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

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

This page contains three views of the steps in the derivation: d3js, graphviz PNG, and a table.

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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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Physics Derivation Graph: Steps for Kepler's Third Law: period squared propto distance cubed
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	3132131132; locally 2340248: $ω = \frac{2 π}{T}$ $p d g_{2321} = \frac{2 p d g_{3141}}{p d g_{9491}}$ 3896798826; locally 9388996: $m_{2} d_{2} ω^{2} = G \frac{m_{1} m_{2}}{r^{2}}$ $p d g_{2321}^{2} p d g_{2798} p d g_{4851} = \frac{p d g_{4851} p d g_{5022} p d g_{6277}}{p d g_{2530}^{2}}$		9070394000; locally 4575586: $m_{2} d_{2} \frac{4 π^{2}}{T^{2}} = G \frac{m_{1} m_{2}}{r^{2}}$ $\frac{4 p d g_{2798} p d g_{3141}^{2} p d g_{4851}}{p d g_{9491}^{2}} = \frac{p d g_{4851} p d g_{5022} p d g_{6277}}{p d g_{2530}^{2}}$	valid	3132131132: 3896798826: 9070394000:	3132131132: 3896798826: 9070394000:
19	substitute LHS of expr 1 into expr 2	2217103163; locally 9110206: $\frac{m_{1} d_{1}}{d_{2}} = m_{2}$ $\frac{p d g_{5022} p d g_{7652}}{p d g_{2798}} = p d g_{4851}$ 4188580242; locally 1709969: $T^{2} = \frac{r^{3} 4 π^{2}}{(m_{1} + (\frac{m_{1}}{d_{2}} d_{1})) G}$ $p d g_{9491}^{2} = \frac{4 p d g_{2530}^{3} p d g_{3141}^{2}}{p d g_{6277} (p d g_{5022} + \frac{p d g_{5022} p d g_{7652}}{p d g_{2798}})}$		5658865948; locally 8711868: $T^{2} = \frac{r^{3} 4 π^{2}}{(m_{1} + m_{2}) G}$ $p d g_{9491}^{2} = \frac{4 p d g_{2530}^{3} p d g_{3141}^{2}}{p d g_{6277} (p d g_{4851} + p d g_{5022})}$	valid	2217103163: 4188580242: 5658865948:	2217103163: 4188580242: 5658865948:
7	declare initial expr			3132131132; locally 2340248: $ω = \frac{2 π}{T}$ $p d g_{2321} = \frac{2 p d g_{3141}}{p d g_{9491}}$	no validation is available for declarations	3132131132:	3132131132:
12	multiply RHS by unity	9170048197; locally 7795202: $T^{2} = d_{2} 4 π^{2} \frac{r^{2}}{G m_{1}}$ $p d g_{9491}^{2} = \frac{4 p d g_{2530}^{2} p d g_{2798} p d g_{3141}^{2}}{p d g_{5022} p d g_{6277}}$	8122039815: $\frac{d_{1} + d_{2}}{d_{1} + d_{2}}$ $1$	1811867899; locally 6577160: $T^{2} = \frac{d_{1} + d_{2}}{d_{1} + d_{2}} d_{2} 4 π^{2} \frac{r^{2}}{G m_{1}}$ $p d g_{9491}^{2} = \frac{4 p d g_{2530}^{2} p d g_{2798} p d g_{3141}^{2}}{p d g_{5022} p d g_{6277}}$	valid	9170048197: 1811867899:	9170048197: 1811867899:
3	change two variables in expr	4393258808; locally 8072137: $F_{c e n t r i p e t a l} = m r ω^{2}$ $p d g_{1687} = p d g_{2321}^{2} p d g_{2530} p d g_{5156}$	8916428651: $m$ $p d g_{5156}$ 1635147226: $m_{2}$ $p d g_{4851}$ 9884115626: $r$ $p d g_{2530}$ 1036530438: $d_{2}$ $p d g_{2798}$	3649797559; locally 6652843: $F_{c e n t r i p e t a l} = m_{2} d_{2} ω^{2}$ $p d g_{1687} = p d g_{2321}^{2} p d g_{2798} p d g_{4851}$	valid	4393258808: dimensions are consistent 3649797559:	4393258808: N/A 3649797559:
11	raise both sides to power	9152823411; locally 7556753: $\frac{1}{T^{2}} = \frac{1}{d_{2} 4 π^{2}} G \frac{m_{1}}{r^{2}}$ $\frac{1}{p d g_{9491}^{2}} = \frac{p d g_{5022} p d g_{6277}}{4 p d g_{2530}^{2} p d g_{2798} p d g_{3141}^{2}}$	7445388869: $- 1$ $- 1$	9170048197; locally 7795202: $T^{2} = d_{2} 4 π^{2} \frac{r^{2}}{G m_{1}}$ $p d g_{9491}^{2} = \frac{4 p d g_{2530}^{2} p d g_{2798} p d g_{3141}^{2}}{p d g_{5022} p d g_{6277}}$	no check is performed	9152823411: 9170048197:	9152823411: 9170048197:
6	substitute LHS of expr 1 into expr 2	3649797559; locally 6652843: $F_{c e n t r i p e t a l} = m_{2} d_{2} ω^{2}$ $p d g_{1687} = p d g_{2321}^{2} p d g_{2798} p d g_{4851}$ 6829281943; locally 4382594: $F_{c e n t r i p e t a l} = G \frac{m_{1} m_{2}}{r^{2}}$ $p d g_{1687} = \frac{p d g_{4851} p d g_{5022} p d g_{6277}}{p d g_{2530}^{2}}$		3896798826; locally 9388996: $m_{2} d_{2} ω^{2} = G \frac{m_{1} m_{2}}{r^{2}}$ $p d g_{2321}^{2} p d g_{2798} p d g_{4851} = \frac{p d g_{4851} p d g_{5022} p d g_{6277}}{p d g_{2530}^{2}}$	valid	3649797559: 6829281943: 3896798826:	3649797559: 6829281943: 3896798826:
17	declare initial expr			5128670694; locally 4476518: $m_{1} d_{1} = m_{2} d_{2}$ $p d g_{5022} p d g_{7652} = p d g_{2798} p d g_{4851}$	no validation is available for declarations	5128670694:	5128670694:
16	simplify	3781109867; locally 6644719: $T^{2} = \frac{r^{3} 4 π^{2}}{(d_{1} + d_{2}) \frac{m_{1}}{d_{2}} G}$ $p d g_{9491}^{2} = \frac{4 p d g_{2530}^{3} p d g_{2798} p d g_{3141}^{2}}{p d g_{5022} p d g_{6277} (p d g_{2798} + p d g_{7652})}$		4188580242; locally 1709969: $T^{2} = \frac{r^{3} 4 π^{2}}{(m_{1} + (\frac{m_{1}}{d_{2}} d_{1})) G}$ $p d g_{9491}^{2} = \frac{4 p d g_{2530}^{3} p d g_{3141}^{2}}{p d g_{6277} (p d g_{5022} + \frac{p d g_{5022} p d g_{7652}}{p d g_{2798}})}$	valid	3781109867: 4188580242:	3781109867: 4188580242:
1	declare initial expr			1292735067; locally 5331094: $F_{g r a v i t a t i o n a l} = G \frac{m_{1} m_{2}}{r^{2}}$ $p d g_{2867} = \frac{p d g_{4851} p d g_{5022} p d g_{6277}}{p d g_{2530}^{2}}$	no validation is available for declarations	1292735067:	1292735067:
10	multiply both sides by	9838128064; locally 6210646: $d_{2} \frac{4 π^{2}}{T^{2}} = G \frac{m_{1}}{r^{2}}$ $\frac{4 p d g_{2798} p d g_{3141}^{2}}{p d g_{9491}^{2}} = \frac{p d g_{5022} p d g_{6277}}{p d g_{2530}^{2}}$	5684907106: $\frac{1}{d_{2} 4 π^{2}}$ $\frac{1}{4 p d g_{2798} p d g_{3141}^{2}}$	9152823411; locally 7556753: $\frac{1}{T^{2}} = \frac{1}{d_{2} 4 π^{2}} G \frac{m_{1}}{r^{2}}$ $\frac{1}{p d g_{9491}^{2}} = \frac{p d g_{5022} p d g_{6277}}{4 p d g_{2530}^{2} p d g_{2798} p d g_{3141}^{2}}$	valid	9838128064: 9152823411:	9838128064: 9152823411:
20	declare final expr	5658865948; locally 8711868: $T^{2} = \frac{r^{3} 4 π^{2}}{(m_{1} + m_{2}) G}$ $p d g_{9491}^{2} = \frac{4 p d g_{2530}^{3} p d g_{3141}^{2}}{p d g_{6277} (p d g_{4851} + p d g_{5022})}$			no validation is available for declarations	5658865948:	5658865948:	period squared propto distance cubed
15	multiply RHS by unity	2906548078; locally 8324356: $T^{2} = \frac{r}{d_{1} + d_{2}} d_{2} 4 π^{2} \frac{r^{2}}{G m_{1}}$ $p d g_{9491}^{2} = \frac{4 p d g_{2530}^{3} p d g_{2798} p d g_{3141}^{2}}{p d g_{5022} p d g_{6277} (p d g_{2798} + p d g_{7652})}$	9524810853: $\frac{1 / d_{2}}{1 / d_{2}}$ $1$	3781109867; locally 6644719: $T^{2} = \frac{r^{3} 4 π^{2}}{(d_{1} + d_{2}) \frac{m_{1}}{d_{2}} G}$ $p d g_{9491}^{2} = \frac{4 p d g_{2530}^{3} p d g_{2798} p d g_{3141}^{2}}{p d g_{5022} p d g_{6277} (p d g_{2798} + p d g_{7652})}$	valid	2906548078: 3781109867:	2906548078: 3781109867:
13	declare assumption			5586102077; locally 8233899: $r = d_{1} + d_{2}$ $p d g_{2530} = p d g_{2798} + p d g_{7652}$	no validation is available for declarations	5586102077:	5586102077:
5	substitute RHS of expr 1 into expr 2	3176662571; locally 2600680: $F_{c e n t r i p e t a l} = F_{g r a v i t y}$ $p d g_{2867} = p d g_{1687}$ 1292735067; locally 5331094: $F_{g r a v i t a t i o n a l} = G \frac{m_{1} m_{2}}{r^{2}}$ $p d g_{2867} = \frac{p d g_{4851} p d g_{5022} p d g_{6277}}{p d g_{2530}^{2}}$		6829281943; locally 4382594: $F_{c e n t r i p e t a l} = G \frac{m_{1} m_{2}}{r^{2}}$ $p d g_{1687} = \frac{p d g_{4851} p d g_{5022} p d g_{6277}}{p d g_{2530}^{2}}$	LHS diff is -pdg1687 + pdg2867 RHS diff is 0	3176662571: 1292735067: 6829281943:	3176662571: 1292735067: 6829281943:
18	divide both sides by	5128670694; locally 4476518: $m_{1} d_{1} = m_{2} d_{2}$ $p d g_{5022} p d g_{7652} = p d g_{2798} p d g_{4851}$	8044416349: $d_{2}$ $p d g_{2798}$	2217103163; locally 9110206: $\frac{m_{1} d_{1}}{d_{2}} = m_{2}$ $\frac{p d g_{5022} p d g_{7652}}{p d g_{2798}} = p d g_{4851}$	valid	5128670694: 2217103163:	5128670694: 2217103163:
4	declare assumption			3176662571; locally 2600680: $F_{c e n t r i p e t a l} = F_{g r a v i t y}$ $p d g_{2867} = p d g_{1687}$	no validation is available for declarations	3176662571:	3176662571:
9	simplify	9070394000; locally 4575586: $m_{2} d_{2} \frac{4 π^{2}}{T^{2}} = G \frac{m_{1} m_{2}}{r^{2}}$ $\frac{4 p d g_{2798} p d g_{3141}^{2} p d g_{4851}}{p d g_{9491}^{2}} = \frac{p d g_{4851} p d g_{5022} p d g_{6277}}{p d g_{2530}^{2}}$		9838128064; locally 6210646: $d_{2} \frac{4 π^{2}}{T^{2}} = G \frac{m_{1}}{r^{2}}$ $\frac{4 p d g_{2798} p d g_{3141}^{2}}{p d g_{9491}^{2}} = \frac{p d g_{5022} p d g_{6277}}{p d g_{2530}^{2}}$	LHS diff is 4pdg2798pdg3141*2(pdg4851 - 1)/pdg9491*2 RHS diff is pdg5022pdg6277(pdg4851 - 1)/pdg2530*2	9070394000: 9838128064:	9070394000: 9838128064:
14	substitute RHS of expr 1 into expr 2	5586102077; locally 8233899: $r = d_{1} + d_{2}$ $p d g_{2530} = p d g_{2798} + p d g_{7652}$ 1811867899; locally 6577160: $T^{2} = \frac{d_{1} + d_{2}}{d_{1} + d_{2}} d_{2} 4 π^{2} \frac{r^{2}}{G m_{1}}$ $p d g_{9491}^{2} = \frac{4 p d g_{2530}^{2} p d g_{2798} p d g_{3141}^{2}}{p d g_{5022} p d g_{6277}}$		2906548078; locally 8324356: $T^{2} = \frac{r}{d_{1} + d_{2}} d_{2} 4 π^{2} \frac{r^{2}}{G m_{1}}$ $p d g_{9491}^{2} = \frac{4 p d g_{2530}^{3} p d g_{2798} p d g_{3141}^{2}}{p d g_{5022} p d g_{6277} (p d g_{2798} + p d g_{7652})}$	LHS diff is 0 RHS diff is 4pdg25302pdg2798pdg31412(-pdg2530 + pdg2798 + pdg7652)/(pdg5022pdg6277(pdg2798 + pdg7652))	5586102077: 1811867899: 2906548078:	5586102077: 1811867899: 2906548078:
2	declare initial expr			4393258808; locally 8072137: $F_{c e n t r i p e t a l} = m r ω^{2}$ $p d g_{1687} = p d g_{2321}^{2} p d g_{2530} p d g_{5156}$	no validation is available for declarations	4393258808: dimensions are consistent	4393258808: N/A

