Editing Relativistic beaming (section)

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

<math display="block">S_o = S_e D^p\,,</math>

where
<math display="block">p = 3 - \alpha\,.</math>

The observed luminosity therefore depends on the speed of the jet and the angle to the line of sight through the Doppler factor, <math>D</math>, 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 ====
Aberration is the change in an object's [[Aberration of light#Apparent and true positions|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 <math>c</math>. 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

<math display="block">D^2.</math>

==== Time dilation ====
Time dilation is a well-known consequence of [[special relativity]] and accounts for a change in observed luminosity of
<math display="block">D^1.</math>

==== Blue- or redshifting ====
[[Blueshift|Blue-]] or [[Redshift|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

<math display="block">\frac{1}{D^\alpha}.</math>

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