# What is the meaning of grain opacity and why does it affect the formation time of gas giants?

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?

The collapse of a body of primordial nebular gas into a protoplanet can be viewed as a contest between gravitation and the so-called supporting agents. These supporting agents include turbulence, rotation, thermal pressure, etc. They are the forces that oppose gravitational collapse.

Your question is related to the latter. In the same manner as in a star, thermal pressure exerts a force that opposes the contraction driven by the self-gravitational force of the gas envelope. Most generally, thermal pressure is proportional to the kinetic temperature of the gas $$T_\mathrm{kin}$$:

$$\begin{equation} P_\mathrm{thermal} = n_\mathrm{H} k_\mathrm{B} T_\mathrm{kin} \end{equation}$$

Where $$n_\mathrm{H}$$ is the hydrogen density and $$k_\mathrm{B}$$ the Boltzmann constant. Hence, the thermal pressure can be reduced as a result of a temperature decrease (i.e. cooling), which brings me to the main point : the radiative cooling efficiency depends on the opacity.

Radiative cooling is the process by which a warm gas emits photons that carry away the thermal energy. The dust opacity $$\kappa_\mathrm{gr}$$ is an absorption cross section per unit mass of dust: it describes the ability of dust grains to absorb incoming radiation. It is typically of order $$0.01-1$$ cm$$^2$$ g$$^{-1}$$. A high dust opacity means that the photons emitted by the warm envelope are more likely to be absorbed by dust grains before they escape. In other words, the opacity governs the thermal balance of the envelope of gas, because it controls the rate at which thermal energy can be removed.

Finally, you can see how the opacity is related to the formation time of a gas giant. Since the thermal pressure supports the envelope against gravity, it is required to remove the thermal energy to allow the collapse to continue. And we have just seen that the rate at which this energy is transported through radiative cooling is controlled by the opacity $$\kappa_\mathrm{gr}$$.

If you want to learn more, I would suggest to read the abstract and introduction of the following paper: Ikoma et al. (2000). Using a numerical approach to model the accretion rate of giant planets, they show that the growth time $$\tau_\mathrm{g}$$ is given by:

$$\begin{equation} \tau_\mathrm{g} = 10^8 \left( \frac{M_\mathrm{core}}{M_\mathrm{earth}} \right)^{-2.5} \left( \frac{\kappa_\mathrm{gr}}{1~\mathrm{cm^2~g^{-1}}} \right)~\mathrm{yr} \end{equation}$$

Where $$M_\mathrm{core}$$ is the core mass.