📂Classical Mechanics

Kepler's First Law: The Law of Elliptical Orbits

Law

Kepler’s First Law. The Law of Elliptical Orbits:

Planets revolve in elliptical orbits with the Sun at one focus.

Explanation

This is the first law among Kepler’s laws of planetary motion.

Proof¹

The equation of orbit for a particle moving under a central force is as follows.

$$ \frac{ d ^{2}u}{ d \theta^{2} } + u=-\frac{1}{ml^{2}u^{2}}F(u^{-1}) $$

Here, $u={\textstyle \frac{1}{r}}$. Since we want to solve the problem with respect to gravity, let’s say $F=-\frac{GMm}{r^{2}}=-\frac{k}{r^{2}}$. $M$ is the mass of the body providing the central force (specifically, this means the mass of the Sun here), and $m$ is the mass of the moving body. Then, the equation of orbit becomes the following.

$$ \frac{ d ^{2}u}{ d \theta^{2} }+u=\frac{k}{ml^{2}} $$

The solution to the above differential equation is as follows.

$$ u=A\cos\theta+\frac{k}{ml^{2}} $$

In this case, $A$ is a constant. Rearranging the above formula with respect to $r$ gives the following equation.

$$ \begin{align*} r=\frac{1}{u}&=\frac{1}{k/ml^{2}+A\cos\theta} \\ &= \frac{ml^{2}/k }{1+(Aml^{2}/k)\cos\theta} \end{align*} $$

This equation is the same as the equation of an ellipse with the focus at the origin represented in polar coordinates. In fact, the equation of an ellipse with a semi-major axis of $\alpha$ and an eccentricity of $\epsilon$ in polar coordinates is as follows.

$$ r=\frac{\alpha}{1+\epsilon \cos\theta } $$

Therefore, the orbit of a planet revolving around the Sun due to gravity is an ellipse with the Sun at one focus, a semi-major axis of $\alpha=\frac{ml^{2}}{k}=\frac{l^{2}}{GM}$, and an eccentricity of $\epsilon=\frac{Aml^{2}}{k}=\frac{Al^{2}}{GM}$.

■