P1: Why the Space Elevator's Center of Mass is not at GEO
I have often encountered people who think that the Space Elevator's center
of mass must be located at the altitude corresponding to geosynchronous
orbit, because it rotates synchronously with the Earth. This assumption
that people make is actually incorrect for two reasons as we shall see.
P2: The Space Elevator is Connected to the Earth
One reason that people easily accept for the center of mass not being at
GEO, is that the Space Elevator is in fact not a free body. It is
attached to the Earth, and there is a significant tension on it at its
base. This is equivalent to having an elevator that is not attached to the
Earth but that has an anchor mass attached at its base, which is sufficient to
provide the appropriate tension in the base of the cable. This mass
effectively lowers the center of gravity of the elevator. At this point,
some people are ready to consider that the center of mass of the
cable-counterweight-anchor mass system has to be at GEO, rather than the
one of the cable-counterweight system. These people are still wrong.
P2: The Space Elevator is not a Point Mass
Kepler's laws of motion, which many people are familiar with, only apply
to masses that are small compared with the variations of the gravity field
they are moving in. In the case of the Space Elevator, this is far from
being the case; gravity goes down by more than a factor of 300 over its
height. For non point masses, the orbital characteristics deviate from
those of point masses (in fact the term of orbit isn't really well suited
anymore). Here is a simple example that proves my point, it comes from one
of my messages on {{[jump="http://groups.yahoo.com/group/space-elevator/"]the
Yahoo Groups space-elevator mailing list}}.
Take two identical masses m attached by a length 2L of
massless cable. Place one mass at a distance R+L from the center of the
Earth, and the other at a distance R-L. Put them in a circular orbit
with angular velocity w in such a way that the axis of the cable always
goes through the center of the Earth.
* The centrifugal acceleration on these masses is:
- (R-L)w^2 for the low one
- (R+L)w^2 for the high one
* The gravity acceleration on these masses is:
- GM/(R-L)^2 for the low one
- GM/(R+L)^2 for the high one
In order for your system to be in equilibrium at angular velocity w you
need these accelerations to compensate each other:
(R-L)w^2+(R+L)w^2=GM/(R-L)^2+GM/(R+L)^2
This leads to w^2=GM/2R * (1/(R-L)^2 + 1/(R+L)^2)
If you set L=0, you get w^2=GM/R^3 which is Kepler's third law.
If L is small then 1/(R+L)^2 is approximately 1/R^2-2L/R^3, so we once again
get Kepler's third law. However, for large values of L you find that w
is greater than Kepler's third law would predict for an object at
altitude R where the center of gravity of the system is located. This is
because the increase in gravity for the low mass is greater than the
decrease for the high mass.
So in general, for an orbit that extends over a great range of
altitudes, the orbit will be faster than the orbit at the center of mass
altitude. In conclusion, even for an elevator not attached to the Earth,
the center of mass should be above GEO.