Editing Aberration (astronomy) (section)

==Explanation==
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[[File:Sun earth relativistic aberration.svg|thumb|300px|Light rays striking the earth in the Sun's rest frame compared to the same rays in the Earth's rest frame according to special relativity. The effect is exaggerated for illustrative purposes.]]

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 [[#Historical_theories_of_aberration|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 [[vector space|addition of velocities]].

===Classical explanation===
In the Sun's frame, consider a beam of light with velocity equal to the speed of light <math>c</math>, with x and y velocity components <math>u_x</math> and <math>u_y</math>, and thus at an angle <math>\theta</math> such that <math>\tan(\theta) = u_y/u_x</math>. If the Earth is moving at velocity <math>v</math> 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 <math>u_x' = u_x + v</math>, and the y velocity is unchanged, <math>u_y' = u_y</math>. Thus the angle of the light in the Earth's frame in terms of the angle in the Sun's frame is

:<math>\tan(\phi) = \frac{u_y'}{u_x'} = \frac{u_y}{u_x+v} = \frac{\sin(\theta)}{v/c + \cos(\theta)}</math>

In the case of <math>\theta = 90^\circ</math>, this result reduces to <math>\tan(\theta - \phi) = v/c</math>, which in the limit <math>v/c \ll 1</math> may be approximated by <math>\theta - \phi = v/c</math>.

===Relativistic explanation===
The reasoning in the relativistic case is the same except that the [[Velocity-addition formula|relativistic velocity addition]] formulas must be used, which can be derived from [[Lorentz transformations]] between different frames of reference. These formulas are

:<math>u_x' = (u_x + v)/(1+u_x v/c^2)</math>
:<math>u_y' = u_y / \gamma (1+u_x v/c^2)</math>

where <math>\gamma = 1/\sqrt{1-v^2/c^2}</math>, 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 <ref name=Mould>{{cite book |title=Basic Relativity |page=8 |url=https://books.google.com/books?id=lfGE-wyJYIUC&pg=PA8 |isbn=0-387-95210-1 |date=2001 |publisher=Springer |author=Richard A. Mould |edition=2nd}}</ref>

:<math>\tan(\phi) = \frac{u_y'}{u_x'} = \frac{u_y}{\gamma(u_x+v)} = \frac{\sin(\theta)}{\gamma(v/c + \cos(\theta))}</math>
or
:<math>\tan\frac{\phi}{2} =\sqrt{\frac{1-v/c}{1+v/c}}\tan\frac{\theta}{2}</math>

In the case of <math>\theta = 90^\circ</math>, this result reduces to <math>\sin(\theta - \phi) = v/c</math>, and in the limit <math>v/c \ll 1</math> this may be approximated by <math>\theta - \phi = v/c</math>. This relativistic derivation keeps the speed of light <math>\sqrt{u_x^2 + u_y^2} = c</math> constant in all frames of reference, unlike the classical derivation above.

===Relationship to light-time correction and relativistic beaming===
[[File:aberrationlighttimebeaming.gif|thumb|400px|Aberration, light-time correction, and relativistic beaming can be considered the same phenomenon depending on the frame of reference.]]

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 simplicity<ref>In 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.</ref>) 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.