# Aberration of light

The apparent position of a star viewed from the Earth depends on the Earth's velocity. The effect is typically much smaller than illustrated.

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 locations dependent on the velocity of the observer. Aberration causes objects to appear to be angled or tilted towards the direction of motion of the observer compared to when the observer is stationary. The change in angle is typically very small, on the order of v/c where c is the speed of light and v the velocity of the observer. In the case of "stellar" or "annual" aberration, the apparent position of a star to an observer on Earth varies periodically over the course of a year as the Earth's velocity changes as it revolves around the Sun, by a maximum angle of approximately 20 arcseconds in right ascension or declination.

### Secular aberration

The Sun and Solar System are revolving around the center of the Galaxy. Aberration due to this motion is known as secular aberration and affects the apparent positions of distant stars and extragalactic objects. However, since the galactic year is about 230 million years the aberration varies very slowly the change in aberration is extremely difficult to observe. Therefore secular aberration is usually ignored when considering the positions of stars. In other words, star maps show the observed apparent positions of the stars, not their calculated true positions after accounting for secular aberration.

For stars significantly less than 230 million light years away, the Solar System may be approximated as an inertial frame and so the effect of secular aberration is equivalent to a light-time correction. This includes stars in the Milky Way, since the Milky Way is about 100,000 light years in diameter. For these stars the true position of the star is then easily computed from the product of its proper motion (in arcseconds per year) and its distance (in light years).

Secular aberration is typically a small number of arcminutes, for example the stationary star Groombridge 1830 is displaced by approximately 3 arcminutes.[8] due to secular aberration. This is roughly 8 times the effect of annual aberration, as one would expect since the velocity of the Solar System relative to the Milky Way is about 8 times the velocity of the Earth relative to the Sun.

## Discovery and first observations

The discovery of the aberration of light was totally unexpected, and it was only by extraordinary perseverance and perspicacity that Bradley was able to explain it in 1727. Its origin is based on attempts made to discover whether the stars possessed appreciable parallaxes. The Copernican theory of the solar system – that the Earth revolved annually about the Sun – had received confirmation by the observations of Galileo and Tycho Brahe and the mathematical investigations of Kepler and Newton.

### Search for stellar parallax

As early as 1573, Thomas Digges had suggested that parallactic shifting of the stars should occur according to the heliocentric model of the Solar System, and consequently if such stellar parallaxes could be observed they would help confirm the heliocentric theory. Many observers claimed to have determined such parallaxes, but Tycho Brahe and Giovanni Battista Riccioli concluded that they existed only in the minds of the observers, and were due to instrumental and personal errors. In 1680 Jean Picard, in his Voyage d’Uranibourg, stated, as a result of ten years' observations, that Polaris, or the Pole Star, exhibited variations in its position amounting to 40″ annually. Some astronomers endeavoured to explain this by parallax, but these attempts were futile, for the motion was at variance with that which parallax would produce. John Flamsteed, from measurements made in 1689 and succeeding years with his mural quadrant, similarly concluded that the declination of the Pole Star was 40″ less in July than in September. Robert Hooke, in 1674, published his observations of γ Draconis, a star of magnitude 2m which passes practically overhead at the latitude of London, and whose observations are therefore free from the complex corrections due to astronomical refraction, and concluded that this star was 23″ more northerly in July than in October.

Bradley's data on the north-south component of the aberration of γ-Draconis in 1727 establishing stellar aberration. .[9]

When James Bradley and Samuel Molyneux entered this sphere of astronomical research in 1725, there consequently prevailed much uncertainty whether stellar parallaxes had been observed or not; and it was with the intention of definitely answering this question that these astronomers erected a large telescope at the house of the latter at Kew.[2] They determined to reinvestigate the motion of γ Draconis; the telescope, constructed by George Graham (1675–1751), a celebrated instrument-maker, was affixed to a vertical chimney stack, in such manner as to permit a small oscillation of the eyepiece, the amount of which (i.e. the deviation from the vertical) was regulated and measured by the introduction of a screw and a plumb line.

The instrument was set up in November 1725, and observations on γ Draconis were made starting in December. The star was observed to move 40″ southwards between September and March, reversing its course from March to September. These results were unexpected and inexplicable by existing theories.

