# What is the bulk Lorentz factor?

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When we are talking about relativistic motions, say from a relativistic jet, what is meant by the term "bulk Lorentz factor", and the bulk speed, $eta$?

I think it is referring to the speed and Lorentz factor $(eta = v/c$ and $gamma = [1-eta^2]^{-1/2})$ of the gas as a whole. Within the gas, there could be particles moving with a variety of velocities.

So if you pick up a ball of gas at 10,000 K (ouch) and throw it at 100 m/s then the bulk speed is 100 m/s, but obviously the particles in the gas have their own individual velocities.

## What is the bulk Lorentz factor? - Astronomy

### Abstract

We use a sample of radio-loud Active Galactic Nuclei (AGNs) with measured black hole masses to explore the jet formation mechanisms in these sources. We find a significant correlation between black hole mass and the bulk Lorentz factor of the jet components for this sample, while no significant correlation is present between the bulk Lorentz factor and the Eddington ratio. Recent investigations suggested that the most super-massive black holes in elliptical galaxies have on average higher spins than the black holes in spiral galaxies. The correlation between black hole mass and bulk Lorentz factor of the jet components found in this work implies that the motion velocity of the jet components is probably governed by the black hole spin. The faster moving jets are magnetically accelerated by the magnetic fields threading the horizon of more rapidly rotating black holes

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## Lorentz transformation derivation

The first derivation is similar to here.

Lorentz transformations for space with time

Let unprimed x and t be from inertial frame K and primed x′ and t′ be from inertial frame K′. Since space is assumed to be homogeneous, the transformation must be linear. The most general linear relationship is obtained with four constant coefficients, A, B, C, and D:

A body at rest in the K′ frame at position x′ = 0 moves with constant velocity v in the K frame. Hence the transformation must yield x′ = 0 if x = vt. Therefore, B = −Av and the first equation becomes

Using the principle of relativity

According to the principle of relativity, there is no privileged Galilean frame of reference: therefore the inverse transformation for the position from frame K′ to frame K should have the same form as the original but with the velocity in the opposite direction, i.e., replacing v with −v:

Determining the constants of the first equation

Since the speed of light is the same in all frames of reference, for the case of a light signal, the transformation must guarantee that t = x/c when t′ = x′/c, with speed of light c. Substituting for t and t′ in the preceding equations gives:

Multiplying these two equations together gives,

At any time after t = t′ = 0, xx′ is not zero, so dividing both sides of the equation by xx′ results in

which is the “Lorentz factor”.

When the transformation equations are required to satisfy the light signal equations in the form x = ct and x′ = ct′, by substituting the x and x′-values, the same technique produces the same expression for the Lorentz factor.

Determining the constants of the second equation

The transformation equation for time can be easily obtained by considering the special case of a light signal, again satisfying x = ct and x′ = ct′, by substituting term by term into the earlier obtained equation for the spatial coordinate

which determines the transformation coefficients C and D as

So C and D are the unique constant coefficients necessary to preserve the constancy of the speed of light in the primed system of coordinates.

Lorentz transformations for time with space

Let unprimed x and t be from the timeframe K and primed x′ and t′ be from the timeframe K′. Since time is assumed to be homogeneous, the transformation must be linear. The most general linear relationship is obtained with four constant coefficients, A, B, C, and D:

A body at rest in the K′ frame at position x′ = 0 moves with constant lenticity w in the K frame. Hence the transformation must yield x′ = 0 if x = t/w. Therefore, B = −A/w and the first equation becomes

Using the principle of relativity

According to the principle of relativity, there is no privileged Galilean frame of reference: therefore the inverse transformation for the position from frame K′ to frame K should have the same form as the original but with the lenticity in the opposite direction, i.e., replacing w with −w:

Determining the constants of the first equation

Since the speed of light is the same in all frames of reference, for the case of a light signal, the transformation must guarantee that t = kx when t′ = kx′, with pace of light k. Substituting for t and t′ in the preceding equations gives:

Multiplying these two equations together gives,

At any time after t = t′ = 0, xx′ is not zero, so dividing both sides of the equation by xx′ results in

which is the “Lorentz factor”.

When the transformation equations are required to satisfy the light signal equations in the form x = t/k and x′ = ct′/k, by substituting the x and x′-values, the same technique produces the same expression for the Lorentz factor.

