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review derivation: radius for satellite in geostationary orbit

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Notes for this derivation:
https://en.wikipedia.org/wiki/Geostationary_orbit#Derivation_of_geostationary_altitude

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Index Inference Rule Input latex Feeds latex Output latex step validity dimension check unit check notes
3 change four variables in expr
  1. 9226945488; locally 8242154:
    \(F = \frac{m v^2}{r}\)
    \(pdg_{4202} = \frac{pdg_{1357}^{2} pdg_{5156}}{pdg_{2530}}\)
  1. 5089196493:
    \(F\)
    \(pdg_{4202}\)
  2. 1333474099:
    \(F_{\rm centripetal}\)
    \(pdg_{1687}\)
  3. 3342155559:
    \(m\)
    \(pdg_{5156}\)
  4. 2114570475:
    \(m_{\rm satellite}\)
    \(pdg_{3569}\)
  5. 7912578203:
    \(v\)
    \(pdg_{1357}\)
  6. 9789485295:
    \(v_{\rm satellite}\)
    \(pdg_{4082}\)
  1. 4627284246; locally 6845877:
    \(F_{\rm centripetal} = \frac{m_{\rm satellite} v_{\rm satellite}^2}{r}\)
    \(pdg_{1687} = \frac{pdg_{3569} pdg_{4082}^{2}}{pdg_{2530}}\)
failed 9226945488:
4627284246:
9226945488:
4627284246:
12 multiply both sides by
  1. 3906710072; locally 2871066:
    \(G \frac{m_{\rm Earth}}{r} = \frac{4 \pi^2 r^2}{T_{\rm orbit}^2}\)
    \(\frac{pdg_{5458} pdg_{6277}}{pdg_{2530}} = \frac{4 pdg_{2530}^{2} pdg_{3141}^{2}}{pdg_{8762}^{2}}\)
  1. 6238632840:
    \(r T_{\rm orbit}^2\)
    \(pdg_{2530} pdg_{8762}^{2}\)
  1. 7010294143; locally 7188516:
    \(T_{\rm orbit}^2 G m_{\rm Earth} = 4 \pi^2 r^3\)
    \(pdg_{5458} pdg_{6277} pdg_{8762}^{2} = 4 pdg_{2530}^{3} pdg_{3141}^{2}\)
valid 3906710072:
7010294143:
3906710072:
7010294143:
14 raise both sides to power
  1. 4858693811; locally 6238570:
    \(\frac{T_{\rm orbit}^2 G m_{\rm Earth}}{4 \pi^2} = r^3\)
    \(\frac{pdg_{5458} pdg_{6277} pdg_{8762}^{2}}{4 pdg_{3141}^{2}} = pdg_{2530}^{3}\)
  1. 4319544433:
    \(1/3\)
    \(\frac{1}{3}\)
  1. 2617541067; locally 7139326:
    \(\left(\frac{T_{\rm orbit}^2 G m_{\rm Earth}}{4 \pi^2}\right)^{1/3} = r\)
    \(\frac{\sqrt[3]{2} \sqrt[3]{\frac{pdg_{5458} pdg_{6277} pdg_{8762}^{2}}{pdg_{3141}^{2}}}}{2} = pdg_{2530}\)
no check is performed 4858693811:
2617541067:
4858693811:
2617541067:
10 divide both sides by
  1. 4072200527; locally 4948724:
    \(\frac{m_{\rm satellite} v_{\rm satellite}^2}{r} = G \frac{m_{\rm Earth} m_{\rm satellite}}{r^2}\)
    \(\frac{pdg_{3569} pdg_{4082}^{2}}{pdg_{2530}} = \frac{pdg_{3569} pdg_{5458} pdg_{6277}}{pdg_{2530}^{2}}\)
  1. 5359471792:
    \(\frac{m_{\rm satellite}}{r}\)
    \(\frac{pdg_{3569}}{pdg_{2530}}\)
  1. 1994296484; locally 2009493:
    \(v_{\rm satellite}^2 = G \frac{m_{\rm Earth}}{r}\)
