The anchor is past geosynchronous orbit so it’s applying a constant upwards force. The energy for horizontal motion comes from the rotation of the earth, much like how a rotating ice skater slows down when they spread their arms.
Station keeping may be used to dampen oscillations, but managing climbing rates works just as well.
So the constant upwards force is providing enough tension to keep the cable in the realm of a rigid body approximation? I would have assumed the force applied to accelerate the payload would also deflect the tether and anchor backwards (in a miniscule amount) and would have built up over successive payloads.
Not arguing, just curious. It's been at least a decade since I've messed with orbital mechanics.
It does defect the anchor from pure vertical, but like a ball on a pendulum the vertical force from the anchor translates into a horizontal force.
EX: https://www.researchgate.net/publication/312934526/figure/do...
Of course if you really dig into things it’s more complicated, but the basic principle is very similar.
I've not done the mechanics in a long time (not since college), but I believe the tether itself would be providing all horizontal propulsion. Essentially by riding the tether up, it starts pushing you faster and faster. Since the payload's mass would be small compared to the anchor, the drag on the anchor would be negligible.
There's likely some need for thrust compensation on the anchor over time to counteract the delta V lost to the lifting of payloads, but that would all be part of station-keeping and would be there for tether drag as well.