X-RAY RADIATION MECHANISMS AND BEAMING EFFECT OF HOT SPOTS AND KNOTS IN ACTIVE GALACTIC NUCLEAR JETS

Hot spots and knots in large-scale jets have been observed in many active galactic nuclei (AGNs). Hot spots are often found near the outmost boundaries of radio lobes. They are regarded as jet termination (Fanaroff & Riley 1974; Blandford & Rees 1974; Begelman et al. 1984; Bicknell 1985; Meisenheimer et al. 1989). Knots are usually thought to be a part of a well-collimated jet (e.g., Harris & Krawczynski 2006). Some of them are also detected in the optical and X-ray bands. With high spatial resolution and sensitivity, the Chandra X-ray Observatory opened a new era to study the X-ray emission of hot spots and knots. It revealed that bright radio knots and hot spots in radio galaxies and quasars are also often detected in the X-ray band (Wilson et al. 2001; Hardcastle et al. 2004; Kataoka & Stawarz 2005; Tavecchio et al. 2005).

The X-ray radiation mechanisms of hot spots and knots are highly debated. It is generally believed that synchrotron radiation of relativistic electrons is responsible for the radio emission. The detection of high polarization in the optical emission indicates that the optical emission is also from synchrotron radiation (Roser & Meisenheimer 1987; Lähteenmäki & Valtaoja 1999). It is uncertain whether the X-ray emission is a simple extrapolation of the synchrotron radiation for the radio-to-optical emission. Indeed, the X-rays of some hot spots (such as hot spots N1 and N2 in 3C 33; Kraft et al. 2007) and knots (such as the knots in 3C 371 and M87; Sambruna et al. 2007; Liu & Shen 2007) can be interpreted as synchrotron radiation from the same electron population responsible for the radio and optical emission. However, the X-ray emission of some hot spots and knots is apparently not a simple extrapolation of the radio-to-optical component (Kataoka & Stawarz 2005). An inverse-Compton scattering (IC) component is necessary to model the X-rays. As shown by Stawarz et al. (2007), the X-ray emission of hot spots in Cygnus A is well interpreted with the synchrotron-self-Compton (SSC) scattering model. The IC scattering of cosmic microwave background (IC/CMB) may also significantly contribute to the observed X-ray emission, if the bulk flow of the material in hot spots and knots is relativistic (Georganopoulos & Kazanas 2003).

Broadband spectral energy distribution (SED) places strong constraints on the radiation mechanism models. Systematical analysis on the SEDs in the radio and X-ray bands of hot spots, knots, and lobes were present by Hardcastle et al. (2004) and Kataoka & Stawarz (2005). Note that the optical data are critical to characterize the SEDs and may give more constraints on the models. In this paper, we compile a large sample of the observed SEDs in the radio-optical-X-ray band of hot spots and knots from the literature, and fit them with various models in order to study the radiation mechanisms of the X-rays and to reveal the differences of the two kinds of sources.

The magnetic field and Doppler boosting effect are crucial ingredients in SED modeling. The energy equipartition assumption between magnetic field and radiating electrons is usually accepted in discussion of the energetics and dynamics of radio sources. Hardcastle et al. (2004) reported that most radio-luminous hot spots can be explained with the SSC model under this assumption. With the equipartition condition, the derived magnetic field of lobes is ∼10⁻⁶ G (Kataoka & Stawarz 2005), being comparable to the strength of the magnetic field in the interstellar medium. Since hot spots are believed to be the terminal of a relativistic jet, the Doppler boosting effect would be less prominent than that in knots, but their magnetic fields may be magnified up to ∼10⁻⁴ G by strong external shocks produced by interaction of a relativistic jet with circum medium (Kataoka & Stawarz 2005). Therefore, the Doppler boosting effect and co-moving magnetic field are essential to discriminate the two kinds of sources, if they are physically different. With our detailed SED fits, we compare their Doppler factor (δ) and co-moving magnetic field (B') between the knots and hot spots in our sample under the equipartition assumption.

