In physics, relativistic beaming (also known as Doppler beaming, Doppler boosting, or the headlight effect) is the process by which relativistic effects modify the apparent luminosity of emitting matter that is moving at speeds close to the speed of light. In an astronomical context, relativistic beaming commonly occurs in two oppositely-directed relativistic jets of plasma that originate from a central compact object that is accreting matter. Accreting compact objects and relativistic jets are invoked to explain x-ray binaries, gamma-ray bursts, and, on a much larger scale, active galactic nuclei (of which quasars are a particular variety).
Beaming affects the apparent brightness of a moving object. Consider a cloud of gas moving relative to the observer and emitting electromagnetic radiation. If the gas is moving towards the observer, it will be brighter than if it were at rest, but if the gas is moving away, it will appear fainter. The magnitude of the effect is illustrated by the AGN jets of the galaxies M87 and 3C 31 (see images at right). M87 has twin jets aimed almost directly towards and away from Earth; the jet moving towards Earth is clearly visible (the long, thin blueish feature in the top image), while the other jet is so much fainter it is not visible.[1] In 3C 31, both jets (labeled in the lower figure) are at roughly right angles to our line of sight, and thus, both are visible. The upper jet actually points slightly more in Earth's direction and is therefore brighter.[2]
Relativistically moving objects are beamed due to a variety of physical effects. Light aberration causes most of the photons to be emitted along the object's direction of motion. The Doppler effect changes the energy of the photons by red- or blue-shifting them. Finally, time intervals as measured by clocks moving alongside the emitting object are different from those measured by an observer on Earth due to time dilation and photon arrival time effects. How all of these effects modify the brightness, or apparent luminosity, of a moving object is determined by the equation describing the relativistic Doppler effect (which is why relativistic beaming is also known as Doppler beaming).
The simplest model 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, but it does illustrate the physical process of beaming quite well.
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
log\nu=3
log\nu=7
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
logS
log\nu
In the simple jet model of a single homogeneous sphere the observed luminosity is related to the intrinsic luminosity as
So=SeDp,
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
The beaming equation can be broken down into a series of three effects:
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
c
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
D2.
Time dilation is a well-known consequence of special relativity and accounts for a change in observed luminosity of
D1.
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
| 1 | |
| D\alpha |
.
A more-sophisticated method of deriving the beaming equations starts with the quantity
| S | |
| \nu3 |
\theta
vj
Se
So
\alpha
S\propto\nu\alpha
c=2.9979 x 108
\beta=
| vj | |
| c |
\gamma=
| 1 | |
| \sqrt{1-\beta2 |
D=
| 1 | |
| \gamma(1-\beta\cos\theta) |