Space Debris
From dead satellites to flecks of paint, there are over 500,000 pieces of space debris currently being tracked. Space debris, and the problems they can cause is a hot topic with everything from a major Hollywood movie to TED talks being given about them. In this post I’m going to go over a less covered facet of the problem, why it’s so hard to track space debris.
What’s so Hard about Tracking Space Debris?
If you’ve been on this blog you know that once we have an objects position and velocity we can propagate it forwards in time. If we just measure it once from the Earth can’t we just propagate it forward in time and know it’s location out forever?
No, and it’s for the same reason that makes astrodynamics so interesting in the first place, the nonlinear dynamics of the problem.
Quick refresher on Numerical Propagation.
Before we jump into space debris, let’s have a quick refresher on simulating an object in space. An object orbiting earth has the following planar equations of motion
where μ is the gravitational constant and
Let’s now take a particle and put it in a circular orbit at about the same height as the international space station
The Real World
That orbit was nice, easy, and clean. Tracking that particle would be easy because we knew it’s initial state perfectly. In the real world our measurements are flawed. Now instead of knowing the states perfectly, we have our states with some error. The accuracy of measurements can vary drastically based on how we take our measurements but for now let’s say our measurements have a 00.1% error in their state. (Note: this is high for demonstration purposes) I’m going to assume the measurement error is Gaussian and then propagate our distribution forward an orbit by simulating 1000 different random particles in the distribution. Note: I’ve colored in a single particle in red to represent where the actual piece of space debris is in the distribution.
Now, I’ve only simulated that out for a single orbit and we can already see that what started as a bunch of states that were extremely close spatially are starting to pull apart from one another. At that altitude a circular orbit is only 90 minutes. What if we simulated it out for about half a day (8 orbits)?
Now some of the particles can be found over basically half the earth. Because of the nonlinear dynamics of the problem, propagating our distribution forward in time has resulted in our particles covering about 25% of the orbit.
Much Better Error
Ok, so in the above section we had some pretty large error in our measurements. Let’s now reduce our error to 1e-6% and propagate the distribution for an orbit. This corresponds to about a 7 meter positional error and 0.01 m/s velocity error. I’ve also randomly chosen a single particle in the distribution to use as the true position of the piece of space debris. After 1 orbit the positional error has grown to 108 meters and the velocity error has grown to .13 m/s.
Both have grown by an order of magnitude in just a single orbit!
After a day the average error has grown another order of magnitude.
The shape of You
There’s one more thing that is hidden in the above gifs. The initial distribution of x values is a Gaussian distribution, but what about after an orbit?
It looks a bit weird but still close to Gaussian,maybe the change is just an artifact of the number of points we chose to propagate. What about after a whole day?
After a while day we can see that we are no longer dealing with a Gaussian distribution.
These two ideas are why it’s so hard to track space debris.
- Even a small uncertainty in the debris state can grow quickly so we need to deal with a distribution when we track space debris
- The distribution may start as Gaussian, which is nice an easy to work with, but quickly changes to a much harder to work with distribution.
So How Do We Do It?
As noted in the beginning of this article, we are tracking 500,000 pieces of space debris, so how do we do it? Our ground based measurement assets (like telescopes and radars) are limited so do we just measure as much as we can and then accept the terrible propagated uncertainty till we can observe it again?
No. We can combine multiple measurements with the particles dynamics in a tool called a filter (Kalman filter, particle filter, PHD filter) to decrease our uncertainty of the particles position, but research into filters is still an open problem. When dealing with tracking an object (from space debris, to UAV’s, to your phone estimating your position) with uncertainty that field is called Estimation When dealing with tracking space debris (and space assets) the field is called Space-Situational Awareness.
Note: This post is extremely basic in terms of content covered, If Space-Situational Awareness sounds interesting here’s a post giving a more broad overview of the duties of the field. This book dives into the math behind some of the more common estimators. If you are more interested in diving into the current state of the art here are three researchers pushing the state of the art in the field.
- Dr Moriba K Jah (He gave the TED talk linked to above)
- Dr Puneet Singlah
- Dr Tomoko Matsuo
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