my understanding of moments is that they refer to distributions about an expected value, which allows us to make the multipole expansion. I read that:
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the zeroth moment of mass refers to the mass of the body itself, first moment refers to the distribution about some centre of mass, and the second moment refers to how skewed the distribution is. I also read in some texts that the second moment refers to the moment of inertia (do correct me if I'm wrong)
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the monopole refers to the mass of the body, the quadrupole refers to how skewed the distribution of mass is.
Thus, my question is — is the gravitational quadrupole moment the same as the moment of inertia? If they aren't, what distinguishes them?
Best Answer
A non-zero dipole term merely indicates that the central point in the multipole expansion is not the center of mass. The standard approach is to make the central point of the expansion be at the center of mass, in which case the dipole terms are zero.
All but the monopole term vanish in the multiple expansion of the gravitational potential of an object with a spherical mass distribution (with the expansion about about the object's center of mass). A non-zero quadrupole term indicate deviations from a spherical mass distribution.
The quadrupole tensor is quite distinct from the moment of inertia. Any non-point mass object will necessarily have a non-zero moment of inertia tensor. On the other hand, any object with a spherical mass distribution looks just like a point mass gravitationally, which means that its quadrupole tensor (expressed with respect to the center of mass) is identically zero.