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'''[[Aberration of light|Stellar aberration]]''' is an astronomical phenomenon "which produces an apparent motion of celestial objects". It can be proven mathematically that stellar aberration is due to the change of the astronomer's [[inertial frame of reference]]. The formula is derived with the use of [[Special_relativity#Reference_frames.2C_coordinates_and_the_Lorentz_transformation|Lorentz transformation]] of the star's ''coordinates''.

As the astronomer [[John Herschel]] has already explained in 1844, the stellar aberration does '''not''' depend on the relative velocity of the star towards the earth.<ref>{{Cite journal|author=Herschel, John Frederick William|title=Schreiben des Herrn Baronets Herschel an den Herausgeber|journal=Astronomische Nachrichten|volume=22|issue=520|year=1844|pages=249–254|url=http://adsabs.harvard.edu//abs/1844AN.....22..249H|bibcode = 1844AN.....22..249H |doi = 10.1002/asna.18450221702 }}</ref> Otherwise [[Binary_star#Eclipsing_binaries|eclipsing binary stars]] would appear to be separated, in stark contrast to observation: both stars are rotating with high speed —and ever changing and different velocity vectors— around each other, but they appear as ''one'' spot all the time.

==Stellar aberration is only due to the change of the astronomer's inertial frame of reference==

In the year 1926 the astrophysicist [[Robert Emden]] published the article ''Aberration und Relativitätstheorie'' in the journal ''[[Naturwissenschaften]]''.<ref>{{Cite journal|author=Emden, R.|title=Aberration und Relativitätstheorie|journal=Die Naturwissenschaften|volume=14|issue=16|year=1926|pages=329–335|doi=10.1007/BF01506966|bibcode = 1926NW.....14..329E }} (14. Jahrgang, Heft 16)</ref> In this article he states that the direction of a light ray isn't influenced by the motion of the star or by the motion of the earth.<ref group="Notes">R. Emden's (rhetorical) question "How will the direction of the light rays of stars incident upon the earth be influenced by the motion of the earth and the motion (not the location) of the star at the time of emission?" is answered by him with "Not at all."</ref> At that time, the opponents of the special theory of relativity reasoned that the theory must be flawed, because it would state that the stellar aberration would depend on the relative velocity of the star — which would be in contradiction to observation — and R. Emden's article explains that the special theory of relativity does not predict this. Today, the special theory of relativity isn't contested anymore but there are still articles that suggest that the aberration would depend on the relative velocity of the star.<ref>See for example equation (4) in {{cite arxiv |eprint=astro-ph/0203056 |title=Why the Bradley aberration cannot be used to measure absolute speeds. A comment |author1=Klaus Kassner |class=astro-ph |year=2002}}</ref>

Although a (relativistic) [[velocity-addition formula]] can be used to explain stellar aberration, (see [[Aberration_of_light#Explanation|Aberration of light]]), another  (relativistic) explanation using only the [[Lorentz transformation]] is also possible, as will be demonstrated. This derivation only uses the star's ''coordinates'' at the time of emission, and therefore has the ''formal'' advantage there is no place for the relative velocity of the star towards the astronomer and therefore it is evident that the observed position position doesn't depend on the star's velocity — provided that the resultant change of position is much smaller than the distance between star and earth.<ref group="Notes">I.e. the stars can rotate very fast like eclipsing binary stars as long as the change in position is much smaller than the star-earth distance.</ref> The observed position of the star wouldn't depend on the earth's motion either, if the astronomer could use the same inertial frame of reference all the time. But of course that is technically impossible,<ref group="Notes">It's easy to (technically) compensate the deviation of the ''direction'' of the axes due to the earth's rotation and earth's orbit but it almost impossible to (technically) compensate a velocity in the order of several 1000&nbsp;m/s. This can only be achieved with help of mathematics.</ref> the astronomer uses his current rest frame and these current rest frames are different at different times as the earth orbits around the sun. It is mathematically convenient to declare the position of star in a rest frame of the sun (more exactly: the center of mass of the solar system) as the "real" position and that the difference to this "real" position derives form the "aberration".<ref group="Notes">Of course every inertial frame of reference is physically equal, but the rest system of the barycenter has the mathematical advantage, that it's the ''mean'' frame of reference and since the time of [[Galileo Galilei]] it's the preferred frame of reference</ref>

