THE ACCELERATING JET OF 3C 279

We begin the analysis by calculating the values of the apparent speed, β_app (see Equation (1)), from the proper motion, μ. The proper motion, μ, is the slope of the linear regression to the component distance from the core versus time, as seen in Figures 2–5. We can clearly see that some of these fits will be closer to linear (such as C8) whereas a component such as C5 shows very complex changes and C6 shows movement from core that clearly implies rapidly increasing apparent speed. As a first approximation, we use just the linear fit for now and apply the standard formulae:

$$\begin{equation} \beta _\mathrm{app}= { 47.4 d_{\theta }\mu (1+z)}, \end{equation} \tag{ 1 }$$

where d_θ is the angular size distance in Mpc and z is the cosmic redshift. A derivation of Equation (1) can be found in Pearson & Zensus (1987) and Porcas (1987).

We then use β_app (see Equations (2) and (3)) to constrain the Lorentz factor, Γ, and the viewing angle, ϑ. Here, the viewing angle refers to the angle between the direction of motion of the relevant component and the line of sight to the observer, as measured from the observer's frame:

$$\begin{equation} \beta _{{\rm app}}={\beta \sin \vartheta \over {1-\beta \cos \vartheta }}, \end{equation} \tag{ 2 }$$

$$\begin{equation} \Gamma ={1 \over \sqrt{1-\beta ^2}}, \end{equation} \tag{ 3 }$$

where β refers to the speed, v, divided by the speed of light, c.

Then, using Equations (4) and (5), we can best determine which range of values of the intrinsic flux density, F_ν, int, and the relativistic Doppler beaming factor, δ, best fit the observed flux densities (Figures 6–9), F_ν, obs, at a given time:

$$\begin{equation} \delta ={1 \over {\Gamma (1-\beta \cos \vartheta })}, \end{equation} \tag{ 4 }$$

$$\begin{equation} { F_{\nu,\, {{\rm obs}}} }= { {F_{\nu,\, {\mathrm{int}}}} {\delta ^{3+\alpha } } }. \end{equation} \tag{ 5 }$$

Here, α refers to the spectral index assuming F_ν∝ν^−α. We have assumed α = 0.8, which is commonly the case, at least approximately, for blazar components that have long ago merged with the core. We note that the kinematics are not at all dependent on this assumption, and the results concerning level and variability of flux density are dependent on this in a predictable way (by inverting Equation (5)). Furthermore, we can see that the general results concerning variability and how that ties in to the kinematics are not highly dependent on spectral index. Here, if we take the ratio of two flux densities from two different times (labeled with subscripts 1 and 2), then

$$\begin{equation} {F_{\nu, \, {{\rm int}}\, 2} \over F_{\nu, \, {{\rm int}} \, 1}} = {{F_{\nu,\,{{\rm obs}}\, 2 } \over {F_{\nu, {{\rm obs}} \, 1 }}} {\biggl ({\delta _2 \over \delta _1}\biggr)^{-(\alpha +3)} }}. \end{equation} \tag{ 6 }$$

If we assume that δ₂/δ₁ is no larger than about 1.65 (the maximum in our set of observations of all of these components), and we assume liberally that α can be in the range of −1 to 1, then, after substituting into Equation (6), the range of intrinsic flux ratios that we calculate can differ by a factor of about 2.7. Therefore, for a typical case, it is unlikely that our results will be critically dependent on assumptions regarding spectral index.

We note that all components show a decreasing observed flux density, which is consistent with the general trend of the bright components having moved far from the core and generally dimmed with time. However, for some components, such as C5, this is clearly not simply monotonic (Figure 7).

Our overall strategy is to fit the proper motion and flux density data by assuming a time dependence for the viewing angle and intrinsic flux density over defined time intervals. The time intervals are chosen to be those that best fit the radial distance versus time curve (Figures 2–5) for each component. As a starting point, following Pearson & Zensus (1987), we see that for a given β_app there are a set of values of Γ and ϑ that can satisfy the equations above, and that we can solve for a minimum value of Γ and the maximum value for ϑ (as Γ → ∞):

$$\begin{equation} {\Gamma _{{\rm min}}}=\sqrt{1+{\beta _{{\rm app}}}^2}, \end{equation} \tag{ 7 }$$

$$\begin{equation} {\tan ({\vartheta _{{\rm max}}}/2)} ={1 \over \beta _{{\rm app}}}. \end{equation} \tag{ 8 }$$

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Kellermann & Owen (1988) aid in visualizing these results by plotting β_app versus ϑ for a set of different values of Γ.

