Math - Galactic Rotation

This calculation models the rotational speed 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 and the rotational speed of different parts of the Galaxy as a function of radius from the Galactic Center.   It uses the Archimedean Spiral to model galactic arms, assumes a Supermassive Black Hole at the Galaxy center, and a Dark Matter halo encompassing the galaxy.  The Dark Matter halo is assumed to be dispersed in the shape of sphere. 

Now lets calculate the Centrifugal Force - due to an object rotating in circular motion about a center point. 

Use Desmos on-line tool to do the calculation. 

This results in rotational speed as a function of radius from the center.  See diagram below.    
    Red - Archimedean Spiral arms of the Galaxy. 
    Orange - Force exerted towards the Galactic Center due to spiral arms, black hole, and dark matter
    Blue - Rotational Velocity

Compare the diagram above with observed velocities of existing spiral galaxies.  See below.