The Curvature of Light Due to Relativistic Aberration

The curvature of light due to relativistic aberration

16-Oct-2012 (Update 26-July-2015)

This paper summarizes the different forms of aberration for nearby and remote stellar objects and provides supporting evidence that the physics of the aberration of light must be a combination of local and remote effects whereby light, from the perspective of the observer, follows a curved path as a consequence of relativistic aberration. This paper includes an illustration that depicts the formula for relativistic aberration.

Table of content Aberration of light – Definition ...... 3 Stellar aberration – A complication related to nearby objects ...... 4 Nearby objects – No aberration ...... 5 The Moon – Diurnal aberration ...... 5 Planets – Planetary aberration ...... 6 Stars within the Milky Way – Secular aberration ...... 7 Remote galaxies – Secular aberration ...... 7 Hubble telescope - Orbital aberration ...... 8 Stellar aberration –Classical explanation for nearby objects ...... 9 Airy’ Water Telescope ...... 10 Where does the aberration of light take place? ...... 11 Occultation of stars and planets ...... 12 Venus transit ...... 13 Relativistic aberration ...... 14 Aberration of light when light passes through a reference frame with a transverse velocity ...... 15 Aberration of light when light enters a preferred reference frame at 90° ...... 15 Aberration of light when θ is different from 90° ...... 17 Aberration of light in a time-lapse perspecive ...... 18 Aberration of light when a telescope moves relative to the local reference frame ...... 19 The aberration of light : principles ...... 20 The preferred reference frame within the Solar System ...... 22 Hypothesis: ...... 22 Michelson-Morley experiment ...... 23 Michelson-Gale-Pearson experiment ...... 23 Global Positioning System ...... 24 Twin paradox ...... 24 Applying the principles of the aberration of light ...... 25 Why can we observe stars of which the true direction is behind the Moon? ...... 25 Venus transit ...... 27 Relativistic beaming ...... 29 The annual residual of the Pioneer anomaly ...... 30 The Hafele-Keating experiment ...... 32 Summary / Conclusion...... 33

Aberration of light – Definition As per Wikipedia:

The aberration of light (also referred to as astronomical aberration or stellar aberration) is an astronomical phenomenon which produces an apparent motion of celestial objects about their real locations.

At the instant of any observation of an object, the apparent position of the object is displaced from its true position by an amount which depends solely upon the transverse component of the velocity of the observer, with respect to the vector of the incoming beam of light (i.e., the line actually taken by the light on its path to the observer). The result is a tilting of the direction of the incoming light which is independent of the distance between object and observer.

In the case of an observer on Earth, the direction of a star's velocity varies during the year as Earth revolves around the Sun (or strictly speaking, the barycenter of the solar system), and this in turn causes the apparent position of the star to vary. This particular effect is known as annual aberration or stellar aberration, because it causes the apparent position of a star to vary periodically over the course of a year. The maximum amount of the aberrational displacement of a star is approximately 20 arcseconds in right ascension or declination.

Light from location 1 will appear to be coming from location 2 for a moving telescope due to the finite speed of light.

The aberration can be calculated as:

There are a number of types of aberration, caused by the differing components of the Earth's motion:

 Annual aberration is due to the revolution of the Earth around the Sun.  Planetary aberration is the combination of aberration and light-time correction.  Diurnal aberration is due to the rotation of the Earth about its own axis.  Secular aberration is due to the motion of the Sun and solar system relative to other stars in the galaxy.

Stellar aberration – A complication related to nearby objects

A complication when considering a more nearby object (whether it is a planet, the Moon, or a satellite) is that the source itself has a significant motion. It is the velocity of the observer relative to the velocity of the source that becomes relevant. As a specific example, if the source and observer move with the same relative transverse velocity, the aberration term is zero.

Considering the above illustration, the question now arises why photons following the same path but originating from different sources can show a different behavior at the moment of entering a telescope.

Nearby objects – No aberration

V = 0 m/s

From the perspective of the observer on Earth, the observed position of the objects located on Earth do not change in the course of a day or in the course of a year.

The Moon – Diurnal aberration

V = + or -464 m/s at the Earth’s equator due to the rotation of the Earth around its axis.

