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review derivation: Kepler's Third Law: period squared propto distance cubed

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Notes for this derivation:
https://en.wikipedia.org/wiki/Kepler%27s_laws_of_planetary_motion#Third_law

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Index Inference Rule Input latex Feeds latex Output latex step validity dimension check unit check notes
8 substitute LHS of expr 1 into expr 2
  1. 3132131132; locally 2340248:
    \(\omega = \frac{2\pi}{T}\)
    \(pdg_{2321} = \frac{2 pdg_{3141}}{pdg_{9491}}\)
  2. 3896798826; locally 9388996:
    \(m_2 d_2 \omega^2 = G \frac{m_1 m_2}{r^2}\)
    \(pdg_{2321}^{2} pdg_{2798} pdg_{4851} = \frac{pdg_{4851} pdg_{5022} pdg_{6277}}{pdg_{2530}^{2}}\)
  1. 9070394000; locally 4575586:
    \(m_2 d_2 \frac{4 \pi^2}{T^2} = G \frac{m_1 m_2}{r^2}\)
    \(\frac{4 pdg_{2798} pdg_{3141}^{2} pdg_{4851}}{pdg_{9491}^{2}} = \frac{pdg_{4851} pdg_{5022} pdg_{6277}}{pdg_{2530}^{2}}\)
valid 3132131132:
3896798826:
9070394000:
3132131132:
3896798826:
9070394000:
19 substitute LHS of expr 1 into expr 2
  1. 2217103163; locally 9110206:
    \(\frac{m_1 d_1}{d_2} = m_2\)
    \(\frac{pdg_{5022} pdg_{7652}}{pdg_{2798}} = pdg_{4851}\)
  2. 4188580242; locally 1709969:
    \(T^2 = \frac{r^3 4 \pi^2}{\left(m_1+\left(\frac{m_1}{d_2}d_1\right)\right)G}\)
    \(pdg_{9491}^{2} = \frac{4 pdg_{2530}^{3} pdg_{3141}^{2}}{pdg_{6277} \left(pdg_{5022} + \frac{pdg_{5022} pdg_{7652}}{pdg_{2798}}\right)}\)
  1. 5658865948; locally 8711868:
    \(T^2 = \frac{r^3 4 \pi^2}{(m_1+m_2)G}\)
    \(pdg_{9491}^{2} = \frac{4 pdg_{2530}^{3} pdg_{3141}^{2}}{pdg_{6277} \left(pdg_{4851} + pdg_{5022}\right)}\)
valid 2217103163:
4188580242:
5658865948:
2217103163:
4188580242:
5658865948:
7 declare initial expr
  1. 3132131132; locally 2340248:
    \(\omega = \frac{2\pi}{T}\)
    \(pdg_{2321} = \frac{2 pdg_{3141}}{pdg_{9491}}\)
no validation is available for declarations 3132131132:
3132131132:
12 multiply RHS by unity
  1. 9170048197; locally 7795202:
    \(T^2 = d_2 4 \pi^2 \frac{r^2}{G m_1}\)
    \(pdg_{9491}^{2} = \frac{4 pdg_{2530}^{2} pdg_{2798} pdg_{3141}^{2}}{pdg_{5022} pdg_{6277}}\)
  1. 8122039815:
    \(\frac{d_1+d_2}{d_1+d_2}\)
    \(1\)
  1. 1811867899; locally 6577160:
    \(T^2 = \frac{d_1+d_2}{d_1+d_2} d_2 4 \pi^2 \frac{r^2}{G m_1}\)
    \(pdg_{9491}^{2} = \frac{4 pdg_{2530}^{2} pdg_{2798} pdg_{3141}^{2}}{pdg_{5022} pdg_{6277}}\)
valid 9170048197:
1811867899:
9170048197:
1811867899:
3 change two variables in expr
  1. 4393258808; locally 8072137:
    \(F_{\rm centripetal} = m r \omega^2\)
    \(pdg_{1687} = pdg_{2321}^{2} pdg_{2530} pdg_{5156}\)
  1. 8916428651:
    \(m\)
