Constraining relativistic beaming model for γ-ray emission properties of jetted AGNs

1 Introduction

Active galactic nuclei (AGNs) denote a special class of galaxies that emit copious amounts of energy due to an actively accreting central supermassive (≳10⁶ M _⊙) black hole (SMBH). They emit across the entire observable electromagnetic spectrum and constitute the largest fraction of sources in any extra-galactic survey (Ackermann et al. 2015). At γ-ray energies covered by the Fermi-Large Area Telescope (Fermi-LAT), AGNs constitute about 50% of the total population in the latest release (4FGL), and more than 75% are high energy (50 MeV to 1 TeV) sources (Abdollahi et al. 2020, Ajello et al. 2020). AGNs show bimodality in the distribution of their radio-loudness parameter R _L defined as the ratio of the radio emission intensity to the optical emission intensity. Few AGNs (∼10–15%) are known as radio-loud with R _L ≥ 10, while ∼85% with R _L < 10 are radio-quiet AGNs (Kellermann et al. 1989, Xu et al. 1999, Zhang et al. 2021). Most radio-loud AGNs have prominent and well-collimated relativistic jets, while radio-quiet AGNs have relatively weak jets with negligible emissions (Tarchi et al. 2011; Doi et al. 2013). Blazars are the most extreme subtype of radio-loud, jetted AGNs whose relativistic jets are aligned to observers’ lines of sight (Urry and Padovani 1995). Blazars are classified into flat-spectrum radio quasars (FSRQs) and BL Lacertae objects (BL Lacs) based on the rest-frame equivalent width (EW) of the optical emission line. FSRQs have broad emission lines with EW >5 Å, while BL Lacs have weak or no emission lines with EW ≤5 Å (see Marcha et al. 1996, Xiong and Xiong 2014). The ratio of broad emission line luminosity (L _BLR) to Eddington luminosity (L _Edd) has also been used to further classify blazars with FSRQs having L _BLR/L _Edd ≥ 5 × 10⁻⁴, while BL Lacs have L _BLR/L _Edd < 5 × 10⁻⁴ (Ghisellini and Celotti 2001, Ghisellini et al. 2011).

Narrow Line Seyfert 1 galaxies (NLSy1s) are another special class of radio-loud AGNs that show powerful relativistic jets, low black hole mass ranging between 10⁶ and 10⁸ M ⊙ and a high accretion rate of 0.1–1 L _Edd (Peterson and Wandel 2000). These sources have gained the attention of researchers in the past three decades due to their blazar-like characteristics with EW of the broad emission larger than 5 Å (Osterbrock and Pogge 1985, Yuan et al. 2008, Oshlack et al. 2001, Rakshit and Stalin 2017). They have prominent γ-ray cores and a reflection-dominated hard X-ray emission (see Fabian et al. 2009). Detailed studies of AGNs in the MeV–GeV–TeV regimes using Fermi-LAT found that blazars are strong γ-ray emitters and possess other outstanding characteristic properties such as rapid variability, high optical polarization, and apparent superluminal motion in the inner radio jets (Bassani et al. 1983, Yang et al. 2019). Thus, Doppler boosting is believed to be one of the important answers to the puzzle about the emission properties of blazars, as evidenced in the results recorded in the radio bands (see Qin and Xie 1998, Odo et al. 2012, 2015). Nevertheless, the information about the origin and evolution of these sources in the high-energy bands is one of the utmost tests that are faced by researchers in extragalactic astronomy. Although the systematic study of γ-ray emissions has not been very successful, with dedicated studies producing a handful of results (e.g., D’Ammando et al. 2019, Odo and Aroh 2020), comprehensive observations by the Fermi-LAT have increased our knowledge about the γ-ray emissions of AGNs (Ajello et al. 2020, Abdollahi et al. 2020). Also, the discovery of the powerful γ-ray emitting NLSy1s with blazar-like properties has provided a shred of substantial evidence about the emission properties of AGNs (see Foschini 2011, Ackermann et al. 2015; D’Ammando 2019, Chen et al. 2021, Foschini et al. 2021, Luashvili et al. 2023, Iyida et al. 2024).

