What would be the shape of a space elevator?

I have never seen a discussion of the shape of a space elevator (as viewed from above the North Pole, let us say). Would it be straight or curved?

I have included a diagram of several potential answers. Assume that the length of each cable/elevator is twice the geosynchronous orbital distance. I have no acquaintance with orbital physics. I am just a curious retired math professor who took some physics in undergraduate school.

The space elevator would rotate with the planet, as a solid object. Gravity is nearly radial, a bit off when you are away from the equator and poles due to the Earth being a bit "squashed" by its rotation. Presumably the tower would be placed with one end at the equator in order to make it as short as possible. That would mean that the tower would be very nearly radial.

Just off hand, I'm not sure what you get if you try to move the Earth end of the tower significantly away from the equator. The space end will want to be near the equator. It seems unlikely since there are some high mountains very close to the equator. Anyway...

There will be perturbations.

The Earth is not quite uniform. That means it is not going to be quite radial, even at the equator. Nor absolutely straight.

https://www.space.com/11849-earth-gravity-map-geoid.html

The moon's orbit isn't quite in line with the Earth's equator.

https://en.wikipedia.org/wiki/Orbit_of_the_Moon

Neither is the Earth's orbit around the sun in line with the equator. These will produce small time dependent forces. So the tower would move a bit. There will also be forces due to lots of other stuff, such as storms and earthquakes.

So at the scale of the drawing, it will be "C" but with perturbations that probably are not visible at this scale.

• One thing that will help keeping a space elevator straight is that it is not a "tower". Instead of "standing" on the ground it is under tension, suspended by the section of it that is above GEO. – hdhondt Aug 28 '18 at 2:30