The resulting diurnal aberration = maximum 0.32" (with a periodicity of 1 day) The light-time correction for the Moon: −0.704".

Light originating from the Moon is not subject to Annual aberration: to calculate the occurrence of a New Moon, we need only to take into account the constant displacement of the Sun (20.5”) and the light-time correction for the Moon (−0.704”). http://en.wikipedia.org/wiki/New_moon )

The observed position of the Moon is displaced with a maximum of 0.32 arcsec (+0.7 arcsec light- time correction) in the course of a day due to the effect of diurnal aberration.

Planets – Planetary aberration

V depends on the relative motion of the observed planet and the Earth.

Planetary aberration is the combination of aberration and light-time correction:

 Light-time correction: the distance the celestial object has moved while light travelled towards the observer once light has been emitted from the source  Aberration: planetary aberration depends on the relative motion between source (planet) and observer (Earth). As a specific example, if source and observer move with the same relative transverse velocity (e.g. the Moon), the aberration term is zero (and only light-time correction applies)

The following table depicts the aberration values (in arcsec) when all planets would be aligned with the Sun and with a distant star:

Light Distance Source > from Orbital Sun Speed Observer on (AU) (km/s) v Sun Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Star 0 0 Sun 0.0 -32.9 -24.1 -20.5 -16.6 -9.0 -6.7 -4.7 -3.7 0.0 0,4 47,9 Mercury 32.9 0.0 8.8 12.4 16.4 23.9 26.3 28.3 29.2 32.9 0,7 35,0 Venus 24.1 -8.8 0.0 *3.6 7.5 15.1 17.4 19.4 20.4 24.1 1 29,8 Earth 20.5 -12.4 -3.6 0.0 3.9 11.5 13.8 15.8 16.8 20.5 1,5 24,1 Mars 16.6 -16.4 -7.5 -3.9 0.0 7.6 9.9 11.9 12.8 16.6 5,2 13,1 Jupiter 9.0 -23.9 -15.1 -11.5 -7.6 0.0 2.3 4.3 5.3 9.0 9,5 9,7 Saturn 6.7 -26.3 -17.4 -13.8 -9.9 -2.3 0.0 2.0 2.9 6.7 19,6 6,8 Uranus 4.7 -28.3 -19.4 -15.8 -11.9 -4.3 -2.0 0.0 0.9 4.7 30 5,4 Neptune 3.7 -29.2 -20.4 -16.8 -12.8 -5.3 -2.9 -0.9 0.0 3.7

Example whereby the observed aberration of 3.6 arcsec for Venus is referred to:

“A Letter from Richard Price, D. D. F. R. S. to Benjamin Franklin, L L. D. F. R. S. on the Effect of the Aberration of Light on the Time of a Transit of Venus Over the Sun. (December 1770):” http://www.jstor.org/stable/105919?seq=1

From the perspective of the observers on Earth, the apparent position of the planets is subject to Diurnal aberration as well.

Stars within the Milky Way – Secular aberration

V = 217 km/s (motion of the Solar System around the Milky Way). The transverse component changes over a period of 226 million years.

The resulting secular aberration is a value up to 149 arcsec.

V = 30 km/s (motion of Earth around the center of the solar system) The transverse component of this velocity changes over the course of a year depending on the position of the Earth.

The resulting Annual aberration is a value up to 20.5 arcsec.

From the perspective of the observers on Earth, the apparent position of the stars is subject to Diurnal aberration as well.

For an observer on Earth, the secular aberration is the largest of the aberration components. The annual aberration causes the star to appear around this average position (at a maximum angular distance of 20.5 arcsec).

Remote galaxies – Secular aberration V = 630 km/s relative to average velocity of galaxies. The resulting secular aberration is a value up to 430 arcsec.

V = 552 km/s relative to CMB (Cosmic Microwave Background) The resulting secular aberration is a value up to 377 arcsec.

From the perspective of the observers on Earth, the apparent position of the galaxies is subject to Diurnal and annual aberration as well.

For an observer on Earth, the secular aberration is the largest of the aberration components but hard to measure because of the extremely long periodicity.

Hubble telescope - Orbital aberration

Every 97 minutes, Hubble completes a spin around Earth, moving at the speed of about five miles per second (8 km per second).