Preliminary investigations on the γ-ray emissions of FSRQs and BL Lacs indicate a strong relationship between radio core dominance and γ-ray luminosity and these are crucial factors in the study of relativistic beaming and orientation effects in AGNs (Browne and Murphy 1987, Kembhavi 1993, Fan et al. 2005, Odo et al. 2015). Subsequently, the study of γ-ray emission from a sample of 80 blazars observed by Fermi-LAT yields a strong correlation between radio core-dominance R_r and the γ-ray luminosity, suggesting that the relativistic beaming effect can play a major role in the detection of radio emissions of AGNs (Liu et al. 2016, Chen et al. 2016, Pei et al. 2020b). Similarly, some authors (see von Montigny et al. 1995, Mattox et al. 1993) suggest that the γ-ray emissions of blazars originate from the relativistic jet and are strongly beamed similar to their radio emission counterparts. Also, Linford et al. (2011) found that the difference between γ-ray loud and radio-quiet FSRQs can be explained by Doppler boosting. Therefore, since astrophysicists appear to explore the fact that γ-ray emissions are associated with Doppler boosting, the relativistic beaming model and γ-ray emissions in jetted AGNs need to be thoroughly investigated on a large scale by using an updated sample of blazars and blazar-like NLSy1s. Inspired by the availability of the observed data of both blazars and NLSy1s by the Fermi-LAT, we aim to use the radio core-dominance parameter and γ-ray luminosity to statistically investigate the hypothesis that γ-ray emissions of blazars and NLSy1s are strongly beamed.

2 Theory of relativistic beaming in AGNs

The relativistic beaming and radio source orientation model predicts that radio sources that are viewed at small angles to the line-of-sight of the observer exhibit Doppler enhancement of the core emission relative to the extended emission component (see Orr and Browne 1982). This general relativistic concept has effectively explained the observational properties of jetted AGNs. The model involves studying the emission mechanisms from two luminosity components: the core (L _C) and extended (L _E) components (Orr and Brown 1982, Fan and Zhang 2003). The ratio of the two components, called the core-dominance parameter (R), is an essential statistical indicator of relativistic beaming and can be expressed as

where ϕ is the angle between the jet and the line of sight, β is the velocity of the radiating plasma (in m/s), R _T is the value of R at ϕ = 90^o, α is the spectral index, and n = 2 is a parameter that depends on the continuous jetted model of AGNs (see Lind and Blandford 1985, Odo et al. 2012, Iyida et al. 2024).

Meanwhile, the relativistic beaming effect, which can be quantified using the bulk Lorentz factor, is primarily described by the Doppler factor at a very small viewing angle (Bai and Lee 2001, Nieppola et al. 2008, Meyer et al. 2011) and is expressed as

where γ = ( 1 − β 2 ) − 1 / 2 is the bulk Lorentz factor of the jets of AGNs. Therefore, using this bulk Lorentz factor, R can be written (see, e.g., Ubachukwu et al. 2002, Odo et al. 2012) as

where δ n + α is the Doppler boosting factor while L _C and L _E are the core and extended luminosity components of AGNs, respectively. Eqs (1) and (3) suggest that the distribution of R should, in principle, provide the range of values of ϕ, which can be used to test the relativistic beaming hypothesis, provided that R _T is known. However, in most cases, particularly in synchrotron self-absorbed AGN cores (e.g., radio-loud quasars and blazars), the observed spectral index can approach zero, producing a flat or nearly flat radio spectrum. Thus, if we assume α = 0 and β ∼ 1, the mean angle to the line-of-sight (ϕ _m) for beamed sources whose radio axes lie closer to the line-of-sight (to a first approximation) can be obtained from Eq. (1):

Here, R _m is the mean value of the R distribution. Similarly, it can be inferred from Eq. (1) that maximum boosting occurs when the angle between the jet and the line of sight is zero. In this case, combining Eqs (1) and (3) yields the value for the maximum core-dominance parameter (R _max) given as

Moreover, numerous authors (see, e.g., Bridle et al. 1994, Fan and Zhang 2003, Pei et al. 2019, 2020, Iyida et al. 2021, 2024) have shown that the core-dominance parameter is a very reliable statistical indicator of the relativistic beaming hypothesis. In particular, it has been argued that R is strongly related to γ-ray luminosity (L _γ) in samples of jetted AGNs in a general form (e.g., Wu et al. 2014, Pei et al. 2020, Iyida et al. 2021) as

where λ is a sample-dependent factor. In effect, the jet model of γ-ray emission from AGNs leverages a correlation between the radio core-dominance parameter and γ-ray luminosity, and a correlation between the two parameters (Liu et al. 2016, Pei et al. 2020b) is an indication that γ-ray emission from these sources is strongly beamed. Thus, R – L _γ-ray correlations can also be used to study relativistic beaming hypotheses in samples of jetted AGNs.

