Aberration Of Light
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astronomy Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...
, aberration (also referred to as astronomical aberration, stellar aberration, or velocity aberration) is a phenomenon which produces an apparent motion of
celestial object An astronomical object, celestial object, stellar object or heavenly body is a naturally occurring physical entity, association, or structure that exists in the observable universe. In astronomy, the terms ''object'' and ''body'' are often u ...
s about their true positions, dependent on the velocity of the observer. It causes objects to appear to be displaced towards the direction of motion of the observer compared to when the observer is stationary. The change in angle is of the order of ''v/c'' where ''c'' is the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
and ''v'' the
velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...
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 Right ascension (abbreviated RA; symbol ) is the angular distance of a particular point measured eastward along the celestial equator from the Sun at the March equinox to the (hour circle of the) point in question above the earth. When paired w ...
or
declination In astronomy, declination (abbreviated dec; symbol ''δ'') is one of the two angles that locate a point on the celestial sphere in the equatorial coordinate system, the other being hour angle. Declination's angle is measured north or south of the ...
. The term ''aberration'' has historically been used to refer to a number of related phenomena concerning the propagation of light in moving bodies. Aberration is distinct from
parallax Parallax is a displacement or difference in the apparent position of an object viewed along two different lines of sight and is measured by the angle or semi-angle of inclination between those two lines. Due to foreshortening, nearby objects ...
, which is a change in the apparent position of a relatively nearby object, as measured by a moving observer, relative to more distant objects that define a reference frame. The amount of parallax depends on the distance of the object from the observer, whereas aberration does not. Aberration is also related to
light-time correction Light-time correction is a displacement in the ''apparent'' position of a celestial object from its ''true'' position (or geometric position) caused by the object's motion during the time it takes its light to reach an observer. Light-time corr ...
and
relativistic beaming Relativistic beaming (also known as Doppler beaming, Doppler boosting, or the headlight effect) is the process by which relativistic effects modify the apparent luminosity of emitting matter that is moving at speeds close to the speed of li ...
, although it is often considered separately from these effects. Aberration is historically significant because of its role in the development of the theories of
light Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 tera ...
,
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of ...
and, ultimately, the theory of
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The laws o ...
. It was first observed in the late 1600s by astronomers searching for stellar parallax in order to confirm the
heliocentric model Heliocentrism (also known as the Heliocentric model) is the astronomical model in which the Earth and planets revolve around the Sun at the center of the universe. Historically, heliocentrism was opposed to geocentrism, which placed the Earth a ...
of the Solar System. However, it was not understood at the time to be a different phenomenon. In 1727,
James Bradley James Bradley (1692–1762) was an English astronomer and priest who served as the third Astronomer Royal from 1742. He is best known for two fundamental discoveries in astronomy, the aberration of light (1725–1728), and the nutation of th ...
provided a classical explanation for it in terms of the finite speed of light relative to the motion of the Earth in its orbit around the Sun, which he used to make one of the earliest measurements of the speed of light. However, Bradley's theory was incompatible with 19th century theories of light, and aberration became a major motivation for the aether drag theories of Augustin Fresnel (in 1818) and
G. G. Stokes Sir George Gabriel Stokes, 1st Baronet, (; 13 August 1819 – 1 February 1903) was an Irish English physicist and mathematician. Born in County Sligo, Ireland, Stokes spent all of his career at the University of Cambridge, where he was the Luc ...
(in 1845), and for Hendrik Lorentz's aether theory of electromagnetism in 1892. The aberration of light, together with Lorentz's elaboration of Maxwell's electrodynamics, the
moving magnet and conductor problem The moving magnet and conductor problem is a famous thought experiment, originating in the 19th century, concerning the intersection of classical electromagnetism and special relativity. In it, the current in a conductor moving with constant vel ...
, the negative aether drift experiments, as well as the
Fizeau experiment The Fizeau experiment was carried out by Hippolyte Fizeau in 1851 to measure the relative speeds of light in moving water. Fizeau used a special interferometer arrangement to measure the effect of movement of a medium upon the speed of light. ...
, led
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
to develop the theory of special relativity in 1905, which presents a general form of the equation for aberration in terms of such theory.


Explanation

Aberration may be explained as the difference in angle of a beam of light in different
inertial frames of reference In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleration ...
. A common analogy is to consider the apparent direction of falling rain. If rain is falling vertically in the frame of reference of a person standing still, then to a person moving forwards the rain will appear to arrive at an angle, requiring the moving observer to tilt their umbrella forwards. The faster the observer moves, the more tilt is needed. The net effect is that light rays striking the moving observer from the sides in a stationary frame will come angled from ahead in the moving observer's frame. This effect is sometimes called the "searchlight" or "headlight" effect. In the case of annual aberration of starlight, the direction of incoming starlight as seen in the Earth's moving frame is tilted relative to the angle observed in the Sun's frame. Since the direction of motion of the Earth changes during its orbit, the direction of this tilting changes during the course of the year, and causes the apparent position of the star to differ from its true position as measured in the inertial frame of the Sun. While classical reasoning gives intuition for aberration, it leads to a number of physical paradoxes observable even at the classical level (see
history History (derived ) is the systematic study and the documentation of the human activity. The time period of event before the History of writing#Inventions of writing, invention of writing systems is considered prehistory. "History" is an umbr ...
). The theory of
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The laws o ...
is required to correctly account for aberration. The relativistic explanation is very similar to the classical one however, and in both theories aberration may be understood as a case of addition of velocities.


Classical explanation

In the Sun's frame, consider a beam of light with velocity equal to the speed of light c, with x and y velocity components u_x and u_y, and thus at an angle θ such that \tan(\theta) = u_y/u_x. If the Earth is moving at velocity v in the x direction relative to the Sun, then by velocity addition the x component of the beam's velocity in the Earth's frame of reference is u_x' = u_x + v, and the y velocity is unchanged, u_y' = u_y. Thus the angle of the light in the Earth's frame in terms of the angle in the Sun's frame is :\tan(\phi) = \frac = \frac = \frac In the case of \theta = 90^\circ, this result reduces to \tan(\theta - \phi) = v/c, which in the limit v/c \ll 1 may be approximated by \theta - \phi = v/c.


Relativistic explanation

The reasoning in the relativistic case is the same except that the relativistic velocity addition formulas must be used, which can be derived from
Lorentz transformations In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation i ...
between different frames of reference. These formulas are :u_x' = (u_x + v)/(1+u_x v/c^2) :u_y' = u_y / \gamma (1+u_x v/c^2) where \gamma = 1/\sqrt, giving the components of the light beam in the Earth's frame in terms of the components in the Sun's frame. The angle of the beam in the Earth's frame is thus :\tan(\phi) = \frac = \frac = \frac In the case of \theta = 90^\circ, this result reduces to \sin(\theta - \phi) = v/c, and in the limit v/c \ll 1 this may be approximated by \theta - \phi = v/c. This relativistic derivation keeps the speed of light \sqrt = c constant in all frames of reference, unlike the classical derivation above.


Relationship to light-time correction and relativistic beaming

Aberration is related to two other phenomena,
light-time correction Light-time correction is a displacement in the ''apparent'' position of a celestial object from its ''true'' position (or geometric position) caused by the object's motion during the time it takes its light to reach an observer. Light-time corr ...
, which is due to the motion of an observed object during the time taken by its light to reach an observer, and
relativistic beaming Relativistic beaming (also known as Doppler beaming, Doppler boosting, or the headlight effect) is the process by which relativistic effects modify the apparent luminosity of emitting matter that is moving at speeds close to the speed of li ...
, which is an angling of the light emitted by a moving light source. It can be considered equivalent to them but in a different inertial frame of reference. In aberration, the observer is considered to be moving relative to a (for the sake of simplicity) stationary light source, while in light-time correction and relativistic beaming the light source is considered to be moving relative to a stationary observer. Consider the case of an observer and a light source moving relative to each other at constant velocity, with a light beam moving from the source to the observer. At the moment of emission, the beam in the observer's rest frame is tilted compared to the one in the source's rest frame, as understood through relativistic beaming. During the time it takes the light beam to reach the observer the light source moves in the observer's frame, and the 'true position' of the light source is displaced relative to the apparent position the observer sees, as explained by light-time correction. Finally, the beam in the observer's frame at the moment of observation is tilted compared to the beam in source's frame, which can be understood as an aberrational effect. Thus, a person in the light source's frame would describe the apparent tilting of the beam in terms of aberration, while a person in the observer's frame would describe it as a light-time effect. The relationship between these phenomena is only valid if the observer and source's frames are inertial frames. In practice, because the Earth is not an inertial rest frame but experiences centripetal
acceleration In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by the ...
towards the Sun, many aberrational effects such as annual aberration on Earth cannot be considered light-time corrections. However, if the time between emission and detection of the light is short compared to the orbital period of the Earth, the Earth may be approximated as an inertial frame and aberrational effects are equivalent to light-time corrections.


Types

The ''
Astronomical Almanac ''The Astronomical Almanac''The ''Astronomical Almanac'' for the Year 2015, (United States Naval Observatory/Nautical Almanac Office, 2014) . is an almanac published by the United States Naval Observatory (USNO) and His Majesty's Nautical Almanac ...
'' describes several different types of aberration, arising from differing components of the Earth's and observed object's motion: * Stellar aberration: "The apparent angular displacement of the observed position of a celestial body resulting from the motion of the observer. Stellar aberration is divided into diurnal, annual, and secular components." ** Annual aberration: "The component of stellar aberration resulting from the motion of the Earth about the Sun." ** Diurnal aberration: "The component of stellar aberration resulting from the observer's diurnal motion about the center of the Earth due to the Earth's rotation." ** Secular aberration: "The component of stellar aberration resulting from the essentially uniform and almost rectilinear motion of the entire solar system in space. Secular aberration is usually disregarded." * Planetary aberration: "The apparent angular displacement of the observed position of a solar system body from its instantaneous geocentric direction as would be seen by an observer at the geocenter. This displacement is caused by the aberration of light and light-time displacement."


