Here are my “back of a napkin” odds. Please correct!
Let’s consider a perfect sphere the size of the earth (500 trillion m^2). On that sphere, there are 10 racecars (each 10 m^2) going lethally fast. What are the chances they will hit each other?
Well, this depends a lot on initial conditions, but if we simply randomize their placement, it should be something on the order of the area of all the cars divided by the area of the sphere. Safe!
Now, let’s assume there are 10000 lethally fast racecars zooming around this earth-sized sphere. Now it is a bit more nerve wracking, but earth is big. And, we can assume they have good initial conditions— all in different lanes (and, IRL, mostly on different spheres!).
So, let’s make things harder. Say I have a gun that shoots 100,000 bullets in all directions on this empty earth, and those bullets keep flying around forever—how long will it take before they hit one of my 10,000 racecars? This is harder to estimate, but it should still be on the order of a 1 in 10 billion chance per orbital cycle (say, of 1 hour each?). So, in 10000 hours (~400 days), you have a chance of 1 in 1 million chance that one of the racecars will be hit by the bullets. So, it will be about a thousand years before you get to a 1 in 1000 chance. Is that about right? Don’t trust my math.
Your math gets you within an order of magnitude or so, check out NASA's work on ORDEM to predict this with Monte Carlos.
Two things:
1) The probability is very sensitive to # of objects and size assumptions. If you decrease the area of the bullets by an order of magnitude, the probability that it you will hit one 10 m^2 satellite in a year goes to 10%+
2) satellite orbits are not randomly distributed. Within 700-900 km altitude, the probabilities are ~2-3 orders of magnitude higher (1/1000/m^2/y)
1. While the orbits might be bunched up, the orbital surface area is bigger than the earth — and satellites are in many different orbital altitudes. So, I wanted to simplify in order to get an order of magnitude estimate.
2. The risk of an individual dying in a car crash is being compared to the risk of any satellite being hit. Of course, people die every day in car crashes and satellites don’t get hit often. The point was to give a sense of the magnitude of acceptable risk.
Because that’s what I’m struggling with — how big of a problem is this at the moment?
The ISS (which always has 3+ humans on it) pretty regularly has to maneuver to avoid debris nowadays (NASA is conservative and will do so if a big window has a ~.01% chance of seeing debris). You can count on your hands the number of times a major collision has occurred since the 90s, but that's probably partly because now more than ever we actively track and avoid debris.
Satellites can also be multi-million to multi-billion dollar assets. They're less like cars and more like oil-rigs.
Let’s consider a perfect sphere the size of the earth (500 trillion m^2). On that sphere, there are 10 racecars (each 10 m^2) going lethally fast. What are the chances they will hit each other?
Well, this depends a lot on initial conditions, but if we simply randomize their placement, it should be something on the order of the area of all the cars divided by the area of the sphere. Safe!
Now, let’s assume there are 10000 lethally fast racecars zooming around this earth-sized sphere. Now it is a bit more nerve wracking, but earth is big. And, we can assume they have good initial conditions— all in different lanes (and, IRL, mostly on different spheres!).
So, let’s make things harder. Say I have a gun that shoots 100,000 bullets in all directions on this empty earth, and those bullets keep flying around forever—how long will it take before they hit one of my 10,000 racecars? This is harder to estimate, but it should still be on the order of a 1 in 10 billion chance per orbital cycle (say, of 1 hour each?). So, in 10000 hours (~400 days), you have a chance of 1 in 1 million chance that one of the racecars will be hit by the bullets. So, it will be about a thousand years before you get to a 1 in 1000 chance. Is that about right? Don’t trust my math.
In contrast, the lifetime odds of dying in a car crash are about 1 in 100 (See cite) https://injuryfacts.nsc.org/all-injuries/preventable-death-o...