Multipole moment

A multipole moment characterizes one angular component of the field produced by a localized source. At distances much larger than the source, a multipole moment organizes what an observer can infer without resolving the source's full spatial structure. The first terms describe total source strength, dipolar asymmetry, and quadrupolar shape.

For a localized static charge density $\rho (\mathbf{x})$ in vacuum, let $r=|\mathbf{r}|$ and $\hat{\mathbf{r}}=\mathbf{r}/r$. Expanding $1/|\mathbf{r}-\mathbf{x}|$ in the electric potential integral givesThe total electric charge $Q$ is the monopole coefficient, and $\mathbf{p}$ is the electric dipole moment. The next term uses the electric quadrupole moment. More generally the degree-$\ell$ angular contribution to this exterior electric potential falls as $r^{-\ell -1}$ and can be resolved into spherical harmonics.

For two opposite electric charges $+q$ and $-q$ separated by vector $\mathbf{d}$ from negative to positive, $Q=0$ and $\mathbf{p}=q\mathbf{d}$. Their leading distant electric potential therefore falls as $r^{-2}$ rather than $r^{-1}$. Translating the coordinate origin by $\mathbf{a}$ gives ${\mathbf{p}}^{\prime }=\mathbf{p}-Q\mathbf{a}$, so this neutral pair's electric dipole moment is origin independent. For nonzero total electric charge, the choice of origin matters.

The same organization is useful for a localized mass density, giving mass multipole moments. The relevant source, field equation and normalization must be stated: electric and gravitational multipole moments encode different physical quantities.

Spatial moments of a mass density distribution organize its far gravitational field. Symmetric trace-free parts isolate angular multipoles; the second moment gives the mass quadrupole moment. In slow-motion linearized gravity, changes of the quadrupole produce the leading gravitational radiation; the conserved monopole and centre-of-mass dipole do not radiate independently.