Hohmann Transfer Orbit

The Hohmann transfer, published by Walter Hohmann in 1925, moves a spacecraft between two circular, coplanar orbits using two engine burns and an elliptical bridge. Burn one, at the departure orbit, adds velocity to raise the far side of the orbit — apogee — to the destination altitude, creating a transfer ellipse that just kisses both circles. The spacecraft coasts half a revolution, then burn two at apogee adds the velocity needed to circularise. For destination radii less than about 11.9 times the starting radius it is the most delta-v-efficient two-impulse transfer, which made it the backbone of practical astronautics: the geostationary transfer orbit is precisely a Hohmann first leg, and interplanetary trajectories generalise the idea to orbits around the Sun, complete with the launch windows that arise from waiting for planets to align. The economy has a cost — time (the transfer takes half the ellipse's period, months to Mars) — and the pattern breaks where three-body dynamics, low-thrust spirals or gravity assists offer cheaper if slower routes.