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FECore Errors in LASER Level Experiments

Saturday, September 29, 2018 - 23:43 | Author: wabis | Topics: FlatEarth, Science
FECore's official conclusion from their LASER Level Experiments "no curve is detectable" and "FECORE measurements disprove the WGS84 model" (WGS84 is the official globe model) is based on errors in the calculations, using Snell's law wrong, assuming a flat earth and a wrong physical model of the atmosphere. In this article I show all errors made by FECore to come to the false conclusion.

Introduction

FECore does not explain how they did their refraction calculations. The correct refraction calculation is the most essential part of the experiment. I figured out, how they did their refraction calculation. And I found fatal errors in FECore's document concerning refraction.

So their results and conclusions are all wrong.

 Link to FECore's final results of the LASER Level Experiments

Data used for the Examples

I will use the values of the 4th measurement on the lake Ijssel as an example as published, but all their refraction calculations are wrong in the same way!

Distance 28.68 km
Laser height 2.85 m
Laser visible height at target 0.85 m
Temperature at 2.85 m height 11.8°C
Temperature at 0.85 m height 7.6°C
Refraction index n1 at 11.8°C 1.000283508
Refraction index n2 at 7.6°C 1.000287811

Published Refraction calculation

This is an excerpt from the FECore document:

Angle of incidence (theta1): 0.0040°
Refractive index calculation
(based on Edlén Equation)

n1 = 1.000283508 (445 nm, 11.8°C, 74%)
n2 = 1.000287811 (445 nm, 7.6°C, 85%)

Angle of refraction is calculated with Snell's law:

sin theta2 = (n1 * sin theta1)/n2 = 0.003999983 degrees

Angle of deviation = 0.000000017°

We concluded the ambient conditions refracted the laser beam downward by maximum of 0.494 mm (0.0194 inches)

FECore's Refraction Calculations

Here is what I figured out about the refraction calculation of FECore:

  1. FECore's Refraction Model
  2. Calculating the Refraction Indexes of Air
  3. Calculating the Angle of Incidence
  4. Applying Snell's law
  5. Calculating the Angle of Deviation
  6. Calculating the Refraction Correction

FECore's Refraction Model

According to the refraction math that FECore applied, they assumed a flat earth and that the air consists of 2 flat layers with constant refractive indices each, derived from 2 atmospheric measurements at the LASER site and the target site respectively.

  1. The lower layer with refractive index n2, calculated from atmospheric measurements at the Target, is assumed to go from ground to h1 = 2.85 m, which is just below the height of the laser. Above that, where the LASER is placed, they assumed a layer with refractive index n1, calculated from atrmospheric measurements at the LASER, see Calculating the Refraction Indexes of Air.
  2. The angle of incidence θ1 was calculated from from h1, h2 and d as described at Calculating the Angle of Incidence.
  3. They applied Snell's law to calculate the angle of refraction θ2 from the angle of incidence θ1 and the refraction indexes n1 and n2, see Applying Snell's law.
  4. The difference between the angle of incidence θ1 and the angle of refraction θ2 from Snell's law gives their angle of deviation γ, see Calculating the Angle of Deviation.
  5. Using the perspective equation they calculated the refraction correction c from d and γ, see Calculating the Refraction Correction.

Wrong: Inappropriate Refraction Model

The FECore refraction model is not an appropriate model for the atmosphere at all.

Refraction in the atmosphere is continuously, even if the earth is flat. It depends mainly on the temperature gradient (continuous change of the temperature with altitude), which results in a corresponding density gradient. In a density gradient a light ray is never straight but bent the whole way. If you have a density gradient, you cannot apply Snell's law like FECore did.

Wikipedia explains in simple terms how terrestrial refraction can be derived and used in practice:

 Terrestrial Refraction; Wikipedia

Calculating the Refraction Indexes of Air

FECore states they calculated the refraction indexes n1 and n2 based on Modified Edlén Equation.