Determining the constants of the second equation

The transformation equation for time can be easily obtained by considering the special case of a light signal, again satisfying x = t/k and x′ = t′/k, by substituting term by term into the earlier obtained equation for the spatial coordinate

which determines the transformation coefficients C and D as

So C and D are the unique constant coefficients necessary to preserve the constancy of the speed of light in the primed system of coordinates.

## 2. ONE-ZONE CASE

Consider a GRB ejecta with bulk LF Γ and radius R. Assume that the photons in the comoving frame of the ejecta are isotropic, with photon number density per photon energy dn'/d'. Hereafter, unless specified otherwise, quantities with prime signs denote the comoving frame and quantities without prime signs the frame of observer on the Earth.

In the (comoving-frame) dynamical time Rc, a photon travels a path of R/Γ. For a photon of energy ε' = ε(1 + z)/Γ (with z the GRB redshift), the optical depth due to γγ collisions during a dynamical time is given by Gould & Schréder (1967)

where and Θ' is the angle between the colliding photon pair. The cross-section is given by

where and are the velocity and energy, respectively, of the generated electron in the center of the momentum frame of the collision. The radius R of the emission region can be related to the angular spreading time δtang, due to a geometric effect, by R = 2Γ 2 cδtang/(1 + z). As the angular spreading time is related to the observed variability time δt by δtang = δt, we have

For a GRB with the observed photon number per unit time per unit photon energy per unit detector area, denoted by N(), the photon number density per unit photon energy in the comoving frame can be given by

where dL is the GRB luminosity distance, and = Γ'/(1 + z).

It is important to note a difference from previous works. In Equation (1), we did not take the upper limit of the integration to be infinity but a certain photon energy ε'max , because the HE tail is expected to be cut off due to γγ absorption. The cutoff energy is just where τ(ε'max ) = 1 happens. To self-consistently solve for the cutoff energy εmax = Γε'max /(1 + z) for given Γ, we need to take the upper limit of the integration to be ε'max , and solve τ(ε'max ) = 1 using Equations (1)–(4) and observed GRB spectrum N().

It is well known that the GRB spectrum can be fitted by the Band function (Band et al. 1993)

where c = p(α − β)/(2 + α), and A, α, β and p are the normalized coefficient, low-energy slope, the HE slope, and the νFν peak energy, respectively. In some Fermi-LAT GRBs, an extra spectral component beyond the Band function is claimed to exist, especially in the HE end (Abdo et al. 2009b, 2009c). This extra component can be described as a power law,

with APL, the normalization at 1 GeV and βPL the spectral index.

It is helpful to solve the Γ–εmax relation with some approximations first. Typically, the HE, 100 MeV, photons mainly interact with photons above the peak energy. Let us approximate the target photon distribution as a single power-law N() = N0s in the following analytical derivation.

In Equation (1), the upper limit of the first integral is usually taken to be . This is valid for εmax Γ 2 m 2 ec 4 /[εmax (1 + z) 2 ] and the spectrum slope s > 1. In this case, using δ-approximation for the crosssection at target photon energy above the threshold, σ ≈ (3/16)σT, τ(εmax ) = 1 can be solved to give Γ as function of εmax ,

However, when εmax Γ 2 m 2 ec 4 /[εmax (1 + z) 2 ], i.e., the energy of annihilated photons is compared with that of target photons, the upper limit cannot be taken as any more. In this case, Γ is given by Li (2010)

Next, we carry out a numerical calculation to solve τ(εmax ) = 1. For the observations, we take the three bright Fermi-LAT GRBs 080916C, 090510, and 090902B, and consider the same time intervals in the GRBs where the LFs have been constrained by Abdo et al. (2009a, 2009b, 2009c), as well as section "a" in GRB 080916C. The properties of spectra and flux for these GRBs are shown in Table 1. The calculated results are given in Figure 1, where we compare the results of the self-consistent calculation and the previous method using a target photon spectrum without the HE cutoff. We see that the results deviate from each other for εmax 100 MeV or Γ a few hundreds. In the case of section "a" in GRB 080916C, where the maximum observed photon energy is lower (see Figure 1), the LF limit with the self-consistent calculation is much smaller than that with the previous method. Thus, to be self consistent, the upper bound of the integration in Equation (1) should be carefully taken as the maximum photon energy in this case. We also note that the LF constraints using an upper limit of infinity are still valid for those time segments that have been used by Abdo et al. (2009a, 2009b, 2009c).

