Radiative transfer on perturbations in protoplanetary disks

We present a method for calculating the radiative transfer on a protoplanetary disk perturbed by a protoplanet. We apply this method to determine the effect on the temperature structure within the photosphere of a passive circumstellar disk in the vicinity of a small protoplanet of up to 20 Earth masses. The gravitational potential of a protoplanet induces a compression of the disk material near it, resulting in a decrement in the density at the disk's surface. Thus, an isodensity contour at the height of the photosphere takes on the shape of a well. When such a well is illuminated by stellar irradiation at grazing incidence, it results in cooling in a shadowed region and heating in an exposed region. For typical stellar and disk parameters relevant to the epoch of planet formation, we find that the temperature variation due to a protoplanet at 1 AU separation from its parent star is about 4% (5 K) for a planet of 1 Earth mass, about 14% (19 K) for planet of 10 Earth masses, and about 18% (25 K) for planet of 20 Earth masses, We conclude that even such relatively small protoplanets can induce temperature variations in a passive disk. Therefore, many of the processes involved in planet formation should not be modeled with a locally isothermal equation of state.