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# Earth as a rigid axisymmetric body

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The earth maybe considered as a rigid axisymmetric body with a small quadrupole deformation. (There are two problems (a) and (b))

(a) If the exterior gravitational potential is written as:

V(r)=-M_e*G*1/r*[1-J(R_e/r)^2* P_2(costheta)]

Here, M_e is the mass of the earth, R_e is the equatorial radius and theta the colatitude, show that J=(I_3-I_1)/M_e*R_e^2)
P_2(costheta)=3costheta^2-1 as easily found in any quantum book.

----------> I have tried to take the derivative of this potential with r to find the "external force" in order to use the equation
(external Torque)=(dL/dt)_body+wXr

I'm not sure how to take the next step.
Could you please show me the full process until I can arrive to the correct solution?
Another related problem(b) is attached as file.

##### Solution Summary

We derive an approximate formula for the gravitational potential outside the Earth in terms of its quadrupole moment about its axis of symmetry.

##### Solution Preview

The earth may be considered a rigid axisymmetric body with a small quadrupole deformation.

(a) If the exterior gravitational potential is written ...

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