© ESO, 2013

1. Introduction

A wide variety of processes in relativistic outflows are sensitive to causal connection, or the ability of a disturbance at the edge of an axisymmetric flow to communicate with the symmetry axis¹. In the standard magnetic model of relativistic jet production (Beskin 2010), the global jet structure determines the nature of bulk acceleration, implying that the jet must be causally connected for such acceleration to take place (e.g., Tchekhovskoy et al. 2009; Komissarov et al. 2009). Causal connection is determined by the half opening angle of an axisymmetric flow, θ_j, and the flow Lorentz factor, Γ, through their product Γθ_j, where Γθ_j ≲ 1 implies that the jet is causally connected. Other important aspects of relativistic jet physics that depend on Γθ_j include jet stability (Narayan et al. 2009), magnetic reconnection (Giannios 2013), recollimation shock energy dissipation (Nalewajko & Sikora 2009), and recollimation shock structure (Kohler et al. 2012).

Relativistic outflows have a wide range of values of Γθ_j. On the one hand, pulsar wind nebulae contain uncollimated (equitorial) outflows that are inferred to reach very high bulk Lorentz factors of Γ ~ 10⁶ prior to the termination shock (Kennel & Coroniti 1984), implying that these outflows have values of Γθ_j ~ 10⁶ and are not causally connected. On the other hand, narrow relativistic outflows (i.e., jets) associated with X-ray binaries (XRBs), gamma-ray bursts (GRBs), and active galactic nucleus (AGN) typically have much lower values of Γθ_j; GRB light curve analyses lead to typical inferred values of Γθ_j of 10−30 (Panaitescu & Kumar 2002), while the narrowness and moderate apparent speeds in XRB jets support the assumption that Γθ_j ≲ 1 (e.g., Miller-Jones et al. 2006). Thus, while the central engines of the abovementioned objects are similar in that they involve compact magnetized spinning objects, their different values of Γθ_j suggest that different physical processes are at work.

There have been two past measurements of the characteristic value of Γθ_j for AGN jets. Using 7 mm Very Long Baseline Array (VLBA) data from 15 different AGN jets, Jorstad et al. (2005) measured Γθ_j by assuming that the observed pattern speed of moving jet components corresponds to the jet bulk flow speed, and that the component variability times are equal to the jet frame light crossing times of the resolved components. From these assumptions they determined the component’s Lorentz factor and jet half-opening angle, and found an anti-correlation between the derived values of θ_j and Γ, with Γθ_j = 0.17. With a larger sample of 56 AGN jets from 15 GHz VLBA data, Pushkarev et al. (2009) performed the same analysis, except they used jet parameters from Hovatta et al. (2009), who derived these values from variability time, maximum flux density of flares, and equipartition derived brightness temperature arguments. The Pushkarev et al. (2009) analysis found a similar anti-correlation with Γθ_j = 0.13.

Motivated by the above theoretical concerns, we construct a very different method of inferring Γθ_j by using apparent opening angles obtained from a flux density limited sample of AGN jets. We construct a theoretical probability density function for apparent opening angles in a flux density limited sample, which we derive in Sect. 2, and which has Γθ_j as a free parameter to be fixed by finding the best fit to an empirical distribution of apparent opening angles. Our data consist of the stacked images from 135 AGN jets that make up the MOJAVE-I sample, a 15 GHz flux density limited survey conducted by the VLBA of radio sources in the northern sky with flux densities above 1.5 Jy, and above 2 Jy for sources with − 20 < dec < 0 (Lister et al. 2009a). The stacked images and opening angles are also discussed and analyzed in Pushkarev et al. (2012). After analyzing the data in Sect. 3, we discuss in Sect. 4 the physical significance of the parameter Γθ_j for AGN in the context of jet instabilities, GRB jet acceleration vs. AGN jet acceleration, and jet parameter estimation. We conclude in Sect. 5.

2. Statistical model of θ_app

Here we model a given jet’s value of θ_app as a random variable that is drawn from the probability density function (PDF) P(θ_app). That is, P(θ_app)dθ_app represents the probability that a given jet in a flux density limited sample will have an observed apparent half opening angle between θ_app and θ_app + dθ_app. First, however, we motivate our model by estimating Γθ_j for blazars, and discuss the effect that velocity shear may have on jet appearance.

Blazars are oriented such that the angle between the jet symmetry axis and the line sight, θ_ob, is x/Γ, where x ≈ 0.5 on average in flux density limited samples (Vermeulen & Cohen 1994) such as MOJAVE. This value for x implies an upper limit on Γθ_j of $\begin{array}{ccc}\mathrm{\Gamma }{\mathit{\theta }}_{\mathrm{j}}\lesssim \mathrm{0.5.}& & \end{array}$This is based on the simple argument that most MOJAVE jets are not observed “down the pipe”, (= θ_ob < θ_j), since a down-the-pipe AGN jet would not display jet-like morphology. In fact, most MOJAVE sources do display a jet-like morphology, which implies that typically θ_ob > θ_j (Clausen-Brown et al. 2011), and therefore that 0.5/Γ > θ_j according to the typical value of θ_ob for flux density limited samples. Also, an estimate of Γθ_j can be made, $\begin{array}{ccc}\mathrm{\Gamma }{\mathit{\theta }}_{\mathrm{j}}& & \mathrm{~}\mathrm{0.10}\left(\frac{\mathrm{⟨}{\mathit{\theta }}_{\mathrm{app}}\mathrm{⟩}}{\mathrm{0.2}\mathrm{rad}}\right)\mathrm{(}{\frac{\mathit{x}}{\mathrm{1}\mathit{/}\mathrm{2}}}^{\mathrm{)}}\mathit{,}\end{array}$(1)where ⟨ θ_app ⟩  ≈ 0.2 rad is the average apparent opening angle in the MOJAVE-I sample that we use in this work. From geometrical considerations, as long as all the relevant angles are small, θ_j = θ_obθ_app, thus if ⟨ θ_app ⟩ is used for θ_app and 0.5/Γ for θ_ob, then we obtain Eq. (1). An interesting feature of the above estimate is that it does not significantly depend on the actual value of Γ, which is useful since jets possess a wide range of Γ values (Lister et al. 2009b). In Sect. 2.1 we will make a more rigorous analysis of the likely value of Γθ_j for blazars in which we will also find that this estimate is mostly independent of blazar values of Γ.

A possibility we explore below is that there is a viewing angle effect related to Doppler beaming and velocity shear that affects the appearance of blazars. Velocity shear is included in a variety of AGN jet models, including parsec-scale models (Swain & Bridle 1998; Attridge et al. 1999; Tavecchio & Ghisellini 2008; Perucho et al. 2012), kiloparsec-scale models (e.g. Owen et al. 1989; Swain & Bridle 1998; Perlman et al. 1999; Laing & Bridle 2004), and more general jet models (Aloy et al. 2000; Chiaberge et al. 2000; McKinney 2006). We assume velocity shear affects very long baseline interferometry (VLBI) measurements of a jet’s apparent opening angle θ_app. While all jets may have the same value of Γθ_j, for jets viewed with very small viewing angles the emission might originate from a fast (beamed) narrow spine that is a fraction of the true jet opening angle θ_j, while for jets with larger viewing angles the emission from a slow outer sheath with half-opening angle θ_j may be more detectable. This effect is described in Sect. 2.2.