Figure 1. Relation between the observed maximum photon energy and the lower limit to the bulk LF in the one-zone case for the three bright GRBs. The adopted parameters of the GRBs are shown in Table 1. As marked in the plot, the dashed lines correspond to results using target photons without a spectral cutoff, while the solid lines correspond to our self-consistent calculations using truncated target photon spectra. The stars denote the observed highest energy of photons in the relevant time intervals.

Table 1. Parameters of Three Bright LAT-GRBs

GRB Name Time Interval p α β A βPL APL z δt εhighest
(s) (keV) (cm −2 s −1 keV −1 ) (cm −2 s −1 keV −1 ) (ms) (GeV)
GRB 080916C-a 0.004–3.58 440 −0.58 −2.63 0.055 ⋅⋅⋅ ⋅⋅⋅ 4.35 100 a 0.02 b
GRB 080916C-b 3.58–7.68 1170 −1.02 −2.21 0.035 ⋅⋅⋅ ⋅⋅⋅ 4.35 100 a 3
GRB 090510 0.8–0.9 1894 −0.86 −3.09 0.028 −1.54 6.439 × 10 −9 0.903 12 30.5
GRB 090902B 9.6–13 821 −0.26 −5.0 0.082 −1.98 4.3 × 10 −10c 1.822 53 11.2

Notes. a Greiner et al. (2009). b The spectral flux of this section at >20 MeV is only an upper limit, as shown in the supporting material of Abdo et al. (2009a). c Private communication with Francesco de Palma the other parameters are taken from Abdo et al. (2009a, 2009b, 2009c).

## Numerical values

In the table below, the left-hand column shows speeds as different fractions of the speed of light (i.e. in units of c). The middle column shows the corresponding Lorentz factor, the final is the reciprocal. Values in bold are exact.

Speed (units of c) Lorentz factor Reciprocal
0.000 1.000 1.000
0.050 1.001 0.999
0.100 1.005 0.995
0.150 1.011 0.989
0.200 1.021 0.980
0.250 1.033 0.968
0.300 1.048 0.954
0.400 1.091 0.917
0.500 1.155 0.866
0.600 1.250 0.800
0.700 1.400 0.714
0.750 1.512 0.661
0.800 1.667 0.600
0.866 2.000 0.500
0.900 2.294 0.436
0.990 7.089 0.141
0.999 22.366 0.045
0.99995 100.00 0.010

## Astronomers Observe Bright Relativistic Jet from Distant Blazar

VLBA image of PSO J0309+27. Image credit: Spingola et al. / Bill Saxton / NRAO / AUI / NSF.

Blazars are active galactic nuclei with relativistic jets of matter speeding towards us almost head-on at the speed of light.

They glow not just with visible light, but with every kind of radiation, from radio waves to gamma rays.

Despite intensive observational and theoretical efforts over several decades, the details of blazar astrophysics remain elusive.

Discovered in 2019, PSO J0309+27 is located some 12.8 billion light-years away from Earth.

The object is seen as it was when the Universe was less than a billion years old, or just over 7% of its current age.

It is the brightest radio-emitting blazar yet seen at such a distance. It also is the second-brightest X-ray emitting blazar at such a distance.

In new research, University of Bologna astronomer Cristiana Spingola and colleagues carried out follow-up observations of PSO J0309+27 with the Karl G. Jansky Very Large Array (VLA) and the Very Long Baseline Array (VLBA).

“In the new image, the brightest radio emission comes from the galaxy’s core, at bottom right,” the researchers said.

“The jet is propelled by the gravitational energy of a supermassive black hole at the core, and moves outward, toward the upper left.”

“The jet seen here extends some 1,600 light-years, and shows structure within it.”

The analysis of PSO J0309+27’s properties provides support for some theoretical models for why blazars are rare in the early Universe.

“If PSO J0309+27 is a genuine blazar, as suggested by its X-ray properties, then we find that its bulk Lorentz factor must be relatively low,” the scientists wrote in their paper.

“This value would be in favor of a scenario currently proposed to reconcile the paucity of high-redshift blazars with current predictions.”

“Nevertheless, we cannot exclude that PSO J0309+27 is seen under a larger viewing angle, which would imply that the X-ray emission must be enhanced, for example, by inverse Compton scattering with the Cosmic Microwave Background.”

