# What are the product terms shown here representing an orbit?

I can compute Kepler parameters from a given TLE using Skyfield. I am checking the satellite information from this website and the orbit is represented as $$546 \times 548 \text{ km} \times 53.1 \text{ deg}$$ I understand that 53.1 is the inclination and I can get that from Skyfield as well. What are the 546 and 548 values? And how can I compute them?

• Surely apogee and perigee altitude? Sep 12, 2022 at 18:23
• Most likely minor and major axis of the orbit ellipse. Sep 12, 2022 at 18:37
• @GregMiller wow, that's a tiny orbit then. Somewhere down at the core of the Earth. Sep 12, 2022 at 18:37
• @OrganicMarble, good point. It shows the current altitude as 565km, so it's likely the geocenter isn't the basis for those numbers. Either way, I'd recommend getting the orbital elements from space-track or celestrak instead, if possible. Sep 12, 2022 at 18:43

Those numbers are the altitudes of perigee (lowest) and apogee (highest). They are given as heights, $$h_p$$ and $$h_a\$$, above Earth's surface. This requires you to assume a value for Earth's radius, $$r_e$$ , to convert them to radial distances from the center, $$r_p$$ and $$r_a$$ . In terms of the semi-major axis, $$a$$, and eccentricity, $$e$$, we have $$h_a + r_e = r_a = a(1+e)\\h_p + r_e = r_p = a(1-e)$$ which is easy to invert, yielding $$(r_a + r_p)/2 = a\\\frac{r_a - r_p}{r_a + r_p} = e$$ The conventional number to use for $$r_e$$ is generally assumed to be the maximum equatorial radius, 6378137 m. Computing the actual instantaneous height would require the WGS84 ellipsoid flattening, and the difference between geocentric and geodetic latitude. If you want to check your calculations against the ones built into the TLE propagator, SGP4, consult Failing at getting apogee and perigee from TLE

When we start making comparisons between different representations of the orbit, things get murky. The main offender is the height $$\times$$ height representation itself: it depends on too many assumptions, some of which conflict. I don't know what calculation that web site is doing, but I suspect the problem is that the apogee and perigee heights are based on the equatorial radius, but the separate "height" of 551 km includes the curvature in determining distance. A much better choice is to use semi-major axis (or mean motion) and eccentricity, as the TLE format does. Another problem may be the difference between mean and osculating values for the same component.

The ultimate source for almost all publicly available orbit data is the U.S. Space Force's https://space-track.org , which is the only reason people ever use TLEs and SGP4 for anything. TLEs are very complicated to describe, but exist to be easy to distribute; SGP4 exists to turn TLEs into something easier to understand and work with, like instantaneous position and velocity. Their accuracy is limited, but they're the only available data for almost everything. Satellite operators generally know their own orbits to better than a TLE, but most don't share those with the general public. A good source for top-quality orbits on navigation satellites and some science missions is CDDIS at NASA GSFC.

• thx for the answer. Is $r_e$ the equatorial radius of earth (6378137)? Am I supposed to use $r_e$ for both apogee and perigee calculations or is the eccentricity value from WGS84 somehow supposed to enter the picture?
– jkt
Sep 12, 2022 at 19:12
• thx for the fast edit. Basically using skyfield I can compute the altitude of the satellite over a period (~90ish minutes for LEO). My problem is I don't think the product representation corresponds to maximum and minimum altitude. Right now aforementioned website shows $546x548$ but the altitude is shown as 551km.
– jkt
Sep 12, 2022 at 19:28
• thanks for further clarification. I have come across the height x height representation in multiple places, for instance see and see for another example satellite. Due to its prevalence, I assumed it might be a commonly utilized measure that can be easily calculated using something like Skyfield. Do you maybe know where Gunther's space page is getting/parsing their values from? Some FCC/ITU application maybe?
– jkt
Sep 12, 2022 at 20:15
• thanks again for your answer. Kind of a related continuation of the question, therefore I don't want to open up a new question. Here orbit is described as Orbit : #704798.73 @ 577 x 658 km x 97.6 deg, what is the first part? (i.e. #704798.73)
– jkt
Sep 17, 2022 at 6:33