Osculating elements in OMM seem to change when creating an EarthSatellite object

[scottshambaugh]

I am comparing orbital elements before and after creation of an EarthSatellite from an OMM, and am getting slightly different values. I expect this is benign, but am not sure what's going on.

Here's a quick example script that shows what's happening:

from skyfield.api import EarthSatellite, load
from skyfield.elementslib import osculating_elements_of

ts = load.timescale()
omm_data = {
    'OBJECT_NAME': 'TEST SAT',
    'OBJECT_ID': '2024-001A',
    'NORAD_CAT_ID': '12345',
    'EPOCH': '2024-01-01T00:00:00.000000',
    'MEAN_MOTION': 15.5,      # rev/day
    'ECCENTRICITY': 0.01,     # unitless
    'INCLINATION': 51.6,      # degrees
    'RA_OF_ASC_NODE': 180.0,  # degrees
    'ARG_OF_PERICENTER': 0.0, # degrees
    'MEAN_ANOMALY': 0.0,      # degrees
    'BSTAR': 0.0,             # 1/earth radius
    'MEAN_MOTION_DOT': 0.0,   # rev/day^2
    'MEAN_MOTION_DDOT': 0.0,  # rev/day^3
    'CLASSIFICATION_TYPE': 'U',
    'EPHEMERIS_TYPE': '0',
    'ELEMENT_SET_NO': '999',
    'REV_AT_EPOCH': '0',
}
sat = EarthSatellite.from_omm(ts, omm_data)

pos = sat.at(sat.epoch)
osc_elements = osculating_elements_of(pos)  # ICRF frame

print(sat.epoch.utc)
print(osc_elements.eccentricity, omm_data['ECCENTRICITY'])
print(osc_elements.inclination.degrees, omm_data['INCLINATION'])
print(osc_elements.longitude_of_ascending_node.degrees, omm_data['RA_OF_ASC_NODE'])
print(osc_elements.argument_of_periapsis.degrees, omm_data['ARG_OF_PERICENTER'])
print(osc_elements.mean_anomaly.degrees, omm_data['MEAN_ANOMALY'])

# All values are at least somewhat different.
'''
CalendarTuple(year=2024, month=1, day=1, hour=0, minute=0, second=0.0)
0.010881909723842008 0.01
51.62267299309298 51.6
179.58708119720018 180.0
4.705040984693307 0.0
355.46545250154924 0.0
'''

I thought this might have to do with the frame conversion between TEME and ICRF, but when I try to check it the results are very roughly the same as before.

from skyfield.sgp4lib import TEME
osc_elements = osculating_elements_of(pos, reference_frame=TEME.rotation_at(sat.epoch))  # TEME frame

'''
CalendarTuple(year=2024, month=1, day=1, hour=0, minute=0, second=0.0)
0.010881909723847644 0.01
51.61995468574305 51.6
179.99991187961967 180.0
4.535383674828455 0.0
355.46545250155054 0.0
'''

[brandon-rhodes]

The snag is that you are comparing an element set from one theory of orbital motion, the simple Kepler elliptical-orbit model, with elements from a different model, the SGP4 theory of perturbed Earth satellite motion. The elements don't mean quite the same thing and so don't ever match.

Is there somewhere in the documentation where, if it mentioned this more prominently, you would have been spared the confusion about the two element sets? If you let me know which document and section, I could try adding a paragraph!

[scottshambaugh]

Thank you for the quick response, and for the wonderful library! I'm finding it incredibly useful.

I was ultimately looking to generate expected OMM messages after propagation. It's possible to do that for the initial epoch with elements matching the input with sgp4lib:

from sgp4 import exporter
exporter.export_omm(sat.model, sat.name)

'''
'OBJECT_NAME' = 'TEST SAT'
'OBJECT_ID' = '2024-001A'
'CENTER_NAME' = 'EARTH'
'REF_FRAME' = 'TEME'
'TIME_SYSTEM' = 'UTC'
'MEAN_ELEMENT_THEORY' = 'SGP4'
'EPOCH' = '2024-01-01T00:00:00.000000'
'MEAN_MOTION' = 15.5
'ECCENTRICITY' = 0.01
'INCLINATION' = 51.6
'RA_OF_ASC_NODE' = 180.0
'ARG_OF_PERICENTER' = 0.0
'MEAN_ANOMALY' = 0.0
'EPHEMERIS_TYPE' = 0
'CLASSIFICATION_TYPE' = 'U'
'NORAD_CAT_ID' = 12345
'ELEMENT_SET_NO' = 999
'REV_AT_EPOCH' = 0
'BSTAR' = 0.0
'MEAN_MOTION_DOT' = 0.0
'MEAN_MOTION_DDOT' = 0.0
'''

But I'm still unclear how to generate TLEs/OMMs at future epochs based on the propagated state. #1021 is a similar ask. Understood that it's a bad idea to do that in real-life scenarios since these messages are by definition only valid at their initial epoch. But there are some legit uses:

• It would be useful for making test data for other tools to ingest.
• Being able to generate propagated orbital elements would be valuable for some use cases without the same pitfalls as generating TLEs/OMMs.
• Being able to generate EarthSatellite objects from an epoch and position/velocity state vectors would be broadly useful

For the original ask, I think it's ultimately on me being confused that the TLE/OMM data was delivered with mean elements instead of osculating! But happy to provide some suggestions on improving the docs:

• https://rhodesmill.org/skyfield/api-elements.html#skyfield.elementslib.OsculatingElements doesn't list the object attributes with their definitions, which are only found here: https://rhodesmill.org/skyfield/api.html#osculating-orbital-elements. Probably makes more sense for that info to live on the object page than the API page.
• I was searching the docs for how to generate orbital elements, and the osculating ones were the only ones listed. Some info on how to generate mean elements like is done in export_omm above would be extremely useful. (Update, I see these are available here, though it's missing some useful ones like semi-major axis: https://rhodesmill.org/skyfield/api-satellites.html#skyfield.sgp4lib.EarthSatellite).
• I think a section in https://rhodesmill.org/skyfield/earth-satellites.html on generating elements would be a good place for that, and having information on both mean and osculating elements present in the same place would also reinforce that there is a difference.
• On https://rhodesmill.org/skyfield/earth-satellites.html there is a disclaimer about Celestrak TLEs/OMMs that "Finally, note that even though several satellite elements have familiar names like inclination and eccentricity, they are not simple Kepler elements, and the SGP4 routine will not predict the same position as a simple Kepler orbit would." Some clarification and definitions on what they are (mean elements generated with sgp4 using the WSG-72 gravity model in the TEME frame) instead of what they are not would be helpful.