We apply the new gravitational wave equation (GWE) derived from the General
Theory of Relativity to determine quantization states in solar systems. The GWE has one ad hoc
assumption: gravitational quantization states depend solely on the gravitationally-bound system's total
angular momentum and its total mass. From the predicted planet and satellite equilibrium orbital
distances we make comparisons to the empirical values. To our surprise, we find that the angular
momentum in the Oort Cloud determines the planetary spacings! We derive also a mass relationship
for orbiting bodies in any planetary system, including exoplanetary systems. We suggest a laboratory
experiment with a torsion bar near a rotating mass.
Researching change in the fine structure constant...
The square law rn = r1n2 for orbital sizes rn (r1 is a constant dependent on the
particular system, and n are consecutive integer numbers) is applied to the recently
discovered planets of Andromedae and to pulsars PSR B1257+12 and PSR
1828-11. A comparison with the solar planetary system is made. The product nvn of the
orbital velocity vn with the corresponding orbital number n for planets of Andromedae
is in good agreement with those for terrestrial planets, demonstrating the
generality of the square law in dynamics of diverse planetary systems. "Quantized
velocity" of nvn is very close to 24 kms-1, i.e. to the step found in the quantized
redshifts of galaxies. A definite conclusion for planetary systems of pulsars requires
Mean orbital distances of planets from the Sun and of major satellites from the parent
planets Jupiter, Saturn and Uranus are described by the square law, where the
values of are consecutive integers, and is the mean orbital distance expected at
for a particular system. Terrestrial planets and Jovian planets are analysed as separate systems.
Thus, five independent solar-like systems are considered. The basic assumption is
that specific orbital angular momentum is "quantized". Consequently, all orbital parameters
are also discrete. The number relates to the law of orbital spacing. An additional
discretization, related to, i.e. to the scale of orbits, accounts for the detailed structure of
planar gravitational systems. Consequently, it is also found that orbital velocity fi-multiplied
by is equal to the multiple of a fundamental velocity, valid for all
subsystems in the Solar System. This velocity is equal to one of the "velocity" increments
of quantized redshifts of galaxies.
Three anomalies associated with the solar system, namely Pioneer anomaly
, the evidence for shrinking of planetary orbits [7, 8, 9], and flyby anomaly
 are discussed. The first anomaly is explained by a universal 1/r distribution
of dark matter, second anomaly finds a trivial explanation in TGD
based quantum model for planetary orbits as Bohr orbits with Bohr quantization
reflecting macroscopically quantum coherent character of dark matter
with a gigantic value of Planck constant . Fly-by anomaly can be understood
if planetary orbits are surrounded by a flux tube containing quantum
coherent dark matter. Also spherical shells can be considered.