fractal hierarchical structure in solar system arrangement 2004

8/13/2019 Fractal Hierarchical Structure in Solar System Arrangement 2004

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VIII Triennial International SAUM Conference

on Systems, Automatic Control and Measurements

Belgrade, Serbia, November 5-6., 2004

pp.118-121.

Fractalhierarchical Structure in Solar System Arrangement

A.S. Tomic

Peoples Observatory, Kalemegdan, Gornji grad 16, 11000 Belgrade, Serbia & Monteneg roE-mail:[email protected]

Abstract In the paper [1] were demonstrated the ordering byFibon acci golden number 2/)51( += in Solar system, and

developed general form of Titius-Bode rule: knkn

rr = ,

...3,2,1, =kn In this paper are analyzed some relation between Fib o-

nacci and Lu cas numbers. We extracted as important property

ratio 5/ nn FL , which only for 3,2,1=n differ and posses whole

number values just 1,3,2. The recurrent formulae for linear and

squared form for both, the Fibo nacci and Lucas numbers were

developed, and hierarchy generic form ulae which gives higher order

terms via lower: 2)1( +

+=k

n

knk

nknLkLL what suggest fractal

hierarchical structure in ordering of distances in Solar system, up to

nearest star.

Key words : Fractals, hierarchy, Fibonacci numbers, Lucasnumbers, planetary di stance s, Solar system.

I. INTRODUCTION

It is known that the unifying concept underlying fractals,

chaos and power laws is self-similarity, i.e. invariance

against changes in scale (or size). Many laws of nature and

innumerable phenomena in world around us have this at-

tribute self-similarity [2]. This property of the Solar sys-

tem were demonstrated into ordering of planetary distances

from the Sun by the Fibonacci golden mean 2/)51( += ,

in our previous paper [1]. We shown that planetary dis-

tances can be expressed by terms of Fibonacci numerical

series nF , defined as:

nnn FFF += ++ 12 , 121 ==FF (1)

by formula:

)( 100 +== nnn

n FFrrr (2)

where are 0r distance of Mercury. With regard to nature

of gravity potential it can be expressed by distance to the

Sun of each planet ( kr ), as:

kn

kn rr = , ...3,2,1, =kn (3)

This is typical self-similarity invariance, which confirmed

fractal structure in Solar system ordering. The universe as

whole posses same property, but to the distance of ap-

proximately9103 light years [3], [4]. Regarding this fol-

lows that Solar system must be fully ordered fractal and

hierarchical. We made attempt to show it.

II PROPERTIES OF NUMBERS FIBONACCI AND LUCAS

For most properties of the Fibonaccis number see for ex-

ample [5], [6]. Here we wish extract interesting fact that each3rdterm of Fibonacci numerical series can be divided with-

out of residue by 2, each 4thterm by 3, each 5thterm by 5,

each 6thterm by 8each

thn term by nF , i.e.

INTEGERFF nnk = / . (4)

For k=1 follows Fibonacci series, for k =2 obtain Lucas

numerical series, de fined as:

nnn LLL += ++ 12 , 3,1 21 == LL (5)

Started from n=3 , for k=2,3,4 we obtain integers

given in Table 1. From Table 1 it is clean thatnnn

LFF =

/2

etc. as it is given in last row.

Table 1.

nnn FF /2 nn FF /3 nn FF /3 nn FF /4

34 4

2+1 18-1 4

3+8

47 7

2-1 47+1 7

3-14

511 11

2+1 123-1 11

3-22

618 18

2-1 322+1 18

3-36

729 29

2+1 843-1 29

3-54

847 47

2-1 2207+1 47

3-94

976 76

2+1 5778-1 76

3-153

10123 123

2-1 15127+1 123

3-246

11199 199

2+1 39603-1 199

3+398

12322 322

2-1 103682+1 322

3-644

nL n

nL )1(2

n

nL )1(

2 +

nn LL 23


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Mutual ratio of each successive 2nd

, 3rd

, 4th Fibonacci number

With small discrepancy, or really, are valid next relations,

to:

13

3 )1(3 nn

nn LLL +=

24

4 )1(4 nn

nn LLL += (6)

etc, or in general case:

2)1(

+

=

k

n

nk

nk

k

n LkLL (7)

This formula contents important new quality it gives

higher order terms expressed by very beginning terms. The

hierarchy appears in fully brightness because second term

in (7) for bigger k very rapidly converges to zero:

nk

k

n LL = (8)

It is useful find the ratio of Lucas and Fibonacci numbers

with same index, nn FL / , (Fig. 1.)