2.1. Derivation of P(θ_app)

To test the viability of the simplest case scenario, we assume that Γθ_j is constant for all relativistic jets, and that these jets are conical and non-accelerating. In general, however, jets are not conical and the jet flow is either accelerating or decelerating, although in some jet models Γθ_j nevertheless remains constant (e.g., Zakamska et al. 2008). Individual observed jet component acceleration is consistent with only very small changes in Lorentz factor, (Homan et al. 2009), although an individual jet does posses a range of component speeds (Lister et al. 2009b, 2013). Thus, our assumption of conical non-accelerating jets clearly introduces uncertainty to our model.

As we show below, Γθ_j is a free parameter of P(θ_app), and thus will be determined in the fit to the empirical distribution of θ_app. To derive P(θ_app), we first derive the PDF for viewing angles, P(θ_ob). If the sample of AGN jets is unbiased with respect to orientation, P(θ_ob) = sinθ_ob. However, because it is flux density limited, it will take the form $\begin{array}{ccc}\mathit{P}\mathrm{\left(}{\mathit{\theta }}_{\mathrm{ob}}\mathrm{\right)}\mathrm{=}\mathrm{Dopplerbiasfactor}\mathrm{\times }\sin {\mathit{\theta }}_{\mathrm{ob}}\mathit{.}& & \end{array}$This additional factor takes into account that more sources are directed at the observer in a flux density limited sample because Doppler beamed jets are detectable at greater distances than unbeamed ones. Cohen (1989) and Vermeulen & Cohen (1994) computed this term and found that it depends on the bulk Lorentz factor distribution in a flux density limited sample, the integral source count index, and the beaming index of the jet. The beaming index is defined from the relation F = δⁿF′, where n is the beaming index, δ is the Doppler factor, F is the observed flux density density, and F′ is the intrinsic flux density. The observed integral source count index is defined in the expression N(>F) ∝ F^− q, representing the number of sources N observed with a flux density above F, which is a power law in F with source count index q. Including the Doppler bias factor in the viewing angle PDF gives $\begin{array}{ccc}\mathit{P}\mathrm{\left(}{\mathit{\theta }}_{\mathrm{ob}}\mathit{,}\mathrm{\Gamma }\mathrm{\right)}\mathrm{=}\mathit{A}{\left(\mathrm{1}\mathrm{-}\mathit{\beta }\cos {\mathit{\theta }}_{\mathrm{ob}}\right)}^{\mathrm{-}\mathit{a}\mathrm{-}\mathrm{1}}\sin {\mathit{\theta }}_{\mathrm{ob}}\mathit{P}\mathrm{\left(}\mathrm{\Gamma }\mathrm{\right)}\mathit{,}& & \end{array}$(2)where a = nq − 1, P(Γ) is the PDF for jet bulk Lorentz factor, and A is the normalization constant. An important assumption made in calculating the Doppler bias term is that the log-log slope of N(>F′) vs. F′ and N(>F) vs. F are the same, which Vermeulen & Cohen (1994) justify based on previous studies of AGN jet luminosity functions (Urry & Shafer 1984; Urry & Padovani 1991). The MOJAVE selection criteria were designed so that source inclusion in the sample is based on beamed emission only (Lister & Homan 2005). These MOJAVE sources are typically dominated by core flux density, which usually has a flat spectrum (Kovalev et al. 2005; Pushkarev & Kovalev 2012). Thus, the beaming index n is most likely ~2 for steady jets (Lind & Blandford 1985), while the integral source count index is approximately 1.5, indicating that the fiducial value for the a-parameter should be approximately (Vermeulen & Cohen 1994) $\begin{array}{ccc}{\mathit{a}}_{\mathrm{fiducial}}\mathrm{=}\mathrm{2.}& & \end{array}$We note that if the MOJAVE sources were typically dominated by optically thin flux density, which typically has a spectral index of α ~ 0.7 (F_ν ∝ ν^− α), then a ≈ 3.

The opening angle distribution may now be derived from P(θ_ob,Γ) by a change of variables from θ_ob to θ_app and marginalizing over Γ, $\begin{array}{ccc}\mathit{P}\mathrm{\left(}{\mathit{\theta }}_{\mathrm{app}}\mathrm{\right)}\mathrm{=}\mathrm{\int }\mathrm{d\Gamma }\mathit{P}\mathrm{(}{\mathit{\theta }}_{\mathrm{ob}}\mathrm{\left(}{\mathit{\theta }}_{\mathrm{app}}\mathit{,}\mathrm{\Gamma }\mathrm{\right)}\mathit{,}{\mathrm{\Gamma }}^{\mathrm{)}}\left|\frac{\mathit{\partial }{\mathit{\theta }}_{\mathrm{ob}}}{ \mathit{\partial }{\mathit{\theta }}_{\mathrm{app}}}\right|\mathit{,}& & \end{array}$(3)where θ_ob and ∂θ_ob/∂θ_app are functions of θ_app and Γ, and can be determined by assuming a particular jet geometry that we take to be conical here. These relationships are often derived by treating conical jets as triangles projected onto the plane of the sky, implying that tanθ_app = R_j/ℓ′ = R_j/(ℓsinθ_ob), where θ_ob is the jet viewing angle, R_j is the jet radius, ℓ is the jet length, and ℓ′ is the jet length projected onto the sky. If we assume θ_j ≈ R_j/ℓ, then $\begin{array}{ccc}\tan {\mathit{\theta }}_{\mathrm{j}}\mathrm{=}\tan {\mathit{\theta }}_{\mathrm{app}}\sin {\mathit{\theta }}_{\mathrm{ob}}\mathit{.}& & \end{array}$(4)For cases where both θ_app and θ_j are ≪1, this reduces to a commonly used relation for jets, θ_j = θ_appsinθ_ob (Jorstad et al. 2005; Pushkarev et al. 2009). Equation (4) also implies a maximum apparent half-opening angle of π/2. We assume that jets viewed down-the-pipe where θ_ob < θ_j are rare, since such jets would have θ_app > π/2. This dearth of down-the-pipe jets, sometimes used to justify the cylindrical approximation in jet models (Clausen-Brown et al. 2011), also indicates that typically Γθ_j < 1. If the typical viewing angle of a jet is θ_ob = 0.5/Γ (Vermeulen & Cohen 1994), and most MOJAVE jets exhibit a jet-like morphology (i.e., θ_app < π/2) such that θ_ob > θ_j, then Γθ_j < 0.5. Here, because many jets have large apparent opening angles, but are often viewed with small observing angles and small intrinsic opening angles, we most often use the approximation that $\begin{array}{ccc}{\mathit{\theta }}_{\mathrm{ob}}\mathrm{\approx }\frac{\mathit{\rho }}{\mathrm{\Gamma }\tan {\mathit{\theta }}_{\mathrm{app}}}\mathrm{·}& & \end{array}$(5)This approximation is mostly appropriate for the blazar dominated MOJAVE sample; below in Sect. 2.2 we show that this approximation is useful for categorizing jets by their apparent opening angles.

