The first weeks off a Hohmann transfer are nearly free

Two burns and half an ellipse are the least velocity change that moves a craft between two circular orbits, and for a journey between the Earth’s orbit and Mars’s that half-ellipse takes 259 days. Hohmann’s transfer is the reference against which every interplanetary trajectory is priced. It is optimal among two-burn transfers, and only a three-burn route that first overshoots its target can beat it, and then only for orbits whose radii differ by more than a factor of twelve.

It is also slow, and the slowness matters to anyone travelling in it. The natural question is what a faster route costs, and the answer has a shape worth understanding before any number is read off it: near the minimum, going faster is nearly free.

A minimum found for a hobby #

Walter Hohmann worked out the transfer in 1925, in a book on how the planets might be reached, while employed as an architect in Essen. He was looking for the least fuel, because the rockets of his imagination were barely capable of the journey at all, and a trajectory that saved fuel at any cost in time was the only kind worth considering. Every early study of interplanetary flight inherited that priority, and the first robotic missions, launched on rockets with little margin, followed it closely.

The priority is not a law. It is a statement about which resource is scarce. For a probe whose instruments do not age in a year and whose operators can wait, velocity change is the scarce resource and Hohmann’s answer is right. For a crew whose bodies accumulate dose and whose supplies are consumed by the day, time is scarce too, and the right answer is somewhere along the curve rather than at its end. The geometry of that curve — flat at the end, steep further along — does not depend on which is scarce; it only decides how cheaply the shift from one priority to the other can be made.

A longer ellipse, cut short #

The faster transfers in the figure are the simplest family there is. The craft leaves the Earth’s orbit tangentially, as on the Hohmann transfer, but with more speed, so its ellipse reaches further from the Sun than Mars’s orbit. It does not go to the far end of that ellipse. It crosses Mars’s orbit on the way out, before reaching the far point, and meets the planet there.

Drawn to scale, the transfers look like a spray of arcs from one departure point. The Hohmann arc sweeps round half the Sun before touching Mars’s orbit tangentially on the far side. The faster arcs cut across, reaching Mars’s distance after sweeping through only 80° to 112°, and they meet the orbit at an angle rather than along it. A faster transfer is a longer ellipse cut short: it pays for the energy to go much further than it needs to, and uses only the first part of the trip that energy buys.

Each point on the cost curve is one such ellipse. Its heliocentric cost is two numbers: the speed that must be added at the Earth’s orbit to put the craft on the ellipse, and the speed that must be removed at Mars’s orbit to match the planet’s velocity. The vis-viva relation gives both: the speed at any distance on an ellipse depends only on the distance and the ellipse’s size. The flight time comes from Kepler’s equation, solved for the moment the ellipse reaches Mars’s radius.

Why the first days are free #

The Hohmann transfer is the minimum of cost against flight time along this family of ellipses, and a smooth curve is flat at its minimum. That is the whole reason the first weeks are nearly free, and it is worth seeing how flat.

On logarithmic axes, a cost that grows as the square of the days saved is a straight line of slope two, and for the first month or so the computed cost follows it closely. Ten days saved costs twenty metres a second; thirty days costs about two hundred; sixty days costs about eight hundred. The square law is not a coincidence of these particular orbits. Stretch the ellipse slightly beyond Mars’s orbit and two things happen at different rates: the extra launch speed grows in proportion to the stretch, while the time saved grows as its square root — because the arrival point moves back from the far end of the ellipse, where the craft is moving slowly and the flight time is most sensitive to where it stops. Cost linear in the stretch and time saved as its square root is cost quadratic in the time saved.

Beyond two months saved the ellipse is long enough that the approximation fails, and the cost turns steeply upward: 3.4 km/s extra for a 150-day transfer, more than doubling the total, and over twenty kilometres a second below ninety days. A mission that insists on the exact minimum-energy transfer pays in weeks for savings measured in metres a second, and one that insists on a very short transfer pays in kilometres a second for weeks.

Real missions sit on that flat part. Mars landers launched in the 2010s and 2020s have taken about 200 to 260 days, not the idealised 259, because the planets’ real, eccentric and inclined orbits move the minimum around from one launch opportunity to the next, and because a few dozen metres a second is worth spending to arrive when the landing site is in daylight or a relay orbiter overhead. The porkchop plots that price real windows are the same trade drawn with real ephemerides: their contours of launch energy are nearly flat around the minimum along the direction of flight time, which is why a launch window lasts weeks rather than a day.

