From: goldenfieldquaternions@gmail.com   
      
   On Sunday, May 13, 2018 at 5:23:24 AM UTC-5, Keith Stein wrote:   
   > On 05/05/2018 23:57, Lawrence Crowell wrote:   
   > > 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.   
   >   
   > This expression clearly assumes the density of dark matter is uniform.   
   > Why would the dark matter not increase towards the center of the   
   > galaxy, as does the visible matter ?   
      
   Yes the actual matter distribution is more complicated. What I presented   
   here is an approximation. The distribution of matter actually decreases   
   closer to the center. This is why the velocity of stars actually slows   
   near the center.   
      
   LC   
      
      
   >   
   > > 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   
   >   
   > Can the dark matter hypothesis explain the empirical relationship,   
   >   
   >                      v^4 = K * M   
      
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