This page contains three views of the steps in the derivation: d3js, graphviz PNG, and a table.
Index |
Inference Rule |
Input latex |
Feeds latex |
Output latex |
step validity |
dimension check |
unit check |
notes |
3
|
change four variables in expr |
-
9226945488; locally 8242154:
\(F = \frac{m v^2}{r}\)
\(pdg_{4202} = \frac{pdg_{1357}^{2} pdg_{5156}}{pdg_{2530}}\)
|
-
5089196493:
\(F\)
\(pdg_{4202}\)
-
1333474099:
\(F_{\rm centripetal}\)
\(pdg_{1687}\)
-
3342155559:
\(m\)
\(pdg_{5156}\)
-
2114570475:
\(m_{\rm satellite}\)
\(pdg_{3569}\)
-
7912578203:
\(v\)
\(pdg_{1357}\)
-
9789485295:
\(v_{\rm satellite}\)
\(pdg_{4082}\)
|
-
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 |
-
3906710072; locally 2871066:
\(G \frac{m_{\rm Earth}}{r} = \frac{4 \pi^2 r^2}{T_{\rm orbit}^2}\)
\(\frac{pdg_{5458} pdg_{6277}}{pdg_{2530}} = \frac{4 pdg_{2530}^{2} pdg_{3141}^{2}}{pdg_{8762}^{2}}\)
|
-
6238632840:
\(r T_{\rm orbit}^2\)
\(pdg_{2530} pdg_{8762}^{2}\)
|
-
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 |
-
4858693811; locally 6238570:
\(\frac{T_{\rm orbit}^2 G m_{\rm Earth}}{4 \pi^2} = r^3\)
\(\frac{pdg_{5458} pdg_{6277} pdg_{8762}^{2}}{4 pdg_{3141}^{2}} = pdg_{2530}^{3}\)
|
-
4319544433:
\(1/3\)
\(\frac{1}{3}\)
|
-
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 |
-
4072200527; locally 4948724:
\(\frac{m_{\rm satellite} v_{\rm satellite}^2}{r} = G \frac{m_{\rm Earth} m_{\rm satellite}}{r^2}\)
\(\frac{pdg_{3569} pdg_{4082}^{2}}{pdg_{2530}} = \frac{pdg_{3569} pdg_{5458} pdg_{6277}}{pdg_{2530}^{2}}\)
|
-
5359471792:
\(\frac{m_{\rm satellite}}{r}\)
\(\frac{pdg_{3569}}{pdg_{2530}}\)
|
-
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 |
-
2617541067; locally 7139326:
\(\left(\frac{T_{\rm orbit}^2 G m_{\rm Earth}}{4 \pi^2}\right)^{1/3} = r\)
\(\frac{\sqrt[3]{2} \sqrt[3]{\frac{pdg_{5458} pdg_{6277} pdg_{8762}^{2}}{pdg_{3141}^{2}}}}{2} = pdg_{2530}\)
|
-
3846345263:
\(T_{\rm orbit}\)
\(pdg_{8762}\)
-
5208737840:
\(T_{\rm geostationary\ orbit}\)
\(pdg_{5595}\)
-
5770088141:
\(r\)
\(pdg_{2530}\)
-
7053449926:
\(r_{\rm geostationary\ orbit}\)
\(pdg_{7110}\)
|
-
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 |
-
9262596735; locally 5369477:
\(d = 2 \pi r\)
\(pdg_{1943} = 2 pdg_{2530} pdg_{3141}\)
-
5426308937; locally 5114041:
\(v = \frac{d}{t}\)
\(pdg_{1357} = \frac{pdg_{1943}}{pdg_{1467}}\)
|
|
-
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 |
-
6935745841; locally 2820438:
\(F = G \frac{m_1 m_2}{x^2}\)
\(pdg_{4202} = \frac{pdg_{4851} pdg_{5022} pdg_{6277}}{pdg_{4037}^{2}}\)
|
-
3398368564:
\(F\)
\(pdg_{4202}\)
-
3594626260:
\(F_{\rm gravity}\)
\(pdg_{2867}\)
-
9794128647:
\(m_1\)
\(pdg_{5458}\)
-
4153613253:
\(m_{\rm Earth}\)
\(pdg_{5458}\)
-
3088463019:
\(m_2\)
\(pdg_{4851}\)
-
3486213448:
\(m_{\rm satellite}\)
\(pdg_{3569}\)
-
4830480629:
\(x\)
\(pdg_{4037}\)
-
7819443873:
\(r\)
\(pdg_{2530}\)
|
-
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 |
-
4245712581; locally 8090893:
\(v = \frac{2 \pi r}{t}\)
\(pdg_{1357} = \frac{2 pdg_{2530} pdg_{3141}}{pdg_{1467}}\)
|
-
3722461713:
\(t\)
\(pdg_{1467}\)
-
9346215480:
\(T_{\rm orbit}\)
\(pdg_{8762}\)
|
-
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 |
-
3614055652; locally 2392562:
\(v = \frac{2 \pi r}{T_{\rm orbit}}\)
\(pdg_{1357} = \frac{2 pdg_{2530} pdg_{3141}}{pdg_{8762}}\)
|
-
2754264786:
\(2\)
\(2\)
|
-
8059639673; locally 6390693:
\(v^2 = \frac{4 \pi^2 r^2}{T_{\rm orbit}^2}\)
\(pdg_{1357}^{2} = \frac{4 pdg_{2530}^{2} pdg_{3141}^{2}}{pdg_{8762}^{2}}\)
|
no check is performed |
3614055652:
8059639673:
|
3614055652:
8059639673:
|
|
11
|
LHS of expr 1 equals LHS of expr 2 |
-
1994296484; locally 2009493:
\(v_{\rm satellite}^2 = G \frac{m_{\rm Earth}}{r}\)
\(pdg_{4082}^{2} = \frac{pdg_{5458} pdg_{6277}}{pdg_{2530}}\)
-
8059639673; locally 6390693:
\(v^2 = \frac{4 \pi^2 r^2}{T_{\rm orbit}^2}\)
\(pdg_{1357}^{2} = \frac{4 pdg_{2530}^{2} pdg_{3141}^{2}}{pdg_{8762}^{2}}\)
|
|
-
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 |
|
|
-
3920616792; locally 9978909:
\(T_{\rm geostationary orbit} = 24\ {\rm hours}\)
\(pdg_{5595}\)
|
no validation is available for declarations |
3920616792:
|
3920616792:
|
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5
|
substitute LHS of two expressions into expr |
-
5563580265; locally 1917654:
\(F_{\rm gravity} = G \frac{m_{\rm Earth} m_{\rm satellite}}{r^2}\)
\(pdg_{2867} = \frac{pdg_{3569} pdg_{5458} pdg_{6277}}{pdg_{2530}^{2}}\)
-
4627284246; locally 6845877:
\(F_{\rm centripetal} = \frac{m_{\rm satellite} v_{\rm satellite}^2}{r}\)
\(pdg_{1687} = \frac{pdg_{3569} pdg_{4082}^{2}}{pdg_{2530}}\)
-
3176662571; locally 2154616:
\(F_{\rm centripetal} = F_{\rm gravity}\)
\(pdg_{2867} = pdg_{1687}\)
|
|
-
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 |
-
7010294143; locally 7188516:
\(T_{\rm orbit}^2 G m_{\rm Earth} = 4 \pi^2 r^3\)
\(pdg_{5458} pdg_{6277} pdg_{8762}^{2} = 4 pdg_{2530}^{3} pdg_{3141}^{2}\)
|
-
7556442438:
\(4 \pi^2\)
\(4 pdg_{3141}^{2}\)
|
-
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 |
|
|
-
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 |
-
6785303857; locally 1115424:
\(C = 2 \pi r\)
\(pdg_{3034} = 2 pdg_{2530} pdg_{3141}\)
|
-
1823570358:
\(C\)
\(pdg_{3034}\)
-
3236313290:
\(d\)
\(pdg_{1943}\)
|
-
9262596735; locally 5369477:
\(d = 2 \pi r\)
\(pdg_{1943} = 2 pdg_{2530} pdg_{3141}\)
|
valid |
6785303857:
9262596735:
|
6785303857:
9262596735:
|
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4
|
declare assumption |
|
|
-
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
Clicking on the step index will take you to the page where you can edit that step.
symbol ID |
category |
latex |
scope |
dimension |
name |
value |
Used in derivations |
references |
4202 |
variable |
F
\(F\) |
['real'] |
|
force
|
|
|
|
21
|
7110 |
variable |
r_{\rm geostationary\ orbit}
\(r_{\rm geostationary\ orbit}\) |
real |
|
geostationary orbital radius
|
|
|
|
2
|
1467 |
variable |
t
\(t\) |
['real'] |
|
time
|
|
|
|
120
|
5022 |
variable |
m_1
\(m_1\) |
real |
|
mass
|
|
|
|
18
|
1357 |
variable |
v
\(v\) |
['real'] |
|
velocity
|
|
|
|
80
|
2530 |
variable |
r
\(r\) |
['real'] |
|
radius
|
|
|
|
34
|
3141 |
constant |
\pi
\(\pi\) |
['real'] |
dimensionless
|
pi
|
3.1415 dimensionless
|
|
|
55
|
4037 |
variable |
x
\(x\) |
['real'] |
|
position
|
|
|
|
53
|
6277 |
constant |
G
\(G\) |
real |
|
gravitational constant
|
6.67430*10^{-11} m^3 * kg^-1 * s^-2
|
|
|
46
|
5458 |
constant |
m_{\rm Earth}
\(m_{\rm Earth}\) |
real |
|
mass of Earth
|
5.97237*10^24 kg
|
|
|
34
|
3569 |
variable |
m_{\rm satellite}
\(m_{\rm satellite}\) |
real |
|
mass of satellite
|
|
|
|
6
|
4851 |
variable |
m_2
\(m_2\) |
real |
|
mass
|
|
|
|
17
|
8762 |
variable |
T_{\rm orbit}
\(T_{\rm orbit}\) |
real |
|
orbital period
|
|
|
|
9
|
2867 |
variable |
F_{\rm gravity}
\(F_{\rm gravity}\) |
real |
|
force due to gravity
|
|
|
|
3
|
4082 |
variable |
v_{\rm satellite}
\(v_{\rm satellite}\) |
real |
|
velocity of satellite
|
|
|
|
4
|
1943 |
variable |
d
\(d\) |
['real'] |
|
displacement
|
|
|
|
25
|
1687 |
variable |
F_{\rm centripetal}
\(F_{\rm centripetal}\) |
real |
|
centripetal force
|
|
|
|
3
|
5595 |
variable |
T_{\rm geostationary\ orbit}
\(T_{\rm geostationary\ orbit}\) |
real |
|
geostationary orbital period
|
|
|
|
3
|
5156 |
variable |
m
\(m\) |
['real'] |
|
mass
|
|
|
|
57
|
3034 |
variable |
C
\(C\) |
['real'] |
|
circumference
|
|
|
|
5
|