===Sample calculation===
[[File:PathLightSignalS.PNG|thumb|left|path of light signal in reference frame S]]
[[File:PathLightSignalSdash.PNG|thumb|path of light signal in reference frame S']]
S and S' are (quasi-)inertial frames of reference and the reference frame S' is in uniform motion with v<sub>x</sub> = 0,5c relative to S, such that in the future the star comes nearer to the origin of the coordinate system (and consequently further afar in the past). The x-,y- and z-axes of both systems ought to be parallel and at time t=t'=0 the origins of both systems ought to coincide. 
Therefore one gets according to Lorentz transformation: <math>\scriptstyle \beta=v/c </math>, <math>\scriptstyle \gamma=1/\sqrt{1-\beta^{2}}</math>:  y'=y ; z=z';  <math>\scriptstyle x'=\gamma\cdot(x-\beta\cdot ct)</math>
 
Suppose now that the star emitted a light signal at time <math>\scriptstyle c\,t_1=-5\,Ly</math> at location <math>\scriptstyle x_1=4\,Ly\;,\;y_1= 3\,Ly\;,\;z_1=0 </math> (S coordinates) and that this light signal is received by an astronomer at time  <math>\scriptstyle c\,t_2=0</math> at location <math>\scriptstyle x_2=0\;,\;y_2=0\;,\;z_2=0</math>.

In S the star's position and the x-axis form an angle <math>\delta</math> with <math>\scriptstyle \tan \delta =3/4 \; \rightarrow \;\delta =36.87^\circ </math>
But in S' <math>\scriptstyle x_1'=\gamma \cdot (x_1-\beta \cdot c\,t_1)=1,1547\cdot (4 Lj - 0,5\cdot(-5 Lj))=7,50555 Lj; \quad y_1'=y_1=3 Lj</math> and therefore the angle <math>\delta'</math> between the star's position and x'-axis<ref group="Notes">The lines of x-axis and x'-axis lie on top of each other</ref> is <math>\scriptstyle \tan \delta'=y_1'/x_1'=3/7,50555\;\rightarrow\;\delta' = 21,79^\circ</math>
 
Calculation with help of the formula in [[aberration of light#Explanation]] gives the same result: <math>\scriptstyle \tan (\delta'/2) = \tan (\delta /2) \cdot \sqrt {(1-0,5)/(1+0,5)} = \tan (36,87^\circ/2) \cdot \sqrt{1/3} = 0,19245\;</math> <math>\scriptstyle\;\rightarrow\;\delta'/2= 10,89^\circ\;\rightarrow\;\delta'=21,79^\circ</math>.

==Two-dimensional problem==

===Derivation of the formula for motion along the x-axis===

For the derivation it is assumed, that the light signal only travels through space regions where the gravitation field is negligible. Hence is suffices to use [[special relativity|'''special''' relativity]] and the path of the light signal is a straight line in any inertial frame of reference.

'''Observation in the [[rest frame]] S of the [[center_of_mass#Astronomy|center of mass]] of  our solar system'''

''The rest frame of the center of mass (barycenter) is a very good<ref group="Notes">An even better quasi-inertial frame of reference would be a frame of reference in which the center of mass of the Milky Way would rest or be in uniform motion.</ref> quasi-inertial frame of reference for periods of time in the order of thousands of years, since our solar system needs about 230 million years ([[galactic year]]) to move completely around the center of the [[Milky Way]]. The space coordinates of this frame of reference form a [[Cartesian coordinate system]].''

In the reference frame S <math>(x_1|y_1|0|c\,t_1)</math> with <math>c\,t_1<0</math> and <math>(x_1|y_1) \ne (0|0)</math> are the (space-time) coordinates at which the star emits a light signal and <math>(0|0|0|0)</math> are the coordinates at which the astronomer receives the light signal.

In reference frame S the light signal starts at <math>t_1<0</math> and stops at time <math>t_2=0</math> and therefore the light signal did cover the distance <math>d = c\cdot(0-t_1) = -c\,t_1</math> .