To compute the refraction indexes n1 and n2, depending on pressure (standard 101,300 Pa used), temperature, humidity and LASER wavelength 445 nm, they seem to have used the calculator from the following website (I checked it for 3 measurements and got exactly the values they used):

Refractive Index of Air Calculator
Based on Modified Edlén Equation
https://emtoolbox.nist.gov/wavelength/Edlen.asp

Edlén published 1966 empirical equation for n of standard dry air and corrections for water vapour, based on experimental data. To get an impression of the Edlén formula see here:

https://pdfs.semanticscholar.org/348d/af60a54e53f09ee3899b784e8a37df70aab4.pdf

Calculating the Angle of Incidence

To compute the angle of refraction θ2 by Applying Snell's law you need the angle of incidence θ1:

Wrong: angle of incidence not measured

FECore did not measure the angle of incidence but calculated it by making the following assumptions:

They assumed a flat earth and calculated the dip angle = angle of incidence θ1 measured down from eye level at h1 = 2.85 m LASER height to h2 = 0.85 m target height in d = 28.68 km distance, which yields:

(1)
\theta_1 = \arctan \left( { h_1 - h_2 \over d } \right) = 0{.}0039955216° \approx 0{.}00400°
where'
\theta_1 ' =' 'angle of incidence = dip angle
h_1 ' =' '2.85 m = LASER height, see Data used for the Examples
h_2 ' =' '0.85 m = target height
d ' =' '28,680 m = target distance

Applying Snell's law

Even if we grant the 2 layer model, where Snell's law can be applied, FECore did it wrong. Snell's law says:

(2)
{ \sin\left( \theta_2^{\,\prime} \right) \over \sin\left( \theta_1^{\,\prime} \right) } = { n_1 \over n_2 }

Snell's law

where'
\theta_1^{\,\prime} ' =' 'angle of incidence measured from the perpendicular to the boundary layer
\theta_2^{\,\prime} ' =' 'angle of refraction measured from the perpendicular to the boundary layer
n_1, n_2 ' =' 'refraction indexes of the two layers

Note: Both angles must be measured with respect to the perpendicular to the boundary layer.

FECore calculated the angle of refraction as follows:

(3)
\theta_2 = \arcsin\left( { n_1 \cdot \sin( \theta_1 ) \over n_2 } \right) = 0{.}00399998279°
where'
\theta_2 ' =' 'angle of refraction
\theta_1 ' =' '0.00400° = angle of incidence
n_1 ' =' '1.000283508 = refraction index of top layer, see Calculating the Refraction Indexes of Air
n_2 ' =' '1.000287811 = refraction index of bottom layer, see Calculating the Refraction Indexes of Air

Fatal Error: wrong angles used in applying Snell's law

FECore did not use the correct angle of incidence measured from the perpendicular to the boundary layer, but the angle measured up from the boundary layer. So they should have used \theta_1^{\,\prime} = 90° - 0{.}00400° instead, which yields a much bigger angle of refraction \theta_2 = 90° - \theta_2^{\,\prime}:

(4)
\theta_2 = 90° - \arcsin\left( \sin( 90° - \theta_1 ) \cdot { n_1 \over n_2 } \right) = 0{.}168°

So with the right application of Snell's law the angle of refraction would be θ2 = 0.168°, which results in a angle of deviation of γ = 0.004° − 0.168° = −0.164°, and not only 0.000000017° as published!

Using the correct angle of deviation γ, the refraction correction would be c = −82.1 m, and not c = 0.494 mm. This is 166,000 times more than what they published.

A refraction correction of c = −82.1 m would mean, that the target is 81.3 m under the surface of the flat earth. This result proves that the refraction model FECore uses is wrong.

Sloppy: wrong direction of refraction stated

FECore stated: "We concluded the ambient conditions refracted the laser beam downward by maximum of 0.494 mm."

But if the angle of refraction θ2 = 0.00399998279° is smaller than the angle of incidence θ1 = 0.00400° then the laser beam is refracted upward.

Sloppy: invalid equation

The publised formula for Snell's law:

sin theta2 = (n1 * sin theta1) / n2 = 0.003999983 degrees

looks different than (4). The above published formula is not a valid equation, because sin theta2 (= sin 0.003999983°) is not equal 0.003999983 degrees. What they ment to write is the above formula (4).

Calculating the Angle of Deviation

Their "angle of deviation" is simply:

(5)
\gamma = \theta_1 - \theta_2 = 0{.}00000001721°
where'
\gamma ' =' 'wrong angle of deviation
\theta_1 ' =' '0.00400° = angle of incidence
\theta_2 ' =' '0.00399998279° = wrong angle of refraction

Using the correct angles we get:

(6)
\gamma = \theta_1 - \theta_2 = -0{.}164°
where'
\gamma ' =' 'correct angle of deviation
\theta_1 ' =' '0.00400° = angle of incidence
\theta_2 ' =' '0.168° = correct angle of refraction

This would be the correct angle of deviation if the FECore model would be correct, which it is not!

Calculating the Refraction Correction

The refraction correction c is simply the deviation of the laser at the target due to refraction compared with the unrefracted beam. An upward bended beam causes a positive refraction correction value. FECore computed the refraction correction using the following equation, which I have reconstructed from the published values. This formula would be correct if the FECore's refraction model would be correct:

(7)
c = d \cdot \gamma = 0{.}494\ \mathrm{mm}
where'
c ' =' 'wrong refraction correction
d ' =' '28,680 m = distance between LASER and target
\gamma ' =' '0.00000001721° = wrong angle of deviation, see Calculating the Angle of Deviation

Error: the angle of deviation γ must be given in radian units in formula (7), not in degrees!

So without this error the refraction correction would be:

(8)
c = d \cdot \gamma \cdot ( \pi / 180° ) = 0{.}00861\ \mathrm{mm}
where'
c ' =' 'still wrong refraction correction
d ' =' '28,680 m = distance between LASER and target
\gamma ' =' '0.00000001721° = wrong angle of deviation, see Calculating the Angle of Deviation

So the refraction correction should even be about 57 times smaller. Using the correct angle of deviation, obtained by applying Snell's law with the correct angles, we would get a refraction correction:

(9)
c = d \cdot \gamma \cdot ( \pi / 180° ) = -82{.}1\ \mathrm{m}
where'
c ' =' 'correct refraction correction
d ' =' '28,680 m = distance between LASER and target
\gamma ' =' '−0.164° = correct angle of deviation, see Calculating the Angle of Deviation

Note: this refraction is 7.5 times bigger than the refraction correction c = −11.0 m using the mainstream refraction model at standard refraction.

Summary

FECore used the difference between the (wrong) angle of incidence θ1 and the (hence also wrong) angle of refraction θ2 as the angle of deviation γ. In their model refraction takes place only once directly at the laser.

In reality the density gradient of the air acts on the whole way to the target and bends the light to a curve. Such a curve can be approximated by an arc of a certain radius for practical purposes. Over long distances the angle of deviation γ is considerably bigger than what FECore has calculated. In the mainstream refraction model this angle is called refraction angle ρ = −γ. A positive refraction angle means the target appears lifted. The greater the distance, the greater the refraction angle ρ.

In FECore's refraction model the angle of deviation γ does not depend on distance d and the refraction correction c increases linear with distance. In the accepted mainstream refraction model the refraction angle ρ grows linear with distance and the refraction correction c (the negative of the value called Lift absolute in the Advanced Earth Curvature Calculator) grows with the square of the distance d. This is the case for flat earth and globe earth.

Lets recap what we have so far:

Angle of Deviation
published angle of deviation using FECore's refraction model with errors γ = 0.00000001721°
angle of deviation using FECore's refraction model correct γ = −0.164°
angle of deviation (refraction angle) using the mainstream refraction model, standard refraction, at 28.68 km γ = −ρ = −0.022°
Refraction Correction
Published refraction correction using FECore's refraction model with errors c = 0.000494 m
Refraction correction using FECore's refraction model with using radian units but Snell's law error c = 0.00000861 m
Refraction correction using FECore's refraction model without errors c = −82.1 m (not 0.000494 m)
Refraction correction using the mainstream refraction model1), with standard refraction, at 28.68 km c = −11.0 m

1) Show refraction calculation in the Advanced Curvature Calculator

Using the wrong refraction angle, a not working refraction model (target would be 81.3 m below the surface) and calculation errors, FECore concluded, refraction can be neglegted. And therefor the WGS84 model is proven wrong.

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