Kepler's Laws of Planetary Motion/Third Law
== Physical Law == '''Kepler's third law of planetary motion''' is one of the three physical laws of celestial mechanics deduced by {{AuthorRef|Johannes Kepler}}:  :The square of the period of the orbit of a planet around the sun is proportional to the cube of its average distance from the sun.  == Proof == Consider a planet $p$ of mass $m$ orbiting a star $S$ of mass $M$ under the influence of the gravitational field which the two bodies give rise to. From Kepler's First Law of Planetary Motion, $p$ travels in an elliptical orbit around $S$: :$(1): \quad r = \dfrac {h^2 / k} {1 + e \cos \theta}$ where $k = G M$. From Equation of Ellipse in Reduced Form: Cartesian Frame, the equation of the orbit can also be given as: :$(2): \quad \dfrac {x^2} {a^2} + \dfrac {y^2} {b^2} = 1$ where the foci are placed at $\tuple {\pm c, 0}$. From Focus of Ellipse from Major and Minor Axis: :$a^2 - b^2 = c^2$ and also: :$e = \dfrac c a$ {{explain|The above needs to be established -- I'm not sure if this has been done}} Thus: {{begin-eqn}} {{eqn | l = e^2       | r = \dfrac {a^2 - b^2} {a^2}       | c =  }} {{eqn | n = 3       | ll= \leadsto       | l = b^2       | r = a^2 \paren {1 - e^2}       | c =  }} {{end-eqn}} :400px Then mean distance $a$ of $p$ from the focus $F$ is half the sum of the least and greatest values of $r$. So $(1)$ and $(3)$ give: {{begin-eqn}} {{eqn | l = a       | r = \dfrac 1 2 \paren {\dfrac {h^2 / k} {1 + e} + \dfrac {h^2 / k} {1 - e} }       | c =  }} {{eqn | r = \dfrac {h^2} {k \paren {1 - e^2} }       | c =  }} {{eqn | r = \dfrac {h^2 a^2} {k b^2}       | c =  }} {{eqn | n = 4       | ll= \leadsto       | l = b^2       | r = \dfrac {h^2 a} k       | c =  }} {{end-eqn}} Let $T$ be the orbital period of $p$. From Area of Ellipse, the area $\mathcal A$ of the orbit is given by: :$\mathcal A = \pi a b$ From Kepler's Second Law of Planetary Motion it follows that: :$\dfrac {h T} 2 = \pi a b$ and so from $(4)$: {{begin-eqn}} {{eqn | l = T^2       | r = \dfrac {4 \pi^2 a^2 b^2} {h^2}       | c =  }} {{eqn | r = \paren {\dfrac {4 \pi^2} k} a^3       | c =  }} {{end-eqn}} We have that: :$k = G M$ where $G$ is the gravitational constant and $M$ is the mass of $S$. Hence the result.  == Astronomical Units == {{:Kepler's Laws of Planetary Motion/Third Law/Astronomical Units}} == Examples == {{:Kepler's Laws of Planetary Motion/Third Law/Examples}} == Also see == * Kepler's First Law of Planetary Motion * Kepler's Second Law of Planetary Motion {{Namedfor|Johannes Kepler|cat = Kepler}} == Historical Note == {{:Kepler's Laws of Planetary Motion/Historical Note}} == Sources == * {{BookReference|Men of Mathematics|1937|Eric Temple Bell|prev = Kepler's Second Law of Planetary Motion|next = Newton's Law of Universal Gravitation}}: Chapter $\text{VI}$: On the Seashore * {{BookReference|Differential Equations|1972|George F. Simmons|prev = Eccentricity of Orbit indicates its Total Energy|next = Mathematician:Leonhard Paul Euler}}: $\S 3.21$: Newton's Law of Gravitation: $(22)$ * {{BookReference|Calculus Gems|1992|George F. Simmons|prev = Book:Johannes Kepler/Epitome Astronomiae Copernicanae|next = Book:Johannes Kepler/Harmonices Mundi}}: Chapter $\text {A}.10$: Kepler ({{DateRange|1571|1630}}) * {{BookReference|Calculus Gems|1992|George F. Simmons|prev = Kepler's First Law of Planetary Motion|next = Definition:Astronomical Unit}}: Chapter $\text {B}.25$: Kepler's Laws and Newton's Law of Gravitation * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev = Kepler's Second Law of Planetary Motion|next = Definition:Kernel of Group Homomorphism|entry = Kepler's laws}} * {{BookReference|The Penguin Dictionary of Mathematics|2008|David Nelson|ed = 4th|edpage = Fourth Edition|prev = Kepler's Second Law of Planetary Motion|next = Definition:Kernel of Group Homomorphism|entry = Kepler's laws}} * {{BookReference|Taming the Infinite|2008|Ian Stewart|prev = Kepler's Second Law of Planetary Motion|next = Book:Johannes Kepler/Mysterium Cosmographicum}}: Chapter $8$: The System of the World: Kepler * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Kepler's Second Law of Planetary Motion|next = Symbols:K/kg|entry = Kepler's laws of planetary motion|subentry = iii}}