Annual aberration

Annual aberration is caused by the motion of an observer on
Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surfa ...
as the planet revolves around the
Sun The Sun is the star at the center of the Solar System. It is a nearly perfect ball of hot plasma, heated to incandescence by nuclear fusion reactions in its core. The Sun radiates this energy mainly as light, ultraviolet, and infrared radi ...
. Due to orbital eccentricity, the orbital velocity v of Earth (in the Sun's rest frame) varies periodically during the year as the planet traverses its
elliptic orbit In astrodynamics or celestial mechanics, an elliptic orbit or elliptical orbit is a Kepler orbit with an eccentricity of less than 1; this includes the special case of a circular orbit, with eccentricity equal to 0. In a stricter sense, i ...
and consequently the aberration also varies periodically, typically causing stars to appear to move in small ellipses. Approximating Earth's orbit as circular, the maximum displacement of a star due to annual aberration is known as the ''constant of aberration'', conventionally represented by \kappa. It may be calculated using the relation \kappa = \theta-\phi \approx v/c substituting the Earth's average speed in the Sun's frame for v and the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
c. Its accepted value is 20.49552  arcseconds (sec) or 0.000099365 
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before tha ...
s (rad) (at
J2000 In astronomy, an epoch or reference epoch is a instant, moment in time used as a reference point for some time-varying astronomical quantity. It is useful for the celestial coordinates or orbital elements of a Astronomical object, celestial body, ...
). Assuming a circular orbit, annual aberration causes stars exactly on the
ecliptic The ecliptic or ecliptic plane is the orbital plane of the Earth around the Sun. From the perspective of an observer on Earth, the Sun's movement around the celestial sphere over the course of a year traces out a path along the ecliptic again ...
(the plane of Earth's orbit) to appear to move back and forth along a straight line, varying by \kappa on either side of their position in the Sun's frame. A star that is precisely at one of the
ecliptic pole An orbital pole is either point at the ends of an imaginary line segment that runs through the center of an orbit (of a revolving body like a planet, moon or satellite) and is perpendicular to the orbital plane. Projected onto the celestial sphe ...
s (at 90° from the ecliptic plane) will appear to move in a circle of radius \kappa about its true position, and stars at intermediate ecliptic latitudes will appear to move along a small ellipse. For illustration, consider a star at the northern ecliptic pole viewed by an observer at a point on the
Arctic Circle The Arctic Circle is one of the two polar circles, and the most northerly of the five major circles of latitude as shown on maps of Earth. Its southern equivalent is the Antarctic Circle. The Arctic Circle marks the southernmost latitude at w ...
. Such an observer will see the star
transit Transit may refer to: Arts and entertainment Film * ''Transit'' (1979 film), a 1979 Israeli film * ''Transit'' (2005 film), a film produced by MTV and Staying-Alive about four people in countries in the world * ''Transit'' (2006 film), a 2006 ...
at the
zenith The zenith (, ) is an imaginary point directly "above" a particular location, on the celestial sphere. "Above" means in the vertical direction (plumb line) opposite to the gravity direction at that location (nadir). The zenith is the "highest" ...
, once every day (strictly speaking
sidereal day Sidereal time (as a unit also sidereal day or sidereal rotation period) (sidereal ) is a timekeeping system that astronomers use to locate celestial objects. Using sidereal time, it is possible to easily point a telescope to the proper coor ...
). At the time of the
March equinox The March equinox or northward equinox is the equinox on the Earth when the subsolar point appears to leave the Southern Hemisphere and cross the celestial equator, heading northward as seen from Earth. The March equinox is known as the ve ...
, Earth's orbit carries the observer in a southwards direction, and the star's apparent
declination In astronomy, declination (abbreviated dec; symbol ''δ'') is one of the two angles that locate a point on the celestial sphere in the equatorial coordinate system, the other being hour angle. Declination's angle is measured north or south of the ...
is therefore displaced to the south by an angle of \kappa. On the
September equinox The September equinox (or southward equinox) is the moment when the Sun appears to cross the celestial equator, heading southward. Because of differences between the calendar year and the tropical year, the September equinox may occur anyt ...
, the star's position is displaced to the north by an equal and opposite amount. On either
solstice A solstice is an event that occurs when the Sun appears to reach its most northerly or southerly excursion relative to the celestial equator on the celestial sphere. Two solstices occur annually, around June 21 and December 21. In many countr ...
, the displacement in declination is 0. Conversely, the amount of displacement in
right ascension Right ascension (abbreviated RA; symbol ) is the angular distance of a particular point measured eastward along the celestial equator from the Sun at the March equinox to the (hour circle of the) point in question above the earth. When paired w ...
is 0 on either
equinox A solar equinox is a moment in time when the Sun crosses the Earth's equator, which is to say, appears directly above the equator, rather than north or south of the equator. On the day of the equinox, the Sun appears to rise "due east" and se ...
and at maximum on either solstice. In actuality, Earth's orbit is slightly elliptic rather than circular, and its speed varies somewhat over the course of its orbit, which means the description above is only approximate. Aberration is more accurately calculated using Earth's instantaneous velocity relative to the
barycenter In astronomy, the barycenter (or barycentre; ) is the center of mass of two or more bodies that orbit one another and is the point about which the bodies orbit. A barycenter is a dynamical point, not a physical object. It is an important con ...
of the Solar System. Note that the displacement due to aberration is orthogonal to any displacement due to
parallax Parallax is a displacement or difference in the apparent position of an object viewed along two different lines of sight and is measured by the angle or semi-angle of inclination between those two lines. Due to foreshortening, nearby objects ...
. If parallax is detectable, the maximum displacement to the south would occur in December, and the maximum displacement to the north in June. It is this apparently anomalous motion that so mystified early astronomers.


Solar annual aberration

A special case of annual aberration is the nearly constant deflection of the Sun from its position in the Sun's rest frame by \kappa towards the ''west'' (as viewed from Earth), opposite to the apparent motion of the Sun along the ecliptic (which is from west to east, as seen from Earth). The deflection thus makes the Sun appear to be behind (or retarded) from its rest-frame position on the ecliptic by a position or angle \kappa. This deflection may equivalently be described as a light-time effect due to motion of the Earth during the 8.3 minutes that it takes light to travel from the Sun to Earth. The relation with \kappa is : .000099365 rad / 2 π radx 65.25 d x 24 h/d x 60 min/h= 8.3167 min ≈ 8 min 19 sec = 499 sec. This is possible since the transit time of sunlight is short relative to the orbital period of the Earth, so the Earth's frame may be approximated as inertial. In the Earth's frame, the Sun moves, at a mean velocity v = 29.789 km/s, by a distance \Delta x = vt ≈ 14,864.7 km in the time it takes light to reach Earth, t=R/c ≈ 499 sec for the orbit of mean radius R = 1 AU = 149,597,870.7 km. This gives an angular correction \tan(\theta) \approx \theta = \Delta x/R ≈ 0.000099364 rad = 20.49539 sec, which can be solved to give \theta = v/c = \kappa ≈ 0.000099365 rad = 20.49559 sec, very nearly the same as the aberrational correction (here \kappa is in radian and not in arcsecond).


Diurnal aberration

Diurnal aberration is caused by the velocity of the observer on the surface of the rotating Earth. It is therefore dependent not only on the time of the observation, but also the
latitude In geography, latitude is a coordinate that specifies the north– south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north pol ...
and
longitude Longitude (, ) is a geographic coordinate that specifies the east–west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek letter l ...
of the observer. Its effect is much smaller than that of annual aberration, and is only 0.32 arcseconds in the case of an observer at the
Equator The equator is a circle of latitude, about in circumference, that divides Earth into the Northern and Southern hemispheres. It is an imaginary line located at 0 degrees latitude, halfway between the North and South poles. The term can als ...
, where the rotational velocity is greatest.


Secular aberration

The secular component of aberration, caused by the motion of the Solar System in space, has been further subdivided into several components: aberration resulting from the motion of the solar system barycenter around the center of our Galaxy, aberration resulting from the motion of the Galaxy relative to the
Local Group The Local Group is the galaxy group that includes the Milky Way. It has a total diameter of roughly , and a total mass of the order of . It consists of two collections of galaxies in a "dumbbell" shape: the Milky Way and its satellites form ...
, and aberration resulting from the motion of the Local Group relative to the
cosmic microwave background In Big Bang cosmology the cosmic microwave background (CMB, CMBR) is electromagnetic radiation that is a remnant from an early stage of the universe, also known as "relic radiation". The CMB is faint cosmic background radiation filling all spac ...
. Secular aberration affects the apparent positions of stars and extragalactic objects. The large, constant part of secular aberration cannot be directly observed and "It has been standard practice to absorb this large, nearly constant effect into the reported" positions of stars. In about 200 million years, the Sun circles the galactic center, whose measured location is near right ascension (α = 266.4°) and declination (δ = −29.0°). The constant, unobservable, effect of the solar system's motion around the galactic center has been computed variously as 150 or 165 arcseconds. The other, observable, part is an acceleration toward the galactic center of approximately 2.5 × 10−10 m/s2, which yields a change of aberration of about 5 µas/yr. Highly precise measurements extending over several years can observe this change in secular aberration, often called the secular aberration drift or the acceleration of the Solar System, as a small apparent
proper motion Proper motion is the astrometric measure of the observed changes in the apparent places of stars or other celestial objects in the sky, as seen from the center of mass of the Solar System, compared to the abstract background of the more dista ...
. Recently, highly precise
astrometry Astrometry is a branch of astronomy that involves precise measurements of the positions and movements of stars and other celestial bodies. It provides the kinematics and physical origin of the Solar System and this galaxy, the Milky Way. His ...
of extragalactic objects using both
Very Long Baseline Interferometry Very-long-baseline interferometry (VLBI) is a type of astronomical interferometry used in radio astronomy. In VLBI a signal from an astronomical radio source, such as a quasar, is collected at multiple radio telescopes on Earth or in space. T ...
and the ''Gaia'' space observatory have successfully measured this small effect. The first VLBI measurement of the apparent motion, over a period of 20 years, of 555 extragalactic objects towards the center of our galaxy at equatorial coordinates of α = 263° and δ = −20° indicated a secular aberration drift 6.4 ±1.5 μas/yr. Later determinations using a series of VLBI measurements extending over almost 40 years determined the secular aberration drift to be 5.83 ± 0.23 μas/yr in the direction α = 270.2 ± 2.3° and δ = −20.2° ± 3.6°. Optical observations using only 33 months of ''Gaia'' satellite data of 1.6 million extragalactic sources indicated an acceleration of the solar system of 2.32 ± 0.16 × 10−10 m/s2 and a corresponding secular aberration drift of 5.05 ± 0.35 µas/yr in the direction of α = 269.1° ± 5.4°, δ = −31.6° ± 4.1°. It is expected that later ''Gaia'' data releases, incorporating about 66 and 120 months of data, will reduce the random errors of these results by factors of 0.35 and 0.15. The latest edition of the
International Celestial Reference Frame The International Celestial Reference System (ICRS) is the current standard celestial reference system adopted by the International Astronomical Union (IAU). Its origin is at the barycenter of the Solar System, with axes that are intended to "sho ...
(ICRF3) adopted a recommended galactocentric aberration constant of 5.8 µas/yr and recommended a correction for secular aberration to obtain the highest positional accuracy for times other than the
reference epoch In chronology and periodization, an epoch or reference epoch is an instant in time chosen as the origin of a particular calendar era. The "epoch" serves as a reference point from which time is measured. The moment of epoch is usually decided by ...
2015.0.


Planetary aberration

Planetary aberration is the combination of the aberration of light (due to Earth's velocity) and light-time correction (due to the object's motion and distance), as calculated in the rest frame of the Solar System. Both are determined at the instant when the moving object's light reaches the moving observer on Earth. It is so called because it is usually applied to planets and other objects in the Solar System whose motion and distance are accurately known.


Discovery and first observations

The discovery of the aberration of light was totally unexpected, and it was only by considerable perseverance and perspicacity that
Bradley Bradley is an English surname derived from a place name meaning "broad wood" or "broad meadow" in Old English. Like many English surnames Bradley can also be used as a given name and as such has become popular. It is also an Anglicisation of t ...
was able to explain it in 1727. It originated from attempts to discover whether stars possessed appreciable parallaxes.


