Move from an inner circular orbit to an outer one. The route that spends the least fuel was worked out by Hohmann in .
It takes two acts. While on the inner circle, burn along the direction of travel, and the craft rises onto an ellipse that reaches the outer circle. Half a lap later, at the top, burn again to circularise.
Nothing happens in between. The engine runs at those two points only, and the rest is coasting under gravity.
Both burns are straight along the direction of travel. Burning at an angle wastes whatever goes into turning.
The intermediate ellipse touches the inner circle at its perihelion and the outer circle at its aphelion. That condition alone fixes it.
The perihelion distance is the inner radius and the aphelion distance the outer radius , so the semi-major axis is their average: .
Fixing fixes the period, and half of it is the travel time. Only half the ellipse is used; the other half is never flown.
The speeds follow from the vis-viva equation: the first burn is the gap between the inner circular speed and the perihelion speed of the ellipse, the second the gap between its aphelion speed and the outer circular speed.
From the orbit of the Earth to the orbit of Mars: km per second for the first burn, for the second, in all.
Push the destination farther out and the first burn grows. The second, though, shrinks, because the circular speed out there is itself small.
Their sum does not simply grow with distance. It peaks where the target radius is times the starting one, and falls away beyond that.
So going farther can genuinely cost less. From the orbit of the Earth, times out is beyond Saturn; Mars is far short of it.
Arriving where Mars is not is pointless, so the alignment at departure is fixed.
Mars moves during the crossing. The trip takes about days, in which Mars covers degrees of its orbit. At departure, therefore, Mars must be some degrees ahead of the Earth.
That alignment comes round only once in months, because the two periods differ and it takes that long for the offset to work its way round. It is called the launch window.
Miss it and the wait is over two years. That is why launches to Mars cluster at two-year intervals.
The Hohmann transfer is cheapest only in fuel. In time it is the dearest there is.
To go faster, ride an ellipse whose aphelion reaches beyond the orbit of Mars. Arrival happens where the path crosses that orbit, without waiting for the half lap. The velocity change needed is correspondingly larger.
For a crewed flight the balance shifts. Weighing radiation and supplies for the journey, cutting days to is worth the fuel.
The next article shows how to change speed while spending no fuel at all. Passing close to a planet is enough.