The Impact of the Earth ’ s Movement through the Space on Measuring the Velocity of Light

Goal of this experiment is basically measuring the velocity of light. As usual we will measure twoway velocity of light (from A to B and back). In contrast to the similar experiments we will not assume that speeds of light from A to B and from B to A are equal. To achieve this we will take into account Earth’s movement through the space, rotation around its axis and apply “least squares method for cosine function”, which will be explained in Section 9. Assuming that direction EastWest is already known, one clock, a source of light and a mirror, is all equipment we need for this experiment.


Introduction
Observe the planet Earth.The Earth orbits the Sun.For this motion we will join the vector v 1 .Sun orbits the center of the Milky Way.For this motion we will join the vector v 2 .In relation to the center of the Milky Way, we can join to the Earth movement sum of vectors 1 2 .

+ v v
It is also known that our Galaxy is moving relative to other galaxies (or to a point in the space outside the Milky Way Galaxy).Similarly, to this motion we could join the vector v 3 .
Denote by v the sum of all these vectors At the end of the sum three points are left, because eventually there may be some other movements.
In the period of 24 h vectors v 2 , v 3 can be taken as constants, while the vector v 1 by making a certain error could also be taken as constant.
Thus for the Earth's motion through the space within 24 h, we can join the constant vector v.
The speed and direction Earth orbits the Sun are known, and let v 0 represent its avarage speed.Suppose that some approximate values for vectors v 2 and v 3 are known as well.On the basis of these values, let suppose that we have inequality (2)

Planning an Experiment
Suppose that an arbitrary point A is given.Earth rotation axis will be taken as the z coordinate, and as the plane xy we will take the plane passing through point A and perpendicular to the z axis.In this case it is natural to take section of the plane xy and z axis as the center of the coordinate system.In addition to point A let the points B and D are given.Line AB lies in the plane xy and parallel to the direction of the Earth's rotation.Distance AB will be marked with L. For the x axis, at some initial time t 0 , we will take the line in the plane xy, parallel to AB.The projection of the vector v in the plane xy denote by v xy .Due to the Earth's rotation the direction of AB will be changed, so that it will be changed the angle, marked by Φ, between the x axis (which remained fixed) and the line AB.Let at point A we have a clock and some source of light.Suppose that speed of light in the direction AB is given by equation ( ) Point D will be chosen so the line AD is parralel to direction South-North.Distance AD is marked by L 1 .
Angle between line AD and z axis we will denote by ϕ.Angle ϕ actually represents Latitude of point A on the Earth's surface, thus it remains unchanged during the experiment.The projection of the vector v on z axis denote by v z (actually v 2 + v 3 , because v 1 is perpendicular on z axis).Assume that the speed of signal in the direction AD is given by equation where c represents "velocity of light in vacuum for a body at rest".Our aim is to find the constant c, vectors v xy and v z .

Conducting an Experiment
In some moment T 0 we will send signal from point A to point B. The angle between the axis x and v xy is marked by Θ.
Once the signal arrived at point B it will be reflected back to point A.
Difference between the time when the signal was being sent from point A, and the time when the signal reached to the point A is denoted by t 0 .
At the same time we will send signal from point A to D and return back to point A. Difference between the time when signal was being sent and reached to point A we will denote by τ 0 .
The same procedure will be within 24 h repeated N (N > 4) times, whereas the time between the two sets of consecutive procedure to be same and equal to 24 h/N.
In that way we will get the series {t i } and {τ i } To the each t i we can join an angle α i between x axis and line AB.
In that way we get the series By assumption (3.1) the speed of the signal c i in the direction AB is equal to If we swap the roles of the points A and B, we would get the same formula as in (6).Therefore it is completely irrelevant whether direction of the vector v xy is equal to direction AB or BA.
We assume that ( ) It would be in principle our experiment.

Computing the Values of c, xy v and Θ
In this section we will deal only with the measurements in direction East-West.Let t i is given by (3.6) and denote the average speed c i (from point A to point B and back to A).
It follows that i c can be written as ( ) where e i represents some experimental error.Replacing The coefficients A, B and Θ will be chosen so the sum of squares has a minimum value.
To acheive our goal we are going to apply Theorem 1 for k = 2.
For the sake of simplicity we've only considered cases when ( ) .
In the similiar way we can get Function Atan () takes values at interval (−Π/2, Π/2).6) it folows that between the values Θ 1 and Θ 2 we have to choose that one for which A 0 > 0. From ( 5) and (6) we can derive values for c and xy v .
We don't know exact direction of vector v xy , thus positive and negative value are assigned to xy v .

