LOG#034. Stellar aberration.

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<<c, 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 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 l \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}.

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