    \(pdg_{5156}\)
  2. 1635147226:
    \(m_2\)
    \(pdg_{4851}\)
  3. 9884115626:
    \(r\)
    \(pdg_{2530}\)
  4. 1036530438:
    \(d_2\)
    \(pdg_{2798}\)
  1. 3649797559; locally 6652843:
    \(F_{\rm centripetal} = m_2 d_2 \omega^2\)
    \(pdg_{1687} = pdg_{2321}^{2} pdg_{2798} pdg_{4851}\)
valid 4393258808: dimensions are consistent
3649797559:
4393258808: N/A
3649797559:
11 raise both sides to power
  1. 9152823411; locally 7556753:
    \(\frac{1}{T^2} = \frac{1}{d_2 4 \pi^2} G \frac{m_1}{r^2}\)
    \(\frac{1}{pdg_{9491}^{2}} = \frac{pdg_{5022} pdg_{6277}}{4 pdg_{2530}^{2} pdg_{2798} pdg_{3141}^{2}}\)
  1. 7445388869:
    \(-1\)
    \(-1\)
  1. 9170048197; locally 7795202:
    \(T^2 = d_2 4 \pi^2 \frac{r^2}{G m_1}\)
    \(pdg_{9491}^{2} = \frac{4 pdg_{2530}^{2} pdg_{2798} pdg_{3141}^{2}}{pdg_{5022} pdg_{6277}}\)
no check is performed 9152823411:
9170048197:
9152823411:
9170048197:
6 substitute LHS of expr 1 into expr 2
  1. 3649797559; locally 6652843:
    \(F_{\rm centripetal} = m_2 d_2 \omega^2\)
    \(pdg_{1687} = pdg_{2321}^{2} pdg_{2798} pdg_{4851}\)
  2. 6829281943; locally 4382594:
    \(F_{\rm centripetal} = G \frac{m_1 m_2}{r^2}\)
    \(pdg_{1687} = \frac{pdg_{4851} pdg_{5022} pdg_{6277}}{pdg_{2530}^{2}}\)
  1. 3896798826; locally 9388996:
    \(m_2 d_2 \omega^2 = G \frac{m_1 m_2}{r^2}\)
    \(pdg_{2321}^{2} pdg_{2798} pdg_{4851} = \frac{pdg_{4851} pdg_{5022} pdg_{6277}}{pdg_{2530}^{2}}\)
valid 3649797559:
6829281943:
3896798826:
3649797559:
6829281943:
3896798826:
17 declare initial expr
  1. 5128670694; locally 4476518:
    \(m_1 d_1 = m_2 d_2\)
    \(pdg_{5022} pdg_{7652} = pdg_{2798} pdg_{4851}\)
no validation is available for declarations 5128670694:
5128670694:
16 simplify
  1. 3781109867; locally 6644719:
    \(T^2 = \frac{r^3 4 \pi^2}{(d_1+d_2) \frac{m_1}{d_2}G}\)
    \(pdg_{9491}^{2} = \frac{4 pdg_{2530}^{3} pdg_{2798} pdg_{3141}^{2}}{pdg_{5022} pdg_{6277} \left(pdg_{2798} + pdg_{7652}\right)}\)
  1. 4188580242; locally 1709969:
    \(T^2 = \frac{r^3 4 \pi^2}{\left(m_1+\left(\frac{m_1}{d_2}d_1\right)\right)G}\)
    \(pdg_{9491}^{2} = \frac{4 pdg_{2530}^{3} pdg_{3141}^{2}}{pdg_{6277} \left(pdg_{5022} + \frac{pdg_{5022} pdg_{7652}}{pdg_{2798}}\right)}\)
valid 3781109867:
4188580242:
3781109867:
4188580242:
1 declare initial expr
  1. 1292735067; locally 5331094:
    \(F_{gravitational} = G \frac{m_1 m_2}{r^2}\)
    \(pdg_{2867} = \frac{pdg_{4851} pdg_{5022} pdg_{6277}}{pdg_{2530}^{2}}\)
no validation is available for declarations 1292735067:
1292735067:
10 multiply both sides by
  1. 9838128064; locally 6210646:
    \(d_2 \frac{4 \pi^2}{T^2} = G \frac{m_1}{r^2}\)
    \(\frac{4 pdg_{2798} pdg_{3141}^{2}}{pdg_{9491}^{2}} = \frac{pdg_{5022} pdg_{6277}}{pdg_{2530}^{2}}\)