### Early hypotheses

This motion was evidently not due to parallax nor was it due to observational errors. Bradley and Molyneux discussed several hypotheses in the hope of finding the solution.

Bradley first hypothesized that the apparent motion could be due to oscillations in the orientation of the Earth's axis relative to the celestial sphere – a phenomenon known as nutation. This could be tested using the fact the apparent position of stars on the opposite side of the celestial sphere would be affected by an equal and opposite amount. Bradley tested this using a star with a right ascension nearly exactly opposite to that of γ Draconis. This star was seen to possess an apparent motion which could be consistent with nutation, but since its declination varied only one half as much as in the case of γ Draconis, it was obvious that nutation did not supply the requisite solution. Although nutation could not explain the observed stellar motion, Bradley later went on to discover that the Earth does indeed nutate.

Bradley also investigated the possibility that the motion was due to an irregular distribution of the Earth's atmosphere, thus involving abnormal variations in the refractive index, but again obtained negative results.

On August 19, 1727, Bradley then embarked upon a further series of observations using a telescope of his own erected at the Rectory, Wanstead. This instrument had the advantage of a larger field of view and he was able to obtain precise positions of a large number of stars over the course of about two years. This established the existence of the phenomenon of aberration beyond all doubt, and also allowed Bradley to formulate a set of rules that would allow the calculation of the effect on any given star at a specified date.

### Development of the theory of aberration

Bradley eventually developed the explanation of aberration in about September 1728 and his theory was presented to the Royal Society in mid January the next year. Based on his early calculations, Bradley was able to estimate the constant of aberration at 20", and with this was able to estimate the speed of light at Template:Convert per second.[10] One well-known story was that he saw the change of direction of a wind vane on a boat on the Thames, caused not by an alteration of the wind itself, but by a change of course of the boat relative to the wind direction. [8] However, there is no record of this incident in Bradley's own account of the discovery, and it may therefore be apocryphal.

The discovery and elucidation of aberration is now regarded as a classic case of the application of scientific method, in which observations are made to test a theory, but unexpected results are sometimes obtained that in turn lead to new discoveries. It is also worth noting that part of the original motivation of the search for stellar parallax was to test the Copernican theory that the Earth revolves around the Sun, but of course the existence of aberration also establishes the truth of that theory.

## Historical theories of aberration

The phenomenon of aberration became a driving force for many physical theories during the 200 years between its observation and the conclusive explanation by Albert Einstein.

The first classical explanation was provided in 1729 by James Bradley as described above, who attributed it to the finite speed of light and the motion of Earth in its orbit around the Sun.[1][2] However, this explanation proved inaccurate once the wave nature of light was better understood, and correcting it became a major goal of the 19th century theories of luminiferous aether. Augustin-Jean Fresnel proposed a correction due to the motion of a medium (the aether) through which light propagated, known as "partial aether drag". He proposed that objects partially drag the aether along with them as they move, and this became the accepted explanation for aberration for some time. George Stokes proposed a similar theory, explaining that aberration occurs due to the flow of aether induced by the motion of the Earth. Accumulated evidence against these explanations combined with new understanding of the electromagnetic nature of light led Hendrik Lorentz to develop an electron theory which featured an immobile aether, and he explained that objects contract in length as they move through the aether. Motivated by these previous theories, Albert Einstein then developed the theory of special relativity in 1905, which provides the modern account of aberration.

Figure 2: As light propagates down the telescope, the telescope moves requiring a tilt to the telescope that depends on the speed of light. The apparent angle of the star φ differs from its true angle θ

Bradley conceived of an explanation in terms of a corpuscular theory of light in which light is made of particles unaffected by gravity.[4] His classical explanation appeals to the motion of the earth relative to a beam of light-particles moving at a finite velocity, and is developed in the Sun's frame of reference, unlike the classical derivation given above.

