Fractal scaling models of natural oscillations in chain systems and the mass distribution of the celestial bodies in the solar system.
Fractal scaling models  of natural oscillations in chain systems of harmonic oscillators are not based on any statements about the nature of the link or interaction between the elements of the oscillating chain system. Therefore the model statements are quite generally, what opens a wide field of possible applications.
In comparison with empty cosmic space, celestial bodies (stars, planets, moons, asteroids) are compressed matter and the contribution of nucleons to the bodies mass is about 99%. In the framework of the standard particle model, protons and neutrons are baryons, in which the proton connects to a lower quantum energy level and a much more stable state than the neutron. In addition, the proton and neutron have similar rest masses, what permits us to interpret protons and neutrons as similar quantum oscillators with regard to their rest masses.
Based on a fractal scaling model  of natural oscillations in this paper we will interpret matter as a chain system of many oscillating protons and find out spectral ranges where the oscillation process stability and energy efficiency are relative high or low.
On the base of continued fraction method  we will search the natural frequencies of a chain system of many vibrating protons on the lowest energy level (ground stage) in this form:
f = [f.sub.p]exp(S), (1)
f is a natural frequency of a chain system of vibrating protons, [f.sub.p] is the natural oscillation frequency of one proton, S is a finite continued fraction with integer elements:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)
where [n.sub.0], [n.sub.1], [n.sub.2], ..., [n.sub.k] [member of] Z. The continued fractions (2) are in the canonical form and have a discrete spectrum of eigenvalues. With the help of the Lagrange transformation  every continued fraction with integer partial denominators can be represented as a continued fraction with natural partial denominators, that's always convergent. In this paper we will investigate spectra generated by convergent continued fractions (2). The present paper follows the Terskich  definition of a chain system, where the interaction between the elements proceeds only in their movement direction.
Model spectra (2) are not only logarithmic-invariant, but also fractal, because the discrete hyperbolic distribution of natural frequencies repeats itself on each spectral level k. We investigate continued fractions (2) with a finite quantity of layers k, which generate discrete spectra, because in this case all continued fractions S represent rational numbers. Therefore the free link [n.sub.0] and the partial denominators [n.sub.1] can be interpreted as "quantum numbers".
The partial denominators [n.sub.1] run through positive and negative integer values. Maximum spectral density areas (spectral nodes) arise automatically on the distance of one logarithmic unit, where [absolute value of [n.sub.1]] [right arrow] [infinity]. Fig.1 shows the spectrum on the first layer k = 1 for [absolute value of [n.sub.1]] = 2, 3, 4, ... and [absolute value of [n.sub.0]] = 0, 1, 2, ... (logarithmic representation). Integer S-values are labeled.
[FIGURE 1 OMITTED]
Ranges of relative low spectral density (spectral gaps) and ranges of relative high spectral density (spectral nodes) arise on each spectral layer. In addition to the first spectral layer, Fig. 2 shows the second spectral layer k = 2 for [absolute value of [n.sub.2]] = 2, 3, 4, ... and [absolute value of [n.sub.1]] = 2 (logarithmic representation).
[FIGURE 2 OMITTED]
In the spectral node ranges, where the spectral density reachs local maximum, natural frequencies are distributed maximum densely, so that near a spectral node almost each frequency is a natural frequency. The energy efficiency of natural oscillations is very high. Therefore, if a frequency of an oscillation process is located near a node of the fractal spectrum (2), the process energy efficiency (degree of effectiveness) should be relative high. More detailed this topic is described in .
