## FANDOM

153 Pages

Consider a source that is emitting isotropically in its own rest frame (denoted by primes):

$\frac{dL'}{d\Omega'}=\frac{L'}{4\pi}$

where $d\Omega'$ is a unit solid angle and $L'$ is the luminosity.

Consider an observer moving relativistically at a speed $\beta c$ compared to the source and at an angle $\theta$ above the line of sight. If the source emits two photons at a time difference $\Delta t$ in the observer frame. The difference in their arrival times will be:$\Delta t _{obs}=(1-\beta cos(\theta))\Delta t$.

The difference of their emission times in the source frame is:

$\Delta t'=\frac{\Delta t}{\Gamma}=\frac{\Delta t _{obs}}{\Gamma(1-\beta cos(\theta))}=D \Delta t _{obs}$

where $\Gamma$ is the Lorentz factor of the source and $D$ is the Doppler factor.

Since the phase difference of an electromagnetic wave must be relativistically invariant: $\Delta \Phi=\omega' \Delta t'=\omega_{obs} \Delta t_{obs}$, we find that:

$D=\frac{\nu}{\nu'}=\frac{\omega_{obs}}{\omega'}=\frac{\Delta t'}{\Delta t_{obs}}$.

Now, consider a photon emitted by the source towards the observer. Its speed is:

$\beta_x=cos(\theta) \equiv \mu$ in the observer frame and $\beta_x'=cos(\theta') \equiv \mu'$ in the source frame. Using the relativistic transformation of velocities, we obtain:

$\mu= \frac{\mu'+\beta}{1+\beta \mu'}$

From which it follows that: $d\mu=D^{-2} d\mu'$. Since $d\phi=d\phi'$ we get that: $d\Omega=D^{-2} D\Omega'$.

Finally we can transform the luminosity from the source to the observer frame:$L_{\nu}\equiv\frac{dL}{d\nu}=\frac{4\pi dE}{dt d\nu d\Omega}=L'_{\nu'}\frac{dE d\Omega}{dE' d\Omega'}=D^3 L'_{\nu'}$.

Notice that $D$ is maximal for $\theta=0$ for which $D\approx 2\Gamma$ (assuming $\Gamma \gg 1$). In addition, for $\theta=1/\Gamma$ we get $cos (\theta)=\beta$ and:

$D=\frac{1}{\Gamma(1-\beta^2)}=\frac{\sqrt{1-\beta^2}}{1-\beta^2}=\frac{1}{\sqrt{1-\beta^2}}=\Gamma$.

So at this angle the luminosity drops by a factor $2^3=8$ compared to that for the forward direction. However, for significantly larger angles, $1-\beta cos(\theta) \ll 1$ and the Doppler factor drops rapidly leading to an even sharper drop in luminsoity.

We conclude that the emission that was isotropic in the source frame, becomes confined to a cone of opening angle $1/\Gamma$ around the direction of motion of the source in the observer frame. This effect is known as relativistic beaming, and may lead to various selection effects when observing relativistic sources.