Math - Gravity Dark Matter Halo

This calculation models the gravitational attraction for a  Grand Spiral Galaxy using the Archimedean 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. 

On this page we model the Spiral Galaxy, its galactic arms, Black Hole at the center, and a  Dark Matter halo.  The Dark Matter halo is assumed to be dispersed in the shape of sphere around the galaxy.  

Navarro-Frenk-White (NFW) have created a density profile for Dark Matter Halos.  

Use that to model the force exerted on the point mass in the X direction.  

Find the total mass of Dark Matter contained within a sphere of radius Rmax. 

 
As the position of the point mass changes along the X axis, the amount of Dark Matter contained within a sphere of radius Rmax changes.  Thus the mass of Dark Matter contained within the sphere increases and the force exerted by Dark Matter on the point mass changes as the distance of the point mass from the galactic center changes.  That is... the force exerted by Dark Matter on the point mass is dynamic and depends upon the point mass's position. 

Calculate the force exerted by the Dark Matter on the point mass.  Apply Newton's shell theorem to the mass of Dark Matter to calculate the force.  Add that force to forces calculated on previous pages for Galactic Arms and Supermassive Black Hole.

Graphic below shows the total force exerted on the point mass by the Dark Matter halo, Supermassive Black Hole, and Galactic arms (orange).   Compare to previous force without Dark Matter halo (green).