At the instant of any observation of an object, the apparent position of the object is displaced from its true position by an amount which depends upon the transverse component of the velocity of the observer, with respect to the vector of the incoming beam of light (i.e., the line actually taken by the light on its path to the observer). In the case of an observer on Earth, the direction of its velocity varies during the year as Earth revolves around the Sun (or strictly speaking, the barycenter of the solar system), and this in turn causes the apparent position of the object to vary. This particular effect is known as annual aberration or stellar aberration, because it causes the apparent position of a star to vary periodically over the course of a year. The maximum amount of the aberrational displacement of a star is approximately 20 arcseconds in right ascension or declination. Although this is a relatively small value, it was well within the observational capability of the instruments available in the early eighteenth century.
Aberration should not be confused with stellar parallax, although it was an initially fruitless search for parallax that first led to its discovery. Parallax is caused by a change in the position of the observer looking at a relatively nearby object, as measured against more distant objects, and is therefore dependent upon the distance between the observer and the object.
In contrast, stellar aberration is independent of the distance of a celestial object from the observer, and depends only on the observer's instantaneous transverse velocity with respect to the incoming light beam, at the moment of observation. The light beam from a distant object cannot itself have any transverse velocity component, or it could not (by definition) be seen by the observer, since it would miss the observer. Thus, any transverse velocity of the emitting source plays no part in aberration. Another way to state this is that the emitting object may have a transverse velocity with respect to the observer, but any light beam emitted from it which reaches the observer, cannot, for it must have been previously emitted in such a direction that its transverse component has been "corrected" for. Such a beam must come "straight" to the observer along a line which connects the observer with the position of the object when it emitted the light.
Aberration should also be distinguished from light-time correction, which is due to the motion of the observed object, like a planet, through space during the time taken by its light to reach an observer on Earth. Light-time correction depends upon the velocity and distance of the emitting object during the time it takes for its light to travel to Earth. Light-time correction does not depend on the motion of the Earth—it only depends on Earth's position at the instant when the light is observed. Aberration is usually larger than a planet's light-time correction except when the planet is near quadrature (90° from the Sun), where aberration drops to zero because then the Earth is directly approaching or receding from the planet. At opposition to or conjunction with the Sun, aberration is 20.5" while light-time correction varies from 4" for Mercury to 0.37" for Neptune (the Sun's light-time correction is less than 0.03").
The apparent position of a star or other very distant object is the direction in which it is seen by an observer on the moving Earth. The true position (or geometric position) is the direction of the straight line between the observer and star at the instant of observation. The difference between these two positions is caused mostly by aberration.
Aberration occurs when the observer's velocity has a component that is perpendicular to the line traveled by light between the star and observer. In Figure 1 to the right, S represents the spot where the star light enters the telescope, and E the position of the eye piece. If the telescope does not move, the true direction of the star relative to the observer can be found by following the line ES. However, if Earth, and therefore the eye piece of the telescope, moves from E to E’ during the time it takes light to travel from S to E, the star will no longer appear in the center of the eye piece. The telescope must therefore be adjusted so that the star light enters the telescope at spot S’. Now the star light will travel along the line S’E’ and reach E’ exactly when the moving eye piece also reaches E’. Since the telescope has been adjusted by the angle SES’, the star's apparent position is hence displaced by the same angle.
Now imagine that you start to walk. Although the rain is still falling vertically (relative to a stationary observer), you find that you have to hold the umbrella slightly in front of you to keep off the rain. Because of your forward motion relative to the falling rain, the rain now appears to be falling not from directly above you, but from a point in the sky somewhat in front of you.
The deflection of the falling rain is greatly increased at higher speeds. When you drive a car at night through falling rain, the rain drops illuminated by your car's headlights appear to (and actually do) fall from a position in the sky well in front of your car.
This quantity is known as the constant of aberration, and is conventionally represented by κ. Its precise accepted value is 20".49552 (at J2000).
The plane of the Earth's orbit is known as the ecliptic. Annual aberration causes stars exactly on the ecliptic to appear to move back and forth along a straight line, varying by κ on either side of their true position. A star that is precisely at one of the ecliptic's poles will appear to move in a circle of radius κ about its true position, and stars at intermediate ecliptic latitudes will appear to move along a small ellipse (see figure 2).
A special case of annual aberration is the nearly constant deflection of the Sun from its true position by κ towards the west (as viewed from Earth), opposite to the apparent motion of the Sun along the ecliptic. This constant deflection is often erroneously explained as due to the motion of the Earth during the 8.3 minutes that it takes light to travel from the Sun to Earth: this is a valid explanation provided it is given in the Earth's reference frame, whereas in the Sun's reference frame the same phenomenon must be described as aberration of light. Hence it is not a coincidence that the angle of annual aberration be equal to the path swept by the Sun along the ecliptic in the time it takes for light to travel from it to the Earth (8.316746 minutes divided by one sidereal year (365.25636 days) is 20.49265", very close to κ). Similarly, one could explain the Sun's apparent motion over the background of fixed stars as a (very large) parallax effect.
