Light aberration

The difference between the observed position of a star and its actual position is called light aberration or Bradley aberration due to the combination of the speed of the star. observer and the speed of light.

Discovery

In 1725 , James Bradley, then Saviliano Astronomy professor at Oxford University, tried to measure the distance to a star observing his orientation in two different times of the year. The position of the earth changed while orbiting around the sun and, therefore, provided a large baseline for the triangulation of the star. To his surprise, he found that the fixed stars showed an apparent systematic movement, related to the direction of the movement of the earth in their orbit and did not depend, as it had anticipated, on the position of the earth in space.

Concept

The phenomenon of aberration of light (also known as astronomical aberration, star aberration or Bradley aberration) is an astronomical phenomenon that produces an apparent movement of the positions of celestial objects dependent on the velocity of the observer. The aberration causes the position of these objects to present a small angular deviation in the direction of the observer movement compared to when the observer is at rest. The variation of the angle is usually very small, of the order V/CWhere C is the speed of light and V the speed of the observer. In the case of "stellar" aberration or "annual" aberration, the apparent position of a star for an observer on Earth varies periodically over the course of one year due to changes in the orientation of the Earth's velocity in its orbit around the Sun, with a maximum angle of approximately 20 archseconds in straight ascension or decline.

The study of astronomical aberration is of great historical importance due to its role in the development of theories of light, electromagnetism and, ultimately, the theory of special relativity. It was first observed in the late 1600s by astronomers searching for stellar parallax in order to confirm the heliocentric model of the solar system. To his surprise, in 1729 James Bradley provided an explanation for this phenomenon in the terms of classical mechanics, considering a finite speed of light with respect to the motion of the Earth in its orbit around the Sun. allowing one of the first calculations of the speed of light to be carried out. However, Bradley's theory was incompatible with light theories of the 19th century, and the aberration became a of the main motivations for the aether drag hypothesis of Augustin Fresnel (in 1818) and G. G. Stokes (in 1845), and for the ether theory in electromagnetism of Hendrick Lorentz in 1892. The aberration of light, together with the elaboration of Maxwell's equations, the problem of the moving magnet and the Lorentz conductor, the unsuccessful experiments to detect the drift of the ether, as well as the Fizeau experiment, led Albert Einstein to develop the theory of Special Relativity in 1905, providing a conclusive explanation for the phenomenon of aberration.

The term 'aberration' It has historically been used to refer to a series of phenomena related to the propagation of light from moving bodies. It should not be confused with the concept of stellar parallax, caused by a change in the position of the observer located on Earth (when it occupies opposite points of its orbit around the sun) with respect to a relatively close object in terms of astronomical (in theory, with respect to any object outside the solar system). Instead, stellar aberration is related to light-time correction and relativistic radiation, although it is often considered separately from these effects.

The term aberration is also used to refer to phenomena related to optical systems (Aberration in optical systems).

Explanation

Aberration can be explained as the difference in the angle of a light beam in different inertial reference frames. A common analogy is the apparent direction of falling rain: if the rain is falling vertically for a person at rest, for a person moving forward the rain will show a certain angle, leading them to tilt their umbrella forward.. The faster you move, the greater the tilt of the umbrella.

The net effect is that light rays reaching an observer moving through a stationary reference frame will appear at a forward angle in the moving observer's reference frame. This effect is sometimes called "reflector" or "lighthouse" effect.

In the case of annual starlight aberration, the direction of starlight incident on the Earth's moving reference frame is inclined with respect to the angle observed in the sun's frame. Since the Earth's motion during its orbit around the sun changes orientation, the direction of this tilt changes over the course of the year, causing the apparent position of a star to differ from its true position.

Although this reasoning to explain aberration is quite intuitive, it leads to a series of paradoxes observable with the assumptions of Newtonian physics. The theory of special relativity does allow us to correctly explain the aberration. However, the relativistic explanation is very similar to the classical one, and in both theories the aberration can be understood as a case of sum of velocities.