An apparent weakness in our model is that P(Γ) is not well constrained. This is not the case, however, since P(θ_app) is insensitive to P(Γ), which we demonstrate here. In a flux density limited VLBI sample, jets with small viewing angles will dominate, so we assume sinθ_ob ≈ θ_ob, and approximate (2) as $\begin{array}{ccc}\mathit{P}\mathrm{\left(}{\mathit{\theta }}_{\mathrm{ob}}\mathit{,}\mathrm{\Gamma }\mathrm{\right)}\mathrm{=}\mathit{A}\mathrm{\left(}\mathrm{2}{\mathrm{\Gamma }}^{\mathrm{2}}{\mathrm{)}}^{\mathit{a}\hspace{0.17em}\mathrm{+}\hspace{0.17em}\mathrm{1}}\mathrm{\right(}\mathrm{1}\mathrm{+}{\mathrm{\Gamma }}^{\mathrm{2}}{\mathit{\theta }}_{\mathrm{ob}}^{\mathrm{2}}{\mathrm{)}}^{\mathrm{-}\mathit{a}\mathrm{-}\mathrm{1}}{\mathit{\theta }}_{\mathrm{ob}}\mathit{P}\mathrm{\left(}\mathrm{\Gamma }\mathrm{\right)}\mathit{.}& & \end{array}$(6)For simplicity, we now evaluate Eq. (3) in light of the geometry implied by Eq. (5), and obtain $\begin{array}{ccc}\mathit{P}\mathrm{\left(}{\mathit{\theta }}_{\mathrm{app}}\mathrm{\right)}& \mathrm{=}\mathit{A}{\left(\mathrm{1}\mathrm{+}\frac{{\mathit{\rho }}^{\mathrm{2}}}{{\tan }^{\mathrm{2}}{\mathit{\theta }}_{\mathrm{app}}}\right)}^{\mathrm{-}\mathit{a}\mathrm{-}\mathrm{1}}\frac{\cos {\mathit{\theta }}_{\mathrm{app}}}{{\sin }^{\mathrm{3}}{\mathit{\theta }}_{\mathrm{app}}}& \\ & \mathrm{\times }\mathrm{[}{\mathrm{2}}^{\mathit{a}\hspace{0.17em}\mathrm{+}\hspace{0.17em}\mathrm{1}}{\mathit{\rho }}^{\mathrm{2}}\mathrm{\int }{{\mathrm{\Gamma }}^{\mathrm{2}\mathit{a}}\mathit{P}\mathrm{\left(}\mathrm{\Gamma }\mathrm{\right)}\mathrm{d\Gamma }}^{\mathrm{]}}& \\ & & \mathrm{=}{\mathit{A}}^{\mathrm{\prime }}{\left(\mathrm{1}\mathrm{+}\frac{{\mathit{\rho }}^{\mathrm{2}}}{{\tan }^{\mathrm{2}}{\mathit{\theta }}_{\mathrm{app}}}\right)}^{\mathrm{-}\mathit{a}\mathrm{-}\mathrm{1}}\frac{\cos {\mathit{\theta }}_{\mathrm{app}}}{{\sin }^{\mathrm{3}}{\mathit{\theta }}_{\mathrm{app}}}\mathit{,}\end{array}$(7)where ρ = Γθ_j, and we have absorbed the term in brackets into the new normalization, A′. Thus, it is apparent from Eq. (7) that P(θ_app) does not depend significantly on the form of P(Γ). Equation (7) is an accurate approximation of P(θ_app) as long as ρ ≪ 1, which is a valid assumption as shown by our best value of ρ ≈ 0.2 discussed below.

2.2. Velocity shear and Doppler beaming

As suggested by the very approximate estimate made above in which radio galaxies appear to have larger θ_j than blazars, velocity shear and Doppler beaming may affect the distribution of θ_app. Unfortunately, modeling the effect of velocity of shear on jet appearance is sensitive to a variety of unknown details regarding the jet structure such as how the density of non-thermal electrons scales with jet radius and the particular functional form of the velocity shear. Thus, in an effort to capture only the most basic effect velocity shear has on jet appearance, we develop a minimalist model.

We assume the velocity field of a jet consists of an ultra-relativistic inner spine and a surrounding shear layer that is mildly relativistic (see Fig. 2). The relativistic spine, which dominates the core emission, is what primarily determines whether a jet is included in a flux density limited sample like the blazar dominated MOJAVE sample. We note, however, that this assumption may sometimes be violated since a few MOJAVE sources such as M87 may have significant sheath emission (e.g., Kovalev et al. 2007). The optically thin jet downstream from the core is where apparent opening angles are measured, and where the degree to which the shear layer is observable is important. For jets aligned close to the line of sight, the jets will be more dominated by the fast spine where Γ ≫ 1 and the slower outer layers will remain unobserved, while more misaligned the jets will have a slower outer sheath of Γ_shear≲ few that is more likely visible. See Fig. 1 for an illustration of this effect.

Now, jets can be divided into two categories based on whether a jet’s viewing angle θ_ob is less than or greater than 1/Γ, where Γ is the value of the Lorentz factor in the fast inner spine. This categorization can be mapped onto θ_app by using Eq. (5), resulting in $\begin{array}{ccc}{\mathit{\theta }}_{\mathrm{app}}& \mathit{>}\arctan \mathrm{\left(}\mathit{\rho }\mathrm{\right)}\mathrm{\Leftarrow }\mathrm{\Rightarrow }{\mathit{\theta }}_{\mathrm{ob}}\mathit{<}\mathrm{1}\mathit{/}\mathrm{\Gamma }& \\ {\mathit{\theta }}_{\mathrm{app}}& \mathit{<}\arctan \mathrm{\left(}\mathit{\rho }\mathrm{\right)}\mathrm{\Leftarrow }\mathrm{\Rightarrow }{\mathit{\theta }}_{\mathrm{ob}}\mathit{>}\mathrm{1}\mathit{/}\mathrm{\Gamma }\mathit{.}& \end{array}$(8)This categorization is useful since the critical angle 1/Γ defines when beaming is important. When θ_ob > 1/Γ, then Earth is outside of the inner jet’s beaming cone, thus the jet’s slower outer layers are more likely to be visible, since the fast inner spine’s beaming is less dominant. The hypothesis that highly beamed jets (θ_ob < 1/Γ) and not highly beamed jets (θ_ob > 1/Γ) can be separated by their observed θ_app has some observational support, which we discuss in Sect. 4.3.

Fig. 1 Meridional slice of a jet illustrating two cases: (i) jets aligned close to the line of sight where emission is dominated by the fast inner spine; and (ii) more misaligned jets where the emission from outer slower layers contributes as well.

