User:JohnOwens/Orbital equations

Variables

Time-related

• ω 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. $NT\equiv 1$
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. $Tv=2\pi r$
11. $T^{2}a=4\pi ^{2}r$
12. $T^{2}MG=4\pi ^{2}r^{3}$
13. $Ta=2\pi v$
14. $Tv^{3}=2\pi MG$
15. $T^{4}a^{3}=16\pi ^{4}MG$
16. $2\pi Nr=v$
17. $4\pi ^{2}N^{2}a=r$
18. $4\pi ^{2}N^{2}r^{3}=MG$
19. $2\pi Nv=a$
20. $2\pi NMG=v^{3}$
21. $16\pi ^{4}N^{4}MG=a^{3}$
22. $ra\equiv v^{2}$
23. $rv^{2}=MG$
24. $ar^{2}\equiv MG$
25. $v^{4}=aMG$

Isolated variable equations

Time-related

ω

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

Distance-related

r

1. $r={v \over \omega }\equiv {v \over 2\pi N}\equiv {Tv \over 2\pi }$
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. $v=\omega r\equiv 2\pi Nr\equiv {2\pi r \over T}$
2. $v={a \over \omega }\equiv {a \over 2\pi N}\equiv {Ta \over 2\pi }$
3. $v={\sqrt[{3}]{\omega MG}}\equiv {\sqrt[{3}]{2\pi NMG}}\equiv {\sqrt[{3}]{2\pi MG \over T}}$
4. $v={\sqrt {ra}}$
5. $v={\sqrt {MG \over r}}$
6. $v={\sqrt[{4}]{aMG}}$

a

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

Gravitational

MG

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