Comparison between Two Methods
In this section we will make comparison between "the least squares method" and "the least squares method for cosine function".
Let consider { } i c given by (4.1) as the series of mutually independent measurements.
Let c m represents the mean value of serial { } i c .
( ) If we apply Least squares method, Variance V 1 is given by ( ) and standard deviation σ 1 by 1 1 .

V N σ =
(3) Suppose that to the each c i we joined the time when measurement took place, or rather the angle between the direction of AB and vector v xy .Expected value E 2 (α i ) for "The Least squares method for cosine function" is given by ( ) ( ) ( ) where Let us find Variance V 2 for this method Standard deviation σ 2 for this method is given by If standard deviation σ 2 is bigger then some expected value it means either our measurement are not accurate enough or our method (curve) doesn't suit to our data.

Analysys of South-North Measurements
In this chapter we will deal with the series { } i τ given by (3.1).Just to remind that τ i represents time it takes for signal to travel from A to D and back to A in direction South-North.
( ) ( ) ( ) denote the average speed γ i .In that way we get the series {γ i } ( ) where e i represents some experimental error.
Since angle ϕ kept constant value during the experiment we could apply Least squares method to the series given by (4).
Let denote γ m by ( ) mean value of the series {γ i }.
We can calculate Variance V 1 ( ) and standard deviation σ If standard deviation σ 1 is bigger then some expected value we should declare the experiment failed.
Combining equations (4) and (5) we get We don't know exact direction of vector v z , thus positive and negative value were assigned to z v .

Conclusions
From (5.13) and (7.8) it follows that length of vector v is given by while vector v is given by .
Recall (from 2.1) that vector v can be written also as Suppose that during one year the same experiments have been repeated 2 * K times.In that way we will get the series where represents length of vector given by Equation (2) or (3) at i-th try.Let ( ) ( ) denote velocity at which Earth orbits the Sun at ( ) i K + -th and i-th try.Suppose also that origins of vectors ( ) lay on the diameter of Earth orbit around the Sun, so they are parallel but in oposite directions.
Mean value m v of the serial (3) is given by Depending on m v we will consider following cases: In other words m v is significantly less than 0 v what is in contradiction to our hypotesis (2.2).In this case we have to reject hypothesis given by (3.1) and declare that velocity of light is not effected by Earth's movement through the space.
This results is consistent with some other experiments, for example with Michelson-Morley experiment.
2) m 0 v v > During the experiments in period of one year v 1 is changing, while v 2 + v 3 is keeping the constant value.Recall that vector v 1 is perpendicular to z axis.Denote vector u by If we replace ( ) ( ) v represents average speed Earth orbits the Sun).From ( 9) and (10) we can get approximate value for ( ) We can form serial Mean value xy u of the serial (12) is given by ( ) Let find standard deviation σ 1 for serial (13).
If σ 1 is bigger then some expected value we have to decline our hypothesis (2.1) and declare the experiment failed.
If σ 2 is bigger then some expected value we have to decline our hypothesis (2.1) and declare the experiment failed.
Otherwise hypothesis given by (3.1) holds and we can conclude that velocity of light depends on Earth's movement through space.In other words velocity of light depends on the direction in which has been measured, what would be in contradiction with Michelson-Morley experiment [1].
The speed that Solar system moves in the space in this case is given by equation Note that while performing the experiment we committed some mistakes.
It was not taken into account the speed of Earth's rotation.This problem can be solved by conducting an experiment at place closer to the Earth's poles, and thus the speed of Earth's rotation taken as small as we want.On other hand this would be counter-productive to our conditions for South-North measurement.Ideally, E-W experiment should be performed on the North/South Pole and S-N experiment at some place on equator.
In addition, within 24 h the Earth changes its direction and the speed at which it revolves around the Sun.We can't solve this problem but we can assume that this speed is relatively small comparing to total speed at which Earth moves through the space.

Theorem 1. Least Squares Method for Cosine Function
Suppose we are given the series {c i }, c i > 0, and there are at least two p, q thus c p <> c q Let take arbitrary coefficients B, A, Θ and form equations ( ) ( ) We will prove that in case 0 ≠ 0 A , function g() has a minimum value at point ( ) In that way we can reduce function g() from function of three variables to fuction of two variables A and Θ, keeping coefficent B fixed and equal to c m .Now we can write the function g() in the form In order to find minimum for function g(), first we have to find partial derivates with respect to A and Θ and critical point (A 0 , Θ 0 ) ( ) ( ) Let us find the first partial derivatives It's easy to prove that g() has minimum at Let us look at the Equations ( 10) and ( 12) For A 0 ≠ we will consider three cases: 1) From (12) it follows A = 0. We will reject this posibility because A 0 ≠ .
In this case we would have

A
Now we have to find the second order partial derivatives of g() with respect to A and Θ. )