How to sort orbital positions in the order of following by spacecraft?

Question

I have an orbiting spacecraft which keplerian elements is known. Also, I have a random-ordered list of true anomalies that represents spacecraft's positions on that orbit.

How to sort this list to get all true anomalies in the order in which it will be followed by spacecraft?

Answer 1

True Anomaly is the angle from periapsis, through the center of the body being orbited, to a designated orbital position, measured in the plane of the orbit, with the positive direction designated as the direction of travel around the orbit.

If your true anomaly values are in the range $(-\pi, \pi]$ in radians or $(-180°, 180°]$ in degrees, sort them in ascending order. On an elliptical orbit, the body will visit these true anomalies once each orbit in that order. On a parabolic or hyperbolic trajectory, it will visit them once, in that order.

If your calculations have resulted in true anomaly values outside those ranges, you can convert the angles to put them in that range. There are many ways to do this, I typically use the arctan2 functon. Given an initial true anomaly value $\theta$:

$$f = \mathrm{arctan2}(\sin \theta, \cos \theta)$$

$f$ will be in the range $(-\pi, \pi]$ or $(-180°, 180°]$ as appropriate, and you can sort in ascending order, as above.

Answer 2

If we have a random list of true anomalies in range [-PI;PI] we can sort it using next algorithm:

1. Substract from each anomaly the current position's anomaly.
2. Add 2*PI to each anomaly value that is lesser than 0
3. Sort list of anomalies in ascending order
4. Add current position's anomaly to each anomaly in list.
5. Substract 2*PI from each anomaly value that greater than PI

After this steps we will receive the anomalies list sorted in following order.