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Review Kepler's Third Law: period squared propto distance cubed

step inference rule input feed output step validity (as per SymPy)
1
  • 0000111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\ref{eq:#1} is an initial equation.
  1. 1292735067
    \(F_{\rm gravity} = G \frac{m_1 m_2}{r^2}\)
 no validation is available for declarations
2
  • 0000111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\ref{eq:#1} is an initial equation.
  1. 4393258808
    \(F_{\rm centripetal} = m r \omega^2\)
 no validation is available for declarations
3
  • 0000111984: change two variables in expression
  • number of inputs: 1; feeds: 4; outputs: 1
  • Change variable $#1$ to $#2$ and $#3$ to $#4$ in Eq.~\ref{eq:#5}; yields Eq.~\ref{eq:#6}.
  1. 4393258808
    \(F_{\rm centripetal} = m r \omega^2\)
  1. 1036530438
    \(d_2\)
  2. 9884115626
    \(r\)
  3. 1635147226
    \(m_2\)
  4. 8916428651
    \(m\)
  1. 3649797559
    \(F_{\rm centripetal} = m_2 d_2 \omega^2\)
 LHS diff is 0 RHS diff is pdg0002321**2*(pdg0002530*pdg0005156 - pdg0002798*pdg0004851)
4
  • 0000111104: declare assumption
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\ref{eq:#1} is an assumption.
  1. 3176662571
    \(F_{\rm centripetal} = F_{\rm gravity}\)
 no validation is available for declarations
5
  • 0000111634: substitute RHS of expr 1 into expr 2
  • number of inputs: 2; feeds: 0; outputs: 1
  • Substitute RHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.
  1. 1292735067
    \(F_{\rm gravity} = G \frac{m_1 m_2}{r^2}\)
  2. 3176662571
    \(F_{\rm centripetal} = F_{\rm gravity}\)
  1. 6829281943
    \(F_{\rm centripetal} = G \frac{m_1 m_2}{r^2}\)
 LHS diff is -pdg0001687 + pdg0002867 RHS diff is pdg0001687 - pdg0004851*pdg0005022*pdg0006277/pdg0002530**2
6
  • 0000111556: substitute LHS of expr 1 into expr 2
  • number of inputs: 2; feeds: 0; outputs: 1
  • Substitute LHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.
  1. 6829281943
    \(F_{\rm centripetal} = G \frac{m_1 m_2}{r^2}\)
  2. 3649797559
    \(F_{\rm centripetal} = m_2 d_2 \omega^2\)
  1. 3896798826
    \(m_2 d_2 \omega^2 = G \frac{m_1 m_2}{r^2}\)
 LHS diff is -pdg0002321**2*pdg0002798*pdg0004851 + pdg0004851*pdg0005022*pdg0006277/pdg0002530**2 RHS diff is pdg0002321**2*pdg0002798*pdg0004851 - pdg0004851*pdg0005022*pdg0006277/pdg0002530**2
7
  • 0000111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\ref{eq:#1} is an initial equation.
  1. 3132131132
    \(\omega = \frac{2\pi}{T}\)
 no validation is available for declarations
8
  • 0000111556: substitute LHS of expr 1 into expr 2
  • number of inputs: 2; feeds: 0; outputs: 1
  • Substitute LHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.
  1. 3896798826
    \(m_2 d_2 \omega^2 = G \frac{m_1 m_2}{r^2}\)
  2. 3132131132
    \(\omega = \frac{2\pi}{T}\)
  1. 9070394000
    \(m_2 d_2 \frac{4 \pi^2}{T^2} = G \frac{m_1 m_2}{r^2}\)
 LHS diff is pdg0002321 - 4*pdg0002798*pdg0003141**2*pdg0004851/pdg0009491**2 RHS diff is 2*pdg0003141/pdg0009491 - pdg0004851*pdg0005022*pdg0006277/pdg0002530**2
9
  • 0000111457: simplify
  • number of inputs: 1; feeds: 0; outputs: 1
  • Simplify Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.
  1. 9070394000
    \(m_2 d_2 \frac{4 \pi^2}{T^2} = G \frac{m_1 m_2}{r^2}\)
  1. 9838128064
    \(d_2 \frac{4 \pi^2}{T^2} = G \frac{m_1}{r^2}\)
 LHS diff is 4*pdg0002798*pdg0003141**2*(pdg0004851 - 1)/pdg0009491**2 RHS diff is pdg0005022*pdg0006277*(pdg0004851 - 1)/pdg0002530**2
10
  • 0000111182: multiply both sides by
  • number of inputs: 1; feeds: 1; outputs: 1
  • Multiply both sides of Eq.~\ref{eq:#2} by $#1$; yields Eq.~\ref{eq:#3}.
  1. 9838128064
    \(d_2 \frac{4 \pi^2}{T^2} = G \frac{m_1}{r^2}\)
  1. 5684907106
    \(\frac{1}{d_2 4 \pi^2}\)
  1. 9152823411
    \(\frac{1}{T^2} = \frac{1}{d_2 4 \pi^2} G \frac{m_1}{r^2}\)
 valid
11
  • 0000111483: raise both sides to power
  • number of inputs: 1; feeds: 1; outputs: 1
  • Raise both sides of Eq.~\ref{eq:#2} to $#1$; yields Eq.~\ref{eq:#3}.