The resulting orbital aberration has a maximum angular displacement of around 5.5 arcseconds.

The orbital aberration applies to any stellar object observed from the Hubble telescope.

Stellar aberration –Classical explanation for nearby objects The classical interpretation of stellar aberration for nearby objects can be illustrated as follows:

In this illustration, the situation for ‘Full Moon’ is chosen whereby we know that stars are showing an aberration of 20.4arcsec.

Once starlight has passed the Moon, starlight and moonlight travel side-by-side to the observer.

During the time it takes to reach the observer on Earth, both the Moon and the Earth continue to travel at 30 km/s.

At the moment when light reaches the Earth:

 the actual direction of the observed star is 20.4 arcseconds behind the Moon  the actual direction of the Moon continues to point in the direction of the telescope because the Moon has travelled along with the Earth

Airy’ Water Telescope In 1871, George Biddell Airy filled a telescope with water to observe how it would affect stellar aberration.

Since the speed of light in water is about 75% of the speed of light in the air, light needs more time to reach the bottom of the telescope. As a consequence, one would expect the aberration angle to increase with 1.33 since the telescope would have moved over a larger distance.

The ‘Airy’ experiment demonstrated that the observed aberration angle did not change.

The fact that filling a telescope with water does not change the magnitude of the observed aberration implies that the effect of aberration must take place before light enters the telescope. As a consequence, the classical explanation for the stellar aberration for nearby objects cannot be valid which leaves the question where the aberration of light really takes place.

Where does the aberration of light take place?

Photons travelling in the exact same direction but originating from different sources can’t show a different behavior when entering a telescope.

As described in the previous sections, the aberration of light can be very different dependent on the observed stellar object. The only “constant aberration term” is the annual aberration which changes the position of a star relative to its average position with a value up to 20.5 arcsec. The average position of a star or galaxy includes a much larger aberration component due the velocity of the Solar System and the Milky Way.

The aberration of light must therefore take place while light is travelling from the source to the observer.

At the point where light that is originating from a more remote object passes a closer object, the photons from the remote and the closer object must continue to travel side-by-side and continue to show the same behavior from that point onwards.

As a consequence, the ‘remaining’ aberration that takes place at the point of entering the telescope is the same for any light source, independent of the distance.

This ‘remaining’ aberration is the aberration as observed for nearby objects: for stellar objects as observed by the Hubble telescope, this ‘remaining aberration’ is the orbital aberration.

Occultation of stars and planets

Light originating from planets is subject to aberration whereas light originating from the Moon isn’t.

Nevertheless, when planets are occulted by the Moon, their images never overlap with the image of the Moon. This implies that photons originating from the planets must have been subject to aberration before passing the Moon.

Considering the above pictures of Saturn and Jupiter:

 Photons originating from these planets must be travelling side-by-side with the photons originating from the border of the Moon  Any effect of diurnal or orbital aberration must apply equally to both the image of observed planet and the Moon

Venus transit The following image depicts the observed (apparent) positions of Venus and the Sun during the transit of June 2004:

Considering the effect of the aberration of light:

 Venus is observed with -3.6 arcsec aberration (to left on the above picture)  The Sun is observed with 20.5 arcsec aberration (to the right on the above picture)

Without the aberration of light, the image of Venus would have been observed 24.1 arcsec to the right relative to the image of the Sun (on the above picture) and would have looked as follows:

The image of Venus (the black disk) cannot have moved to the left (relative to the image of the Sun) while light (originating from the Sun) travelled from Venus to the Earth. The black disk is the ‘absence of light’ and the only photons observed on Earth are the photons emitted by the Sun ….

Knowing that we observe the black disk of Venus 3.6 arcsec left of its true direction, we can deduct that the image of the Sun must have been 3.6 arcsec more to the right near Venus.

Relativistic aberration From Wikipedia:

Relativistic aberration is described by Einstein's special theory of relativity, and in other relativistic models such as Newtonian emission theory. It results in aberration of light when the relative motion of observer and light source changes the position of the light source in the field of view of the observer. The effect is independent of the distance between observer and light source.

