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 :
: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:
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 universal 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 $\mathbf r$.
:
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 $(2)$:
Let $z = \dfrac 1 r$.
Then:
Then:
Substituting from $(4)$ into $(3)$:
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 $z = \dfrac 1 r$ 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 Concave Real Function is Non-Positive:
:$\dfrac {\d^2 z} {\d \theta^2} < 0$
Thus:
:$A = 0$
:$B > 0$
Substituting these and $z = \dfrac 1 r$ back into $(6)$:
:$r = \dfrac 1 {B \cos \theta + \dfrac k {h^2} }$
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
== Historical Note ==
== Sources ==
* : Chapter $\text{VI}$: On the Seashore
* : $\S 3.21$: Newton's Law of Gravitation
* : Chapter $\text {A}.10$: Kepler ()
* : Chapter $\text {B}.25$: Kepler's Laws and Newton's Law of Gravitation
* 
* 
* : Chapter $8$: The System of the World: Kepler
* 
Category:Kepler's Laws of Planetary Motion