  The polarization of the $\Lambda$ particle offers the unique opportunity to
study the hydrodynamic gradients in the Quark-Gluon Plasma formed in heavy-ion
collisions. However, the theoretical formula commonly used to calculate
polarization is only a linear order expansion in thermal vorticity and neglects
higher-order corrections. Here, I present an exact calculation to all orders in
(constant) thermal vorticity at global equilibrium, obtaining the analytic form
of the spin density matrix and the polarization vector for massive particles of
any spin. Finally, I extend these results to local equilibrium and assess their
phenomenological impact by numerically calculating the polarization vector in a
3+1 hydrodynamic simulation.
