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The solution to the radial equation for two celestial bodies with eccentricity $\varepsilon$ greater than 1 can be expressed as

$$ \frac{(x-\delta)^2}{\alpha^2} - \frac{y^2}{\beta^2}=1, $$ where \begin{align} \alpha &= \frac{c}{\varepsilon^2-1},\\ \beta &= \frac{c}{\sqrt{\varepsilon^2-1}},~~~\text{and}\\ \delta &= \varepsilon\alpha. \end{align}

I get that $\alpha$ is the distance from the origin to the closest approach of the hyperbolic orbit and $\delta$ is the distance from the sun to the origin. Can someone help me out by explaining the physical meanings of $c$ and $\beta$?

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  • $\begingroup$ $\beta$ seems to be something like the "horizontal" width of the hyperbola, seeing as if $x=\delta$ then $y$ is $\pm \beta$ $\endgroup$ Nov 24, 2023 at 23:34

1 Answer 1

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Your equation is close to the standard form, rewrite it with $\delta$ of zero, and $\alpha$ and $\beta$ as $a$ and $b$. Here is a figure from Wikipedia (Hyperbola article):

Hyperbola with a, b, c, p and asymptotes

The distance $MF_1$ is labeled $c$, the eccentricity is $\varepsilon = c/a$. What you called $c$ is $p$ in the figure. What you call $\delta$ is $c$ in the figure.

Going back to your equation, the line through $F_1$ and $F_2$ is the x axis and the $\delta$ shifts the y axis through $F_1$.

The quantity $p$ is important in orbits because it is directly related to angular momentum (which is constant). Mathematicians call it semi-latus rectum and astronomers say parameter.

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