From: goldenfieldquaternions@gmail.com   
      
   On Wednesday, May 2, 2018 at 6:50:09 AM UTC-5, Keith Stein wrote:   
   > I suspect there could be a simple relationship of the form:   
   >   
   >      ( constant orbital speed )^2 = K * mass of galaxy   
   >   
   > I arrive at this by assuming that for spiral galaxies there is, in   
   > addition to the usual Newtonian inverse square gravitation, an   
   > additional inverse linear relationship.   
   >   
   > At normal distances the linear term is clearly very small and can be   
   > ignored, but obviously at very large distances the linear term will   
   > dominate, and at sufficiently large distances the inverse square term   
   > can be ignored.   
   >   
   > At large distance from the galactic center (R) we may therefore write:   
   >   
   >         Centripetal Force = m * v^2 / R = K * M * m / R   
   >   
   > Which gives:                   v^2     = K * M   
   >   
   > Note v is independent of R, so this would nicely explain the   
   > flat orbital velocity curves of spiral galaxies.   
   >   
   > keith stein   
      
   To understand the rotation of galaxies and its connection to dark   
   matter we need to look at Newton's second law of motion with   
   gravitational   
      
   mdp/dt = F = -GMm/r^2   
      
   for a mass m around a larger mass M. For a circular motion the force   
   is F = -mv^2/r = -mw^2r, for w a stand in for omega the angular   
   frequency. We then derive easily the relationship between the angular   
   velocity and the radius of the orbit   
      
   w^2 = GM/r^3.   
      
   This is Kepler's third law.   
      
   Now consider the case where the satellite is in a region with a   
   density D of matter. The mass M above is then M = 4piDr^3/3. We now   
   use Gauss' law that the force when integrated over the surface   
   surrounding a region with radius R the mass is then int F*dS.. We   
   assume complete symmetry and simplify this with letting   
      
      
   int F*dS = 2piR^2F = 8pi^2GDR.   
      
   This means the force is proportional to the radius of the orbit R   
   and the angular velocity is constant and not dependent on the radius.   
   The motion is the same for a harmonic oscillator.   
      
   A galaxy has this blob or halo of dark matter it is embedded in.   
   Dark matter is about 75-80% of matter in the galaxy. The motion of   
   a star is then due to this dark matter which would cause galactic   
   rotation to be similar to the rotation of a disk. The occurrence   
   of stars that are more concentrated towards the center will give   
   more of a Kepler law of motion result. The actual dynamics is then   
   a summation of the two. The Keplerian dynamics does result in a   
   small decrease in rotation with radius, but not as pronounced as   
   would happen without dark matter.   
      
   LC   
      
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