What are the odds that an alien spaceship entering a low Earth orbit would collide with an orbiting body?

Question

After centuries of careful search and travel, some intrepid extraterrestrial explorers have finally found another inhabited world! Elated with their discovery and exhausted after their long journey, they swing into a high orbit around Earth and start running their scans. Thankfully their vessel is very hard for our primitive telescopes to detect, so they aren't worried about any interference from the surface until they're well and ready to reach out to us.

However, might their chief apprehension be an accidental high-impact collision with one of our 934,000 orbiting objects over 1cm in length?

When we send things into orbit, we obviously have a big leg up in that we've already catalogued the orbital trajectories of our satellites. We currently track over 22,300 total orbiting objects.

Assumptions:

1. It would be difficult for these aliens to gain that same level of knowledge from their high orbit, because of the small size and high speed of all of the objects. Let's say they can only detect the largest 20% of our 22,300 tracked orbiting bodies initially.
2. They need to send a stealth probe down into a low Earth orbit for 3 weeks to complete their scans. This is the most crowded area (63% of satellites operate here, and we can assume that's the distribution of junk as well.)
3. The probe is a 5-meter radius sphere.
4. The aliens do not possess any handwavium-powered shields or unobtanium-plated armor, so a collision with even a small orbiting body could be disastrous.
5. The alien's method of interstellar travel effectively avoids space debris, but that technology cannot be used in orbit.

If the probe remained in a random low-Earth orbit for three weeks, what are the odds that it would collide with an object over 1cm in length?

BONUS 1: Is there an optimal orbit to minimize the chance of these collisions, given their initial level of knowledge of our orbital patterns?

BONUS 2: Assume further that in this low orbit they can detect an additional 44,000 orbiting objects every day, and adjust the probe's orbit accordingly at the end of each day. By the end of 3 weeks they know where nearly all of our 934,000 dangerous objects are. Exactly how much does this effect the probe's overall survival chance?

Answer 1

Luckily for your intrepid explorers, the odds are infinitesimal.

According to this study, the chances of something twice as large as your probe hitting another object with a cross section larger than 1 cm is between 1:100 and 1:1000 over a ten year period. Insofar as an ideal altitude (bonus #1), figures 4.2 and 4.3 in that same study seem to indicate that an altitude below 200 km would be best. After that, between 1200 and 1400 km has the lowest chance of collision. (Most satellites operate between 500 and 1000 km.) These are very simplified estimates, though; the article has more details on various considerations.

This article corroborates that altitudes of 800-1000 km are the most congested. It also gets into micrometeoroid data.

Note the odds go up quickly the smaller the cross section you’re concerned about. It might hit one or two objects in the mm range, but in the lowest altitudes even the odds of that are tiny. But after a three week period (bonus #2), they should be able to avoid just about anything as large as a grain of sand, if they wanted to.

Answer 2

I would say that if the alien race is advanced enough for Interstellar travel than they would have the capabilities to detect granular sized objects at a decent safe distance. They needed to be able to detect these if they were traveling at any amount of a relativistic speed otherwise a grain of sand could have destroyed their ship.

Answer 3

The odds are very, very, very low. For reference, the International Space Station (110 meters long, currently) has lived in low Earth orbit for over 20 years and there have been no significant collisions. The ISS is subject to micro-strikes all the time, as can be seen on this equipment which flew on the ISS for 19 years. But these are essentially grains of sand.

The ISS does have the benefit of Earth-based tracking of debris, and does occasionally maneuver to avoid potential obstacles. But the odds are still very low even if they did not do that. The ISS has to re-boost itself periodically anyway, so maneuvering to avoid obstacles is essentially free.

Your assumption 2 is wrong. Low Earth orbit is cleaner than higher orbits, and this is one of the reasons we put things there. The reason is that there's still enough atmosphere in low Earth orbit to produce significant drag, so any debris left there will eventually de-orbit on its own (in maybe 6-24 months, depending on initial conditions). When LEO satellites reach the end of their useful life they are either de-orbited, or boosted into a higher stable orbit where they are unlikely to cause problems in the future.

Answer 4

Regarding your first bonus point, of minimising the chances of collision (and hoping my shaky Kerbal-based understanding of orbital mechanics is correct):

If two objects are orbiting at the same altitude and the same inclination, then they'll be moving at the same velocity. So their relative velocity will be zero, and they'll never collide as long as they maintain their orbits. The risk would be of something in a higher orbit losing altitude due to atmospheric drag (and thus gaining relative velocity) and potentially colliding with a lower object.

If they're at different inclinations, there would also be the risk of collision at the points where their orbits intersect.

So your safest orbit would seem to be just high enough that drag isn't a factor, and at the same inclination as the majority of objects orbiting at that altitude. Assuming that altitude isn't too high for your probe's instruments. At which point your potential collisions are other objects at the same altitude but a different inclination.

The problem with that is, if that turns out to be equatorial, then that's not going to be much help for your probe scanning the entire planet, since it's always going to be over the equator. The optimal inclination is obviously going to depend on how long you have to scan the planet and how far either side of the probe's path it can scan on each pass.