Covariance matrix of a satellite position

Question

In order to estimate the probability of collision with space debris, the covariance matrix of the primary and secondary objects should be calculated (JSPoC paper). I'm going to calculate the matrix using the publicly available data (NORAD TLE).

To calculate the matrix, I'm taking TLE files of an object for the previous 2 weeks. Then I propagate all the TLE's to the most actual TLE date. Propagation of the most actual TLE to its epoch will be the most accurate prediction and will be used as the true value ($[x_0,y_0,z_0]$) in the covariance matrix calculations:

$$\sigma(x,y)=\frac{1}{n-1}\sum_{i=1}^{n-1}(x_i-x_0)(y_i-y_0)^T$$ $$C= \begin{pmatrix} \sigma(x,x) & \sigma(x,y) & \sigma(x,z) \\ \sigma(y,x) & \sigma(y,y) & \sigma(y,z) \\ \sigma(z,x) & \sigma(z,y) & \sigma(z,z) \\ \end{pmatrix} $$

1. Is this the right way?
2. It's said, that the elements of the matrix should be in RSW (Radial, Along-track, and Cross-track) frame. Why?

Answer 1

Here's a paper that describes what I think you're trying to do with generating an estimated covariance matrix from historical data.

Peterson, G.; Gist, R.; Oltrogge, D., “Covariance Generation for Space Objects Using Public Data”, Proceedings of the 11th Annual AAS/AIAA Space Flight Mechanics Meeting, Santa Barbara, CA; UNITED STATES; 11-15 Feb. 2001. pp. 201-214. 2001 (PDF downloadable here).

Note that any maneuvers included in the historical span will tend to greatly inflate this estimated covariance.

Answer 2

What you're describing is what JSpOC does. This is their job. They monitor everything, and warn people who operate satellites they can see may be at risk of collision. That paper is just intended to give satellite operators a good feeling that JSpOC is taking care of them and knows what they are doing. It does not enable you to take over JSpOC's job yourself, primarily because doing so at all well involves having access to sources of information they are not going to share with you.

Instructions for signing up to receive conjunction assessment warnings are posted on the web page at https://www.space-track.org/documentation#odr . They say

United States Space Command (USSPACECOM) is committed to promoting a safe, stable, sustainable, and secure space environment through Space Situational Awareness (SSA) information sharing. There are three levels of SSA services: basic, emergency, and advanced. Advanced services are available to all entities who sign an SSA Sharing Agreement with USSPACECOM.

USSPACECOM/J535 Strategy, Plans, and Policy negotiates SSA Sharing Agreements. The SSA Sharing Agreement establishes the parameters within which data will be exchanged by both signing parties to facilitate ongoing cooperation and advance spaceflight safety. All members of the space community, including satellite operators, launching agencies, commercial service providers, and research/academic institutions, are welcome to contact [email protected] for more information on the agreement process. U.S. government (USG) organizations and their contractors have implied agreements and do not need to pursue additional documentation with USSPACECOM.

Whether you are putting your first satellite on orbit, or adding to an existing constellation, we encourage you to register your asset with 18 SPCS so that we can plan for optimal tracking and identification, and provide you with conjunction assessment services as soon as possible. As soon as you register, a member of 18 SPCS will contact you to discuss the details of your mission and coordinate conjunction assessment and other required support.

Interestingly, however, in the same zip file in which those who register for a free account on space-track.org (the public web site of the US Space Command's 18th Space Control Squadron (18 SPCS)) can download a working copy of SGP4, there is also a copy of the release notes for the entire Standardized Astrodynamics Algorithms Library (SAAL), including both SGP4 and also the parts of the SAAL that you can only obtain if you are doing official business of the U.S. Government. One of those pieces is the tool they actually use for this purpose. It is called COMBO --- Computation Of Miss Between Orbits. The release notes say (emphasis mine)

Combo computes close approaches between satellite orbits based on user-specified screening volume, exclusion volume, and warning and alert thresholds expressed in either standoff radius or asset-centered UVW (radial, in-track, and cross-track) miss-distance criteria. Precomputed SGP4, SGP4-XP, or SP-based ephemerides are used in the evaluation. Note: SGP4 GP-based Combo results are appropriate for general understanding of the frequency of approaches between objects and coarse assessment of expected miss distances between objects, but not appropriate for collision-avoidance decisions. Only high-accuracy SP ephemerides that include propagated covariance resulting in computed Probability of Collision $P_c$ should be used for collision-avoidance decisions.

"SP" is the companion piece of software, the not-SGP4 propagator they use for precision work. Its name, unfortunately, is simply "Special Perturbations", which is terrible because everything that numerically integrates the equations of motion technically deserves the same name (see this answer for the difference between special and general perturbations). SP uses a different kind of data, which does include covariance matrices, and computes motion from them in a totally different way from SGP4. SGP4's job is to be fast, and SP's job is to be accurate. The new "SGP4-XP" is supposed to be a complete rewrite of SGP4, using a much improved theory and much higher-order approximation than was possible 40 years ago, because computing hardware is now so much more powerful, but still using only a TLE-sized input string as control parameters while producing a much more accurate result. Unfortunately, I have yet to find out how to obtain this new data ("TLE Type 4"), so I am unable to test the "eXtended Perturbations" for myself. I suspect what I need is a properly-worded and approved Orbit Data Request form.

The reason for working in radial, in-track, and cross-track coordinates is that for most operational satellites, the position uncertainty as output by the orbit determination software is not spherical, but instead is considerably larger in-track than the other two directions. Determining whether one cigar (or perhaps potato) passes through another is a different problem than determining whether one sphere passes through another, and orientation of those shapes in space is important. In fact, the covariances JSpOC distributes to the small number of people who have been authorized to receive them, are always at least six dimensional (position and velocity), plus sometimes extra things solved for in the fit by the batch differential corrector that computed the state vectors and their associated estimated uncertainties.