The observed SEDs and our data analysis are present in Section 2. Models and our SED fits are shown in Section 3. Conclusions and discussion are present in Section 4. Throughout, H₀ = 70 km s⁻¹ Mpc⁻¹, Ω_Λ = 0.7, and Ω_m = 0.3 are adopted.

Twenty-two hot spots and 45 knots from 35 AGNs (15 radio galaxies, 16 quasars, three BL Lac objects, and one Seyfert galaxy; see Table 1) are included in our sample. Most of them are taken from XJET Home Page.⁶ Observations for these hot spots and knots are summarized in Table 2, and their SEDs are displayed in Figure 1. We show the distributions and the correlations of the spectral indices in the radio and X-ray bands (α_r and α_X) in Figure 2. It is found that α_r is not correlated with α_X. The α_r of both the knots and hot spots are smaller than 1. The α_X narrowly clusters at 0.7–1.2 for the hot spots, but it ranges in 0.2–2.4 for the knots. We test whether α_r and α_X distributions show statistical differences between the two kinds of sources with the Kolmogorov–Smirnov test (K–S test), which yields a chance probability p_KS. A K–S test probability larger than 0.1 would strongly suggest no statistical difference between two distributions. We get p_KS = 0.37 and p_KS = 0.41 for the radio spectral indices and the X-ray spectral indices between the two samples, respectively, indicating that no statistical difference is found for the α_r and α_X distributions of two kinds of sources.

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Figure 3(a) shows the correlations between the observed luminosities at 5 GHz and 1 keV (L_5 GHz and L_1 keV) for the hot spots and the knots. It is found that L_1 keV is tightly correlated with L_5 GHz. We measure the correlations with the Spearman correlation analysis, which yields log L_1 keV = (6.2 ± 4.7) + (0.84 ± 0.11)log L_5 GHz with a correlation coefficient r = 0.86 and a chance probability p < 10⁻⁴ for the hot spots and log L_1 keV = (6.9 ±  2.2) + (0.85 ±  0.05)log L_5 GHz with r = 0.92 and p < 10⁻⁴ for the knots. The slopes of the two correlations are the same in the error scopes, but averagely speaking, the X-ray luminosity of the knots is larger than that of the hot spots with ∼0.7 order of magnitude, indicating systematical difference between the two kinds of sources. As seen in Section 3.3, this correlation is much tighter by correcting with the Doppler boosting effect (see Figure 3(b) and details in Section 3.3). The knots and the hot spots in the L_5 GHz–L_1 keV plane are roughly separated with a division line of L_1 keV = L_5 GHz. Therefore, the ratio of R_L ≡ L_5 GHz/L_1 keV is a characteristic to distinguish the hot spots and the knots. This ratio should be an intrinsic parameter independent of the Doppler boosting effect and the cosmological effect. It may reflect the properties of the radiation regions.

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The tight L_5 GHz–L_1 keV correlation indicates that the radiations in the two energy bands may be produced by the same electron population. For few cases, such as hot spots N1 and N2 in 3C 33 and knots in M87 (Kraft et al. 2007; Liu & Shen 2007), their X-ray spectra are soft with α_X > 1, and smoothly connect to the spectrum in the radio and optical band. The X-rays of these sources may be the high energy tail of the synchrotron radiation by the same electron population for the radio and optical emission. The synchrotron radiation model is preferred to fit the X-rays of these sources.

Some well-sampled SEDs in Figure 1 roughly show two bumps similar to that observed in blazers. The two-bump feature is generally interpreted with the synchrotron radiation and IC scattering by the relativistic electrons. Therefore, we fit these SEDs with single-zone synchrotron + IC scatting models. The IC seed photons may be originated from the synchrotron radiation itself (SSC) or from the CMB. The photon field energy density of synchrotron radiation in the co-moving frame is given by U'_syn = L_syn/(4πR²cδ⁴) ≈ 2.65 × 10⁻¹²L_syn,40R⁻²₂₀δ⁻⁴ erg cm⁻³, where Q_n = Q/10ⁿ in cgs units, δ is the beaming factor, R is the radius of the radiation region, and c is the speed of light. The energy density of the CMB is U_CMB ≈ 4 × 10⁻¹³(1 + z)⁴Γ² erg cm⁻³, which dramatically increases with the redshift z of the sources and the bulk Lorentz factor Γ (taking Γ ≃ δ) of the radiation site. Without considering the Doppler boosting effect, the IC component should be dominated by the SSC process, but it may be dominated by the IC/CMB, if the source is relativistic motion. We take the two scenarios into account.

