The cheapest circle to stop in

The speed left over after escaping a planet combines with the escape speed as the two sides of a right triangle: a spacecraft that must leave with an excess $v_\infty$ has to reach $\sqrt{v_{\rm esc}^2 + v_\infty^2}$ at its departure point, and because the combination is quadratic, the excess is cheap to buy deep in a gravitational well. At the far end of the transfer the same triangle runs backwards. The spacecraft arrives at its target planet on a hyperbola with an excess it cannot lose by coasting, and to stay it must brake by enough to fall below escape speed.

What the departure calculation never had to ask is where to do that, because leaving was from a parking orbit chosen by the launch. Arriving has no parking orbit. The spacecraft can aim its hyperbola to pass the planet at any distance, brake at that closest point, and end in any orbit consistent with the braking. The choice of where and into what turns out to have a clean answer in one case and a set of trades in all the others, and the answers explain why a Jupiter orbiter skims the cloud tops, why Mercury was the last planet anyone orbited, and why nearly every orbiter begins its mission on an orbit it does not want.

A minimum set by the arrival speed #

The cheapest place to brake is at periapsis, where the spacecraft is moving fastest and a given change in speed changes its energy most. Suppose the aim is a circular orbit of radius $r$: the hyperbola is aimed to pass at $r$, and one burn there brings the spacecraft from its hyperbolic speed $\sqrt{v_\infty^2 + 2\mu/r}$ down to circular speed $\sqrt{\mu/r}$. The cost is the difference,

$$\Delta v = \sqrt{v_\infty^2 + \frac{2\mu}{r}} - \sqrt{\frac{\mu}{r}},$$

and it has a minimum. Setting its derivative with respect to $r$ to zero gives

$$r = \frac{2\mu}{v_\infty^2}, \qquad \Delta v_{\min} = \frac{v_\infty}{\sqrt{2}}.$$

The cheapest circular orbit to capture into is at a radius set by the arrival speed and the planet’s mass alone, and the cost there is exactly the excess divided by the square root of two, whatever the planet. For a minimum-energy arrival at Mars, at 2.65 kilometres a second, the optimum is 3.6 Mars radii from the centre and the cost is 1.87. For an arrival at 4 kilometres a second it moves in to 1.6 radii and costs 2.83. At 6 kilometres a second the optimum would be at 0.7 radii — inside the planet — and every real circular orbit is on the expensive side of it.

The minimum is a balance between the two things the Oberth effect trades. Lower orbits let the braking happen deeper in the well, where the spacecraft is faster and a kilometre a second of braking removes more energy — but the circular speed at the bottom of the well is also higher, and the spacecraft has to be left moving at it. Higher orbits need the spacecraft left moving more slowly, but the braking is done where the Oberth effect is weaker. The two effects cross at $2\mu/v_\infty^2$, which is also the periapsis at which the hyperbola’s turning angle is 120 degrees. Nothing about the planet’s size enters, except through whether that radius is above its surface.

The optimum has a tidy description in terms of speeds. At $r = 2\mu/v_\infty^2$ the circular speed is exactly $v_\infty/\sqrt{2}$, and the speed at periapsis of the arriving hyperbola is $\sqrt{v_\infty^2 + v_\infty^2}$, which is $\sqrt{2}\,v_\infty$ — exactly twice the circular speed. The spacecraft arrives at the optimum radius moving at twice the speed of the orbit it wants, and the burn halves its speed. At any other radius the ratio is different: deeper in, the arrival is less than twice the circular speed but the circular speed is higher; further out, the ratio exceeds two. The factor of two is a signature of the optimum that can be checked from a mission’s numbers without knowing the planet’s mass.

The curves are flat near their minima. At Mars, capturing into a circle anywhere between 2 and 8 radii costs within five per cent of the minimum, so the optimum is a guide rather than a requirement. What the curves do say firmly is that capturing directly into a low circular orbit — the one most useful for mapping — is never the cheapest way to arrive.

Stopping just short of escape #

Most orbiters do not capture into circles at all, and the reason is the second calculation.

A burn at periapsis that leaves the spacecraft on a long ellipse, rather than a circle, only has to bring it from above escape speed to just below. A circle requires bringing it all the way down to circular speed, which is escape speed divided by the square root of two. The difference is large. At Mars, with a periapsis 300 kilometres up, capture into a circle costs 2.09 kilometres a second; capture into an ellipse whose apoapsis is a hundred times further out costs 0.70. The first factor of ten in apoapsis buys almost all of the saving, because the speed at periapsis of an orbit with a distant apoapsis is already close to escape speed.

