R$ , $d$ $\rightarrow$ $a_{tid}$ . Roche limit when a_tid ~ self-gravity. Moons shred into rings inside it.

Purpose

Purpose: compute

a_{tid}

from

M

R

d

in Astrophysics via

a_{\mathrm{tid}}=2GMR/d^3

a_tid = 2 G M R / d³. Differential gravity across a body of radius 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_oplus}

R = 1737.000\,\mathrm{km}

d = 384000.000\,\mathrm{km}

, the governing relation

a_{\mathrm{tid}}=2GMR/d^3

yields

a_{tid} = 24.455\,\mathrm{\mu m/s^{2}}

. A planet, two ends of a moon, two arrows. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Perturber mass M1 M_\oplusTarget radius R1737 kmSeparation d384000 km

Outputs

Tidal acceleration a_{tid}24.455 µm/s²

Reading speed

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Narration of this film

A planet, two ends of a moon, two arrows.

Roche limit when a_tid ~ self-gravity. Moons shred into rings inside it.

Reading speed

Watch on YouTube