In astronomy, aberration (also referred to as astronomical aberration, stellar aberration, or velocity aberration) is a phenomenon where celestial objects exhibit an apparent motion about their true positions based on the velocity of the observer: It causes objects to appear to be displaced towards the observer's direction of motion. The change in angle is of the order of where is the speed of light and the velocity of the observer. In the case of "stellar" or "annual" aberration, the apparent position of a star to an observer on Earth varies periodically over the course of a year as the Earth's velocity changes as it revolves around the Sun, by a maximum angle of approximately 20 arcseconds in right ascension or declination.
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, 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 and relativistic beaming, 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, 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. However, it was not understood at the time to be a different phenomenon. In the 1720s Italian astronomer Eustachio Manfredi carried out several observations of the phenomenon. He was one of the first to realize that aberration was not the effect of parallax, but he still interpreted it within a geocentric framework. It was Manfredi who coined the term aberration. In 1727, James Bradley 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, initially 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 (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 negative aether drift experiments, as well as the Fizeau experiment, led Albert Einstein to develop the theory of special relativity in 1905, which presents a general form of the equation for aberration in terms of such theory. However the results of the Michelson Morley experiment were misunderstood at the time and since then it has been realised that their experiment did not not rule out an aether medium for a wave only model of light. But rather showed only that the aether, now referred to as the vacuum, did not impede or deflect the direction or velocity of light in the source frame. Ruling out the necessity to invoke a relativistic explanation of aberration.
Explanation
Aberration may be explained as the difference in angle of a beam of light in different inertial frames of reference. 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). The theory of special relativity 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 , with x and y velocity components and , and thus at an angle such that . If the Earth is moving at velocity 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 , and the y velocity is unchanged, . Thus the angle of the light in the Earth's frame in terms of the angle in the Sun's frame is
In the case of , this result reduces to , which in the limit may be approximated by .
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 between different frames of reference. These formulas are
where , 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
or
In the case of , this result reduces to , and in the limit this may be approximated by . This relativistic derivation keeps the speed of light 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, which is due to the motion of an observed object during the time taken by its light to reach an observer, and relativistic beaming, 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 simplicityIn fact, the light source doesn't need to be stationary, consider for example eclipsing binary stars: they are rotating with high speed —and ever changing and different velocity vectors— around each other, but they appear as one spot all the time.) 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 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 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 as the planet revolves around the Sun. Due to orbital eccentricity, the orbital velocity of Earth (in the Sun's rest frame) varies periodically during the year as the planet traverses its elliptic orbit 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 . It may be calculated using the relation substituting the Earth's average speed in the Sun's frame for and the speed of light . Its accepted value is 20.49552 arcseconds (sec) or 0.000099365 radians (rad) (at J2000).
Assuming a circular orbit, annual aberration causes stars exactly on the ecliptic (the plane of Earth's orbit) to appear to move back and forth along a straight line, varying by on either side of their position in the Sun's frame. A star that is precisely at one of the ecliptic poles (at 90° from the ecliptic plane) will appear to move in a circle of radius 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. Such an observer will see the star transit at the zenith, once every day (strictly speaking sidereal day). At the time of the March equinox, Earth's orbit carries the observer in a southwards direction, and the star's apparent declination is therefore displaced to the south by an angle of . On the September equinox, the star's position is displaced to the north by an equal and opposite amount. On either solstice, the displacement in declination is 0. Conversely, the amount of displacement in right ascension is 0 on either equinox and at maximum on either solstice.