Classic explanation

In the frame of the Sun, it is considered a beam of light with a speed equal to the speed of light c, with components of speed x e and ${displaystyle u_{x}}$ ${displaystyle u_{y}}$, and with an angle of incidence ${displaystyle tan(theta)=u_{y}/u_{x}}$. If Earth is moving at speed ${displaystyle v}$ in the direction x regarding the Sun, then by the sum of velocities of the component x of the speed of light beam in the frame of reference of the Earth, you have to ${displaystyle u_{x}'=u_{x}+v}$. The speed on the shaft has not been modified: ${displaystyle u_{y}'=u_{y}}$ (Please note that what needs to be known is the speed of the Sun vis-à-vis the Earth, which is the negative of the speed of the Earth with respect to the Sun, also bearing in mind that only vectors are being used, without indicating their direction.) Therefore the angle of light in the Earth's reference frame regarding the angle of light in the Sun's reference frame is:

${displaystyle tan(phi)={frac {u_{y}'}{u_{x}}}}}}{frac {u_{y}}{(u_{x}+v)}}}}}}}{{frac {sin(theta)}{(cos(theta)+v/c)}}}}}}}}}}$

In the case of ${displaystyle theta =90^{circ }}$, this result is reduced to ${displaystyle tan(theta -phi)=v/c}$.

Relativistic explanation

The reasoning in the relativistic case is the same, except that the formulas for the sum of relativistic velocities must be used, which can be derived from the Lorentz Transformation between the different reference frames. These formulas are:

${displaystyle u_{x}'=(u_{x}+v)/(1+u_{xv}/c^{2})}}$

${displaystyle u_{y}'=u_{y}/gamma (1+u_{x}v/c^{2}}}}}$

where ${displaystyle gamma =1/{sqrt {1-v^{2}/c^{2}}}}}}$, with the light beam components in the Earth reference framework expressed in terms of its components in the Sun reference framework. The angle of the beam within the framework of the Earth is, therefore,

${displaystyle tan(phi)={frac {u_{y}'}{u_{x}}}}}}{frac {u_{y}}{gamma (u_{x}+v)}}}}={frac {sin(theta)}{gamma (cos(theta+v/c)}}}}}}}}}}}}}}}$

In the case of ${displaystyle theta =90^{circ }}$, this result is reduced to ${displaystyle sin(theta -phi)=v/c}$and at the limit when ${displaystyle v/cll 1}$, this can approach ${displaystyle theta -phi =v/c}$. This relativistic deduction keeps the speed of constant light in all frames of reference ${displaystyle {sqrt {u_{x}^{2}+u_{y}{2}}}{2}}}}}=c$unlike the classic deduction previously exposed.

Relation to correction in light time and relativistic radiation

aberration is related to two other phenomena, the correction in light time, which is due to the movement of an object observed during the time it takes to light to reach the observer, and the relativistic radiation, related to the movement of a light source. It can be considered equivalent to both, but in an inertial system of different reference. In the case of aberration, the observer is considered to move with respect to one (to simplify the problem) stationary light source, while in terms of the correction of light time and relativistic radiation it is considered that the source of light is the one that moves with respect to a stationary observer.

Consider the case of an observer and a light source that moves at a constant speed with respect to the first, with a beam of light that moves from the source to the observer. At the time of the emission, in the observer's resting system the light beam is inclined compared to the reference reference of the source, as understood considering the phenomenon of relativistic radiation. During the time it takes for the light beam to reach the observer, the light source moves within the framework of the observer, and the " true position " The light source moves in relation to the apparent position that the observer sees, as explained by the correction of light time. Finally, the beam within the framework of the observer at the time of observation is inclined compared to the lightning in the framework of the source, which can be understood as an aberration effect. Therefore, an observer within the framework of the light source would describe the apparent inclination of the lightning light in terms of aberration, while a person within the framework of the observer would describe it as a light time correction effect.