In our models, the radiation region is assumed to be a homogeneous sphere with radius R. The radius is derived from the angular radius θ (see Table 2), which is obtained from the optical or the X-ray observations. Considering the beaming effect, R and volume V of the emitting region are needed to take a relativistic transformation. We simply assume that the emitting region still is a sphere with V' = V/δ and derive the radius of the emitting region in co-moving frame by R' = (3V'/4π)^1/3. The electron distribution as a function of electron energy (γ) is taken as a single power law or a broken power law,

$$\begin{equation} N(\gamma)= N_{0}\left\lbrace \begin{array}{@{}ll} \gamma ^{-p_1} & \mbox{ $\gamma \le \gamma _b$}, \\ \gamma _b^{p_2-p_1} \gamma ^{-p_2} & \mbox{ $\gamma > \gamma _b$,} \end{array} \right. \end{equation} \tag{ 1 }$$

where p_1,2 = 2α_1,2 + 1 are the energy indices of electrons below and above the break energy γ_b, and α_1,2 are the observed spectral indices.

In our calculations, the Klein–Nishina effect for the radiation in the GeV–TeV band is considered, but the absorption in the GeV–TeV band by the infrared background light and by CMB photon during the gamma-ray photons propagating to the Earth (Stecker et al. 2006) is not taken into account.

3.1. Equipartition Magnetic Field and the Synchrotron Radiation Model

As mentioned in Section 1, the equipartition condition, which assumes that the magnetic field energy density U_B is equal to the electron energy density U_e, is usually adopted in discussion of the X-ray origin. We first derive the magnetic field strength B^{δ = 1}_eq (see Appendix A) under this condition for the hot spots and knots in our sample without considering the beaming effect (δ = 1). The calculation of B^{δ = 1}_eq depends on γ_min (see Equations (A5) and (A10)). The γ_min is quite uncertain (e.g., Harris & Krawczynski 2006). The γ_min values of 12 hot spots and all the knots in our sample are constrained with the observed SEDs via a method reported by Tavecchio et al. (2000). The average of γ_min for the 12 hot spots is ∼200. For those hot spots that their γ_min lost constraints from the observed SEDs, we take γ_min = 200 in our calculation.

We fit the observed SEDs in the radio-optical band with the synchrotron radiation model to derive B^{δ = 1}_eq. Our results are reported in Table 2. They are roughly consistent with the results derived from the formulae given by Brunetti et al. (1997). The distributions of B^{δ = 1}_eq for both the hot spots and knots are shown in Figure 4(a). They range in 10–700 μG. No systematical difference of B^{δ = 1}_eq is found between the two kinds of sources.

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The observed X-ray spectra of the knots in 4C 73.18 (K-A), PKS 0521 (K), M87 (K-A, B, C1, D, E, F), 3C 31 (K), 3C 66B (K-B), and 3C 120 (K-K7) smoothly connect to the spectra in the radio and optical bands. They are well fit with the synchrotron radiation model (the thin solid line in Figure 1), indicating that the X-rays of these sources are the high energy tail of the synchrotron radiation by the same electron population for the radio and optical emission. We do not take these sources into account in our following analysis.

3.2. Representing the X-rays with SSC Model for the Sources at Rest

The observed X-ray spectral indices and SEDs shown in Table 2 and Figure 1 indicate that the X-rays of most hot spots and knots should be contributed by IC scattering. We model the SEDs with the synchrotron + IC model assuming that the sources are at rest. In this scenario, the SSC process should dominate the IC process, as mentioned above. Although the contribution of IC/CMB to the X-ray emission is not negligible for some sources at high redshift, we only consider the SSC component in this section.

