Tracking Arc and Time for Precise Orbit Determination

Question

It is said that for Precise Orbit Determination (POD) of interplanetary spacecrafts, generally, during the cruise phase for aiding accurate TCM (Trajectory Correction Maneuvers) planning, long arc of tracking observables is required. During Cruise Phase of ISRO's MOM mission, its Facebook fan page updated:

TCM has been executed and tracking details suggest successful TCM. However, we will have to wait for another 5-6 hours for accurate orbit details. (Result of POD)

Is there any such record of distinct tracking time required in missions? Since Apollo program has been quite elaborately known, I hope there must be record of tracking details or any other mission for that case. I am just looking at a general trend of interplanetary spacecrafts' orbits and consequent time intervals required to get sufficiently accurate orbit details (by employing tracking by common Delta-DOR or Single station Doppler), to draw raw relation between orbits and measurement arcs required for POD without having to go into error and accuracy analysis.

Answer 1

Is there any such record of distinct tracking time required in missions?

I was unable to find any mission specific examples of utilized tracking time following a TCM or otherwise. I believe this is simply because it is a highly technical aspect of the mission and is not generally publicized. That being said, there are a few common methods of orbit determination in use that may reveal a requisite tracking time.

Perhaps the most advanced technique used is called "Admissible Regions". This method seeks to avoid the TSA problem (too short arc) which arises when the three instances used for orbit determination are too close to one another and the position vectors become nearly parallel. When this is the case, the commonly cited Gibb's Method, breaks down as the cross products become undefined. To quantify this within the context of your question, Gibb's method requires a minimum track length of about 5 degrees, which correlates to roughly 10 days of observation for a martian orbit. The Herrick-Gibbs method does better, requiring a minimum of about 1 degree, which shortens the observation time to just 2 days.

The Admissible Regions technique uses a probabilistic approach to estimate the most likely orbit given much closer observations. Using this, a track length of just a few tenths of a degree is all that is required before a comparably accurate orbit can be determined. On the martian orbit time scale, a 0.2 degree arc length represents Just 10 hours of observation, which is similar to the time span cited by ISRO.

It is important to note that these times are dependent on the distance of the object from the sun/earth; for missions beyond mars, the time required to sweep the same arc will increase. It is also important to note that these times can be reduced by supplementing the analysis with position and velocity data from on-board sensors or by considering the expected trajectory, both of which will reduce the error envelope and, consequently, the time required for an accurate determination.