We thus begin with these values from Equations (7) and (8), and search parameter space until χ² is minimized (discussed below). The same time intervals are then used for the remainder of the fits (described below) for each component. For simplicity, we have chosen a linear dependence with time (in each interval) for each of these parameters. The results of these fits are given in Table 1.

In this table, Column 1 refers to the component being considered. The parameters of the tested models are the bulk Lorentz factor, Γ, viewing angle, ϑ, and the intrinsic flux density (F_{ν, int}). Columns 2 and 3 give the start and end times of the fitted intervals. Columns 4 and 5 give the starting and ending bulk Lorentz factors. Columns 6 and 7 give the starting and ending viewing angles. Columns 8 and 9 give the beginning and ending Doppler factors using Equation (4). Columns 10 and 11 give the beginning and ending intrinsic flux density values. The results of our χ² analysis of these models are given in Table 2. Column 1 refers to the component being considered and Column 2 refers to the time-dependent quantity that is being fitted. The explanation and calculation of χ² in Column 3 can be found in Chapter 13 of Press et al. (1986). The probability given in Column 4 is such that the model could lead to a χ² value as poor as that calculated by chance. Generally speaking, very low probability values (<1 × 10⁻³) would indicate a poor fit. If the probability is given as 0, it is less than the lowest value that can be calculated by our χ² routine. The number of degrees of freedom (dof) given in Column 5, as defined in Chapter 13 of Press et al. (1986), is also given in Table 2.

Though we leave the magnitude of intrinsic flux density as a free parameter in our model above, we can get a sense of whether or not our model results are realistic or at least partly explained if we consider a model such as the shocked jet model of Marscher & Gear (1985). We first consider that the intrinsic flux density of a homogeneous source is proportional to the electron energy normalization factor, K, and the magnetic field, B, in the following way:

$$\begin{equation} F_{\nu, {\rm int}} \propto { K B^{(1+p)/2}}R^3, \end{equation} \tag{ 9 }$$

where the relativistic electron energy distribution is defined by

$$\begin{equation} N(E)=K E^{-p} \end{equation} \tag{ 10 }$$

and R is the radius of the jet.

In Marscher & Gear (1985), broadband flux variations are modeled by a shock passing through a relativistic jet. If we assume a plasma of relativistic electrons and subrelativistic protons with adiabatic index of 13/9 and if the compression ratio of the shock (in the strong shock limit) can be approximated as

$$\begin{equation} \eta \simeq {{13\Gamma ^\prime +9} \over 4 }, \end{equation} \tag{ 11 }$$

then the energy normalization factor can be enhanced by a factor of ηξ^{p − 1}, where ξ is a constant factor by which the electron energies are enhanced. For the purposes of order-of-magnitude analysis, we can, as Marscher & Gear (1985) discuss, assume that ξ has a lower limit related to the compression ratio of η^1/3. Similar considerations lead to a magnetic field enhancement of approximately (η² + 1/2)^1/2. We note that Γ' is the bulk Lorentz factor of the shocked medium as viewed from the un-shocked medium and can be assumed to have a value between approximately 1 and 2 (Marscher & Gear 1985).

However, we also need to take into consideration the radial decrease of electron density and magnetic field as the shock progresses down the jet. First, we can assume that these quantities decrease following a power law

$$\begin{equation} K(r)=K_0 (r/r_0)^{-n}, \end{equation} \tag{ 12 }$$

and assuming that the jet flow is adiabatic (Marscher & Gear 1985), then

$$\begin{equation} n= {{2(p+2)} \over 3}, \end{equation} \tag{ 13 }$$

and also

$$\begin{equation} B(r)=B_0 (r/r_0)^{-b}, \end{equation} \tag{ 14 }$$

and

$$\begin{equation} R=ar^\epsilon, \end{equation} \tag{ 15 }$$

where, as before, R refers to the radius of the jet, r refers to the distance down the jet from the apex, and r₀ is a reference point some radial distance from the apex of the jet, and usually taken to be physically related to the observed VLBI core. For a conical jet, = 1 and a = tanλ, where λ refers to the jet opening angle.