We first calculate the X-ray flux density at 1 keV (F^eq_1 keV) with the synchrotron + SSC model under the equipartition assumption, i.e., B = B^{δ = 1}_eq. Our results are reported in Table 2. We measure the consistency between F^eq_1 keV and F^obs_1 keV with ratio R_F ≡ F^obs_1 keV/F^eq_1 keV, where F^obs_1 keV is the observed flux density at 1 keV. It is found that R_F is much larger than 1 for almost all the hot spots and knots in our sample, indicating that the observed X-ray flux density is much larger than the model prediction. The distributions of R_F, and R_F as a function of L_5 GHz, L_1 keV, and R_L for the hot spots and knots are shown in Figure 5. A tentative correlation presented in the R_F–L_5 GHz plane shows that the brighter sources in the radio band tend to be more consistent with the equipartition condition (see Figure 5(b)). However, no similar feature is seen for the X-ray bright sources (see Figure 5(c)). It is interesting that R_F is anti-correlated with R_L, and both the hot spots and knots shape a well sequence (see Figure 5(d)). The hot spots are at the lower end of the sequence and they tend to be closer to the equipartition condition than the knots.

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As shown above, the derived X-ray flux densities from the SSC model under the equipartition condition significantly deviate the observations, especially for the knots. In order to model the observed SEDs with the synchrotron + SSC model, we have to get rid of this assumption. Keeping the model parameters the same as that used above, we fit the SEDs with this model and derive the magnetic field strengths (B^{δ = 1}_ssc). The fits are shown in Figure 1 (the thick solid line). Although the SSC model can represent the observed flux at 1 keV for the knots in 4C 73.18 (K-A), PKS 0521, M87 (K-A, B, C1, D, E, F), 3C 31, 3C 66B (K-B), 3C 120 (K-K7), 3C 273 (K-C1, C2, D1, D2H3), the bow ties defined by the errors of the X-ray spectral indices clearly rule out this model for these knots. As discussed in Section 3.1, the X-rays of these knots are well fitted by the synchrotron radiation model except knots in 3C 273. We do not include these knots in our following statistics.

The derived B^{δ = 1}_ssc are listed in Table 2, and their distributions are shown in Figure 4(b). It is found that the B^{δ = 1}_ssc of the knots are much smaller than the hot spots, with medians of 1 μG and ∼30 μG for the knots and hot spots, respectively. Comparing B^{δ = 1}_ssc with B^{δ = 1}_eq, it is found that they are roughly consistent for the hot spots, indicating that the X-rays of the hot spots can be roughly fitted with the SSC model under the equipartition condition. However, the B^{δ = 1}_ssc of the knots are much smaller than B^{δ = 1}_eq, even unreasonably smaller than the magnetic field strength of the interstellar medium for some knots. Therefore, the X-rays of these knots may not be dominated by the SSC component.

To investigate the deviation of B^{δ = 1}_ssc to B^{δ = 1}_eq for individual source, we define ratio R_B ≡ B^{δ = 1}_eq/B^{δ = 1}_ssc, which is physically the same as R_F. Similar to R_F, R_B is much larger than 1 for almost all the sources, especially for the knots. The distributions of R_B and R_B as a function of L_5 GHz, L_1 keV, and R_L are shown in Figure 6. We find the same features as shown in Figure 5.

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3.3. Modeling the X-rays by Considering Relativistic Bulk Motion

It is generally believed that the knots should have relativistic motion. The hot spots may also be relativistic (Dennett-Thorpe et al. 1997; Tavecchio et al. 2005; Harris & Krawczynski 2006). In this section, we fit the SEDs with the synchrotron + IC model by considering the beaming effect under the equipartition condition. In this scenario, the IC/CMB process should dominate the IC component. Although the contribution of SSC is negligible comparing with IC/CMB in this case, we still take SSC process into account in our calculation.

