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review derivation: radius for satellite in geostationary orbit

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Index Inference Rule Input latex Feeds latex Output latex step validity dimension check unit check notes
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:
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:
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:
4 declare assumption
  1. 3176662571; locally 2154616:
    \(F_{\rm centripetal} = F_{\rm gravity}\)
    \(pdg_{2867} = pdg_{1687}\)
no validation is available for declarations 3176662571:
3176662571:
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:
4072200527:
5563580265:
4627284246:
3176662571:
4072200527:
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:
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:
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:
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:
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:
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:
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:
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: inconsistent dimensions
4858693811:
2617541067: N/A
15 declare assumption
  1. 3920616792; locally 9978909:
    \(T_{\rm geostationary orbit} = 24\ {\rm hours}\)
    \(pdg_{5595}\)
no validation is available for declarations 3920616792:
3920616792:
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: inconsistent dimensions
1559688463: inconsistent dimensions
2617541067: N/A
1559688463: N/A
Physics Derivation Graph: Steps for radius for satellite in geostationary orbit

Symbols for this derivation

See also all 212 symbols
symbol ID category latex scope dimension name value Used in derivations references
1687 variable F_{\rm centripetal}
\(F_{\rm centripetal}\)
real
  • length: 1
  • time: -2
  • mass: 1
centripetal force 3
1467 variable t
\(t\)
['real']
  • time: 1
time 115
1357 variable v
\(v\)
['real']
  • length: 1
  • time: -1
velocity 80
4037 variable x
\(x\)
['real']
  • length: 1
position 47
7110 variable r_{\rm geostationary\ orbit}
\(r_{\rm geostationary\ orbit}\)
real
  • length: 1
geostationary orbital radius 2
6277 constant G
\(G\)
real
  • length: 3
  • time: -2
  • mass: -1
gravitational constant 6.67430*10^{-11}   m^3 * kg^-1 * s^-2
46
3569 variable m_{\rm satellite}
\(m_{\rm satellite}\)
real
  • mass: 1
mass of satellite 6
2867 variable F_{\rm gravity}
\(F_{\rm gravity}\)
real
  • length: 1
  • time: -2
  • mass: 1
force due to gravity 3
4082 variable v_{\rm satellite}
\(v_{\rm satellite}\)
real
  • length: 1
  • time: -1
velocity of satellite 4
3034 variable C
\(C\)
['real']
  • length: 1
circumference 5
1943 variable d
\(d\)
['real']
  • length: 1
displacement 25
8762 variable T_{\rm orbit}
\(T_{\rm orbit}\)
real
  • time: 1
orbital period 9
5595 variable T_{\rm geostationary\ orbit}
\(T_{\rm geostationary\ orbit}\)
real
  • time: 1
geostationary orbital period 3
4202 variable F
\(F\)
['real']
  • length: 1
  • time: -2
  • mass: 1
force 21
5156 variable m
\(m\)
['real']
  • mass: 1
mass 49
5458 constant m_{\rm Earth}
\(m_{\rm Earth}\)
real
  • mass: 2
mass of Earth 5.97237*10^24   kg
34
2530 variable r
\(r\)
['real']
  • length: 1
radius 34
5022 variable m_1
\(m_1\)
real
  • mass: 1
mass 18
3141 constant \pi
\(\pi\)
['real']
pi 3.1415   dimensionless
55
4851 variable m_2
\(m_2\)
real
  • mass: 1
mass 17
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