The relationship between these phenomena is only valid if the observer and the light source are located on inertial reference systems. In practice, because the earth is not an inertial rest system, because it experiences a centripetal acceleration towards the sun, annual aberration on Earth cannot be considered as a correction in time of light. However, if the time between the emission and the detection of the light is short compared to the orbital period of the Earth, the land itself can approximate an inertial system, and the effects of aberration are equivalent to the corrections in the corrections in Light time

Types of aberrations

There are a series of types of aberration, caused by the different components of the Earth's movement:

• Annual is due to the rotation of the Earth around the Sun.

• Annual solar aberration is the almost constant deflection of the apparent position of the Sun regarding its own resting system.

• Planetary aberration is the combination of aberration and light time correction.
• Daytime aberration is due to the rotation of the Earth on its own axis.
• Secular aberration is due to the movement of the Sun and the solar system in relation to other stars in the galaxy.

Annual aberration

The annual aberration is caused by the movement of an observer located on Earth, which revolves around the Sun. The Speed of the Earth ${displaystyle v}$ (considering a system with the Sun at rest) varies periodically in the course of a year in which the Earth runs through its orbit, and therefore aberration also varies periodically, causing the apparent movement of stars in small ellipses.

Considering the orbit of the Earth practically circular, the maximum displacement of a star due to annual aberration is known as the constant aberrationconventionally represented by ${displaystyle kappa }$. Can be calculated using the ratio ${displaystyle kappa =theta -phi approx v/c}$replacing the average speed of the Earth within the Sun ${displaystyle v}$ and the speed of light ${displaystyle c}$. Its value is 20,49552" archseconds (in J2000).

Assuming a circular orbit of the Earth, the annual aberration causes the stars located exactly in the ecliptic (the plane of the Earth's orbit) to seem to move backward and forward along a straight line, which varies according to ${displaystyle kappa }$ on both sides of its position in the sun frame. A star that is precisely in one of the poles of the ecliptic (90° compared to the plane of the ecliptic) will appear to move in a circle of radio ${displaystyle kappa }$ around their true position; and the stars in intermediate ecliptic latitudes will seem to move along a small ellipse.

For example, consider a star at the north pole of the ecliptic, seen by an observer at the summit of the earth (respect to the very plane of the ecliptic), at a point of the Arctic polar circle. At the time of the March equinox, the Earth orbit takes the observer in a direction southward, and therefore the apparent decline of the star moves southwards at an angle of ${displaystyle kappa }$. In the September equinox, the position of the star moves northward in an equal and opposite amount. In the solstices of June and December, the displacement in decline is zero. On the contrary, the amount of displacement in straight ascension is zero in equinoxes and maximum in solstices.

In practice, the Earth's orbit is slightly elliptical rather than circular, and therefore with slight changes in speed throughout its orbit, meaning that the above description is only approximate. The aberration is most accurately calculated using the relative instantaneous velocity of the Earth with respect to the center of mass of the Solar System.

It should be noted that displacement due to aberration is orthogonal to any displacement due to parallax. When parallax is detectable, the maximum southward shift would occur in December, and the maximum northward shift in June. It was this apparently anomalous motion that baffled the first astronomers who observed it.

Annual solar aberration

A special case of annual aberration is the almost constant deflection of the apparent position of the Sun regarding its own system at rest, of value ${displaystyle kappa }$ towards West (as seen from Earth), opposed to the apparent movement of the Sun along the ecliptic (which is from west to east, seen from Earth). Thus, this deviation makes the Sun seeming to be behind (or retarded) in the ecliptic at an angle ${displaystyle kappa }$ regarding its position in the frame of reference of the Sun itself at rest.

This deviation can be described in an equivalent way as a light-time correction effect due to the Earth's movement for the 8.3 minutes it takes the light to travel from the Sun to Earth. This is assumed because the transit time of the Sun's light is short in relation to the Earth's orbital period, so the Earth's frame can be considered as inertial in this case. Within the framework of the Earth, the Sun has shifted a distance ${displaystyle Delta x=vt}$ in the time it takes the light to reach Earth, being calculated ${displaystyle t=R/c}$ radio ${displaystyle R}$ of Earth's orbit around the Sun. This gives a angular correction ${displaystyle tan(theta)approx theta =Delta x/R}$which can be operated to give ${displaystyle theta =v/c=kappa }$the same as the correction of star aberration.