The arrival pays more than the departure #

The extra cost is not shared equally between the two ends.

On the Hohmann transfer the craft leaves the Earth with about 2.9 kilometres a second of excess speed and reaches Mars with about 2.7, both small because the transfer touches both orbits tangentially. Shortening the transfer raises the departure speed only as fast as the ellipse lengthens. The arrival speed rises faster, because the transfer now crosses Mars’s orbit at an angle: at 150 days the path is inclined 17° to the planet’s direction of motion, and the craft arrives not only faster than Mars along its orbit but with a large component across it. It leaves at 4.2 kilometres a second and arrives at 7.0.

That arrival speed must be removed somehow, and a burn to remove it is expensive. How expensive depends on where the burn is made: the cheapest orbit to capture into is not the lowest one but one whose radius depends on the arrival speed, and for a fast arrival the optimum moves inward, closer to the planet, where the Oberth effect described below does the most good. Even at its optimum, capturing a 7-kilometre-a-second arrival into a circular orbit costs more than the entire departure. The alternative is the planet’s atmosphere. A craft entering Mars’s thin atmosphere on a carefully chosen path can shed several kilometres a second in a single pass, protected by a heat shield, and emerge on a captured orbit — aerocapture. It has not yet been flown as a capture into orbit, though every Mars lander has used the atmosphere to lose its entire arrival speed on the way down. For fast transfers it is close to a requirement: without it, the arrival burn alone would cost more than the whole Hohmann transfer.

The planets soften the price #

The figure’s two curves differ, and the difference is a correction that runs in the traveller’s favour. The dashed curve is the heliocentric cost: the change in velocity relative to the Sun at each end. The solid curve is the cost from a low orbit round the Earth to a low orbit round Mars, which is what a mission actually pays, and at high speeds it is lower.

The reason is that the departure and arrival burns are made deep in the planets’ gravity wells, where the craft is already moving fast. A burn is worth more when made at high speed: kinetic energy grows as the square of speed, so adding a given velocity change to a craft already moving fast adds more energy than adding it to one moving slowly. From a low Earth orbit, a burn of $\Delta v$ sends a craft away with an excess speed of

$$v_\infty = \sqrt{(v_c + \Delta v)^2 - v_{\rm esc}^2},$$

where $v_c$ and $v_{\rm esc}$ are the orbital and escape speeds at that height; for large excess speeds this exceeds $\Delta v$ itself. On the Hohmann transfer the effect is modest, and the orbit-to-orbit cost is a little above the heliocentric one because the craft must first climb out of the wells. On the fastest transfers it reverses the comparison: at 100 days the orbit-to-orbit cost is several kilometres a second below the heliocentric sum. The gravity wells that cost energy to climb out of also multiply the value of every burn made inside them.

A cost the velocity does not show #

For a robotic craft the only costs are propellant and time, and the time costs only patience. For a crew, time is paid in something else.

A radiation detector carried inside the capsule of a Mars lander in 2011–12 measured the dose on the way: 1.84 millisieverts a day, almost all from galactic cosmic rays — atomic nuclei accelerated to near light speed elsewhere in the Galaxy, too energetic for any practical thickness of shielding to stop. At that rate the Hohmann transfer’s 259 days deliver about 480 millisieverts one way. For comparison, the career limit set for astronauts by the American space agency is 600 millisieverts, so a round trip on minimum-energy transfers, before any time spent at Mars, would use most of a career.

Against that measure the trade looks different. The first 39 days saved cost 337 metres a second and remove 72 millisieverts: about two hundred millisieverts for each kilometre a second. The exchange rate worsens as the transfer shortens, because the dose falls in proportion to the time while the cost rises faster than in proportion, but the flat start of the cost curve means that the first weeks are the cheapest radiation reduction available — far cheaper, per millisievert, than the shielding mass that would buy the same reduction. The cheapest transfer in velocity is not the cheapest in the unit a crew is paying in.

