Relativistic beaming

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Relativistic beaming is the process by which the relativistic effect modifies the apparent luminosity of a relativistic jet. Beaming is common in many Active Galactic Nuclei (AGN) galaxies. When relativistic beaming occurs in such galaxies, a central supermassive black hole is the ultimate source of energy for twin jets of intensely energetic plasma. Relativistic beaming is important in astronomy to understanding the nature and evolution of galaxies.

Relativistic beaming is of particular importance for a jet which is oriented close to the line of sight from the AGN to Earth. In the simplest case the jet can be considered to be a series of spherical clouds or blobs, each emitting a high luminosity. In the rest frame of the Earth each blob is approaching at speeds which can be in the range of 95% to 98% of the speed of light and, because of Special Relativity and some physical properties of the jet, the luminosity observed on Earth will be higher than the intrinsic luminosity measured in the rest frame of the jet clouds.

At very small angles to the line of sight such beaming effects become quite large and the observed luminosity of a jet may be a few hundred or a thousand times higher than the intrinsic luminosity. The jet on the other side, however, moving away from Earth at relativistic speeds is affected quite differently. The observed luminosity from this “counter-jet” is less than the intrinsic luminosity.

Relativistic beaming effects can greatly alter the appearance of a distant galaxy. Two AGN which are identical except for different angles to the line of sight can seem to us on Earth to be very different, unrelated galaxies. This is, in fact, what seems to be the case with blazars.

1 A simple jet model
- 1.1 Synchrotron spectrum and the spectral index
- 1.2 Beaming equation
2 Terminology
- 2.1 Physical Quantities
- 2.2 Mathematical Expressions
3 External links

[edit] A simple jet model

The simplest mode for a jet is one where a single, homogeneous sphere is travelling towards the earth at nearly the speed of light. This simple model is also an unrealistic one, although it does illustrate the physical process of beaming quite well.

Simple Jet Model. A simple model for a jet is a relativistic sphere emitting synchrotron radiation. This simple model hides the complexity of a real jet but can still be used to illustrate the principles of relativistic beaming.

[edit] Synchrotron spectrum and the spectral index

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.

Synchrotron Spectrum. A sketch of the synchrotron spectrum from a homogeneous, non-thermal source of electrons in non-regular magnetic fields.

The figure of the sample spectrum shows basic features of a simple synchrotron spectrum. At low frequencies the jet sphere is opaque. The amount luminosity increases with frequency until it peaks and begins to decline. In the sample image this peak frequency occurs at $logν = 3$ . 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 $logν = 7$ . 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 $log S$ vs. $logν$ . (Of course for α to have real meaning the spectrum must be very nearly a straight line across the range in question.)

[edit] Beaming equation

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

$S_o = S_e D^p\,,$

where

$p = 3 - \alpha\,.$

The observed luminosity therefore depends on the speed of the jet and the angle to the line of sight through the Doppler factor, $D$ , 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 Red-) shifts

[edit] Relativistic and nonrelativistic aberration

Relativistic Aberration. Photons emitted by an object moving close to the speed of light will appear to be emitted in a cone around the direction of motion. When viewed along this direction, the object appears brighter.

Generally aberration is the apparent change of an object's direction of movement relative to an observer's frame of reference. In everyday life, when relative speeds are much less than $c$ (within the 'classical limit') 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 moving at an angle, from the front to back. This change in direction is aberration.

The degree of aberration depends on the relative speed between an object and an 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 $c$ , but it can drastically change the aberration angle.

In the case of a relativistic jet, 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.

Time Dilation.

Quantitatively, aberration accounts for a change in luminosity of

$D^2\,.$

[edit] Time dilation

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

Blueshift.

$D^1\,.$

[edit] Blue- (Red-) shifting

Blue shifting (or red shifting if θ > 90 degrees) can change the observed luminosity at a particular frequency, but this is not a beaming effect.

Blue-shifting accounts for a change in observed luminosity of

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

[edit] Lorentz invariants

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

[edit] Terminology

beamed, beaming: shorter terms for ‘relativistic beaming’
beta: the ratio of the jet speed to the speed of light, sometimes called ‘relativistic beta’
core: region of a galaxy around the central black hole
counter-jet: the jet on the far side of a source oriented close to the line of sight, can be very faint and difficult to observe
Doppler factor: a mathematical expression which measures the strength (or weakness) of relativistic effects in AGN, including beaming, based on the jet speed and its angle to the line of sight with Earth
flat spectrum: a term for a non-thermal spectrum that emits a great deal of energy at the higher frequencies when compared to the lower frequencies
intrinsic luminosity: the luminosity from the jet in the rest frame of the jet
jet: a relativistic jet of plasma emanating from the polar direction of an AGN
observed luminosity: the luminosity from the jet in the rest frame of Earth
spectral index: of measure of how a non-thermal spectrum changes with frequency, the smaller α is the more significant is the energy at higher frequencies, typically α is in the range of 0 to 2
steep spectrum: a term for a non-thermal spectrum that emits little energy at the higher frequencies when compared to the lower frequencies

[edit] Physical Quantities

angle to the line-of-sight with Earth: $\theta\,\!$
jet speed: $v_j\,\!$
intrinsic luminosity: $S_e\,\!$ (sometimes called emitted luminosity)
observed Luminosity: $S_o\,\!$
spectral index: $\alpha\,\!$ where $S \propto \nu^{\alpha}\,\!$
Speed of light: $c\,\! = 2.9979 \times 10^{8}$ m/s

[edit] Mathematical Expressions

relativistic beta: $\beta = \frac{v_j}{c}$
Lorentz factor: $\Gamma = \frac{1}{\sqrt{1 - \beta^2}}$ (often written as $\gamma\,\!$ and referred to as relativistic gamma)
Doppler factor: $D = \frac{1}{\Gamma (1 - \beta \cos\theta)}$