Our fits are shown in Figure 1 (the dashed line). The distributions of δ for the knots and hot spots are shown in Figure 7. It is found that, averagely, δ ∼ 10 for most of the knots and δ ∼ 5 for most of the hot spots. Some sources in our sample are included in Kataoka & Stawarz (2005). We compare our results of δ for these sources with that reported by Kataoka & Stawarz (2005) (δ^KS05) in Figure 8. They are roughly consistent.

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The B'_eq distributions with comparison to B^{δ = 1}_eq and B^{δ = 1}_SSC are shown in Figure 4(c). The B'_eq distributions are more consistent with B^{δ = 1}_eq than B^{δ = 1}_SSC. Both B'_eq and B^{δ = 1}_eq distributions approximately span an order of magnitude, much narrower than that of B^{δ = 1}_SSC, especially for the knots. All B'_eq are larger than the magnetic filed strength of the interstellar medium, implying that the magnetic filed of the interstellar medium would be amplified in the knots and hot spots by the turbulence of the relativistic shocks. The B'_eq of the knots are smaller than that of the hot spots, with typical values of 10 μG and 40 μG for the knots and the hot spots, respectively, favoring the idea of different origins of the shocks (internal versus external) for the two kinds of sources (e.g., Harris & Krawczynski 2006).

As our discussed in Section 2, R_L is an intrinsic characteristic for the sources. It may be a representative of the co-moving magnetic field and the Doppler boosting effect of the sources. We show R_L as a function of B'_eq and δ in Figure 9. It is found that R_L is correlated with B'_eq for the knots, with a linear coefficient of r = 0.77 and chance probability p < 10⁻⁴. We obtain log R_L = (−2.23 ± 0.24) + (1.45 ± 0.23)log B'_eq. No significant correlation between R_L and B'_eq is found for the hot spots. However, both the knots and hot spots form a sequence in the R_L–B'_eq plane, with a best linear fit log R_L = (−2.55 ± 0.23) + (1.92 ± 0.17)log B'_eq. The hot spots locate at the higher end of the sequence. Similar feature is also observed in the R_L–δ plane, as shown in Figure 9(b). These results imply that R_L would be determined by both B'_eq and δ. The strong anti-correlation of R_F–R_L (or R_B–R_L) shown in Figure 5 (or Figure 6) may be due to neglect of the beaming effect. This effect plays important role on the observed flux since the observed flux is proportional to δ⁴. Correcting by the Doppler boosting effect, we show the L'_5 GHz–L'_1 keV relations in Figure 3(b). The relations are tighter than the observed ones, with a linear coefficient of r = 0.98 and r = 0.90 for the knots and hot spots, respectively.

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We have present extensive analysis and SED fits for 22 hot spots and 45 knots in 35 AGN jets. We find that L_5 GHz and L_1 keV are tightly correlated. The two kinds of sources can be roughly separated with a division of L_1 keV = L_5 GHz. Our SED fits show that the mechanisms of the X-rays are diverse. While the X-ray emission of a small fraction of the sources is a simple extrapolation of the synchrotron radiation for the radio-to-optical emission, the IC component may dominate the observed X-rays for most of the sources. Without considering the relativistic bulk motion, the SSC model can explain the X-rays for some hot spots with B^{δ = 1}_ssc being consistent with the equipartition magnetic field B^{δ = 1}_eq in 1 order of magnitude, but an unreasonably low magnetic field strength is required in modeling the X-rays for all knots with this model. Considering relativistic bulk motion for the sources, the IC/CMB dominated model well explains the X-ray emission for most sources under the equipartition condition. Although the derived B'_eq and δ for some hot spots are comparable to that of the knots, the B'_eq value for the knots tends to be smaller than that of the hot spots and the δ tends to be larger, favoring the idea that the hot spots are jet termination and knots are a part of a well-collimated jet. Corrected by the beaming effect, the L'_5 GHz–L'_1 keV relations for the two kinds of sources are even tighter than the observed ones, indicating that the correlations are intrinsic. The ratio R_L is correlated with B'_eq and δ. These facts suggest that, under the equipartition condition, the differences on the X-ray observations for the knots and hot spots would be mainly due to the differences of the Doppler boosting effect and the co-moving magnetic field, although some hot spots have similar feature to the knots.

