Mass multipole moment ($I_{i_1\cdots i_{\ell }}=\int \rho x_{i_1}\cdots x_{i_{\ell }}{d}^{3}x$)

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.