Coagulation of small grains in disks: the influence of residual infall and initial small-grain content

¹ Sterrenkundig Instituut “Anton Pannekoek”, Kruislaan 403, 1098 SJ Amsterdam, The Netherlands e-mail: dominik@science.uva.nl
² Afdeling Sterrenkunde, Radboud Universiteit Nijmegen, Postbus 9010, 6500 GL Nijmegen, The Netherlands
³ Leader of Junior Research Group “Formation of planetary building blocks” at the Max-Planck-Institut für Astronomie, Königstuhl 17, 69117 Heidelberg, Germany e-mail: dullemon@mpia.de

Received: 16 March 2008
Accepted: 4 August 2008

Abstract

Turbulent coagulation in protoplanetary disks is known to operate on a timescale far shorter than the lifetime of the disk. In the absence of mechanisms that replenish the small dust grain population, protoplanetary disks would rapidly lose their continuum opacity-bearing dust. This is inconsistent with infrared observations of disks around T Tauri stars and Herbig Ae/Be stars, which are usually optically thick at visual wavelengths and show signatures of small (a ≲ 3 μm) grains. A plausible replenishing mechanism of small grains is collisional fragmentation or erosion of large dust aggregates, which model calculations predict to play an important role in protoplanetary disks. If optically thick disks are to be seen as proof for ongoing fragmentation or erosion, then alternative explanations for the existence of optically thick disks must be studied carefully. In this study we explore two scenarios. First, we study the effect of residual, low-level infall of matter onto the disk surface. We find that infall rates as low as 10^-11 /yr can, in principle, replenish the small grain population to a level that keeps the disk marginally optically thick. However, it remains to be seen if the assumption of such an inflow is realistic for star+disk systems at the age of several Myrs, at which winds and jets are expected to have removed any residual envelope. The effectiveness of even a low level infall can be understood by the strongly non-linear behavior of the coagulation equation: a high, fine-grain, dust density at any given time leads to very fast, effective removal of these small grains, while a low fine-grain density lasts for a far longer time. We then consider a second scenario in which, during the buildup phase of the disk, an intermediate fine-grain dust abundance is generated that is sufficiently low to ensure longevity of the state yet sufficiently high for the disk to remain optically thick. While our models confirm that such an “initial condition” can be constructed, we argue that these special initial conditions cannot be achieved during the disk build-up phase. In summary, fragmentation or erosion still appear to be the most promising processes to explain the abundant presence of small grains in old disks.