Aberration of light

Light from location 1 will appear to be coming from location 2 for a moving telescope due to the finite speed of light, a phenomenon known as the aberration of light.

The aberration of light (also referred to as astronomical aberration or stellar aberration) is an astronomical phenomenon which produces an apparent motion of celestial objects about their real locations. It was discovered in 1725 and later explained by the third Astronomer Royal, James Bradley, who attributed it to the finite speed of light and the motion of Earth in its orbit around the Sun.^[1][2]

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 solely 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). The result is a tilting of the direction of the incoming light which is independent of the distance between object and observer.

In the case of an observer on Earth, the direction of a star's 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 star 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.^[2] 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.^[2]

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.^[2]

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").

Explanation

It has been stated above that aberration causes a displacement of the apparent position of an object from its true position. However, it is important to understand the precise technical definition of these terms.

Apparent and true positions

Figure 1. Diagram illustrating stellar aberration

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 by parallax and by aberration. When the star is a distant object, parallax is negligible and the difference is due mostly to aberration.

Aberration occurs when the observer's velocity has a component that is perpendicular to the line traveled by the light incoming from the star. Let us suppose (as is the practical case) that the star is sufficiently distant that all light from the star travels in parallel paths to the Earth observer, regardless of where the Earth is in its orbit. That is, there is zero parallax. On the left side of Figure 1, the case of infinite light speed is shown. S represents the spot where the star light enters the telescope, and E the position of the eye piece. If light moves instantaneously, the telescope does not move, and the true direction of the star relative to the observer can be found by following the line ES. However, if light travels at finite speed, the 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. Consequently, the star will no longer appear in the center of the eye piece. The telescope must therefore be adjusted. When the telescope is at position E it must be oriented toward spot S’ so that the star light enters the telescope at spot S’. Now the star light will travel along the line S’E’ (parallel to SE) 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.

For the simple case where the direction of the light beam along the line SE is perpendicular to the motion of the observer then only light emitted at an angle equal to the arc cosine of the ratio of the observer's speed to the speed of light will reach the telescope. Consequently, the telescope too must be oriented at this same angle away from the light beam's trajectoery in the direction of the motion to capture this emitted light.^[3]

Moving in the rain

Many find aberration to be counter-intuitive, and a simple thought experiment based on everyday experience can help in its understanding. Imagine you are standing in the rain. There is no wind, so the rain is falling vertically. To protect yourself from the rain you hold an umbrella directly above you.

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.

Relative frame of reference

According to the special theory of relativity, the aberration only depends on the relative velocity v between the observer and the light from the star. The formula from relativistic aberration can be simplified to^[4]

$\tan{\frac{\theta^'}{2}} = \tan{\frac{\theta}{2}}\sqrt{\frac{c-v}{c+v}}$

where θ is the true angle SEE’, θ’ is the apparent angle S’EE’, and v is the relative speed between the starlight and the observer. Thus, the aberration of light does not imply an absolute frame of reference, as when one moves in the rain. The speed of the rain perceived in a running car is increased as it hits the windscreen more heavily. Instead, according to the special theory of relativity, the speed of light is constant and only its direction changes. The above formula accounts for that while the simpler tan(θ-θ’)=v/c does not.

In most cases the transverse velocity of the star is unknown. However, for some binary systems where a high rotating speed can be inferred, it doesn't cause an aberration as apparently implied by the relativity principle. As discussed above, aberration occurs because the observer moves relative to parallel beams of light coming from the star. In contrast to the case of the observer, the star moves with the divergent beams of light that it emits in all directions, and its motion just selects which one is destined to hit the observer.^[5] Indeed, dependency on the source is paradoxical: Consider a second source of light that on a given instant coincides with the star, but is not at rest with it. Suppose that two rays of light reach the observer, one emitted by the star and the other by the second source in the instant when they coincide. If rays are straight, since they share two points (the coinciding sources and the observer) then they must coincide. However, since the velocities of the sources differ, the observer would see those rays coming from different directions, if aberration depended on the source's motion.^[6]

Although the velocity of the star may be unknown, from the above formula one can derive the relation between the angles θ₁ and θ₂ seen by two arbitrary observers moving with velocities v₁ and v₂, and then use the velocity addition theorem to subtract the unknown velocity w of the star in order to express v₁ and v₂ relative to an arbitrary frame:^[4]

$\tan{\frac{\theta_2}{2}} = \tan{\frac{\theta_1}{2}} \sqrt{\frac{(c-w)(c+v_1)\;(c+w)(c-v_2)}{(c+w)(c-v_1)\;(c-w)(c+v_2)}} = \tan{\frac{\theta_1}{2}} \sqrt{\frac{c-v_{21}}{c+v_{21}}}$

Provided that the observers were actually looking at the same star and its velocity didn't change between their observations, the formula shows how w cancels out. Then, using again the velocity addition theorem to express the relative velocity of the two observers as v₂₁ = (v₂ − v₁) / (1 − v₁v₂ / c²) one finds the relative aberration. It only uses the latter relative velocity and c to equate the angles observed in different frames of reference.

