Eclipse & Beta-Angle Timeline

The beta angle (β) is the angle between the Sun vector and the orbit plane — the Sun's elevation above the plane the satellite sweeps. It is the single number that governs how much of each orbit is spent in sunlight versus in Earth's shadow. A beta angle near 0° means the Sun lies in the orbit plane and eclipses are at their longest; a high |β| tilts the orbit edge-on to the Sun and shortens — eventually eliminates — the shadow pass. Beta angle is not constant: it cycles as the Sun moves along the ecliptic and as the orbit's node drifts, so power and thermal margins must be sized for the worst day, not the launch day.

This tool computes β from the standard closed form (Vallado; SMAD Ch. 5): sin β = cos δ_s · sin i · sin(Ω − Ω_s) + sin δ_s · cos i, where i is the orbit inclination, Ω the Right Ascension of the Ascending Node (RAAN), and δ_s / Ω_s the Sun's declination and right ascension. The Sun is modelled as moving uniformly along the ecliptic (~0.9856°/day), and its declination and right ascension follow from the obliquity of the ecliptic.

Each simulated day the orbit's RAAN is advanced by the J2 secular nodal regression rate dΩ/dt = −1.5 · n · J2 · (R_e / a)² · cos i, the dominant perturbation from Earth's oblateness. It is negative (westward drift) for prograde orbits, positive for retrograde, and ~0 near 90° inclination. This is the same effect that, when tuned to ~0.9856°/day, makes a Sun-synchronous orbit hold its beta angle nearly fixed.

Eclipse duration uses the cylindrical-shadow approximation — Earth's shadow is treated as an infinite cylinder of Earth's radius. The fraction of an orbit spent in shadow is f_E = (1/π) · acos( √(h² + 2·R_e·h) / ((R_e + h)·cos β) ), which collapses to zero once |β| exceeds the beta-star threshold β* = asin(R_e / (R_e + h)) — the elevation above which the orbit clears the shadow entirely and the satellite is in full sun. Higher orbits have a smaller β*, so they reach the full-sun regime more easily.

What this tool does not capture: the penumbra (the model uses a hard umbral cylinder, so eclipse entry/exit are instantaneous), the conical taper of the real shadow, orbit eccentricity (the orbit is assumed circular), higher-order geopotential and lunisolar perturbations beyond J2, and the equation-of-time correction to the Sun's mean longitude. It is an orbit-geometry trade study for power-budget and thermal-cycle scoping — flight design needs a full ephemeris propagator with a penumbra-aware shadow model.