So the standard arrival is in two stages. The spacecraft is captured into a loose ellipse for the smallest burn it can afford, and the orbit is brought down afterwards. At Mars and Venus the bringing down is done by the atmosphere: periapsis is lowered into the upper atmosphere and each pass removes a little speed, aerobraking, over months, until the apoapsis has fallen to where the mission wants it. The Mars orbiters of the past three decades saved hundreds of kilograms of propellant this way, and the most extreme version — capturing directly in one atmospheric pass without any burn at all — has been studied since the 1970s and never flown. At the giant planets the bringing down is done by the moons.

Diving to the cloud tops #

The Oberth effect is proportional to how fast the spacecraft is moving when it burns, and at a giant planet the speeds at small periapsis are enormous.

Jupiter’s escape speed at its cloud tops is sixty kilometres a second. A spacecraft arriving with an excess of 5.6 — the minimum-energy arrival from the Earth — and aimed to pass just above the clouds is moving at more than sixty kilometres a second there, and a burn of 0.55 kilometres a second is enough to capture it into an orbit reaching out to 110 radii. The same capture made at 10 radii, where it is moving at under twenty kilometres a second, costs 1.62. The difference is a factor of three in propellant for the same captured orbit, and it is the reason the most recent Jupiter orbiter was aimed at a periapsis of 1.06 Jupiter radii, four thousand kilometres above the clouds, and the Saturn orbiter before it made its capture burn just outside the main rings.

Both paid for the saving. A periapsis that close to Jupiter passes through the most intense part of its radiation belts, and the spacecraft’s electronics had to be shielded in a titanium vault; a periapsis just outside Saturn’s rings required the spacecraft to pass through the gap between two ring regions with its antenna turned forward as a shield against dust. The navigation needed to aim at a periapsis a few thousand kilometres from the cloud tops, after a journey of hundreds of millions of kilometres, is itself a demonstration of aiming at a plane rather than at a planet. The Oberth effect sets the target; the planet sets the price of reaching it.

The numbers the missions flew #

The two-body arithmetic can be checked against the burns that were actually made, and it holds up to within the few per cent that finite burns and real orbits add. The Jupiter orbiter that dived to 1.06 radii made a capture burn of 542 metres a second, into an orbit with a period of 53 days — the figure’s 0.55 kilometres a second for an apoapsis of about 110 radii, to within two per cent. The Saturn orbiter’s capture burn, made just outside the rings with an arrival excess of about 5.5 kilometres a second, was 626 metres a second into an orbit of 116 days, beside the 0.6 in the comparison of planets. At Mars, capture burns have ranged from about 0.8 to 1.2 kilometres a second depending on how loose the first orbit was allowed to be, bracketing the 0.7 drawn for an apoapsis a hundred times the periapsis; the orbiters that used most propellant were those whose instruments needed a tighter orbit sooner.

The first orbits are enormous. A capture that stops just short of escape leaves the spacecraft with an apoapsis millions of kilometres out and a period of weeks to months, and the first months of every giant-planet mission are spent coming back to periapsis a few times before the science orbit is reached. The saving in propellant is paid for in calendar time, which for a mission with a fixed lifetime of radiation dose or power supply is a real cost, and one the two-body minimum does not see.

Five planets, compared #

The arithmetic can be applied to each planet after the cheapest possible transfer from the Earth, and the results spread over a factor of twenty.

Venus and Mars are cheap for a loose capture — under a kilometre a second — and have atmospheres that can do the rest. Jupiter and Saturn are cheap for a loose capture too, despite arriving faster than the inner planets, because their wells are so deep that the Oberth effect overwhelms the excess; but a low circular orbit at either costs ten to seventeen kilometres a second, more than launching from the Earth, and no mission has attempted one. At the giants the orbit is shaped instead by repeated flybys of the large moons, each of which changes the spacecraft’s orbit about the planet for nothing, and a tour of dozens of such flybys is how every giant-planet orbiter has explored its system.

Mercury is the outlier, and for two reasons that compound. A spacecraft falling from the Earth’s orbit towards the Sun gains speed all the way, and arrives at Mercury’s orbit on a minimum-energy transfer with an excess of 9.6 kilometres a second relative to the planet — the fastest arrival of the five. And Mercury’s gravitational well is shallow: its escape speed is 4.3 kilometres a second, so there is little Oberth effect to exploit. Even a loose capture costs 6.4 kilometres a second, more than leaving low Earth orbit for Mars. Mercury has no atmosphere to help, and no moons.

