Example: Planetary Gaps¶

In this example we demonstrate how to add gaps carved by planets into the gas.
There are in principal three ways to achieve this: setting the gas surface density to the desired gap profile and not evolving it, adding torques to the gas evolution, or changing the viscosity.
In this example we focus on adding a torque profile.

The torque profile that is needed to impose a gap on the gas is given by Lau (2024):

$\Large \Lambda = -\frac{3}{2} \nu \Omega_\mathrm{K} \left( \frac{1}{2f} - \frac{\partial \log f}{\partial \log r} - \frac{1}{2} \right)$,

where $\large f$ is the pertubation in the gas surface density

$\Large f = \frac{\Sigma_\mathrm{g} \left( r \right)}{\Sigma_0}$.

Gap Profiles¶

Since we are not self-consistently creating gaps, we have to impose a known gap profile. In this demonstration we are using the numerical fits to hydrodynamical simulations found by Kanagawa et al. (2017):

$\Large \Sigma\mathrm{g} \left( r \right) = \begin{cases} \Sigma_\mathrm{min} & \text{for} \quad \left| r - a_\mathrm{p} \right| < \Delta r_1, \\ \Sigma_\mathrm{gap}\left( r \right) & \text{for} \quad \Delta r_1 < \left| r - a_\mathrm{p} \right| < \Delta r_2, \\ \Sigma_0 & \text{for} \quad \left| r - a_\mathrm{p} \right| > \Delta r_2, \end{cases}$

where $\large a_\mathrm{p}$ is the semi-major axis of the planet and $\large \Sigma_0$ is the unperturbed surface density, and

$\Large \frac{\Sigma_\mathrm{gap}\left(r\right)}{\Sigma_0} = \frac{4}{\sqrt[4]{K'}}\ \frac{\left|r-a_\mathrm{p}\right|}{a_\mathrm{p}}\ -\ 0.32$

$\Large \Sigma_\mathrm{min} = \frac{\Sigma_0}{1\ +\ 0.04\ K}$

$\Large \Delta r_1 = \left( \frac{\Sigma_\mathrm{min}}{4\Sigma_0} + 0.08 \right)\ \sqrt[4]{K'}\ a_\mathrm{p}$

$\Large \Delta r_2 = 0.33\ \sqrt[4]{K'}\ a_\mathrm{p}$

$\Large K = \left( \frac{M_\mathrm{p}}{M_*} \right)^2 \left( \frac{H_\mathrm{p}}{a_\mathrm{p}} \right)^{-5} \alpha^{-1}$

$\Large K' = \left( \frac{M_\mathrm{p}}{M_*} \right)^2 \left( \frac{H_\mathrm{p}}{a_\mathrm{p}} \right)^{-3} \alpha^{-1}$

$\large M_\mathrm{p}$ and $\large M_*$ are the masses of the planet and the star, $\large H_\mathrm{p}$ the pressure scale height at the planet location, and $\large \alpha$ the viscosity parameter.

We therefore need to write a function that calculates the pertubation $\large f = \Sigma_\mathrm{g}\left(r\right)/\Sigma_0$.

A planet of $30$ Earth masses at $10$ AU and the following parameters would induce the following pertubation onto the gas surface density.

Adding planets¶

For this example we want to add two planets: Jupiter and Saturn. We therefore add new groups and subgroups to organize them.

In addition to that we want to refine the grid close to the planet locations. For this we use the function similar to the one discussed in an earlier chapter.

The planets are now set up.

Adding torque profile¶

We now have to write a funtion that returns the $\large \Lambda$ torque profile that imposes the desired gap profile onto the gas surface density from those two planets.

This function now produces the $\large \alpha$ viscosity profile that creates planetary gaps in the gas surface density.

We could now assign this profile to the $\large \Lambda$ torque parameter.

Growing planets¶

Right now we could already start the simulation. But in this demonstration we want to let the planets grow linearly to their final mass. we therefore write a function that returns the planetary mass as a function of time.

We want to start with planet masses of $1$ Earth mass and they should reach their final masses after $100\,000$ years. We have to write functions for each planet that returns their mass as a function of time.

These functions have to be assigned to the updaters of their respective fields.

We also have to oragnize the update structure of the planets.

And we have to tell the main simulation frame to update the planets group. This has to be done before the gas quantities, since Simulation.gas.alpha needs to know the planetary masses. So we simply update it in the beginning.

Since the $\large \Lambda$ torque profile is not constant now, we have assign our $\Lambda$ function to its updater.

To bring the simulation frame into consistency we have to update it.

The planetary masses are now set to their initial values.

We can now start the simulation.

This is the result of our simulation.

As can be seen, the core of Jupiter and Saturn carved gaps in the gas disk, which in turn affected the dust evolution.

To see that the planetary masses actually changed we can plot them over time.