Fig. 1. Ratio nn FL / , have only tree integer : 1, 3, 2.

Only tree values in this ratio are integers, after n=4 ratio

rapidly converges to 5 , and:

5= nn FL . (9)

For our purposes it is useful transform Fibonacci numbers

into Lucas numbers:

5/5/ knnknk LLF == . (10)

On this way we introduced hierarchy in fractal structure of

planetary distances from the Sun. From Fig. 1 it follows that

first, second and third ratio presents separate whole, differ-

ent from all other values.1

1 To compare with quantum physics we can think on the differenceof first two Lucas ( 212 =LL ) and Fibonacci number

( 012 =FF ) in sense of spin numbers. In this case, both numerical

series have charact eristics of the same Bose - Einstein statistics. It

can be used as argument for

second hierarchical ordering, into first - fractal ordering in plane-

Combined all cited properties we can conclude on the re-

ality of triplets of Lucas numbers: (1,3,4), (7,11,18),

(29,47,76)Because the parity independence to the left

right transformation is present in gravity law (2

r ), wecan propose that these triplets would be doubled.

III DISTANCE OREDERING IN GRAVITY SYSTEM

We developed minimal distance ( 0r ) which really can be

used in spiral ordered growth of distances, expressed by

Schwarzschilds radius (0

R ) of central body- the Sun, and

from realized position of exis ting planetary bodies [7]:

30 2.2400

= eRr = 0.000 4584 (A.U.) (11)

By simple calculation from values of Fibonacci and Lucas

numbers nn LF, of quantities nn LL 1 , 11 + + nn LL and

111 )( + + nnn FLL it is easy develop the formulas:

nnnnnn FLLLL )1(2)( 1111 += + (12)

n

nnn LLL )1(112 += + (13)

n

nnn LLL )1(22 = (14)

where second term on right side by increasing exponent can

be ignored.

Formula (3) contents fundamental property of symmetri-

cal potential, that only ratio is given and that borders cond i-

tion gives eventually minimal distance. In purpose to utilize

this property it is useful the properties (2) and (9) transform

to:

5/)( 11 + += nnn

LL (15)

and using:

1111 )( + += nnnnn FLLLL (16)

further transform to:

nnL = (17)

Now, using formula (17) we can each new distance o btain

as:

n

n

n Lrrr == 00 (18)

For odd values of index it gives using (13):

nnnn

n LLrLrrr ===

1012012

012 (19)

for even values using (14):

nnn

n

n LLrLrrr === 0202

02 (20)

tary distances.


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Calculated values are presented in Table 2. In last column

were used hierarchical form of Lucas numbers, given as (8),

for Lucas numbers 7, 18, 29, where is denoted

1717 E , 2727 E , etc.

Table 2.a

Possible planetary distances by the fractal hierarchical model

Table 2.b

Distances in dual triads 8 and 9

Bearing in the mind duality as property of substance, in

quantum physics as particle-wave, we can consider that

here duality is possibility of position state determination

by odd even number. In this case it is justified except dual

triplets of Lucas numbers as distance entry, what is made in

the Table 2.

