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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,

    $\displaystyle P + \frac{P^{\,3}}{3}$ $\displaystyle = \left(\frac{G\,M}{2\,r_p^{\,3}}\right)^{1/2} (t-\tau),$ (4.102)
    $\displaystyle r$ $\displaystyle = r_p\,(1+P^{\,2}),$ (4.103)
and   $\displaystyle \tan(\theta/2)$ $\displaystyle = P.$ (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 Exercise 19.) However, a numerical solution is generally more convenient.

Richard Fitzpatrick 2016-03-31