Combining this with the flux density dependence on the aforementioned parameters in Equation (9), we can then show that the optically thin flux density will be proportional to the radius in the following way:

$$\begin{equation} F_{\nu } \propto (r/r_0)^{\kappa }, \end{equation} \tag{ 16 }$$

where

$$\begin{equation} \kappa =\epsilon (-2n +6 -b(p+1))/2. \end{equation} \tag{ 17 }$$

If we assume, as do Marscher & Gear (1985), that the distance down the jet can be determined observationally based on the observed progression of the component, and inferred velocity and viewing angle, then we can further use

$$\begin{equation} r \propto { {(t_{{\rm obs}} -t_{ej}) \beta c} \over {(1-\beta \cos \vartheta)(1+z)}}. \end{equation} \tag{ 18 }$$

Substituting into Equation (16), we get

$$\begin{equation} F_{\nu } \propto \left[ {({t_{{\rm obs}}-t_{{\rm ej}})\beta c} \over {(1-\beta \cos \vartheta)(1+z)} }\right]^{\kappa }, \end{equation} \tag{ 19 }$$

where t_obs refers to the time of observation and t_ej refers to the time of ejection of the component from the core. If we can then compare the expected radial amplification factor between the two intervals, and then multiply by the shock amplification factor discussed above, we arrive at a new combined amplification factor, A_f:

$$\begin{eqnarray} {A_f} &=& \left[{{13\Gamma ^\prime +9}\over 4}\right]^{(p+2)/3} \left[{(13 \Gamma ^\prime +9)^2 +16} \over 32\right]^{(p+1)/4)} \nonumber\\ &&\times \left[{{(t_{{\rm obs},\,2}-t_{{\rm ej}})} \over {(t_{{\rm obs},\,1}-t_{{\rm ej})}}} {{(1-\beta \cos \vartheta _1)} \over {(1-\beta \cos \vartheta _2)}}\ \right]^{-\kappa }. \end{eqnarray} \tag{ 20 }$$

We consider this model in interpreting our results in our discussion in Section 4 below.

Here we summarize the details of what is seen for each component.

C1. In Figure 2, we see that the plot of radial distance from the core versus time shows a significant transition in 1998 due to an increase in β_app (the slope of this diagram) from near 8 to as high as 11 in 2003, decreasing again thereafter. The 1998 event has previously been interpreted as due to an impulse acceleration (Homan et al. 2003) related to a sudden bend in the jet. In Figure 6, we see that the observed flux density is primarily decreasing, which we have modeled as being due to a decrease of intrinsic flux density (Table 1). The relativistic Doppler factor, δ, is also primarily decreasing over the period of observations (with some small fluctuations; see Table 1), contributing to the decrease in observed flux density (Figure 6).

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C5. From Figure 3, we can see that this source exhibits very complex motion, especially between 2002.5 and 2005.5, during which time we see a sudden increase in apparent speed, β_app, followed by a decrease. Though proper motion, μ (and thus β_app), is generally well fitted for all components, C5 clearly has the poorest fit. These results could indicate a relatively fast-moving unresolved component that is leading us to misinterpret the radial distance curve around 2002.5 to 2005.5. The observed flux density (Figure 7) is decreasing during the range of observations, primarily due to a decrease in intrinsic flux density.

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C6. The visible dramatic change in the motion of this component (Figure 4) is best explained by both an increase in Γ from 29 to 41 and an increase in viewing angle from 02 to 11, accounting for the change in β_app from 5c in 2002.0 to 40c in 2003.9. The observed flux density (Figure 8) is decreasing during this time, which we have modeled as being caused mainly by a decrease in intrinsic flux density (Table 1).

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C8. The increase in slope of the plot in Figure 5 indicates that this component has gone from an apparent speed of 1.4c to 12c due to a dramatic change in viewing angle from 05 to 50 and an increase in Lorentz factor from Γ = 9 to 12. The observed flux density (Figure 9) is decreasing steadily during this same time frame, which we have modeled as being caused by a decrease in Doppler beaming factor, δ, from approximately 19 down to near 11 (see Table 1), while the intrinsic flux density remains roughly constant (see Table 1).

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We can follow up this analysis by additionally constraining Lorentz factors and viewing angles with theoretical modeling of the measured accelerations. After fitting Gaussian circular models to the (u, v) data using the MODELFIT routine within DIFMAP we have determined the positions of distinct features as a function of time. Following Homan et al. (2001), we have first determined the x and y positions (relative right ascension and declination, respectively) of each robust modeled component, and then used the LFIT fitting routine from Numerical Recipes (Press et al. 1986) to parameterize the x and y positions in terms of angular velocity and acceleration in the following way:

$$\begin{equation} x(t)={ {\mu _x}(t-t_0)+{{\dot{\mu }_x} \over {2}}(t-t_{\mathrm{mid}})^2 }, \end{equation} \tag{ 21 }$$

$$\begin{equation} y(t)={{\mu _y}(t-t_0)+{{\dot{\mu }_y} \over {2}}(t-t_{\mathrm{mid}})^2 }. \end{equation} \tag{ 22 }$$

Here, μ_x and μ_y refer to the proper motions in the x and y directions, t₀ is the time of component ejection, and t_mid is the midpoint of the time span of the observations of the given component.