Types of aberration

There are a number of types of aberration, caused by the differing components of the Earth's motion:

• Annual aberration is due to the revolution of the Earth around the Sun.
• Planetary aberration is the combination of aberration and light-time correction.
• Diurnal aberration is due to the rotation of the Earth about its own axis.
• Secular aberration is due to the motion of the Sun and solar system relative to other stars in the galaxy.

Annual aberration

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

Figure 3: Bradley's data on the north-south component of the aberration of γ-Draconis in 1727 establishing stellar aberration.

Figure 4. Diagram illustrating the effect of annual aberration on the apparent position of three stars at ecliptic longitude 270 degrees, and ecliptic latitude 90, 45 and 0 degrees, respectively

Figure 5. Diagram illustrating aberration of a star at the north ecliptic pole

Suppose a star is observed with a telescope idealized as a narrow tube. The light enters the tube from a star at angle θ and travels at speed c taking a time h/c to reach the bottom of the tube, where our eye detects the light. 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 photon to reach the bottom of the tube, the tube must be inclined at an angle φ different from θ, resulting in an apparent position of the star at angle φ. As the Earth proceeds in its orbit, the velocity changes direction, so φ changes with the time of year the observation is made, allowing the speed of light to be determined. The two angles are related by the speed of light and the speed of the tube, but actually do not depend upon the length of the tube, as explained next. The apparent angle and true angle are related using trigonometry as:^[7]

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

independent of the path length h traversed by the light. It may be more useful to express the correction (θ−φ) to the observed angle φ in terms of the observed angle itself:

$\sin (\theta-\phi) \approx \frac{v}{c} \sin (\phi) \ ,$

which, because small v/c leads to small corrections, becomes:

$\theta-\phi \approx \frac{v}{c} \sin (\phi) \ ,$

where use is made of the small-angle approximation to the sine function sin x ≈ x.

As an example, if v is the component of the Earth's velocity along the direction of the light rays, this velocity changes month-to-month as the Earth traverses its orbit, making v a periodic function of the time of year, and consequently the aberration also varies periodically. This effect was used in 1727 by J Bradley to determine the speed of light as approximately 183,000 miles/s.^[8][9] A facsimile of his observations on the star γ-Draconis is shown in Figure 3.^[10] More detail is provided below.

As the Earth revolves around the Sun, it is moving at a velocity of approximately 30 km/s. The speed of light is approximately 300,000 km/s. In the special case where the Earth is moving perpendicularly to the direction of the star (that is, if angle θ in Figure 2 is 90 degrees), the angle of displacement, θ − φ, would therefore be (in radians) the ratio of the two velocities, or 1/10000, or about 20.5 arcseconds.

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 4).

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 5 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.

Solar annual aberration

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 (which is from west to east, as seen from Earth). The deflection thus makes the Sun appear to be behind (or retarded) from its actual position on the ecliptic by a position or angle κ. This constant deflection is often 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 (where it becomes purely a light-time correction for the position of the eastward-moving Sun as seen from a stationary Earth), whereas in the Sun's reference frame the same phenomenon must be described as aberration of light when seen by the westward-moving Earth, which involves having Earth's telescopes pointed "forward" (westward, in a direction toward the Earth's motion relative to the Sun) by a slight amount.

Since this is the same physical phenomenon, simply described from two different reference frames, it is not a coincidence that the angle of annual aberration of the Sun is 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 nearly κ). Similarly, one could explain the Sun's apparent motion over the background of fixed stars as a (very large) parallax effect.

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). 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.

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 and longitude of the observer. Its effect is much smaller than that of annual aberration, and is only 0".32 in the case of an observer at the equator, where the rotational velocity is greatest.

Secular aberration

The Sun and solar system are revolving around the center of the Galaxy, as are other nearby stars. Therefore the aberrational effect affects the apparent positions of other stars and on extragalactic objects: if a star is two thousand light years from Earth, we don't see it where it is now, but where it was two thousand years ago (in a reference frame moving with the solar system).

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.

Historical background

The discovery of the aberration of light in 1725 by James Bradley was one of the most important in astronomy. It was totally unexpected, and it was only by extraordinary perseverance and perspicacity that Bradley was able to explain it in 1727. Its origin is based on attempts made to discover whether the stars possessed appreciable parallaxes. The Copernican theory of the solar system – that the Earth revolved annually about the Sun – had received confirmation by the observations of Galileo and Tycho Brahe (who, however, never accepted heliocentrism), and the mathematical investigations of Kepler and Newton.

