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The science
Gravity and the orbit
The satellite moves under a = -GM r / r3 with GM = 3.986×1014 m3/s2, integrated with a fourth-order Runge-Kutta step. A bound orbit is an ellipse with semi-major axis a. Its period is T = 2π√(a3/GM), about 92.6 minutes at 408 km.
Kepler and vis-viva
The speed at distance r is v = √(GM (2/r - 1/a)): fastest at perigee, slowest at apogee. The line from Earth to the satellite sweeps equal areas in equal times (angular momentum conservation), shown by the yellow wedge. The specific energy -GM/(2a) stays constant.
Ground track
For inclination i and orbit angle u the latitude is asin(sin i sin u) and the inertial longitude atan2(cos i sin u, cos u); subtracting the Earth's rotation ωE t = 7.292×10-5 rad/s · t shifts each pass about 22.9 degrees west.
Honest simplifications: point-mass spherical Earth (no J2 flattening, so no nodal regression), no atmospheric drag or boosts, and the orbit starts at perigee over longitude 0.
What to try
- Use Equatorial: the ground track is a horizontal line and the satellite stays over the equator.
- Use Polar LEO and watch successive passes shift westward.
- Pick Molniya-like and compare the speed at perigee and apogee in the third graph.
- Raise the perigee altitude: the period readout grows with a^(3/2).