The accelerations can then be transformed to a coordinate system with components perpendicular and parallel to the radial motion (if this direction is presumed constant):

$$\begin{equation} {\dot{\mu }_\parallel }={\dot{\mu }_x}{\sin \phi }+{\dot{\mu }_y}{\cos \phi }, \end{equation} \tag{ 23 }$$

$$\begin{equation} {\dot{\mu }_\perp } ={\dot{\mu }_x}{\cos \phi }-{\dot{\mu }_y}{\sin \phi }. \end{equation} \tag{ 24 }$$

Here, ϕ refers to the angle between the velocity vector and the y-axis.

In this context, the parallel accelerations correspond to either a speeding up or a slowing down in the radial direction or a change in the angle to the line of sight (or both), and the perpendicular accelerations are those that cause bending in the azimuthal direction (in reference to the line of sight). We caution readers not to confuse this geometry with the geometry of the jet itself. A more detailed description, including diagrams, can be found in Homan et al. (2009).

The uncertainties of the accelerations are calculated in a manner similar to that of Homan et al. (2001). That is, there are no initial assumptions regarding the uncertainties of the input component positions. Rather, the positions that go into the fit are initially weighted by unity, and then iteratively weighted by whichever uncertainty values are needed to bring the reduced χ² of the fit down to approximately one. Though this method does not allow us to conclude that the assumed model is the correct one, the estimated uncertainties are valid if the model outlined above is the appropriate choice (Lister et al. 2009b).

We can determine the consistency of observed values with theoretically determined accelerations following the procedure of Homan et al. (2009). They use the following formulae to predict parallel and perpendicular accelerations and transform them into expressions for changes in the proper motion:

$$\begin{equation} {d \beta _{\parallel, {{\rm obs}} }\over dt_{{\rm obs}}}={{\dot{\beta }{\sin } \, \vartheta +\beta \dot{\vartheta }({\cos }\,\vartheta -\beta)} \over {(1-\beta {\cos } \, \vartheta)^3}}, \end{equation} \tag{ 25 }$$

$$\begin{equation} {d \beta _{\perp,{{\rm obs}}} \over dt_{{\rm obs}}}={{ \beta \dot{\varphi }{\sin }\, \vartheta } \over {(1-\beta {\cos } \, \vartheta)^2}}, \end{equation} \tag{ 26 }$$

$$\begin{equation} { \dot{\mu }_{\parallel }}={\mu \, {{d \beta _{\parallel, {{\rm obs}}} \over dt_{{\rm obs}}}}}{\left[(1+z){\beta _{{\rm app}}}\right]}^{-1}, \end{equation} \tag{ 27 }$$

$$\begin{equation} {\dot{\mu }_{\perp }}={\mu \, {{d \beta _{\perp,{{\rm obs}}} \over dt_{{\rm obs}}}}}{\left[(1+z){\beta _{{\rm app}}}\right]}^{-1}. \end{equation} \tag{ 28 }$$

Here, β is the speed as a fraction of the speed of light, $\dot{\beta }$ is the time derivative of β, ϑ is the viewing angle, $\dot{\vartheta }$ is the time derivative of this angle, and $\dot{\varphi }$ is the time derivative of the azimuthal angle relative to an axis along the direction of the viewing angle. $\dot{\mu }_{\parallel }$ and $\dot{\mu }_{\perp }$ refer to the parallel and perpendicular components of acceleration in angular units (mas yr⁻²), and μ is the proper motion in mas yr⁻¹.

Understanding these formulae above requires some constraints on the time derivatives $\dot{\beta }$, $\dot{\vartheta }$, and $\dot{\varphi }$. These cannot be directly observed but further constraints can be put on these values by examining the proper motion diagrams (Figures 2–5) and light curves (Figures 6–9), as we discussed above. To aid in the interpretation of the results we note that a non-zero perpendicular acceleration only corresponds to a bend of the component direction in the azimuthal direction, with respect to the radial velocity vector. There are no consequences for the bulk speed. Interpreting the parallel acceleration is more complex. In the limiting case for which $\dot{\beta }=0$ (no changes in bulk speed), an increase in apparent parallel velocity can still occur if $\dot{\vartheta }>0$ and β < cos  ϑ or when $\dot{\vartheta }< 0$ and β > cos  ϑ. For the limiting case of $\dot{\vartheta }=0,$ an increase in apparent velocity can still occur for $\dot{\beta }>0$. For the case of decreasing apparent velocity, the same relationships hold with a simple reversal of the inequality sign above for $\dot{\vartheta }$.

