The Complete Nature of Stellar Aberration
Presented is a detailed explanation of the complete nature of stellar aberration that goes beyond the basic principles associated with the effect in direct relation to Earth's orbital motion. The intent is to make the entire process clear in order to understand its full implications with regard to the principles of astronomy, cosmology, and physics as applied to constant relative motion and acceleration. Only then can the effect be properly integrated into the principles of relativistic physics.
Stellar aberration1 is a process whereby the velocity of the Earth in its orbit around the Sun causes a small displacement of where the stars and other celestial objects in space are seen in the Earth frame. The problem with stellar aberration is that it appears to violate the principle of relative motion2 because it appears to be absent in a reciprocal manner in relation to the transverse motion of the stars and other celestial objects observed from Earth. This behavior will be explored in detail in this present work and shown not to be in violation of the principle of relative motion at all when the behavior is properly understood.
2. The Standard Theory of Stellar Aberration
According to the standard theory of stellar aberration, the observed location of celestial objects is affected by the objects' directional relationship to the Earth's orbital plane and the speed of the Earth along its orbital path. Though very small, the affect is consistent and is supposedly greatest for objects located perpendicular to the Earth's orbital plane as illustrated in Figure 1.
As shown in Figure 1, when viewing the same star located in a direction perpendicular to the Earth's orbital plane, the telescope always leans slightly in the direction of the Earth's orbital motion as the Earth travels around the Sun. This, of course, means that the direction of the telescope at any point in the orbit will be exactly opposite to what it was ½ an orbit previously. And that, along with the fact that the angle of tilt is always in the direction of the Earth's orbital motion, was the behavior that made discovery of the effect possible. By associating the speed of light with the Earth's orbital speed a precise relationship to the angle of tilt is found as will be shown next.
Referring to Figure 2-A, a telescope in the Northern Hemisphere of Earth is shown tilted in the direction of the Earth's orbit around the Sun as starlight enters in a perpendicular direction to Earth's orbital plane. Since the telescope is moving along with the Earth in the direction shown, the light that enters the telescope travels the dotted line path through the telescope and exits in a straight line path as illustrated in Figure 2-B. Although the angle of tilt shown is greatly exaggerated for illustrative purposes, such angle is significant enough to be measured and is found to be in agreement with the triangular relationships defined in Figure 2-C. Using a revised version of those relationships as will be discussed next in relation to Figure 3, we can derive a formula for finding the angle of stellar aberration.
For clarity purposes, the angle of tilt, θ, has been exaggerated even more in Figure 3 than it was in Figure 2. But an even more important change to the relationships originally shown in Figure 2-C has been incorporated as will be discussed next.
Contrary to what was shown in Figure 2-C, side c of the triangle has been changed to the hypotenuse of the triangle shown in Figure 3 and side L is now the perpendicular side. This switch in designators between the two sides of the triangle brings the relationships into agreement with the principles of relativity with virtually no effect in regard to the evidence supported principles of stellar aberration. More specifically, the angle at which the light passes through the center of the telescope, represented by the hypotenuse of the right triangle, has been designated as side c to bring the principle of stellar aberration into agreement with the relativistic principle that light travels at speed c relative to the reference frame in which it is observed. If the light is traveling in a perpendicular direction to the Earth's orbital plane, then according to the principles of relativity, it is traveling in that direction relative to the stationary frame of the Sun, and not the moving frame of the Earth. Thus it will have a speed c in that direction relative to the stationary frame of the Sun and not the moving frame of the Earth. Since speed c in the frame of the Sun is different from speed c in the frame of the Earth, and it is the Earth's frame in which we are interested, the switching of speeds L and c in Figure 3 is necessary. In the moving fame of the Earth, the light travels at speed c along the vectored path defined by the L designated Sun frame light speed, Earth's orbital speed v, and the angle of observation θ. This means the correct formula for determining the angle θ associated with stellar aberration is
where v is the speed of the Earth along its orbital path and c is the speed of light in the Earth frame.
Using v = 29783 m/s and c = 299792458 m/s in the above formulas gives
θ = 0.00569207182 degrees for formula (1)
θ = 20.4914586 arc seconds for formula (1)
for the corresponding results that, as can be seen, are in close agreement with each other. Thus it is reasonable to claim that formula (1) is supported by the evidence while being consistent with the relativistic principle that the speed of light is c in the frame of the observer.
