Orbit Eclipse and Battery

What is modelled

A satellite on a circular orbit passes through the cylindrical shadow of the Earth once per orbit unless the Sun is high above the orbit plane. The beta angle is the angle between the sunlight direction and the orbit plane. The longer the eclipse, the more energy the battery must supply and the deeper it discharges on every orbit.

Equations

• Period T = 2π √(a3 / μ) with a = R + h.
• Eclipse fraction f = (1/π) arccos( √(h2 + 2 R h) / ((R + h) cos β) ), zero for sin β ≥ R / (R + h).
• Break-even array output Pload [1 + f / ((1 − f) ηc)].
• Battery energy stepped through the orbit with charge efficiency ηc, capped at the capacity.

Checks run in the page

• A point-by-point shadow test against the closed-form eclipse fraction.
• An RK4 two-body orbit against the Kepler period.
• The full-sun beta angle found by bisection.
• The energy bookkeeping of the battery over six orbits.

Honest simplifications

The shadow is a sharp cylinder with no penumbra or atmosphere, the Earth is a sphere, the orbit is circular and the Sun direction does not change during the run. The array is assumed to track the Sun with a constant output, the load is constant, the battery has a single charging efficiency and no ageing, and the cell chemistry and thermal limits are ignored.

• Press Equatorial sun (beta 0) for the longest eclipse, then Full-sun orbit for none.
• Press Starved array and watch the battery fail to recharge from orbit to orbit.
• Raise the altitude: the period grows and so does the eclipse in minutes.