Search for stellar parallax

As early as 1573, Thomas Digges had suggested that this theory should necessitate a parallactic shifting of the stars, and, consequently, if such stellar parallaxes existed, then the Copernican theory would receive additional confirmation. Many observers claimed to have determined such parallaxes, but Tycho Brahe and Giovanni Battista Riccioli concluded that they existed only in the minds of the observers, and were due to instrumental and personal errors. In 1680 Jean Picard, in his Voyage d’Uranibourg, stated, as a result of ten years' observations, that Polaris, or the Pole Star, exhibited variations in its position amounting to 40" annually. Some astronomers endeavoured to explain this by parallax, but these attempts were futile, for the motion was at variance with that which parallax would produce.

John Flamsteed, from measurements made in 1689 and succeeding years with his mural quadrant, similarly concluded that the declination of the Pole Star was 40" less in July than in September. Robert Hooke, in 1674, published his observations of γ Draconis, a star of magnitude 2^m which passes practically overhead at the latitude of London, and whose observations are therefore free from the complex corrections due to astronomical refraction, and concluded that this star was 23" more northerly in July than in October.

Bradley's observations

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

The instrument was set up in November 1725, and observations on γ Draconis were made on the 3rd, 5th, 11th, and 12 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".

Aberration vs nutation

This motion was evidently not due to parallax, for the reasons given in the discussion of Figure 2, and neither was it due to observational errors. Bradley and Molyneux discussed several hypotheses in the hope of finding the solution. The idea that immediately suggested itself was that the star's declination varied because of short-term changes in the orientation of the Earth's axis relative to the celestial sphere – a phenomenon known as nutation. Because this is a change to the observer's frame of reference (i.e. the Earth itself), it would therefore affect all stars equally. For instance, a change in the declination of γ Draconis would be mirrored by an equal and opposite change to the declination of a star 180 degrees opposite in right ascension.

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.

Development of the theory of aberration

Bradley eventually developed the explanation of aberration in about September 1728 and his theory was presented to the Royal Society in mid January the next year. One well-known story (quoted in Berry, page 261) 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 apocryphal.

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

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.

See also

• Aberration
• Nutation
• Proper motion
• Bradley, James
• Fresnel, Augustin-Jean
• List of astronomical topics
• Stokes, George Gabriel
• Timeline of electromagnetism and classical optics

References

• P. Kenneth Seidelmann (Ed.), Explanatory Supplement to the Astronomical Almanac (University Science Books, 1992), 127-135, 700.
• Simon Newcomb, A Compendium of Spherical Astronomy (Macmillan, 1906 – republished by Dover, 1960), 160-172.
• Arthur Berry, A Short History of Astronomy (John Murray, 1898 – republished by Dover, 1961), 258-265.
• Stephen Peter Rigaud, Memoirs of Bradley (1832)
• Charles Hutton, Mathematical and Philosophical Dictionary (1795).
• H. H. Turner, Astronomical Discovery (1904).

Notes

1. ^ {{cite book |last=Bradley |first=James |authorlink= |title=An account of the new discovered motion of the fixed stars |year=1729 |publisher=Phil. Trans. R. Soc.
2. ^ ^{a b c d e} Hirschfeld, Alan (2001). Parallax:The Race to Measure the Cosmos. New York, New York: Henry Holt. ISBN 0-8050-7133-4. 
3. ^ Tolman, R.C., "The theory of the relativity of motion", University of California Press, Berkeley (1917), p.59
4. ^ ^{a b} Klaus Kassner (2002). "Why the Bradley aberration cannot be used to measure absolute speeds. A comment". arXiv:astro-ph/0203056 [astro-ph]. 
5. ^ Paul Marmet (1996). "Stellar Aberration and Einstein's Relativity". Newton Physics. http://www.newtonphysics.on.ca/aberration/index.html. Retrieved 2008-10-18. 
6. ^ The argument of two coincident sources in relative motion is attributed to prof. T. Gold of Cornell University, in Edward Eisner (1966-12-12). "Aberration of Light from Binary Stars—a Paradox?". doi:10.1119/1.1974259. http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=AJPIAS000035000009000817000001&idtype=cvips&gifs=yes. Retrieved 2008-10-20. 
7. ^ Richard A. Mould (2001). Basic Relativity (2 ed.). Springer. p. 8. ISBN 0-387-95210-1. http://books.google.com/?id=lfGE-wyJYIUC&pg=PA8. 
8. ^ James Bradley (1729). "An account of a new discovered motion of the fixed stars". Phil Trans Roy Soc 35: 637. 
9. ^ Encyclopaedia Britannica
10. ^ A copy of the actual data is Figure 2-3 in AP French (1968). Special Relativity. CRC Press. p. 43. ISBN 0-7487-6422-4. http://books.google.com/?id=udA0Gjq8vHIC&pg=PA43. 

External links

• Stellar Aberration at MathPages
• Courtney Seligman on Bradley's observations

Attribution

 This article incorporates text from a publication now in the public domain: Chisholm, Hugh, ed (1911). Encyclopædia Britannica (11th ed.). Cambridge University Press.