Modeling Collisional Cascades in Debris Disks: Steep Dust-Size Distributions

Abstract

We explore the evolution of the mass distribution of dust in collision-dominated debris disks, using the collisional code introduced in our previous paper. We analyze the equilibrium distribution and its dependence on model parameters by evolving over 100 models to 10 Gyr. With our numerical models, we confirm that systems reach collisional equilibrium with a mass distribution that is steeper than the traditional solution by Dohnanyi (1969). Our model yields a quasi steady-state slope of n(m) ∼ m −1.88 [n(a) ∼ a −3.65 ] as a robust solution for a wide range of possible model parameters. We also show that a simple power-law function can be an appropriate approximation for the mass distribution of particles in certain regimes. The steeper solution has observable effects in the submillimeter and millimeter wavelength regimes of the electromagnetic spectrum. We assemble data for nine debris disks that have been observed at these wavelengths and, using a simplified absorption efficiency model, show that the predicted slope of the particle mass distribution generates SEDs that are in agreement with the observed ones.

Interpreting the extended emission around three nearby debris disc host stars

“…We obtain a minimum grain size, s min = 2.15 µm, a few times larger than the blow-out radius (0.5 µm). The particle size distribution exponent, γ = 3.9, is comparable to that of a system in steady state collisional cascade (3.5 to 3.7, Dohnanyi 1969;Thébault & Augereau 2007;Gáspár et al 2012) and within the range of expected values for typical debris discs (3 to 4) (see e.g. Pan & Schlichting 2012).…”

HD 104860 and HD 192758: Two Debris Disks Newly Imaged in Scattered Light with the Hubble Space Telescope

“…Without a prior on the dust separation to the star and assuming that the disks are in radiative and collisional equilibrium, these blackbody emissions correspond, respectively, to a ∼10 −5 M Moon disk at a radius of 3au from the star, and to a massive ∼1.5 M Moon cold disk at a radius of 366au, based on silicate spherical grain compositions. By measuring the cold disk spectral index in the millimeter from VLA and ATCA observations, MacGregor et al (2016b) estimated a dust size distribution in the system with a power-law q=3.64±0.15, consistent with steadystate collisional cascade models (Dohnanyi 1969;Gáspár et al 2012;Pan & Schlichting 2012). Assuming a disk composed of astro-silicates (Draine 2003), a stellar luminosity of L å =1.16 L e , and mass of M å =1.04 M e , the radiative pressure blowout grain size limit is estimated as a blow ∼0.4 μm, but the minimum grain size is inferred to be ∼7 μm from joint modeling of the system's SED and Herschel images (Pawellek et al 2014).…”

Alma Observations of the Debris Disk of Solar Analog Τ Ceti

“…Recent SCUBA-2 observations at 850 µm yield a total flux density of 4.5 ± 0.9 mJy, including a contribution from the star of ∼ 1 mJy (Holland et al, in prep.). An extrapolation of this measurement using the typical spectral index of 2.58 for debris disks at (sub)millimeter wavelengths (Gáspár et al 2012), yields an expected flux density of the disk at 1.3 mm of 1.2 ± 0.2 mJy. This more robust single-dish flux measurement allows us to constrain the total flux density of our models with a Gaussian prior, 0.6 mJy ≤ F belt ≤ 1.6 mJy, accounting for uncertainty in both the single-dish 850 µm flux measurement and the extrapolation to 1.3 mm.…”

“…To first order, the dust temperature also depends on the grain opacity, T ∝ r −2/(4+β) , where β is the power law index of the grain opacity as a function of frequency, κ ν ∝ ν β . Gáspár et al (2012) measure β = 0.58, from observations of debris disks, which implies a temperature power law index of ∼ −0.44. Thus, the expected change in the temperature profile due to β is much smaller than the uncertainty in our resulting model fits and we choose to ignore this effect.…”