Kepler2 - ellipse In fact the foci are both distance $c$...

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Kepler's First Law 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
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Unformatted text preview: 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....
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