Add in error propagation from finite differencing lower energy cutoff in Lockwood formulas

[iMurfyD]

What this is and why it's needed:

The derivative of the lower energy cutoff is used to estimate the reconnection rate. However, due to the sparsity of data the derivative itself can be quite noisy. This change aims to quantify the error present in this derivative and propagate that error into the Lockwood formulas.

This hasn't been tested yet, so please do not merge.

Formulation

For a function $f(x)$, its the finite difference approximate of the derivative can be expanded as

$$\frac{f(x+h)-f(x)}{h} = \frac{1}{h}\left(f(x)+f'(x)h+ \frac{1}{2} f''(x)h^2 + ... - f(x) \right),$$

where $h$ is the step size of the grid. This can be rearranged to find the error in the derivative. This error is given by

$$\frac{f(x+h)-f(x)}{h} - f'(x) = \frac{h}{2} f''(x) + \mathcal{O}\left( h^{2}\right),$$

where $\mathcal{O}\left( h^{2}\right)$ are higher order terms in the Taylor expansion of $f(x+h)$. By numerically calculating the second derivative using finite difference, we can estimate the magnitude of $\frac{h}{2} f''(x)$.

[iMurfyD]

@ddasilva Is there an example driver for lib_lockwood1992.py to test and plot the error? I didn't see one in the repo.

[ddasilva]

@ddasilva Is there an example driver for lib_lockwood1992.py to test and plot the error? I didn't see one in the repo.

Thanks for this! Here's a self-contained directory with a notebook where you can play with plotting the reconnection rate for several events:
https://drive.google.com/file/d/1zUW7FBECpoT1Q2pZU6YQpfzV2e9dCZf6/view?usp=sharing