Search for stellar parallax

The Copernican heliocentric theory of the
Solar System The Solar SystemCapitalization of the name varies. The International Astronomical Union, the authoritative body regarding astronomical nomenclature, specifies capitalizing the names of all individual astronomical objects but uses mixed "Solar S ...
had received confirmation by the observations of Galileo and
Tycho Brahe Tycho Brahe ( ; born Tyge Ottesen Brahe; generally called Tycho (14 December 154624 October 1601) was a Danish astronomer, known for his comprehensive astronomical observations, generally considered to be the most accurate of his time. He was k ...
and the mathematical investigations of
Kepler Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws o ...
and Newton. As early as 1573,
Thomas Digges Thomas Digges (; c. 1546 – 24 August 1595) was an English mathematician and astronomer. He was the first to expound the Copernican system in English but discarded the notion of a fixed shell of immoveable stars to postulate infinitely many s ...
had suggested that parallactic shifting of the stars should occur according to the heliocentric model, and consequently if stellar parallax could be observed it would help confirm this theory. Many observers claimed to have determined such parallaxes, but Tycho Brahe and
Giovanni Battista Riccioli Giovanni Battista Riccioli, SJ (17 April 1598 – 25 June 1671) was an Italian astronomer and a Catholic priest in the Jesuit order. He is known, among other things, for his experiments with pendulums and with falling bodies, for his discussion ...
concluded that they existed only in the minds of the observers, and were due to instrumental and personal errors. However, in 1680 Jean Picard, in his ''Voyage d’ Uranibourg,'' stated, as a result of ten
year A year or annus is the orbital period of a planetary body, for example, the Earth, moving in its orbit around the Sun. Due to the Earth's axial tilt, the course of a year sees the passing of the seasons, marked by change in weather, the hou ...
s' observations, that Polaris, the Pole Star, exhibited variations in its position amounting to 40″ annually. Some astronomers endeavoured to explain this by parallax, but these attempts failed because the motion differed from that which parallax would produce.
John Flamsteed John Flamsteed (19 August 1646 – 31 December 1719) was an English astronomer and the first Astronomer Royal. His main achievements were the preparation of a 3,000-star catalogue, ''Catalogus Britannicus'', and a star atlas called '' Atlas C ...
, from measurements made in 1689 and succeeding years with his mural quadrant, similarly concluded that the declination of Polaris was 40″ less in July than in September.
Robert Hooke Robert Hooke FRS (; 18 July 16353 March 1703) was an English polymath active as a scientist, natural philosopher and architect, who is credited to be one of two scientists to discover microorganisms in 1665 using a compound microscope that ...
, in 1674, published his observations of γ Draconis, a star of
magnitude Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (mathematics), the relative size of an object *Norm (mathematics), a term for the size or length of a vector *Order of ...
2m which passes practically overhead at the latitude of London (hence its observations are largely free from the complex corrections due to
atmospheric refraction Atmospheric refraction is the deviation of light or other electromagnetic wave from a straight line as it passes through the atmosphere due to the variation in air density as a function of height. This refraction is due to the velocity of ligh ...
), and concluded that this star was 23″ more northerly in July than in October.


James Bradley's observations

Consequently, when Bradley and
Samuel Molyneux Samuel Molyneux FRS (16 July 1689 – 13 April 1728) was an amateur astronomer and politician who sat in the British House of Commons between 1715 and 1728 and in the Irish House of Commons from 1727 to 1728. His work with James Bradley attempt ...
entered this sphere of research in 1725, there was still considerable uncertainty as to whether stellar parallaxes had been observed or not, and it was with the intention of definitely answering this question that they erected a large telescope at Molyneux's house at
Kew Kew () is a district in the London Borough of Richmond upon Thames. Its population at the 2011 census was 11,436. Kew is the location of the Royal Botanic Gardens ("Kew Gardens"), now a World Heritage Site, which includes Kew Palace. Kew is a ...
. They decided to reinvestigate the motion of γ Draconis with a telescope constructed by
George Graham George Graham (born 30 November 1944), nicknamed "Stroller", is a Scottish former Association football, football player and manager (association football), manager. In his successful playing career, he made 455 appearances in England's Football ...
(1675–1751), a celebrated instrument-maker. This was fixed 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, and then reversed its course from March to September. At the same time, 35 Camelopardalis, a star with a right ascension nearly exactly opposite to that of γ Draconis, was 19" more northerly at the beginning of March than in September. These results were completely unexpected and inexplicable by existing theories.


