# Meaning of a Constant in the Unbound Orbit Equation

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 $$\beta$$?

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

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):
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.