The R_L may be an indicator of $B_{\rm eq}^{{\prime }}$ and δ. It is an intrinsic parameter independent of the Doppler boosting and the cosmological effects. Our results show that the X-rays of a hot spot with larger R_L are better to be fitted with the SSC model under equipartition condition without considering the beaming effect. This is consistent with that reported by Hardcastle et al. (2004), who found that the X-rays of the radio-bright hot spots can be explained with the SSC model under equipartition condition. The strong anti-correlation between R_B (or R_F) and R_L may also offer a tool to discriminate the two kinds of sources. We find in the Figures 3, 5, and 6 that the hot spot H-A and the knot K-B⁷ in PKS B1421 − 490 are significant outliers. They were reported as knots by Gelbord et al. (2005). The knot K-B is very peculiar for its extreme optical output, with a ratio of knot/core optical flux ∼300. Gelbord et al. (2005) suspected that it is a core between components A and C. We find that K-B is a significant outlier in the figures and K-A resembles a hot spot. Most recently, Godfrey et al. (2009) confirmed that the K-B is a core and the K-A is a hot spot with very long baseline interferometry (VLBI) observations.

The B'_eq value for the knots tends to be smaller than that of the hot spots and the δ tends to be larger, favoring the idea that the hot spots are jet termination and knots are a part of a well-collimated jet. However, the B'_eq and δ for some hot spots are comparable to that of the knots, making uncertainty on identifying a component as a knot or a hot spot. For example, the northeast double hot spots in 3C 351, which locate at the outer boundary of a lobe, have relativistic motion feature, hence may be identified as knots of the jet (Harris & Krawczynski 2006).

The synchrotron radiation in the optical band indicates that there are relativistic electrons existed in these extended regions (Roser & Meisenheimer 1987; Lähteenmäki & Valtaoja 1999). Moreover, the X-ray emission of some sources may be also synchrotron radiation as mentioned in Section 3.1. Assuming a magnetic field strength B ∼ 10⁻⁵ G, one can estimate the energy of relativistic electrons is γ ∼ 10⁶, which contribute to the optical emission by the synchrotron process. These electrons may interact with the synchrotron photons and external field photons to produce very high-energy γ-ray photons by IC scattering. As shown in Figure 1, both the SSC and IC/CMB models predict a prominent GeV–TeV component in the SEDs of some sources. We check if the predicted GeV–TeV emission can be detectable with H.E.S.S. and Fermi/LAT, and also show the sensitivity curves of H.E.S.S. and Fermi/LAT in Figure 1 for these sources.⁸ The detections of these high energy emission would place much stronger constraints on the radiation mechanisms and on the physical parameters of these sources. The origin of the high energy TeV gamma-ray emission is also a debating issue, and detections of these high energy emission would drastically improved our view of the universe (see Cui 2009 for a review).

Note that our one-single zone lepton models cannot explain the observed SEDs for the four knots in 3C 273 (K-C1, K-C2, K-D1, and K-D2H3). Jester et al. (2006) had reported that the X-ray spectra rule out the single-zone model of X-ray emission for some jet knots in 3C 273. It is possible that these sources may have a complex structure as the western hot spot in Pictor A (Zhang et al. 2009).

We thank the anonymous referee for his/her valuable suggestions. This work was supported by the National Natural Science Foundation of China (grants 10778702, 10533050, 10873002), the National Basic Research Program ("973" Program) of China (2009CB824800), and the West PhD project of the training Programme for the Talents of West Light Foundation of the CAS. J.M.B. thanks supports of the Bai-Ren-Ji-Hua and the Zhong-Yao-Fang-Xiang (grant KJCX2-YW-T21) projects of the CAS.

