Introduction

FRECKLL (Full and Reduced Exoplanet Chemical Kinetics DistiLLed) is a Python Chemical Kinetic Solver for Exoplanetary atompsheres. The code attempts to solve the equation:

\[ \frac{\partial n_i}{\partial t} = P_i -L_i - \frac{\partial \phi_i}{\partial z} \]

\[ \phi_i = -n_i D_i \left( \frac{1}{n_i}\frac{\partial n_i}{\partial z} + \frac{1}{H_i} + \frac{1}{T}\frac{\partial T}{\partial z} \right) \]

where \(n_i\), \(P_i\) and \(L_i\) are the number density, production rate and loss rate of species \(i\), \(z\) is the vertical coordinate of the atmosphere, and \(\phi_i\) is the vertical flux for species \(i\) which has the form of a diffusion equation given below

\[ \phi_i = -n_i D_i \left( \frac{1}{n_i}\frac{\partial n_i}{\partial z} + \frac{1}{H_i} + \frac{1}{T}\frac{\partial T}{\partial z}\right) - n_i K_{zz}\left(\frac{1}{y_i}\frac{\partial y_i}{\partial z}\right) \]

Here, \(D_i\) is the molecular diffusion coefficient, \(H_i\) is the scale height and \(y_i\) the mixing ratio for species \(i\), T is the temperature (K) and \(K_{zz}\) is the eddy diffusion coefficient.

Distillation

So what is FRECKLL doing differently? Its best to explain which an example. What is the output of this code?

sum([1e16, 1, -1e16])

This should be easy enough. Let see what python says:

>>> sum([1e16, 1, -1e16])
0.0

Wait a minute... this should be one. what is going on? Well this example hits the limit of floating point numbers. First 1e16 + 1 since the magnitude differece is so large, the 1 is lost. Then 1e16 - 1e16 is zero. So the result is zero. This is called catastrophic cancellation. For our case, reaction rates often have these type of magnitude differences. Not handling this properly can lead to discontinuous function which is not good for numerical solvers.

This is where FRECKLL comes in. It uses a K-sum, distillation method to avoid this problem:

>>> from freckll.distill import ksum
>>> data = np.array([1e16, 1, -1e16])
>>> ksum(data)
np.float64(1.0)

Smooth Jacobians! Perfect for solving.