Falling without ever hitting the ground
A satellite keeps falling towards the Earth, but it goes so fast sideways that the ground curves away beneath it: that is an orbit. Gravity weakens with the square of the distance (1/r²), the speed it takes follows v = √(μ/r), μ = 398,600 km³/s².
- At 420 km (the ISS): 7.66 km/s, a lap in an hour and a half. Slower, the satellite falls back into the air; at 10.83 km/s (√2 times as fast), it escapes the Earth for ever.
- The launcher first climbs straight up to get out of the thick air, then tilts little by little towards the horizontal: the pitch program. Flat too early, the air brakes it; straight up too long, it spends its propellant against gravity.
- Cutting the engine sets the apogee; at the apogee, a second burn raises the perigee and rounds the orbit off. A low orbit takes about 9.3 km/s of Δv in all, losses included.
- The sun-synchronous orbit (500 to 800 km) passes near the poles, slightly backwards: the Earth's rotation no longer helps the launcher, it even brakes it a little.
- The Hohmann transfer: from 250 km, +2.44 km/s at perigee start an ellipse rising to 35,786 km in 5.3 h; up there, +1.47 km/s make it a circle at 3.07 km/s. A lap there takes 1,436 min, a sidereal day: QO-100 stays over 25.9° E.
The mark: how far the perigee and the apogee are from the target orbit. PERFECT within the mission's tolerance (20 km in low orbit, 250 km in geostationary), then GREAT, GOOD, OK; a bonus for the propellant left.
Simplified, but right Gravity, orbital speed and the Hohmann transfer are the real ones. The rest is simplified: a single stage of constant thrust, an autopilot that only keeps it from sinking while the engine burns, a flight in the orbit's plane, an atmosphere halving every 6 km, the Earth's rotation as a push at the start.