The simplest way to model the effect that velocity shear and beaming have on jet appearence is to postulate that jets that have θ_ob < 1/Γ have an effective jet opening angle θ_j,eff = ρ_eff/Γ, where θ_j,eff is some fraction of the true jet opening angle such that θ_j,eff = f_shθ_j, or equivalenty, ρ_eff = f_shρ. Thus, $\begin{array}{ccc}{\mathit{\theta }}_{\mathit{j,}\mathrm{eff}}& \mathrm{\to }{\mathit{\theta }}_{\mathrm{j}}& \mathrm{if}{\mathit{\theta }}_{\mathrm{app}}\mathrm{\ll }\arctan \mathrm{\left(}\mathit{\rho }\mathrm{\right)}\mathit{,}\\ {\mathit{\theta }}_{\mathit{j,}\mathrm{eff}}& \mathrm{\to }{\mathit{f}}_{\mathrm{sh}}{\mathit{\theta }}_{\mathrm{j}}& \mathrm{if}{\mathit{\theta }}_{\mathrm{app}}\mathrm{\gg }\arctan \mathrm{\left(}\mathit{\rho }\mathrm{\right)}\mathit{,}\end{array}$where f_sh is a free parameter. We note that the inner spine Doppler factor of a jet with θ_app ~ θ_j (i.e., a radio galaxy) is δ ~ 1/Γ, a jet with θ_app = ρ has δ ~ Γ, and a jet with θ_app ~ 1 has δ ~ 2Γ. In other words, the most drastic change in δ occurs in a narrow range of θ_app, for 0 < θ_app < ρ, while δ only changes by a factor of 2 in the large range ρ < θ_app ≲ 1. To illustrate this point, we plot the Doppler factor as a function of θ_app in Fig. 2. Thus, the effect of velocity shear on jet appearance should be strongest for θ_app = 0 to ρ. To reproduce this behavior, we choose the following arbitrary function, $\begin{array}{ccc}\frac{{\mathit{\rho }}_{\mathrm{eff}}}{\mathit{\rho }}& & \mathrm{=}\mathrm{(}\mathrm{1}\mathrm{-}{\mathit{f}}_{\mathrm{sh}}\mathrm{)}\exp \left(\mathrm{-}{\left(\frac{{\mathit{\theta }}_{\mathrm{app}}}{\arctan \mathrm{\left(}\mathit{\rho }\mathrm{\right)}}\right)}^{\mathrm{2}}\right)\mathrm{+}{\mathit{f}}_{\mathrm{sh}}\mathit{.}\end{array}$(9)We plot this function in Fig. 2. This equation can easily be inserted into our theoretical PDF described in Eq. (3), which can then be evaluated numerically, where ρ and f_sh are free parameters to be found in the fit. If this model is correct, then the best fit value of f_sh should be less than unity. In the case of no shear, then f_sh = 1 and ρ_eff = ρ.

Fig. 2 Doppler factor and ρeff as a function of a jet’s apparent half-opening angle θapp. The semi-logarithmic Doppler factor plot in the upper panel is for a jet with Γ = 10 and ρ = 0.21, and uses Eq. (4) to convert θapp to θob. The lower panel plot shows ρeff as a function of θapp from Eq. (9) using ρ = 0.21 and fsh = 0.33, the best fit values found in Sect. 3.2.

3. Data analysis and results

3.1. Apparent opening angles

The apparent opening angles used here are derived from stacked images of 133 sources from the MOJAVE-I catalogue of 135 sources². For two sources opening angles could not be derived. To produce a stacked image of a given source, all single-epoch maps were aligned by their VLBI core components, and then averaged together. The resulting opening angle data originates from the analysis in Pushkarev et al. (2012), who derived θ_app by taking the median value of $\begin{array}{ccc}{\mathit{\theta }}_{\mathrm{app}}\mathrm{=}\arctan \left(\frac{\sqrt{{\mathit{d}}^{\mathrm{2}}\mathrm{-}{\mathit{b}}_{\mathit{\phi }}^{\mathrm{2}}}}{\mathrm{2}\mathit{r}}\right)\mathit{,}& & \end{array}$(10)where “d is the full width half maximum (FWHM) of the Gaussian transverse profile, r is the distance to the core along the jet axis, b_φ is the beam size along the position angle φ of the jet-cut, and the quantity $\mathrm{(}{\mathit{d}}^{\mathrm{2}}\mathrm{-}{\mathit{b}}_{\mathit{\phi }}^{\mathrm{2}}{\mathrm{)}}^{\mathrm{1}\mathit{/}\mathrm{2}}$ is the deconvolved FWHM transverse size of the jet” (for more details, see Pushkarev et al. 2012). Note that in this work we use half opening angles, while Pushkarev et al. (2012) used full opening angles, which merely differ by a factor of 2.

3.2. Best fits and goodness of fit

We now compare the opening angle data to Eq. (3), where P(Γ) ∝ Γ^-1.5 with Γ_min = 2 and Γ_max = 50. We note, however, that Eq. (3) is insensitive to the form of P(Γ) as we demonstrated in Eq. (7).

We find the best fits using maximum likelihood estimation (MLE) by minimizing the negative log-likelihood function $\begin{array}{ccc}\mathit{h}\mathrm{\left(}{X}\mathit{,}{m}\mathrm{\right)}\mathrm{=}\mathrm{-}\mathrm{2}\sum _{\mathit{i}\mathrm{=}\mathrm{1}}^{\mathit{N}}\ln \mathit{P}\mathrm{\left(}{\mathit{X}}_{\mathit{i}}\mathit{,}{m}\mathrm{\right)}\mathit{,}& & \end{array}$(11)where our data set is X = (θ_app,1,...,θ_app,N), P(X_i,m) represents P(θ_app) evaluated at θ_app = X_i, and m is a vector representing the free parameters of the distribution P(θ_app). As discussed below, we fit our data set of N = 133 for six different cases in which the distribution’s free parameters ranges from three, m = (ρ,f_sh,a), to only one, m = ρ. When f_sh is not a free parameter it is fixed at 1, and when a is not free it is fixed at either 2 or 3, as specified below. In all of these cases, we obtain the best fit parameters by numerically minimizing h (Eq. (11)).

To correctly model the fitting error and assess the goodness of fit, we used the Kolmogorov-Smirnov (KS) statistic L_N in conjunction with the nonparametric bootstrap as described in Feigelson & Babu (2012). Recall that the KS statistic gives a measure of the distance between the data and the model by finding the maximum distance between the empirical cumulative distribution function F_N(θ_app) and theoretical cumulative distribution function , i.e., $\begin{array}{ccc}\frac{{\mathit{L}}_{\mathit{N}}}{\sqrt{\mathit{N}}}\mathrm{=}\underset{{\mathit{\theta }}_{\mathrm{app}}}{\sup }\left|{\mathit{F}}_{\mathit{N}}\mathrm{\left(}{\mathit{\theta }}_{\mathrm{app}}\mathrm{\right)}\mathrm{-}\mathit{F}\mathrm{\left(}{\mathit{\theta }}_{\mathrm{app}}\mathit{,}\mathit{m̂}\mathrm{\right)}\right|\mathit{.}& & \end{array}$(12)Here, for each bootstrap realization, we generate the simulated data X^∗ via sampling with replacement, find the best fit parameters for the simulated data X^∗ using the MLE procedure described above, and then calculate the KS statistic ${\mathit{L}}_{\mathit{N}}^{\mathrm{\ast }}$ from X^∗ and by using Eq. (12) with an additional bias correction factor taken into account (see Eq. (3.48) of Feigelson & Babu 2012).

After iterating the bootstrap B = 2000 times, we obtain confidence intervals around by analyzing the distribution of simulated best-fit parameters and directly compute the 68% and 95% confidence intervals and error contours. The resulting distribution of the statistic ${\mathit{L}}_{\mathit{N}}^{\mathrm{\ast }}$ can be used for model selection by finding the probability p that a value of L_N or greater is observed, assuming that X is drawn from . This is done by defining k as the number of ${\mathit{L}}_{\mathit{N}}^{\mathrm{\ast }}$ values that fulfill the criterion ${\mathit{L}}_{\mathit{N}}^{\mathrm{\ast }}\mathrm{\ge }{\mathit{L}}_{\mathit{N}}$, and then computing the p-value as p = k/B. Thus, for a significance level of α = 0.05, models with p-values of 0.05 and above are favored by the data (i.e. they cannot be rejected). As discussed below, we consider six different cases, thus we perform different bootstrap simulations for each case.

