Kepler's Laws of Planetary Motion/First Law
== Physical Law == '''Kepler's first law of planetary motion''' is one of the three physical laws of celestial mechanics deduced by {{AuthorRef|Johannes Kepler}}:  : Planets move around the sun in elliptical orbits, with the sun at one focus.  == 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 $\polar {r, \theta}$. Let $\mathbf F$ be expressed as: :$\mathbf F = F_r \mathbf u_r + F_\theta \mathbf u_\theta$ where: :$\mathbf u_r$ is the unit vector in the direction of the radial coordinate of $p$ :$\mathbf u_\theta$ is the unit vector in the direction of the angular coordinate of $p$ :$F_r$ and $F_\theta$ are the magnitudes of the components of $\mathbf F$ in the directions of $\mathbf u_r$ and $\mathbf u_\theta$ respectively. Let $\mathbf r$ be the radius vector from the origin to $p$. From Motion of Particle in Polar Coordinates, the second order ordinary differential equations governing the motion of $m$ under the force $\mathbf F$ are: {{begin-eqn}} {{eqn | n = 1       | l = F_\theta       | r = m \paren {r \dfrac {\d^2 \theta} {\d t^2} + 2 \dfrac {\d r} {\d t} \dfrac {\d \theta} {\d t} }       | c =  }} {{eqn | n = 2       | l = F_r       | r = m \paren {\dfrac {\d^2 r} {\d t^2} - r \paren {\dfrac {\d \theta} {\d t} }^2}       | c =  }} {{end-eqn}} By Newton's Law of Universal Gravitation, $\mathbf F$ is a central force of value: :$\mathbf F = -G \dfrac {M m} {r^3} \mathbf r$ where: :$G$ is the gravitational constant :$M$ and $m$ are the masses of the bodies :the minus sign indicates that the central force is in the opposite direction to that of the radius vector $\mathbf r$. :600px Hence we have: :$F_\theta = 0$ :$F_r = -G \dfrac {M m} {r^2} \dfrac {\size {\mathbf r} } r = -G \dfrac {M m} {r^2}$ As $G$ and $M$ are both constants, so we can express this as: :$F_r = -\dfrac {k m} {r^2}$ So from $(2)$: {{begin-eqn}} {{eqn | l = -\dfrac {k m} {r^2}       | r = m \paren {\dfrac {\d^2 r} {\d t^2} - r \paren {\dfrac {\d \theta} {\d t} }^2}       | c =  }} {{eqn | n = 3       | ll= \leadsto       | l = \dfrac {\d^2 r} {\d t^2} - r \paren {\dfrac {\d \theta} {\d t} }^2       | r = -\dfrac k {r^2}       | c =  }} {{end-eqn}} Let $z = \dfrac 1 r$. Then: {{begin-eqn}} {{eqn | l = \dfrac {\d r} {\d t}       | r = \dfrac \d {\d t} \dfrac 1 z       | c =  }} {{eqn | r = -\dfrac 1 {z^2} \dfrac {\d z} {\d t}       | c = Power Rule for Derivatives }} {{eqn | r = -\dfrac 1 {z^2} \dfrac {\d z} {\d \theta} \dfrac {\d \theta} {\d t}       | c = Chain Rule for Derivatives }} {{eqn | r = -\dfrac 1 {z^2} \dfrac {\d z} {\d \theta} \dfrac h {r^2}       | c = Derivative of Angular Component under Central Force }} {{eqn | r = -h \dfrac {\d z} {\d \theta}       | c = as $r = \dfrac 1 z$ }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \dfrac {\d^2 r} {\d t^2}       | r = -h \map {\dfrac \d {\d t} } {\dfrac {\d z} {\d \theta} }       | c =  }} {{eqn | r = -h \map {\dfrac \d {\d \theta} } {\dfrac {\d z} {\d \theta} } \dfrac {\d \theta} {\d t}       | c = Chain Rule for Derivatives }} {{eqn | r = -h \dfrac {\d^2 z} {\d \theta^2} \dfrac {\d \theta} {\d t}       | c = Chain Rule for Derivatives }} {{eqn | r = -h \dfrac {\d^2 z} {\d \theta^2} \dfrac h {r^2}       | c = Derivative of Angular Component under Central Force }} {{eqn | n = 4       | r = -h^2 z^2 \dfrac {\d^2 z} {\d \theta^2}       | c = as $r = \dfrac 1 z$ }} {{end-eqn}} Substituting from $(4)$ into $(3)$: {{begin-eqn}} {{eqn | l = -h^2 z^2 \dfrac {\d^2 z} {\d \theta^2} - r \paren {\dfrac {\d \theta} {\d t} }^2       | r = -\dfrac k {r^2}       | c =  }} {{eqn | ll= \leadsto       | l = -h^2 z^2 \dfrac {\d^2 z} {\d \theta^2} - r \paren {\dfrac h {r^2} }^2       | r = -\dfrac k {r^2}       | c = Derivative of Angular Component under Central Force }} {{eqn | ll= \leadsto       | l = -h^2 z^2 \dfrac {\d^2 z} {\d \theta^2} - \frac 1 z \paren {\dfrac h {z^2} }^2       | r = -k z^2       | c = substituting $r = \dfrac 1 z$ }} {{eqn | n = 5       | ll= \leadsto       | l = \dfrac {\d^2 z} {\d \theta^2} + z       | r = \dfrac k {h^2}       | c = simplifying }} {{end-eqn}} From Linear Second Order ODE: $y'' + y = K$, $(5)$ has the general solution: :$(6): \quad z = A \sin \theta + B \cos \theta + \dfrac k {h^2}$ , we shift the polar axis so as to make $r$ a minimum when $\theta = 0$. That is, when $\theta = 0$, $p$ is at its closest point to the origin. This means that $z = \dfrac 1 z$ is a  maximum at this point. Thus at $\theta = 0$: From Derivative at Maximum or Minimum: :$\dfrac {\d z} {\d \theta} = 0$ From Second Derivative of Strictly Concave Real Function is Strictly Negative: :$\dfrac {\d^2 z} {\d \theta^2} < 0$ Thus: :$A = 0$ :$B > 0$ Substituting these and $z = \dfrac 1 r$ back into $(6)$: :$r = \dfrac 1 {B \cos \theta + \dfrac k {h^2} }$ {{begin-eqn}} {{eqn | l = r       | r = \dfrac 1 {B \cos \theta + k / h^2}       | c =  }} {{eqn | r = \dfrac {h^2 / k} {1 + \paren {B h^2 / k} \cos \theta}       | c =  }} {{end-eqn}} Setting $B h^2 / k = e$: :$r = \dfrac {h^2 / k} {1 + e \cos \theta}$ where $e \in \R_{>0}$ and is constant. This can be expressed as: :$(7): \quad r = \dfrac {p e} {1 + e \cos \theta}$ by setting $p = \dfrac 1 B$. From Equation of Conic Section in Polar Form, $(7)$ is the equation of a conic section with one focus at the origin. As the planets remain in the solar system it follows that their orbits are stable and elliptical.  == Also see == * Kepler's Second 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 = Kepler's Laws of Planetary Motion|next = Definition:Equidistance Property of Ellipse}}: Chapter $\text{VI}$: On the Seashore * {{BookReference|Differential Equations|1972|George F. Simmons|prev = Definition:Gravitational Constant|next = Definition:Eccentricity of Conic Section}}: $\S 3.21$: Newton's Law of Gravitation * {{BookReference|Calculus Gems|1992|George F. Simmons|prev = Definition:Focus/Linguistic Note|next = Kepler's Laws of Planetary Motion/Historical Note}}: Chapter $\text {A}.10$: Kepler ({{DateRange|1571|1630}}) * {{BookReference|Calculus Gems|1992|George F. Simmons|prev = Definition:Gravitational Constant|next = Kepler's Third Law of Planetary Motion}}: 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 Laws of Planetary Motion/Historical Note|next = Kepler's Second 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 Laws of Planetary Motion/Historical Note|next = Kepler's Second Law of Planetary Motion|entry = Kepler's laws}} * {{BookReference|Taming the Infinite|2008|Ian Stewart|prev = Kepler's Laws of Planetary Motion|next = Kepler's Second 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 Laws of Planetary Motion|next = Kepler's Second Law of Planetary Motion|entry = Kepler's laws of planetary motion|subentry = i}}