  By including the recently introduced thermal shear term that contributes to
the spin polarization vector at local equilibrium, we determine longitudinal
polarization of $\Lambda$ hyperons emitted from a hot and rotating hadronic
medium using the thermal model with single freeze-out. In our analysis, we
consider the RHIC top energies and use the model parameters which were
determined in the earlier analyses of particle spectra and elliptic flow. We
confirm that, unlike the previous calculations done by using only the thermal
vorticity, the thermal shear term alone leads to the correct sign of the
quadrupole structure of the longitudinal component of the polarization
three-vector measured in experiments. However, we find almost complete
cancellation between thermal shear and vorticity terms, which eventually leads
to disagreement with the data. To clarify the role played by velocity and
temperature gradient terms, we present a systematic analysis of different
contributions to the longitudinal polarization.
