The Complete Nature of Stellar Aberration
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The Complete Nature of Stellar Aberration Copyright © 2010 Joseph A. Rybczyk Abstract 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. 1 The Complete Nature of Stellar Aberration Copyright © 2010 Joseph A. Rybczyk 1. Introduction 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. Starlight from a perpendicular direction to the Earth’s orbital plane Telescope tilted toward reader Telescope tilted away from reader Sun Telescope Earth FIGURE 1 Starlight Perpendicular to Earth’s Orbital Plane 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 2 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. Direction of light Direction of light Apparent direction to source c Light path through θ Telescope telescope L L θ (Tilt angle) Direction of Earth Direction of Earth v Direction of light A B C FIGURE 2 Stellar Aberration – Standard Theory 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. Apparent direction to source θ c L c θ Light path through L is direction of light telescope in Sun Frame v FIGURE 3 Stellar Aberration - Revised 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 3 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 asin 1 and not atan 2 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 = 0.00569207179 degrees for formula (2) or θ = 20.4914586 arc seconds for formula (1) θ = 20.4914585 arc seconds for formula (2) 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 4 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. Starlight in a direction Starlight in a direction along the Earth’s along the Earth’s orbital plane orbital plane Sun Telescope Earth FIGURE 4 Starlight along Earth’s Orbital Plane 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.