I working on a story, and there is a space elevator. I know that moon and sun have a gravitational effect on bodies in orbit. And if a heavy asteroid was used for counterweight, wouldn't it require a massive amount of propellant (or power depending on propulsion) to keep the north-south station keeping? Anyway, what I want to ask is if anyone knows how much influence would moon and sun be on the counterweight. Same, more or less or than on a large unanchored station in geostationary orbit.
I don't have the math or orbital mechanics to prove it (despite playing hundreds or hours of Kerbal Space Program), but I'm inclined to think that if your ground anchor is smack on the equator, in one of the ground locations corresponding to the most stable points in GEO (the reason Taprobane -- essentially a fictional Sri Lanka but on the equator -- was chosen in Fountains of Paradise), station keeping for the counterweight could be nearly zero.
First, a geostationary satellite over one of these stable locations needs almost none to begin with, because these locations are potential wells in the geosynchronous environment -- and secondly, the counterweight is, in terms of forces, hanging from the ground anchor. Any perturbation by the Moon or Sun, forward, backward, north or south, will create a restoring force that wants to return the tether to a vertical position. This will, of course, result in some pendulum effect, but it should be possible to damp this (especially in the fore-and-aft plane) by timing transits of mass up and down the tether (momentum to bring those to orbital velocity is also supplied by the ground anchor through the tether).
In the end, I would suppose that having the ability to actually build a Beanstalk tends to presuppose the ability to stabilize it.