In S the path of the light signal is a straight line and it forms an angle <math>\delta</math> with the x-axis with: <math>\sin \delta = \frac{y_1}{d}=\frac{y_1}{-c\,t_1}</math>&nbsp; and <math>\cos\delta=\frac{x_1}{-c\,t_1}</math>

'''Observation in the inertial frame of reference S' which is in uniform motion (relative to S) along the x-axis'''
  
The origin of the reference frame S' is in uniform motion relative to S with <math>(v|0|0)</math>, i.e. moves along the x-axis, and the x-,y- und z-axes of S' and S are parallel to each other and at time <math>\scriptstyle t=t'=0</math> the origins of S and S' coincide. Let <math>\beta = \frac{v}{c} , \gamma = \frac{1}{\sqrt{1-\beta^2}}</math>

S' now is an equally good quasi-inertial frame of reference as S: the space coordinates form a [[Cartesian coordinate system]] and the path of the light signal is a straight line.
 
According to the Lorentz transformation one gets:  <math> x_1' = \gamma \cdot(x_1-\beta \cdot c\,t_1) , y_1'=y_1 , z_1'=0 , c\,t_1' = \gamma \cdot (c\,t_1-\beta \cdot x_1)</math>

In reference frame S' the light signal starts at <math>t_1'<0</math> and stops at time <math>t_2'=0</math> and therefore the light signal did cover the distance <math>d\,' = c\cdot(0-t_1') = -c\,t_1'</math> .

In S' the path of the light signal is a straight line, too. It forms an angle <math>\delta'</math> with the x'-axis and one gets:

<math> \sin \delta' = \frac{y_1'}{d\,'}=\frac{y_1}{-c\,t_1'}
=\frac{y_1}{-\gamma \cdot (c\,t_1-\beta \cdot x_1)}
=\frac{y_1}{-c\,t_1 \cdot \gamma\cdot \left(1-\beta\cdot\frac{x_1}{c\,t_1}\right)}=\frac{\frac{y_1}{-c\,t_1}}{\gamma\cdot\left(1+\beta\cdot\frac{x_1}{-c\,t_1}\right)}
=\frac{\sin \delta}{\gamma\cdot(1+\beta\cdot\cos\delta)}</math>

<math>\cos\delta'=\frac{x_1'}{-c\,t_1'}
=\frac{\gamma \cdot(x_1-\beta \cdot c\,t_1)}{-\gamma \cdot (c\,t_1-\beta \cdot x_1)}
=\frac{-c\,t_1 \cdot \gamma \cdot \left ( \frac{x_1}{-c\,t_1}+\beta \right)}{-c\,t_1 \cdot \gamma \cdot \left( 1 + \beta \cdot \frac{x_1}{-c\, t_1} \right) }
=\frac{\frac{x_1}{-c\,t_1}+\beta}{1 + \beta \cdot \frac{x_1}{-c\, t_1}}
=\frac{\cos \delta +\beta}{1 + \beta \cdot \cos \delta}</math>

Hence: <math>\tan \delta' = \frac{\sin \delta'}{\cos \delta'}=\frac{\sin \delta}{\gamma\cdot(\cos \delta +\beta)}</math>

These are the same formulas as in [[aberration of light#Explanation]].

===Approximate formula for motion along the x-axis in case of v/c <<1===

<math>\triangle\delta\;=\;\delta' - \delta\; </math> is the change of the angle &delta;. As &beta;<<1 this change is also very small.

'''Case I: '''<math>\;\delta\ne\pm 90^\circ\; \rightarrow \;\cos \delta \ne 0</math>

As &Delta;&delta;<<1 one gets: <math>\frac{\tan \delta' - \tan \delta}{\triangle\delta} \approx\frac{d}{d\delta}\tan \delta = \frac{1}{(\cos \delta)^2}\; \rightarrow \; \triangle\delta\;=\;(\cos \delta)^2 \cdot (\tan \delta' - \tan \delta)</math>

