In this entry, we are going to study a relativistic effect known as “stellar aberration”.
From the known Lorentz transformations of velocities (inverse case), we get:
![\[ v_x=\dfrac{v'_x+V}{1+\dfrac{v'_xV}{c^2}}\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-cca94a0e5b4ae5dd0b2e9c6bd6682563_l3.png "Rendered by QuickLaTeX.com")
![\[ v_y=\dfrac{v'_y\sqrt{1-\beta^2}}{1+\dfrac{v'xV}{c^2}}\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-847bdb9efa731600ec651be70e335ad4_l3.png "Rendered by QuickLaTeX.com")
![\[ v_z=\dfrac{v'_z\sqrt{1-\beta^2}}{1+\dfrac{v'zV}{c^2}}\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-ee47865945c41a73ee697b2c7b605584_l3.png "Rendered by QuickLaTeX.com")
The classical result (galilean addition of velocities) is recovered in the limit of low velocities 
or sending the light speed get the value “infinite” 
. Then,
![\[ v_x=v'_x+V\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-cb1df20c57c2a37719f3f53d4eda086d_l3.png "Rendered by QuickLaTeX.com")
![\[v_y=v'_y\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-7362993a01a0a7fced222b3536150bc8_l3.png "Rendered by QuickLaTeX.com")
![\[v'_z=v_z\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-e1ba1413a51a861300f78be5b28d10f8_l3.png "Rendered by QuickLaTeX.com")
Let us define
![\[ \theta =\mbox{angle formed by x}\; \mbox{and}\; v_x\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-72d0790692c819ca6780d1efbb30c6c2_l3.png "Rendered by QuickLaTeX.com")
![\[ \theta' =\mbox{angle formed by x'}\; \mbox{and}\; v'_x\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-ac8cdafdbfc4d2f74f57c30c94b5cc14_l3.png "Rendered by QuickLaTeX.com")
Thus, we get the component decomposition into the xy and x’y’ planes:
![\[ v_x=v\cos\theta\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-dc9885417e661a10f9b9db116dc7c198_l3.png "Rendered by QuickLaTeX.com")
![\[v_y=v\sin\theta\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-9bc6a883d8becc33da9b0875fbbe47d6_l3.png "Rendered by QuickLaTeX.com")
![\[ v'_x=v'\cos\theta'\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-73624170c9b9da259d581612d040bc5c_l3.png "Rendered by QuickLaTeX.com")
![\[ v'_y=v'\sin\theta'\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-ff4ffa5d5b6da8a528eea4e00d65d051_l3.png "Rendered by QuickLaTeX.com")
From this equations, we get
![\[ \tan \theta=\dfrac{v'\sin\theta'\sqrt{1-\beta^2}}{v'\cos\theta'+V}\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-ee800e0c982eceeed4fad3b03d5f51ea_l3.png "Rendered by QuickLaTeX.com")
If 
![\[ \boxed{\tan \theta=\dfrac{\sin\theta'\sqrt{1-\beta_V^2}}{\cos\theta'+\beta_V}}\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-52ac93aead64b76857cd14a7ea0ef415_l3.png "Rendered by QuickLaTeX.com")
and then
![\[ \boxed{\cos\theta=\dfrac{\beta_V+\cos\theta'}{1+\beta_V\cos\theta'}}\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-a59653d9a6599a595b3c43c8c49fbe4f_l3.png "Rendered by QuickLaTeX.com")
![\[ \boxed{\sin\theta=\dfrac{\sin\theta'\sqrt{1-\beta_V^2}}{1+\beta_V\cos\theta'}}\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-5f3bd90b8da23c664f66039277148b22_l3.png "Rendered by QuickLaTeX.com")
From the last equation, we get
![\[ \sin\theta'\sqrt{1-\beta_V^2}=\sin\theta\left(1+\beta_V\cos\theta'\right)\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-9de41f10841e317684578f466914f4c1_l3.png "Rendered by QuickLaTeX.com")
From this equation, if 
, i.e., if 
and 
with 
, we obtain the result
![\[ \Delta \theta=\theta'-\theta=\beta_V\sin\theta\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-9fe1d74011c4560b8dd2fcf4f5ede764_l3.png "Rendered by QuickLaTeX.com")
By these formalae, the angle of a light beam propagating in space depends on the velocity of the source respect to the observer. We can observe this relativistic effect every night (supposing a good approximation that Earth’s velocity is non-relativistic, as it shows). The physical interpretation of the above aberration formulae (for the stars we watch during a sky night) is as follows: due to the Earth’s motion, a star in the zenith is seen under an angle 
.
Other important consequence from the stellar aberration is when we track ultra-relativistic particles (
). Then, 
and then, the observer moves close to the source of light. In this case, almost every star (excepting those behind with 
) are seen “in front of” the observer. If the source moves with almost the speed of light, then the light is “observed” as it were concentrated in a little cone with an aperture 
.
