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While doing research for my presentation on the formation of gas giants, more specifically the "core-accretion model", I have been stumbling across the term "grain opacity" and don't quite understand it's meaning. From what I've already read, it's a quantity that heavily influences the time it takes for giant gas planets to form and that a lower grain opactiy leads to shorter formation times. But why does the opacity of dust grains in the protoplanetary disk have such an effect? Is it because of the protoplanet cooling off faster?

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The cooling of a gas giants outer envelope (which in turn cools the convective interior) is dominated by radiative transport.

The radiative flux at the photospheric boundary is $$F_{rad}\propto- \frac{ T^3}{\rho \kappa_R}\nabla T,$$ where $\kappa_R$ is the Rosseland-mean opacity (for more, see e.g. these open lecture notes, by K. Dullemond, in 5.5.3). This is what is often called 'the' opacity in the context of gas giant formation.
The opacity quantifies how opaque a medium is to the transport of photons. High opacity means low transport of photons, hence low cooling.

Now depending on the exact density and temperature conditions of a gas layer, it can be that either atomic, molecular or dust-grain contributions dominate the opacity (see a more detailed account for the collapse of protostars, a related problem, in Vaytet et al., (2013), particularly their fig. 2). As the outer layers of gas giants, which set $F_{rad}$, can be quite cold (T<2000K) it is the strong continuum opacity of dust grains that sets the cooling rate, therefore the contraction rate of the envelope, and hence the mass accretion into the inner layers of the proto-gas giant. Note, that when talking about $F_{rad}$ it is equivalent to talk about the gas giants cooling luminosity $L$, as they are trivially related via $L=4\pi r^2_{\tau=1}F_{rad, \tau=1}$, i.e. the radiative flux at the protogiants photosphere.

The actual self-consistent dust-dependent cooling problem was adressed in Movshovitz et al. (2010), while you are probably aware of the seminal work by Pollack et al., (1996).

For 3-D radiative hydrodynamic simulations of this process see e.g. SPH-based in Ayliffe & Bate (2009), long SPH-based in D'Angelo and Bodenheimer (2013) and grid-based simulations in Schulik et al., (2019).

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