A series of reflections on the physical and philosophical nature of power electronics, where the transformation of energy reveals the fundamental rules of the world.
A rocket does not fly to another orbit. It is thrown, once, and then it falls the rest of the way — every maneuver after ignition is a single instant of thrust bookended by long silent coasting, because coasting is free and burning is not.
The Hohmann transfer is the cheapest of these throws between two circular orbits: two burns, and an ellipse in between that is tangent to both. Burn once at the inner orbit to raise your apogee out to meet the outer orbit's radius — you are now on the transfer ellipse, falling outward and slowing as you climb, the same way a ball thrown up slows before it turns over. At apogee, exactly where the ellipse kisses the target circle, burn again to circularize: raise your perigee up to match, and the ellipse becomes the circle you wanted. Two burns, one coast, minimum total delta-v of any two-impulse transfer between two circular orbits — proven, not guessed, by Hohmann in 1925, decades before anything left the ground to test it.
The cost is time. A transfer to Mars takes the shape of half an ellipse whose far point sits where Mars will be — not where Mars is now, but where the arithmetic says it will have traveled to, months later. Miss the window and you are not late; you are burning fuel you don't have to force a costlier path, because orbital mechanics has no such thing as hurrying without paying. A bi-elliptic transfer trades an even longer coast — flying out past the target orbit and back — for a lower total delta-v when the ratio of the two radii is large enough, a shape that looks like waste and calculates as thrift.
Every rendezvous is an appointment with a moving room whose location you can predict exactly but cannot see. The phase angle at launch — how far ahead the target already sits in its own orbit — is not a detail, it is the whole difficulty: burn at the wrong moment and the ellipse that would have found the target instead crosses empty space where the target used to be. Patience is not a virtue applied to spaceflight. It is spaceflight's actual method — the fast way is a myth that costs fuel nobody launched with, and the slow way is simply the one arithmetic already chose for you, whether or not you like waiting.
Seed: Orbital Mechanics & Astrodynamics (Hohmann Transfer, Bi-elliptic Transfer, Phase Angle, Rendezvous)