Planetary aberration

Planetary aberration is the combination of light aberration (due to the Earth's speed) and light-time correction (due to object motion and distance), as calculated in the system at rest Of the solar system. Both are determined at the instant when the light from the moving object reaches the moving observer on Earth. It is so called because it is generally applied to planets and other objects in the solar system whose motion and distance are precisely known.

Diurnal aberration

Diurnal aberration is caused by the speed of the observer over the rotating Earth's surface. Therefore, it depends not only on the moment in which the observation is made, but also on the latitude and longitude of the observer. Its effect is much smaller than that of the annual aberration, and is only 0.32" in the case of an observer at the equator, where the tangential velocity derived from the Earth's rotation is greater.

Secular aberration

The solar system revolves around the center of the Galaxy. The 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 so slowly that its change is extremely difficult to observe. Therefore, secular aberration tends to be ignored when considering star positions. In other words, star maps show the apparent observed positions of stars, not their true positions calculated after accounting for secular aberration.

For stars significantly less than 230 million light-years away, the solar system can be considered an inertial system, and therefore the effect of secular aberration is equivalent to a light-time correction. This includes the stars of the Milky Way, since the Milky Way is about 100,000 light years in diameter. For these stars, their true position is then easily calculated by the product of their proper motion (in arc seconds per year) and their distance (in light years).

Secular aberration is usually a small number of minutes of arc. For example, the stationary star Groombridge 1830 moves about 3 minutes of arc due to secular aberration. This is approximately 8 times the effect of annual aberration, as it would be expected, given that the speed of the Solar System in relation to the Milky Way is about 8 times the speed of the Earth with respect to the Sun.

Discovery and first observations

The discovery of light aberration was totally unexpected. Only Bradley's extraordinary perseverance and insight was able to give the first explanation of the phenomenon in 1727. Its origin is based on attempts made to discover whether stars possessed appreciable parallaxes. The Copernican theory of the solar system – stating that the Earth revolves around the Sun every year; had received confirmation from 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 the parallactic shift of stars must occur in accordance with the heliocentric model of the solar system, and therefore, if such stellar parallaxes could be observed this could help confirm the heliocentric theory.. Many observers claimed to have determined some form of parallax, but Tycho Brahe and Giovanni Riccioli concluded that they existed only in the observers' minds, and were due to instrumental and procedural errors. In 1680 Jean Picard, in his Voyage d'Uranibourg, declared as a result of his observations for ten years, that the North Star exhibited variations in its position of 40 & # 34; annually. Some astronomers tried to explain this fact as a case of parallax, but their attempts were useless, because the motion was in disagreement with the cause that should produce the aforementioned parallax. John Flamsteed, from measurements made in 1689 and subsequent years with his mural quadrant, similarly concluded that the declination of the North Star was 40 & # 34; lower in July than in September. Robert Hooke in 1674 published his observations of gamma Draconis, a star of magnitude 2^m which passes practically above the latitude of London, and whose observations are therefore free of the complex corrections due to astronomical refraction, and concluded that this star was 23" more northerly in July than in October.

The observations of James Bradley

When James Bradley and Samuel Molyneux approached this area of astronomical research in 1725, great uncertainty still prevailed as to whether stellar parallaxes had been observed or not; and it was undoubtedly the intention to answer this question that these two astronomers erected a large telescope in the latter's house, in Kew, with which they decided to re-investigate the movement of gamma Draconis. The telescope, built by George Graham (1675-1751), a famous instrument maker, was fixed to a vertical chimney, in such a way as to allow a small oscillation of the eyepiece, the variation of which (i.e. the deviation from the vertical) It was regulated by a screw and measured with a plumb line.

