Math - Gravity Galaxy

This calculation attempts to model the gravitational attraction 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 above the plane of the disk.  

Set up the basic equations for the Archimedean Spiral. 

Calculate the Arclength along the Spiral.  

Find the Force Vector R which points from the point mass to a point on the Spiral.  

Find a Unit Vector from the point mass to an arbitrary point on the spiral.  

Calculate the infinitesimal force on the point mass.  

Pick a formula for the mass density of the spiral as a function of radius from the center (theta) and which also adheres reasonably well to reality and also makes calculations easier :)   

Two graphics below show that our function p(theta) is not too far off from reality.  

The density function includes the mass of visible stars, white dwarfs, neutron stars, galactic dust, black holes, interstellar hydrogen, gas, molecules, stellar remnants, and Dark Matter.  

Plug in our mass density function p(theta) and simplify. 

Now compute the total force on the point mass from the Galactic Spiral. 

Pause here to do a calculation on the total mass of the Spiral M 

OK, now replace Theta with M and we get the formula below.   

With more revolutions and with the spiral more spread out, the attractive force weakens. 

Let's expand the denominator as a series.   

Take just the first term ... which is a gross simplification... but lets do it for fun!  

After many simplifications, this reduces to Newton's equation for gravitational attraction.