Astronomy & Celestial Mechanics
the sky, computed
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Gravity assist — the series

6 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. A gravity assist with a 70° turn. The velocity triangle of a flyby. In the planet's frame the spacecraft's speed is unchanged and only its direction turns; adding the planet's own velocity converts that turn into a gain in speed measured from the Sun.

    Stealing speed from a planet, which does not notice

    A flyby cannot change a spacecraft's speed relative to the planet. It changes its direction — and adding the planet's own motion back turns that into free velocity.

    part 1 · spaceflight
  2. Along a contour is free; across one costs, and 6 resonances sit on this one. Perihelion against aphelion, in units of Jupiter's orbit, with contours of constant v∞ — which is the Tisserand parameter through v∞² = (3 − T)v_p². Every contour crosses the line r_p = r_a = 1, because a spacecraft has to be at the planet's orbit to have an encounter there at all, and every point on one contour is reachable from every other point on it with no propellant: a flyby rotates the v∞ vector without changing its length, which moves the pump angle and slides the spacecraft along the curve. A burn is the only thing that moves it between curves, and that is the whole economy of a gravity-assist tour. The diagonals are resonant orbits — 1:1, 3:2, 2:1, 5:2, 3:1, 4:1 with the planet — and each is a straight line of slope −1 because a resonance fixes the semi-major axis and r_p + r_a = 2a. They matter because a spacecraft on one comes back to the same place at the same time as the planet, which is what makes a second encounter possible without waiting for a chance alignment. The marked points are where the v∞ = 0.3 contour meets each: a tour is a walk along the highlighted curve from one to the next, and the arithmetic that decides whether it can be walked is the turn one flyby delivers. At 1.35 Jupiter radii and 3.9 km s⁻¹ that turn is 163°, so the longest step drawn here needs 0.1 encounters — which is why a real tour has dozens of flybys and why the ones with the largest steps are the ones that need a deep-space manoeuvre in between.

    The same planet, three times

    A flyby cannot change the encounter speed, only its direction, so a tour has to be designed in the space of what is conserved. One pass moves a spacecraft along a single curve and no further than the planet can bend it — and reaching a distant target means walking that curve, returning to the same planet again and again.

    part 2 · spaceflight
  3. A gravity assist with a 68° turn. The velocity triangle of a flyby. In the planet's frame the spacecraft's speed is unchanged and only its direction turns; adding the planet's own velocity converts that turn into a gain in speed measured from the Sun.

    The planet pays, and it shows

    A gravity assist takes energy from a planet and gives it to a spacecraft, and the planet's loss is exactly the spacecraft's gain. For a two-tonne probe past Jupiter that loss is unmeasurable. Do it with a hundred Earth masses of icy debris and the same bookkeeping moves Neptune outward by several astronomical units.

    part 3 · spaceflight
  4. Propellant a moon saves, against how close the flyby comes. The reduction in the Jupiter orbit-insertion burn bought by one flyby of a Galilean moon on the way in, in metres per second, against the flyby altitude, for an arrival at 5.6 km/s, a periapsis of five Jupiter radii and a capture orbit of 200 days, whose insertion alone costs 825 m/s. At 300 km: Io 268 m/s, Europa 122 m/s, Ganymede 191 m/s, Callisto 124 m/s. The saving falls with altitude because the turn angle does, roughly as one over the flyby distance once that is well above the moon's surface. Io saves most: it is deep in Jupiter's well, where it moves fastest, so a given turn of the relative velocity takes more energy from the spacecraft. Ganymede, the most massive, is second.

    Braking on a moon before braking with the engine

    A gravity assist is usually told as a way to gain speed, but the same exchange runs the other way. A spacecraft falling toward Jupiter can pass behind one of its moons on the way in, lose a few hundred metres per second of its speed to the moon's orbit, and arrive at its capture burn needing a fifth less propellant. Which moon brakes best depends on three things at once — its mass, how fast it is met and how fast it moves — and the best one sits in radiation no spacecraft wants to fly through.

    part 4 · spaceflight
  5. Each kilometre of aim error, turned into a velocity error. The error in the outgoing velocity per kilometre of error in where a spacecraft passes a moon, in metres per second per kilometre, against the flyby altitude on a logarithmic scale, for the four large moons of Jupiter at an encounter speed of 6 km/s. The flyby turns the velocity through an angle δ with tan(δ/2) = aₕ/b, where b is the aim point's distance from the moon's centre and aₕ = μ/v∞²; missing the aim by a kilometre along b changes δ, and missing it by a kilometre across b rotates the plane of the turn, and both move the outgoing velocity by about 2v∞aₕ/b² per kilometre. At 300 km above Ganymede that is 0.32 m/s per km along and 0.32 across. The closer and more massive the flyby, the more it turns — and the more it amplifies whatever error the approach carried.

    A flyby that multiplies its own aiming error

    A close flyby of a moon turns a spacecraft's path by ten degrees for nothing, and it turns every error in the aim with it. A kilometre of miss at three hundred kilometres above Ganymede becomes a third of a metre per second of velocity error, and a week later two hundred kilometres of miss at the next moon. A tour of flybys is an amplifier that must be re-zeroed after every pass, and whether a small spacecraft can use one in place of an engine burn depends on how many kilometres it can aim to.

    part 5 · spaceflight
  6. The last correction: accurate if late, cheap if early. For a spacecraft approaching a moon at 6 km/s with a 30 km aim error left by navigation from the Earth, which locates the moon by camera to 20 microradians and acts an hour after its last image: the error remaining at the flyby after one final correction (blue, km) and that correction's cost (orange, m/s), against how long before the flyby it is made, all on logarithmic scales. The error falls in proportion to the time-to-go, because the moon is seen from closer, until it meets a floor: 3.1 km from how well the limb of a Europa-sized moon can be fitted once it fills the camera's view, with the burn's own two-per-cent error, 0.6 km, beneath it. The cost rises as one over the time-to-go, because the same error has to be removed in less time. The floor is reached 6.2 hours out, and a correction later than that costs more and buys nothing. A correction a day out costs 0.35 m/s and delivers 10.8 km; one six hours out, 1.4 m/s for 3.2 km. With 1 m/s to spend, the best is 4.2 km, at 8.7 hours out.

    An aim bought with the last day of the approach

    A spacecraft that photographs the moon it is approaching can aim its flyby far more precisely than the ground can tell it to, because it sees the target from closer every hour. But each hour closer leaves less time for a correction to act, so every kilometre of aim is bought with propellant — and below a few kilometres the camera stops helping at all, because a moon that fills the view can be located only as well as its limb can be fitted. For a small craft with a metre a second to spend, the aim comes out at four kilometres, set by geometry and not by the antenna.

    part 6 · spaceflight

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