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,

$$e\,\sinh H - H = \left(\frac{G\,M}{a^{\,3}}\right)^{1/2} (t-\tau),$$ (4.105)

$$r = a\,(e\,\cosh H - 1),$$ (4.106)

$$\tan(\theta/2) = \left(\frac{e+1}{e-1}\right)^{1/2} \tanh (H/2).$$ (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.