“More stringent constraints on the bulk Lorentz factor in PSO J0309+27 and on these factors in the other high-redshift blazars are necessary to test whether their properties are intrinsically different from those of the low-redshift blazar population.”

The findings were published in the journal Astronomy & Astrophysics.

C. Spingola et al. 2020. Parsec-scale properties of the radio brightest jetted AGN at z > 6. A&A 643, L12 doi: 10.1051/0004-6361/202039458

## What Governs Lorentz Factors of Jet Components in Blazars?

We use a sample of radio-loud Active Galactic Nuclei (AGNs) with measured black hole masses to explore the jet formation mechanisms in these sources. We find a significant correlation between black hole mass and the bulk Lorentz factor of the jet components for this sample, while no significant correlation is present between the bulk Lorentz factor and the Eddington ratio. Recent investigations suggested that the most super-massive black holes in elliptical galaxies have on average higher spins than the black holes in spiral galaxies. The correlation between black hole mass and bulk Lorentz factor of the jet components found in this work implies that the motion velocity of the jet components is probably governed by the black hole spin. The faster moving jets are magnetically accelerated by the magnetic fields threading the horizon of more rapidly rotating black holes.

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## SOLUTIONS OF THE WIND EQUATION IN RELATIVISTIC MAGNETIZED JETS

We study the bulk acceleration in relativistic axisymmetric magnetized outflows, by solving the momentum equation along the flow, the so-called wind equation. The solutions for the bulk Lorentz factor depend on the geometry of the field/streamlines through the "bunching function" S. We investigate the general characteristics of the S function and how its choice affects the acceleration. In our study, various fast rise and slow decay examples are selected for S, with a global maximum near the fast magnetosonic critical point, as required from the regularity condition. For each case we determine the terminal Lorentz factor γ and the acceleration efficiency γ/μ, where μ is the total energy-to-mass flux ratio (which equals the maximum possible Lorentz factor of the outflow). With proper choices of S we can achieve efficiencies greater than 50%. Last, we examine the shape of the field/streamlines with respect to the choice of the S function. The results of this work, depending on the choices of μ, can be applied to relativistic GRB or AGN jets.

This is an Open Access article published by World Scientific Publishing Company. It is distributed under the terms of the Creative Commons Attribution 3.0 (CC-BY) License. Further distribution of this work is permitted, provided the original work is properly cited.

## Doppler factor, Lorentz factor and viewing angle of superluminal quasars

We carried out the investigation of the properties of features seen within superluminal sources often referred to as components. Our result indicates a fairly strong correlation of r∼0.76 between component radial distance L and component size ℜ. Assumption of simple ballistic motion and free adiabatic expansion, enabled us to use the observed jet component parameters to constrain the Doppler factor, Lorentz factor and the lower limit to the viewing angle with respect to a distant observer. The estimated average Doppler factor, Lorentz factor and viewing angle respectively are 10.3±5.0, 18.3±6.2 and 3.7±2.3 for Γ=4/3 while the values obtained for Γ=5/3 are 12.2±5.9,17.2±5.1 and 2.9±1.6, where Γ is the adiabatic index. The large scatter in our results may be due to the uncertainties introduced by the assumptions made.

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## The terminal bulk Lorentz factor of relativistic electron–positron jets

We present a numerical simulation of the bulk Lorentz factor of a relativistic electron–positron jet driven by the Compton rocket effect from accretion disc radiation. The plasma is assumed to have a power-law distribution ne(γ) ∝ γ −s with 1 < γ < γmax and is continuously reheated to compensate for radiation losses. We include the full Klein–Nishina (hereafter KN) cross-section, and study the role of the energy upper cut-off γmax, spectral index s and source compactness. We determine the terminal bulk Lorentz factor in the cases of supermassive black holes, relevant to AGN, and stellar black holes, relevant to galactic microquasars. In the latter case, Klein–Nishina cross-section effects are more important and induce a terminal bulk Lorentz factor smaller than in the former case. Our result are in good agreement with bulk Lorentz factors observed in Galactic (GRS 1915+105, GRO J1655−40) and extragalactic sources. Differences in scattered radiation and acceleration mechanism efficiency in the AGN environment can be responsible for the variety of relativistic motion in those objects. We also take into account the influence of the size of the accretion disc if the external radius is small enough, the bulk Lorentz factor can be as high as 60.