Why propulsion changes the picture #

The shape of the cost curve assumes that velocity change is bought with chemical rockets, whose exhaust speed of about four and a half kilometres a second makes each extra kilometre a second cost a quarter more propellant for everything that follows. The rocket equation turns the steep part of the curve into an exponential wall in mass. A rocket with twice the exhaust speed — the nuclear thermal engines tested in the 1960s reached about that — halves the exponent, and pushes the wall towards shorter flight times by weeks. That, rather than the total velocity change, is what the case for such engines for crewed missions rests on: they make the flat part of the curve longer.

Engines that thrust continuously at low power, electric propulsion, change the problem more fundamentally — a drive that is not bound by the rocket equation in the same way — since the trajectory is no longer an ellipse between two burns but a spiral shaped by the thrust throughout. The argument here does not carry over to them directly, although the conclusion does: the minimum is flat, and a little extra effort buys a lot of time before it buys very little.

The wait that the transfer does not shorten #

The cruise is not the whole trip, and for a crew it is not the longest part. The Earth and Mars return to the same relative position only once every 780 days, their synodic period, and the launch windows that come back with it fix not only when a crew can leave the Earth but when it can leave Mars. A crew that arrives on a minimum-energy transfer must wait about a year and a half at Mars before the planets are placed for a minimum-energy return; the whole mission lasts about two and a half years, of which the two transfers are well under half.

Shortening the transfers does not shorten that wait, because the wait is set by the planets’ positions rather than by the craft’s speed. The alternative is to leave Mars after a few weeks, on a return transfer that is far from minimum-energy — typically one that swings in towards the Sun and past Venus to be turned homeward — which cuts the total mission to about a year and a half but costs far more velocity change and spends months deeper inside the Earth’s orbit, closer to the Sun and its storms. The two classes of mission are called conjunction and opposition, after where Mars is relative to the Sun during the stay.

For the radiation, the result is that a fast outbound transfer removes a few tens of per cent of the dose of the cruise but a smaller fraction of the dose of the mission. On the surface of Mars the planet shields half the sky and its thin atmosphere a little more, so the dose rate there is roughly a third of the cruise rate; a year and a half on the surface adds about as much as one cruise. The flat start of the transfer’s cost curve still makes the first weeks off each leg the cheapest reduction in dose available — but the total a crew receives is set more by the calendar than by the engine.

Two ways round the Sun #

The family drawn here is only half of the possible transfers. A transfer that sweeps through less than 180° before meeting Mars — every one in the figures — is called type I. A transfer that sweeps through more than 180°, passing the far side of the Sun from the departure point before reaching Mars, is type II, and for the same launch energy it is slower. Real launch windows usually offer both: two basins in the porkchop plot, separated by the ridge where the transfer would have to sweep exactly 180° and its plane would be forced to stand perpendicular to the ecliptic. Crewed studies prefer type I for the obvious reason; many robotic missions have flown type II, accepting a longer cruise for a cheaper launch or a better arrival season.

What the idealisation leaves out #

Both planets are placed on circular orbits in one plane. Mars’s real orbit has an eccentricity of 0.09 and is inclined by nearly two degrees, which moves the minimum-energy flight time between about 150 and 300 days depending on the opportunity and makes some years’ windows much cheaper than others. The transfers are one-tangent ellipses; the fastest transfers are better built from ellipses that are not tangent at either end, or from hyperbolas, which trade departure speed against arrival speed differently. The planetary ends use a patched-conic calculation — the craft on a hyperbola about each planet, the burn instantaneous at the lowest point — and ignore the gravity losses of a finite burn and the atmosphere’s contribution to capture. The radiation dose is the average cruise rate measured during one solar cycle’s rising phase; the cosmic-ray flux is higher at solar minimum and lower at maximum, by a factor of about two, and solar particle storms add doses that shielding can stop but a short transfer cannot avoid.

Still open: what a crewed trajectory should minimise #

For robotic missions the answer is settled: minimise velocity change, subject to arriving when the landing site allows. For crewed missions it is not, because the costs are in different units — propellant mass, radiation dose, time away from Earth, the consequences of an abort — and there is no agreed exchange rate between them. The flat start of the cost curve says that some shortening is nearly free and should always be taken. How much further to go depends on how much a millisievert is worth against a tonne of propellant, and on engines that do not yet exist. The geometry sets the shape of the trade and the physics of cosmic rays sets its units; the choice of where to stand on the curve is still being argued in mission studies, and will be until someone has to fly it.