####
Ellipses and foci

To understand Kepler's First Law completely it is necessary to introduce
some of the mathematics of ellipses. In standard form the equation for an
ellipse is:
\begin{equation}
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \end{equation}
where $a$ and $b$ are the semimajor and semiminor axes respectively. This is
illustrated in the figure below:

Figure %: Semiminor and semimajor axes of an ellipse.

The semimajor axis is the distance from the center of the ellipse to the most
distant point on its perimeter, and the semiminor axis is the distance from the
center to the closest point on the perimeter.

The foci of an ellipse both lie along its major axis and are equally spaced
around the center of the ellipse. In fact, the foci are both distance $c$ from
the center of the ellipse where $c$ is given by $c = \sqrt{a^2 - b^2}$. As
shown in , each foci is placed such that semiminor axis (of
length $b$), part of the semimajor axis (of length $c$) form a right-angled
triangle of hypotenuse length $a$, the semimajor axis.

The eccentricity of an ellipse, can then be defined as:
\begin{equation}
\epsilon = \sqrt{1 - \frac{b^2}{a^2}} \end{equation}
For a circle (which is a special case of an ellipse), $a=b$ and thus $\epsilon =
0$. The eccentricity is a measure of how "elongated," or stretched out an
ellipse is.

####
Statement of Kepler's First Law

We can now state Kepler's First Law clearly:

* Planets orbit the sun in ellipses with the sun at one focus. *

This statement means that if a point $P$ represents the position of a planet on
an ellipse, then the distance from this point to the sun (which is at one focus)
plus the distance from $P$ to this other focus remains constant as the planet
moves around the ellipse. This is a special property of ellipses, and is
illustrated clearly in . In this case $d_1 + d_2 = l_1 +
l_2 = $ a constant as the planet moves around the sun.

Figure %: Sum of distances to each focus is a constant.

As marked on the figure, the closest point that the planet comes to
the sun is known as the aphelion and the farthest point that the planet
moves from the sun is called the perihelion.