Monte Carlo Radiation for Extrasolar Planet atmospheres
We would like to determine the limits of reflectivity of extrasolar giant planets (EGP) in order to determine the best observation practices attempting to detect these EGP. To determine the reflectivity, we will model EGP atmospheres with Monte Carlo radiation transfer simulations.
Impetus
In specifics, we would like to test a fractal cloud model against the one-dimensional models used by Sudarsky et al to model extrasolar giant planet atmospheres. I will model my work on their paper from 2005, Phase Functions and Light curves of Wide-separation Extrasolar Giant Planets . I am also using a paper from the American Meteorological Society by Christopher Kuchinke, The Angular Distribution of UV-B Sky Radiance under Cloudy Conditions: A Comparison of Measurements and Radiative Transfer Calculations Using a Fractal Cloud Model. He compares in his paper a multifractal and a plane-parallel homogeneous (PPH) cloud field with transmission data from Australia, and finds that the multifractal model is superior to the PPH. Thus, I would like to try to make this comparison between a fractal cloud model and a one dimensional cloud model for extrasolar planet atmospheres.
Monte Carlo Code
The code which I use, developed by Kenny Wood and Barb Whitney and Jon Bjorkman is made up of individual cells in which there is a measure of mass. Photons are then injected into the model, and react depending on which cells they enter. The photons continue to fly through the cells until they hit some mass (some cell which is not empty) and are either scattered in some direction or absorbed. We assign albedos to the mass in order to determine if the photon is absorbed or scattered. If the photon is scattered, then the photon’s direction is determined by a scattering phase function. Currently this phase function is a Henyey-Greenstein phase function. In the spherical grid, we have also implemented a subroutine to input another phase function, which can be calculated with Mie scattering routines. This could be tidied up by incorporating the Mie scattering subroutines into the program.
The mass is set up in a static density distribution based on a fractal component and a smooth component. For instance, if we have a smooth component of 0.1, then 10% of the mass is smooth, and the remaining 90% of the mass is contained in the fractal structure. It is static in that for each model it does not change based on the atmosphere heating up or any other force. Once the mass distribution is determined at the beginning of the simulation, it is set for the duration. This means that for the models to be valid, we need realistic density models to give the simulation. The fractal model has several parameters which determine the distribution of the fractal mass, weather it is broken up into somewhat discreet clouds or a largely homogenous cloud (we vary the first fractal seed between 8 [where the cloud is very fractious] and 128, [when there are so many seeds (and the fractal mass distributed well enough) that the model acts a little like a smooth model]. The smooth component of the mass is distributed equally into all the cells, unless another model (like an exponential model) is assigned to the distribution.
An improvement which has been made to the spherical grid but not to the plane parallel grid is implementing multiple species of dust. As stated before, the mass needs an assigned albedo and phase function to determine the action of the photon that hits it. By implementing multiple species of dust, we can assign different albedos and different phase functions to more accurate represent the different types of dust that would be found in a planetary atmosphere.
Spherical Code
The spherical code was what we first used to attempt to model planets. It has a rho, theta and phi components and we had set the values of each to 100. So, there were 1 x 10^6 cells to represent the planet. We were able to use this code to generate phase functions from our simulations, and tested cloudy models. However, early in this work we decided that we should switch to the plane parallel model to increase our resolution.
Benefits of spherical code:
•More natural representation of spherical planet
•Using one grid for the entire planet, so will represent global phenomenon (red spot on Jupiter)
•Automatically calculates flux for the entire planet
•Multispecies model implemented
•Mie scattering is partially implemented (phase function file can be input)
Detriments of Spherical grid
•Very coarse resolution, because it represents the entire planet, and much more computationally expensive to have fine detail
•Written to model stellar clouds, so it doesn’t take advantage of the fact that we are only interested in a narrow envelope of atmosphere around a much larger planet
Plane Parallel Code
To get around some of the drawbacks of using the spherical code, we started using the plane-parallel code to model small sections on the surface of the planet. Then, we can use this model to test the reflection of the atmosphere at multiple incident angles and emergent angles. Using geometry, we can tile these small sections onto the surface of a sphere and determine the incident and emergent radiation at each longitude and latitude. Then, we can integrate the total reflectivity of the planetary surface based on these tiles and and compute a phase angle. Right now the code uses 8 x 10^6 cells, which are tiled ~64000 times onto the surface of the planet. Obviously, this allows us much finer detail than the spherical code.
Benefits of PP Code
•Much finer detail than the spherical code, can more accurately model real clouds. Can increase the resolution relatively easily with finer numerical integration
•Only uses computational time on the area we are interested in, the atmosphere
Detriments of PP code
•More difficult to implement global features, such as enormous storm systems (Could be done by writing a file which maps storms, etc. to specific longitudes and latitudes)
•Multiple species have yet to be implemented
•Mie scattering yet to be implemented
Some preliminary results
Native results from the PP code
The PP code outputs flux data between zero and 180 degrees. The different phi components (20) are represented with different lines on the graph. In the following movies, different fractal models are represented all on one graph in different colors. The red lines are the smooth model, and the purple lines are the most fractured models (n1 of the fractal =8). The rest of the lines throughout the graph are the intermediate fractal models (n1 = 16,32,64, etc.). Finally, the movie starts with an incident angle of radiation of 5° (measured from parallel) and progresses through to an incident radiation of 90° (light coming from directly overhead). The movies all have a smooth component of 10^-2 (so only 0.01 of the mass is uniformly distributed throughout the grid, and most of the mass is represented by the fractal component) but their tau values differ by factors of ten.
Here are some graphs of the same information but presented three-dimensionally. These are the values at each incident angle (the title of the graph), and the emergent angle (mu, measured from the x coordinate) and the azimuthal angle between the two (phi, measured from the y coordinate) for specific Tau values and fractal models. This is similar to figure 2 in the Sudarsky paper.
fractal n1=8 fractal n1=16 fractal n1=32 fractal n1=128
Cloud coverage
We have run some of our fractal models and compared the percentage of cloud coverage that results. The pictures shown are the column integrations of the grid. It is as if we are looking down on the grid from above. The black spots are where there is lowest mass
n1=128 results in 100% cloud coverage (clouds at every x and y coordinate, integrated over all z) and 56.72% volume filled
n1=64 results in 99.98% cloud coverage and 34.41% volume filled
n1=32 results in 92.23% cloud coverage and 19.13% volume filled
n1=16 results in 69.77% cloud coverage and 10.20% volume filled
n1=8 results in 43.35% cloud coverage and 5.20% volume filled
Tiled results showing entire phase function
The final step was to consolidate the above results into a phase function, since I had coarsely covered all incident angles and the code had output the emergent angles. I used the same functions from Sobelev that are mentioned in the Sudarsky paper.
The phase functions are created by tiling our plane parallel fractal code on a sphere and calculating the phase angle. This is similar to Figure 3 in Sudarsky’s paper. These models are all for fsmooth = 0.1, which means that there is a significant smooth component (10% of the mass is smooth, and 90% fractal). These are the figures for different Tau values: