Copyright © Klaus Piontzik  
German Version 
Our solar system also has a concentric structure with its planetary orbits. The distances in astronomical units (AU) for the first 8 planets are compiled from the common literature. The individual planets are arranged and numbered according to their distance. 
Planet  Nr  Distance  ln(Distance) 
(AU)  
Mercury 
0 
0,3871 
0,94907222 
Venus 
1 
0,723 
0,32434606 
Earth 
2 
1 
0 
Mars 
3 
1,524 
0,42133846 
Jupiter 
4 
5,203 
1,64923538 
Saturn 
5 
9,582 
2,25988634 
Uranus 
6 
19,201 
2,95496236 
Neptune 
7 
30,047 
3,40276282 
AU = astronomical unit = 149,597,870 km = distance Earth  Sun The table gives the following function y for the logarithms of the distances: 
Illustration 8.2.1 – Logarithmic planetary orbits
The dashed line in the diagram represents an approximate straight line.
As can be seen, there is already good linearity of the values.

Illustration 8.2.2 – Linearization
The parameters of the approximation
line are determined using the same procedure as described for the sun shells.
The equation for the approximation line is: y = 0.62169 · x – 0,94907 The following applies to the planetary orbits: Distance = 0.3871 · e^{0.62169x} [AU] This results in the following values: 
Planet  Nr  Distance  Distance 
[AU]  [AU]  
calculated  
Mercury 
0 
0,3871 
0,3871 
Venus 
1 
0,723 
0,7208087 
Earth 
1,5 
1 
0,98359983 
Mars 
2 
1,524 
1,34219887 
Jupiter 
4 
5,203 
4,65383058 
Saturn 
5 
9,582 
8,66577519 
Uranus 
6 
19,201 
16,1363114 
Neptune 
7 
30,047 
30,047 
The entire situation for the planetary orbits then looks like this:
Illustration 8.2.3 – Planetary orbits as an efunction
The gap for n = 3 is remarkable.
The distance to it is 2.5 AU.
The asteroid belt extends from 2.0 to 3.4 AE. The average is 2.7 AU. The model therefore corresponds very well to real events. And it shows that the asteroid belt is part of the oscillation system of the planets. And it also shows that the procedure of logarithmization and linearization is not an arbitrary act, but rather reveals the harmonic structures of a system. 
Planet  Nr  Distance  ln(Distance) 
(AU)  
Mercury 
0 
0,3871 
0,94907222 
Venus 
1 
0,723 
0,32434606 
Earth 
2 
1 
0 
Mars 
3 
1,524 
0,42133846 
Asteroid belt 
4 
2,7 
0,99325177 
Jupiter 
5 
5,203 
1,64923538 
Saturn 
6 
9,582 
2,25988634 
Uranus 
7 
19,201 
2,95496236 
Neptune 
8 
30,047 
3,40276282 
Pluto 
9 
39,482 
3,67584487 
The table gives the following function for the logarithms of the distances:
Illustration 8.2.4 – Logarithmic planetary orbits
The dashed line in the diagram again represents an approximation straight line. As can be seen, there is already a very good linearity behavior of the values. The values ??are linearized and result in the following diagram: 
Illustration 8.2.5 – Linearization
The equation for the approximation line is:
y = 0.52856 · x – 0.94907 The following applies to the planetary orbits: Distance = 0.3871 · e^{0.52856 · x}[AU] This results in the following values: 
Planet  Nr  Distance 
new  [AU]  
Mercury 
0 
0,3871 
Venus 
1,3 
0,723 
Earth 
1,9 
1 
Mars 
2,6 
1,524 
Asteroid belt 
3,75 
2,7 
Jupiter 
5 
5,203 
Saturn 
6,1 
9,582 
Uranus 
7,5 
19,201 
Neptune 
8,25 
30,047 
Pluto 
8,75 
39,482 
The new situation for the planetary orbits then looks like this:
Illustration 8.2.6 – Planetary orbits as an efunction
There is still the possibility
of better adjustment for the numbering values. Based on the previous linearization,
exact numbers can be calculated, just as was already seen with the sun shells.
The table below shows the calculated numbering values ??as well as the approximate numbers and the initial numbers. 
Planet  Nr  Distance  Nr  Nr 
alt  [AU]  calculated  approximately  
Mercury 
0 
0,3871 
0 
0 
Venus 
1 
0,723 
1,3 
1,182 
Earth 
2 
1 
1,9 
1,796 
Mars 
3 
1,524 
2,6 
2,593 
Asteroid belt 
4 
2,7 
3,75 
3,675 
Jupiter 
5 
5,203 
5 
4,916 
Saturn 
6 
9,582 
6,1 
6,071 
Uranus 
7 
19,201 
7,5 
7,386 
Neptune 
8 
30,047 
8,25 
8,233 
Pluto 
9 
39,482 
8,75 
8,750 
The more precise function for the planetary orbits is shown in the following illustration 8.2.7
Illustration 8.2.7 – Planetary orbits
There is a mathematical relationship between orbital radii and orbital times for the planets that also allows the orbital times to be represented as an efunction 
There is a mathematical relationship between the orbital radii and orbital times of the planets, which Johannes Kepler (*Dec. 27, 1571  ?Nov. 15, 1630) discovered at the beginning of the 17th century, a German natural philosopher, mathematician, and astronomer. It also allows the circulation times to be represented as an efunction. 
The general rule is: 

The same applies to Mercury: 

By rearranging, the equation for the orbital periods of the planets results:
The following applies: r/r_{0} = e^{ax} und x = calculated numbering 
Inserting all sizes provides: 
In 2005, US astronomers discovered a new planet behind Pluto
made of ice and rock. It is the tenth planet in our solar system.
Based on the existing model, the distance to the sun can be determined. The following applies to the planetary orbits: Distance = 0.3871 · e^{0.52856x}[AU] The following applies: x = 10. Based on the previous values, a tolerance of ±0.25 is assumed. This gives the range for the distance of the tenth planet between 67 AU und 87 AU. DThe exact value for x = 10 is 76,44 AE. 
200 sides, 23 of them in color 154 pictures 38 tables Production und Publishing: ISBN 9783735738547 Price: 25 Euro 