Aberration can be resolved into east-west and north-south components on the celestial sphere, which therefore produce an apparent displacement of a star's right ascension and declination, respectively. The former is larger (except at the ecliptic poles), but the latter was the first to be detected. This is because very accurate clocks are needed to measure such a small variation in right ascension, but a transit telescope calibrated with a plumb line can detect very small changes in declination.
Figure 3, above, shows how aberration affects the apparent declination of a star at the north ecliptic pole, as seen by an imaginary observer who sees the star transit at the zenith (this observer would have to be positioned at latitude 66.6 degrees north – i.e. on the arctic circle). At the time of the March equinox, the Earth's orbital velocity is carrying the observer directly south as he or she observes the star at the zenith. The star's apparent declination is therefore displaced to the south by a value equal to κ. Conversely, at the September equinox, the Earth's orbital velocity is carrying the observer northwards, and the star's position is displaced to the north by an equal and opposite amount. At the June and December solstices, the displacement in declination is zero. Likewise, the amount of displacement in right ascension is zero at either equinox and maximum at the solstices.
Note that the effect of aberration is out of phase with any displacement due to parallax. If the latter effect were present, 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 Bradley and his contemporaries.
However, the change in the solar system's velocity relative to the center of the Galaxy varies over a very long timescale, and the consequent change in aberration would be extremely difficult to observe. Therefore, this so-called secular aberration is usually ignored when considering the positions of stars. In other words, star maps show the observed apparent positions of the stars, not their calculated true positions.
To estimate the true position of a star whose distance and proper motion are known, just multiply the proper motion (in arcseconds per year) by the distance (in light years). The apparent position lags behind the true position by that many arcseconds. Newcomb gives the example of Groombridge 1830, where he estimates that the true position is displaced by approximately 3 arcminutes from the direction in which we observe it. Modern figures give a proper motion of 7 arcseconds/year, distance 30 light years, so the displacement is 3 arcminutes and a half. This calculation also includes an allowance for light-time correction, and is therefore analogous to the concept of planetary aberration.
John Flamsteed, from measurements made in 1689 and succeeding years with his mural quadrant, similarly concluded that the declination of the Pole Star was 40" less in July than in September. Robert Hooke, in 1674, published his observations of γ Draconis, a star of magnitude 2m which passes practically overhead at the latitude of London, and whose observations are therefore free from the complex corrections due to astronomical refraction, and concluded that this star was 23" more northerly in July than in October.
The instrument was set up in November 1725, and observations on γ Draconis were made on the 3rd, 5th, 11th, and 12th of December. There was apparently no shifting of the star, which was therefore thought to be at its most southerly point. On December 17, however, Bradley observed that the star was moving southwards, a motion further shown by observations on the 20th. These results were unexpected and inexplicable by existing theories. However, an examination of the telescope showed that the observed anomalies were not due to instrumental errors.
The observations were continued, and the star was seen to continue its southerly course until March, when it took up a position some 20" more southerly than its December position. After March it began to pass northwards, a motion quite apparent by the middle of April; in June it passed at the same distance from the zenith as it did in December; and in September it passed through its most northerly position, the extreme range from north to south, i.e. the angle between the March and September positions, being 40".
Observations of such a star were made difficult by the limited field of view of Bradley and Molyneux's telescope, and the lack of suitable stars of sufficient brightness. One such star, however, with a right ascension nearly equal to that of γ Draconis, but in the opposite sense, was selected and kept under observation. This star was seen to possess an apparent motion similar to that which would be a consequence of the nutation of the Earth's axis; but since its declination varied only one half as much as in the case of γ Draconis, it was obvious that nutation did not supply the requisite solution. Whether the motion was due to an irregular distribution of the Earth's atmosphere, thus involving abnormal variations in the refractive index, was also investigated; here, again, negative results were obtained.
On August 19, 1727, Bradley then embarked upon a further series of observations using a telescope of his own erected at the Rectory, Wanstead. This instrument had the advantage of a larger field of view and he was able to obtain precise positions of a large number of stars that transited close to the zenith over the course of about two years. This established the existence of the phenomenon of aberration beyond all doubt, and also allowed Bradley to formulate a set of rules that would allow the calculation of the effect on any given star at a specified date. However, he was no closer to finding an explanation of why aberration occurred.
The discovery and elucidation of aberration is now regarded as a classic case of the application of scientific method, in which observations are made to test a theory, but unexpected results are sometimes obtained that in turn lead to new discoveries. It is also worth noting that part of the original motivation of the search for stellar parallax was to test the Copernican theory that the Earth revolves around the Sun, but of course the existence of aberration also establishes the truth of that theory.
In a final twist, Bradley later went on to discover the existence of the nutation of the Earth's axis – the effect that he had originally considered to be the cause of aberration.
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