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The path of a light’s signal is one and the same in the universal space regardless of the inertial frame by which it is identified. However, only one frame can be taken stationary and identified with the universal space while all other frames are moving. The direction of the path of a light’s pulse in a moving frame is determined in terms of its direction in the stationary one; the result is utilized to explain stellar aberration and show that the tilted direction in the moving frame depends only on its velocity. The aberration increment vector is introduced and employed to determine the apparent position of a star at each point of the earth orbit. Aberration in an earth satellite relative to the geocentric frame is presented. The direction’s change of a light beam between graded inertial frames promotes explaining aberration in an earth’s satellite in parallel to stellar aberration on earth.

In earlier works [

The light path is seen in the moving frame tilted from its direction in the stationary frame by the aberration angle. In the heliocentric frame the direction of the earth’s velocity, due to its orbital motion, keeps changing, resulting in every received starlight continuously changing its direction in the geocentric frame. The latter fact is manifested in a periodic change in the apparent position of distant stars throughout the year. The phenomenon of the annual apparent motion of celestial objects about their locations, named stellar aberration, was discovered by Bradley in 1727 [

The following items summarize the goals that we seek in this work and the plan of its presentation.

We start by an outline of the structure and relevant results of UST theory.

Find the relation between the path of a light’s pulse in the stationary and moving frames.

Utilize the relation that we found to explain the aberration phenomenon and obtain a new expression for stellar aberration angle. Although different from Bradley’s and the relativistic expressions, our formula is in accord with experiment results and it determines the aberration angle at each instant of the year.

The concepts of the aberration increment and aberrated vector are introduced and used to determine the apparent motion of a given star and the direction at which the telescope should be pointed at various times of the year.

The direction’s change of light beams between graded inertial frame is introduced and employed to quantify aberration in an earth’s satellite.

Basics and discussions of stellar aberration can be found in many textbooks and articles [

Let S be an inertial frame in which a source of light b is moving at a constant velocity u, and s be another inertial frame in standard configuration with S and co-moving with b. We found in [

where R

This shows that no matter was the magnitude of the source’s velocity in S, or equivalently, regardless of the frame S in which the source b was viewed, the direction of the light’s path as observed within S is identical to its direction as observed within the frame s in which the source b is stationary (the word “within” here implies that the frame concerned is considered stationary). This means that all observers conjugate to o, each within his frame, are in accord with the direction’s measurements obtained by o, and consequently with each other. Therefore, the direction of the path of a light’s pulse, as measured within each inertial frame, is independent of the relative velocity between the source and the receiver; the same values of the directional angles are obtained by all inertial conjugate observers, each within his inertial frame (

In the rest of this section we brief some basic concepts and results of UST that is relevant to our current work.

Suppose that at an instant

where

It was shown [

Note that if the pulse emanates from B, then as before, the pulse follow

Taking the cross product of both sides with

or

where

is the Euclidean factor [

In the UST theory, time is an absolute entity and hence the duration of a light trip is the same in all inertial frames, but its direction and spatial length may differ. Therefore, if S is the universal frame, the duration of the trip

If the frame s is chosen the universal frame (stationary), then in the first case the time duration of the trip

In parallel to the relations (2, 4 - 6) we have when s is stationary

or

which all are obtained through replacing

Consider a parallel beam of light propagating in the universal frame S parallel to the direction

The path of the ray in M is also determined by two points in M. Let m be a point in M and assume that when at

Thus the direction of the ray is observed in M tilted from a fixed direction

which yields

where

The source of light b (a star for instance) can be seen by a telescope, with ocular and objective lenses r and o respectively, only if the telescope ro is set along the direction

Consider a “distant” star b, in the sense that the radius of the earth’s orbit is negligible in comparison with the distance between the star and the sun. In this context, the phrase “the vicinity of the sun at some instant

All S-observers (in vicinity of the sun) see the rays from the star b throughout the year coming along the

negative direction of the vector

and

The vector velocity

where we assume in writing (4.2b) that the earth orbit is approximately circular. The cosine of the angle

Identifying

where

aberration angle

By (4.5) the aberration angle attains its maximal values

for

for

Substituting for v and c in (4.6) by

where

Recalling that the telescope in s is tilted by

tude of the star is highest for

remains unchanged because the telescope is tilted horizontally (by

Two Special Cases: (i) For

(ii) The distant star b is in the ecliptic: Setting

The aberration occurs in the ecliptic attaining its maximal absolute value (4.6) at

the earth’s velocity and minimum when the earth directly approach the star or recedes from it.

The aberrated direction is the momentary direction along which the star is seen from Earth. Being in the aberration plane, its unit vector

Since

The aberration vector

Two particular independent directions of

(i) At

which shows that, when the Earth’s velocity is perpendicular to the line of sight to b, the telescope is tilted horizontally in the direction of motion by

(ii) For

As the z-component is positive for

It is convenient to decompose the aberration increment into a horizontal (east-west) and “vertical” (north- south) components,

It is clear that the triad

where f is a unit vector defined by the quantity in the parentheses. It is clear that f is perpendicular to

From the parametric representation

Likewise, the aberrated vector

Aberration in Graded Inertial Frames:

When S and M are equally inertial the star is seen in each frame when taken universal along the same direction

The aberrated direction in the geocentric frame M is determined with reference to a fixed direction in the heliocentric frame S. But it is known [

In reality no frame is exactly inertial, but there are graded inertial frames in which one frame is more inertial than another. For instance, the set of non-rotating frames with origins at the center of mass of, the galaxy (G), the solar system (S), the earth-moon system (M), an earth satellite, is graded. Indeed, the motion of S can be counted uniform for thousands of years, whereas the duration of uniformity of earth’s motion in S ranges from minutes to hours depending on the required degree of accuracy. The apparent position

Suppose that a satellite is orbiting the earth in a low circular orbit that lies in the ecliptic. During the orbital period

where

where

The apparent motion of the star b repeats itself once a year when observed from the earth, and

The UST theory provides a neat explanation of the stellar aberration phenomenon that highlights its independence from the velocity of the light source. By introducing the concept of the aberration increment vector we were able to find an approximate formula for the apparent position of the celestial object at each moment of the year. The concept of graded inertial frames makes it clear why annual stellar aberration depends only on the velocity of the earth relative to the heliocentric frame. Employing the same concept enables us to treat aberration when observed from a satellite in parallel to its treatment when observed from the earth.

Caesar P.Viazminsky,Piere K.Vizminiska, (2015) Rays’ Change of Directions between Inertial Frames and Stellar Aberration. Applied Mathematics,06,1553-1562. doi: 10.4236/am.2015.69138