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User:JohnOwens/Orbital equations - Wikipedia, the free encyclopedia

User:JohnOwens/Orbital equations

From Wikipedia, the free encyclopedia

< User:JohnOwens

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Contents

1 Variables
2 Cumulative equations
3 Isolated variable equations

[edit] Variables

[edit] Time-related

ω angular velocity
N rotational speed
T time (of period)

[edit] Distance-related

r radius
v velocity (tangential)
a acceleration
- $a c$ centripetal acceleration

[edit] Gravitational

MG product of central or total mass and gravitational constant

[edit] Cumulative equations

$\omega \equiv 2\pi N$
$\omega T \equiv 2\pi$
$N T \equiv 1$
$\omega r \equiv v$
$\omega^2 r \equiv a$
$ω 2 r 3 = M G$
$ω v = a$
$ω M G = v 3$
$ω 4 M G = a 3$
$T v = 2π r$
$T 2 a = 4π 2 r$
$T 2 M G = 4π 2 r 3$
$T a = 2π v$
$T v 3 = 2π M G$
$T 4 a 3 = 16π 4 M G$
$2π N r = v$
$4π 2 N 2 a = r$
$4π 2 N 2 r 3 = M G$
$2π N v = a$
$2π N M G = v 3$
$16π 4 N 4 M G = a 3$
$r a \equiv v^2$
$r v 2 = M G$
$a r^2 \equiv MG$
$v 4 = a M G$

[edit] Isolated variable equations

[edit] Time-related

[edit] ω

$\omega \equiv {2 \pi \over T} \equiv 2 \pi N$
$\omega = {v \over r}$
$\omega = {a \over v}$
$\omega = \sqrt{a \over r}$
$\omega = {v^3 \over MG}$
$\omega = \sqrt{MG \over r^3}$
$\omega = \sqrt[4]{a^3 \over MG}$

[edit] N

$N \equiv {\omega \over 2\pi} \equiv {1 \over T}$
$N = {v \over 2\pi r}$
$N = {a \over 2\pi v}$
$N = {\sqrt{r} \over 2\pi\sqrt{a}} \equiv \sqrt{r \over 4\pi^2a}$
$N = {v^3 \over 2\pi MG}$
$N = {\sqrt{MG} \over 2\pi\sqrt{r^3}} \equiv \sqrt{MG \over 4\pi^2r^3}$
$N = {\sqrt[4]{a^3 \over MG} \over 2\pi} \equiv \sqrt[4]{a^3 \over 16\pi^4 MG}$

[edit] T

$T \equiv {2\pi \over \omega} \equiv {1 \over N}$
$T = {2\pi r \over v}$
$T = {2\pi v \over a}$
$T = 2\pi\sqrt{r \over a} \equiv \sqrt{4\pi^2 r \over a}$
$T = {2\pi MG \over v^3}$
$T = 2\pi\sqrt{r^3 \over MG} \equiv \sqrt{4\pi^2 r^3 \over MG}$
$T = 2\pi\sqrt[4]{MG \over a^3} \equiv \sqrt[4]{16\pi^4 MG \over a^3}$

[edit] Distance-related

[edit] r

$r = {v \over \omega} \equiv {v \over 2\pi N} \equiv {T v \over 2\pi}$
$r = {a \over \omega^2} \equiv {a \over 4\pi^2 N^2} \equiv {T^2 a \over 4\pi^2}$
$r = \sqrt[3]{MG \over \omega^2} \equiv \sqrt[3]{MG \over 4\pi^2 N^2} \equiv \sqrt[3]{T^2 MG \over 4\pi^2}$
$r = {v^2 \over a}$
$r = {MG \over v^2}$
$r \equiv \sqrt{MG \over a}$

[edit] v

$v = \omega r \equiv 2\pi N r \equiv {2\pi r \over T}$
$v = {a \over \omega} \equiv {a \over 2\pi N} \equiv {T a \over 2\pi}$
$v = \sqrt[3]{\omega MG} \equiv \sqrt[3]{2\pi N MG} \equiv \sqrt[3]{2\pi MG \over T}$
$v = \sqrt{r a}$
$v = \sqrt{MG \over r}$
$v = \sqrt[4]{a MG}$

[edit] a

$a = \omega r^2 \equiv {4\pi^2 r \over T^2} \equiv 4\pi^2N^2 r$
$a = \omega v \equiv {2\pi T \over v} \equiv 2\pi N v$
$a = \sqrt[3]{\omega^4 MG} \equiv \sqrt[3]{16\pi^4 MG \over T^4} \equiv \sqrt[3]{16\pi^4N^4 MG}$
$a = {v^2 \over r}$
$a \equiv {MG \over r^2}$
$a = {v^4 \over MG}$

[edit] Gravitational

[edit] MG

$MG = \omega^2 r^3 \equiv {4\pi^2 r^3 \over T^2} \equiv 4\pi^2N^2 r$
$MG = {v^3 \over \omega} \equiv {T v^3 \over 2\pi} \equiv {v^3 \over 2\pi N}$
$MG = {a^3 \over \omega^4} \equiv {T^4 a^3 \over 16\pi^4} \equiv {a^3 \over 16\pi^4N^4}$
$M G = r v 2$
$M G = r 2 a$
$MG = {v^4 \over a}$

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