Before going deeper into the subject of stellar aberration there appears to be another false notion associated with the standard theory that needs to be dispelled. It has to do with the mistaken claim that the effect is greatest for objects observed at a perpendicular angle to the Earth's orbital plane and decreases to zero for objects observed along the orbital plane. This view is totally inconsistent with the principles of stellar aberration just discussed as will be shown next. Referring to Figure 4 we have the condition where the telescope has been repositioned to observe objects directly along the Earth's orbital plane. In analyzing what is shown it should be quite obvious that for light approaching from any direction along the orbital plane, the telescope oriented in that direction will be subjected to the same orbital motion related effects as the telescope used in the previous discussion. For example, assume that the light is coming from an object to the far right of the orbital plane as illustrated and is being observed through the telescope shown on the far right in the illustration. If the orbit is occurring in the counterclockwise direction shown, then the Earth and telescope are moving in a perpendicular direction to that incoming light, and if the principle of stellar aberration is as presently believed, the telescope will have to be tilted in the direction of the Earth's orbit by the same amount that it was in the previous example.
There are, however, several differences in this example that makes it more difficult to prove and that is probably why it has not been emphasized in the past. As is readily seen, when the Earth and telescope are moving directly toward, or directly away from that same object to the far right, the motion is not transverse and subsequently there will be no stellar aberration affect. This would not be true, however, for the case where the Earth and telescope are at the completely opposite position of the original position, i.e., to the far left in the illustration, instead of to the far right as originally discussed. However, in that case, the Sun is in the direct path of observation and so the object cannot be observed at all. Otherwise, if the principles being discussed are correct, they would require the telescope to be tilted in the opposite direction as in the case with the previous example of Figure 1. More specifically, if the telescope is tilted in the direction of the Earth's orbital motion, it will be tilted away from the reader when the Earth is to the far right in its illustrated orbit, and will be tilted toward the reader when the Earth is at the far left in its illustrated orbit. As is well known and was previously stated, it was this kind of behavior that led to the discovery of the principle of stellar aberration to begin with.
Before moving on, there is one last issue to discuss involving the standard theory. It involves the combined effects of the Earth's orbit and rotation speeds. Referring to Figure 5 it is readily seen that for a telescope located near the equator, the total speed through space is a combination of the Earth's orbital speed and its rotational speed. When the physical telescope location is rotating in the direction of the Earth's orbit, the total speed is the sum of the orbital speed and rotational speed. When the physical telescope location is rotating in the opposite direction a ½ rotation later, the total speed is the difference between the two speeds. Although the speed of the Earth's rotation is very small compared to its orbital speed, it is still a factor to consider because the ability to measure the effect of stellar aberration is considered to be of almost unlimited accuracy.
Using r = 465.1 m/s for the Earth's rotational speed along with v = 29783 m/s for its orbital speed and c = 299792458 m/s as given earlier, we can substitute vmax = v + r = 30248.1 m/s for v in formulas (1) and (2) given previously to obtain
for the corresponding results which remain in close agreement with each other.
Then, in a like manner, using vmin = v – r = 29317.9 m/s for v in formulas (1) and (2) we obtain
θ = 20.171458 arc seconds for formula (1)
for the corresponding results which continue to be close agreement with each other.
3. The Inverse Skinny Triangle Principle
The skinny triangle principle is perhaps best known for the limitation it places on the determination of nearby objects in space through the process of comparing the difference in observation angles from the two extreme positions of Earth's orbit. In this usage the distance between the two extreme points in the Earth's orbit serves as the base of a triangle whose pinnacle is the object in space under observation. Even for relatively close objects in space this method of determining distance is very limited because as the distance to the object increases, the triangle becomes so skinny that its two sides begin to merge into a single line. Nonetheless, for a nearby star in orbit about another star this triangulation principle can theoretically be used in reverse along with the principle of stellar aberration to determine the speed of the star along its orbital path about the other star. In practice, however, the same skinny triangle limitation just discussed makes its use for this alternate purpose virtually impossible, even for star orbits relatively close to the Earth. This limitation is illustrated in Figure 6 where it is readily seen that for relatively small orbits with periods similar to Earth's, the difference in the angle of the received starlight even from the two most extreme points in the star's orbit is virtually impossible to distinguish.
