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Geometry of the Solar System, part 2
by Frank Hoogerbeets

15 January 2017

In part 1 we primarily focussed on geometry and numerical sequences and how we can recognize them in the Solar System. We have also seen the significance of number 15 and that it defines the orbital radii of the planets as a mathematical model. However, there is one question mark we have not addressed yet. Is there a planet at about 15 million kilometers from the Sun? Apparently, astronomers have been aware of this possibility.

In 1859 the French astronomer Urbain Jean Joseph Leverrier suspected an unknown planet between Mercury and the Sun to be responsible for the deviation in the orbit of Mercury. Leverrier also discovered Neptune 13 years earlier that was responsible for the deviation in the orbit of Uranus. For this hypothetical planet between Mercury and the Sun, he had already come up with the name Vulcan, the Roman God of fire and forgery, because of its close distance to the Sun. However, the planet was never found and the deviation in Mercury's orbit was later explained by Einstein’s theory of relativity.

The absense of a planet between the Sun and Mercury probably explains why astronomers have overlooked the significance of number 15 and its key role in the mathematical model of our Solar System, because truly the orbital velocity and period of all planets relate to this number, as we shall see.

Distance, Velocity and Time

Again, we will go back to basics to see how velocity relates to distance and time and also how the vector of velocity of an orbiting body relates to the vector of acceleration i.e. the orbital radius.

The first relation is not so difficult to understand. We know that with a specific velocity it takes a certain amount of time to cover a specific distance, be it by foot, by car or by any other means of transportation. If we know the velocity by which we travel and we know the distance then we can easily find out how long it will take us to reach our destination.

For example, if we travel 100km/h and the distance is 200 kilometers, then it will take us two hours to reach our destination. We can also put this in formula:

200 100 = 2 {200} over {100} = 2

In physics symbols are used to express these quantities. For velocity the letter v is used, for distance the capital letter S and for time the letter t. We will use the same symbols here. The formula for the relation between distance, time and velocity will then look like this:

S v = t {S} over {v} = t
formula (2.1)

Depending on what we want to know, we can rewrite this formula. For example, if we know the distance and the time we want or need to travel, then the matching speed or velocity will be:

v = S t v = {S} over {t}
formula (2.2)

Acceleration

A somewhat more complicated concept is acceleration (symbol a) and how it relates to velocity and distance. An orbiting planet has an orbital velocity. This is the fairly constant speed by which that planet revolves around the Sun. At the same time there is a constant acceleration along the imaginary line from the planet to the Sun, also called the orbital radius vector, for which we use the symbol r. This constant acceleration is what keeps the planet revolving.

<image>
figure (2.1)

From this we can deduce that the velocity relates to the acceleration as the orbital radius relates to the velocity, or in formula:

v : a = r : v v:a=r:v
formula (2.3)

This means that the acceleration is inversely proportional to the orbital radius:

a = v 2 r a = { v^{2}} over {r}
formula (2.4)

I cover these basics so that you know where these formulae (2.2 and 2.4) come from, because we need both to find out the orbital velocity and period of any given planet.

Time Relativity

The quantities that we have discussed so far, are distance (S), velocity (v), acceleration (a) and time (t). These are the basic quantities of the Universe and they allow us to describe and measure the motion of objects anywhere, be it on Earth or in Space. But everything begins with observation. What can we observe and what not? Of these quantities, we can observe distance as the space between two objects. We can observe velocity and acceleration as the motion of an object relative to another object.

But what about time? Can we see or observe time? No. All we can do is create a time-scale with different markers and indicators, like a clock, and calibrate its mechanism to the frequency of an existing cycle, such as the motions of Earth and the Moon. We can then observe the indicators on their cyclical path along the markers and count the cycles as a definition of time. We could in fact create any time-scale we would like and use it to make schedules and appointments, as long as everyone uses the same clock. In essence there are only cycles of moving objects in the Universe. It is up to the observer to pick out the cycle that is most practical as the basis for a time-scale definition.

