Orbital planes are so close together that it doesn't make much difference. However, now that I've actually calculated it, your sqrt(2) factor seems to be about

Orbital planes are so close together that it doesn't make much difference. However, now that I've actually calculated it, your sqrt(2) factor seems to be about right for the average distance -- there's too few satellites per plane in the current phase. In this phase Starlink uses 72 orbital planes, with 22 satellites per plane, so 1440 satellites in total (they're almost there). It orbits at 550km above Earth's surface, so the orbit has radius 6921km, which gives an orbital length of 43486km. Separation between orbital planes varies depending on your latitude, but assume the worst case, where it is 43486km / 72 / 2 = 302km¹. Thus, the nearest orbital plane is at most 302km / 2 = 151km away from the orbital plane directly overhead. However, since the planes process, on average the nearest orbital plane is only half that, or 76km away from the plane overhead. Satellites within each plane have a separation of 43486km / 22 = 1976km. Thus, there's always a satellite at most 1976km / 2 = 988km away¹ from any point in each orbital plane, and on average there's a satellite half that away, or 494km. Adding all this together, the nearest satellite is on average √(550^2 + 76^2 + 494^2) = 743 km away (at the worst latitude). [EDIT: Actually, that's improper averaging, the correct average is obtained with ∫√(550^2 + x^2 + y^2) dx dy / ∫ dx dy on x=0..151, y=0..988, which yields 777km]. The original plan used 24 planes with 66 satellites, which reduces average distance to 617km. At more favorable latitudes the difference with the current design would be even larger. [EDIT: This should be 635km.] ¹ This is distance on the surface of the orbital sphere, straight-line distance is a bit less. It probably doesn't make much difference.