Hohmann transfer orbit explained
A spacecraft circles in a low orbit and has to move to a higher one. It could keep the engine running and spiral outward, but the most fuel-efficient route rides a single ellipse that touches each circle once. One burn on the inner circle puts the craft on that ellipse; it coasts halfway around, and a second burn as it reaches the outer circle settles it into the new orbit. Changing orbits with two burns this way is called a Hohmann transfer. This lesson starts with escape speed, the speed that leaves orbit altogether, then builds the transfer ellipse and follows it all the way to the fuel bill.
Before switching orbits, it is worth asking how fast a craft must go to leave altogether. A circular orbit of radius r has speed vcirc = √(GM/r); launch faster than that from the white dot on the dashed circle and the orbit stretches into an ever longer ellipse. At exactly vesc = √(2GM/r) the ellipse can no longer close and opens into a parabola. That value is the escape speed, and the two are related simply by vesc = √2 · vcirc. Raise the speed with the slider and, at √2 times circular speed, the trajectory's label switches to the escape threshold; beyond it lies a hyperbola.
Now build the transfer ellipse yourself. It needs its periapsis on the inner circle (radius r₁) and its apoapsis on the outer circle (radius r₂), so its semi-major axis is a = (r₁ + r₂) / 2, and the speed anywhere on it follows from the vis-viva equation v = √(GM(2/r − 1/a)). The extra speed needed at periapsis beyond circular speed is Δv₁ at the green dot, and the speed needed at apoapsis to match the outer circular orbit is Δv₂ at the orange dot. Raise r₂ with the slider and the gold ellipse redraws to touch both circles while both burns and their total change. As the outer circle moves away Δv₁ keeps growing, but past a certain point Δv₂ actually shrinks.
Between the two burns the spacecraft shuts its engine off and simply coasts. After one Δv₁ at periapsis puts it on the ellipse, it does nothing for half an orbit and moves on momentum alone. Press play and the white dot follows the gold ellipse; just as Kepler's laws say, it is fast near periapsis and slows as it nears apoapsis. The green and orange dots swell at the moment of each burn, and applying Δv₂ exactly on arrival at apoapsis settles the craft onto the outer circle. The transfer takes exactly half the period of this ellipse.
The sum of all the burns is the mission's Δv budget, and what matters is that this budget translates directly into propellant. The link is the Tsiolkovsky rocket equation Δv = ve · ln(m₀ / mf), where ve is the exhaust velocity, m₀ the mass at departure, and mf the mass once the propellant is spent. The upper bar stacks the budget for launch, orbit transfer, and capture, and the gold part of the lower bar is the share of the departure mass that is propellant. Because of the logarithm, the required mass ratio grows exponentially with Δv; lower ve with the slider and, for the same budget, the propellant share climbs toward the end of the bar.
Is the Hohmann transfer always the cheapest? When the radius ratio r₂ / r₁ of the two circles is very large, another route can cost less. A bi-elliptic transfer swings far beyond the target, comes back, and burns three times. Going so far out looks wasteful, but high up it takes very little change of speed to reshape an orbit, so the detour pays off. Switch between the two methods with the buttons and raise the radius ratio with the slider: once it passes about 11.94, the advantage marker jumps from Hohmann to bi-elliptic. The blue dashed path is a shrunk-down sketch of the trip out to a very distant apoapsis.