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 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 $\mathbf r$.


 * CentralForce.png

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 $z = \dfrac 1 z$ 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 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