Two burns on opposite sides of an ellipse is the minimum-energy route between circular orbits, and LEO to geostationary costs about 3.9 km/s over five and a quarter hours. Past a radius ratio of roughly 11.94, a three-burn bi-elliptic route out to a distant apoapsis is genuinely cheaper.
Both burns in a Hohmann transfer fight the same trade: raising apoapsis is cheap deep in the well, circularising at the top is not. Going far higher first makes the second burn almost free, because a spacecraft crawling at apoapsis needs very little push to change its orbit — and past about a twelvefold ratio that saving beats the cost of going out there. The price is time: the bi-elliptic route can take years where Hohmann takes days.
"Minimum energy" is not "minimum time" — Hohmann is already the slow option, and bi-elliptic is far slower still. Both assume impulsive burns, coplanar circular orbits and a single dominant body; a plane change, a real finite burn, or a third body makes the neat answer an estimate.
Estimate with the rule, then check it against the calculator that models it properly.
Open Orbital Mechanics →Two burns on opposite sides of an ellipse is the minimum-energy route between circular orbits, and LEO to geostationary costs about 3.9 km/s over five and a quarter hours. Past a radius ratio of roughly 11.94, a three-burn bi-elliptic route out to a distant apoapsis is genuinely cheaper. Both burns in a Hohmann transfer fight the same trade: raising apoapsis is cheap deep in the well, circularising at the top is not.