# Eccentricity of TLE orbit from velocity and radius in TEME

I was reading this question: Calculation of Eccentricity of orbit from velocity and radius and tried to use the proposed equation to reconstruct the ecco field of TLE by it's R,V vectors. The formula is simple:

$$\mathbf{e} = \frac{\mathbf{v} \times \mathbf{h}}{\mu} - \frac{\mathbf{r}}{\|\mathbf{r}\|}$$

Where $$\mathbf{h} = \mathbf{r} \times \mathbf{v}$$

The tested TLE:

1 41386U 16016A   23053.13950576  .00028887  00000-0  32773-3 0  9993
2 41386  97.1800 138.9553 0010664 243.3776 116.6390 15.62259129388863

It's ecco is 0.0010664 while the formula gives me 0.0001432 for TEME/GCRF vectors. And for ITRF vectors it results in 0.0202059.

I've adapted my code for python to print all 3 values:

import numpy as np
from skyfield.framelib import itrs
from sgp4.api import Satrec, WGS84

line1 = '1 41386U 16016A   23053.13950576  .00028887  00000-0  32773-3 0  9993'
line2 = '2 41386  97.1800 138.9553 0010664 243.3776 116.6390 15.62259129388863'

satrec = Satrec.twoline2rv(line1, line2, WGS84)
print('TLE ecco: ', satrec.ecco)

e, r, v = satrec.sgp4(satrec.jdsatepoch, satrec.jdsatepochF)
res = np.linalg.norm(np.cross(np.cross(v,r), v)/398600.8 - r/np.linalg.norm(r))
print('TEME ecco: ', res)

sat = EarthSatellite.from_satrec(satrec, ts)
geocentric1 = sat.at(sat.epoch)

r = geocentric1.position.km
v = geocentric1.velocity.km_per_s
res = np.linalg.norm(np.cross(np.cross(v,r), v)/398600.8 - r/np.linalg.norm(r))

print('GCRF ecco: ', res)

pos1, vel1 = geocentric1.frame_xyz_and_velocity(itrs)
r = pos1.km
v = vel1.km_per_s
res = np.linalg.norm(np.cross(np.cross(v,r), v)/398600.8 - r/np.linalg.norm(r))

print('ITRF ecco: ', res)

The question is: What frames this formula is valid for? Is it possible at all to get eccentricity of near-circular orbit by RV vectors with at least 1e-5 precession?

• The data in the TLE aren't really orbital elements, even though many fields have the same name. They are just values generated to reproduce the actual position and velocity within some tolerance when using the SGP algorithm. So it's best to assume anything in a TLE is useless outside of the SGP algorithm. Jul 25, 2023 at 19:18
• just curious - you're already posting questions in Space Exploration SE where a TLE/artificial satellite orbit propagation question would be centrally on-topic; was there a reason you asked this in Astronomy SE instead? I don't think it's off-topic here necessarily, just wondering.
– uhoh
Jul 26, 2023 at 12:51
• @uhoh That's just because the original question was in Astronomy: astronomy.stackexchange.com/questions/29005 I don't mind if someone move my question to a proper place.
– truf
Jul 26, 2023 at 13:31