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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.

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