Radial trajectory

In astrodynamics and celestial mechanics a radial trajectory is a Kepler orbit with zero angular momentum. Two objects in a radial trajectory move directly towards or away from each other in a straight line.

There are three types of radial trajectories (orbits).

Unlike standard orbits which are classified by their orbital eccentricity, radial orbits are classified by their specific orbital energy, the constant sum of the total kinetic and potential energy, divided by the reduced mass:

where x is the distance between the centers of the masses, v is the relative velocity, and ${\displaystyle \mu ={G}(m_{1}+m_{2})}$ is the standard gravitational parameter.

Another constant is given by:

Given the separation and velocity at any time, and the total mass, it is possible to determine the position at any other time.

The first step is to determine the constant w. Use the sign of w to determine the orbit type.

where ${\displaystyle \textstyle x_{0}}$ and ${\displaystyle \textstyle v_{0}}$ are the separation and relative velocity at any time.

...
Wikipedia