If you find it difficult to follow this reasoning, then imagine a star, like our Sun, with only one planet. There are no other planets, no moons, no background stars, nothing. Just one planet revolving around one star. How would we define time? How do we know when the planet has completed one revolution? There is no way to know, because there is nothing we could use as a reference to build a time-scale. This is important to understand because it helps us realize that everything in the Universe is in essence only based on cycles and that time is but a definition of cycles intertwined.

Velocity of the Planets

We can use formula (2.2) to find the orbital velocity of a planet. In order to do so, we would need the distance (S) and the time (t). The distance is the path of the planet around its star. But what is the time? As dicussed earlier, with only one star and one planet there is no reference, so let us simply define the time as 1. We could use any value here, but 1 is simplest and it allows us to rewrite formula (2.2):

v = S 1 v = S v = {S} over {1} dlrarrow v = S
formula (2.5)

As we can see, if the orbital period of a planet is 1, then the orbital velocity equals the size of the orbital path. As in our mathematical model this orbital path is a circle, the size or distance equals the circumference, which in mathematics is calculated by 2 times π times r. Therefore we can rewrite the formula to:

v = 2 π r v= 2 cdot %pi cdot r
formula (2.6)

From this we can conclude that the orbital velocity is related only to the orbital radius. In truth, this velocity is the result of the constant acceleration along the orbital radius, as we have seen with formula (2.4). For the first planet we could rewrite this formula to see how the acceleration relates to the orbital radius and circumference:

a = 4 π 2 r 2 r a = {4 cdot {%pi^{2}} cdot r^{2}} over {r}
formula (2.7)

Keep in mind that formula (2.6) and (2.7) only apply to the first planet. So what if there is more than one planet? How do we determine the amount of acceleration of the second, third or fourth planet? We can do this by setting up a matrix with formula (2.7) to see how the acceleration of any planet n relates to the acceleration of planet 1. Through substitution we can determine the velocity:

v n 2 r 1 = 4 π 2 r 1 2 r n v n 2 = 4 π 2 r 1 3 r n v n = 4 π 2 r 1 3 r n {v_{n}^{2}} over {r_{1}} = {4 cdot %pi^{2} cdot r_{1}^{2}} over r_{n} dlrarrow v_{n}^{2} = {4 cdot %pi^{2} cdot r_{1}^{3}} over {r_{n}} dlrarrow v_{n} = sqrt{{4 cdot %pi^{2} cdot r_{1}^{3}} over {r_{n}}}
formula (2.8)

As you can see from formula (2.8) we only need to fill in the orbital radius of planet 1 and planet n to find out the orbital velocity of planet n. The mathematical model as discussed in part 1 gives us the orbital radius of each planet. We learned that the numerical sequence is based on number 15, which would be the orbital radius of the first planet in this mathematical model. In reality in our Solar System, this planet does not exist, but we use this number to find out the orbital velocities of the other planets:

v n = 4 π 2 15 3 r n v_{n} = sqrt{{4 cdot %pi^{2} cdot 15^{3}} over {r_{n}}}
formula (2.9)

As you can see, our mathematical model gives us a constant, which in formula (2.9) is the numerator. Calculating this numerator gives us:

133239.659415

From table 1.1 in part 1 we can take the orbital radius of each planet and calculate the orbital velocity with formula (2.9). The results are shown in the following table:

planet math.orbit radius astr.orbit radius math.orbit velocity astr.orbit velocity
?
15
-
94.25
-
Mercury
60
57.9
47.12
47.87
Venus
105
108.2
35.62
35.02
Earth
150
149.6
29.80
29.78
Mars
240
227.9
23.56
24.13
Ceres
420
414.0
17.81
17.94
Jupiter
780
778.4
13.07
13.06
Saturn
1500
1426.7
9.42
9.64
Uranus
2940
2871.0
6.73
6.79
Neptune
4380
4498.3
5.52
5.43
Pluto
5820
5906.2
4.78
4.67
Eris
11580
10123.4
3.39
3.44
(table 2.1)

In part 3 we will discuss the orbital time.

Copyright © 2006-2017 Frank Hoogerbeets. You have my permission to copy and distribute this article as long as you do not change its content including this copyright notice.

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