The instrument was ready in November 1725, and observations on gamma Draconis were made beginning in December. The star was observed moving 40" towards the south between September and March, reversing its course from March to September. These results were unexpected and inexplicable by existing theories.

Starting hypothesis

This movement was evidently not due to parallax nor was it due to observation errors. Bradley and Molyneux analyzed several hypotheses in the hope of finding the solution.

Bradley's first hypothesis was that the apparent motion could be due to oscillations in the orientation of the Earth's axis with respect to the celestial sphere - a phenomenon known as nutation. This could be proven using the fact that the apparent position of the stars on the opposite side of the celestial sphere would be affected by an equal and opposite amount. Bradley tested this idea using a star with a right ascension almost exactly opposite that of gamma Draconis. It was observed that this star had an apparent motion that could be consistent with nutation, but its declination varied only half as much as in the case of gamma Draconis, so it was obvious that nutation did not facilitate the required solution. Although nutation could not explain the observed stellar motion, Bradley obtained clues to later discover the pitching of the Earth's axis.

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

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

The development of the theory of aberration

Bradley finally developed the explanation for the aberration around September 1728 and his theory was presented to the Royal Society in mid-January of the following year. Based on his first calculations, Bradley was able to estimate the aberration constant in the 20's, and with this he was able to estimate the speed of light at 183,300 miles (294,992 km) per second. A well-known story says that the explanation of the phenomenon came to him when he saw the change of direction of a weather vane on a ship in the Thames, not caused by an alteration of the wind itself, but by a change in the direction of the ship with respect to the direction of the wind. Curiously, there is no record of this incident in Bradley's own account of its discovery.

The discovery and elucidation of the phenomenon of astronomical aberration is now considered a classic case of the application of the scientific method, in which observations are made to test a theory, but the results obtained are sometimes unexpected, which in turn leads to new discoveries. It is also worth noting that part of the original motivation for the search for stellar parallax was to test Copernicus's theory that the Earth revolves around the Sun, making it clear that the existence of the aberration also establishes the truth of that theory.

Historical theories of aberration

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

As described above, the first classical explanation was provided in 1729 by James Bradley, who attributed it to the finite speed of light and the motion of the Earth in its orbit around the Sun. However, this explanation proved inaccurate once the wave nature of light was better understood, and correction became an important goal of 19th century theories. /span> of the luminous ether. Augustin Fresnel proposed a correction due to the movement of a medium (the ether) through which light had to propagate, known as "partial ether drag". He proposed that objects (especially planets) partially drag aether along with them as they move, and this became the accepted explanation for the aberration for some time. George Stokes proposed a similar theory, explaining that the aberration occurs due to the aether current induced by the movement of the Earth. The evidence accumulated against these explanations, combined with a new understanding of the electromagnetic nature of light, led Hendrik Lorentz to develop an electronic theory that featured an immobile ether, positing the contraction of the length of objects as they move through of the ether. Motivated by these earlier theories, Albert Einstein developed the theory of Special Relativity in 1905, which provided the definitive explanation of the phenomenon of aberration.

Bradley's classic explanation

Bradley conceived an explanation in terms of the corpuscular theory of light, in which light is composed of particles unaffected by gravity. His classic explanation refers to the motion of the Earth relative to a beam of light of particles that move at a finite speed, and takes place in the reference system of the Sun, unlike the explanations given previously.

