ner_tags: O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O
tokens: efficiency of sublimation in this context, compared with the incidence of “dry rain” in the Earth’s atmosphere, can be ascribed to a combination of two factors: the relatively high silicate vapor mass fraction (typically X sil ∼ (0.3–1) X H when refluxing of pebble debris from the envelope begins to balance pebble accretion) and an infall time that is a factor ∼10 3 times longer than for raindrops. Of course, much larger planetesimals that survive transport through the envelope may still inject solid/liquid silicates into the core. 3. Envelope Expansion in Response to Planetesimal Heating and Dust Loading: Semianalytic Treatment As a prelude to the numerical investigation described in Sections 4 and 5 , we now present a semianalytic model of an envelope with a low dust density and outer radiative structure. This model connects smoothly to a uniform medium of a fixed temperature T disk and gas density ρ g ,disk . As the dust abundance rises, the inner convective layer grows in size, and eventually overwhelms the Bondi radius. Assuming that the dust abundance in the outer radiative layer is tied directly to the solid accretion rate, we can work out the critical value of above which convection extends beyond the Bondi radius, some refluxing of solids will occur, and the self-consistency of the model breaks down. We next quantify the reduction in dust size due to convective stirring, and demonstrate how a convective layer that traps accreted solids may expand in response to deep heating by large planetesimals. 3.1. Two-layer Convective-radiative Envelope Even a small amount of embedded dust acts as a powerful coolant, in the sense that any excess of gas temperature over ambient radiation temperature is rapidly converted into thermal radiation via collisions between gas molecules and grains (e.g., Chiang Goldreich 1997 ). The presence of an ambient blackbody radiation field of temperature T disk is therefore assumed, with an internal energy amounting to a small fraction of the ambient gas energy density. The temperature is buffered near T disk in the outer bound envelope, around the Bondi radius, and grows in the inner, optically thick envelope. In a state of spherical hydrostatic and radiative equilibrium at local temperature T g and uniform luminosity ,The opacity is per unit mass of dust. We first consider the case of uniform dust-mass fraction X d (although the equations presented here do not depend on this assumption). We rescale radius to R B , temperature to T disk , and gas density to ρ g ,disk . Then the preceding equations can be writtenwhereand the luminosity is expressed in terms of a core-accretion time,Taking cm 2 g −1 (see Appendix A ), the characteristic optical depth at the Bondi radius isand The solution to Equation ( 25 ) is shown in Figure 2 for β = 0 and a range of values of the parameter ε rad (10 −4 , 10 −3.5 , 10 −3 , ... 10 0.5 , 10). When ε rad ≪ 1, the temperature remains flat some distance inside R B , whereas the