Calculate the posterior distribution of the "minimum atmospheric height" (MAH) of an exoplanet by inputting the joint posterior distribution of the mass and radius.

README file by David Kipping, Columbia University, Dept. of Astronomy (d.kipping at columbia.edu). Please feel free to contact me regarding questions on MAH or any bugs you may spot.

1. What is MAH?
2. Compiling & Executing
3. How to use with your data
4. Outputs
5. Caveats

Fortran 90 code for calculating the Minimum Atmospheric Height (MAH) of
an exoplanet. This calculation only requires knowledge of the planet's mass and radius.

For an inputted array of planetary masses and planetary radii, MAH returns R_MAH = the minimum atmospheric height. Please see Kipping, Spiegel & Sasselov (2013) for details on how this calculation is made and the applications. If you use this code or the equations within, please cite Kipping, Spiegel & Sasselov (2013) and Zeng & Sasselov (2013).

This tarball contains two Fortran 90 files: {1} MAH.f90 = the module which computes R_MAH {2} MAH_call.f90 = example calling code.

The code may be compiled together using... g95 -O3 -c MAH.f90 && g95 -O3 -o MAH MAH_call.f90 MAH.o

and the code can be executed using... ./MAH

or, to dump the screen output into a summary file... ./MAH > summary.txt

There is an example set of data included, which comes from Anglada-Escude et al. 2013: http://adsabs.harvard.edu/abs/2013A%26A...551A..48A) for the planet GJ 1214b, to whom we are grateful for sharing their data. Needless to say if you use this data please cite Anglada-Escude et al. 2013. The current example file should run in ~2 mins on a typical single core.

Consider that you have fitted some exoplanet observations with a Monte Carlo technique, such as MCMC, and have a derived joint posterior distribution of the planet's mass (MP) and the planet's radius (RP). To use MAH directly without any modification to code itself, simply export a ASCII file called "example_data.dat" into the same directory as the MAH code containing two columns (MP and RP, in Earth units, respectively) and 10^5 rows. If you want to use a different number of rows, simply change "n" in MAH_call.f90 from 1D5 to whatever you want. Then, simply compile and execute MAH to complete a run.

Advanced users may want to import higher dimensional posteriors directly into MAH or simply call the MAH.f90 module themselves without the wrapper provided here and of course you are welcome to do so!

MAH produces two types of output. Firstly, a set of summary statistics are dumped onto the screen at the end of the calculation. These statistics follow the same terminology used in our paper. The numbers are outputted in LaTeX ready format to copy and paste into your papers. However, depending upon the number of significant figures and decimal places desired, you may need to change the formatting rules defined within the MAH_call.f90 code.

The second type of output from MAH are two ASCII files exported to the same directory in which the code is executed. The file names of these two are:

100_H2O.dat = Output assuming a 100%-water planet for the RH20 75_H2O_25_MgSiO3.dat = Output assuming a 75%-water-25%-silicate planet for RH20

The difference between the two files is typically quite small, but I recommend using the 75%-water-25%-silicate model, in general.

Each file has three columns and the same number of rows as in the original inputted array. The i'th row of the output files corresponds to the i'th row of the input file. The columns are:

{1} R_H20 = Radius of the planet, in Earth radii, assuming a water composition {2} R_MAH = Minimum atmospheric height of the planet, in Earth radii {3} valid = Logical flag (T or F) dictating whether this row is trustable or not

The first two columns are self-explanatory to anyone who has read our paper. The last column is less obvious and essentially checks whether the inputted planetary mass falls within the regime covered by the Zeng & Sasselov (2013) models. If the inputted mass is extreemly small or large, the models become unreliable and so we flag such cases in this column. All summary statistics calculated in MAH_call.f90 know about this and exclude such rows in their calculations.

A typical use of these outputs would be to create a histogram of the second column of the file 75_H2O_25_HgSiO3.dat, which would represent the posterior distribution of the minimum atmospheric height of the planet (in units of Earth radii). One could also divide this column by the inputted RP to get (R_MAH/R_P) i.e. a relative measure of R_MAH.

• It is very important that the inputted masses and radii are in units of Earth masses and Earth radii respectively.

• It is also important that users understand that the following statement is false: Probability of planet being atmosphere-less = 1 - P(RMAH>0)

• In fact if your posterior distribution of RMAH peaks to negative values, then the MAH method says nothing about whether the planet does or does not have an atmosphere. Therefore, please do not interpret such a result as indicating a rocky planet because this is fundamentally wrong! However, if P(RMAH>0) is high, then you may interpret this as indicating that the planet has an extended atmosphere.