  1. 5684907106:
    \(\frac{1}{d_2 4 \pi^2}\)
    \(\frac{1}{4 pdg_{2798} pdg_{3141}^{2}}\)
  1. 9152823411; locally 7556753:
    \(\frac{1}{T^2} = \frac{1}{d_2 4 \pi^2} G \frac{m_1}{r^2}\)
    \(\frac{1}{pdg_{9491}^{2}} = \frac{pdg_{5022} pdg_{6277}}{4 pdg_{2530}^{2} pdg_{2798} pdg_{3141}^{2}}\)
valid 9838128064:
9152823411:
9838128064:
9152823411:
20 declare final expr
  1. 5658865948; locally 8711868:
    \(T^2 = \frac{r^3 4 \pi^2}{(m_1+m_2)G}\)
    \(pdg_{9491}^{2} = \frac{4 pdg_{2530}^{3} pdg_{3141}^{2}}{pdg_{6277} \left(pdg_{4851} + pdg_{5022}\right)}\)
no validation is available for declarations 5658865948:
5658865948:
period squared propto distance cubed
15 multiply RHS by unity
  1. 2906548078; locally 8324356:
    \(T^2 = \frac{r}{d_1+d_2} d_2 4 \pi^2 \frac{r^2}{G m_1}\)
    \(pdg_{9491}^{2} = \frac{4 pdg_{2530}^{3} pdg_{2798} pdg_{3141}^{2}}{pdg_{5022} pdg_{6277} \left(pdg_{2798} + pdg_{7652}\right)}\)
  1. 9524810853:
    \(\frac{1/d_2}{1/d_2}\)
    \(1\)
  1. 3781109867; locally 6644719:
    \(T^2 = \frac{r^3 4 \pi^2}{(d_1+d_2) \frac{m_1}{d_2}G}\)
    \(pdg_{9491}^{2} = \frac{4 pdg_{2530}^{3} pdg_{2798} pdg_{3141}^{2}}{pdg_{5022} pdg_{6277} \left(pdg_{2798} + pdg_{7652}\right)}\)
valid 2906548078:
3781109867:
2906548078:
3781109867:
13 declare assumption
  1. 5586102077; locally 8233899:
    \(r = d_1 + d_2\)
    \(pdg_{2530} = pdg_{2798} + pdg_{7652}\)
no validation is available for declarations 5586102077:
5586102077:
5 substitute RHS of expr 1 into expr 2
  1. 3176662571; locally 2600680:
    \(F_{\rm centripetal} = F_{\rm gravity}\)
    \(pdg_{2867} = pdg_{1687}\)
  2. 1292735067; locally 5331094:
    \(F_{gravitational} = G \frac{m_1 m_2}{r^2}\)
    \(pdg_{2867} = \frac{pdg_{4851} pdg_{5022} pdg_{6277}}{pdg_{2530}^{2}}\)
  1. 6829281943; locally 4382594:
    \(F_{\rm centripetal} = G \frac{m_1 m_2}{r^2}\)
    \(pdg_{1687} = \frac{pdg_{4851} pdg_{5022} pdg_{6277}}{pdg_{2530}^{2}}\)
LHS diff is -pdg1687 + pdg2867 RHS diff is 0 3176662571:
1292735067:
6829281943:
3176662571:
1292735067:
6829281943:
18 divide both sides by
  1. 5128670694; locally 4476518:
    \(m_1 d_1 = m_2 d_2\)
    \(pdg_{5022} pdg_{7652} = pdg_{2798} pdg_{4851}\)
  1. 8044416349:
    \(d_2\)
    \(pdg_{2798}\)
  1. 2217103163; locally 9110206:
    \(\frac{m_1 d_1}{d_2} = m_2\)
    \(\frac{pdg_{5022} pdg_{7652}}{pdg_{2798}} = pdg_{4851}\)
valid 5128670694:
2217103163:
5128670694:
2217103163:
4 declare assumption
  1. 3176662571; locally 2600680:
    \(F_{\rm centripetal} = F_{\rm gravity}\)
    \(pdg_{2867} = pdg_{1687}\)
no validation is available for declarations 3176662571:
3176662571:
9 simplify
  1. 9070394000; locally 4575586:
    \(m_2 d_2 \frac{4 \pi^2}{T^2} = G \frac{m_1 m_2}{r^2}\)