    \(pdg_{4082}^{2} = \frac{pdg_{5458} pdg_{6277}}{pdg_{2530}}\)
valid 4072200527:
1994296484:
4072200527:
1994296484:
16 change two variables in expr
  1. 2617541067; locally 7139326:
    \(\left(\frac{T_{\rm orbit}^2 G m_{\rm Earth}}{4 \pi^2}\right)^{1/3} = r\)
    \(\frac{\sqrt[3]{2} \sqrt[3]{\frac{pdg_{5458} pdg_{6277} pdg_{8762}^{2}}{pdg_{3141}^{2}}}}{2} = pdg_{2530}\)
  1. 3846345263:
    \(T_{\rm orbit}\)
    \(pdg_{8762}\)
  2. 5208737840:
    \(T_{\rm geostationary\ orbit}\)
    \(pdg_{5595}\)
  3. 5770088141:
    \(r\)
    \(pdg_{2530}\)
  4. 7053449926:
    \(r_{\rm geostationary\ orbit}\)
    \(pdg_{7110}\)
  1. 1559688463; locally 4507350:
    \(\left(\frac{T_{\rm geostationary\ orbit}^2 G m_{\rm Earth}}{4 \pi^2}\right)^{1/3} = r_{\rm geostationary\ orbit}\)
    \(\frac{\sqrt[3]{2} \sqrt[3]{\frac{pdg_{5458} pdg_{5595}^{2} pdg_{6277}}{pdg_{3141}^{2}}}}{2} = pdg_{7110}\)
valid 2617541067:
1559688463:
2617541067:
1559688463:
7 substitute LHS of expr 1 into expr 2
  1. 9262596735; locally 5369477:
    \(d = 2 \pi r\)
    \(pdg_{1943} = 2 pdg_{2530} pdg_{3141}\)
  2. 5426308937; locally 5114041:
    \(v = \frac{d}{t}\)
    \(pdg_{1357} = \frac{pdg_{1943}}{pdg_{1467}}\)
  1. 4245712581; locally 8090893:
    \(v = \frac{2 \pi r}{t}\)
    \(pdg_{1357} = \frac{2 pdg_{2530} pdg_{3141}}{pdg_{1467}}\)
valid 9262596735:
5426308937:
4245712581:
9262596735:
5426308937:
4245712581:
1 change four variables in expr
  1. 6935745841; locally 2820438:
    \(F = G \frac{m_1 m_2}{x^2}\)
    \(pdg_{4202} = \frac{pdg_{4851} pdg_{5022} pdg_{6277}}{pdg_{4037}^{2}}\)
  1. 3398368564:
    \(F\)
    \(pdg_{4202}\)
  2. 3594626260:
    \(F_{\rm gravity}\)
    \(pdg_{2867}\)
  3. 9794128647:
    \(m_1\)
    \(pdg_{5458}\)
  4. 4153613253:
    \(m_{\rm Earth}\)
    \(pdg_{5458}\)
  5. 3088463019:
    \(m_2\)
    \(pdg_{4851}\)
  6. 3486213448:
    \(m_{\rm satellite}\)
    \(pdg_{3569}\)
  7. 4830480629:
    \(x\)
    \(pdg_{4037}\)
  8. 7819443873:
    \(r\)
    \(pdg_{2530}\)
  1. 5563580265; locally 1917654:
    \(F_{\rm gravity} = G \frac{m_{\rm Earth} m_{\rm satellite}}{r^2}\)
    \(pdg_{2867} = \frac{pdg_{3569} pdg_{5458} pdg_{6277}}{pdg_{2530}^{2}}\)
LHS diff is 0 RHS diff is pdg3569*pdg6277*(pdg5022 - pdg5458)/pdg2530**2 6935745841:
5563580265:
6935745841:
5563580265:
8 change variable X to Y
  1. 4245712581; locally 8090893:
    \(v = \frac{2 \pi r}{t}\)
    \(pdg_{1357} = \frac{2 pdg_{2530} pdg_{3141}}{pdg_{1467}}\)
  1. 3722461713:
    \(t\)
    \(pdg_{1467}\)
  2. 9346215480:
    \(T_{\rm orbit}\)
    \(pdg_{8762}\)
  1. 3614055652; locally 2392562:
    \(v = \frac{2 \pi r}{T_{\rm orbit}}\)
    \(pdg_{1357} = \frac{2 pdg_{2530} pdg_{3141}}{pdg_{8762}}\)
valid 4245712581:
3614055652:
4245712581:
3614055652:
9 raise both sides to power
