  While there have been multiple observational programs aimed at detecting
linear polarization of optical radiation emitted by ultracool dwarfs, there has
been comparatively less rigorous theoretical analysis of the problem. The
general expectation has been that the atmospheres of those substellar-mass
objects with condensate clouds would give rise to linear polarization due to
scattering. Because of rotation-induced non-sphericity, there is expected to be
incomplete cancellation of disk-integrated net polarization and thus a finite
polarization. For cloudless objects, however, only molecular Rayleigh
scattering will contribute to any net polarization and this limit has not been
well studied. Hence in this paper we present a detailed multiple scattering
analysis of the polarization expected from those T-dwarfs whose spectra show
absence of condensates. For this, we develop and solve the full radiative
transfer equations for linearly polarized radiation. Only atomic and molecular
Rayleigh scattering are considered to be the source of polarization. We compute
the local polarization at different angular directions in a plane-parallel
atmospheres calculated for the range of effective temperatures of T dwarfs and
then average over the whole surface of the object. The effects of gravity and
limb darkening as well as rotation induced non-sphericity are included. It is
found that the amount of polarization decreases with the increase in effective
temperature. It is also found that significant polarization at any local point
in the atmosphere arises only in the optical (B-band). However, the disk
integrated polarization--even in the B-band--is negligible. Hence we conclude
that, unlike the case for cloudy L dwarfs, polarization of cloudless T-dwarfs
by atomic and molecular scattering may not be detectable.