Using Equations (27) and (28) above, we determine that the predicted theoretical acceleration over the entire interval of observations is approximately equal to those observed if the values of the parameters in Equations (25) and (26) equal those presented in Table 3. In Table 3, Column 1 identifies the component being considered, $\dot{\mu }_\parallel$ in Column 2 and σ_∥ in Column 3 refer to the acceleration of proper motion in the direction parallel to the proper motion vector and the associated uncertainty. $\dot{\mu }_\perp$ in Column 4 and σ_⊥ in Column 5 refer to the acceleration of proper motion in the direction perpendicular to the proper motion vector and the associated uncertainty. t₁ in Column 6 and t₂ in Column 7 refer to the relevant time interval of the data being modeled, β in Column 8, $\dot{\beta }$ in Column 9, ϑ in Column 10, $\dot{\vartheta }$ in Column 11, and $\dot{\varphi }$ in Column 12 have the same meaning as in Equations (25) and (26) above. ϑ is taken to be the initial value from Table 1 and β is calculated from Γ (given in Table 1) using Equation (3).

Table 3. Acceleration Model Parameters

Component	$\dot{\mu }_\parallel$	σ∥	$\dot{\mu }_\perp$	σ⊥	t1	t2	β	$\dot{\beta }$	ϑ	$\dot{\vartheta }$	$\dot{\varphi }$
	(mas yr−1)	(mas yr−1)	(mas yr−1)	(mas yr−1)				(yr−1)		(rad yr−1)	(rad yr−1)
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)	(12)
C1	0.0004	0.0037	0.0130	0.0037	1995.57	2010.46	0.998	2.0 × 10−8	0.019	−9.3× 10−8	1.6 × 10−4
C5	−0.0189	0.0042	−0.0001	0.0042	1999.37	2010.46	0.9996	−5.0 × 10−9	0.005	−1.0 × 10−7	−1.8 × 10−7
C6	0.2916	0.0290	0.0284	0.0290	2000.04	2003.89	0.9994	9.9 × 10−7	0.003	1.5 × 10−7	1.7 × 10−4
C8	0.1245	0.0755	0.3821	0.0755	2003.45	2005.45	0.9945	5.3 × 10−5	0.008	1.0 × 10−4	6.7× 10−2

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We can check for some consistency between the methods in Sections 3.1, 3.2, and 3.5 in the following manner. Using the results in Section 3.1 (Table 1) we can estimate the overall change in viewing angle, Δϑ, for each component. We can then divide by the change in time, Δt, to estimate $\dot{\vartheta }$ determined in Section 3.3; however, we must correct for the relativistic motion of the component into the line of sight and divide the observed time interval by (1 + z)(1 − βcos ϑ) now to get Δt'. We can estimate $\dot{\beta }$ in similar manner. There are some caveats. For instance, both β and ϑ may be changing rapidly over the range of observations. Therefore, the values used for these quantities in further calculations are likely only to be rough averages over the entire time range of our observations. Performing these estimations leads to mixed results. For instance, for C5, we estimate Δϑ/Δt' to be approximately −2.3 × 10⁻⁷ rad yr⁻¹. As compared to $\dot{\vartheta }$ in Table 3, there is a 130% difference. In repeating for Δβ/Δt', we estimate −5.8 × 10⁻⁹. As compared to $\dot{\beta }$ in Table 3, there is a 16% difference. However, this is the case with the best agreement between these methods. At the other extreme, for C1, we estimate (Δϑ/Δt') = 2.5 × 10⁻⁶, which has a 2600% difference with $\dot{\vartheta }$ in Table 3. Comparison of (Δβ/Δt') = −5.6 × 10⁻⁷ with $\dot{\beta }$ shows a 2900% difference. Though the results are mixed, they are promising in that they can be self-consistently used to make a first estimate of $\dot{\vartheta }$ and $\dot{\beta }$, to be refined by the methods in Section 3.5. Such a method may also be used to assist in rejecting our simple acceleration model, for cases such as C5, for which discrepancies are large.