Self-similarity and scaling behaviour of infrared emission from radiatively heated dust - I. Theory

Dust infrared emission possesses scaling properties that yield powerful results with far-reaching observational consequences. Scaling was first noticed by Rowan-Robinson for spherical shells and is shown here to be a general property of dust emission in arbitrary geometries. Overall luminosity is never an input parameter of the radiative transfer problem; spectral shape is the only relevant property of the heating radiation when the inner boundary of the dusty region is controlled by dust sublimation. Similarly, the absolute scales of densities and distances are irrelevant; the geometry enters only through angles, relative thicknesses and aspect ratios, and the actual magnitudes of densities and distances enter only through one independent parameter, the overall optical depth. That is, as long as the overall optical depth stays the same, the system dimensions can be scaled up or down by an arbitrary factor without any effect on the radiative transfer problem. Dust properties enter only through dimensionless, normalized distributions that describe the spatial variation of density and the wavelength dependence of scattering and absorption efficiencies. Scaling enables a systematic approach to modelling and classification of IR spectra. We develop a new, fully scale-free method for solving radiative transfer, present exact numerical results, and derive approximate analytical solutions for spherical geometry, covering the entire range of parameter space relevant to observations. For a given type of grains, the spectral energy distribution (SED) is primarily controlled by the profile of the spatial dust distribution and the optical depth - each density profile produces a family of solutions, with position within the family determined by optical depth. From the model SEDs presented here, the density distribution and optical depth can be observationally determined for various sources. Scaling implies tight correlations among the SEDs of various members of the same class of sources such as young stellar objects, late-type stars, etc. In particular, all members of the same class occupy common, well-defined regions in colour-colour diagrams. The observational data corroborate the existence of these correlations.