The same burn is worth more when moving fast

A rocket’s engine does not know where it is. Given a fixed quantity of propellant and a fixed exhaust velocity, the rocket equation returns a change of speed, and that number is the same at the bottom of a gravity well as in interstellar space.

What is not the same is what the change of speed is worth.

Where the factor comes from #

The vis-viva relation gives the specific orbital energy as

$$\varepsilon = \frac{v^2}{2} - \frac{GM}{r} = -\frac{GM}{2a},$$

and every manoeuvre is an attempt to change $\varepsilon$, because $\varepsilon$ is what decides the size and the openness of the orbit.

Now apply an impulse $\Delta v$ along the velocity. The potential term does not change — the vehicle has not moved — and the kinetic term becomes $(v+\Delta v)^2/2$. Subtracting,

$$\Delta\varepsilon = v\,\Delta v + \tfrac12 \Delta v^2.$$

The first term is the whole of it for a small burn, and it is proportional to the speed the vehicle already has. Doubling the speed at which the burn is made doubles the energy it buys.

This is the Oberth effect, named for Hermann Oberth, who set it out in Die Rakete zu den Planetenräumen in 1923 as an argument about how to arrange an interplanetary mission. It is not a rocketry effect at all — it is a statement about kinetic energy being quadratic, which is why the same manoeuvre is available to a comet.

The numbers, and why they are so large #

The gain factor is $v_{\rm deep}/v_{\rm far}$, and near a massive body that ratio is not modest.

At Jupiter’s cloud tops the escape speed is 59.5 kilometres a second, so a vehicle arriving on a hyperbola with an excess of 5.6 is moving at nearly 60 there. Four million kilometres out — about fifty-six Jupiter radii — it is moving at 8.3. The same one-kilometre-a-second burn therefore buys about six times as much energy at the bottom, and the figure computes 5.9.

Translating that into what a mission cares about: the burn made deep leaves the vehicle with a hyperbolic excess of 12.3 kilometres a second, and the burn made far out leaves it with 7.2. The difference is more than the entire departure energy of a Hohmann transfer to Saturn.

What it is used for #

Escaping from a planet. A vehicle in low Earth orbit is moving at 7.8 kilometres a second and needs 3.2 more to escape. A vehicle in a high orbit at ten Earth radii is moving at 2.5 and needs only 1.0 more. The second sounds cheaper and is not: getting to that high orbit from low orbit costs about 2.4 kilometres a second in the first place. Burning where the speed is highest is the cheapest single-impulse escape available. The powered flyby. A spacecraft passing close to a planet can burn at periapsis and convert a modest quantity of propellant into a large change in its heliocentric orbit. Juno performed two deep-space manoeuvres of this kind on its way to Jupiter; Parker Solar Probe uses the same physics in reverse, burning to reduce its perihelion where its speed is greatest.

The most extreme proposals of this kind use the Sun. A vehicle dropped to a perihelion of a few solar radii is moving at several hundred kilometres a second, and a burn there is multiplied enormously — a “solar Oberth manoeuvre” is the standard suggestion for reaching the interstellar medium in a human lifetime. The difficulty is not the physics but the thermal engineering of surviving the perihelion.

Where the opposite strategy wins #

The Oberth effect is about adding energy. Not every manoeuvre is trying to.

Changing the direction of an orbit — a plane change, or moving the periapsis — is cheapest where the vehicle is moving slowest, because the impulse needed to rotate a velocity vector by a given angle is proportional to the vector’s length. That is exactly the opposite prescription, and it comes from exactly the same equation. The rule that covers both cases is worth stating once: energy changes are cheap where the vehicle is fast; direction changes are cheap where it is slow. Every trajectory design is an arrangement of those two facts, and the two point in opposite directions along the same orbit.

The three-burn escape, where both rules are used at once #

Putting the two halves together produces a manoeuvre that looks perverse and is occasionally optimal.

Suppose a vehicle in low orbit needs to escape and change its orbital plane substantially — a departure to a target well out of the ecliptic, say. Doing both at once from low orbit is expensive, because a plane change at 7.8 kilometres a second costs 7.8 times twice the sine of half the angle, which for 30° is four kilometres a second on its own.

The alternative is three burns. Raise the apoapsis to a great height with a burn made low, where energy is cheap. Perform the plane change out at apoapsis, where the vehicle is crawling and the same rotation costs a few tens of metres a second. Then return to periapsis and make the escape burn there, deep, where the Oberth multiplier is largest.

Two of those three burns are placed by the rule that energy is cheap where the vehicle is fast, and the middle one by the rule that direction is cheap where it is slow. The total can beat the direct manoeuvre by a kilometre a second or more for a large plane change, at the price of days or weeks of extra flight time — and the crossover angle at which it starts to win is computed the same way the bi-elliptic transfer’s radius ratio of 11.94 is.

