This work is a theoretical exploration facilitating the interpretation of polarimetric observations in terms of cloudiness, rotational velocities, and effective temperatures of brown dwarfs (BDs). An envelope of scatterers like free electrons, atoms/molecules, or haze/clouds affects the Stokes vector of the radiation emitted by oblate bodies. Due to high rotation rates, BDs can be considerably oblate. We present a conics-based radiative transfer scheme for computing the disk-resolved and disk-integrated polarized emission of an oblate BD or extrasolar giant planet bearing homogeneous or patchy clouds. Assuming a uniform gray atmosphere, we theoretically examine the sensitivity of photopolarimetry to the atmosphere's scattering properties, like cloud optical thickness and grain size, concurrently with BD properties, like oblateness, inclination, and effective temperature, which are all treated as free parameters. Additionally, we examine the potential effects of gravitational darkening (GD), revealing that it could significantly amplify disk-integrated polarization. GD imparts a nonlinear inverse temperature dependence to the resulting polarization. Photopolarimetric observations are sensitive to oblateness and inclination. The degree of polarization increases in response to both, making it potentially useful for assessing the spatial orientation of the BD. Under our model assumptions, increasing droplet size in optically thick clouds causes a blueward shift in the near-infrared colors of BDs, which is interesting in light of the observed J – K brightening in the L/T transition. For large cloud grains, polarization decreases sharply, while the transmitted intensity shows a steady increase. BD polarization is thus a potential indicator not only of the presence of clouds but also provides information on cloud grain size
