Parabolic orbits

For the case of a parabolic orbit about the Sun, characterized by $e=1$, similar analysis to that in Section 4.11 yields,

$$P + \frac{P^{\,3}}{3} = \left(\frac{G\,M}{2\,r_p^{\,3}}\right)^{1/2} (t-\tau),$$ (4.102)

$$r = r_p\,(1+P^{\,2}),$$ (4.103)

$$\tan(\theta/2) = P.$$ (4.104)

Here, $P$ is termed the parabolic anomaly and varies between $-\infty$ and $+\infty$, with the perihelion point corresponding to $P=0$. Note that Equation (4.102) is a cubic equation, possessing a single real root, that can, in principle, be solved analytically. (See Section 4.17, Exercise 19.) However, a numerical solution is generally more convenient.