Absolute Velocity and Total Stellar Aberration(II)

In this paper, we will show that in addition to measuring annual and diurnal stellar aberration it is also possible directly to measure the angle of secular aberration caused by the motion of the solar system relative to other stars. In the manuscript [1] we dealt with this problem and gave a short description of a special telescope. Using such a telescope we would be able to measure the exact position of the cosmic objects and thus eliminate errors that occur due to the stellar aberration. Assuming that the tube of the telescope is filled with some optical medium [2], we will show that this does not significantly affect the measurement of the stellar aberration angle, but also that these differences are still large enough to enable us to determine the velocity at which the solar system moves relative to the other stars.


Introduction
Suppose we observe an arbitrarily chosen star, which we denote by (Z). Starlight moves in straight line and will remain in the same direction regarding to the ecliptic plane. Photons enter in a perpendicular direction to the top plane of the telescope.
The starlight represents an inertial frame of reference marked by (K) and the telescope represents a moving frame of reference that is marked by (T ). We are assuming that [1] : P 1 -speed of light in vacuum c is constant and equal in all inertial frames (K) P 2 -there is a one common time for the all frames (K) and the moving frame (T ) P 3 -frame (T ) is moving uniformly in a straight line regarding the frame (K) Suppose we have a telescope whose tube is filled with some matter whose index of refraction is denoted by n. The   Velocity at which the telescope moves relative to the starlight is decomposed to the two components. The first component noted by v is perpendicular to the starlight and the second one noted by u is parallel to starlight.
We will first find the distance that the photon has traveled in the direction of u, regarding the (K). Referring to the [ Figure 2] we can write that: After that we are going to find the distance that the photon has traveled in the direction of v, regarding the (K). In order to simplify the calculations, we assume that u = 0. Let S (t 0 ) denotes the position of the point S in the instant t 0 , and S (t 1 ) denotes the position of the point S in the instant t 1 , regarding the (K).
S (t 0 )S (t 1 ) = v * ∆t (the total distance traveled by the point S', regarding the (K) ) (6) S (t 0 )A = n − 1 n * v * ∆t (the photon is dragged by the medium in the moving telescope -our hypothesis) (7) From the Equation (8) it follows that angle of the total stellar aberration [1] denoted by θ is equal to:

Coordinate Systems
In this section are given the descriptions of the four Coordinate Systems that will be used in a further discussion. https://rajpub.com/index.php/jap T 1 . We will assume that the Coordinate Systems (K) and (P ) are defined as "The Heliocentric-Ecliptic Coordinate System" except that their origins O k and O p are different. We assume that the Coordinate System (P ) is moving with a uniform, rectilinear space motion u relative to the Coordinate System (K). O p https://rajpub.com/index.php/jap measured northward from the equator to the line of sight, we would say that is an angle between the plane of equator and the direction of the starlight [1].

Figure 5: Equatorial Coordinate System (Q) and Telescope Coordinate System (T)
We will assume that the center line SS of the telescope lies in the plane defined by the Earth's axis of rotation marked by z q and the star Z(α, δ) Figure[5]. In addition, we will assume that SS is parallel to the light rays coming from the star. We will define a right-handed Telescope Coordinate System (T ) system as follows. The origin of the Coordinate System (T ) is point S . The bottom plane of the telescope represents the (x t y t ) plane of the Coordinate System (T ) . The positive z t − axis lies in the S S direction and the x t − axis is parallel to the equatorial plane

