What is the probability of impact?

Space Track is a website dedicated to passing along information about satellites to consumers, primarily satellite operators. The information comes from the US military. In addition to providing the Two Line Elements that inform operators of exactly where their satellites are, they also produce conjunction reports, of predicted near misses to other objects in space.

I am trying to use the information they provide in order to predict with a medium fidelity the probability of impact. Some of the information they provide is:

• The In Track, Radial, and Cross Track miss distances.
• A 3x3 correlation matrix with the above values
• Radar Cross Sectional data for each object.
• Other miscellaneous data.

I'm trying to figure out the following:

1. How can I determine a minimum distance apart? It's clear if the two objects are 1 m diameter spheres, if they are less than 1 meter from each other, it is a hit, but is there a good rule of thumb for an arbitrary object, and a known object (Our satellite) without extensive modeling applied?
2. Given that I have a concept on how much of a miss distance is considered a hit, how do I determine that miss distance?
• I think you are talking about radar data from NORAD, right? It is usually a few hours old, at least, sometimes days. The standard propagation algorithm is SGP4, if I remember it correctly, which is also used in STK. Both, the old & radar-based TLEs and the inaccuracy of SGP4 should make it incredibly hard to actually predict a collision of most objects ... Usually, you calculate a probability based on the mentioned inaccuracies for an geometric primitive - a sphere. Arbitrary object - you are asking about odd-shaped stuff with solar panels etc? – s-m-e Jul 24 '13 at 23:55
• They actually provide direct conjugation reports to satellite operators, which I have. I have the radar cross sections, and other data provided. By arbitrary, I'm not sure what the shape looks like. – PearsonArtPhoto Jul 24 '13 at 23:59

I... like this question. First things first, TLE sets are nearly useless for precise probability of collision estimation, because they do not contain an estimate of the uncertainty in the orbit, which is ultimately what affects your probability of collision.

So, assuming you get a full message with a high-precision state vector and an associated covariance matrix (full 6x6 state is better, but 3x3 can work) for each satellite, there are a few things you can do. There are quite a few different methods for estimating this, look up work by Alfano and Patera for a couple examples.

What the problem mostly boils down to is estimating the overlap of the uncertainty at the time of closest approach. Imagine "uncertainty" (in 3 dimensions) as an ellipsoid - if you're familiar with concepts of probability, it's your multivariate probability density. Now, the probability of collision can be thought of as the amount of overlap between those two ellipsoids. In your specific situation, you do not have full state and uncertainty information for both objects, but similar concepts apply here (they essentially make some assumptions and combine the uncertainty of the objects).

1. I assume you're asking what minimum distance you (as an operator) should tolerate. This is hard to put a specific value on. Normally, you get a probability of collision estimate and assign a threshold in that space, such as "anything higher than a 1e-4 probability of collision and we will maneuver". As far as what distance constitutes a "hit", what's usually done is assuming both objects are spherical. This can be hard if you don't know much about the other object.

2. See above for the basic concepts, but implementation obviously requires knowledge of probabilistic concepts, orbit dynamics, and (preferably) state estimation techniques.

Finally, this is very much an active topic of research, and there are often papers describing a "new" way to do this. I should also mention that while what I explained is often done in practice, there are several reasons to believe that it does not, in fact, provide an accurate probability of collision estimate.

Unfortunately, open literature on this topic is sparse, but searching for the authors I mentioned above, and in fact just the terms "satellite probability of collision" will get you some very useful hits if you have the right access (AIAA journal papers, for instance).

However, you can estimate this numerically instead of analytically using Monte Carlo methods. Here's a quick and dirty way to do this:

1. Sample from your "primary" satellite (the one you got the conjunction report for). To do this, generate a 3x1 normally-distributed random vector with zero mean and unit variance (the standard normal. Now, take your 3x3 covariance for this object, find its Cholesky decomposition (you should be able to find a routine to do this in your software package - 'chol' in Matlab), and multiply it by your random vector. You've now sampled your probability space.

2. Repeat #1 for the conjuncting object. However, once you sample, you need to translate each point according to its position, which is given to you in RIC coordinates. Just add your position vector to the random vector.

3. Calculate the distance between your two samples. Record said distance.

4. Repeat steps 1-3 for some very large number of samples (think tens of thousands - generally, the more, the merrier).

5. Now, you have a list of "miss distances". There are a couple things you can do with this, but one thing that's particularly helpful is to construct an empirical cumulative distribution function. You can look up how to do this, but you should be able to find a function to do this in most software packages (in Matlab, it's 'cdfplot'). This will give you a plot that looks like this (only with x>0):

Now, one way you can read this is by choosing a "miss distance" you're comfortable with (say, 20m), and finding the value of $F(x)$ at this value of $x$. That's the probability that you will come within 20m or less of the offending satellite. If it's greater than some threshold you've decided upon with your management, you begin the fun task of planning an avoidance maneuver. If not, you wait and watch. Alternatively, you can specify a probability you're comfortable with, and read off a miss distance associated with it.

One thing you'll notice is that you're really focused on the tail of your plot... and you'll see that to get a smooth representation at the tail, you need a whole hell of a lot of samples. So like I said, the more, the merrier.

One last thing: if you've followed these steps, the probability number you have is $1\sigma$. For, say, a $3\sigma$ estimate, just multiply your two covariance matrices by your factor (in this case 3).

• I'm trying to estimate the probability of collision. Part of that is figuring out how close a collision is. Assuming the object we don't know about is a sphere is fine, so long as I can get a conversion from RCS to diameter of the sphere. – PearsonArtPhoto Jul 25 '13 at 0:18
• Due to the nature of radar you're not going to get a simple or even accurate conversion like that. You are better off estimating something large, since that is going to be soaked up by error in the calculation anyway. – user29 Jul 25 '13 at 0:22
• @PearsonArtPhoto - If being completely arbitrary doesn't gel with you, I suppose you could look up the object and get a feel for at least the order of magnitude of the size of the object. Debris objects will still be problematic. – user29 Jul 25 '13 at 0:59