Math - Gravity Galactic Arms & Black Hole

This calculation models the gravitational attraction in-between arms of a Grand Spiral Galaxy by using the Archimedian Spiral.  It uses several simplifications to calculate the gravitational force of a galaxy on a point mass that lies slightly above the plane of the disk. This is a continuation of the model on the previous page.  Here we use the Desmos calculator to do a summation of the gravitational force in-between galactic arms.  I won't repeat all the set up calculations here.  They are on the original page modeling a Spiral Galaxy by using the Archemidean Spiral.  

Calculate the force experienced by the point mass at a position along different points of the X-axis. 
Not an easy integral to calculate so use Desmos.   N is the number of revolutions of the spiral.  

The diagram below shows the force experienced by the point mass in the X-direction.
Z-position is slightly above the Galactic plane.  The Archimedean spiral in this model has the same density function as used in the calculation on the previous web page. 

Notice the ripples of gravitational attraction in-between spiral arms.  

Add in a term for a Supermassive Black Hole at the center of the Spiral Galaxy.  Below shows the force exerted on a point mass along the X-axis, but slightly above the plane of the Spiral Galaxy with a Supermassive central Black Hole.  Now there are ripples of gravitational attraction between spiral arms, but the overall force of attraction always points to the galaxies center.