  1. 3614055652; locally 2392562:
    \(v = \frac{2 \pi r}{T_{\rm orbit}}\)
    \(pdg_{1357} = \frac{2 pdg_{2530} pdg_{3141}}{pdg_{8762}}\)
  1. 2754264786:
    \(2\)
    \(2\)
  1. 8059639673; locally 6390693:
    \(v^2 = \frac{4 \pi^2 r^2}{T_{\rm orbit}^2}\)
    \(pdg_{1357}^{2} = \frac{4 pdg_{2530}^{2} pdg_{3141}^{2}}{pdg_{8762}^{2}}\)
no check is performed 3614055652:
8059639673:
3614055652:
8059639673:
11 LHS of expr 1 equals LHS of expr 2
  1. 1994296484; locally 2009493:
    \(v_{\rm satellite}^2 = G \frac{m_{\rm Earth}}{r}\)
    \(pdg_{4082}^{2} = \frac{pdg_{5458} pdg_{6277}}{pdg_{2530}}\)
  2. 8059639673; locally 6390693:
    \(v^2 = \frac{4 \pi^2 r^2}{T_{\rm orbit}^2}\)
    \(pdg_{1357}^{2} = \frac{4 pdg_{2530}^{2} pdg_{3141}^{2}}{pdg_{8762}^{2}}\)
  1. 3906710072; locally 2871066:
    \(G \frac{m_{\rm Earth}}{r} = \frac{4 \pi^2 r^2}{T_{\rm orbit}^2}\)
    \(\frac{pdg_{5458} pdg_{6277}}{pdg_{2530}} = \frac{4 pdg_{2530}^{2} pdg_{3141}^{2}}{pdg_{8762}^{2}}\)
input diff is -pdg1357**2 + pdg4082**2 diff is 0 diff is 0 1994296484:
8059639673:
3906710072:
1994296484:
8059639673:
3906710072:
15 declare assumption
  1. 3920616792; locally 9978909:
    \(T_{\rm geostationary orbit} = 24\ {\rm hours}\)
    \(pdg_{5595}\)
no validation is available for declarations 3920616792:
3920616792:
5 substitute LHS of two expressions into expr
  1. 5563580265; locally 1917654:
    \(F_{\rm gravity} = G \frac{m_{\rm Earth} m_{\rm satellite}}{r^2}\)
    \(pdg_{2867} = \frac{pdg_{3569} pdg_{5458} pdg_{6277}}{pdg_{2530}^{2}}\)
  2. 4627284246; locally 6845877:
    \(F_{\rm centripetal} = \frac{m_{\rm satellite} v_{\rm satellite}^2}{r}\)
    \(pdg_{1687} = \frac{pdg_{3569} pdg_{4082}^{2}}{pdg_{2530}}\)
  3. 3176662571; locally 2154616:
    \(F_{\rm centripetal} = F_{\rm gravity}\)
    \(pdg_{2867} = pdg_{1687}\)
  1. 4072200527; locally 4948724:
    \(\frac{m_{\rm satellite} v_{\rm satellite}^2}{r} = G \frac{m_{\rm Earth} m_{\rm satellite}}{r^2}\)
    \(\frac{pdg_{3569} pdg_{4082}^{2}}{pdg_{2530}} = \frac{pdg_{3569} pdg_{5458} pdg_{6277}}{pdg_{2530}^{2}}\)
failed 5563580265:
4627284246:
3176662571: dimensions are consistent
4072200527:
5563580265:
4627284246:
3176662571: N/A
4072200527:
13 divide both sides by
  1. 7010294143; locally 7188516:
    \(T_{\rm orbit}^2 G m_{\rm Earth} = 4 \pi^2 r^3\)
    \(pdg_{5458} pdg_{6277} pdg_{8762}^{2} = 4 pdg_{2530}^{3} pdg_{3141}^{2}\)
  1. 7556442438:
    \(4 \pi^2\)
    \(4 pdg_{3141}^{2}\)
  1. 4858693811; locally 6238570:
    \(\frac{T_{\rm orbit}^2 G m_{\rm Earth}}{4 \pi^2} = r^3\)
    \(\frac{pdg_{5458} pdg_{6277} pdg_{8762}^{2}}{4 pdg_{3141}^{2}} = pdg_{2530}^{3}\)
valid 7010294143:
4858693811:
7010294143:
4858693811:
2 declare initial expr