As &beta;<<1 one gets: <math>\tan\delta'=\frac {\sin \delta}{\gamma(\cos \delta +\beta)} \approx \frac {\sin \delta}{\cos \delta +\beta} = \frac{\sin \delta}{\cos \delta \cdot \left(1+\frac{\beta}{\cos \delta}\right)} \approx \tan \delta \cdot \left(1-\frac{\beta}{\cos \delta}\right)</math>

Therefore <math>\triangle\delta\;=\;(\cos \delta)^2 \cdot (\tan \delta' - \tan \delta) = (\cos \delta)^2 \cdot \tan \delta \left(1-\frac{\beta}{\cos \delta}\;-1\right) = - \cos \delta \cdot \tan \delta \cdot \beta = -\beta \cdot \sin \delta </math>

'''Case IIa: '''<math>\;\delta\;=90^\circ\;</math>, hence: <math>\tan \delta' =\frac{\sin 90^\circ}{\gamma\cdot(\cos 90^\circ + \beta)}= \frac{1}{\gamma \cdot \beta}\;\approx \frac {1}{\beta} \quad \rightarrow \quad \cot \delta' = \beta \quad \rightarrow \quad \delta' = \arccot \beta \approx \frac{\pi}{2} - \beta </math>

and therefore: <math>\triangle\delta = \delta' - \delta = -\beta \quad \left(= -\beta \cdot \sin(90^\circ)\right)</math>

'''Case IIb: '''<math>\;\delta\;=-90^\circ\;</math>, and hence: <math>\tan \delta' = \frac{-1}{\gamma \cdot \beta}\;\approx -\frac {1}{\beta} \quad \rightarrow \quad \cot \delta' = -\beta \quad \rightarrow \quad \delta' = \arccot \beta \approx -\frac{\pi}{2} + \beta \quad</math> &nbsp; &nbsp;<ref group="Notes">Here the solution with cot 0 = -&pi;/2 has to be used, see [http://www.google.de/imgres?imgurl=http://www.efunda.com/math/trig_functions/images/arccot_plot.gif&imgrefurl=http://www.efunda.com/math/trig_functions/display.cfm?name%3Darccot&h=329&w=485&sz=3&tbnid=a0hRF-qzOibryM:&tbnh=82&tbnw=121&zoom=1&usg=__rTZwdunutqIAiG-l4WbgNeG82pY=&docid=KYfggGmcAxqigM&sa=X&ei=Z52XUZTAK6nk4QSDxIEQ&ved=0CEMQ9QEwAw&dur=197 Plot of arccot (x)]</ref>

Hence: <math>\triangle\delta = \delta' - \delta = +\beta\quad \left(= -\beta \cdot \sin(-90^\circ)\right)</math>

'''Conclusion:'''The change of the angle &Delta;&delta; = &delta;'-&delta;  in the case of &beta; = v/c << 1 can be described by the approximate formula <math>\triangle\delta = -\frac{v}{c} \cdot \sin \delta\;</math> resp. in the degree measure <math>\triangle\delta = -\frac{v}{c} \cdot \sin \delta  \cdot \frac{180^\circ}{\pi}</math>

===Symmetric form of the (exact) formula for motion along the x-axis===

With help of [[tangent half-angle formula]] <math>\scriptstyle \tan (\alpha/2)=\sin\alpha/(1+\cos\alpha)</math>&nbsp; one can prove the symmetric form: <math>\tan \frac{\delta'}{2}=\sqrt{\frac{1-\beta}{1+\beta}}\cdot\tan\frac{\delta}{2}</math> &nbsp;  (derivation found in a SR-textbook<ref>''Spezielle Realtivitätstheorie für Studenanfänger'', Jürgen Freund, vdf-Verlag (Zürich), 2nd. Ed. (2005)</ref>)

<math>\tan \frac{\delta'}{2}=\frac{\sin \delta'}{1+\cos \delta'}
=\frac{\frac{\sin \delta}{\gamma\cdot(1+\beta\cdot\cos\delta)}}{\frac{1 + \beta \cdot \cos \delta}{1 + \beta \cdot \cos \delta}+\frac{\cos \delta +\beta}{1 + \beta \cdot \cos \delta}}
=\frac{\sin \delta}{\gamma\cdot(1 + \beta \cdot \cos \delta + \cos \delta +\beta)}=\frac{\sin \delta}{\gamma\cdot(1 + \beta)\cdot(1+ \cos \delta)}
=\frac{\tan \frac{\delta}{2}}{\gamma\cdot(1+ \beta)}</math>

And as <math>\frac{1}{\gamma\cdot(1+\beta)}
=\frac{\sqrt{1-\beta^2}}{1+\beta}
=\frac{\sqrt{(1+\beta)(1-\beta)}}{1+\beta}
=\sqrt{\frac{1-\beta}{1+\beta}}</math> the symmetric form follows.

