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?