Statistical Theory of Planetary Orbits Stability by A.M. Krot

1. Development of the statistical theory of forming cosmogonical bodies to explain a stability of the orbital movements of planets and the forms of planetary orbits Alexander M. Krot The Laboratory of Self-organization System Modeling, United Institute of Informatics Problems of the National Academy of Sciences of Belarus E-mail: alxkrot@newman.bas-net.by Tel.: (+375 17) 284-20-86

2. Statistical Theory of Formation of Gravitating Cosmogonical Bodies

3. Fundamental Physics: Gravity, Relativity & Quantum Theory http://www.zarm.uni-bremen.de/ 2forschung/gravi/gravity_main/htm

4. The Development of Statistical Model of CosmogonicalBody Forming 1.The gravitating body under consideration is homogeneous in its chemical structure, i.e. it consists of N identical particles with the mass m0. 2. The gravitating body is not subjected to influence of external fields and bodies. 3. The gravitating body is isothermal and has a low temperature T, besides Tdeg<T, where Tdeg = (h2/m0kB)n2/3 is a degeneration temperature, h is the Planck constant, kB is the Boltzmann constant, n is concentration of particles. 4.The process of gravitational interaction of particles is a slow-flowing one with time both immovable and rotating gravitating bodies.

5. The Derivation of Function of Particle Distribution in Space Based on StatisticalModel of Rotating and Gravitating Body We use a cylindrical frame of reference (h, , z): x = h∙cos ; y = h∙sin ; z = z

6. Analogously ; ; .

7. The density mass for rotating spheroidal body

8. The mass density function for an immovable spheroidal body: Thus, a great number of particles form a spheroidal body. The mass density of gravitating spheroidal body with the point of view of General Relativity:

9. antidiffusion equation for an immovable spheroidal body: gravitational compression function (GCF): The general antidiffusion equation of a slowly evolving process of initial gravitational condensation of a rotating spheroidal body from an infinitely distributed substance general antidiffusion equation for an rotating spheroidal body: antidiffusion mass flow density antidiffusion acceleration antidiffusion velocity

10. On the universal stellar law for extrasolar systems

11. Estimation of parameter of gravitational compression of the Sun on the basis of an estimation of the linear size of its core, i.e. thethickness of a visible part of the solar corona Graphic dependence of relative brightness of components of spectrum of the solar corona on distance up to edge of a disk The solar corona embodied during the solar eclipse in 1999

12. Calculations of the orbit of Mercury and estimation of angular shift of the Mercury’ perihelion based on the statistical theory of gravitating spheroidal bodies The equation of the “disturbed” ellipse

13. Dependence of luminosity of a star on temperature of its stellar surface (the diagram of Hertzsprung–Russell) and dependence of luminosity of a star on its mass Whether there exist like the Kepler’s laws a universal law for the planetary systems connecting temperature, size and mass of each of stars?

14. Derivation of the universal stellar law Statistical theory of gravitating spheroidal bodies Poincaré’s virial theorem

15. We introduce a new constant called the universal stellar constant: We obtain the equation of state of an ideal stellar substance: named so by analogy to the known Clapeyron–Mendeleev’ equation of state of a ideal gas . The universal stellar law (USL)

16. Estimation of mean relative molecular weight of a highly ionized stellar substance The first approximation under condition of full ionization for an element with atomic serial number and relative atomic weight (the formula of Сhandrasekhar) The secondapproximation takes into account the ionization state of stellar substance (the formula of Strömgren)

17. In case of stellar substance of the Sun

18. The modified universal stellar law relatively the modified USL We would like to verify correctness of the modified USL for extrasolar systems belonging to the different spectral classes O, B, A, F, G, K,M.

20. Estimation of temperature of the stellar corona , where factor of an integral absorptivity of a body, besides for a real body Estimation of the modified USL for the different classes of stars with regard to the errors in measurements

22. On the stability of the orbital movements of planets and the forms of planetary orbits

23. The derivation of the combined Kepler ’s 3rd law with the universal stellar law (3KL-USL) and explanation of stability of planetary orbits through 3KL-USL , • USL connects the temperature, the size and the mass of a star: • Kepler’s 3rd law (3KL) connects the major semi-axis of a planetary orbit and the period of motion of planet around a star: the universal stellar constant/ where The combined Kepler’s 3rd law with the universal stellar law (3KL-USL): a specific entropy

24. On the nature of Alfvén-Arrhenius’ small oscillations modifying circular orbit with point of view of the statistical theory . Alfvén-Arrhenius’ approach Application of statistical theory Acceleration or gravitational field strength of a forming spheroidal body An additional specific force acts upon the body at modification of the circular orbit: where The frequency of the radial oscillations: where The frequency of the axial oscillations: is the Keplerian angular velocity.

25. An approximate description of the motion using the guiding-center method Statistical theory considers two important cases of dynamical states of a slowly rotating spheroidal body: 1) the condition of mechanical equilibrium , i.e. when the induced acceleration becomes the regular gravitational field strength 2) the condition of mechanical quasiequilibrium when the small fluctuations ofgravitational field strength are possible, i.e. and

26. Dynamical and statisticalcharacteristics of the stars of different multi-planet extrasolar systems

27. The plot of dependence of the stabilization value of gravitational compression function on the mass of star

28. The plot of dependence of the stabilization value of gravitational compression function onthe effective temperature

29. The plot of dependence of the main circular frequency of inner oscillations on the mass of star

30. The comparative analysis of orbital characteristics of the Solar system planets

31. Axial and radial oscillations of the orbital motion in the gravitational field of a rotating and gravitating spheroidal body Alfvén-Arrhenius’approach Application of the statistical theory The condition of the stabilization of GCF Alfvén–Arrhenius’s inequalities The constraining force

32. The comparative analysis of the orbital motion characteristics of planets for the Solar system

33. The plot of dependence of the deviation ofKeplerian angular velocity, s-1on the planetary distances , AU

34. The plot of dependence of the angular velocity of orbital pericenter motion , s-1 on the planetary distances , AU

35. The plot of dependence of the angular velocity of ascending node motion, ,s-1 on the planetary distances , AU

36. Conclusion Using the statistical theory of gravitating spheroidal bodies we show the following: 1.The temporal deviation of GCF of a spherically symmetric cosmogonical body in a vicinity of its equilibrium value leads to the origin of the additional periodic forcemodifying forms of the circular orbits of moving bodies to the slightly elliptical orbits. 2. The spatial deviationof the gravitational potential in the remote zone of a rotating non-spherically symmetric cosmogonical body leads to a precessing elliptic orbitof planet moving in a vicinity of the core of this body. 3. The temporal deviation of GCF of a spherically symmetric cosmogonical body (under the condition of its mechanical quasiequilibrium) is determined by an oscillation behavior of its derivative: this implies the special case when thesquared generalized circular frequency can be negative,i.e. the additional periodic force becomes oriented opposite to the gravitational force. It means that the mentioned principle of an anchoring mechanism is realized in any exoplanet system. (116b). , .