Consider the case where a distant star is motionless relative to the Sun, and the star is extremely far away, so that parallax may be ignored. In the rest frame of the Sun, this means light from the star travels in parallel paths to the Earth observer, and arrives at the same angle regardless of where the Earth is in its orbit. Suppose the star is observed on Earth with a telescope, idealized as a narrow tube. The light enters the tube from the star at angle ${\displaystyle \theta }$ and travels at speed ${\displaystyle c}$ taking a time ${\displaystyle h/c}$ to reach the bottom of the tube, where it is detected. Suppose observations are made from Earth, which is moving with a speed ${\displaystyle v}$. During the transit of the light, the tube moves a distance ${\displaystyle vh/c}$. Consequently, for the particles of light to reach the bottom of the tube, the tube must be inclined at an angle ${\displaystyle \phi }$ different from ${\displaystyle \theta }$, resulting in an apparent position of the star at angle ${\displaystyle \phi }$. As the Earth proceeds in its orbit it changes direction, so ${\displaystyle \phi }$ changes with the time of year the observation is made. The apparent angle and true angle are related using trigonometry as:

${\displaystyle \tan(\phi )={\frac {h\sin(\theta )}{hv/c+h\cos(\theta )}}={\frac {\sin(\theta )}{v/c+\cos(\theta )}}}$.

In the case of ${\displaystyle \theta =90^{\circ }}$, this gives ${\displaystyle \tan(\theta -\phi )=v/c}$. While this is different from the more accurate relativistic result described above, in the limit of small angle and low velocity they are approximately the same, within the error of the measurements of Bradley's day. These results allowed Bradley to make one of the earliest measurements of the speed of light.[11][12]

### Luminiferous aether

Young reasoned that aberration could only be explained if the aether was immobile in the frame of the Sun. On the left, stellar aberration occurs if an immobile aether is assumed, showing that the telescope must be tilted. On the right, the aberration disappears if the aether moves with the telescope, and the telescope does not need to be tilted.

In the early nineteenth century the wave theory of light was being rediscovered, and in 1804 Thomas Young adapted Bradley's explanation for corpuscular light to wavelike light traveling through a medium known as the luminiferous aether. His reasoning was the same as Bradley's, but it required that this medium be immobile in the Sun's reference frame and must pass through the earth unaffected, otherwise the medium (and therefore the light) would move along with the earth and no aberration would be observed. [13] He wrote:

Upon consideration of the phenomena of the aberration of the stars I am disposed to believe that the luminiferous aether pervades the substance of all material bodies with little or no resistance, as freely perhaps as the wind passes through a grove of trees.

—Thomas Young, 1804[4]

However, it soon became clear Young's theory could not account for aberration when materials with a non-vacuum index of refraction were present. An important example is of a telescope filled with water. The velocity of the light in such a telescope will be slower than in vacuum, and is given by ${\displaystyle c/n}$ rather than ${\displaystyle c}$ where ${\displaystyle n}$ is the index of refraction of the water. Thus, by Bradley and Young's reasoning the aberration angle is given by

${\displaystyle \tan(\phi )={\frac {\sin(\theta )}{v/(c/n)+\cos(\theta )}}}$.

which predicts a medium-dependent angle of aberration. When refraction at the telescope's objective is taken into account this result deviates even more from the vacuum result. In 1810 François Arago performed a similar experiment and found that the aberration was unaffected by the medium in the telescope, providing solid evidence against Young's theory. This experiment was subsequently verified by many others in the following decades, most accurately by Airy in 1871, with the same result.[13]

### Aether drag models

#### Fresnel's aether drag

In 1818 Augustin Fresnel developed a modified explanation to account for the water telescope and for other aberration phenomena. He explained that the aether is generally at rest in the Sun's frame of reference, but objects partially drag the aether along with them as they move. That is, the aether in an object of index of refraction ${\displaystyle n}$ moving at velocity ${\displaystyle v}$ is partially dragged with a velocity ${\displaystyle (1-1/n^{2})v}$ bringing the light along with it. This factor is known as "Fresnel's dragging coefficient". This dragging effect, along with refraction at the telescope's objective, compensates for the slower speed of light in the water telescope in Bradley's explanation. [nb 1] With this modification Fresnel obtained Bradley's vacuum result even for non-vacuum telescopes, and was also able to predict many other phenomena related to the propagation of light in moving bodies. Fresnel's dragging coefficient became the dominant explanation of aberration for the next decades.

Conceptual illustration of Stokes' aether drag theory. In the rest frame of the Sun the Earth moves to the right through the aether, in which it induces a local current. A ray of light (in red) coming from the vertical becomes dragged and tilted due to the flow of aether.