Consider the case in which a distant star is still relative to the Sun (the star is far away, so the parallax can be ignored). In the reference system with the Sun at rest, this means that the light of the star moves along parallel paths to the observer on Earth, and arrives at the same angle, regardless of where the Earth is in its orbit. Suppose the star is seen from Earth with a telescope, idealized as a narrow tube. The light of the star enters the tube with the angle ${displaystyle theta }$ and travel at speed ${displaystyle c}$ during the time ${displaystyle h/c}$ to reach the bottom of the tube, where it is detected. Also suppose that the observations are made from Earth, which moves with speed ${displaystyle v}$. During the traffic of light, the tube moves a distance ${displaystyle vh/c}$. Therefore, for light particles to reach the bottom of the tube, it must be inclined at an angle ${displaystyle phi }$ different from ${displaystyle theta }$, which translates into position apparent of the star at the angle ${displaystyle phi }$. As the Earth advances in its orbit changes direction, so ${displaystyle phi }$ changes with the time of the year when the observation was made. The apparent angle and the true angle are related using trigonometry as:

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

In the case of ${displaystyle theta =90^{circ }}$This gives ${displaystyle tan(theta -phi)=v/c}$. The value is different from the most accurate relativistic result described above, but at the limit of small angles corresponding to low speeds, the difference is within the Bradley measurements appreciation error. These results allowed Bradley to make one of the first estimates of light speed.

Luminíco ether

See also: Bright age

at the beginning of the century XIX The undulating theory of light was being rediscovered, and in 1804 Thomas Young adapted the explanation of Bradley for the corpuscular light, in light in the form of waves that travel through a medium known as the light ether. His reasoning was the same as Bradley's, but required that this medium be motionless in the frame of reference of the sun, and should be crossed by the earth without being affected. Otherwise, the medium (and therefore light) would move with the earth, and no aberration would be observed. He wrote:

After the examination of the phenomena of the aberration of the stars I am willing to believe that the luminum ether permeates the substance of all material bodies with little or no resistance, as freely as the wind passes through a forest of trees.

Thomas Young1804

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

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

which predicts an aberration angle dependent on the properties of the medium crossed. When the refraction of light in the telescope objective is taken into account, this result deviates even further from the result in a vacuum. In 1810 François Arago performed a similar experiment and found that the aberration was not affected by the medium in the telescope tube, providing solid evidence against Young's theory. This experiment was subsequently verified by many other scientists in the following decades, even more precisely by Airy in 1871, with the same result.

Ether drag models

See also: Ether (physics)

Fresnel ether drag

In 1818, Augustin Fresnel developed a modified explanation to take into account the water telescope and other aberration phenomena. He postuled that the ether is generally at rest with the Sun's reference system, but that it is partially dragged by the objects that cross it as they progress. This is, a refractive index object ${displaystyle n}$ that moves at speed ${displaystyle v}$partially drag the ether with a speed ${displaystyle (1-1/n^{2})v}$, taking the light with him. This factor is known as Fresnel's drag coefficient. This drag effect, together with the refraction in the telescope's objective, offsets the lower speed of light in the studied water telescope. With this modification, Fresnel obtained the result for Bradley's vacuum even for telescopes with the purpose of filling water or other substances, and was also able to predict many other phenomena related to the propagation of light in moving bodies. Fresnel's drag coefficient became the dominant explanation of aberration in the following decades.

Stokes ether drag

However, the fact that light can be polarized (discovered by Fresnel himself) led other scientists such as Cauchy and George Green to argue that ether was a fully elastic immobile solid, rather than the fluid ether of Fresnel. Therefore, the need for an explanation of aberration consistent with both Fresnel's predictions (and Arago's observations) and polarization was not renewed.

In 1845, Stokes proposed that ether was a 'putty-like' substance, acting as a liquid on a large scale, but as a solid in small flakes, thus justifying both the transverse vibrations necessary for explain polarized light, as the flow of ether required to understand the phenomenon of aberration. Making only the assumptions that the fluid is irrotational, and that the boundary conditions of the flow are such that the ether has zero velocity away from the Earth, and moves at the velocity of the Earth on its surface and in its interior, was able to completely explain the aberration. The speed of the ether outside the Earth could decrease as a function of the distance from the Earth, so light rays from the stars would be progressively dragged as they approached the Earth's surface. The movement of the Earth would be affected by the ether due to D'Alembert's paradox.

Both Fresnel and Stokes' theories were very popular at the time. However, the question of aberration was neglected for much of the second half of the 19th century, and the focus of the research was directed at the electromagnetic properties of ether.