Coordinate Transformations
In the previous paper [1], we have already derived transformation matrices that map Cartesian coordinates from one Coordinate System to another, but we will do it again in a different way.
We will first find simple formulas for transforming the arbitrarily chosen unit vector a = a(α, δ) from a spherical to a Cartesian Coordinate System.
Where δ measures elevation from the xy − plane.
Referring to the Figure[6] we have the following definitions and equations Let the triplets [î k ,ĵ k ,k k ] r [î p ,ĵ p ,k p ] , [î q ,ĵ q ,k q ] and [î t ,ĵ t ,k t ] represent the orthonormal bases for the Coordinate Systems (K), (P ), (Q) and (T ) , respectively.
It is obvious that the unit matrix I 3 is transformation matrix from the basis [î k ,ĵ k ,k k ] to the basis [î p ,ĵ p ,k p ], which means that we have following equality: Let the unit vectorsî q ,ĵ q andk q are expressed in spherical coordinates regarding the Ecliptic Coordinate System (P).
In that way we can define the matrix noted by κ(Q, P ) that transforms the basis [î p ,ĵ p ,k p ]) to the basis [î q ,ĵ q ,k q ].
If the [î,ĵ ,k] represents orthonormal basis , then it is obvious that we have the following equalities: In other words, if two vectors are known then the third can be expressed as a cross product of the other two known vectors.
where κ T is transpose of matrix κ Journal of Advances in Physics Vol 17 (2020) ISSN: 2347-3487 https://rajpub.com/index.php/jap Now we can find the matrix noted by κ(P, Q) that transforms the basis [î q ,ĵ q ,k q ] to the basis [î p ,ĵ p ,k p ].
After that we are going to determine a transformation matrix noted by κ(T, Q) that maps basis [î q ,ĵ q ,k q ] to the basis [î t ,ĵ t ,k t ].
Let the unit vectorsî t ,ĵ t andk t be expressed in spherical coordinates regarding the Equatorial Coordinate System (Q).
Referring to the Figure In that way we can define the matrix noted by κ(T, Q) that transforms [î q ,ĵ q ,k q ] to the basis [î t ,ĵ t ,k t ].
It is easy to prove following equation: Now we can determine the matrix noted by κ(T, P ) that transforms the basis [î p ,ĵ p ,k p ] to the basis [î t ,ĵ t ,k t ] κ(T, P ) = κ(T, Q) * κ(Q, P ) (45) and the matrix noted by κ(P, T ) that transforms [î t ,ĵ t ,k t ] to the [î p ,ĵ p ,k p ].
Let a matrix noted by κ transforms an orthonormal basis [î 1 ,ĵ 1 ,k 1 ] to an orthonormal basis [î 2 ,ĵ 2 ,k 2 ] and suppose that we have some arbitrary vector denoted by v. Then we have the following equations : Using the Equation (49), it is easy to map the vector v from one basis to another. O q Figure 7: Determining the actual right ascension α, declination δ and total aberration of the star We will assume that at point S there is a splitter, so that the light ray emitted from star Z is sent simultaneously in two directions SX and SY where (SX = SY and SX⊥SY ), Figure[7]. The rays SX and SY are perpendicular to SS which implies that the points Z, S and S lie on one line. Now we can measure the angles θ x , θ y and θ z between S S and the positive x q , y q and z q axes, respectively. Applying the Equations (23) − (24) it follows that : The angles α and δ are known and using the Equations (37) − (39) we are able to define a basis [î t ,ĵ t ,k t ] regarding the Telescope Coordinate System (T).
Let the telescope remains in the same position, which means that points Z, S and S still lie on the same line. Instead of sending beam to points X, Y we will let the light signal continue moving toward bottom surface of the telescope.
Due to stellar aberration instead of point S , the light signal will hit point A.
Now, knowing the vector S'A our goal is to determine a velocity at which the telescope moves relative to Coordinate System (K).

5.
Determining a velocity at which the telescope moves relative to Coordinate System (K) The velocity at which the telescope moves with respect to Coordinate System (K) will be denoted by U(t). We have the following equations: Refer to Figure[3] it follows that the Coordinate System (P ) moves at a uniform velocity u relative to the Coordinate System (K) , which will be noted in the following way.
Where u x , u y , u z are the coordinates of u over [î k ,ĵ k ,k k ]. Now we are going to transform the coordinates u x , u y , u z of the vector u with respect to the basis [î k ,ĵ k ,k k ] to the coordinates u x [T ], u y [T ], u z [T ] with respect to the basis [î t , Refer to Figure[4] it follows that the Coordinate System (Q) moves at velocity v(t) relative to the Coordinate System (P ) , which will be noted in the following way.
Now we are going to transform the coordinates v x , v y , v z of the vector v(t) with respect to the basis [î p ,ĵ p ,k p ] to the Journal of Advances in Physics Vol 17 (2020) ISSN: 2347-3487 https://rajpub.com/index.php/jap As can be seen from the Figure[5], the Coordinate System (T ) moves at velocity w(t) relative to the Coordinate System (Q), which will be noted in the following way.
After that we are going to transform the coordinates w x , w y , w z of the vector w(t) with respect to the basis [î q ,ĵ q ,k q ] to the coordinates w x [T ], w y [T ], w z [T ] with respect to the basis [î t ,ĵ t ,k t ].
In order to find the vector u we first need to determine the vector U[T].
6. Determining a velocity that solar system moves regarding the Coordinate System (K) We will first define the Coordinate System (K ) whose origin is at the point O k and the axes x k , y k , z k determined Suppose we have two telescopes filled with two different materials whose indices of refraction are equal to n 0 and n 1 respectively. We can actually say that we have one telescope with two tubes. The tube may be located in the primary or secondary position. When the tube is in the primary position then using that tube it is possible to make measurements.
If |S A|= 0 we can conclude that the proposed method did not produce the expected results and therefore we can say that the experiment failed.
Otherwise from the Equation (8) and the Figure[8] we can find that: S A x (n 0 ) − S A x (n 1 ) n 0 * S A x (n 0 ) − n 1 * S A x (n 1 ) = S A y (n 0 ) − S A y (n 1 ) n 0 * S A y (n 0 ) − n 1 * S A y (n 1 ) From the Equations (60) − (62) it follows that: In this way we have found the velocity u at which the Coordinate System (P ) moves with respect to Coordinate System (K).

Analysis of the results
It remains to determine what velocity u actually represents.
Suppose we observed n different stars from our Galaxy and performed all the necessary measurements and calculations.
In that way we got a sequence of vectors: Let u E denotes some expected minimal value of the u i , and ε denotes some small positive number.
Based on the obtained values of u i , we will consider three cases: 1. u i < u E {i = 1, 2, ..n} This means that we have a following identity: In this case, we are not able to detect any movement of the solar system regarding to other stars. Therefore, this movement has no effect on stellar aberration.
2. u i − u j ≤ ε {i, j = 1, 2, ..n} In this case we will define a vector u as follows: Obviously, in this case we can say that velocity u has an absolute value and the motion of the solar system regarding to other stars affects stellar aberration.