Getting to Mars the cheap way

A spacecraft cannot steer like a car. To climb from one orbit to a bigger one it fires its engine twice, coasts along half an ellipse in between, and pays the smallest fuel bill there is. The price is patience.

Two burns and a coast

Picture two circular orbits around the same body, one inside the other, in the same plane. A Hohmann transfer is the route between them that uses an ellipse touching both: its lowest point sits on the inner circle, its highest point on the outer one. The spacecraft fires once to leave the inner circle and join the ellipse, coasts round exactly half of it, and fires again at the top to settle into the outer circle. Both burns point straight along the direction of travel.

Why it is the cheap way

For two circular orbits in one plane, this one usually needs the least total change in speed, and the total speed change is what sets the fuel a mission has to carry. That is why it is the default for raising a satellite and the baseline for reaching another planet. But the coast is a fixed half-ellipse: you arrive at one place, the far end, and your destination has to be there when you do. The cheap route does not let you choose when to leave.

Keeping track of the speeds

Everything comes down to comparing speeds at two points. In a circular orbit of radius R the speed is √(μ/R), where μ is a constant for the body you orbit. The transfer ellipse has a semi-major axis, half its long diameter, of (R₁+R₂)/2. From that you get the ellipse's speed at its low point. It is faster than the inner circle's speed, and the difference is the first burn, ΔV₁. At the high point the ellipse is moving slower than the outer circle needs, so the second burn, ΔV₂, is also a push forwards. Add the two and you have the total.

Time of flight is half the ellipse's period: π√(a³/μ); the widget computes both. Burns are treated as instantaneous, which the lecture allows because a burn lasts far less than one lap.

Now the trade. Burn harder at the start and the ellipse grows past the outer orbit. You cross it early, before the top, and arrive at an angle, so the second burn has to bend your path as well as speed it up. That one-tangent transfer — the general case the lecture calls Lambert's problem — gets there sooner and costs more fuel. Drag the stretch slider in the interactive and both numbers move at once, in opposite directions. One exception: this is the least-fuel route between two circular orbits, but when the outer orbit is much larger — past about twelve times the inner radius — a three-burn route wins. And Mars is not quite in our plane — the two orbits differ by about 1.8°, so this is a good model, not an exact one.

In short

A Hohmann transfer is an ellipse that just touches both orbits, joined to them by two burns along the direction of travel. Because every burn is a plain speed difference at a tangent point, no other two-burn route between the two circles needs less total speed change, and so less fuel. What you give up is time and choice of departure: the coast is half an ellipse, and the target must be waiting at the far end.