===Formula for motion along the y-axis===

Let <math>\theta </math> be the angle between the light ray (=path of the light signal which is a straight line) and the y-axis whereby &theta; is positive if the y-axis would have to rotate counter-clockwise to coincide with the light ray. Then the derivation of the formula of angle &theta;' for the motion along the y-axis is the same as the derivation of the formula of angle &delta;' for the motion along the x-axis.

Hence: <math>\tan\theta\,' = \frac{\sin\theta}{\gamma\cdot(\cos\theta+\beta)}</math>

The symmetric form is: <math>\tan \frac{\theta\,'}{2}=\sqrt{\frac{1-\beta}{1+\beta}}\cdot\tan\frac{\theta}{2}</math> and the approximate formula is: <math>\triangle\theta = -\beta\cdot\sin\theta</math>

As <math>\theta=\delta-90^\circ</math> and as <math>\scriptstyle\sin(\delta-90^{\circ})=-\sin(90^\circ-\delta)=-\cos\delta\;,\;\cos(\delta-90^\circ)=\cos(90^\circ-\delta)=\sin\delta</math> and <math>\scriptstyle\tan(\delta-90^\circ)=-\tan(90^\circ-\delta)=-\cot\delta</math>&nbsp; one gets:

<math>\tan(\delta'-90^\circ) = \frac{\sin(\delta-90^\circ)}{\gamma\cdot\left(\cos(\delta - 90 ^\circ)+\beta\right)}</math> and therefore:&nbsp; <math>-\cot\delta' = \frac{-\cos\delta}{\gamma\cdot(\sin\delta+\beta)}</math> and hence <math>\cot\delta' = \frac{\cos\delta}{\gamma\cdot(\sin\delta+\beta)}</math>

Since <math>\cot\delta'=\frac{1}{\tan\delta'}</math>&nbsp; one also gets: <math>\tan\delta' = \frac{\gamma\cdot(\sin\delta+\beta)}{\cos\delta}</math>

And the symmetric form is: <math>\tan \frac{\delta'-90^\circ}{2}=\sqrt{\frac{1-\beta}{1+\beta}}\cdot\tan\frac{\delta-90^\circ}{2}</math>

And as  <math>\scriptstyle\triangle\theta=(\delta'-90^\circ)-(\delta-90^\circ)=\triangle\delta</math>&nbsp; the approximate formula is: <math>\triangle\delta = \triangle\theta =-\beta\cdot\sin(\delta-90^\circ)=+\beta\cdot\cos\delta</math>&nbsp; &nbsp;

===Formula for motion along a ray lying in the x-y-plane with direction vector (cos &alpha; | sin &alpha;) ===

With the same reasoning as above one gets the formula: <math>\tan(\delta'-\alpha) = \frac{\sin(\delta-\alpha)}{\gamma\cdot\left(\cos(\delta-\alpha)+\beta\right)}</math>

The symmetric form is: <math>\tan \frac{\delta'-\alpha}{2}=\sqrt{\frac{1-\beta}{1+\beta}}\cdot\tan\frac{\delta-\alpha}{2}</math>

The approximate formula is: <math>\triangle\delta = - \beta\cdot\sin(\delta-\alpha)</math>&nbsp; and in the degree measure: <math>\triangle\delta = -\frac{v}{c}\cdot\sin(\delta-\alpha)\cdot\frac{180^\circ}{\pi}</math>

==Three-dimensional problem==

[[File:Heliocentric rectangular ecliptic.png|thumb|Heliocentric ecliptic coordinates]]
[[File:Ecliptic grid globe.png|thumb|Earth-centered Ecliptic coordinates]]

