There are now well over 20,000 objects larger than a softball in orbit around the Earth. We are poised to get even more objects, as several companies are planning on launching mega-constellations of satellites (750-4,000) to provide internet across the world from low-Earth orbiting platforms. While space appears to be quite empty, there are a lot of satellites (and random stuff) out there. The problem is that there is no one “driving” the majority of these objects, and collisions between objects that are moving at 17,000 MPH typically produce even more pieces of debris that could potentially collide with other objects. And the satellites that are being “driven” have only a certain amount of fuel – once that runs out, the satellite is essentially dead. Therefore, satellite operators are not really wanting to move their satellites all of the time.
The United States Air Force (USAF) is in charge of keeping track of the locations and tracks of all objects in orbit around the Earth. There are several websites and apps where you can see which satellites are going to pass over your location at any time. These are pretty cool, because you can often very clearly see orbiting satellites. If you go outside on a clear day just before sunrise or after sunset, while the sun is still shining in the upper atmosphere, but is not shining on the ground (say 10-45 minutes after sunset), and you look up for a long time, you can see little dots of light moving across the sky that are not airplanes. Those are satellites. An app will tell you exactly which satellites those are and how they are moving. You can even see the shape of the International Space Station (ISS) with good binoculars!
These websites and apps get this orbit information from USAF, which tracks them using radars. There are many tracking stations on the ground that are similar to radar guns that police officers use. Simplistically, the radar sends out a pulse that bounces off the satellite and returns back to the radar. The radar can record the time that it took to go from the radar, to the satellite, and back to the radar, which specifies the distance to the satellite. The radar can also record the Doppler shift of the signal, which indicates the speed of the satellite. The strength of the return signal indicates the size of the satellite. This is all a simplification, but in general it is the case. The radar takes a few of these measurements, and then moves on to another object. With a bunch of measurements, the orbital characteristics of the satellite can be determined. The Air Force does this for the more than 20,000 objects on a daily basis. They make the (rough) orbital characteristics available to the public.
(As an aside, you could actually do the same thing with a telescope at home, just by tracking how the objects that you see move across the sky. It wouldn’t be as accurate as a radar, but just knowing how the object moves across the sky gives you a pretty good estimate of the orbit. Kepler did this with the moons of Jupiter, and you could do it with the ISS!)
The USAF uses an orbit propagator that takes these radar measurements and calculates the position and velocity of the 20,000 objects every second for the next five days (roughly). They then look for any objects that come within a certain distance from each other. Those satellites are flagged to be examined more carefully. There are multiple ways to do this, but I will discuss the easiest to understand here.
First, let’s back up a bit. Let’s say that you want to predict whether two cars that are moving towards the same intersection will collide. One is traveling east and one is traveling south. Consider some scenarios:
- The cars are one block apart and both traveling 30 MPH (44 feet/sec) and are exactly 0.1 miles (528 feet) apart. They will meet in the middle of the intersection. A crash is definite, and will take place in about 12 seconds!
- Imagine the same scenario (one), but one car is 0.11 miles away. The cars will miss each other by 0.01 miles (roughly 53 feet).
- Imagine scenario one, but one car is moving at a slightly slower speeds (say 29 MPH, or 42.5 feet/sec, instead of 30 MPH). In 12 seconds, the car will fall behind its expected distance traveled by about 18 feet, which is about the length of a car, and therefore will end up missing the other car.
- Imagine scenario one again, and both cars start off on a perfect collision course, but one of the cars starts to decelerate due to there being a hill in the way. If that deceleration is enough, the speed that the car is traveling decreases enough, and the collision will be avoided.
In predicting whether the cars will crash, there are many things that cause uncertainty: estimating the initial position and velocity as well as figuring out all of the forces on the cars that can cause accelerations. While there are several differences between collisions between cars and satellites, the concepts are the same. Satellites are moving much faster than cars, and there are significantly fewer objects in orbit than cars in, say, downtown New York City. In addition, satellite operators need a couple of days to figure out whether there will be a collision, so they would like to know well ahead of time if they need to move the satellite out of the way.
