# LOG#034. Stellar aberration.

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:

$v_x=\dfrac{v'_x+V}{1+\dfrac{v'_xV}{c^2}}$

$v_y=\dfrac{v'_y\sqrt{1-\beta^2}}{1+\dfrac{v'xV}{c^2}}$

$v_z=\dfrac{v'_z\sqrt{1-\beta^2}}{1+\dfrac{v'zV}{c^2}}$

The classical result (galilean addition of velocities) is recovered in the limit of low velocities $V\approx 0$ or sending the light speed get the value “infinite” $c\rightarrow \infty$. Then,

$v_x=v'_x+V$ $v_y=v'_y$ $v'_z=v_z$

Let us define

$\theta =\mbox{angle formed by x}\; \mbox{and}\; v_x$

$\theta' =\mbox{angle formed by x'}\; \mbox{and}\; v'_x$

Thus, we get the component decomposition into the xy and x’y’ planes:

$v_x=v\cos\theta$ $v_y=v\sin\theta$

$v'_x=v'\cos\theta'$ $v'_y=v'\sin\theta'$

From this equations, we get

$\tan \theta=\dfrac{v'\sin\theta'\sqrt{1-\beta^2}}{v'\cos\theta'+V}$

If $v=v'=c$

$\boxed{\tan \theta=\dfrac{\sin\theta'\sqrt{1-\beta_V^2}}{\cos\theta'+\beta_V}}$

and then

$\boxed{\cos\theta=\dfrac{\beta_V+\cos\theta'}{1+\beta_V\cos\theta'}}$

$\boxed{\sin\theta=\dfrac{\sin\theta'\sqrt{1-\beta_V^2}}{1+\beta_V\cos\theta'}}$

From the last equation, we get

$\sin\theta'\sqrt{1-\beta_V^2}=\sin\theta\left(1+\beta_V\cos\theta'\right)$

From this equation, if $V<, i.e., if $\beta_V<<1$ and $\theta'=\theta+\Delta\theta$ with $\Delta \theta<<1$, we obtain the result

$\Delta \theta=\theta'-\theta=\beta_V\sin\theta$

By these formaulae, 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 skynight) is as follows: due to the Earth’s motion, a star in the zenith is seen under an angle $\theta\neq \dfrac{\pi}{2}$.

Other important consequence from the stellar aberration is when we track ultra-relativistic particles ($\beta\approx 1$). Then, $\theta'\rightarrow \pi$ and then, the observer moves close to the source of light. In this case, almost every star (excepting those behind with $\theta=\pi$) 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 $\Delta\theta\sim\sqrt{1-\beta_V^2}$