  1. 9152823411
    \(\frac{1}{T^2} = \frac{1}{d_2 4 \pi^2} G \frac{m_1}{r^2}\)
  1. 7445388869
    \(-1\)
  1. 9170048197
    \(T^2 = d_2 4 \pi^2 \frac{r^2}{G m_1}\)
 valid
12
  • 0000111646: multiply RHS by unity
  • number of inputs: 1; feeds: 1; outputs: 1
  • Multiply RHS of Eq.~\ref{eq:#2} by 1, which in this case is $#1$; yields Eq.~\ref{eq:#3}
  1. 9170048197
    \(T^2 = d_2 4 \pi^2 \frac{r^2}{G m_1}\)
  1. 8122039815
    \(\frac{d_1+d_2}{d_1+d_2}\)
  1. 1811867899
    \(T^2 = \frac{d_1+d_2}{d_1+d_2} d_2 4 \pi^2 \frac{r^2}{G m_1}\)
 valid
13
  • 0000111104: declare assumption
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\ref{eq:#1} is an assumption.
  1. 5586102077
    \(r = d_1 + d_2\)
 no validation is available for declarations
14
  • 0000111634: substitute RHS of expr 1 into expr 2
  • number of inputs: 2; feeds: 0; outputs: 1
  • Substitute RHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.
  1. 1811867899
    \(T^2 = \frac{d_1+d_2}{d_1+d_2} d_2 4 \pi^2 \frac{r^2}{G m_1}\)
  2. 5586102077
    \(r = d_1 + d_2\)
  1. 2906548078
    \(T^2 = \frac{r}{d_1+d_2} d_2 4 \pi^2 \frac{r^2}{G m_1}\)
 LHS diff is pdg0002530 - pdg0009491**2 RHS diff is (-4*pdg0002530**3*pdg0002798*pdg0003141**2 + pdg0005022*pdg0006277*(pdg0002798 + pdg0007652)**2)/(pdg0005022*pdg0006277*(pdg0002798 + pdg0007652))
15
  • 0000111646: multiply RHS by unity
  • number of inputs: 1; feeds: 1; outputs: 1
  • Multiply RHS of Eq.~\ref{eq:#2} by 1, which in this case is $#1$; yields Eq.~\ref{eq:#3}
  1. 2906548078
    \(T^2 = \frac{r}{d_1+d_2} d_2 4 \pi^2 \frac{r^2}{G m_1}\)
  1. 9524810853
    \(\frac{1/d_2}{1/d_2}\)
  1. 3781109867
    \(T^2 = \frac{r^3 4 \pi^2}{(d_1+d_2) \frac{m_1}{d_2}G}\)
 valid
16
  • 0000111457: simplify
  • number of inputs: 1; feeds: 0; outputs: 1
  • Simplify Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.
  1. 3781109867
    \(T^2 = \frac{r^3 4 \pi^2}{(d_1+d_2) \frac{m_1}{d_2}G}\)
  1. 4188580242
    \(T^2 = \frac{r^3 4 \pi^2}{\left(m_1+\left(\frac{m_1}{d_2}d_1\right)\right)G}\)
 valid
17
  • 0000111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\ref{eq:#1} is an initial equation.
  1. 5128670694
    \(m_1 d_1 = m_2 d_2\)
 no validation is available for declarations
18
  • 0000111975: divide both sides by
  • number of inputs: 1; feeds: 1; outputs: 1
  • Divide both sides of Eq.~\ref{eq:#2} by $#1$; yields Eq.~\ref{eq:#3}.
  1. 5128670694
    \(m_1 d_1 = m_2 d_2\)
  1. 8044416349
    \(d_2\)
  1. 2217103163
    \(\frac{m_1 d_1}{d_2} = m_2\)
 valid
19
  • 0000111556: substitute LHS of expr 1 into expr 2
  • number of inputs: 2; feeds: 0; outputs: 1
  • Substitute LHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.
  1. 4188580242
    \(T^2 = \frac{r^3 4 \pi^2}{\left(m_1+\left(\frac{m_1}{d_2}d_1\right)\right)G}\)
  2. 2217103163
    \(\frac{m_1 d_1}{d_2} = m_2\)
  1. 5658865948
    \(T^2 = \frac{r^3 4 \pi^2}{(m_1+m_2)G}\)
 LHS diff is -pdg0009491**2 + pdg0005022*pdg0007652/pdg0002798 RHS diff is (-4*pdg0002530**3*pdg0003141**2 + pdg0004851*pdg0006277*(pdg0004851 + pdg0005022))/(pdg0006277*(pdg0004851 + pdg0005022))
Note about step 20: period squared propto distance cubed
20
  • 0000111341: declare final expression
  • number of inputs: 1; feeds: 0; outputs: 0
  • Eq.~\ref{eq:#1} is one of the final equations.
  1. 5658865948
    \(T^2 = \frac{r^3 4 \pi^2}{(m_1+m_2)G}\)
no validation is available for declarations

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