Early hypotheses

Bradley and Molyneux discussed several hypotheses in the hope of finding the solution. Since the apparent motion was evidently caused neither by parallax nor observational errors, Bradley first hypothesized that it could be due to oscillations in the orientation of the Earth's axis relative to the celestial sphere – a phenomenon known as
nutation Nutation () is a rocking, swaying, or nodding motion in the axis of rotation of a largely axially symmetric object, such as a gyroscope, planet, or bullet in flight, or as an intended behaviour of a mechanism. In an appropriate reference frame ...
. 35 Camelopardalis was seen to possess an apparent motion which could be consistent with nutation, but since its declination varied only one half as much as that of γ Draconis, it was obvious that nutation did not supply the answer (however, Bradley later went on to discover that the Earth does indeed nutate). He also investigated the possibility that the motion was due to an irregular distribution of the
Earth's atmosphere The atmosphere of Earth is the layer of gases, known collectively as air, retained by Earth's gravity that surrounds the planet and forms its planetary atmosphere. The atmosphere of Earth protects life on Earth by creating pressure allowing for ...
, thus involving abnormal variations in the refractive index, but again obtained negative results. On August 19, 1727, Bradley 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 twenty years. During his first two years at Wanstead, he established the existence of the phenomenon of aberration beyond all doubt, and this also enabled him 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 his explanation of aberration in about September 1728 and this theory was presented to the
Royal Society The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a learned society and the United Kingdom's national academy of sciences. The society fulfils a number of roles: promoting science and its benefits, re ...
in mid January the following year. 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. However, there is no record of this incident in Bradley's own account of the discovery, and it may therefore be
apocrypha Apocrypha are works, usually written, of unknown authorship or of doubtful origin. The word ''apocryphal'' (ἀπόκρυφος) was first applied to writings which were kept secret because they were the vehicles of esoteric knowledge considered ...
l. The following table shows the magnitude of deviation from true declination for γ Draconis and the direction, on the planes of the solstitial
colure Colure, in astronomy, is either of the two principal meridians of the celestial sphere. Equinoctial colure The ''equinoctial colure'' is the meridian or great circle of the celestial sphere which passes through the celestial poles and the two ...
and ecliptic prime meridian, of the tangent of the velocity of the Earth in its orbit for each of the four months where the extremes are found, as well as expected deviation from true ecliptic longitude if Bradley had measured its deviation from right ascension: Bradley proposed that the aberration of light not only affected declination, but right ascension as well, so that a star in the pole of the ecliptic would describe a little ellipse with a diameter of about 40", but for simplicity, he assumed it to be a circle. Since he only observed the deviation in declination, and not in right ascension, his calculations for the maximum deviation of a star in the pole of the ecliptic are for its declination only, which will coincide with the diameter of the little circle described by such star. For eight different stars, his calculations are as follows: Based on these calculations, Bradley was able to estimate the constant of aberration at 20.2", which is equal to 0.00009793 radians, and with this was able to estimate the speed of light at per second. By projecting the little circle for a star in the pole of the ecliptic, he could simplify the calculation of the relationship between the speed of light and the speed of the Earth's annual motion in its orbit as follows: :\cos\left(\frac\pi-0.00009793\right) = \sin(0.00009793) = \frac Thus, the speed of light to the speed of the Earth's annual motion in its orbit is 10,210 to one, from whence it would follow, that light moves, or is propagated as far as from the Sun to the Earth in 8 minutes 12 seconds. The original motivation of the search for stellar parallax was to test the Copernican theory that the Earth revolves around the Sun. The change of annual aberration in the course of the year demonstrates both the annual motion of the Earth around the Sun and the relative motion of the Earth and the stars.


Retrodiction on Descartes' lightspeed argument

In the prior century,
René Descartes René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Mathem ...
argued that if light were not instantaneous, then shadows of moving objects would lag; and if propagation times over terrestrial distances were appreciable, then during a lunar eclipse the Sun, Earth, and Moon would be out of alignment by hours' motion, contrary to observation. Huygens commented that, on Rømer’s lightspeed data (yielding an earth-moon round-trip time of only seconds), the lag angle would be imperceptible. What they both overlooked is that aberration (as understood only later) would exactly counteract the lag even if large, leaving this eclipse method completely insensitive to light speed. (Otherwise, shadow-lag methods could be made to sense absolute translational motion, contrary to a basic principle of relativity.)


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 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 The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
and the motion of
Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surfa ...
in its orbit around the
Sun The Sun is the star at the center of the Solar System. It is a nearly perfect ball of hot plasma, heated to incandescence by nuclear fusion reactions in its core. The Sun radiates this energy mainly as light, ultraviolet, and infrared radi ...
. 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 Luminiferous aether or ether ("luminiferous", meaning "light-bearing") was the postulated medium for the propagation of light. It was invoked to explain the ability of the apparently wave-based light to propagate through empty space (a vacuum), so ...
.
Augustin-Jean Fresnel Augustin-Jean Fresnel (10 May 1788 – 14 July 1827) was a French civil engineer and physicist whose research in optics led to the almost unanimous acceptance of the wave theory of light, excluding any remnant of Newton's corpuscular th ...
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 Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
then developed the theory of
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The laws o ...
in 1905, which provides the modern account of aberration.


Bradley's classical explanation

Bradley conceived of an explanation in terms of a
corpuscular theory of light In optics, the corpuscular theory of light states that light is made up of small discrete particles called " corpuscles" (little particles) which travel in a straight line with a finite velocity and possess impetus. This was based on an alternate ...
in which light is made of particles. 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 \theta and travels at speed c taking a time 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 v. During the transit of the light, the tube moves a distance vh/c. Consequently, for the particles of light to reach the bottom of the tube, the tube must be inclined at an angle \phi different from \theta, resulting in an ''apparent'' position of the star at angle \phi. As the Earth proceeds in its orbit it changes direction, so \phi changes with the time of year the observation is made. The apparent angle and true angle are related using trigonometry as: :\tan(\phi) = \frac =\frac . In the case of \theta = 90^\circ, this gives \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 The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
.Encyclopædia Britannica


Luminiferous aether

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.
He wrote: 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 c/n rather than c where n is the index of refraction of the water. Thus, by Bradley and Young's reasoning the aberration angle is given by :\tan(\phi) = \frac . which predicts a medium-dependent angle of aberration. When refraction at the telescope's
objective Objective may refer to: * Objective (optics), an element in a camera or microscope * ''The Objective'', a 2008 science fiction horror film * Objective pronoun, a personal pronoun that is used as a grammatical object * Objective Productions, a Brit ...
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.