Suppose, in the reference frame of the observer, the source is moving with speed at an angle relative to the vector from the observer to the source at the time when the light is emitted. Then the following formula, which was derived by Einstein in 1905, describes the aberration of the light source, , measured by the observer:

In this circumstance, the rays of light from the source which reach the observer are tilted towards the direction of the source's motion (relative to the observer).

The relativistic aberration can also be calculated as:

Where ,

Relativistic aberration associates the effect of aberration with the relative movements of reference frames instead of just of movement of the observer.

While relativistic aberration allows to calculate the net effect of aberration, it does not prescribe the exact path light is following between source and observer. The exact path can only be calculated when knowing the relative velocities of the references frames.

Aberration of light when light passes through a reference frame with a transverse velocity

Aberration of light when light enters a preferred reference frame at 90°

When a photon enters a reference frame with a different transverse velocity, light will continue its journey at the speed of light ‘c’ but now relative to the frame B it just entered. This implies that the photon must be at a distance proportional to the speed of light ‘c’ relative to the point where the photon entered frame B. This distance is depicted by the half-circle above. The center point of this circle moves along with frame B at velocity ‘v’. So the photon must be positioned somewhere on this half-circle.

At the same time, the photon will continue to travel on the same straight line that is was following in frame A.

We can therefore find the photon at the intersection of the half-circle with the straight line. This point of intersection provides us with the aberration angle which is the direction in which light is moving at velocity ‘c’ relative to Frame B.

From the perspective of Frame A, the speed of light appears to have reduced from ‘c’ to √푐2 − 푣2 due to the superposition of velocity vectors v and c. The wave front of the light has changed direction.

In the following illustration, the velocities have been divided by c to simplify the calculation:

푐표푠(∅) = 푠푖푛(휓) = −푣/푐 푠푖푛(∅) = 푐표푠(휓) = √1 − (푣/푐)²

√1 − (푣/푐)² 푡푎푛(∅) = −푣/푐

This formula (derived from the above illustration) is consistent with the formula for relativistic aberration:

푠푖푛⁡(휃) 1 푡푎푛(∅) = 푣 with 훾 = 훾(푐표푠(휃)− ) √1−(푣/푐)² 푐 For 휃=90°:

푣 2 푠푖푛⁡(90°) √1−(푣/푐)² 푡푎푛(∅) = √1 − ( ) . 푣 = 푐 푐표푠(90°)− −푣/푐 푐

Aberration of light when θ is different from 90°

As per the law of sines:

−푣/푐 1 휇 = = 푠푖푛⁡(휃 − ∅) 푠푖푛(휋 − 휃) 푠푖푛⁡(∅ ) 푣 푣 푠푖푛⁡(∅ ) 푠푖푛(휃 − ∅) = − . 푠푖푛(휋 − 휃) = − . 푐 푐 휇 풗 풔풊풏(휽 − ∅) = 풔풊풏(흍) = − . 풔풊풏(휽) 풄

As per the law of cosines: (with negative v/c)

푣 2 푣 휇 = √1 + ( ) + 2. . 푐표푠(∅) 푐 푐

For 휃=90°: 푐표푠(∅)=-v/c :

휇 = √1 − (푣/푐)²

Aberration of light in a time-lapse perspecive The following illustration provides a time-lapse perspective:

From the perspective of frame A:

 light shifts to the right with velocity ‘v’ after having entered reference frame B.  The aberration angle changes the direction of light in such a way that it exactly compensates for this shift ‘v/c.’ As a result of this compensation, light appears to continue its original direction from the perspective of reference frame A.  The speed of light appears to have reduced with a factor √1 − (푣/푐)²

From the perspective of frame B, light travels with the speed of light ‘c’ in the direction of the aberration angle.

