  We study the dynamical formation of scalar monopole and dipole hair in scalar
Gauss-Bonnet theory and dynamical Chern-Simons theory. We prove that the
spherically-symmetric mode of the dipole hair is completely determined by the
product of the mass of the spacetime and the value of the monopole hair. We
then show that the dynamics of the $\ell=1$ mode of the dipole hair is
intimately tied to the appearance of the event horizon during axisymmetric
collapse, which results in the radiation of certain modes that could have been
divergent in the future of the collapse. We confirm these analytical
predictions by simulating the gravitational collapse of a rapidly rotating
neutron star in the decoupling limit, both in scalar Gauss-Bonnet and dynamical
Chern-Simons theory. Our results, combined with those of Ref.~\cite{R:2022cwe},
provide a clear physical picture of the dynamics of scalar monopole and dipole
radiation in axisymmetric and spherical gravitational collapse in these
theories.