Under the equipartition condition, we have

$$\begin{equation} \frac{B_{\rm eq}^{2}}{8\pi }=U_{e}=m_{e}c^{2}\int ^{\gamma _{\max }}_{\gamma _{\min }}N(\gamma)(\gamma -1)d\gamma. \end{equation} \tag{ A1 }$$

The peak frequency of the synchrotron radiation is given by

$$\begin{equation} \nu _{s}=\frac{4}{3}\nu _B\gamma _{b}^{2}\frac{\delta }{1+z}, \end{equation} \tag{ A2 }$$

where ν_B = 2.8 × 10⁶B Hz is the Larmor frequency in the magnetic field B. The luminosity of the synchrotron radiation is derived from

$$\begin{equation} L_{\rm syn}=\int ^{\gamma _{\max }}_{\gamma _{\min }}N(\gamma)P(\gamma)Vd\gamma, \end{equation} \tag{ A3 }$$

where P is the radiation power of single electron, P = 1.1 × 10⁻¹⁵γ²B² erg s⁻¹, and $V=\frac{4}{3}\pi R^{3}$ is the volume of radiation region. Without considering the beaming effect (δ = 1) and assuming p₂ > 3>p₁ > 2, B_eq is expressed as

$$\begin{equation} B_{\rm eq}=\bigg (\frac{A_{1}L_{\rm syn}}{A_{3}A_{2}^{(3-p_{1})/2}}\bigg)^{\frac{2}{5+p_{1}}}, \end{equation} \tag{ A4 }$$

where A₁, A₂, and A₃ are given by

$$\begin{equation} A_{1}=8\pi m_{e}c^{2}\bigg (\frac{\gamma _{\min }^{2-p_{1}}}{p_{1}-2}-\frac{\gamma _{\min }^{1-p_{1}}}{p_{1}-1}\bigg), \end{equation} \tag{ A5 }$$

$$\begin{equation} A_{2}=\nu _{s}/3.7\times 10^{6}, \end{equation} \tag{ A6 }$$

and

$$\begin{equation} A_{3}=1.1\times 10^{-15}V\frac{p_{1}-p_{2}}{(3-p_{1})(3-p_{2})}. \end{equation} \tag{ A7 }$$

For the case of p₂ > 3 and p₁ < 2, B_eq can be calculated by

$$\begin{equation} B_{\rm eq}^{7/2}=\frac{L_{\rm syn}A_{1}A_{2}^{(p_{1}-3)/2}}{A_{3}}B_{\rm eq}^{\frac{2-p_{1}}{2}}+\frac{A_{0}L_{\rm syn}}{A_{3}A_{2}^{1/2}}, \end{equation} \tag{ A8 }$$

where

$$\begin{equation} A_{0}=8\pi m_{e}c^{2}\frac{p_{1}-p_{2}}{(2-p_{1})(2-p_{2})}. \end{equation} \tag{ A9 }$$

For the case of p₁ = 2, A₁ in Equation (A4) is

$$\begin{equation} A_{1}=8\pi m_{e}c^{2}\bigg (\frac{1}{p_2-2}-\ln \gamma _{\rm min}-\frac{1}{\gamma _{\rm min}}\bigg). \end{equation} \tag{ A10 }$$

If the beaming effect is considered, we have L'_syn = L_syn/δ⁴, V' = V/δ. The equipartition magnetic field hence is obtained with

$$\begin{equation} B_{eq}^{{\prime }}=B_{eq}/\delta ^{(p_{1}+3)/(p_{1}+5)}. \end{equation} \tag{ A11 }$$

This is consistent with the result presented by Stawarz et al. (2003), B'_eq = B_eq/δ^5/7 for p₁ = 2. It is generally believed the radio emission is produced by synchrotron radiation. We obtain the values of α_1,2, ν_s, and L_syn by fitting the observed radio (and optical) data using synchrotron radiation and calculate B_eq with Equations (A4), (A8), and (A11).