That is why Mercury, the nearest planet to the Sun and the second nearest to the Earth at its closest, was the last of the classical planets to be orbited. The two missions that did it never flew the minimum-energy transfer. They spent six and seven years on flybys of the Earth, Venus and Mercury itself, each flyby lowering the arrival excess by a few kilometres a second, until the capture burn fell to something a spacecraft could carry. The cheapest planet to reach by energy was among the most expensive to stop at, and the difference was paid in time.

Moons as the second half of the braking #

After the capture burn, a giant-planet orbiter lowers and reshapes its orbit without propellant, by passing close to the large moons. Each flyby is a two-body encounter in the moon’s frame — the spacecraft’s speed relative to the moon is unchanged and only its direction turns — but in the planet’s frame the turn changes the spacecraft’s orbital energy, just as a planetary flyby does about the Sun. A pass behind a moon in its orbit slows the spacecraft relative to the planet and lowers its apoapsis. Titan, massive and at a convenient distance from Saturn, provided changes of up to about 0.8 kilometres a second per flyby, and the Saturn orbiter’s thirteen-year tour was built on more than a hundred of them; the moon did the work that a propellant load several times the spacecraft’s mass would otherwise have had to do.

At Jupiter the first orbiter used a flyby of Io on its approach, a few hours before the capture burn, to lower the arrival excess before braking — the moon taking part of the arrival itself. The same strategy runs in reverse for leaving a moon system, and it is the reason the orbits of giant-planet missions look like tangled rosettes: each petal is the spacecraft’s orbit between two moon encounters, and each encounter was chosen, years in advance, to hand the orbit to the next one.

A kilometre a second, paid twice #

The last curve prices the temptation to arrive faster.

A transfer to Mars can be made faster than the minimum-energy one by leaving the Earth with more excess, and the shorter journey is attractive for any mission with a crew or a deadline. The penalty at departure is modest, because the departure excess adds in quadrature to the Earth’s escape speed. The penalty at arrival is not. While the arrival excess is small beside Mars’s escape speed at periapsis, 4.8 kilometres a second, it too adds in quadrature and costs little; above that, the capture cost rises almost linearly, and each extra kilometre a second of arrival costs 0.84 of braking. A fast transfer is paid for at both ends, and at the arrival end nearly one for one.

That is the arithmetic behind every proposal for fast crewed transfers to Mars: they are possible only if the arrival speed can be shed without propellant, which at Mars means in the atmosphere, with a heat shield sized to the excess. The trade between a shield’s mass and a propellant load’s decides the architecture, and it is the arrival end of the triangle that forces it.

What the two-body picture leaves out #

Every number here treats the arrival as a two-body problem: the spacecraft on a hyperbola about the planet, the burn instantaneous at periapsis. Real capture burns last tens of minutes, during which the spacecraft moves through a substantial arc of its hyperbola and some of the burn is made away from periapsis, where it is less efficient; the loss is a few per cent for a well-designed burn and grows with the burn’s length relative to the time spent near periapsis. Finite burns also have to be aimed off the ideal direction to keep the orbit’s shape, and the planet’s oblateness perturbs the capture orbit before the first apoapsis.

The two-body picture also assumes the spacecraft arrives with a fixed excess. The excess itself is set by the transfer, and the transfer can be designed to arrive slowly — by flybys, as Mercury’s orbiters did, or by approaching through the region where the Sun’s pull and the planet’s are comparable, where the neck of the zero-velocity surface lets a spacecraft drift into a loosely bound orbit with almost no excess at all. In that regime the capture burn falls towards zero and the hyperbola that this essay is built on stops being a good description of the arrival.

Still open: arriving without a burn #

The circle of radius $2\mu/v_\infty^2$ is the cheapest place to stop for a spacecraft that must stop by braking. The frontier is not stopping by braking. Aerocapture removes the excess in one atmospheric pass and has never flown; ballistic capture removes it by threading the region where the planet’s and the Sun’s gravity balance, and has been flown to the Moon but not yet to a planet; low-thrust spirals remove it continuously over months and make the Oberth effect an integral rather than a factor. Each of them replaces the arithmetic of this essay with something slower, riskier or less predictable, and each is being developed because the arithmetic, applied honestly to a fast arrival at a planet without an atmosphere, says the braking will cost almost as much as the arrival speed itself.