Gravity and the orbit. The satellite is falling around the Earth. The pull of gravity is GM/r² (GM = 3.986×10⁵ km³/s²), and it is always towards the centre. For a circular orbit the speed is exactly the one for which gravity supplies the centripetal force: v = √(GM/r).

KE = ½v², PE = −GM/r, E = KE + PE = −GM/2a (per kilogram)

Energy. The potential energy is negative (zero is at infinity). In a circular orbit the kinetic and potential energy stay fixed. In an elliptical orbit they swap: the satellite goes fastest at the lowest point and slowest at the highest, and the total energy stays the same while the engine is off. The size of the orbit depends on the total energy only. If it reaches zero, the satellite has escaped and it will not come back. The unit of energy per kilogram is the MJ/kg (1 km²/s² = 1 MJ/kg).

What a burn does. Speeding up along the direction of motion (a prograde burn) raises the total energy. The orbit does not simply get bigger: the point of the orbit on the opposite side from the burn goes up. So a burn at the lowest point raises the highest point, and a burn at the highest point raises the lowest. A retrograde burn does the reverse. Burn twice, once at each end, to change a circular orbit into a bigger circular one (the Hohmann transfer).

Keplerian orbits. The orbit is always a conic section: an ellipse (E < 0), a parabola (E = 0) or a hyperbola (E > 0). The period depends only on the long axis, T = 2π√(a³/GM) (Kepler’s third law).

The engine pushes at 1 m/s² (about a tenth of g), which is a strong upper-stage engine. The time is sped up by the warp you choose. The satellite is not to scale, and the Earth is drawn as seen from above the north pole. If the orbit dips below 100 km the satellite burns up.