We now apply the above analysis to the data for two different types of models:

No shear model: three cases are considered for our model with no shear, i.e., f_sh = 1, depending how the parameter a is treated: (i) a is set to 2; (ii) a = 3; and (iii) a is a free parameter found in the best fit. As it turns out, in case (iii) the best-fit value of a, 6.3, is much higher than the expected fiducial value of 2, and the distribution of best-fit values of a in bootstrap simulations routinely ranges much higher (several tens). In addition, the best values of ρ in the bootstrap simulations is tightly correlated with a, and also ranges widely, suggesting that a and ρ are highly degenerate.

Velocity shear model: here we perform the same analysis and consider the same three cases as above, but with f_sh as a free parameter. Thus, ρ and f_sh are free parameters and we consider three different cases regarding a: (i) a = 2; (ii) a = 3; and (iii) a as a free parameter to be found in the best fit.

Table 1 gives a summary of our best-fit results, and Fig. 3 shows a comparison between the data and two different best-fit models, each with the same number of free parameters (two). The models that include relativistic velocity shear are clearly favored by the data, as they all have p-values above 0.05. When shear is included and all the parameters are varied (the shear (iii) model), the best fit produces a reasonable value of $\mathit{a}\mathrm{=}\mathrm{1.}{\mathrm{6}}_{-0.3}^{\mathrm{+}\mathrm{0.5}}$ (and ρ = 0.21    ±    0.03 and f_sh = 0.33    ±    0.1), which is close to the expected value of a = 2 from Doppler beaming models and integral source counts of radio-selected AGN. Encouragingly, the best-fit values for all of the models is ρ = 0.1−0.2, which is consistent with the values of 0.17 and 0.13 reported in Jorstad et al. (2005) and Pushkarev et al. (2009), respectively. Figure 4 shows the two dimensional error contours for the shear (iii) model. Since our shear models produce better fits with high p-values and a reasonable value of a, we conclude that it is likely that relativistic shear and Doppler beaming play a role in jet appearance. As our final result for a measurement of Γθ_j, we report 0.21 ± 0.03 from the shear (iii) model. However, for a more direct comparison between our value of ρ and that of other researchers, we calculate the expected value of ρ_eff(θ_app) $\begin{array}{ccc}\mathrm{⟨}{\mathit{\rho }}_{\mathrm{eff}}\mathrm{⟩}& \mathrm{=}\mathrm{\int }{\mathit{\rho }}_{\mathrm{eff}}\mathrm{\left(}{\mathit{\theta }}_{\mathrm{app}}\mathrm{\right)}\mathit{P}\mathrm{\left(}{\mathit{\theta }}_{\mathrm{app}}\mathrm{\right)}\mathrm{d}{\mathit{\theta }}_{\mathrm{app}}& \\ & & \mathrm{\approx }\mathrm{0.13}\mathrm{\pm }\mathrm{0.02}\mathit{,}\end{array}$(13)where the parameters of P(θ_app) are those for the shear (iii) model listed in Table 1, and the confidence interval comes from the bootstrap simulations used to derive the confidence intervals for the shear (iii) model. Indeed, this value of ⟨ ρ_eff ⟩  ≈ 0.13 ± 0.02 is consistent with both Pushkarev et al. (2009) and Jorstad et al. (2005).

Fig. 3 Example best fits for two cases are shown in the form of cumulative distribution functions (CDF, upper panel) and probability density functions (PDF, lower panel). The two cases shown, “no shear (iii)” (dash-dotted line) and “shear (i)” (dotted line), both have the same number of free parameters (two), but the shear (i) model clearly fits the data better. The data is represented as a solid line (upper panel) or histogram (lower panel).

Fig. 4 68% and 95% confidence contours from Monte Carlo error analysis for the best-fit parameters ρ, a, and fsh, from the case of shear (iii). The central star shows the MLE of the parameters. We emphasize the a vs. ρ plot since there is a fiducial value for a (=2, vertical dotted line), and ρ has constraints set on it by Jorstad et al. (2005) and Pushkarev et al. (2009).

A more rigorous comparison between our best-fit value for ρ and those of other researchers requires a proper error analysis for all of the different measured/inferred values for ρ, both in this paper and in other works. However, in the case of Jorstad et al. (2005) and Pushkarev et al. (2009), the error in their estimated values for θ_j and Γ for each jet is unknown, as these estimates rely upon highly uncertain model assumptions regarding, for example, equipartition brightness temperature arguments and the equation of component variability with light-crossing times. In addition to this error, it is also possible that AGN jets possess a range of values of ρ, as opposed to our assumption that all jets have the same ρ. These same issues apply to our simple model. More specifically, our confidence limits (see Fig. 4) are probably underestimated since there is considerable uncertainty in our model of velocity shear, our assumption of jet conical geometry, and our assumption that all MOJAVE jets posses the same value of ρ.

4. Discussion

4.1. Relativistic jet physics and Γθ_j

A variety of physical processes in jets are sensitive to Γθ_j. Causal connection implies that θ_jℳ ≤ 1, where ℳ = βΓ/(Γ_sβ_s) is the relativistic Mach number, which is the ratio of the jet proper speed to proper signal speed. (In the Appendix we explain the relationship between θ_jℳ, Γθ_j, and causality). For jets with a dynamically important magnetic field, the signal speed is the fast magnetosonic speed, while for jets with no significant magnetic field the signal speed is the sound speed, which is ${\mathit{\beta }}_{\mathrm{s}}\mathrm{=}\mathrm{1}\mathit{/}\sqrt{\mathrm{3}}$ for a relativistically hot jet. The fast magnetosonic proper speed is Γ_msβ_ms = σ^1/2, where σ is the magnetization parameter, which is the ratio of Poynting flux to kinetic flux (Kennel & Coroniti 1984). Beyond the acceleration zone the jet is likely to be in equipartition such that σ is of the order of unity (e.g., Komissarov et al. 2007). Thus, fiducial jet signal speeds imply that the causal connection condition for jets is $\begin{array}{ccc}\mathrm{\Gamma }{\mathit{\theta }}_{\mathrm{j}}& \lesssim \mathrm{0.7} & \mathrm{relativisticsoundspeed}\\ \mathrm{\Gamma }{\mathit{\theta }}_{\mathrm{j}}& \lesssim \mathrm{1} & \mathrm{equipartitionfastmagnetosonicspeed}\mathit{,}\end{array}$indicating that AGN jets with the value we have determined here, Γθ_j ~ 0.2, are probably causally connected. This verifies the standard picture of jet production (see Sect. 4.2 and Komissarov et al. 2009), and also implies that AGN jets are susceptible to various instabilities and reconnection, and are also sensitive to conditions at the boundary between the jet and the interstellar medium.

The particular value of Γθ_j is also important for instability development (Narayan et al. 2009), because it gives a measure of the extent to which jet sidewise expansion inhibits instability growth. Most global instabilities grow on some comoving signal crossing timescale, t_dyn = Γθ_jz/(β_sc), where β_s is the signal speed and z is the jet height above the launching region. For Kelvin-Helmholtz instabilities β_s is the sound speed (Perucho et al. 2004; Hardee et al. 2005), while for current driven kink instabilities β_s is the Alfvén speed (Giannios & Spruit 2006). This so-called dynamical timescale for the growth of instabilities must be compared to the jet expansion time t_exp = z/(βc), where if t_exp > t_dyn then jet expansion will quench instability growth (Begelman 1998; Giannios & Spruit 2006; Moll et al. 2008; Spruit 2010). This criterion for instability growth then becomes $\begin{array}{ccc}\frac{{\mathit{t}}_{\mathrm{dyn}}}{{\mathit{t}}_{\exp }}\mathrm{\approx }\frac{\mathrm{\Gamma }{\mathit{\theta }}_{\mathrm{j}}}{{\mathit{\beta }}_{\mathrm{s}}}\lesssim \mathrm{1}\mathrm{·}& & \end{array}$(14)Since Γθ_j ~ 0.2, then AGN jet instabilities can grow despite jet expansion, unless β_s ≲ 0.2, which would be below the relativistic adiabatic sound speed of ~0.6.

