Physics/Essays/Anonymous/Solar planets as gravitational resonators

Solar planets as gravitational resonators form some law like the s.c. Titzius-Bode law for the planets mass-radius characteristics.

History
The theory of the quantum gravitational resonator (QGR) is based on the Maxwell-like gravitational equations and similar in many relation to the theory of quantum electromagnetic resonator (QER), therefore the QGR history is close connected with the QER history.

Gravitational resonators
Due to McDonald first who used Maxwell equations to describe gravity was Oliver Heaviside The point is that in the weak gravitational field the standard theory of gravity could be written in the form of Maxwell equations

In the 90-ties Kraus first introduced the gravitational characteristic impedance of free space, which was detaled later by Kiefer, and now Raymond Y. Chiao    who is developing the ways of experimental determination of the gravitational waves.

Velocity circulation quantum
First the VCQ was proposed in the early 50-th for the quantum superfluids in the general form by R.Feynman, :
 * $$\oint_{L} \mathbf{v}\cdot\,d\mathbf{l} = n\frac{h}{m}, \ $$

where $$n $$ could be integer or fractional in the general case.

Further developments this approach was made by Yakymakha (1994) for inversion layers in MOSFETs.

General resonator characteristics
Geometrical properties of a planet define the following resonance frequency:
 * $$\omega_{pl} = \frac{c}{R_{pl}}, \ $$

where $$c $$ is velocity of light, and $$R_{pl} $$ is the planet radius. This frequency could be connected with the "minimal mass conseption":
 * $$m_{pl} = \frac{\hbar \omega_{pl}}{c^2} = \frac{\hbar}{cR_{pl}}, $$

where $$\hbar $$ is the reduced Planck constant. Considering that total planet mass $$M_{pl} $$ is replaced on the resonator surface:
 * $$M_{pl} \rightarrow S_{pl} = 4\pi R_{pl}^2, \ $$

and therefore the "minimal mass" should be placed on the minimal surface:
 * $$m_{pl} \rightarrow s_{pl} = 4\pi r_{pl}^2. \ $$

Thus, the minimal radius will be:
 * $$r_{pl} = \sqrt{\frac{m_{pl}}{M_{pl}}}. \ $$

Velocity circulation quantum approach
In the general case the velocity circulation quantum is defined as:
 * $$\omega \cdot S = n_x \frac{h}{m}, \ $$

where $$n_x $$ is integer number. This equation could be rewritten in the "mass form":
 * $$m\cdot S = \frac{2\pi n_x}{c^2}\frac{\hbar^2}{m}. \ $$

For $$n_x = 2 $$ this equation defines the minimal mass as:
 * $$m = \frac{\hbar}{cR}. \ $$

Note that this definition is compatible with the gravitational resonato approach presented in the above section.

Solar system gravitational characteristics
The full sets of the planetary data are presented in the Table 1.

Note that all planetary data were taken from the textbook .