It also gives the clearest statement of what the Oberth effect is not. It is not a claim that burning low is always better. It is a claim about one particular quantity — orbital energy — and any manoeuvre that is not about that quantity obeys the opposite rule.

Where the energy comes from #

The obvious objection to all of this is that it looks like something for nothing. The same propellant, burned in two places, produces different amounts of energy — and energy is conserved.

The resolution is that the accounting has left out the exhaust.

A rocket ejects propellant at a fixed speed relative to itself. In an inertial frame, what matters is the propellant’s speed relative to that frame, and that depends on how fast the rocket was going. A rocket moving fast ejects its exhaust slower in the inertial frame — sometimes backwards, sometimes merely less forwards — and therefore leaves less kinetic energy behind in the exhaust. The energy the vehicle gains is the energy the exhaust does not carry away.

The total, vehicle plus exhaust, is the same in both cases, and it is the chemical energy of the propellant. What changes is the split. A deep burn is efficient because it dumps less energy into the exhaust, not because it extracts more from the fuel.

That is the honest version, and it also explains why the effect has a ceiling: the maximum possible benefit is reached when the exhaust is left exactly at rest in the inertial frame, which happens when the vehicle’s speed equals the exhaust velocity. Beyond that the exhaust is thrown forward and starts carrying energy away again.

A number worth carrying: the ratio at the Earth #

For a quick estimate of what the effect is worth anywhere, the useful quantity is the ratio of the speed at the burn point to the speed far away, and near a planet that ratio is set by how deep the burn is.

At the Earth, a vehicle escaping with a modest excess is moving at about 11 kilometres a second in low orbit and at 3 or 4 near the Moon’s distance. A burn made low is therefore worth roughly three times as much energy as one made out there. At Jupiter the same comparison gives six; at the Sun, for a vehicle dropped to a few solar radii, it gives fifty or more.

The pattern is that the multiplier goes as $\sqrt{GM/r}$, so it rewards deep far more than it rewards massive: halving the burn radius buys a factor of $\sqrt2$, and there is no equivalent lever on the mass of the body one happens to be passing.

That is the practical form of the rule and it is the one a trajectory designer carries. Everything else in this essay is a derivation of it.

What was actually measured #

Nothing here is inferred from a distant observation; it is flown, and the check is Doppler tracking.

A spacecraft’s radial velocity relative to a ground station is measured from the shift of its telemetry carrier, routinely to better than a millimetre a second. After a burn, the tracking gives the achieved change and the resulting orbit, and the achieved hyperbolic excess is compared against the planned one. Missions are navigated by exactly this loop, and the residuals are small enough that the interesting quantity has become the unmodelled acceleration rather than the modelled one.

The most instructive case is one where a residual appeared and did not go away for a decade. Doppler tracking of Galileo, NEAR, Cassini and Rosetta during Earth flybys showed unexplained changes in hyperbolic excess speed of a few millimetres a second — the flyby anomaly — with an empirical formula fitting the sign and magnitude across several encounters. Later flybys of Rosetta showed no anomaly at the same level, and the current view is that it was a modelling artefact rather than physics. The episode is worth remembering as a scale marker: the arithmetic in this essay is checked, in flight, at the level of parts in $10^{7}$ of the speeds involved.

The transfer that goes the wrong way first #

The rule has a mirror image, and it produces a manoeuvre that looks like a mistake until the arithmetic is done.

A change of orbital plane costs $2v\sin(\Delta i/2)$, which is proportional to the speed at which it is performed. So the same logic that makes a burn along the velocity vector worth more when moving fast makes a burn across it cost more when moving fast — and the cheapest place to turn is the slowest point of the orbit.

Put those two together and a transfer between two circular orbits of very different sizes can be improved by leaving the plane of the problem entirely. The bi-elliptic transfer raises apoapsis far beyond the target, changes the orbit cheaply out there where the speed is a few hundred metres a second, and then drops back to the target radius. It uses three burns where a Hohmann transfer uses two, spends far longer in flight, and for radius ratios above about 11.94 it costs less total velocity.

A manoeuvre that overshoots its target by a factor of ten and takes twice as long is the cheaper one, and the reason is entirely the two halves of this essay’s rule applied in opposite directions: cheap turning happens where the speed is low, and cheap energy change happens where it is high.

The same reasoning explains a common feature of real missions that otherwise looks wasteful. A geostationary spacecraft launched into an inclined transfer orbit does not correct its inclination near perigee, where it is moving at ten kilometres a second; it waits until apogee, where it is moving at one and a half, and combines the plane change with the circularisation burn into a single manoeuvre whose vector sum is shorter than the two performed separately.

The deepest well available #

If the multiplier goes as the square root of $GM/r$, the largest one in the solar system is at the Sun, and the idea of using it has a long history and one serious obstacle.