4.2. Jet acceleration for GRBs vs. AGNs

Long duration GRBs have typical values of Γθ_j of the order of 10−30 (Panaitescu & Kumar 2002), which is two orders of magnitude higher than the value we find for AGN jets: $\begin{array}{ccc}\frac{\mathrm{(}\mathrm{\Gamma }{\mathit{\theta }}_{\mathrm{j}}{\mathrm{)}}_{\mathrm{GRB}}}{\mathrm{(}\mathrm{\Gamma }{\mathit{\theta }}_{\mathrm{j}}{\mathrm{)}}_{\mathrm{AGN}}}\mathrm{~}\mathrm{100.}& & \end{array}$Thus, it is possible that different physics are at work in GRB jets. Tchekhovskoy et al. (2010) found that a jet acceleration mechanism (first discovered by Aloy & Rezzolla 2006 and Mizuno et al. 2008) can operate in GRBs in which a brief period of bulk acceleration occurs upon jet break out into the circumstellar medium and allows the jet to take on values of Γθ_j ≫ 1. Komissarov et al. (2010) call this process rarefaction acceleration, and contrast it to the standard jet acceleration model of collimation acceleration described in Li et al. (1992) and many other works. Collimation acceleration entails jet acceleration over an extended distance along the jet (Vlahakis & Konigl 2004) and implies that Γθ_j ≤ 1 (Komissarov et al. 2009; Tchekhovskoy et al. 2009). In contrast, rarefaction acceleration occurs in GRBs because they are initially confined by the shocked boundary layer in the star until the jet breaks out into the circumstellar medium, becomes unconfined, and launches a rarefaction wave toward the center of the jet. If the jet is still magnetically dominated at that point, then this process further accelerates the jet, and can produce Γθ_j ≫ 1. Thus, the dichotomy between GRB jets and AGN jets is nicely explained by the different physical processes at work in GRB jets (rarefaction acceleration due to jet break out) and AGN jets (collimation acceleration). Furthermore, as Komissarov et al. (2010) explain, rarefaction acceleration increases the Γθ_j parameter primarily by increasing Γ alone (The increase in θ_j is less than 1/Γ). Thus, to the degree that the typical AGN jet value of Γθ_j is equal to the pre-breakout GRB jet value of Γθ_j, one can infer that (Γθ_j)_GRB is so much larger than (Γθ_j)_AGN because of the increase of the GRB jet’s Γ during the rarefaction acceleration process. Interestingly, this suggests that GRBs have Γ ≳ 100, which is consistent with the lower limit obtained by requiring that GRB prompt emission regions be optically thin to gamma-rays with respect to photon-photon pair production (e.g., Piran 2004).

4.3. Doppler beaming and θ_app

If Γθ_j is approximately constant in the AGN jet population, then θ_app is an important observable quantity related to Doppler beaming for two reasons. First, it can serve as a dividing line between highly beamed (θ_ob < 1/Γ) jets and not highly beamed jets (θ_ob > 1/Γ), a property we exploit in our model of velocity shear in Sect. 2.2. This division conveniently maps onto θ_app as follows: $\begin{array}{ccc}{\mathit{\theta }}_{\mathrm{app}}& \mathit{>}\arctan \mathrm{\left(}\mathit{\rho }\mathrm{\right)}\mathrm{\Leftarrow }\mathrm{\Rightarrow }\mathrm{highlybeamedjets}& \\ {\mathit{\theta }}_{\mathrm{app}}& \mathit{<}\arctan \mathrm{\left(}\mathit{\rho }\mathrm{\right)}\mathrm{\Leftarrow }\mathrm{\Rightarrow }\mathrm{nothighlybeamedjets}\mathit{.}& \end{array}$This implies that jets with θ_app ~ Γθ_j ~ 0.2 are observed at the critical angle, thus maximizing the apparent speed of their superluminal components. This division provides a concise way of explaining the Pushkarev et al. (2009) argument that large-opening angle jets are more highly Doppler beamed: large opening angle jets with θ_app ≳ 0.2 are all observed within the critical angle and therefore more highly beamed than smaller θ_app jets. Pushkarev et al. (2009) also find that AGN jets with larger θ_app have a higher Fermi-LAT detection rate, and all jets with θ_app > 0.35 are detected by Fermi-LAT, implying that Fermi-LAT detected jets tend to have higher Doppler factors. Notably, this finding is also supported by Kovalev et al. (2009), Savolainen et al. (2010), and Lister et al. (2009c), who found evidence that radio jets of Fermi-detected AGN are more likely to have high Doppler factors than are non Fermi-LAT detected sources.

Second, by measuring both a jet’s θ_app and its typical apparent speed β_app = βsinθ_ob(1 − βcosθ_ob)^-1, we can derive that jet’s Doppler factor, Lorentz factor, and therefore also the viewing angle, $\begin{array}{ccc}\mathit{\delta }& & \mathrm{=}\frac{{\mathit{\beta }}_{\mathrm{app}}\tan {\mathit{\theta }}_{\mathrm{app}}}{\mathit{\beta }{\mathit{\rho }}_{\mathrm{eff}}}\mathrm{\approx }\frac{{\mathit{\beta }}_{\mathrm{app}}\tan {\mathit{\theta }}_{\mathrm{app}}}{{\mathit{\rho }}_{\mathrm{eff}}}\\ \mathrm{\Gamma }& & \mathrm{\approx }\frac{{\mathit{\beta }}_{\mathrm{app}}\left(\mathrm{1}\mathrm{+}{\mathit{\rho }}_{\mathrm{eff}}^{\mathrm{2}}{\cot }^{\mathrm{2}}{\mathit{\theta }}_{\mathrm{app}}\right)}{\mathrm{2}{\mathit{\rho }}_{\mathrm{eff}}\cot {\mathit{\theta }}_{\mathrm{app}}}\\ {\mathit{\theta }}_{\mathrm{ob}}& & \mathrm{\approx }\frac{\mathrm{2}{\mathit{\rho }}_{\mathrm{eff}}^{\mathrm{2}}{\cot }^{\mathrm{2}}{\mathit{\theta }}_{\mathrm{app}}}{{\mathit{\beta }}_{\mathrm{app}}\mathrm{(}\mathrm{1}\mathrm{+}{\mathit{\rho }}_{\mathrm{eff}}^{\mathrm{2}}{\cot }^{\mathrm{2}}{\mathit{\theta }}_{\mathrm{app}}\mathrm{)}}\mathit{,}\end{array}$where $\mathit{\delta }\mathrm{=}\mathrm{(}\mathrm{\Gamma }\mathrm{-}\sqrt{{\mathrm{\Gamma }}^{\mathrm{2}}\mathrm{-}\mathrm{1}}\cos {\mathit{\theta }}_{\mathrm{ob}}{\mathrm{)}}^{-1}$ is the Doppler factor and β is the jet velocity in units of the speed of light. Except for Eq. (15), which shows both the exact and approximate form of δ, the above equations are approximations that assume Γ ≫ 1 and θ_ob ≪ 1.

