What model do you want recognized? Two-Body mechanics? 3-body? n-body? Solar radiation, drag, which earth model? The question seems easy at the surface, and for some back-on-the-envelope math is is, but you can easily go nuts with complexity here. Also, which PE/AP for the ISS should one assume?

@Polygnome see edit

Don't forget to account for the quadrupole moment!

@OrganicMarble I had you in mind, it's there as J2!

Which one is which color?

@Polygnome I've asked this separately, let's ignore it here. How much of a drag is it, orbiting the Earth in a space suit?

@OrganicMarble Gallant makes it back to the ISS in just over 20 minutes whereas Zippy is way out there!

Alas, poor Zippy. Thanks.

This doesn't seem right. The guy 100m in front of ISS is still exactly in the same orbit, just with a slight phase shift. His distance shouldn't change. The guys above and sideways are in elliptical orbit and at a different inclination, repsectively, these curves seem about right.

@asdfex It's directly in front in the cartesian sense, not along track. I've added the initial vectors in the text. For ISS and Goofus it's [a, 0, 0] and [a, 100, 0] meters. However Goofus is still moving with ISS' velocity. At least that's what I've simulated.

@asdfex btw those are cartesian coordinates before I rotated to an inclination of 51 degrees (rotated everybody at once, pivoting about the x-axis (the ISS) at t=0).

Are this the very smart astronauts you mention in the other question? :-P .

@DiegoSánchez yes indeed! Gallant always does the right thing, so he represents the very smart astronaut. Goofus always does the wrong thing, and Zippy the Pinhead is in a completely different world!

I was finally motivated to run one of your Python codes. Thanks for posting it! I mainly wanted to learn how to plot stuff without using a spreadsheet.

@OrganicMarble that's great news! Python feels a little different in the beginning, but it's the gift that keeps on giving because there are so many packages out there you can do just about anything! Strip images out of PDFs or PPTs or web pages, play music, calculate or plot stuff as you mention... I learned it just by typing "How to X with python" sentences in to google; there are so many blogs and tutorials out there. Ping me in chat anytime, pastebin is an easy way to exchange snippets.

I have written many programs in Python but they were all manipulating text / strings. For example I wrote one to extract all the SRB segment data from the pages in the Space Shuttle Almanac and process it into tables so I could get the complete flight history by booster segment by typing in a shuttle flight number. But I did not know how to plot anything. imgur.com/O49pcSm

@asdfex I'll have another look today, thanks for the feedback! I certainly won't require any answer to match my calculation, it's offered as background to the question and "my attempt to solve the problem".

The tiny difference between straight ahead and ahead in orbit shouldn't result in a large change like this.

@asdfex the reason I resist making plots look too good is that somebody may come along later and use it out of context as if it's absolutely correct. I make them deliberately less-than-user-friendly because they're not peer-reviewed. FYI I've taken DavidHammen's advice and used the simple yet symplectic method semi-implicit Euler...

@asdfex I've removed J2 and inclination and fancy SciPy and NumPy from the looped calculation, and I get the same result. I'm gonna stand by it until proven wrong with math! script: pastebin.com/qM2Y3iMW result: i.stack.imgur.com/I17iq.png

@uhoh Leaving out information makes it for all of us much more complicated to understand what you did. Essentially we have to recreate all your work to get started. With labels and such one can start right ahead.

@asdfex I've said This is what I got and also I certainly won't require any answer to match my calculation, it's offered as background to the question and "my attempt to solve the problem".. If the attempt bothers you, please ignore it. The problem is completely described in the text above the separator line.

@asdfex But to "we can't understand it but we simultaneously think it's wrong" I can only say I disagree.

@uhoh Are you aware that there is no analytical solution to your problem? I don't think you can do better than solving numerically.

@Polygnome first three words of the title: "How to estimate..." If nobody posts an answer, then I'll post one myself. This is not a hard question.

@Polygnome no analytical solution? Why? Any attraction between astronauts and the ISS is negligible compared to the Earth gravity, so it's essentially three separate two-body problems, not a four-body one, which would be non-analytical indeed.

@IMil So you ignore the other bodies, drag, assume earth is a point mass and so on. Lots of simplifications to make it fit that model. Sure, but for some reason the way the question is asked the obvious solution of treating it keplerian was off the table. Maybe that wasn't intended, but that was how it appeared to me.