Let's assume that the oscillation amplitudes are low, the oscillations are harmonic and the energy level [E.sub.f] of the vibrating protons depends only on their oscillation frequency (h is the Planck constant):
[E.sub.f] = hf. (3)
Atomic nucleuses arise in the result of high energy processes of nucleosynthesis. Einstein's formula defines not only the connection between the rest energy and rest mass of nucleons, but also between binding energy and the mass defect of an atomic nucleus. Therefore we assume that the rest mass m of our model matter corresponds to the energy [E.sub.m]:
[E.sub.m] = m[c.sup.2]. (4)
Let's assume that the basis of nucleosynthesis is harmonic oscillations of protons and the energy (4) is identically with (3). In this case we can write:
m = f[h/[c.sup.2]]. (5)
[FIGURE 3 OMITTED]
In the framework of our oscillation model (1) the equation (5) means not only that mass can be changed into energy, but also that quantum oscillations generate the mass spectrum of our model matter. Under consideration of (1) now we can create a fractal scaling model of the natural mass spectrum of our model matter of vibrating protons. This mass spectrum is described by the same continued fraction (2), for [m.sub.p] = [f.sub.p][h/[c.sup.2]]:
ln[m/[m.sub.p]] = [[n.sub.0];[n.sub.1], [n.sub.2], ..., [n.sub.k]]. (6)
Consequently, the frequency spectrum (2) and the mass spectrum (6) are isomorphic, and [m.sub.p] is the proton rest mass 1.672621637(83) x [10.sup.-27] kg . As mentioned already, we assume that mass generation processes are based on quantum natural oscillation processes. Celestial bodies are compressed matter, which consist of nucleons over 99%. Therefore we expect that the distribution of the celestial bodies in the proton resonance mass spectrum is not random and near spectral nodes the formation probability of massive bodies is maximum. Like in the Kundt's tube , near resonance nodes the matter accumulation reachs maximum intensity. The mass spectrum (6) is fractal and consequently it has a clear hierarchical structure, in which continued fractions (2) of the form [[n.sub.0]] and [[n.sub.0]; 2] define main spectral nodes, as Fig. 2 shows.
[FIGURE 4 OMITTED]
In the present paper we will compare the scaling mass spectrum (6) of our model matter in the range of [10.sup.19] kg to [10.sup.30] kg with the mass distribution of well-known celestial bodies. These are asteroids, planetoids, moons and planets of the Solar System (including the Sun), which masses were measured precisely enough and which are massive enough to be rounded by their own gravity.
For example, to locate the mass of the planet Venus in the scaling mass spectrum (6) of our model matter, one divides the Venus body mass by the proton rest mass and represents the logarithm as a continued fraction:
[S.sub.venus] = ln[[[m.sub.venus]]/[m.sub.p]] = ln([[4.869 x [10.sup.24] kg]/[1.67262 x [10.sup.-27]kg]]) [congruent to] [congruent to] 118.50 = 118 + [1/2]. (7)
[FIGURE 5 OMITTED]
The analysis (6) of the Venus body mass takes the result [n.sub.0] = 118, [n.sub.1] = 2. This means, that the Venus body mass corresponds to a spectral node on the first layer k = 1 of the spectrum (6). The Sun mass is near the spectral node [131; 2]. It's also correct for the Alpha Centauri A and B masses. The Alpha Aquilae (Altair) mass is about 1.7 solar masses, that's near the node . Table 1 shows the logarithms (6) calculated from the measured masses m of the celestial bodies and the S-values of the nearest spectral nodes.
Table 1 shows, that spectral nodes are occupied by bodies which have maximum mass in a local group or family. For example, the spectral node  is occupied by Ganymede and Titan, the most massive moons of Jupiter and Saturn, the spectral node  is occupied by Triton, the most massive moon of Neptune, the body mass of Eris, the largest defined dwarf planet, is also near the spectral node , but the spectral node  is occupied by Ceres, the most massive body of the asteroid belt. Mercury's mass is near the node . Possibly, not Eris, but Mercury is the most massive dwarf planet in the Solar System. Actually, Mercury behaves like a dwarf planet, because it has the highest eccentricity of all the Solar System planets and it has the smallest axial tilt.
[FIGURE 6 OMITTED]
For the nodes [[n.sub.0]] and [[n.sub.0]; 2] the finite continued fraction
(2) is S = [n.sub.0]+1/[n.sub.1] and the corresponding discrete mass values can be defined by linear S-trajectories, in which N[member of]Z:
S = [S.sub.0] + [N/2]. (8)
The prime divisibility of N = pn, in which p is a prime factor of N, defines sets of S-trajectories which form different sequences of mass-values m of the discrete spectrum (6).