Symbols for this derivation

See also all 227 symbols

symbol ID	category	latex	scope	dimension	name	value	Used in derivations	references
4851	variable	m_2 $m_{2}$	real	mass: 1	mass		Kepler's Third Law: period squared propto distance cubed escape velocity Newton's Law of Gravitation radius for satellite in geostationary orbit mass of the Earth velocity at distance r of object dropped from infinity	Mass	31
3141	constant	\pi $π$	['real']	dimensionless	pi	3.1415 dimensionless	frequency relations Kepler's Third Law: period squared propto distance cubed angle of maximum distance for projectile motion Newton's Law of Gravitation upper limit on velocity in condensed matter derivation of Schrodinger Equation radius for satellite in geostationary orbit Euler equation to e^(i pi) + 1 = 0 particle in a 1D box speed of Earth around Sun		72
2798	variable	d_2 $d_{2}$	real	length: 1	distance		Kepler's Third Law: period squared propto distance cubed	Distance	18
1687	variable	F_{\rm centripetal} $F_{c e n t r i p e t a l}$	real	length: 1 mass: 1 time: -2	centripetal force		radius for satellite in geostationary orbit Kepler's Third Law: period squared propto distance cubed Newton's Law of Gravitation	Centripetal_force	8
5156	variable	m $m$	['real']	mass: 1	mass		Kepler's Third Law: period squared propto distance cubed equation of motion for a spring escape velocity Schwarzschild radius for non-rotating black hole Newton's Law of Gravitation work and force and energy derivation of Schrodinger Equation radius for satellite in geostationary orbit mass of the Earth velocity at distance r of object dropped from infinity time invariant force conserves energy	Mass Q11423 Mass	69
5022	variable	m_1 $m_{1}$	real	mass: 1	mass		Kepler's Third Law: period squared propto distance cubed escape velocity Newton's Law of Gravitation radius for satellite in geostationary orbit mass of the Earth velocity at distance r of object dropped from infinity	Mass	35
6277	constant	G $G$	real	length: 3 mass: -1 time: -2	gravitational constant	6.6743010^{-11} m^3 kg^-1 * s^-2	Kepler's Third Law: period squared propto distance cubed escape velocity Schwarzschild radius for non-rotating black hole Newton's Law of Gravitation radius for satellite in geostationary orbit mass of the Earth velocity at distance r of object dropped from infinity equations of motion in 2D (calculus)	Gravitational_constant	60
7652	variable	d_1 $d_{1}$	real	length: 1	distance		Kepler's Third Law: period squared propto distance cubed	Distance	8
2867	variable	F_{\rm gravity} $F_{g r a v i t y}$	real	length: 1 mass: 1 time: -2	force due to gravity		radius for satellite in geostationary orbit Newton's Law of Gravitation Kepler's Third Law: period squared propto distance cubed	Gravity	12
9491	variable	T $T$	['real']	time: 1	period		frequency relations Kepler's Third Law: period squared propto distance cubed equation of motion for a spring Newton's Law of Gravitation frequency and period		20
2530	variable	r $r$	['real']	length: 1	radius		Kepler's Third Law: period squared propto distance cubed escape velocity Schwarzschild radius for non-rotating black hole Newton's Law of Gravitation radius for satellite in geostationary orbit speed of Earth around Sun velocity at distance r of object dropped from infinity	Radius	60
2321	variable	\omega $ω$	['real']	time: -1	angular frequency		frequency relations Kepler's Third Law: period squared propto distance cubed equation of motion for a spring derivation of Schrodinger Equation electric field wave equation: from time dependent to time independent	Angular_frequency	26

MESSAGE:

local variable 'all_df' referenced before assignment