Mean and Osculating Elements¶

Real satellite orbits are perturbed by Earth's oblateness (\(J_2\)). The osculating elements describe the instantaneous orbit at a given moment, while mean elements average out the short-period \(J_2\) oscillations. Mean elements are useful for mission design, constellation maintenance, and long-term orbit characterisation because they vary smoothly over time.

The conversion between mean and osculating elements uses first-order Brouwer-Lyddane theory. The perturbation parameter is:

For a typical LEO orbit (\(a \approx 6878\) km), \(\gamma_2 \approx 5 \times 10^{-4}\), so the first-order corrections are on the order of hundreds of metres in semi-major axis and milliradians in angular elements.

import jax.numpy as jnp
from astrojax.constants import R_EARTH
from astrojax.orbits import state_koe_mean_to_osc, state_koe_osc_to_mean

# Define mean elements: [a, e, i, Omega, omega, M]
mean = jnp.array([R_EARTH + 500e3, 0.01, 45.0, 30.0, 60.0, 90.0])

# Mean -> Osculating
osc = state_koe_mean_to_osc(mean, use_degrees=True)

# Osculating -> Mean (roundtrip)
mean_back = state_koe_osc_to_mean(osc, use_degrees=True)

The forward and inverse transformations are not exact inverses — the roundtrip error is of order \(J_2^2\), which is small but nonzero.