Titius-bode series galaxy group red shift differences calculated from roots of the bessel equation

It has been shown in detail by Wells (Wells, 1986) that the Bode numbers and measured velocity ratios of the planets are accurately predicted by the eigenvalues of the Euler-Lagrange equations resulting from the variation of the free energy of the generic plasma that formed the Sun and planets. This theory is reviewed and extended to show that the equations make accurate predictions for all the major planets out to and including Pluto. The semimajor axes and velocity ratios of Pluto and Neptune are predicted exactly. The Bode numbers are shown in Table I to correspond to the roots of the first-order Bessel functions. The extrema of the roots of the zeroth-order Bessel function predict the ratios of the measured planetary velocities almost without error for the outer planets. Both sets of roots correspond to the same eigenvalue solution of the forcefree equation. The eigenvalues are set by the initial energy input to the plasma nebula. Both the Titius-Bode series and Kepler's harmonic law are predicted by the relaxed state solution of the free-energy equation for the generic plasma that formed the Sun and planets. Newton's law of gravitation is not used in the calculations. The solution makes exact predictions for the outer planets where the Titius-Bode series fails completely.The work of Arp (Arp, 1985) adds to the growing body of observable evidence of objects which appear to be attached to galaxies or galaxy systems but display red shifts, sometimes quite large, differing from those of the associated galaxies. Adding to the mystery and confusion are a series of objects that have quantized red shifts.It is widely recognized that the history of these objects involves extremely high energy processes. The solutions of the equations of the relaxed state of the resulting high energy plasmas is discussed and it is shown that the predictions of red shift frequencies are quantized and agree numerically with many of the quantized shifts reported by Arp and Sulentic.