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Several space probes have used gravitational slingshots around Earth as part of their mission plan to get to other places in our solar system. Some examples I could find quickly are Galileo, Messenger and Cassini. Many more probes have, to the same end, used gravitational slingshot maneuvers around other planets and moons.

Borrowing from Wikipedia's explanation of gravity assists, my boldface:

To increase speed, the spacecraft flies with the movement of the planet (taking a small amount of the planet's orbital energy); to decrease speed, the spacecraft flies against the movement of the planet. The sum of the kinetic energies of both bodies remains constant (see elastic collision).

Since gravitational slingshots, when used to increase the velocity of a spacecraft, due to conservation of momentum transfer energy from the astronomical object (the planet or moon) to the spacecraft, this leads to a tiny decrease in the rotational rate (or a tiny increase in the rotational period length) of the astronomical object, and/or correspondingly for the orbital period.

Given that particularly on Earth we have systems that rely on highly accurate timekeeping, but that this effect is almost certainly tiny to begin with:

  • What is the extent of this effect?
  • To what extent is this effect worth considering in mission planning?
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    $\begingroup$ Turns out Randall Munroe has also looked at this, half a year later. XKCD What-If: Stop Jupiter $\endgroup$
    – user
    Feb 17, 2016 at 9:38

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It is completely negligible. Mass of spacecraft: 1000 kg (Thereabouts). Mass of Earth: $5.9e24$. The difference is 21 orders of magnitude. The variation of the rotation of the Earth from day to day is orders of magnitude larger than the difference caused by a gravitational flyby.

Also, it's the angular momentum of the orbit, not the rotation, which is affected. The momentum of any planet is huge, as momentum is m*v, and the velocity is quite high. You can safely ignore the gravitational affect of a flyby on the host planet/moon completely, unless you are talking about a spaceship the weight of a large asteroid.

How big of an asteroid? Here I have to make a few assumptions, which are that the asteroid is spherical, 1/3rd of a second/ year would be problematic, and that the amount varied is proportional to the weight difference. Also assuming 5 g/cm^3 density. Given all of that, an asteroid with a diameter of 130 km would be a candidate for causing a small difference to the year of the Earth. The most problematic assumption is the amount varied is proportional to the weight difference, which I'm sure isn't exactly correct, but is a reasonable assumption.

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    $\begingroup$ One thing probably also worth mentioning is that if you're using gravity slingshot not just to increase heliocentric velocity, but to also reduce it, say for a dive deeper into the Solar system, you're adding a tiny bit of momentum to Earth's orbit. So even any cumulative effects of doing this over and over again with small masses will be negligible. Similar for effects of launches on Earth's rotation on it's axis, tho that's a different matter. $\endgroup$
    – TildalWave
    Sep 21, 2015 at 15:48
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    $\begingroup$ @SarahBourt: My very rough order of magnitude guess says it would be around 130 km in diameter to have some effect (About 1/3rd of a second longer of a year, assuming a dense asteroid (5 g/cm^3)) $\endgroup$
    – PearsonArtPhoto ♦
    Sep 21, 2015 at 15:56
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    $\begingroup$ @PearsonArtPhoto Anything, anything to get rid of that annoying leap second!! :D $\endgroup$
    – TildalWave
    Sep 21, 2015 at 15:58
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    $\begingroup$ For an analysis of a smack on space craft and small body can be found in The Orbital History of Comet 9P/Tempel 1: "The 370 kg impactor will impart a very modest 0.0001 mm/s velocity change in the comet's orbital motion and by so doing decrease the comet's perihelion distance by 10 meters and decrease its orbital period by far less than a second of time." $\endgroup$
    – user5892
    Sep 21, 2015 at 18:46
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    $\begingroup$ @MichaelT Gravity assist doesn't work quite like kinetic impactors. With impactors there's the initial absorbed momentum of the collision itself, and an additional gain in momentum from impact ejecta. So it depends on multiple physical properties of the bodies colliding, and can also change body's rotation on own axis (again from impact at an angle or off axis and/or by losing angular momentum on ejecta). Powered flyby gains momentum from Oberth effect, and change in vector direction with a hyperbolic flyby through primary's gravity well. They're just too different to be directly comparable. $\endgroup$
    – TildalWave
    Sep 21, 2015 at 20:33
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This can be worked out through conservation of energy: if a spaceship gains kinetic energy from a flyby, the planet must have lost the same amount of energy.

The Wikipedia article on gravity assists shows Cassini as gaining 4000 m/s from the Earth flyby; assuming a mass at flyby of around 4500 kg, this means it gained about 72 GJ of energy. Earth has a mass of $5.9*10^{24} kg$, so assuming all the energy came from the Earth's orbital velocity rather than from its rotational velocity, the flyby caused the Earth to slow down by 0.00000011 m/s, or about one part in 280,000,000,000.

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