Kepler's three laws
1. Orbits are ellipses with the central body at one focus. 2. A line from the central body to the orbiting body sweeps equal areas in equal times, so objects move fastest at periapsis. 3. The square of the period is proportional to the cube of the semi-major axis: T² = 4π²a³/μ.
Describing an orbit
Six orbital elements fix an orbit and a position on it: semi-major axis a (size), eccentricity e (shape), inclination i (tilt), longitude of the ascending node Ω and argument of periapsis ω (orientation), and an anomaly giving where the object is at a given time.
Energy and speed
The vis-viva equation v² = μ(2/r − 1/a) gives the speed anywhere on an orbit. The specific orbital energy ε = −μ/2a depends only on the semi-major axis. To raise an orbit you must add energy — a prograde burn — and to lower it you burn retrograde.
Counter-intuitively, slowing down at one point can make a spacecraft move faster overall: a retrograde burn drops it to a lower orbit, where circular speed is higher.