# Planetary motion

Corresponding Wikipedia article: Celestial mechanics

The stars and the galactic centers emit the streams of the synthesized heavy particles into their equatorial plane. Consequently:

• The planets were formed near Sun’s equatorial plane.
• The planets and Sun are rotating in the same direction.
• The large galaxies have a spiral shape. The spiral arms are rotating in the same direction.

The planets rotate around the Sun in elliptic orbit, which is slowly shifting. This shift is called the perihelion precession, and it is caused by the perturbations sum due to the gravity of other planets as well as the anomalous component.

A system of two material particles, which go away from each other at velocity $$v$$, can be considered at any point as an aetheric vortex, where the condition ("Mass and inertia", 8) is not valid, and the mean velocities are determined by ("Mass and inertia", 7). Thus, the relativistic correction of the product of gravitational mass-energy of two particles at small velocities $$v_1$$ and $$v_2$$, ignoring the small values of $$\frac{v^2}{c^2}$$, is estimated as $\sqrt{1-\frac{2v}{c}-\frac{v_1^2}{c}}\sqrt{1-\frac{2v}{c}-\frac{v_2^2}{c}}\approx(1-\frac{v}{c})^2\tag{1}$

Such correction of the gravitational potential is used in Paul Gerber's theory, which gives the same correction of the anomalous precession of perihelion as the General relativity gives.

The axial precession of planets (precession of the equinoxes) is caused by the external gravitational torques. The anomalous component is called the geodetic (de Sitter, Fokker) effect.

A particle of a precessing body has mass of $$m$$ and moves with velocity $$\mathbf{\overrightarrow{v}}$$ within the gravitational field of a particle with mass $$M$$ and the angular momentum $$\mathbf{\overrightarrow{L}}$$. The following equation can be assumed for analysis of a system of two particles separated by a distance $$\mathbf{\overrightarrow{R}}$$ from each other $\mathbf{\overrightarrow{L}}=\mathbf{\overrightarrow{R}}\times M\mathbf{\overrightarrow{V}}\tag{2}$

The equation ("Mass and inertia", 7) for a system of two combined particles is $c^2=(\mathbf{\overrightarrow{c_1}}+\mathbf{\overrightarrow{V}}-\mathbf{\overrightarrow{v}})^2={c_1}^2+V^2+v^2+2\mathbf{\overrightarrow{c_1}}\mathbf{\overrightarrow{V}}-\mathbf{\overrightarrow{c_1}}\mathbf{\overrightarrow{v}}-2\mathbf{\overrightarrow{V}}\mathbf{\overrightarrow{v}}\tag{3}$

Since the motion is circular $\mathbf{\overrightarrow{c_1}}\mathbf{\overrightarrow{V}}=\mathbf{\overrightarrow{c_1}}\mathbf{\overrightarrow{v}}=0\tag{4}$

The relativistic correction of the gravitational mass product is decomposed into two factors, ignoring the 4th degree terms, is $\sqrt{1-\frac{V^2-2\mathbf{\overrightarrow{V}}\mathbf{\overrightarrow{v}}+v^2}{c^2}}\approx \sqrt{1-\frac{V^2+v^2}{c^2}}\sqrt{1+\frac{2\mathbf{\overrightarrow{V}}\mathbf{\overrightarrow{v}}}{c^2}}\tag{5}$

In this case, the first factor in (5) consists of the constant corrections, one of which is compensated by the same correction of inertial mass-energy, and the other one is included in the mass of planet $\sqrt{1-\frac{V^2+v^2}{c^2}}\approx \sqrt{1-\frac{V^2}{c^2}}\sqrt{1-\frac{v^2}{c^2}}\tag{6}$

The resulting variable correction of the gravitational force is $\sqrt{1+\frac{2\mathbf{\overrightarrow{V}}\mathbf{\overrightarrow{v}}}{c^2}}\approx 1+\frac{\mathbf{\overrightarrow{V}}\mathbf{\overrightarrow{v}}}{c^2}\tag{7}$ $\Delta\mathbf{\overrightarrow{F}}\approx \frac{GMm}{R^3}\mathbf{\overrightarrow{R}}\frac{\mathbf{\overrightarrow{V}}\mathbf{\overrightarrow{v}}}{c^2}\tag{8}$

At the same time, the so-called gravitomagnetic force in case of $$\mathbf{\overrightarrow{L}}\mathbf{\overrightarrow{R}}=0$$ using Lagrange’s formula is $\mathbf{\overrightarrow{F_B}}=\frac{m\mathbf{\overrightarrow{v}}}{c}\times\frac{G\mathbf{\overrightarrow{L}}}{cR^3}=\frac{m\mathbf{\overrightarrow{v}}}{c}\times(\mathbf{\overrightarrow{R}}\times\frac{GM\mathbf{\overrightarrow{V}}}{cR^3})=\frac{GMm}{R^3}(\mathbf{\overrightarrow{R}}\frac{\mathbf{\overrightarrow{V}}\mathbf{\overrightarrow{v}}}{c^2}-\mathbf{\overrightarrow{V}}\frac{\mathbf{\overrightarrow{R}}\mathbf{\overrightarrow{v}}}{c^2})\tag{9}$

The gravitomagnetic force (9) differs from (8) by the transverse vector $$\mathbf{\overrightarrow{V}}\frac{\mathbf{\overrightarrow{R}}\mathbf{\overrightarrow{v}}}{c^2}$$, the symmetric effect of which is compensated by circular motion of the point mass $$m$$.

The weight of a gyroscope depends on its rotational speed with respect to Earth’s rotational speed due to the force (8). The weight deviation is so small ($$< 10^{–10}$$), that some experiments detect[1][2][3] it, but others[4][5][6][7] do not. Kozyrev used a vibration to reduce friction within the beam of the balance scale.

## References

1. Н. А. Козырев (N. A. Kozyrev), Причинная или несимметричная механика в линейном приближении (гл. 5), Пулково, 1958
2. H. Hayasaka and S. Takeuchi, Anomalous weight reduction on a gyroscope’s right rotation around the vertical axis on the earth, Phys. Rev. Lett. 63, 2701, (1989)
3. A. Imanishi, K. Maruyama, S. Midorikawa and T. Morimoto, Observation against the weight reduction of spinning gyro, J. Phys. Soc. Japan 60, 1150,1991
4. J. E. Faller, W. J. Hollander, P. G. Nelson and M. P. McHugh, Gyroscope-weighing experiment with a null result, Phys. Rev. Lett. 64, 825 (1990)
5. T. J. Quinn and A. Picard, The Mass of Spinning Rotors - no Dependence on Speed or Sense of Rotation, Nature 343, 732 (1990)
6. J. M. Nitschke and P. A. Wilmarth, Null Result for the Weight Change of a 8 Spinning Gyroscope, Phys. Rev. Lett. 64, 2115-2116, 1990
7. J. Luo, Y. X. Nie, Y. Z. Zhang and Z. B. Zhou, Null result for violation of the equivalence principle with free-fall rotating gyroscopes, Phys. Rev. D 65, 042005 (2002)