A vehicle that falls to a perihelion of a few solar radii is moving at several hundred kilometres a second when it gets there. A burn of one kilometre a second at that point is worth, in departure energy, what tens of kilometres a second would be worth at the Earth’s distance — enough to leave the solar system at a speed no other manoeuvre delivers, which is why the idea appears in every serious study of an interstellar precursor mission.

The obstacles are two and neither is dynamical. Getting to a low perihelion is itself expensive: removing nearly all of the Earth’s 30 kilometres a second of orbital motion is the most costly thing in the inner solar system, which is why the missions that have gone close to the Sun have used repeated Venus flybys over years to walk their perihelion down. And a spacecraft a few solar radii out is receiving thousands of times the solar constant, so the engine, the propellant and the structure must survive a thermal environment that no chemical stage has been designed for.

The manoeuvre is cheap and getting into position for it is not, which is the general shape of this rule’s practical limits: the Oberth effect rewards depth, and depth in a gravitational well is exactly what a spacecraft has to pay for on the way in.

The comparison between the two is a fair summary of the rule’s practical reach: it is worth a great deal wherever a vehicle is already deep, and worth nothing at all as a reason to go deeper on purpose unless something else pays the fare.

There is a second and harder limit, and it is about time rather than energy. The whole argument assumes the burn happens at periapsis, where the speed is greatest. A real burn takes minutes, during which the vehicle climbs away from the point that made it valuable, so the achieved gain is less than the impulsive calculation promises — the loss is a few per cent for a chemical stage at a giant planet and is the reason such burns are split into arcs centred on periapsis. For a low-thrust vehicle the loss is total: an ion engine producing a hundredth of a gravity cannot deliver a kilometre a second in the hours a periapsis passage lasts, so it accumulates its energy over months at whatever speed it happens to have. The Oberth effect is available only to engines that can spend their propellant faster than the geometry changes, and that is a statement about thrust rather than about efficiency — which is why high-thrust chemical stages survive on missions whose cruise is electric.

What the picture cannot show #

A finite burn. Every equation here treats the impulse as instantaneous. A real burn takes minutes, during which the vehicle moves along its orbit and leaves the point where the speed was highest — so the achieved energy is less than the impulsive calculation gives, by an amount called the gravity loss. For a deep periapsis burn at Jupiter the loss is not negligible, and the correction is one of the reasons a single large burn is sometimes split into several passes.

The mass ratio. The horizontal axis is where the burn happens and nothing on either figure is the amount of propellant. A one-kilometre-a-second burn from a vehicle with a chemical engine costs about a third of its mass; the Oberth effect changes what that third buys, not how much of it there is.

And the arrival. A launch window is a contour on a plane of dates, and the departure energy these figures compute is one coordinate of that plane; The figures compute the departure speed and stop. What a mission actually cares about is arriving somewhere, and a large hyperbolic excess at departure means a large one at arrival too, which has to be removed — usually by another planet, sometimes by an atmosphere, and occasionally by not stopping at all.

And nothing here is about a rocket in an atmosphere. A launch is dominated by drag and by the gravity loss of fighting downward acceleration while still slow, and neither is in any equation above. The Oberth effect belongs to the vacuum part of a mission — the departure burn, the flyby, the arrival — and applying it to a first stage produces the familiar and wrong conclusion that a launch vehicle should stay low and accelerate horizontally from the pad.

One accounting question is worth closing, because the effect looks like something for nothing and is not. The energy the vehicle gains is not created by being deep in the well; it is taken from the propellant, and the propellant’s share depends on where it is dropped. An exhaust stream ejected while the vehicle is moving fast leaves with a smaller speed in the inertial frame — the engine subtracts the exhaust velocity from a large number rather than a small one — so it carries away less kinetic energy, and the difference is exactly what the vehicle keeps. Deep in a well the exhaust can even be left on a bound orbit while the vehicle escapes. Nothing about the total is violated at any point; what changes is how the fixed chemical energy of the propellant is divided between the two pieces, and the division is decided by the frame the burn happens in. The same accounting explains why the effect has no analogue for a manoeuvre that only turns the velocity: a rotation leaves the speed alone, so the exhaust leaves with the speed it would have had anywhere, and there is nothing to divide differently. That is the same asymmetry between energy and direction the section above states as a rule, arrived at from the propellant’s side rather than the vehicle’s. Both statements are the same conservation law read in two frames, and neither is a separate fact about rockets — which is why the effect needs no appeal to relativity, to reference frames chosen carefully, or to anything beyond bookkeeping.

Where the ladder goes next #

The rung above is the low-thrust regime, where the impulsive approximation fails completely: an ion engine thrusting continuously for years follows a spiral rather than a conic, the burn happens everywhere at once, and the Oberth effect becomes an integral rather than a multiplier. The rung beside it is the three-burn escape, in which a vehicle deliberately raises its apoapsis, makes a small cheap plane change out there, and returns for a deep Oberth burn — a manoeuvre that uses both halves of this essay’s rule in one trajectory.