Orbit Propagation Calculator: Kepler, J2 Drift and Satellite Passes

The mean elements are propagated analytically, then Kepler’s equation is solved by a bracketed Newton iteration. With $p=a(1-e^2)$ and $n=\sqrt{\mu/a^3}$, the first-order rates are

Rates use unperturbed $n$ to retain first order in J2; multiplying the plane rates by the corrected mean motion would add second-order terms. Disabling J2 sets all three corrections to zero. The method follows Vallado’s secular element treatment and the open SatelliteToolbox algorithm, truncated consistently to first order.

At elapsed time $t$, update $\Omega,\omega,M$ linearly. Newton solves $E-e\sin E=M$ with a residual below $10^{-13}$ rad (50-iteration safeguard). Perifocal coordinates are $x_p=a(\cos E-e)$ and $y_p=a\sqrt{1-e^2}\sin E$; the standard $R_3(\Omega)R_1(i)R_3(\omega)$ active rotation gives ECI coordinates. A GMST rotation gives ECEF. Eight fixed-point ellipsoid iterations give WGS-84 geodetic latitude and altitude. Observer ECEF coordinates are subtracted and rotated to east–north–up to obtain range, azimuth and geometric elevation.

The pass search samples elevation on the advanced time interval. Each sampled local maximum is refined by bounded optimisation, including maxima between samples whose samples lie below the mask. Brent root finding locates the mask crossings on either side. Search-window boundaries above the mask are retained and labelled. The table's peak is the maximum within that window. Ground-track samples are evenly spaced in eccentric anomaly and mapped back to time, which resolves fast perigee motion better than a uniform-time display. The elevation chart adds 121 samples around each detected pass and includes its refined peak. Ground-track paths are split at ±180° longitude.

The published FreeFlyer example supplies $\mu=398600.442$ km³/s², $R=6378.1363$ km and $J_2=0.00108262668$. WGS-84 observer geometry uses $a_E=6378.137$ km and $1/f=298.257223563$ from NGA.

Earth-centred bound ellipses with perigee at least 100 km above the equatorial surface. Input elements must be consistent mean elements for this secular model, or osculating elements for the two-body option. UTC is treated as UT1 for Earth rotation. The equatorial inertial frame is approximated as mean-of-date; precession, nutation, polar motion and Earth-orientation corrections are omitted. Observer height is ellipsoidal, not height above mean sea level. Elevation is unrefracted above the local ellipsoid tangent plane.

This is a secular J2 / Kepler planning and teaching model, not SGP4 or a numerical force-model ephemeris. Long- and short-period perturbations, atmospheric drag, higher harmonics, third bodies, manoeuvres and radiation pressure are absent. Errors can accumulate into large along-track and pass-time errors; use fresh elements and a validated propagator for tracking operations. The displayed Keplerian speed and conserved specific energy belong to the instantaneous fixed-a, fixed-e ellipse; they are not the derivative and full potential energy of a perturbed trajectory.

RAAN is conventional at equatorial inclinations, and perigee is conventional for a circular orbit; the combined orientation still produces a position. The sun-synchronous target is the published example's 0.98562625°/day; a blank inclination means the requested node rate is infeasible at this a and e. It does not test eclipse, local solar time or mission feasibility.

The ground track has 361 intervals per mean-anomaly cycle by default, sampled in eccentric anomaly. Its lines are sampled directly, without smoothing the trajectory. Pass scanning uses 60 s by default, followed by continuous peak and crossing refinement; very narrow or multiple peaks inside a scan interval can still be missed, especially on eccentric orbits. Reduce the advanced interval and confirm convergence near grazing events. The input condition limits the scan interval near perigee, but is not a mathematical guarantee of detecting every tangent event. Exactly tangent, zero-duration contacts are excluded. At exact zenith, azimuth is indeterminate; at a geographic pole longitude defines the horizontal bearing convention. Passes above the mask at a search boundary are clipped and marked with an asterisk (*). Schedule times are grouped by UTC start date; +1d or +2d denotes a later date within a pass. A pass table describes geometric elevation only; terrain, refraction, sunlight and communications link budgets are not included.