## 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

symbol ID category latex scope dimension name value Used in derivations references
4082 variable v_{\rm satellite}
$$v_{\rm satellite}$$
real
• length: 1
• time: -1
velocity of satellite
4
3141 constant \pi
$$\pi$$
['real'] dimensionless pi 3.1415   dimensionless
72
1943 variable d
$$d$$
['real']
• length: 1
displacement
25
8762 variable T_{\rm orbit}
$$T_{\rm orbit}$$
real
• time: 1
orbital period
14
3034 variable C
$$C$$
['real']
• length: 1
circumference
5
4851 variable m_2
$$m_2$$
real
• mass: 1
mass
31
1687 variable F_{\rm centripetal}
$$F_{\rm centripetal}$$
real
• length: 1
• mass: 1
• time: -2
centripetal force
8
1467 variable t
$$t$$
['real']
• time: 1
time
121
2530 variable r
$$r$$
['real']
• length: 1
60
5595 variable T_{\rm geostationary\ orbit}
$$T_{\rm geostationary\ orbit}$$
real
• time: 1
geostationary orbital period
3
5156 variable m
$$m$$
['real']
• mass: 1
mass
69
7110 variable r_{\rm geostationary\ orbit}
$$r_{\rm geostationary\ orbit}$$
real
• length: 1
2
5022 variable m_1
$$m_1$$
real
• mass: 1
mass
35
4202 variable F
$$F$$
['real']
• length: 1
• mass: 1
• time: -2
force
21
1357 variable v
$$v$$
['real']
• length: 1
• time: -1
velocity
83
6277 constant G
$$G$$
real
• length: 3
• mass: -1
• time: -2
gravitational constant 6.67430*10^{-11}   m^3 * kg^-1 * s^-2
60
3569 variable m_{\rm satellite}
$$m_{\rm satellite}$$
real
• mass: 1
mass of satellite
6
5458 constant m_{\rm Earth}
$$m_{\rm Earth}$$
real
• mass: 2
mass of Earth 5.97237*10^24   kg
34
4037 variable x
$$x$$
['real']
• length: 1
position
53
2867 variable F_{\rm gravity}
$$F_{\rm gravity}$$
real
• length: 1
• mass: 1
• time: -2
force due to gravity
12
MESSAGES:
• local variable 'all_df' referenced before assignment
• in step 1306821: 6
• in step 8659528: 0