This limitation is clearly demonstrated by progressing from view A to view D in Figure 6 where in each succeeding view the distance between the star orbit and stationary observer increases. It is view D that is representative of such orbits relative to Earth whereby the previous views are for illustrative purposes only. Theoretically, if the orbit was of an extremely large diameter, and very close to the Earth, the difference in the angles of starlight received from extreme orbital positions might be decipherable, but then the problem becomes one involving the time required to observe the light from those two different locations. That is, the orbital period is a direct function of the orbital diameter, and therefore it could take far too long of a time between observations to see any difference in the angles of the received light. What is seen here, then, is the reason why the principle of stellar aberration does not appear to work in reverse. That, however, does not mean that the principle is not valid in that reverse application as will be discussed next.
4. The Inverse Principle of Stellar Aberration
When the fundamental principles of stellar aberration are properly understood, it is apparent that contrary to current scientific opinion, the phenomenon is of a relative nature. To apply the principles in an inverse, relative, manner, one must simply look beyond the limitations previously discussed. As a matter of fact, the limitations previously discussed involve motion of a cyclical nature, i.e., orbital motion. The problem with orbital motion in regard to stellar aberration is that it limits the base of the inverse skinny triangle. Such, however, is not the case for linear motion. If, for example, a light source was moving at a constant speed in a transverse direction relative to an observer in the frame of the Sun, then the base of the triangle continuously increases as the light from the source travels to the observer at speed c. In that case the base of the triangle is proportional to the distance between the source and observer as shown in previous works by this investigator3 and therefore the distance to the observer is no longer the limiting factor in the manner that it was involving orbital motion. Subsequently, if we have evidence of uniform transverse motion involving a source, we can apply the principles of stellar aberration in reverse to determine the actual location of the source based upon its observed apparent location in space. Using the Earth's orbital speed as an example, let us see what is meant by this.
Assume that a light source is moving transversely relative to a stationary observer in the vicinity of the Earth as shown in Figure 7. For purposes of illustration, further assume that the light source is 4.243 light years away from the observer. (This is the distance to Alpha Centauri, the nearest star system to Earth excluding the Sun.)
If the speed of the source is v = 29783 m/s (I.e., Earth's orbital speed) and the light is received at speed c = 299792458 m/s along the dotted line path shown in the illustration, and the triangle is a Pythagorean triangle, then the angle between the actual location of the source and its apparent location (I.e., Image) is given by the same formula (1) used previously in relation to Figure 3. And since the two speeds, v and c, have not changed from the previous example, the result will be the same, (I.e., θ = 0.00569207182 degrees, or θ = 20.4914586 arc seconds).
The main difference between this example and the previous example relating to Figure 6 of Section 3 is the increase in side v of the representative triangle. Since the distance to the source is 4.243 light years away, the source traveled a distance of 4.243 Earth orbits along path v as the light traveled distance c to the observer. This equates to a total distance of approximately 4.243π(Earth's Orbital Diameter) or, approximately 13.33 x (Earth's Orbital Diameter). Subsequently, this triangle is significantly easier to solve mathematically than a similar triangle based only on the diameter of the orbit of a similar source located the same distance away from the observer. And as is readily seen, the ratio of side v to side c of this constant speed source will remain unchanged as the distance to the observer increases. That, of course, would not be true of the orbiting source, in which case the ratio of the orbital diameter to observer distance would decrease with an increase in distance to the source.
5. Mathematical Derivation of Inverse Stellar Aberration Formula
Referring to Figure 8 where the trigonometric relationships of an approaching and a receding source relative to a stationary observer are illustrated we can begin the mathematical derivation process for a general formula for inverse stellar aberration.
In reference to Figure 8, the constant c and other variables shown have the following definitions:
c – Constant Speed of Light in the Stationary Frame of the Observer
v – Speed of Source in the Stationary Frame of the Observer
L – Variable Speed of Light in the Stationary Frame of the Observer
R – Angle of Recession (Image – What is actually seen)
A – Angle of Approach (Image – What is actually seen)
Rs – Angle of Recession (Source – The actual physical object)
As – Angle of Approach (Source – The actual physical object)
θ – Angle of Separation between Image and Source – Inverse Stellar Aberration
We begin the derivation process by applying the Law of Cosines to the propagation triangles of Figure 8 to obtain
where the constant and variables have the definitions just given. From equation (3) we then derive
for light propagation distance L. Next we apply the Law of Sines to the same propagation triangles to obtain
that by way of substitution with equation (4) gives
that in turn gives
for the sine of θ. This then gives
where θ is the angle of inverse stellar aberration. In the interest of completeness, it is understood that the relationship between angle of approach A, and angle of recession R is given by
for use with the inverse stellar aberration formula (8) as appropriate.