Telescope perspective:

 The telescope maintains a fixed position relative to reference frame B and moves along with frame B  As of the moment when light has entered reference frame B: o The telescope continuously points to the position in reference frame B where light has entered the frame o The telescope continuously points to the photon that will later enter the telescope  The photon arrives with an aberration angle 푣 푠푖푛(휃 − ∅) = 푠푖푛(휓) = − . 푠푖푛(휃) 푐  The photon arrives with the speed of light ‘c’ from the ‘apparent direction’ 휓

Aberration of light when a telescope moves relative to the local reference frame When the telescope moves relative to the local reference frame, the situation looks like follows:

From the perspective of the telescope/observer:

 The telescope continuously points to the photon that will later enter the telescope  Light appears to arrive with an aberration angle of 휓  Due to the relativistic velocity addition principle, the photon moves with the speed of light ‘c’ (instead of √푐2 + 푣²) through the telescope (from the perspective of the observer who moves along with the telescope and for whom time dilatation applies).  Light appears to arrive from the direction 휓 and the telescope needs to be tilted accordingly: 푣 푠푖푛(휃 − ∅) = 푠푖푛(휓) = − . 푠푖푛(휃) 푐

The aberration of light : principles The above mechanisms translate into a number of principles:

1. The movement of an observed stellar object relative to its surrounding preferred reference frame does not contribute to the observed stellar aberration (figure 1, moving star). 2. Light that leaves a moving reference frame will change direction (figure 4, observer B). When the distance to the observer is large in comparison with the ‘size’ of the reference frame containing the observed object, the change in direction will be very small from the perspective of the distant observer. 3. The movement of an observer relative to the surrounding reference frame leads to stellar aberration (figure 1 and 2). 4. When the observer and the observed object move at the same speed and in the same direction within a preferred reference frame, the effects of stellar aberration and light-time delay cancel each other out. This is because the observed object moves the exact distance to stay aligned with the observer telescope (figure 2, observer A, object x). 5. The movement of a reference frame relative to the vector of the incoming beam of light leads to stellar aberration for observers who move along with the reference frame (figure 3). 6. The effect of stellar aberration is ‘cancelled out’ after light has passed through a reference frame that moves relative to the vector of the incoming beam of light. (figure 3, observer B). 7. It takes extra time for light to travel through the moving reference frame (figure 3) due to apparent decrease of the velocity of light.

8. The magnitude of stellar aberration doesn’t depend on the prior reference frames where light has passed through (Figure 5 and 6; observer A and B). 9. The magnitude of stellar aberration relates to the difference in velocity between the reference frame of the observer and the reference frame containing the observed stellar object. (Figure 7, both telescope A and B)

The preferred reference frame within the Solar System

Hypothesis: The ‘preferred reference frame’ at any point in the plane of the Solar System is one that rotates around the center of the Solar System with the same speed as a planet would have on that position when in circular orbit.

Within the plane of the Solar System, the velocity of the ‘preferred reference frame’ relative to the barycenter of the Solar System can be calculated as :

퐺⁡푥푀푎푠푠⁡푆푢푛 √⁡ 퐷푖푠푡푎푛푐푒⁡푓푟표푚⁡푆푢푛

Distance Orbital from Sun Speed Orbital Speed (km/s) relative to distance (AU) (km/s) from Sun (AU) 0 0 Sun 60 0,4 47,9 Mercury 0,7 35 Venus 50 1 29,8 Earth 40 1,5 24,1 Mars 30 5,2 13,1 Jupiter 20 9,5 9,7 Saturn 19,6 6,8 Uranus 10 30 5,4 Neptune 0 120 0 Solar system 0 20 40 60 80 100 120

For positions above and below the plane of the Solar System, the velocity gradually decreases.

Michelson-Morley experiment With the Michelson-Morley experiment, the relative motion of the Earth relative to space was measured with an interferometer and based on the hypothesis depicted as follows:

The Michelson-Morley experiment did not reveal a difference of the measured speed of light thereby reflecting the motion of the Earth around the Solar System.

The outcome of the Michelson-Morley meets the expected result when re-evaluated from the perspective of a preferred reference frame that rotates around the Sun with the same speed as the planets.

Michelson-Gale-Pearson experiment The Michelson–Gale–Pearson experiment (1925) is a modified version of the Michelson–Morley experiment and the Sagnac-Interferometer. The aim was to find out whether the rotation of the Earth has an effect on the propagation of light in the vicinity of the Earth. The Michelson-Gale experiment was a very large ring interferometer, (a perimeter of 1.9 kilometer), large enough to detect the angular velocity of the Earth. The outcome of the experiment was that the angular velocity of the Earth as measured by astronomy was confirmed to within measuring accuracy.

