# 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[1] first who used Maxwell equations to describe gravity was Oliver Heaviside[2] The point is that in the weak gravitational field the standard theory of gravity could be written in the form of Maxwell equations[3]

In the 90-ties Kraus [4] first introduced the gravitational characteristic impedance of free space, which was detaled later by Kiefer [5], and now Raymond Y. Chiao[6] [7] [8] [9] [10] 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 [11], [12]:

${\displaystyle \oint _{L}\mathbf {v} \cdot \,d\mathbf {l} =n{\frac {h}{m}},\ }$

where ${\displaystyle n}$ could be integer or fractional in the general case.

Further developments this approach was made by Yakymakha (1994) for inversion layers in MOSFETs [13].

## Gravitational resonator approach to the Solar System

### General resonator characteristics

Geometrical properties of a planet define the following resonance frequency:

${\displaystyle \omega _{pl}={\frac {c}{R_{pl}}},\ }$

where ${\displaystyle c}$ is velocity of light, and ${\displaystyle R_{pl}}$ is the planet radius. This frequency could be connected with the "minimal mass conseption":

${\displaystyle m_{pl}={\frac {\hbar \omega _{pl}}{c^{2}}}={\frac {\hbar }{cR_{pl}}},}$

where ${\displaystyle \hbar }$ is the reduced Planck constant. Considering that total planet mass ${\displaystyle M_{pl}}$ is replaced on the resonator surface:

${\displaystyle M_{pl}\rightarrow S_{pl}=4\pi R_{pl}^{2},\ }$

and therefore the "minimal mass" should be placed on the minimal surface:

${\displaystyle m_{pl}\rightarrow s_{pl}=4\pi r_{pl}^{2}.\ }$

Thus, the minimal radius will be:

${\displaystyle r_{pl}={\sqrt {\frac {m_{pl}}{M_{pl}}}}.\ }$

### Velocity circulation quantum approach

In the general case the velocity circulation quantum is defined as:

${\displaystyle \omega \cdot S=n_{x}{\frac {h}{m}},\ }$

where ${\displaystyle n_{x}}$ is integer number. This equation could be rewritten in the "mass form":

${\displaystyle m\cdot S={\frac {2\pi n_{x}}{c^{2}}}{\frac {\hbar ^{2}}{m}}.\ }$

For ${\displaystyle n_{x}=2}$ this equation defines the minimal mass as:

${\displaystyle m={\frac {\hbar }{cR}}.\ }$

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.