Thus, Eqs. ((15)–(17)) demonstrate that, if the spread of ρ (=Γθ_j) is small enough in the jet population, the measurable quantities β_app and θ_app can be useful for calculating intrinsic jet quantities such as the intrinsic brightness temperature and Lorentz factor of jet components. In a future work we intend to explore this new method of deriving a jet’s beaming parameters.

5. Conclusion

We have derived a statistical model of relativistic jet apparent opening angles and fit it to the observed distribution of jet apparent opening angles in the MOJAVE sample. The product of Lorentz factor and intrinsic jet opening angle Γθ_j is a free parameter in our model and was determined by the best fit to be Γθ_j ≈ 0.2. We summarize our conclusions as follows.

• 1.

  Γθ_j ~ 0.2 implies that jets are causally connected (see the Appendix), which is predicted by magnetic jet production models. Causal connection also implies that AGN jets are subject to Kelvin-Helmholtz and current-driven (kink) modes, unless the relevant signal speed is ≲Γθ_jc ~ 0.2c.

• 2.

  The value of Γθ_j for GRB jets is 100 times larger than Γθ_j for AGN jets. This difference is neatly explained by an acceleration process probably unique to GRBs, wherein a rarefaction wave is launched into the jet after the jet breaks out of its stellar envelope and into the lower pressure circumstellar medium. This is consistent with the high Lorentz factors inferred for GRB jets of Γ ≳ 100.

• 3.

  In order to adequately fit the θ_app data, we included the effects of relativistic velocity shear and Doppler beaming. Velocity shear affects jets by making blazars appear narrower as their ultra-relativistic inner spine is all that is visible, while jets viewed outside the critical angle 1/Γ appear to have larger jet opening angles. Distinguishing jets based on their critical angle conveniently creates a division between highly beamed jets and not so highly beamed jets that corresponds to whether an individual jet’s apparent opening angles is θ_app ≲ Γθ_j (not highly beamed) or θ_app ≳ Γθ_j (highly beamed).

• 4.

  Assuming Γθ_j is mostly constant across the AGN jet population, then a jet’s Doppler factor, Lorentz factor, and viewing angle can be calculated if the observable values of apparent jet opening angle θ_app and the apparent speed of the jet components β_app are known. This is shown in Eqs. ((15)–(17)).

Acknowledgments

ECB thanks M. Böck, M. Zamaninasab, M. Lister, M. Lyutikov, and D. Giannios for valuable discussions. A.B.P. was supported by the “Non-stationary processes in the Universe” Program of the Presidium of the Russian Academy of Sciences. Y.Y.K. was supported by the Russian Foundation for Basic Research (project 12-02-33101), the Dynasty Foundation, and the Research Program OFN-17 of the Division of Physics, Russian Academy of Sciences. This research has made use of data from the MOJAVE database that is maintained by the MOJAVE team (Lister et al. 2009a).

References

Appendix A: Causal connection and Γθ_j

Here we discuss two different criteria for defining whether or a not a jet is causally connected for the simplistic case of a jet with radial velocity streamlines of constant speed. For jets with velocity shear, as we posit in this work, causal connection is more complicated than we have presented below, although we expect the following discussion to be approximately correct. First, the relativistic Mach number ℳ = βΓ/(β_sΓ_s) (Konigl 1980) is sometimes used to define causal connection for supersonic jets by requiring θ_jℳ < 1, or Γθ_j < Γ_sβ_s/β (e.g., Komissarov et al. 2009). Second, causally connected jets are sometimes defined as those for which Γθ_j < 1 (e.g., Zakamska et al. 2008). We note that for relativistic jets with β ≈ 1 and an ultra-relativistic equation of state where the proper sound speed is Γ_sβ_s = 2^− 1/2 ≈ 0.71, both of these causality criteria resemble one another: Γθ_j < 0.71 for the relativistic Mach number approach, and Γθ_j < 1 for the other. However, for magnetically dominated plasmas, the fast magnetosonic speed can approach the speed of light, thus no upper bound can be placed on Γ_sβ_s and these two criteria for causal connection give contradictory answers. Thus, a jet can have any value of Γθ_j and still in principle be causally connected, provided it is magnetically dominated enough. In particular, for jets with magnetization σ (Kennel & Coroniti 1984), the proper Alfvén speed is Γ_Aβ_A = σ^1/2, thus the Mach number condition for the jet to be causally connected is Γθ_j < σ^1/2 (e.g., Komissarov et al. 2009). Jets can then be causally connected even though Γθ_j ≫ 1, as long as σ is large enough (high σ implies the jet is Poynting flux dominated). Below, we discuss why these two criteria are different and conclude that the relativistic Mach angle analysis is usually the more appropriate criterion, even though it is only approximate.

θ_jℳ < 1 criterion: This criterion is only relevant for highly supersonic or supermagnetosonic jets, since for subsonic or transonic jets there is no limit on wave propagation.

The relativistic Mach number can be derived by assuming a flow with parallel velocity streamlines and by analyzing the observer frame angle a sound wave can make with respect to the flow direction, tanχ = β_⊥/β_∥. The relativistic Mach angle is then found by maximizing χ by varying χ′, the rest frame angle between the flow direction and the sound wave direction. This procedure gives cosχ′ = −β_s/β and a maximum angle of sinχ_max = 1/ℳ (Konigl 1980). Thus, χ_max represents the largest observer frame angle a sound wave can make with respect to a supersonic flow. For this reason, jets where θ_j > χ_max are assumed to be out of causal contact with themselves. However, this Mach angle analysis assumes a flow of parallel velocity streamlines, something that is not the case for conical jets.