The outcome of the Michelson-Gale-Pearson experiment meets the expected result when evaluated from the perspective of a preferred reference frame that rotates around the Sun.

The experiment can be regarded as a set of mirrors that move relative to the preferred reference frame. The set of mirrors that are closer to the axis of the Earth move more slowly than those separated from the axis of the Earth. The outcome of the experiment can be calculated from the perspective of the Earth Centered Initial Frame (non-rotating) when taking into account the different distances light needs to travel between the moving mirrors.

Global Positioning System (Neil Ashby: The Sagnac Effect in the Global Positioning System).

In the Global Positioning System, the Sagnac effect arises because the primary reference frame of interest for navigation is the rotating Earth-Centered, Earth Fixed frame, whereas the speed of light is constant in a locally inertial frame, the Earth-Centered Inertial frame.

The Earth-Centered Inertial frame in which the speed of light is the constant ‘c’ in the immediate neighborhood of the Earth can be considered to be a ‘special case’ of the general principle whereby the speed of light is the constant ‘c’ relative to the preferred reference frame that rotates around the Solar System with the same speed as the planets.

Twin paradox

Wikipedia: In Albert Einstein's theory of relativity, time dilation can be summarized as:

 In special relativity (or, hypothetically far from all gravitational mass), clocks that are moving with respect to an inertial system of observation are measured to be running more slowly.  In general relativity, clocks at a position with lower gravitational potential – such as in closer proximity to a planet – are found to be running more slowly. In special relativity, the time dilation effect is reciprocal: as observed from the point of view of either of two clocks which are in motion with respect to each other, it will be the other clock that is time dilated. (This presumes that the relative motion of both parties is uniform; that is, they do not accelerate with respect to one another during the course of the observations.) In physics, the twin paradox is a thought experiment in special relativity involving identical twins, one of whom makes a journey into space in a high-speed rocket and returns home to find that the twin who remained on Earth has aged more. This result appears puzzling because each twin sees the other twin as moving, and so, according to an incorrect naive application of time dilation and the principle of relativity, each should paradoxically find the other to have aged more slowly.

From the perspective of a rotating ‘preferred reference frame’, time dilatation is caused by an object moving relative to the surrounding preferred reference frame. E.g.: GPS atomic clocks are subject to an amount of time dilatation proportional to their speed relative to the Earth-Centered Inertial frame.

When we apply this principle to the twin paradox:  the twin who remained at home kept ‘static’ relative to preferred reference frame and was therefore not affected by time dilatation  the twin who made a journey has travelled relative to the reference frames he encountered and was therefore affected by time dilatation (and therefore didn’t age as much as his twin brother)

If the twin brother would have travelled around the Solar in the same orbit as the Earth although in opposite direction, then he would be affected by an amount of time-dilatation proportionally to 60km/s. 24

Applying the principles of the aberration of light

Why can we observe stars of which the true direction is behind the Moon?

With full Moon:

- stars that show right next the Moon are subject to an amount of annual aberration equal to 20.5 arcsec - the Moon itself is not subject to annual aberration - the Moon rotates with 0.5 arcsec/sec around the Earth

As a logical consequence, the actual direction of a star that is closer than 20.5 arcsec to the border of the Moon is in effect behind the Moon at the moment of the observation.

From the perspective of the observer in frame B:

 Light truly comes from the ‘apparent direction’; the change in direction took place when entering Frame B.  The place in frame B where starlight entered frame B moves along with the frame.  The Earth is ‘static’ relative to frame B, as a consequence there is no stellar aberration for objects within frame B (such as the Moon).  The movement of the Moon relative to frame B has no effect on stellar aberration but results into a light-time correction for the Moon of −0.704".