S-trajectories (8) present the discrete scaling mass distribution (6) very clear and can be interpreted as exponential equivalents to linear square-mass trajectories, which are a well-known systematic feature in the hadrons spectrum . Fig. 3 shows the S-trajectory for [S.sub.0] =  and p = 1. Largest bodies are labeled. Possibly, vacant nodes are occupied by extrasolar bodies or bodies still to be discovered in the Solar System.
Possibly, the existence of the discrete spectrum (6) in the range of celestial bodies masses can be interpreted as "macroscopic quantization" . The larger the bodies the more distinctive is this phenomenon. This can be recognized well at the example of the 8 largest planets in the Solar System, as Fig. 4 shows.
For [S.sub.0] =  and every p is [m.sub.0] = [m.sub.p], so that every S-trajectory can be prolonged down to the proton rest mass. Also the electron and W-Z-bosons rest masses lie on the S-trajectory for [S.sub.0] =  and p=3, as Fig. 5 shows. Already within the eighties the scaling exponent 3/2 was found in the distribution of particle masses by Valery A. Kolombet .
Table 2 shows the logarithms (6) calculated from the measured particle rest masses, and the S-values of the nearest spectral nodes.
The S-trajectory in Fig. 5 is the same as the S-trajectory in Fig. 4, but prolonged down to the electron rest mass for S = [-7; -2]. Possibly, there is a fundamental link between particle rest masses and the masses of celestial bodies. Fig. 6 shows the S-trajectory for [S.sub.0] =  and p = 3 in the range of -9 [less than or equal to] S [less than or equal to] 135, of the electron rest mass to the body mass of the Sun.
In the framework of the present model discrete scaling distributions arise as result of natural oscillations in chain systems of harmonic oscillators. Particularly, the observable mass distribution of celestial bodies arise as result of natural oscillations in chain systems of protons, that can be understood as contribution to the fundamental link between quantum- and astrophysics. Possibly, the high energy efficiency of natural oscillations is the cause of the fractal scaling distribution of matter in the universe.
The author is deeply grateful to S.E. Shnoll, V. A. Panchelyuga and V. A Kolombet for valuable discussions and support.
Submitted on October 28, 2009 / Accepted on November 10, 2009
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Global Scaling Research Institute in memoriam Leonhard Euler, Munich, Germany. E-mail: firstname.lastname@example.org
Table 1: The masses of celestial bodies/planets, dwarf planets (P), asteroids (A), moons of Jupiter (J), Saturn (S), Uranus (U), Neptune (N) and Earth (E) and the 5/values (6) of the nearest spectral nodes. The relative deviation d = (ln(m/[m.sub.p])-s)/S is indicated in percents. Celestial body Body mass m, kg x [m.sub.p]) 15 Eunomia (A) 3.12 x [10.sup.19]  106.54 Mimas (S) (3.7493 [+ or -] 0.0031) 106.73 x [10.sup.19]  Miranda (U) (6.59 [+ or -] 0.75) 107.29 x [10.sup.19]  10 Hygiea (A) (8.98 [+ or -] 0.01) 107.60 x [10.sup.19]  Enceladus (S) (1.08022 [+ or -] 0.00101) 107.78 x [10.sup.20]  2 Pallas (A) (2.11 [+ or -] 0.26) 108.50 x [10.sup.20]  4 Vesta (A) (2.67 [+ or -] 0.02) 108.69 x [10.sup.20]  Tethys (S) (6.17449 [+ or -] 0.00132) 109.53 x [10.sup.20]  1 Ceres (P) (9.43 [+ or -] 0.07) 109.95 x [10.sup.20] [8,9] Dione (S) (1.095452 [+ or -] 0.000168) 110.10 x [10.sup.21]  Umbriel (U) (1.172 [+ or -] 0.135) 110.10 x [10.sup.21]  Ariel (U) (1.350 [+ or -] 0.120) 110.23 x [10.sup.21]  Charon (P) (1.52 [+ or -] 0.06) 110.43 x [10.sup.21]  Iapetus (S) (1.805635 [+ or -] 0.000375) 110.60 x [10.sup.21]  Rhea (S) (2.306518 [+ or -] 0.000353) 110.84 x [10.sup.21]  Oberon (U) (3.014 [+ or -] 0.075) 111.12 x [10.sup.21]  Titania (U) (3.53 [+ or -] 0.09) 111.28 x [10.sup.21]  Haumea (P) (4.006 [+ or -] 0.040) 111.40 x [10.sup.21]  Pluto (P) (1.305 [+ or -] 0.007) 112.57 x [10.sup.22]  Eris (P) (1.67 [+ or -] 0.02) 112.83 x [10.sup.22]  Triton (N) 2.14 [+ or -] 113.07 [10.sup.22]  Europa (J) 4.80 [+ or -] 113.88 [10.sup.22]  Moon (E) 7.3477 [+ or -] 114.30 [10.sup.22] Io(J) (8.9319 [+ or -] 0.0003) 114.50 x [10.sup.22]  Callisto (J) (1.075938 [+ or -] 0.000137) 114.69 x [10.sup.23]  Titan (S) (1.3452 [+ or -] 0.0002) 114.91 x [10.sup.23]  Ganymede (J) (1.4819 [+ or -] 0.0002) 115.00 x [10.sup.23]  Mercury (3.3022 [+ or -] 0.0001) 115.81 x [10.sup.23] Mars (6.4185 [+ or -] 0.0001) 116.47 x [10.sup.23] Venus (4.8685 [+ or -] 0.0001) 118.50 x [10.sup.24] Earth (5.9722 [+ or -] 0.0006) 118.69 x [10.sup.24]  Uranus (8.6810 [+ or -] 0.0013) 121.38 x [10.sup.25]  Neptune (1.0243 [+ or -] 0.0015) 121.55 x [10.sup.26] Saturn (5.6846 [+ or -] 0.0001) 123.27 x [10.sup.26] Jupiter (1.8986 [+ or -] 0.0001) 124.47 x [10.sup.27] Sun (1.9884 [+ or -] 0.0002) 131.42 x [10.sup.30]  Celestial body S d, % 15 Eunomia (A) [106; 2] 0.037 Mimas (S) [106; 2] 0.216 Miranda (U) [107; 2] -0.195 10 Hygiea (A) [107; 2] 0.093 Enceladus (S)  -0.204 2 Pallas (A) [108; 2] 0.001 4 Vesta (A) [108; 2] 0.175 Tethys (S) [109; 2] 0.028 1 Ceres (P)  -0.045 Dione (S)  0.091 Umbriel (U)  0.091 Ariel (U)  0.209 Charon (P) [110; 2] -0.064 Iapetus (S) [110; 2] 0.090 Rhea (S)  -0.144 Oberon (U)  0.108 Titania (U) [111; 2] -0.197 Haumea (P) [111; 2] -0.090 Pluto (P) [112; 2] 0.018 Eris (P)  -0.150 Triton (N)  0.062 Europa (J)  -0.105 Moon (E) [114; 2] -0.175 Io(J) [114; 2] 0.001 Callisto (J) [114; 2] 0.166 Titan (S)  -0.078 Ganymede (J)  0.001 Mercury  -0.164 Mars [116; 2] -0.026 Venus [118; 2] 0.001 Earth [118; 2] 0.160 Uranus [121; 2] -0.099 Neptune [121; 2] 0.041 Saturn [123; 2] -0.186 Jupiter [124; 2] -0.024 Sun [131; 2] -0.061 Table 2: The rest masses of the electron, proton and the W/Z-bosons and the 5/values (6) of the nearest spectral nodes. The relative deviation d = (in (m/[m.sub.p])-S)/S is indicated in percent. Particle Rest mass m, MeV/ ln (m/ S d, % [c.sup.2]  [m.sub.p]) electron 0.510998910 [+ or -] -7.515 [-7; -2] -0.206 0.000000013 proton 938.27203 [+ or -] 0.000  0.000 0.00008 W 80398 [+ or -] 25 4,451 [4; 2] 1,089 Z 91187.6 [+ or -] 2.1 4,577 [4; 2] 1,711
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|Publication:||Progress in Physics|
|Date:||Jan 1, 2010|
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