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In a comment to this question, @PcMan suggested my stuffin.space for satellite data. Now I am not quite sure of how to interpret the data, take for example the following screenshot:

enter image description here

  1. Why is the altitude larger than the apogee value in this case? Isn't the apogee value the maximum altitude? Is it because the apogee (and perigee) values reference to some kind of mean surface level (i.e. earth idealized as sphere) and the altitude gives the value above the ground directly under the satellite (i.e. including mountains etc.)?
  2. How to get from this data the correct value of the semimajor axis : The apogee and perigee values seem to be taken from ground level, so the major axis should be the sum of the apogee value, the perigee value and the earth diameter (and the semimajor axis then the half of this value). But what would be the correct value for the earth radius to use? Is it the nominal radius (6,378.1 km), the arithmetic mean radius (6,371.0072 km), volumetric radius or some other conventional radius?
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Answers to this can be found by examining the website's code on GitHub.

Apogee and Perigee are numbers that are valid at the epoch of the active TLE for the satellite in question. They are derived from the mean motion and eccentricity values and reference plain old 6371.0 as earth radius.

The altitude comes from the propagator and is in reference to the geodetic ellipsoid.

So, to answer directly:

  1. The altitude can be larger than the apogee from two sources: propagation drift as time passes from the TLE epoch (though this is probably unlikely), and the difference in reference values -- notably, apogee and perigee being in reference to a sphere with radius 6371 km, while the altitude is based on height above the geodetic ellipsoid, which varies between 6378 km and 6356 km, depending on latitude.

  2. To get semimajor axis, use the apogee and perigee values with an earth radius of 6371 km.

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    $\begingroup$ Another the way to reconcile these data ("apogee, perigee, altitude"): take a perfect circular orbit. Put the orbit on the Equator: Altitude=Apogee=Perigee. Swing the orbit by 90° (polar orbit), the Altitude varies with time, maximum over the poles, minimum over the Equator. The listing pointed to by @Tristan uses 6378.1 km for the Equatorial radius (var a), and 6356.7 Km for the Polar radius (var b). $\endgroup$
    – Ng Ph
    Oct 21 '21 at 21:40

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