    \(\frac{4 pdg_{2798} pdg_{3141}^{2} pdg_{4851}}{pdg_{9491}^{2}} = \frac{pdg_{4851} pdg_{5022} pdg_{6277}}{pdg_{2530}^{2}}\)
  1. 9838128064; locally 6210646:
    \(d_2 \frac{4 \pi^2}{T^2} = G \frac{m_1}{r^2}\)
    \(\frac{4 pdg_{2798} pdg_{3141}^{2}}{pdg_{9491}^{2}} = \frac{pdg_{5022} pdg_{6277}}{pdg_{2530}^{2}}\)
LHS diff is 4*pdg2798*pdg3141**2*(pdg4851 - 1)/pdg9491**2 RHS diff is pdg5022*pdg6277*(pdg4851 - 1)/pdg2530**2 9070394000:
9838128064:
9070394000:
9838128064:
14 substitute RHS of expr 1 into expr 2
  1. 5586102077; locally 8233899:
    \(r = d_1 + d_2\)
    \(pdg_{2530} = pdg_{2798} + pdg_{7652}\)
  2. 1811867899; locally 6577160:
    \(T^2 = \frac{d_1+d_2}{d_1+d_2} d_2 4 \pi^2 \frac{r^2}{G m_1}\)
    \(pdg_{9491}^{2} = \frac{4 pdg_{2530}^{2} pdg_{2798} pdg_{3141}^{2}}{pdg_{5022} pdg_{6277}}\)
  1. 2906548078; locally 8324356:
    \(T^2 = \frac{r}{d_1+d_2} d_2 4 \pi^2 \frac{r^2}{G m_1}\)
    \(pdg_{9491}^{2} = \frac{4 pdg_{2530}^{3} pdg_{2798} pdg_{3141}^{2}}{pdg_{5022} pdg_{6277} \left(pdg_{2798} + pdg_{7652}\right)}\)
LHS diff is 0 RHS diff is 4*pdg2530**2*pdg2798*pdg3141**2*(-pdg2530 + pdg2798 + pdg7652)/(pdg5022*pdg6277*(pdg2798 + pdg7652)) 5586102077:
1811867899:
2906548078:
5586102077:
1811867899:
2906548078:
2 declare initial expr
  1. 4393258808; locally 8072137:
    \(F_{\rm centripetal} = m r \omega^2\)
    \(pdg_{1687} = pdg_{2321}^{2} pdg_{2530} pdg_{5156}\)
no validation is available for declarations 4393258808: dimensions are consistent
4393258808: N/A
Physics Derivation Graph: Steps for Kepler's Third Law: period squared propto distance cubed

Symbols for this derivation

See also all 227 symbols
symbol ID category latex scope dimension name value Used in derivations references
2867 variable F_{\rm gravity}
\(F_{\rm gravity}\)
real
  • length: 1
  • mass: 1
  • time: -2
force due to gravity 12
3141 constant \pi
\(\pi\)
['real'] dimensionless pi 3.1415   dimensionless
72
5156 variable m
\(m\)
['real']
  • mass: 1
mass 69
4851 variable m_2
\(m_2\)
real
  • mass: 1
mass 31
2530 variable r
\(r\)
['real']
  • length: 1
radius 60
5022 variable m_1
\(m_1\)
real
  • mass: 1
mass 35
7652 variable d_1
\(d_1\)
real
  • length: 1
distance 8
2798 variable d_2
\(d_2\)
real
  • length: 1
distance 18
2321 variable \omega
\(\omega\)
['real']
  • time: -1
angular frequency 26
1687 variable F_{\rm centripetal}
\(F_{\rm centripetal}\)
real
  • length: 1
  • mass: 1
  • time: -2
centripetal force 8
9491 variable T
\(T\)
['real']
  • time: 1
period 20
6277 constant G
\(G\)
real
  • length: 3
  • mass: -1
  • time: -2
gravitational constant 6.67430*10^{-11}   m^3 * kg^-1 * s^-2
60
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