At what point do satellites make astronomical images hard to avoid?

Before checking sources

Matt's first pass

One of my assumptions must be way off. I assumed the average satellite is orbiting at about 1,000 km above the surface and traveling at about 5,000 m/s. I also assumed a long exposure covers about 1 square cm of sky at about 1 m from the observer and takes about an hour, or about 3,700 seconds.

First, I figured out what distance a satellite would cover during that time:

distance ~= 3.7 x 10^3 s x 5 x 10^3 m/s
         ~= 1.9 x 10^7 m

Then I estimated the total surface area of a sphere at the satellite altitude. Using a radius of about 7.4 x 10^6 m:

satellite-shell area ~= 4 x pi x r^2
                     ~= 12 x (7.4 x 10^6)^2
                     ~= 6.6 x 10^14 m2

Next, I figured out what portion of the sky is covered by a single exposure using a 1 cm square at a distance of 1 m. A 1 m sphere has surface area about 12 m2, so:

fraction of sky ~= 1 x 10^-4 m2 / 12 m2
                ~= 8.3 x 10^-6

search areas in sky ~= 12 / 1 x 10^-4
                    ~= 1.2 x 10^5

A single search area at the satellite shell would therefore cover:

search-area size ~= 6.6 x 10^14 x 8.3 x 10^-6
                 ~= 5.5 x 10^9 m2

search-area width ~= sqrt(5.5 x 10^9)
                  ~= 7 x 10^4 m

Then I tried to figure out how many of these search areas would be crossed by a satellite in an hour. I divided the distance covered in an hour by the search-area width and got about 3 x 10^2 search areas crossed by a single satellite in an hour.

Comparing that to 1.2 x 10^5 total search areas, I estimated about a 0.25% chance that a single satellite would cross a given search area in an hour.

I thought the probabilities could be summed for each additional satellite, so once there are about 400 satellites in orbit, assuming random or even distribution of their orbits, there should be a 100% chance that a satellite crosses a search area in a given hour.

That seems wrong. In reality, satellite orbits will not be perfectly random because launch mechanics creates preferred orbital families. But it seems I must be making an error if it would only take 400 satellites to bring the chance of a streak in one search area up to 100%.

Post-check reflection

Matt's reflection

Looks like my mental model and assumptions were not terribly far off.

I made a mistake assuming probabilities could be summed, when really the probability approaches 1. This is the second time the suggested peg of expected event probability has come up, and I should really try to understand that better. Correcting that would have pushed the number of satellites necessary before we would expect the crossing event to likely occur upward.

I was way high on the time for a long exposure, more like 5 minutes than 60, and the field width was probably 6 times larger than it should have been, so the probability of a field crossing estimate was high by a factor of about 70.

I also did not exclude satellites for being unlikely to leave light streaks, or factor into my calculations that many are likely to follow a similar path.

But ultimately I think I ended up near the right conclusion: it really does not take tens of thousands of satellites in orbit before there is a risk of streaks in earth-based long exposures, and hundreds of thousands would almost certainly result in multiple unavoidable streaks during exposures.