Goal: Fit order of fractional Laplacian in fractional Poisson equation for gravity to experimental data on orbital velocity versus radius for different galaxies.
Assume the order of the fractional Laplacian, s(r) takes the form $s = \frac{x_0 + x_1 r}{x_2 + r} + x_3 $. We want to find $\vec{x} = [x_0, x_1, x_2, x_3]$.
We use the relation $v(s, r) = \sqrt{r \frac{d\Phi}{dr}}$ where r is radius, v is orbital velocity, and K is gravitational potential. $\Phi$ is given by $G_N M \frac{ \Gamma \left(\frac{n}{2} - s(r) \right)}{4^{s (r)-1} \pi^{\frac{1}{2}} \Gamma \left(s (r) \right)} \frac{l^{2-2 s(r)}}{r^{n-2 s (r)}} $.
We want to choose $s(r)$ such that the relationship between radius and orbital velocity is as close as possible to what we observe in experimental data.
We measure how far $s(r)$ is from capturing the true relationship for a set of radii and orbital velocities as $\epsilon = || \vec{v} (s, \vec{r}) - \vec{v}_{data} ||_2$. The $\vec{x}$ that minimizes $\epsilon$ are that for which $\frac{d \epsilon}{d \vec{x}} = \vec{0}$. To find this $\vec{x}$, we use Newton's Method where each Newton step is given as
$$\vec{x}^{k+1} = \vec{x}^{k}- \left(\nabla^2 \epsilon(\vec{x}^{k})\right)^{-1} \nabla(\epsilon(\vec{x}^{k})).$$
solve_for_s.py : Find $\vec{x}$ for each galaxy by running Newton's method described above on extracted orbital velocity vs radius data data.
extract_galaxies.py : Extract orbital velocity vs radius data from .dat files downloaded from online database.
examine_results.py : Analyze distribution of $\vec{x}$ values for different galaxies.
grid_search_util.py: Use grid search to choose an initial guess for $\vec{x}$.
newton_util.py: Use Newton's method to fit $\vec{x}$ to data using tensorflow's automatic differentiation tools.
NGC_galaxies_dict : Dictionary storing extracted orbital velocity vs radius data for NGC galaxies.