Table 3

Mean planetary distances from the Sun

IV FRACTAL HIERARCHY OF SOLAR SYSTEM

No =)(AJr

Nr=0

ji LL

ji LLN =

+

n

jLN=

0.00000 0x1 0

0.00046 1x1 1

0.00136 1x3 3

0.00410 3x3 97E1

0.00547 3x4 12

0.00638 4x4 16-218 E1

0.01230 4x7 28-1 29 E1

0.02338 7x7 49+2 7 E2

0.03575 7x11 77+1

0. 05455 11x11 1 21 -2

0. 09030 11x18 1 98 -10 .14760 18x18 322+2 7 E3 18 E2

0.23 883 18x29 521+1

0.38643 29x29 843-2 29 E2

0.62 526 29x47 136 4 -1

1.012 47x47 2208+2 7 E4

1.637 47x76 3571+1

2.647 76x76 5779-2 18 E3

4.286 76x123 9350-1

6.934 123x123 15127+2 7 E5

11.22 123x 199 24476+1 29 E3

18.154 1 99x199 39603-2

29.374 199x322 64079-1

47.528 322x322 103682+2 7 E6 18 E4

76.9 322x521 167762+1

124.4 521x521 271448-2

201.3 521x843 439203-1

325. 8 843x 843 710649+2 7 E6 29 E4

527.1 843x1364 1149852+1

852.8 1364x1364 1860496-2 18 E5

1379.9 1364x2207 3010348-1

2218.9 2207x2207 4870849+2 7 E7

3590.3 2207x3571 7881197+1

5809.3 3571x3571 12752041-2

9399.6 3571x5778 20633238-1 29 E5

15208.9 5778x5778 33385284+2 7 E8 18 E6

24608.5 5778x9349 54018522+1

39816.8 9349x9349 87403801-2

64424.9 9349x15127 141422323-1

104241.7

15127x1512

7 228826129+2 7 E9

168666.7

15127x2447

6 370248452+1

272908.1

24476x2447

6 599074576-2 18 E7 29 E6

No =)(AJr

Nr = 0 ji LL ji LLN =

+

=NnjL= )(lyr

441575.1 24476x39603 969323028-1 7.0

714483.5 39603x39603 1568397609+2 11.3

1156058.6 39603x64079 2537720637+1 18.3

1870542.1 64079x64079 4106118241-2 29.7

3026601.0 64079x103682 6643838878-1 50.0

8

4897144.0 103682x103682 10749957124+2 18 E8 80.1

9

18 E9

129.4

209.4

338.9

548.3

887.1

1435.4

Planet )(AUrn

Planet )(AUrn

Mercury 0,387 Jupiter 5,203

Venus 0,723 Saturn 9,555

Earth 1,000 Uran 19,218

Mars 1,524 Neptune 30,111

Asteroids 2,709 Pluto

(Asteroids)

39,530


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Mean planetary distances from the Sun are given in Table

3. The comparison of the data in Table 2 and Table 3 gives

possibil ity for following interesting conclusions.

(1) Planetary distances are determined by dual comb i-

nat ion of Lucas numbers (0,1), (1,1), (1,3), (3,3)

etc.

(2) Six duals can be covered into one triplet of Lucasnumbers (1,3,4), (7,11,18) etc.

(3) Each triplet can be determined as hierarchical jump

of Lucas number 18, at position of last dual ( 118 ,218 , 318 ).

(4) First triplet, except his last dual, is sited into central

body.

(5) Planets exist in triplets 3,4 and few known planetary

bodies in triplet 5.

(6) Asteroid belts are on the position of last dual in

triplet 3,4 and comet cloud at same position in trip-

let 5.(7) By quantization of angular momentum in solar

system positions 1,2 are vacant and first of them

( 1476.0=r AU) is sited at last dual position invacant triplet 2.

(8) The nearest star is at position of 718 , what suggest

that:

(a) 6 triplets determined solar system, and its

radius approximately 15 000 AU,

(b) next, 7thtriplet, presents hierarchical (qual-

ity) jump.

(9) Triplets 8,9 contents interstellar distances in stellar

cluster and association, up to 1500 light year.

(10) Calculated for 10th triplet are obtained dimension

and distances of galaxies (25 000 l.y. to 8.8 million

l.y.), what is in good correlation with [3],[4].

We emphasize that minimal distance used in calculation

is derived from physically minimal possible radius for given

mass of central body, and from existing planetary distances.

From this reason we think that here is not in work random

congruence.

CONCLUSIONS

Starting on Titius Bode rule we found that Solar sys tem

is kinetically ordered by Fibonacci numbers [1]. Here we

continued research to the Lucas numbers, for which found

properties of golden ratio and direct connection with Fib o-

nacci numbers. Transformation between Fibonacci and

Lucas numbers extracted possibility that Lucas numbers

contents property of hierarchical connecting, very impor-

tant for description of the Universe [3]. Except that, ratio

Fibonacci / Lucas number (with same index n) gives integer

(1,3,2) only for first tree, for all other gives 5 . It is arg u-

ment for opinion that fractal hierarchical triplets are in the

base of spatial ordering for matter.

REFERENCES

[1] A.S.Tomic, Phlogist on, No 7, 151, 1998.(in Serbian)

[2] M.Schroeder,Fractals, Chaos, Power Laws, Freeman &Co.New

York 1991.

[3] L. Pietroneto, Physica, 144, A, 257.1987.

[4] P. Grujic, Serbian Astron, J. 165, 45, 2002; Vasiona, 50, 5, 125.

2002.

[5] N.N. Vorobev, Chisla Fibonacci , Nauka, Moskva (in Russian),

1978.

[6] R.Knott, I.Galkin et other, at:

http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci

[7] A.S.Tomic, Discreetisation of state in gravity macroscopic

systems, (in Serbian), JAA 41-01, 2001.

fractal hierarchical structure in solar system arrangement 2004

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