For conical jets, jet sidewise expansion lengthens the signal crossing time compared to a flow with parallel streamlines (e.g., a cylindrical jet). Causal connection can be analyzed by making a simple estimate of a conical jet’s signal crossing time, assuming the signal propagates at a speed β_s in the local fluid rest frame and makes an angle χ′ between the jet comoving frame signal wave propagation direction and the local fluid streamline. We also assume that the jet flow is radial with a half-opening angle of θ_j and a constant Lorentz factor of Γ. In the observer frame the wave has a speed parallel to the local streamline of $\begin{array}{ccc}{\mathit{\beta }}_{\mathit{r}}\mathrm{=}\frac{{\mathit{\beta }}_{\mathrm{s}}\cos {\mathit{\chi }}^{\mathrm{\prime }}\mathrm{+}\mathit{\beta }}{\mathrm{1}\mathrm{+}\mathit{\beta }{\mathit{\beta }}_{\mathrm{s}}\cos {\mathit{\chi }}^{\mathrm{\prime }}}& & \end{array}$(A.1)and a perpendicular speed of $\begin{array}{ccc}{\mathit{\beta }}_{\mathit{\theta }}\mathrm{=}\frac{{\mathit{\beta }}_{\mathrm{s}}\sin {\mathit{\chi }}^{\mathrm{\prime }}}{\mathrm{\Gamma }\mathrm{(}\mathrm{1}\mathrm{+}\mathit{\beta }{\mathit{\beta }}_{\mathrm{s}}\cos {\mathit{\chi }}^{\mathrm{\prime }}\mathrm{)}}\mathrm{·}& & \end{array}$(A.2)For simplicity we assume the emitted wave trajectory is such that χ′ is constant. If the radial coordinate (centered on the jet’s central engine) of the wave is r, then dr = β_rcdt. In a time dt, the wave propagates in the polar direction an arc length of ds = β_θcdt. In terms of polar angle, then, the signal propagation can be written as dθ = β_θcdt/r, which, combined with the dr = β_rcdt, can be solved for the time it takes for a signal to propagate through a polar angle θ_j, $\begin{array}{ccc}{\mathit{t}}_{\mathrm{cross}}\mathrm{=}\frac{{\mathit{r}}_{\mathrm{0}}}{{\mathit{\beta }}_{\mathit{r}}\mathit{c}}\left(\exp \left(\frac{{\mathit{\theta }}_{\mathrm{j}}{\mathit{\beta }}_{\mathit{r}}}{{\mathit{\beta }}_{\mathit{\theta }}}\right)\mathrm{-}\mathrm{1}\right)\mathit{,}& & \end{array}$(A.3)where r₀ is the radial location of the initial wave emission. We note that a different form of Eq. (A.3) was derived in Kinoshita et al. (2004) and a similar result was also found in Nakar et al. (2003). If in the observer frame a sound wave has a trajectory such that χ = 1/ℳ (i.e., χ′ = −β_s/β), then β_r/β_θ ≈ ℳ and β_r ≈ β, yielding the differential equation dθ = (ℳr)^-1dr. This differential equation has the solution for polar angle through which the signal propagates of θ(r) = ℳ^-1ln(r/r₀) with the associated signal crossing time of $\begin{array}{ccc}{\mathit{t}}_{\mathrm{cross}}& & \mathrm{\approx }\frac{{\mathit{r}}_{\mathrm{0}}}{\mathit{\beta c}}\mathrm{(}{\mathrm{e}}^{{\mathit{\theta }}_{\mathrm{j}}\mathrm{ℳ}}\mathrm{-}{\mathrm{1}}^{\mathrm{)}}\mathit{,}\end{array}$(A.4)assuming ℳ ≫ 1. Equation (A.4) implies that jets for which θ_jℳ ≫ 1 have t_cross ≈ (r₀/c)exp(θ_jℳ), effectively making such jets fall out of causal contact in the sense that the dynamic time is longer than the jet expansion time r₀/c by a factor exp(θ_jℳ).

Accelerating conical jets are different in that they can have a causal horizon that depends on the details of jet acceleration (Kinoshita et al. 2004). An accelerating supersonic jet will have an increasing proper speed and a decreasing (or constant) proper signal speed, which we parameterize as ℳ = ℳ₀(r/r₀)^b. For a signal emitted at r₀ that propagates at the Mach angle relative to the local fluid streamline, then dθ = (ℳr)^-1dr has the asymptotic solution $\begin{array}{ccc}{\mathit{\theta }}_{\mathrm{\infty }}\mathrm{=}\frac{\mathrm{1}}{\mathit{b}{\mathrm{ℳ}}_{\mathrm{0}}}\mathrm{·}& & \end{array}$(A.5)That is, disturbances located at radius r₀ will propagate through a polar angle θ_∞ as r → ∞. Thus, in this circumstance, the causal connection criterion becomes θ_jℳ(r) < b^-1.

Γθ_j < 1 criterion: We assume an initially cylindrical jet with speed β ≈ 1 and associated Lorentz factor Γ ≫ 1, and let the radius of the cylindrical flow suddenly begin to expand in the flow rest frame with velocity ${\mathit{\beta }}_{\exp }^{\mathrm{\prime }}$ perpendicular to the symmetry axis. Transforming back into the observer frame then gives ${\mathit{\beta }}_{\mathrm{\perp }}\mathrm{=}{\mathit{\beta }}_{\exp }^{\mathrm{\prime }}\mathit{/}\mathrm{\Gamma }$, the small angle the velocity stream lines make with the jet axis is now $\begin{array}{ccc}\mathit{\theta }\mathrm{=}\frac{{\mathit{\beta }}_{\exp }^{\mathrm{\prime }}}{\mathrm{\Gamma }}\mathrm{·}& & \end{array}$(A.6)The requirement that the jet cross section expands at less than the speed of light ${\mathit{\beta }}_{\exp }^{\mathrm{\prime }}\mathit{<}\mathrm{1}$ implies that (Zakamska et al. 2008) $\begin{array}{ccc}\mathrm{\Gamma }{\mathit{\theta }}_{\mathrm{j}}\mathit{<}\mathrm{1.}& & \end{array}$(A.7)Thus, requiring that Γθ_j < 1 is an important constraint for jet flows that are initially close to cylindrical, and for some reason undergo expansion. We note, however, that this constraint is not important for flows that are initially not collimated, such as the highly relativistic equatorial outflows from pulsars that power pulsar wind nebulae (Kennel & Coroniti 1984).

Alternatively, Γθ_j can be an important quantity if the sound wave emission direction in the local rest frame is defined as perpendicular to the local flow direction, i.e., χ′ = π/2, as could be the case if the wave is restricted to a thin spherical shell (which may be relevant for GRBs, Lyutikov et al. 2003; Kinoshita et al. 2004), so that β_θ = β_s/Γ and β_r = β. In this case, the sound crossing time from Eq. (A.3) becomes $\begin{array}{ccc}{\mathit{t}}_{\mathrm{cross}}\mathrm{=}\frac{{\mathit{r}}_{\mathrm{0}}}{\mathit{\beta c}}\left(\exp \left(\frac{\mathrm{\Gamma }{\mathit{\theta }}_{\mathrm{j}}\mathit{\beta }}{{\mathit{\beta }}_{\mathrm{s}}}\right)\mathrm{-}\mathrm{1}\right)\mathrm{·}& & \end{array}$(A.8)For jets with β ≈ 1, β_s ≈ 1, and Γθ_j ≫ 1, then t_cross ≈ (r₀/c)exp(Γθ_j), showing that in this case Γθ_j plays the same role that θ_jℳ does in the general case for determining an effective causal condition.

All Tables

All Figures

	Fig. 1 Meridional slice of a jet illustrating two cases: (i) jets aligned close to the line of sight where emission is dominated by the fast inner spine; and (ii) more misaligned jets where the emission from outer slower layers contributes as well.
In the text

	Fig. 2 Doppler factor and ρeff as a function of a jet’s apparent half-opening angle θapp. The semi-logarithmic Doppler factor plot in the upper panel is for a jet with Γ = 10 and ρ = 0.21, and uses Eq. (4) to convert θapp to θob. The lower panel plot shows ρeff as a function of θapp from Eq. (9) using ρ = 0.21 and fsh = 0.33, the best fit values found in Sect. 3.2.
In the text

Fig. 3 Example best fits for two cases are shown in the form of cumulative distribution functions (CDF, upper panel) and probability density functions (PDF, lower panel). The two cases shown, “no shear (iii)” (dash-dotted line) and “shear (i)” (dotted line), both have the same number of free parameters (two), but the shear (i) model clearly fits the data better. The data is represented as a solid line (upper panel) or histogram (lower panel).

In the text

	Fig. 4 68% and 95% confidence contours from Monte Carlo error analysis for the best-fit parameters ρ, a, and fsh, from the case of shear (iii). The central star shows the MLE of the parameters. We emphasize the a vs. ρ plot since there is a fiducial value for a (=2, vertical dotted line), and ρ has constraints set on it by Jorstad et al. (2005) and Pushkarev et al. (2009).
In the text