#### Stokes' aether drag

However, the fact that light is polarized (discovered by Fresnel himself) led scientists such as Cauchy and Green to believe that the aether was a totally immobile elastic solid as opposed to Fresnel's fluid aether. There was thus renewed need for an explanation of aberration consistent both with Fresnel's predictions (and Arago's observations) as well as polarization.

In 1845 Stokes proposed a 'putty-like' aether which acts as a liquid on large scales but as a solid on small scales, thus supporting both the transverse vibrations required for polarized light and the aether flow required to explain aberration. Making only the assumptions that the fluid is irrotational and that the boundary conditions of the flow are such that the aether has zero velocity far from the Earth, but moves at the Earth's velocity at its surface and within it, he was able to completely account for aberration.[nb 2] The velocity of the aether outside of the Earth would decrease as a function of distance from the Earth so light rays from stars would be progressively dragged as they approached the surface of the Earth. The Earth's motion would be unaffected by the aether due to D'Alembert's paradox.

Both Fresnel and Stokes' theories were popular. However, the question of aberration was put aside during much of the second half of the 19th century as focus of inquiry turned to the electromagnetic properties of aether.

### Lorentz' length contraction

{{#invoke:see also|seealso}} In the 1880s once electromagnetism was better understood, interest turned again to the problem of aberration. By this time flaws were known to both Fresnel's and Stokes' theories. Fresnel's theory required that the relative velocity of aether and matter to be different for light of different colors, and it was shown that the boundary conditions Stokes had assumed in his theory were inconsistent with his assumption of irrotational flow.[4][13][14] At the same time, the modern theories of electromagnetic aether could not account for aberration at all. Many scientists such as Maxwell, Heaviside and Hertz unsuccessfully attempted to solve these problems by incorporating either Fresnel or Stokes' theories into Maxwell's new electromagnetic laws.

Hendrik Lorentz spent considerable effort along these lines. After working on this problem for a decade, the issues with Stokes' theory caused him to abandon it and to follow Fresnel's suggestion of a (mostly) stationary aether (1892, 1895). However, in Lorentz's model the aether was completely immobile, like the electromagnetic aethers of Cauchy, Green and Maxwell and unlike Fresnel's aether. He obtained Fresnel's dragging coefficient from modifications of Maxwell's electromagnetic theory, including a modification of the time coordinates in moving frames ("local time"). In order to explain the Michelson-Morely experiment (1887), which apparently contradicted both Fresnel's and Lorentz's immobile aether theories, and apparently confirmed Stokes' complete aether drag, Lorentz theorized (1892) that objects undergo "length contraction" by a factor of ${\displaystyle {\sqrt {1-v^{2}/c^{2}}}}$ in the direction of their motion through the aether. In this way, aberration (and all related optical phenomena) can be accounted for in the context of an immobile aether. Lorentz' theory became the basis for much research in the next decade, and beyond. Its predictions for aberration are identical to those of the relativistic theory.[13][15]

### Special relativity

Lorentz' theory matched experiment well, but it was complicated and made many unsubstantiated physical assumptions about the microscopic nature of electromagnetic media. In his 1905 theory of special relativity, Albert Einstein reinterpreted the results of Lorentz' theory in a much simpler and more natural conceptual framework which disposed of the idea of an aether. His derivation is given above, and is now the accepted explanation. Robert S. Shankland reported some conversations with Einstein, in which Einstein emphasized the importance of aberration:[16]

He continued to say the experimental results which had influenced him most were the observations of stellar aberration and Fizeau’s measurements on the speed of light in moving water. “They were enough,” he said.

Other important motivations for Einstein's development of relativity were the moving magnet and conductor problem and (indirectly) the negative aether drift experiments, already mentioned by him in the introduction of his first relativity paper. Einstein wrote in a note in 1952:[3]

My own thought was more indirectly influenced by the famous Michelson-Morley experiment. I learned of it through Lorentz’ path breaking investigation on the electrodynamics of moving bodies (1895), of which I knew before the establishment of the special theory of relativity. Lorentz’ basic assumption of a resting ether did not seem directly convincing to me, since it led to an [struck out: to me artificial appearing] interpretation of the Michelson-Morley experiment, which [struck out: did not convince me] seemed unnatural to me. My direct path to the sp. th. rel. was mainly determined by the conviction that the electromotive force induced in a conductor moving in a magnetic field is nothing other than an electric field. But the result of Fizeau’s experiment and the phenomenon of aberration also guided me.