Lorentz length contraction

In the 1880s, once electromagnetism was better understood, interest again turned to the problem of aberration. In these years the defects of both Fresnel's and Stokes' theories were already known. Fresnel's theory requires that the relative velocity of the ether entrained by matter be different for light of different colors, and the boundary conditions Stokes had assumed in his theory were shown to be incompatible with his assumption of irrotational flow. At the same time, modern theories of the electromagnetic ether could not account for the aberration at all. Many scientists such as Maxwell, Heaviside and Hertz tried unsuccessfully to solve these problems by incorporating Fresnel or Stokes theories into Maxwell's new electromagnetic laws.

Hendrik Lorentz made considerable effort with these assumptions. After working on this problem over a decade, difficulties with Stokes' theory made him abandon this idea and follow the suggestion of a quasi-stationary Fresnel ether (1892, 1895). However, in the Lorentz model the ether was completely immobile, such as the electromagnetic ethers of Cauchy, Green and Maxwell, unlike Fresnel's not altogether immobile ether. He obtained Fresnel's drag coefficient by adapting Maxwell's electromagnetic theory, for which he introduced a change of time for the coordinates of reference systems in motion ("local time"). In order to explain the Michelson-Morley Experiment (1887), which apparently contradicted the two theories of the immobile ether of Lorentz and Fresnel (also confirming in an apparent way Stokes' idea of the complete trawl of the ether), Lorentz theorized in (1892) that the objects are subject to a "Length Contraction" by a factor of the ether. ${displaystyle {sqrt {1-v^{2}/c^{2}}}}}$ in the direction of his movement through the ether. Thus, aberration (and all related optical phenomena) can be explained in the context of an immobile ether. Lorentz's theory became the basis for much research in the following decades. His predictions for aberration are identical to those of relativist theory.

Special relativity

Lorentz's theory explained the results of the experiments, but it was complicated and made many unproven physical assumptions about the microscopic nature of the media of electromagnetic propagation. Albert Einstein, in 1905 with his theory of special relativity, reinterpreted the results of Lorentz's theory in a much simpler and more natural conceptual framework, with which he discarded the idea of the ether. His deduction is now the accepted explanation. Robert S. Shankland remembers some conversations with Einstein, in which the German scientist emphasized the importance of the phenomenon of aberration as a driving force for research with very relevant consequences:

He continued to say that the experimental results that had influenced him most were observations of star aberration and Fizeau measurements of the speed of light in the 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 unsuccessful ether drift experiments, which he already mentioned in the introduction to his first relativity paper. Einstein wrote in a note in 1952:

My own thought was more indirectly influenced by the famous Michelson-Morley experiment. I learned from him through the breaking investigation of Lorentz on the electrodynamics of the moving bodies (1895), which he knew before the establishment of the special theory of relativity. The basic assumption of Lorentz of a resting ether did not seem directly convincing to me, as it gave rise to a [tached: for me artificial appearance] interpretation of the Michelson-Morley experiment, which [tached: did not convince me] seemed unnatural. My route of direct access to the special theory of relativity was determined mainly by the conviction that the electromotric forces induced in a conductor that moves in a magnetic field is nothing but an electric field. But the result of Fizeau's experiment and the aberration phenomenon also led me.

While Einstein's result is the same as that of the original Bradley equation except for an additional factor of ${displaystyle gamma }$It should be emphasized that the result of Bradley is not limited to giving the classic limit of the relativistic case, in the sense that it gives incorrect predictions even at relative low speeds. Bradley's explanation cannot account for situations such as the water telescope, or many other optical effects (such as interference) that can occur within the telescope. This is because within the framework of the Earth it is predicted that the direction of propagation of the beam of light in the telescope is not normal to the wave fronts of the beam, in contradiction to Maxwell's theory of electromagnetism. Likewise, it does not keep the speed of light between reference systems. However, Bradley correctly deduced that the effect was due to the conjunction of relative speeds.