Hyperbolic Trajectories (\(e > 1\))

When \(e > 1\), the geometry of the trajectory is a hyperbola. The shape of a hyperbola is two symmetric, disconnected curves. The body following the trajectory occupies one of the curves. The other is an empty, imaginary orbit present only due to the mathematics.

All interplanetary bodies such as comets or asteroids that approach the earth, or any spacecraft we want to send to other planets, must be on a hyperbolic trajectory. Whereas a parabolic trajectory has zero velocity at infinite radius, the hyperbolic trajectory has some non-zero velocity.

From the orbit equation, Eq. (113), we see that the denominator goes to zero when \(1 + e\cos\nu\) goes to zero. The true anomaly when this happens is called the true anomaly of the asymptote:

(150)\[\nu_{\infty} = \cos^{-1}\left(-1/e\right)\]

As the true anomaly approaches \(\nu_{\infty}\), \(r\) approaches infinity. \(\nu_{\infty}\) is restricted to be between 90° and 180°.

For \(-\nu_{\infty} < \nu < \nu_{\infty}\), the trajectory of \(m_2\) follows the occupied or real trajectory shown on the left in Fig. 40. For \(\nu_{\infty} < \nu < \left({360}^{\circ} - \nu_{\infty}\right)\), \(m_2\) would occupy the virtual trajectory on the figure below. This trajectory would require a repulsive gravitational force for a mass to actually follow it, so it is only a mathematical result.