To compare the results of inverse stellar aberration to the standard version, an alternate formula for the inverse stellar aberration angle θ is needed that uses angle Rs and not angles A or R. Such formula can be derived using the Law of Sines as given by
and solving for angle θ. This gives
as an alternate formula for the inverse stellar aberration angle θ. When angle Rs is given the value of 90° as is obvious from the previous discussions involving Figures 3 and 7, and speeds v and c are given the same values used previously in regard to formula (1); the value given for angle θ by formula (7) is identical to the value originally given by formula (1). This provides convincing evidence as to the validity of the inverse stellar aberration theory being discussed.
6. Other Considerations Involving Inverse Stellar Aberration
From what has been shown in this work, our failure to detect significant effects of stellar aberration of the inverse kind has to do with practical considerations and not because such effects are not actually occurring. When we explore deeper into the inverse form of these effects it can be seen that there are an infinite number of possible variations to consider. In Figure 9 two of the most obvious variations are shown.
In viewing Figure 9 is should be understood that the magnitude of the effects has to be grossly exaggerated in order to make it possible to even illustrate their nature. This is probably one of the reasons why the inverse nature of the effect has missed greater scrutiny in the past. That is the inverse effect is just too difficult to visualize, illustrate, and discuss in an easily understood manner.
As can be seen in Figure 9-A the light propagated by an orbiting source in a perpendicular direction to its orbital plane is confined to the diameter of the orbit in propagating to a distant observer. Thus, the base of the inverted right triangle given by orbital speed v is insignificant compared to the great distance represented by the side showing the propagation distance c that represents many light years even for nearby objects. And this does not even take into consideration the fact that most observers will not be located in a direction perpendicular to the source's orbital plane. Nor does it consider the possibility of the entire orbit being in motion relative to the observer as would be the case involving the combined effects of Earth's motion along its own orbit, if nothing else.
In Figure 9-B the effect of a source's acceleration on the propagated light is shown. Adding this effect to those already discussed involving orbital motion only complicates the problem even more. And, again, even here the motion shown is in a transverse direction relative to the direction of propagation. In reality such motion will most likely be in a variety of directions for a source nearby in the Milky way Galaxy. When we add to all of this the consideration of the expanding universe for objects receding away from Earth in the regions of space outside our galaxy, it is easily seen that such longitudinal direction of the motion will negate the effect it would otherwise have involving the observed effect of inverse stellar aberration as observed from Earth.
From what was just discussed in reference to Figure 9 in addition to the preceding Sections it should be rather apparent that any failure to observe the effects of inverse stellar aberration has more to do with practical considerations and limitations and not because such effect is inconsistent with the principles of relative motion as is apparently presently believed.
The author has done his best to show in this present work that the principle of stellar aberration as expanded upon in his previous works3 is based upon valid scientific principles. With the insights given here and the unlimited accuracy clamed for experimental verification of standard stellar aberration it might now be possible to detect the included effect of inverse stellar aberration. Hopefully, scientists in appropriate fields of physics, astronomy, and cosmology will take what has been shown here and verify it through appropriate observational/experimental means.
1 James Bradley, Astronomer, discovered the principle of stellar aberration in 1727 while searching for evidence of stellar parallax, a completely different phenomenon.
2 Albert Einstein, On the Electrodynamics of Moving Bodies, now known as the Special Theory of Relativity, Annalen der Physik, (1905), It is in this work that the principle of relative motion as presently accepted was given.
3 Joseph A. Rybczyk, The Light Speed Effect, (2009); The True Nature of Light Propagation, (2009); Light Speed Affected by Source Speed Theoretically Proven, (2009); How Light Speed is Affected by Source Speed, (2010); All available on the Millennium Relativity web site at http://www.mrelativity.net