While Einstein's result is the same as Bradley's original equation except for an extra factor of ${\displaystyle \gamma }$, it should be emphasized that Bradley's result does not merely give the classical limit of the relativistic case, in the sense that it gives incorrect predictions even at low relative velocities. Bradley's explanation cannot account for situations such as the water telescope, nor for many other optical effects (such as interference) that might occur within the telescope. This is because in the Earth's frame it predicts that the direction of propagation of the light beam in the telescope is not normal to the wavefronts of the beam, in contradiction with Maxwell's theory of electromagnetism. It also does not preserve the speed of light c between frames. However, Bradley did correctly infer that the effect was due to relative velocities.

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6. In fact, the light source doesn't need to be stationary, consider for example eclipsing binary stars: they are rotating with high speed —and ever changing and different velocity vectors— around each other, but they appear as one spot all the time.
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12. Encyclopædia Britannica
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• P. Kenneth Seidelmann (Ed.), Explanatory Supplement to the Astronomical Almanac (University Science Books, 1992), 127-135, 700.
• Stephen Peter Rigaud, Memoirs of Bradley (1832)
• Charles Hutton, Mathematical and Philosophical Dictionary (1795).
• H. H. Turner, Astronomical Discovery (1904).

## Notes

1. More in detail, Fresnel explains that the incoming light of angle ${\displaystyle \theta }$ is first refracted at the end of the telescope, to a new angle ${\displaystyle \psi }$ within the telescope. This may be accounted for by Snell's law, giving ${\displaystyle \sin(\theta -\phi )=n\sin(\psi -\phi )}$. Then drag must be accounted for. Without drag, the x and y components of the light in the telescope are ${\displaystyle (c/n)\sin(\psi )}$ and ${\displaystyle (c/n)\cos(\psi )}$, but drag modifies the x component to ${\displaystyle (c/n)\cos(\psi )-(1-1/n^{2})v}$ if the Earth moves with velocity ${\displaystyle v}$. If ${\displaystyle \alpha }$ is angle and ${\displaystyle v_{l}}$ is the velocity of the light with these velocity components, then by Bradley's reasoning ${\displaystyle \tan(\phi )={\frac {h\sin(\alpha )}{vt+h\cos(\alpha )}}}$ where ${\displaystyle h}$ is the modified path length through the water and t is the time it takes the light to travel the distance h, ${\displaystyle t=h/v_{l}}$. Upon solving these equations for ${\displaystyle \phi }$ in terms of ${\displaystyle \theta }$ one obtains Bradley's vacuum result.
2. Stokes' derivation may be summarized as follows: Consider a wavefront moving in the downwards z direction. Say the aether has velocity field ${\displaystyle u,v,w}$ as a function of ${\displaystyle x,y,z}$. Now, motion of the aether in the x and y directions does not affect the wavefront, but the motion in the z direction advances it (in addition to the amount it advances at speed c). If the z velocity of the aether varies over space, for example if it is slower for higher x as shown in the figure, then the wavefront becomes angled, by an angle ${\displaystyle \tan(\alpha )=tdw/dx}$. Now, say in time t the wavefront has moved by a span ${\displaystyle dz\approx ct}$ (assuming the speed of the aether is negligible compared to the speed of light). Then for each distance ${\displaystyle dz}$ the ray descends, it is bent by an angle ${\displaystyle \alpha \approx (dw/dx)(dz/c)}$, and so the total angle by which it has changed after travelling through the entire fluid is

${\displaystyle \alpha \approx {\frac {1}{c}}\int {\frac {\partial w}{\partial x}}dz}$

If the fluid is irrotational it will satisfy the Cauchy-Riemann equations, one of which is

${\displaystyle {\frac {\partial w}{\partial x}}={\frac {\partial u}{\partial z}}}$.

Inserting this into the previous result gives an aberration angle ${\displaystyle \alpha =(u_{2}-u_{1})/c}$ where the ${\displaystyle u}$s represent the x component of the aether's velocity at the start and end of the ray. Far from the earth the aether has zero velocity, so ${\displaystyle u_{2}=0}$ and at the surface of the earth it has the earth's velocity ${\displaystyle v}$. Thus we finally get

${\displaystyle \alpha \approx {\frac {v}{c}}}$

which is the known aberration result.