It seems like it would be relatively straightforward to predict a collision, since the Air Force has the position and velocity of each of the objects, and it uses a really highly accurate orbit propagator to determine the future positions of the satellites. Except, as illustrated above, if there are uncertainties in the initial position or the initial velocity, then it is not clear if there will be a collision or not. In addition, if there are uncertainties in the accelerations that the objects undergo, it will cause uncertainties in whether there will be a collision.
So, what are the sources of those uncertainties? Well, there are a lot of them. Let’s go through them all one by one:
- The position could be off due to issues with the radar signals. These signals travel up into the atmosphere, hit the orbiting object, then come back through the atmosphere. As the signal goes through the atmosphere, the speed of the signal actually changes, since the propagation speed is dependent on the medium that it travels through. In fact, the path through the atmosphere is dependent on these atmospheric conditions also. So, moisture in the atmosphere and the ionosphere can both change the path of travel, the propagation speed and therefore the delay. If the atmosphere is not modeled correctly, then this will cause the exact position of the orbiting object to have some uncertainty. And remember that the uncertainty only needs to be about as big as the satellite (a few feet) to cause issues in determining whether there will be a collision.
- The velocity of the orbiting object needs to be known really, really, really well. As an example, let’s say that there may be a collision 24 hours into the future, and the object that is moving is 10 feet across. That means that the speed needs to be known to within 0.000116 feet/sec, or 0.000079MPH. That is crazy small. Considering that the objects are orbiting at about 17,000 MPH, this is an impossible task. This is a source of very large uncertainty!
- The forces that act on the orbiting objects are quite complex and challenging to model. For example, some of the forces include:
- Gravity: While most people think of gravity as 9.8 m/s2, it is more complicated than this. The Earth is to first order a flattened sphere, so that the satellites feel more gravity near the equator than near the poles. To higher orders (literally), the orbit propagators need to take into account many of the features of the Earth, such as mountain ranges and oceans. The equation that is used to (accurately) describe the gravity has several hundred terms in it. If you discount these terms, the positions of the objects can be systematically miscalculated, which is bad. The European Space Agency has launched several satellite missions to accurately model the gravity of the Earth, such as CHAMP, GRACE, and GOCE. Neglecting higher order gravity terms can cause the satellite orbits to be off by several hundred meters after a day.
- Sun and moon Gravity: The sun and the moon both exert forces on objects in orbit around the Earth, so they need to be taken into account. Luckily, the orbits of the Earth around the sun and the moon around Earth are pretty well known, so these are easy forces to account for.
- Sunlight: Sunlight bouncing off the orbiting object actually imparts momentum to it. While this force is quite small, it actually is very important to model correctly. It is dependent on the materials that the orbiting object is made out of (if the material absorbs sunlight it feels a different force than if it reflects the sunlight), and the orientation of the object (if it has a large area pointed at the sun, then there is a lot of force, but if there is a small area, the force is smaller). Neglecting this force can cause an error of 10s of meters after a day.
- Earth shine: Sunlight reflecting off of the Earth adds pressure to the orbiting object, similar to sunlight described above. This only happens on the dayside, and is pretty weak compared to the direct sunlight, but it still needs to be accounted for. Further, the Earth radiates infrared energy (i.e., the Earth glows). Satellites feel this glow, just like they feel the sunlight, but instead of being directed from the sun, it is directed from the Earth. It is complicated to accurately take into account the Earth shine, but luckily it is a pretty weak force, so for collision avoidance, it can be approximated and not modeled exactly. Neglecting this results in errors of a couple of meters at most after a day.
- Drag: Just like a biker with a headwind feels wind resistance, a satellite in orbit feels a tiny bit of atmospheric drag force that causes it to lose energy all of the time. (See a post on drag here!) The drag force is directly dependent on density of the atmosphere and is dependent on the difference between the velocity of the object and the winds in the atmosphere squared. There is significant uncertainty in both the winds and the density of the atmosphere. As described below, it can be one of the main sources of error in the probability of collision, and can cause the positions to be uncertain to hundreds of meters after a day.
It is impossible to definitively say “there will be a collision between these two objects in 24 hours from now”. What is done instead is determine the probability of collision. This information is then passed on to satellite operators, so they can choose whether they want to move their satellite or not. If the decide to move their satellite, the operators typically speed it up or slow it down to definitively avoid the collision.
