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 and later explained by the
third
Astronomer Royal,
James Bradley, in 1725, who attributed it to
the finite
speed of light and the
motion of
Earth in its orbit around the
Sun.
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. 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").
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.
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
- \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. 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.
Although the velocity of the star may be unknown, from the above
formula one can derive the relation between the angles
θ_{1} and
θ_{2} seen by two
arbitrary observers moving with velocities
v_{1}
and
v_{2}, and then use the velocity addition
theorem to subtract the unknown velocity
w of the star in
order to express
v_{1} and
v_{2}
relative to an arbitrary frame:
- \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 v_{21}=(v_2-v_1)/(1-v_1v_2/c^2) 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
Suppose we look at a star 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 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:
- \tan(\phi) = \frac { h\sin(\theta)}{hv/c + h \cos
(\theta)}=\frac { \sin(\theta)}{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) = \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. A
facsimile of his observations on the star γ-Draconis is shown in
Figure 3. 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).
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
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, 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.
The special case of 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. 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 moving Sun), whereas in the
Sun's reference frame
the same phenomenon must be described as aberration of light when
seen by the moving Earth. 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 perspicuity 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. 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 a year later. 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
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.
- S. Rigaud, Memoirs of Bradley (1832)
- Charles Hutton, Mathematical and Philosophical
Dictionary (1795).
- H. H. Turner, Astronomical Discovery (1904).
External links