==Application: Aberration in astronomy==

The stellar aberration is only an effect of the change of the reference frame. The astronomer orbits (with the earth) around the sun in one year and furthermore rotates around the axis of the earth in one day. His current rest frame S' therefore has different velocities relative to the rest frame S of the barycenter of the solar system at different times. 
Hence the astronomer observes, that the position of the star changes. The formula is derived under the condition that the change of the ''position'' of the star and of the earth is negligible in the period of observation <ref group="Notes">For the star this period lies in the past.</ref> That is correct for almost all stars: The amplitude of the [[parallax]] of a star in a distance of ≥ n [[parsec]] is ≤ 1/n&#x202F;".

===Stellar aberration due to the orbit of the earth (around the sun)===

The mean orbital speed of the earth is <math>v_e = \frac{2 \cdot \pi \cdot 1\,AE}{365,25\,d} = 29,78\,km/s </math>, and therefore <math>\frac{v_e}{c} = 0,00009935</math>.

-> <math>\frac{v}{c} \cdot \frac{180^\circ}{\pi} = 20,5^{\prime\prime}</math>.

<math> k_A= 20,5^{\prime\prime}</math> is dubbed ''constant of aberration'' for the annual aberration.<ref>see e.g. [http://adsabs.harvard.edu/full/1964SvA.....7..737M The Constant of Aberration and the Solar Parallax]</ref>

===Stellar aberration due to the earth's rotation===

An astronomer at the [[latitude]] <math>\varphi</math> rotates in 24 hours around the axis of the earth. His speed of rotation is therefore <math>v_r= \frac{\cos \varphi \cdot 40000\,km}{1\,d} = \cos \varphi \cdot 463\,m/s</math>. Hence <math>\frac{v_r}{c} = \cos \varphi \cdot 1,54\cdot10^{-6}</math>. Form this so called ''diurnal'' aberration one gets an additional contribution of (at max.) <math> \cos \varphi \cdot 0,32^{\prime\prime}</math>.

===Stellar aberration due the orbit of our solar system around the center of the Milky Way===

The rest frame of the center of mass of our solar system isn't a perfect inertial frame of reference since our solar system orbits around the center of the [[Milky Way]]. An estimation for the time of period of circulation is 230 million years (estimations vary between 225 and 250 million years).<ref>see [http://hypertextbook.com/facts/2002/StacyLeong.shtml hypertextbook.com]</ref><ref>[http://starchild.gsfc.nasa.gov/docs/StarChild/questions/question18.html NASA - StarChild Question of the Month for February 2000]</ref> As the estimation for the distance between our solar system and the center of the Milky Way is about 28000 [[Light-year|Ly]], the assumed [[orbital speed]] of our solar system is <math>\scriptstyle v = \frac{2\pi \cdot 280000 \cdot 9,461\cdot10^{15}\,m}{230\cdot 10^{6} \cdot 365,25 \cdot 24 \cdot 3600} \approx\, 230\,km/s</math>. This would cause an aberration ellipse with a major semiaxis of 2,6' ([[minute of arc|arcminutes]]). Therefore, in one year the aberration angle could change (at max.)  <math>\scriptstyle 2,6^\prime \cdot \frac{2 \pi \cdot 1\,a}{230000000\,a}</math> = 4,3 µas ([[minute of arc|microarcseconds]]). This very small value isn't detectable now, perhaps it's possible with the planned mission of the [[Gaia (spacecraft)|Gaia spacecraft]].<ref>This is a hint from H.-H. Voigt: ''Abriss der Astronomie'', 6. Auflage. Herausgegeben von H.-J. Röser und W. Tscharnuter. © 2012 Wiley-VCH Verlag GmbH & Co. KGaA., page 35 of [http://www.science-shop.de/sixcms/media.php/370/Leseprobe.1162646.pdf extract]</ref>

==See also==
* [[Aberration of light]]
* [[Parallax#Parallax_in_astronomy|Parallax in astronomy]]

==Notes==
<references group="Notes"/>

==References==
<references/>

[[Category:Special relativity]]
[[Category:Astrometry]]