Table 1: Solar planatery system
Object Radius, m Mass, kg Minimal Mass, kg Minimal Radius, m ${\displaystyle l_{W}/r_{min}}$
Sun ${\displaystyle 6.96\cdot 10^{8}}$ ${\displaystyle 1.989\cdot 10^{30}}$ ${\displaystyle 5.054\cdot 10^{-52}}$ ${\displaystyle 1.1095\cdot 10^{-32}}$ ${\displaystyle 109.4}$
Jupiter ${\displaystyle 7.13\cdot 10^{7}}$ ${\displaystyle 1.899\cdot 10^{27}}$ ${\displaystyle 4.934\cdot 10^{-51}}$ ${\displaystyle 1.149\cdot 10^{-31}}$ ${\displaystyle 10.56}$
Saturn ${\displaystyle 6.01\cdot 10^{7}}$ ${\displaystyle 5.686\cdot 10^{26}}$ ${\displaystyle 5.853\cdot 10^{-51}}$ ${\displaystyle 1.928\cdot 10^{-31}}$ ${\displaystyle 6.3}$
Neptun ${\displaystyle 2.51\cdot 10^{7}}$ ${\displaystyle 1.03\cdot 10^{26}}$ ${\displaystyle 1.402\cdot 10^{-50}}$ ${\displaystyle 2.928\cdot 10^{-31}}$ ${\displaystyle 4.15}$
Uran ${\displaystyle 2.45\cdot 10^{7}}$ ${\displaystyle 8.689\cdot 10^{25}}$ ${\displaystyle 1.436\cdot 10^{-50}}$ ${\displaystyle 3.149\cdot 10^{-31}}$ ${\displaystyle 3.86}$
Earth ${\displaystyle 6.371\cdot 10^{6}}$ ${\displaystyle 5.976\cdot 10^{24}}$ ${\displaystyle 5.521\cdot 10^{-50}}$ ${\displaystyle 6.124\cdot 10^{-31}}$ ${\displaystyle 1.98}$
Venus ${\displaystyle 6.07\cdot 10^{6}}$ ${\displaystyle 4.87\cdot 10^{24}}$ ${\displaystyle 5.795\cdot 10^{-50}}$ ${\displaystyle 6.622\cdot 10^{-31}}$ ${\displaystyle 1.83}$
Mars ${\displaystyle 3.395\cdot 10^{6}}$ ${\displaystyle 6.424\cdot 10^{23}}$ ${\displaystyle 1.036\cdot 10^{-49}}$ ${\displaystyle 1.364\cdot 10^{-30}}$ ${\displaystyle 0.89}$
Mercury ${\displaystyle 2.425\cdot 10^{6}}$ ${\displaystyle 3.311\cdot 10^{23}}$ ${\displaystyle 1.451\cdot 10^{-49}}$ ${\displaystyle 2.185\cdot 10^{-30}}$ ${\displaystyle 0.756}$

Note that all planetary data were taken from the textbook [14].

## References

1. K.T. McDonald, Am. J. Phys. 65, 7 (1997) 591-2.
2. O. Heaviside, Electromagnetic Theory (”The Electrician” Printing and Publishing Co., London, 1894) pp. 455-465.
3. W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism (Addison-Wesley, Reading, MA, 1955), p. 168, 166.
4. J. D. Kraus, IEEE Antennas and Propagation. Magazine 33, 21 (1991).
5. C. Kiefer and C. Weber, Annalen der Physik (Leipzig) 14, 253 (2005).
6. Raymond Y. Chiao. "Conceptual tensions between quantum mechanics and general relativity: Are there experimental consequences, e.g., superconducting transducers between electromagnetic and gravitational radiation?" arXiv:gr-qc/0208024v3 (2002). [PDF
7. R.Y. Chiao and W.J. Fitelson. Time and matter in the interaction between gravity and quantum fluids: are there macroscopic quantum transducers between gravitational and electromagnetic waves? In Proceedings of the “Time & Matter Conference” (2002 August 11-17; Venice, Italy), ed. I. Bigi and M. Faessler (Singapore: World Scientific, 2006), p. 85. arXiv: gr-qc/0303089. PDF
8. R.Y. Chiao. Conceptual tensions between quantum mechanics and general relativity: are there experimental consequences? In Science and Ultimate Reality, ed. J.D. Barrow, P.C.W. Davies, and C.L.Harper, Jr. (Cambridge:Cambridge University Press, 2004), p. 254. arXiv:gr-qc/0303100.
9. Raymond Y. Chiao. "New directions for gravitational wave physics via “Millikan oil drops” arXiv:gr-qc/0610146v16 (2009). PDF
10. Stephen Minter, Kirk Wegter-McNelly, and Raymond Chiao. Do Mirrors for Gravitational Waves Exist? arXiv:gr-qc/0903.0661v10 (2009). PDF
11. Putterman S.J. (1974). Superfluid hydrodynamics. North-Holland, Amsterdam
12. Feynman, R. P. (1955). Application of quantum mechanics to liquid helium. Progress in Low Temperature Physics 1: 17–53. ISSN 00796417
13. Cite error: Invalid <ref> tag; no text was provided for refs named Yakym2
14. Allen C.W.(1973). Astrophysical quantities. 3-d edition. University of London, The Athlone Press