  1. 9226945488; locally 8242154:
    \(F = \frac{m v^2}{r}\)
    \(pdg_{4202} = \frac{pdg_{1357}^{2} pdg_{5156}}{pdg_{2530}}\)
no validation is available for declarations 9226945488:
9226945488:
6 change variable X to Y
  1. 6785303857; locally 1115424:
    \(C = 2 \pi r\)
    \(pdg_{3034} = 2 pdg_{2530} pdg_{3141}\)
  1. 1823570358:
    \(C\)
    \(pdg_{3034}\)
  2. 3236313290:
    \(d\)
    \(pdg_{1943}\)
  1. 9262596735; locally 5369477:
    \(d = 2 \pi r\)
    \(pdg_{1943} = 2 pdg_{2530} pdg_{3141}\)
valid 6785303857:
9262596735:
6785303857:
9262596735:
4 declare assumption
  1. 3176662571; locally 2154616:
    \(F_{\rm centripetal} = F_{\rm gravity}\)
    \(pdg_{2867} = pdg_{1687}\)
no validation is available for declarations 3176662571: dimensions are consistent
 3176662571: N/A
Physics Derivation Graph: Steps for radius for satellite in geostationary orbit

Symbols for this derivation

See also all 227 symbols
symbol ID category latex scope dimension name value Used in derivations references
3569 variable m_{\rm satellite}
\(m_{\rm satellite}\)		 real
  • mass: 1
 mass of satellite 6
5156 variable m
\(m\)		 ['real']
  • mass: 1
 mass 69
4082 variable v_{\rm satellite}
\(v_{\rm satellite}\)		 real
  • length: 1
  • time: -1
 velocity of satellite 4
5595 variable T_{\rm geostationary\ orbit}
\(T_{\rm geostationary\ orbit}\)		 real
  • time: 1
 geostationary orbital period 3
2530 variable r
\(r\)		 ['real']
  • length: 1
 radius 60
2867 variable F_{\rm gravity}
\(F_{\rm gravity}\)		 real
  • length: 1
  • mass: 1
  • time: -2
 force due to gravity 12
4851 variable m_2
\(m_2\)		 real
  • mass: 1
 mass 31
4037 variable x
\(x\)		 ['real']
  • length: 1
 position 53
1357 variable v
\(v\)		 ['real']
  • length: 1
  • time: -1
 velocity 83
1467 variable t
\(t\)		 ['real']
  • time: 1
 time 121
5458 constant m_{\rm Earth}
\(m_{\rm Earth}\)		 real
  • mass: 2
 mass of Earth 5.97237*10^24   kg
 34
1943 variable d
\(d\)		 ['real']
  • length: 1
 displacement 25
6277 constant G
\(G\)		 real
  • length: 3
  • mass: -1
  • time: -2
 gravitational constant 6.67430*10^{-11}   m^3 * kg^-1 * s^-2
 60
3034 variable C
\(C\)		 ['real']
  • length: 1
 circumference 5
1687 variable F_{\rm centripetal}
\(F_{\rm centripetal}\)		 real
  • length: 1
  • mass: 1
  • time: -2
 centripetal force 8
8762 variable T_{\rm orbit}
\(T_{\rm orbit}\)		 real
  • time: 1
 orbital period 14
5022 variable m_1
\(m_1\)		 real
  • mass: 1
 mass 35
4202 variable F
\(F\)		 ['real']
  • length: 1
  • mass: 1
  • time: -2
 force 21
3141 constant \pi
\(\pi\)		 ['real'] dimensionless pi 3.1415   dimensionless
 72
7110 variable r_{\rm geostationary\ orbit}
\(r_{\rm geostationary\ orbit}\)		 real
  • length: 1
 geostationary orbital radius 2
MESSAGES: