User:JohnOwens/Orbital equations

=Variables=

Time-related

 * &omega; angular velocity
 * N rotational speed
 * T time (of period)

Distance-related

 * r radius
 * v velocity (tangential)
 * a acceleration
 * $$a_c$$ centripetal acceleration

Gravitational

 * MG product of central or total mass and gravitational constant

=Cumulative equations=
 * 1) $$\omega \equiv 2\pi N$$
 * 2) $$\omega T \equiv 2\pi$$
 * 3) $$N T \equiv 1$$
 * 4) $$\omega r \equiv v$$
 * 5) $$\omega^2 r \equiv a$$
 * 6) $$\omega^2 r^3 = MG$$
 * 7) $$\omega v = a$$
 * 8) $$\omega MG = v^3$$
 * 9) $$\omega^4 MG = a^3$$
 * 10) $$T v = 2\pi r$$
 * 11) $$T^2 a = 4\pi^2 r$$
 * 12) $$T^2 MG = 4\pi^2 r^3$$
 * 13) $$T a = 2\pi v$$
 * 14) $$T v^3 = 2\pi MG$$
 * 15) $$T^4 a^3 = 16\pi^4 MG$$
 * 16) $$2\pi N r = v$$
 * 17) $$4\pi^2 N^2 a = r$$
 * 18) $$4\pi^2 N^2 r^3 = MG$$
 * 19) $$2\pi N v = a$$
 * 20) $$2\pi N MG = v^3$$
 * 21) $$16\pi^4N^4 MG = a^3$$
 * 22) $$r a \equiv v^2$$
 * 23) $$r v^2 = MG$$
 * 24) $$a r^2 \equiv MG$$
 * 25) $$v^4 = a MG$$

=Isolated variable equations=

&omega;

 * 1) $$\omega \equiv {2 \pi \over T} \equiv 2 \pi N$$
 * 2) $$\omega = {v \over r}$$
 * 3) $$\omega = {a \over v}$$
 * 4) $$\omega = \sqrt{a \over r}$$
 * 5) $$\omega = {v^3 \over MG}$$
 * 6) $$\omega = \sqrt{MG \over r^3}$$
 * 7) $$\omega = \sqrt[4]{a^3 \over MG}$$

N

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

T

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

r

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

v

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

a

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

MG

 * 1) $$MG = \omega^2 r^3 \equiv {4\pi^2 r^3 \over T^2} \equiv 4\pi^2N^2 r$$
 * 2) $$MG = {v^3 \over \omega} \equiv {T v^3 \over 2\pi} \equiv {v^3 \over 2\pi N}$$
 * 3) $$MG = {a^3 \over \omega^4} \equiv {T^4 a^3 \over 16\pi^4} \equiv {a^3 \over 16\pi^4N^4}$$
 * 4) $$MG = r v^2$$
 * 5) $$MG = r^2 a$$
 * 6) $$MG = {v^4 \over a}$$