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:
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)
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?