At which point would two GSO/GEO satellites with similar orbital elements be closest to each other?

Question

While I was answering another question on How closely spaced are satellites at GEO I came to realize I can't really word it with much conviction how two close proximity satellites in GEO/GSO orbits approach each other and at what point would they theoretically come the closest. I will try to explain my current understanding:

When observing GEO/GSO orbited satellites from the surface of the Earth, they would appear to make a circle on the night sky whose radius is defined by the satellite's inclination to the equator. The circle would have its center on the equatorial plane at the longitude the satellite is stationed at, usually defined as an angle to the prime meridian in the Eastern to Western direction (but could be differently). If this is more or less correct, then since these satellites are geosynchronous to the rotation of the Earth on its own axis this circle on the night sky (or as a projection for their ground track on the Earth's surface) they make in their orbit should translate to a wave function with the inclination being the wave's amplitude and frequency at the same time, if the Earth didn't rotate. I can draw both these orbits and distances of two satellites with similar longitude and inclination well enough from the ground up or the bird's-eye view, that's not a problem and so far so good, I hope. Correct me if I'm wrong though.

But here's the thing; If two GEO/GSO satellites have inclined orbit that by definition has to intersect the equatorial plane split in the middle (I'm not thinking here of levitated orbits, just the vanilla GEO/GSO ones), if they were at the same altitude, wouldn't their orbits intersect too, preferably exactly on the equatorial plane so their orbits are stable? So all this got me thinking, which orbital elements (or a combination of them) exactly defines the minimum separation between two GEO/GSO orbital bodies? My intuition says it would have to be the longitudinal separation, since it defines their orbital position at any time, but would I be correct in assuming that, or should I correct my calculations in the mentioned answer?

Additionally, if there is some way to show this in pictures, I'd appreciate if you could include some in your answer. Hand drawings will do just fine, if you can't find any online that were already prepared. In short though, I would like you to describe the orbital path two close proximity GEO/GSO satellites make in relation to one another, and not from the frame of reference of the body they orbit (in our case the Earth's surface) and at which point would these two satellites come the closest to each other?

Calculations would be appreciated, and if you need an example to work with, then you can use orbital elements of Astra 2C and Astra 1KR that were both slotted at longitude of 19.2° East during 2006-2007 and 2010-2013:

Name:      Common name:     Orbit:         Inclination:
-------------------------------------------------------
01025A     ASTRA 2C         19.2174576°    0.078°
06012A     ASTRA 1KR        19.2189152°    0.083°

My simple maths tell me they were theoretically separated by mere 1.1 km at their closest approach, and I based that assumption on their longitudinal separation of 0.0014576° (5.25") and took their altitude from the GSO semi-major axis at exact altitude of GEO, 42,164 km (26,199 mi) from the center of the Earth.

Answer 1

Ok, there's quite a bit to cover here. First off, let's look at the groundtrack of a satellite in GEO. Obviously, a satellite in a perfectly geostationary orbit will project as just a point on the Earth's equator. Because of natural perturbations (mass distribution of Earth and the combined effect of Lunar and Solar gravity, specifically), a satellite in a perfect geostationary orbit will not stay there for long. If we assume only a change in inclination (i.e. still at the ideal GEO altitude), the ground track looks like the red track here (from the ever-helpful T.S. Kelso:

The maximum latitude of this figure-8 is equal to the inclination of the orbit. If you consider deviations in the altitude, you start drifting with respect to the Earth. Consider a slightly lower-than-GEO altitude: your orbital period is now less than a sideral period of Earth rotation, and so in time, you begin drifting Eastward (by the same token, you drift West if you're at a higher altitude).

Now, getting back to relative distance... if you assume you have two satellites separated by some (mean) longitude, and at the ideal circular GEO altitude, then the minimum distance will in fact be the arc length for the difference in longitude. Another way of showing this is by inspecting Hill's equations for relative orbit motion. They describe the motion of an object with respect to another object, assuming the object is in a nearly circular orbit. That assumption fits nicely here.

Without getting too far into it, just take one of your inclined objects, and look at its motion with respect to a "virtual" satellite with the same properties but at zero degrees inclination. If you look at the closed-form solution of the three equations in that paper, the important thing to realize is that cross-track ($z$) motion is totally decoupled from in-track ($y$) and radial ($x$) motion. Adding inclination is essentially only adding a cross-track component to the orbit - no matter what inclination you add, it in itself does not have any effect on radial or in-track motion. Therefore, if you assume circular orbits and look at only the longitudinal separation, no matter what the inclination is, your minimum separation is still given by the arc length of the difference in longitude.

Of course, in reality, orbits are not this perfect. However, taking away that assumption makes things significantly more complex - now you need to worry about where the argument of perigee is with respect to the other object, where the node is, etc. For the situation you name, it's likely that they control the relative motion of these two satellites by controlling just those two things - argument of perigee and the right ascension of the ascending node (or something equivalent). This way, you can ensure one satellite is at perigee while the other is at apogee (thus ensuring some radial distance), and choose the node such that when the altitudes are equal (twice per period), they're at their maximum cross-track separation.

If you're interested in the more general problem of relative motion, I suggest reading up on Hill's equations, as they make describing relative motion fairly straightforward. Given the full state of two spacecraft, though, it would probably be easier to just propagate both over one period and find the point of closest approach that way, because it really does depend on all six orbital elements (and time).