Utilitarian Opacity Model for Aggregate Particles in Protoplanetary Nebulae and Exoplanet Atmospheres

Abstract

As small solid grains grow into larger ones in protoplanetary nebulae, or in the cloudy atmospheres of exoplanets, they generally form porous aggregates rather than solid spheres. A number of previous studies have used highly sophisticated schemes to calculate opacity models for irregular, porous particles with size much smaller than a wavelength. However, mere growth itself can affect the opacity of the medium in far more significant ways than the detailed compositional and/or structural differences between grain constituents once aggregate particle sizes exceed the relevant wavelengths. This physics is not new; our goal here is to provide a model that provides physical insight and is simple to use in the increasing number of protoplanetary nebula evolution, and exoplanet atmosphere, models appearing in recent years, yet quantitatively captures the main radiative properties of mixtures of particles of arbitrary size, porosity, and composition. The model is a simple combination of effective medium theory with small-particle closed-form expressions, combined with suitably chosen transitions to geometric optics behavior. Calculations of wavelength-dependent emission and Rosseland mean opacity are shown and compared with Mie theory. The model's fidelity is very good in all comparisons we have made, except in cases involving pure metal particles or monochromatic opacities for solid particles with size comparable to the wavelength.

Grain opacity and the bulk composition of extrasolar planets

“…This is confirmed by Fig. 10 in Cuzzi et al (2014): from the figure it becomes clear that the wavelength-independent geometrical limit for Q (and therefore our approximation) is applicable for grains of 10 µm if the temperature is larger than 300 K, and for grains of 100 µm (or larger) if T > ∼ 50 K. These conditions are usually met in protoplanetary atmospheres except for the advection regime where the grains remain small in the outer layers of the atmosphere. Fortunately, this regime occurs only when the planet is already at or close to the critical core mass, so that the impact on the total formation history of a planet remains small (Sect.…”

“…2.7. We found that the results are again very similar because the grains are so large and the temperature sufficiently high that we are in the wavelength-independent regime (Cuzzi et al 2014). …”

“…7 During the preparation of this paper, we became aware of the work of Cuzzi et al (2014). In future work it will be possible to use the results In all three cases, λ max corresponds to the wavelength of the maximum in the blackbody spectrum.…”

“…2.5.5) and of convection are also included in some simulations. Once the grain size distribution is calculated, the wavelength-dependent opacity is computed using an approximation to Mie theory (Cuzzi et al 2014). Finally, the Rosseland mean is calculated using the Planck function at the local temperature of the gas.…”

Dust Coagulation with Porosity Evolution

“…We have also derived a semianalytical formula and confirm the fact that the absorption opacity is characterized by af except for the compact grains at the wavelengths of D 2 a (see Kataoka et al 2014). We also note that Cuzzi et al (2014) obtained similar conclusions on the opacity of porous dust aggregates.…”