From the perspective of Frame A:

 Both Moon and telescope move at velocity ‘v’ along with frame B  Light travels a velocity ‘c’ until it reaches Frame B.  The wave front of the starlight reorients itself in the direction 푠푖푛(휓) = −푣/푐⁡at the moment when it enters Frame B  The apparent speed of light becomes √푐2 − 푣² once it has entered Frame B (the superposition of velocity vectors v and c).  At the moment when starlight passes the border of the Moon: Moon o the telescope points to the border of the Moon o the telescope points to actual position of the starlight o the wave front of the starlight points towards the telescope  At the moment when starlight enters the telescope: o The wave front of the starlight is aligned with the tilted orientation of the telescope o the telescope points to retarded position of the Moon (taking into account the −0.704" light-time correction) o the actual position of the Moon is between the telescope and the actual direction of the star with an angle: 푠푖푛(휓) = −푣/푐⁡ relative to the border of the Moon

Venus transit The following image depicts the observed (apparent) positions of Venus and the Sun during the transit of June 2004:

Considering the effect of the aberration of light:

 Venus is observed with -3.6 arcsec aberration (to left on the above picture)  The Sun is observed with 20.5 arcsec aberration (to the right on the above picture)

Without the aberration of light, the image of Venus would have been observed 24.1 arcsec to the right relative to the image of the Sun (on the above picture) and would have looked as follows:

From the perspective of the observer in the frame of the Earth:

 Light travels on a curved path from the Sun to the observer on Earth  Sunlight initially travels through frames with a higher transversal velocity as compared to the reference frame where the Earth is located. Given the principle that only the velocity of the reference frame of the observer matters, sunlight is observed with 20.5 arcsec aberration.  Light originating from the reference frame of Venus will be subject to an aberration of 3.6arcsec when leaving the reference frame of Venus (due to the 5.2 km/s difference in speed with the reference frame of the Earth).  Sunlight will decrease its aberration angle from 24.1arcsec to 20.5arc when travelling from the reference frame of Venus into the reference frame of the Earth (again due to the 5.2 km/s difference in speed).

From the perspective of an observer on the Sun:

 Light travels on a straight path from the Sun to the observer on Earth  The wave front changes direction significantly when leaving the reference frame of the Sun and gradually decreases to 20.5arcsec when arriving at the Earth.  The planet Venus moves at 5.2km/s relative to the Earth while light travels from Venus to the Earth. This causes the apparent position of Venus to be displaced 3.6arcsec from its actual direction.

Relativistic beaming

Stellar objects will appear to be displaced in the direction of:

- The movement of the reference frame - The movement of the observer relative to the reference frame

While the observer cannot move faster than the speed of light ‘c’ within the local reference frame, the combined velocities of the reference frame and the observer relative to the reference frame can exceed ‘c’.

The annual residual of the Pioneer anomaly

John D. Anderson,∗a Philip A. Laing,†b Eunice L. Lau,‡a Anthony S. Liu,§c Michael Martin Nieto,¶d and Slava G. Turyshev: Study of the anomalous acceleration of Pioneer 10 and 11. Page. 40

“In Ref. [13] we reported, in addition to the constant anomalous acceleration term, a possible annual sinusoid. If approximated by a simple sine wave, the amplitude of this oscillatory term is about 1.6 × 10−8 cm/s2.”

Ref [13]: The Apparent Anomalous, Weak, Long-Range Acceleration of Pioneer 10 and 11 Slava G. Turyshev, John D. Anderson, Philip A. Laing, cEunice L.Lau, Anthony S. Liu, and Michael Martin Nieto

“The internal consistency tests indicate that, in addition to the formal uncertainties, there is evidence for a systematic mismodeling which results in an annual periodic term (plot B in Figure 1). This term has been found in the residuals of the both Pioneers and is currently being investigated.”

An annual acceleration residual of 1.6 E-10m/s² translates into an apparent maximum increase in distance of 7734m over the course of a year:

Acceleration / Apparent increase in distance 2E-10 10000 1,5E-10 8000 1E-10 6000 5E-11

0 4000

43 15 29 57 71 85 99

127 141 155 169 183 197 211 225 239 253 267 281 295 309 323 337 351 -5E-11 113 2000 -1E-10 0 -1,5E-10 -2E-10 -2000

Acceleration Distance

The graph below depicts the apparent cumulative increase of apparent distance for a signal sent from the Earth in the direction of the Sun taking into account an apparent velocity of √푐2 − 푣² and assuming: 퐺⁡푥푀푎푠푠⁡푆푢푛 푣 = √⁡ 퐷푖푠푡푎푛푐푒⁡푓푟표푚⁡푆푢푛

The x-axis is the percentage of the total distance travelled from the Earth to the Sun.

