PiMath.de Planetary Systems of the Earth 1
Classisc Systems
 
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3.8 - Layers of the atmosphere and Laplace

Generally, you can put 11 elevant atmospheric layers together. The individual layers are sorted and numbered according to height.

Layers of the atmosphere

Right in the table is the logarithm naturalis for the respective heights.


The logarithm of height is represented as a function of numberin:

 logarithm of height

Illustration 3.8.1 – logarithm of height

 

The function in illustration 3.8.1, seems to be not very linear on first view ex-cept in few parts. If you look but closer to the course so you can see

a)Between the points 2 to 7 the slope is nearly constan

b)Between points 1 and 2 as well as between points 2, 8 and 9 is the slope so large, there can still a point be inserted, to flatten the slope

c)Between points 9 and 10 the slope so large that there still several points can be inserted. Therefore, point 10 is eliminated first

d) Between point 7 and 8 the slope is so small that there the numbering can be used on a half, increasing so the slope

The following table shows the layers that are corrected in the numbering arranged by height:

Layers of the atmosphere

The newly added layers at the numbers 2 and 9 are clear to see in the table. The corrected function looks like this:

Linearisierung

Illustration 3.8.2 – logarithm of height

 

The red line in illustration 3.8.2 represents a linear function that was obtained by linear regression from the corrected table. Like is to see the layers values match well with the approximation function.
It can be used here so a linear function for the atmospheric layers as a solution.

The following applies to the additive constant:

b = ln wMin = ln H0 = ln 20 = 2.995732

The following applies to the slope of the line:

Δ y = ln wMax – ln wMin = ln 320 – ln 20 = 2.772588
Δ x = n = 10

a = Δ y/Δ x = 2.772588/10 = 0.277258

A linear function for the atmospheric layers can be used here as a solution approach. In general, the following applies to the straight line from Figure 3.8.2:

y = ln(Höhe) = ax + b

The values ??found are inserted into the straight line equation:

 

It results: ln(Hight) = 0.277 x + 2.9957

 

By rearranging you get:

 

3.8.1 - Equation: Hight = 20 e0.277x [Km]

 

Putting x = n then following function arises for equation 3.8.1:

Height as e-funktion

Illustration 3.8.3 – Height as e-funktion

 

Equation 3.8.1 has all the properties that are necessary to get as a solution function of Laplace's equation in consideration to 2.11.3.

Thus the atmospheric layers represent a solution of Laplace's equation, specially for the radial part.

In consequence the following sentence can be set up:

 

3.8.2 - Theorem: The atmospheric layers are an expression of an oscillation phenomenon

 

On the equation of 3.8.1 can be still made simplifications.

Es gilt: 0.277 = 3.6-1 = 5/18


When all values are used:

 

3.8.2 - Equation: Equation Hight [Km]

 

It can be made to the following relation (see Theorem 5.1.5):

rik= RE/5 rik = inner core and RE = 6371 Km

and it still applies:

5100 = RE– rik = 4/5 RE = 4rik

5100 = 255 20 ==> 20 = 4/255r
ik


Then you can write for the atmospheric layers:

 

3.8.3 - Equation: Equation Hight [Km]

 

Then you can continue to write for the atmospheric layers

 

3.8.4 - Equation: Equation Hight [Km]

 

The layers arranged by height and the calculated values result in the following table:

Layers of the atmosphere

The mean error of the calculated values for the layers is below 2 percent.

In addition, even 2 layers arise


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 Planetary Systems 1

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Der Autor - Klaus Piontzik