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Hyperbolic orbits

For the case of a hyperbolic orbit about the Sun, characterized by $ e>1$ , similar analysis to that in Section 4.11 gives,

    $\displaystyle e\,\sinh H - H$ $\displaystyle = \left(\frac{G\,M}{a^{\,3}}\right)^{1/2} (t-\tau),$ (4.105)
    $\displaystyle r$ $\displaystyle = a\,(e\,\cosh H - 1),$ (4.106)
and   $\displaystyle \tan(\theta/2)$ $\displaystyle = \left(\frac{e+1}{e-1}\right)^{1/2} \tanh (H/2).$ (4.107)

Here, $ H$ is termed the hyperbolic anomaly, and varies between $ -\infty$ and $ +\infty$ , with the perihelion point corresponding to $ H=0$ . Moreover, $ a=r_p/(e-1)$ . As in the elliptical case, Equation (4.105) is a transcendental equation that is most easily solved numerically.

Richard Fitzpatrick 2016-03-31