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 n moving at velocity v is partially dragged with a velocity (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. 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.


Stokes' aether drag

However, the fact that light is polarized (discovered by Fresnel himself) led scientists such as
Cauchy Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He w ...
and
Green Green is the color between cyan and yellow on the visible spectrum. It is evoked by light which has a dominant wavelength of roughly 495570 nm. In subtractive color systems, used in painting and color printing, it is created by a combi ...
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 In vector calculus, a conservative vector field is a vector field that is the gradient of some function. A conservative vector field has the property that its line integral is path independent; the choice of any path between two points does not c ...
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. 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 In fluid dynamics, d'Alembert's paradox (or the hydrodynamic paradox) is a contradiction reached in 1752 by French mathematician Jean le Rond d'Alembert. D'Alembert proved that – for incompressible and inviscid potential flow – the drag for ...
. 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

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. At the same time, the modern theories of electromagnetic aether could not account for aberration at all. Many scientists such as Maxwell,
Heaviside Oliver Heaviside FRS (; 18 May 1850 – 3 February 1925) was an English self-taught mathematician and physicist who invented a new technique for solving differential equations (equivalent to the Laplace transform), independently developed vec ...
and
Hertz The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), equivalent to one event (or cycle) per second. The hertz is an SI derived unit whose expression in terms of SI base units is s−1, meaning that on ...
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–Morley experiment The Michelson–Morley experiment was an attempt to detect the existence of the luminiferous aether, a supposed medium permeating space that was thought to be the carrier of light waves. The experiment was performed between April and July 1887 ...
(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 Length contraction is the phenomenon that a moving object's length is measured to be shorter than its proper length, which is the length as measured in the object's own rest frame. It is also known as Lorentz contraction or Lorentz–FitzGerald ...
" by a factor of \sqrt 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.


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: Other important motivations for Einstein's development of relativity were the
moving magnet and conductor problem The moving magnet and conductor problem is a famous thought experiment, originating in the 19th century, concerning the intersection of classical electromagnetism and special relativity. In it, the current in a conductor moving with constant vel ...
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: While Einstein's result is the same as Bradley's original equation except for an extra factor of \gamma, 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 In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
. 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.


See also

* Apparent place * Stellar parallax *
Astronomical nutation Astronomical nutation is a phenomenon which causes the orientation of the axis of rotation of a spinning astronomical object to vary over time. It is caused by the gravitational forces of other nearby bodies acting upon the spinning object. Al ...
*
Proper motion Proper motion is the astrometric measure of the observed changes in the apparent places of stars or other celestial objects in the sky, as seen from the center of mass of the Solar System, compared to the abstract background of the more dista ...
*
Timeline of electromagnetism and classical optics Timeline of electromagnetism and classical optics lists, within the history of electromagnetism, the associated theories, technology, and events. Early developments * 28th century BC – Ancient Egyptian texts describe electric fish. They ref ...
*
Relativistic aberration Relativistic aberration is the relativistic version of aberration of light, including relativistic corrections that become significant for observers who move with velocities close to the speed of light. It is described by Einstein's special theory ...


Notes


References


Further reading

* * P. Kenneth Seidelmann (Ed.), ''Explanatory Supplement to the Astronomical Almanac'' (University Science Books, 1992), 127–135, 700. *
Stephen Peter Rigaud Stephen Peter Rigaud (12 August 1774–16 March 1839) FRAS was an English mathematical historian and astronomer. Rigaud was born into a French Protestant family. His father, Stephen (also known as James Stephen) Rigaud, was Observer at ...
, ''Miscellaneous Works and Correspondence of the Rev. James Bradley, D.D. F.R.S.'' (1832). *
Charles Hutton Charles Hutton FRS FRSE LLD (14 August 1737 – 27 January 1823) was a British mathematician and surveyor. He was professor of mathematics at the Royal Military Academy, Woolwich from 1773 to 1807. He is remembered for his calculation of the ...
, ''Mathematical and Philosophical Dictionary'' (1795). * H. H. Turner, ''Astronomical Discovery'' (1904). *
Thomas Simpson Thomas Simpson FRS (20 August 1710 – 14 May 1761) was a British mathematician and inventor known for the eponymous Simpson's rule to approximate definite integrals. The attribution, as often in mathematics, can be debated: this rule had been ...
, ''Essays on Several Curious and Useful Subjects in Speculative and Mix'd Mathematicks'' (1740). * :de:August Ludwig Busch, ''Reduction of the Observations Made by Bradley at Kew and Wansted to Determine the Quantities of Aberration and Nutation'' (1838).


External links


Courtney Seligman
on Bradley's observations {{DEFAULTSORT:Aberration Of Light Electromagnetic radiation Astrometry Radiation