  For mature spiral galaxies, the rotation velocities quickly increase from the
galactic center and achieve a constant velocity from the core to the periphery.
This dynamic behavior is described by models balancing Newtonian gravitational
and centrifugal forces in rotating thin axisymmetric disks. Freeman's disk
assumes a mass density decreasing exponentially with radius which correctly
produces rotational velocities which increase from the galactic center to a
maximum near the outer core, but then decreases out to the periphery contrary
to measurements. Mestel's disk assumes a mass distribution decreasing more
slowly (inversely with radius) that yields a constant rotational velocity
across the entire disk, but has an unrealistic central mass singularity and
does not describe the core rotation properly. Thus combine the Freeman and
Mestel disks to utilize their strengths and eliminate their deficiencies.
Utilize the Freeman formula for the central core, and the Mestel formula beyond
the core to the galactic rim. This combined model produces rotation curves
comparable to the measurements of mature spiral galaxies. For a more general
thin-disk model, we develop an alternative computational method to solve for
the mass density distributions for various measured rotation curves We compute
radial mass densities that balance the Newtonian gravitational and centrifugal
forces at every point. The computational solutions show mass densities which
decrease approximately exponentially in the central core (similar to Freeman),
and transition to a slower inverse radial decrease (similar to Mestel) to the
periphery. Thus these diverse approaches yield similar results and are mutaully
self-consistent with Newtonian gravity/dynamics.