A simple method of calculating the probability of collision is to do what is called a “Monte Carlo” simulation of the interaction. By this, the modelers create about a few million versions of object 1 and a few million versions of object 2. They give these objects slightly different initial positions, velocities, and satellite characteristic (using random numbers to perturb these quantities) and see how many of them collide. This number, divided by the few million scenarios, gives the probability of collision.
The satellite characteristics that are perturbed have to do with the drag. Satellites have different shapes and sizes and masses. While the satellites’ shapes are typically well described, the orbital debris is not well described at all. For example, several years ago, the Chinese blew up one of their own weather satellites, resulting in several thousand pieces of debris of unknown size and mass. There are estimates of the size and mass of each of these objects, but there is significant uncertainty. Therefore, the few million objects in the Monte Carlo simulation are given slightly different sizes and masses (technically ballistic coefficients – the ballistic coefficient also depends on the shape of the object and what the object is made out of, but that is a detail. Well, this whole post is a detail, so it is a detail on a detail!)
The USAF propagates these millions of Monte Carlo satellites through a single atmosphere. While the majority of the time this is fine (since the upper atmosphere is calm most of the time), at times this can give huge systematic errors. For example, when the northern (and southern) lights become active, they add a bunch of energy to the atmosphere, causing it to heat up and expand. This expansion causes the density to increase, which drives a stronger drag force. If this isn’t accounted for, then the probability of collision will be incorrect.
Recently, a paper was published that showed that the behavior of the atmosphere has a large effect on the probability of collision. An event was explored, where the probability of collision was determined to be above the threshold where something should be done. The million objects were then simulated over and over and over again, propagating them through different atmosphere, depending on what was predicted. It was shown that if the sun and the aurora was a tiny bit more active, the probability of collision would be increased, while if the aurora and sun were either a lot more active or any less active, the probability of collision would decrease. It was suggested that this uncertainty in the sun’s brightness and the auroral activity should be taken into account when calculating the probability of collision. (This paper was published by Charles Bussy-Virat, and there is a youtube video of him explaining all of this in a seminar here.)
Finally, the probability of collision that is typically taken as a threshold to do something with the satellites is typically 0.0001%. This is incredibly low! But, considering that a satellite may cost several hundred million dollars and many tens of millions of dollars to launch into space, the operators want to be as cautious as possible.
In summary, calculating the probability of collision between objects in orbit is really hard. When the atmosphere is really calm, the hardest part is figuring out the velocity of the objects – a tiny error in this can cause a large error in the position of the objects at the time of closest approach. When the atmosphere goes a bit crazy, due to the aurora or the sun having more activity than expected, the satellite’s drag force can change pretty dramatically, acting to change the acceleration, velocity, and ultimately the position of the objects at the time of closest approach. While the distribution of the velocity errors is really well understood, so the Air Force can very accurately account for this in determining the probability of collision, the uncertainty in how the atmosphere is behaving is very hard to account for. This lack of knowledge in how to treat the future state of the atmosphere is one of the largest challenges in accurately determining the probability of collision of objects in orbit around the Earth.
A Side Note:
There are a lot of issues in determining the density of the atmosphere. The most accurate models of the upper atmosphere, at this time, are empirical models, meaning that the models were created fitting a ton of data. These models get the mean state of the atmosphere (i.e., the climate) (mostly) correct, but have a hard time with the “weather”. Indeed, the Air Force uses a very old model of the upper atmosphere that is no longer the state of the art, but they have a way of compensating for this. There are over 50 perfect spheres orbiting the Earth at different altitudes. The Air Force can get very good orbital characteristics from these spheres. In addition, they know the shape, mass and what the sphere is made out of, so they can use these to figure out what the drag force actually is, given the change of orbit from one time to another. They can compare the “actual” drag force and the drag force predicted by the model and adjust the model until it matches. Then they use that corrected model to predict the density into the future.
The science community is attempting to improve models of the upper atmosphere all of time. These are like weather models for the troposphere in that they use fluid dynamic equations to simulate the atmosphere. Ultimately, the USAF will have to use these types of models if they really want to improve the forecasts when there are large storms in space that can dramatically alter the trajectory of the satellites. They will make predictions like hurricane forecasters make predictions of landfall – using many different models to better understand the uncertainty.