Relativistic beaming

Relativistic jets emit most of their energy via synchrotron emission. In our simple model the sphere contains highly relativistic electrons and a steady magnetic field. Electrons inside the blob travel at speeds just a tiny fraction below the speed of light and are whipped around by the magnetic field. Each change in direction by an electron is accompanied by the release of energy in the form of a photon. With enough electrons and a powerful enough magnetic field the relativistic sphere can emit a huge number of photons, ranging from those at relatively weak radio frequencies to powerful X-ray photons.

The figure of the sample spectrum shows basic features of a simple synchrotron spectrum. At low frequencies the jet sphere is opaque and its luminosity increases with frequency until it peaks and begins to decline. In the sample image this peak frequency occurs at ${\displaystyle \log \nu =3}$. At frequencies higher than this the jet sphere is transparent. The luminosity decreases with frequency until a break frequency is reached, after which it declines more rapidly. In the same image the break frequency occurs when ${\displaystyle \log \nu =7}$. The sharp break frequency occurs because at very high frequencies the electrons which emit the photons lose most of their energy very rapidly. A sharp decrease in the number of high energy electrons means a sharp decrease in the spectrum.

The changes in slope in the synchrotron spectrum are parameterized with a spectral index. The spectral index, α, over a given frequency range is simply the slope on a diagram of ${\displaystyle \log S}$ vs. ${\displaystyle \log \nu }$. (Of course for α to have real meaning the spectrum must be very nearly a straight line across the range in question.)

In the simple jet model of a single homogeneous sphere the observed luminosity is related to the intrinsic luminosity as

${\displaystyle S_{o}=S_{e}D^{p}\,,}$

where

${\displaystyle p=3-\alpha \,.}$

The observed luminosity therefore depends on the speed of the jet and the angle to the line of sight through the Doppler factor, ${\displaystyle D}$, and also on the properties inside the jet, as shown by the exponent with the spectral index.

The beaming equation can be broken down into a series of three effects:

• Relativistic aberration
• Time dilation
• Blue- or redshifting

Aberration is the change in an object's apparent direction caused by the relative transverse motion of the observer. In inertial systems it is equal and opposite to the light time correction.

In everyday life aberration is a well-known phenomenon. Consider a person standing in the rain on a day when there is no wind. If the person is standing still, then the rain drops will follow a path that is straight down to the ground. However, if the person is moving, for example in a car, the rain will appear to be approaching at an angle. This apparent change in the direction of the incoming raindrops is aberration.

The amount of aberration depends on the speed of the emitted object or wave relative to the observer. In the example above this would be the speed of a car compared to the speed of the falling rain. This does not change when the object is moving at a speed close to ${\displaystyle c}$. Like the classic and relativistic effects, aberration depends on: 1) the speed of the emitter at the time of emission, and 2) the speed of the observer at the time of absorption.

In the case of a relativistic jet, beaming (emission aberration) will make it appear as if more energy is sent forward, along the direction the jet is traveling. In the simple jet model a homogeneous sphere will emit energy equally in all directions in the rest frame of the sphere. In the rest frame of Earth the moving sphere will be observed to be emitting most of its energy along its direction of motion. The energy, therefore, is ‘beamed’ along that direction.

Quantitatively, aberration accounts for a change in luminosity of

${\displaystyle D^{2}\,.}$

Time dilation is a well-known consequence of special relativity and accounts for a change in observed luminosity of

${\displaystyle D^{1}\,.}$

Blue- or redshifting can change the observed luminosity at a particular frequency, but this is not a beaming effect.

Blueshifting accounts for a change in observed luminosity of

${\displaystyle {\frac {1}{D^{\alpha }}}\,.}$

A more-sophisticated method of deriving the beaming equations starts with the quantity ${\displaystyle {\frac {S}{\nu ^{3}}}}$. This quantity is a Lorentz invariant, so the value is the same in different reference frames.

angle to the line-of-sight with Earth

${\displaystyle \theta \,\!}$

jet speed

${\displaystyle v_{j}\,\!}$

intrinsic luminosity

${\displaystyle S_{e}\,\!}$ (sometimes called emitted luminosity)

observed Luminosity

${\displaystyle S_{o}\,\!}$

spectral index

${\displaystyle \alpha \,\!}$ where ${\displaystyle S\propto \nu ^{\alpha }\,\!}$

Speed of light

${\displaystyle c\,\!=2.9979\times 10^{8}}$ m/s

relativistic beta

${\displaystyle \beta ={\frac {v_{j}}{c}}}$

Lorentz factor

${\displaystyle \gamma ={\frac {1}{\sqrt {1-\beta ^{2}}}}}$

Doppler factor

${\displaystyle D={\frac {1}{\gamma (1-\beta \cos \theta )}}}$