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Can rockets launching change Earth's rotation, and can going into orbit do the same?

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Yes, but no.

Angular momentum is conserved, and since we launch rockets roughly tangentially from Earth's edge there will be a counter-spin imparted. While an occasional rocket is launched "backwards" (retrograde, The strange orbit of Ofeq 11 - how does it (actually) do this?) most go roughly prograde while some go polar. I don't yet have my license to practice angular momentum conservation using tensors so let's just assume an equatorial launch to orbit or to escape in the equatorial plane.

If a rocket launches a payload to equatorial LEO at 400 km altitude, the final payload velocity is about 7700 m/s. That angular momentum came from somewhere, and that place is Earth's atmosphere. It launched slow and straight, then turned and started tangential acceleration in the atmosphere, and even while angled towards 400 km altitude and the exhaust was released to "space" (arbitrarily defined as above 100 km) the exhaust velocity was ballpark 3000 to 4000 m/s relative to the rocket, and so in the Earth's frame much of the mass of the exhaust was going either much slower backwards or even in the forward (prograde) direction.

That doesn't matter because the exhaust never gets close to escape velocity. It is all gravitationally bound to the Earth and will eventually collide and mix with other atoms and meld its momentum with the rest of the atmosphere.

And over time, the atmosphere's angular momentum is shared with the Earth's solid body via friction.

So whatever angular momentum is associated with the payload and other orbiting rocket bodies will be subtracted from the Earth, but only until that stuff's orbit decays in a few decades due to drag and it falls back to the Earth's atmosphere, which then reclaims its angular momentum and then shares it with the solid Earth via friction again.

However...

Satellites and spent upper stage rocket bodies in higher orbits (e.g. MEO, GEO) hang on to some angular momentum borrowed from the Earth's atmosphere (and therefore the solid Earth) for much longer, so long you could consider it permanent. Getting to those higher orbits requires emitting some exhaust at very high altitudes. However even at MEO and GEO the escape velocities are 5500 and 4350 m/s so there's no chance the exhaust will escape to infinity. It will very very slowly make its way back to the Earth, and so only the angular momentum of the solid bits (payload, upper stages) will have been long-term borrowed from the solid Earth.

At the "turn of the century"

Satellites got to GEO usually with chemical propulsive maneuvers, one prograde impulse in LEO to achieve GTO (Geostationary Transfer Orbit), and a second prograde impulse at apoapsis to roughly circularize at/near GEO. In this case the exchaust slowly makes its way back to Earth, and the spacecraft keeps some of the Earth's angular momentum.

Taking @RussellBorogove's challenge a 5,000 kg satellite in GEO has an angular momentum

$$L = \frac{mv}{r}$$

of about 6.5E+14 kg m^2/s. The moment of inertia of the Earth from NASA's Earth fact sheet's $I/MR^2 =$ 0.3308 is 8.0E+37 kg m^2/s. That's only about one part in 1.2E+23 change, which is pretty small! The Earth would rotate about 1 millimeter less in ten billion years.

However, what's more interesting is if the satellite moved from LEO to GEO using electric propulsion! As I discussed here and then asked about later in Where do ion propulsion's ions go? Do they remain in the solar system or shoot out into interstellar space?

You can estimate the exhaust velocity of an ion engine using

$$\frac{v}{c} = \sqrt{\frac{2E}{m_0 c^2}}. $$

Choose $E=$ 100 keV and $m_0 c^2=$ 931 MeV times 50 to 200 AMU and you get between 0.2 and 0.1% of the speed of light, which is way beyond escape velocity. Any angular momentum gained by electric propulsion in Earth orbit at or beyond reasonable LEO is compensated by an equal and opposite angular momentum of ions flying out of the back of the spacecraft and into the Earth's magnetic field where it would twist and turn and either hit the poles or escape to the interplanetary magnetic field. Depending on direction that could spin the earth up or slow it down!

So in the case of electric propulsion there would be anywhere between no angular momentum transfer to the same amount or more than from impulsive transfer, when moving from LEO to GEO Getting the 5 tons to LEO removes only 2.4E+14 kg m^2/s of angular momentum from the Earth, which is only about 1/3 the amount of loss of that lost if you had gone all the way chemically, but depending on the ion trajectories the total amount could be anywhere from (6.5-2.4=4.1)E+14 to (6.5+2.4=8.1)kg m^2/s.

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    $\begingroup$ I'll upvote if you can give an order-of-magnitude estimate of how much a typical launch of 5t to GEO slows Earth's rotation. $\endgroup$ – Russell Borogove Feb 25 '19 at 3:53
  • $\begingroup$ @RussellBorogove that's a nice touch, have a look... $\endgroup$ – uhoh Feb 25 '19 at 5:48
  • $\begingroup$ You're mixing up changing our angular momentum with changing our rotation. Launching a spacecraft to orbit most certainly changes our rotation, just by very little. It's the same as an ice skater putting their arms out, but the Earth is huge and the spacecraft is tiny. $\endgroup$ – Loren Pechtel Feb 25 '19 at 6:14
  • $\begingroup$ @LorenPechtel I don't believe I have "mixed up" anything. The skater problem is treated as rigid body rotation, the arms never leave the skater and everything rotates at a single rate (to first order). Here there is the planet, the atmosphere, several stages and a final payload, each rotates mostly independently of the others at a different speed than the others. The skater is not a suitable analogy for spaceflight. $\endgroup$ – uhoh Feb 25 '19 at 9:48
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    $\begingroup$ okay, I just asked this question because I had finished 2001: A Space Odyssey, and I wondered if we could really change planet's velocity. This is a very helpful answer! $\endgroup$ – AndrewMaxwellRockets Feb 25 '19 at 12:15

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