The apparent increase in distance is calculated for small increments of the distance Earth-Sun based on the formula: c.( distance/√푐2 − 푣² − ⁡distance/c⁡).

Cumulative apparent increase in distance (meters) due to the aberration of light 6000

5000

4000

3000

2000

1000

0 0,0% 10,0% 20,0% 30,0% 40,0% 50,0% 60,0% 70,0% 80,0% 90,0% 100,0%

When the Pioneer is located at the opposite side of the Sun, the back-and-return Doppler tracking signal will travel four times the 1-way distance to the Sun. The 7734 meter apparent increase in distance (corresponding with the annual acceleration residual of 1.6 E-10m/s² ) therefore corresponds with 1933 meter on the above graph. As per the graph, this corresponds with a signal that approaches the Sun at 7.3% of the distance Earth-Sun.

In 1987, the annual acceleration residual was 2.4 E-10m/s² corresponding to an apparent increase of 11600 meter. A per the above graph, this corresponds with a signal that approaches the Sun at 1.8% of the distance Earth-Sun.

We can assume that the Pioneer satellite was not located at the exact opposite side of the Sun but slightly above or below the Sun due the difference in inclination.

The apparent decrease of the velocity of light due to the aberration of light may therefore be an explanation of the observed annual variation.

The Hafele-Keating experiment From Wikipedia: “The Hafele–Keating experiment was a test of the theory of relativity. In October 1971, Hafele and Keating took four cesium-beam atomic clocks aboard commercial airliners. They flew twice around the world, first eastward, then westward, and compared the clocks against others that remained at the United States Naval Observatory. When reunited, the three sets of clocks were found to disagree with one another, and their differences were consistent with the predictions of special and general relativity.” The clock aboard the plane moving eastward, in the direction of the Earth's rotation, had a greater velocity (resulting in a relative time loss) than one that remained on the ground, while a clock aboard the plane moving westward, against the Earth's rotation, had a lower velocity than one on the ground.”

For the purpose of the Hafele-Keating experiment, the rotation speed of the Earth and the speed of the planes is measured relative to the Earth Centered Inertial Frame (centered on the Earth and not rotating with respect to the distant stars) which is different from the ‘preferred reference frame’ as used in this paper in the context of stellar aberration.

In the ‘Earth Centered Inertial Frame’, a plane that maintains a speed and orientation such that its position remains the same with respect to distant stars will be ‘at rest’ and will therefore show the highest clock frequency.

In the ‘preferred reference frame’ as per the hypothesis put forward in this document, a plane that maintains a speed and orientation such that its orientation remains the same relative to the Sun will be ‘at rest’ and will show the highest clock frequency.

The difference between both approaches is very small (1/365 = 0.27% ) and would not be measurable through the Hafele–Keating experiment but would be expected to have a small effect on GPS measurements such as described in the following paper.

Anomalous harmonics in the spectra of GPS position estimates (February 2007) J. Ray Æ Z. Altamimi Æ X. Collilieux Æ T. van Dam http://www.ngs.noaa.gov/CORS/Articles/pos-harmonics_gpssoln08.pdf

“We find no confirmation of the anomalous GPS position harmonics (multiples of ~1.04 cpy) in corresponding results from VLBI or SLR, nor in geophysical loadings due to atmospheric pressure, non- tidal ocean bottom pressure, or continental water storage. Because of this and the fact that the anomalous period of ~350 days matches the GPS constellation repeat cycle, it seems likely that the harmonics are a consequence of some technique error.”

Summary / Conclusion

The aberration of light takes place when either of 2 conditions take place:

- light propagates through reference frames with different transverse velocities - the observer moves relative to the local reference frame

The principles used to derive the formula for relativistic aberration:

- Light continues to follow a straight path when it enters a preferred reference frame with a different transverse velocity - Light will propagate with the speed of light ‘c’ relative to the reference frame it entered

The proposed ‘preferred reference frame’ at any point in the plane of the Solar System is one that rotates around the barycenter of the Solar System with the same speed as a planet would have at that position.