Kepler's Laws of Planetary Motion/Second Law
== Physical Law == '''Kepler's second law of planetary motion''' is one of the three physical laws of celestial mechanics deduced by {{AuthorRef|Johannes Kepler}}:  : The line joining the sun and a planet sweeps out equal areas in equal times. :500px  == Proof == Consider a planet $p$ of mass $m$ moving around the sun in the plane under the influence of the force $\mathbf F$ imparted by the gravitational field which the two bodies give rise to. Let the position of $p$ at time $t$ be given in polar coordinates as $\polar {r, \theta}$. By definition of the gravitational field, $\mathbf F$ is a central force. From Derivative of Angular Component under Central Force: :$r^2 \dfrac {\d \theta} {\d t} = h$ for some constant $h$. , assume that $h > 0$, which means that $p$ is travelling in the direction of positive $\theta$. Let $\map A t$ be the area swept out by $\mathbf r$ in time $t$ relative to some fixed point of reference. For a small angle $\delta \theta$, the area $\delta A$ can be approximated to the area of a sector of a circle. Thus: :$\delta A = \dfrac {r^2 \delta \theta} 2$ and so in the limit: {{begin-eqn}} {{eqn | l = \dfrac {\d A} {\d \theta}       | r = \dfrac {r^2} 2       | c =  }} {{eqn | ll= \leadsto       | l = \int \rd A       | r = \dfrac 1 2 \int r^2 \rd \theta       | c =  }} {{eqn | r = \dfrac 1 2 \int r^2 \dfrac {\d \theta} {\d t} \rd t       | c =  }} {{eqn | r = \dfrac 1 2 \int h \rd t       | c =  }} {{eqn | ll= \leadsto       | l = \int_{t_1}^{t_2} \rd A       | r = \dfrac 1 2 \int_{t_1}^{t_2} h \rd t       | c = integrating between times $t_1$ and $t_2$ }} {{eqn | ll= \leadsto       | l = \map A {t_2} - \map A {t_1}       | r = \dfrac h 2 \paren {t_2 - t_1}       | c =  }} {{end-eqn}} That is, given a time interval $t_2 - t_1$, the area $\map A {t_2} - \map A {t_1}$ is the same, whatever the physical position of $p$.  == Also see == * Kepler's First Law of Planetary Motion * Kepler's Third 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 = Definition:Equidistance Property of Ellipse|next = Kepler's Third Law of Planetary Motion}}: Chapter $\text{VI}$: On the Seashore * {{BookReference|Differential Equations|1972|George F. Simmons|prev = Derivative of Angular Component under Central Force|next = Kepler's Laws of Planetary Motion/Historical Note}}: $\S 3.21$: Newton's Law of Gravitation: $(10)$ * {{BookReference|Calculus Gems|1992|George F. Simmons|prev = Book:Johannes Kepler/Astronomia Nova|next = Book:Johannes Kepler/Epitome Astronomiae Copernicanae}}: Chapter $\text {A}.10$: Kepler ({{DateRange|1571|1630}}) * {{BookReference|Calculus Gems|1992|George F. Simmons|prev = Derivative of Angular Component under Central Force|next = Definition:Gravitational Constant}}: 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 First Law of Planetary Motion|next = Kepler's Third Law of Planetary Motion|entry = Kepler's laws}} * {{BookReference|The Penguin Dictionary of Mathematics|2008|David Nelson|ed = 4th|edpage = Fourth Edition|prev = Kepler's First Law of Planetary Motion|next = Kepler's Third Law of Planetary Motion|entry = Kepler's laws}} * {{BookReference|Taming the Infinite|2008|Ian Stewart|prev = Kepler's First Law of Planetary Motion|next = Kepler's Third Law of Planetary Motion}}: 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 First Law of Planetary Motion|next = Kepler's Third Law of Planetary Motion|entry = Kepler's laws of planetary motion|subentry = ii}}