I've been returning to satellite tracking software after a while (I
wrote some early AMSAT tools in the early 1980s) and am wondering if
there has ever been a resolution of the exact definitions of "apogee
height" and "perigee height".
The simple geometric definitions of "perigee" and "apogee" are the
points where the spacecraft is the closest to or farthest from the
center of the earth. This is easy if you assume the earth is a perfect
sphere; all you need is the semimajor axis a and the eccentricity e:
apogee = a * (1+e) - earth_radius
perigee = a * (1-e) - earth_radius
But reality is more complicated than that. For a nonequatorial orbit the
apogee and perigee usually occur over some point off the equator where
the earth's radius is smaller than at the equator. You can correct for
this given the inclination and the argument of perigee, which together
tell you the latitude at which apogee and perigee occur; one will occur
in the northern hemisphere and the other will occur in the southern
hemisphere at the opposite latitude.
There's a complication here in that this is geocentric latitude, while
we more often use geodetic latitude on a daily basis. Converting
geodetic latitude to geocentric is fairly easy, but converting in the
other direction is like Kepler's equation: apparently there's no closed
form solution so you have to iterate.
But this is a relatively minor detail. The real problem comes when you
have a satellite with a relatively high inclination and an argument of
perigee close to 0 or 180 degrees; in this situation the satellite can
easily be farther from the earth's surface than *either* the calculated
apogee or perigee!
The ISS is a case in point at the moment. Using element set 906, which
has an argument of perigee of about 329 degrees I calculate an apogee of
408 km and a perigee of 402.4 km assuming an oblate earth and ignoring
the distinction between geodetic and geocentric latitude (which is
relatively small for this argument of perigee). But near the
southernmost point of its orbit, I calculate an altitude of about 421
km, well above both the perigee and apogee heights because the earth's
surface through the poles is more elliptical than the ISS's orbit.
So what conventions do people use? How meaningful do people expect these
figures to be? For a low altitude orbit like the ISS, the difference in
drag between 402 and 421 km is actually quite significant. At the very
least it would be nice if everybody used the same convention.