3 Data sample and results

3.2 Analysis and results

The statistical analysis of the parameters of AGNs is essential in understanding their intrinsic relationships. We study these properties using the probability density function (PDF) and the two-dimensional Kolmogorov–Smirnov (K–S) test. The PDF is a key hypothesis test tool used to describe the distribution of probabilities over a continuous random value. On the other hand, the K–S test calculates the chance probability (p-value) for independent parameters and measures the null hypothesis that two samples originate from the same underlying distribution. The threshold for rejecting the null hypothesis is a confidence level of 95%, corresponding to a p-value of less than 0.05. The probability density distribution of the radio core-dominance parameter (R) for the sample of blazar on a logarithmic scale is shown in Figure 1a. The distribution is continuous, with more than 80% (∼426) of FSRQs, BL Lacs, and NLSy1s having log R > 0, with the values for FSRQs and BL Lacs extending by more than one magnitude. The values of log R for NLSy1s range from −0.45 to 2.24 and lie in between FSRQs and BL Lacs, while FSRQs and BL Lacs appear to have larger values, spreading from −3.56 to 3.46; this indicates the level of relativistic beaming in FSRQs and BL Lacs. However, we computed the mean FSRQs, BL Lacs, and NLSy1s values, and the results are shown in Table 1. However, to estimate the mean viewing angles for observing the γ-ray emissions of FSRQs, BL Lacs, and NLSy1s, the choice of R _T plays a key role (see Orr and Browne 1982, Kembhavi 1993, Zeng et al. 2022). A number of authors (e.g., Odo et al. 2015, Pei et al. 2020) have argued that R _T < 0.10 is compatible with the relativistic beaming of AGNs. Hence, using R _T = 0.024, which has been used consistently across different wavebands (Orr and Browne 1982, Fan et al. 2005, Pei et al. 2020, Iyida et al. 2021), we computed the mean viewing angles of each subsample using Eq. (4), and the results are provided in Table 1. It can be deduced from the table that FSRQs and BL Lacs with small viewing angles are strongly beamed compared to NLSy1s. Further statistical analysis on the distributions of R for FSRQs, BL Lacs, and NLSy1s was done using the K–S test. The cumulative distribution function (CDF) is shown in Figure 1b. In general, we found that at 95% confidence, there is approximately zero probability p ∼ 0, and there is no fundamental difference between the distributions of these objects in R. The NLSy1s appear not to be significantly different from blazars, which suggests that a relativistic beaming effect is observed in both samples of AGNs.

Figure 1

(a) PDF and (b) CDF of radio core-dominance parameters of FSRQs, BL Lacs, and NLSy1s.

Table 1

Mean values of R and viewing angles of FSRQs, BL Lacs, and NLSy1s

Samples	Mean value (R)	θ m
FSRQs	1.32 ± 0.08	21.4°
BL Lacs	1.21 ± 0.05	18.8°
NLSy1s	0.93 ± 0.21	27.5°

The PDF of the logarithm of the γ-ray luminosity parameter (L _γ-ray) for the present sample of blazars and NLSy1s is shown in Figure 2. There is a sequence that shows the level of emission of high-energy γ-rays among FSRQs, BL Lacs, and NLSy1s, with BL Lacs being the least emitter of γ-rays; however, NLSy1s appear to have the highest values, suggesting that NLSy1s are the highest γ-ray emitters of our sample followed by the FSRQs. Nonetheless, the distribution of our sample yield mean (logarithm) values ∼ 46.40 ± 0.20 for FSRQs, 45.39 ± 0.10 for BL Lacs, and 47.25 ± 0.20 for NLSy1s. A two-sample K–S test was carried out on log L _γ-ray data. The CDF is shown in Figure 2b. It was found that the distribution shows that the samples belong to different parent populations (p ≪ 10⁻⁵). However, the high values of NLSy1s indicate that they emit most in the γ-ray band compared to FSRQs and BL Lacs.

Figure 2

(a) PDF and (b) CDF of γ-ray luminosity of FSRQs, BL Lacs, and NLSy1s.

To further investigate the evolutionary relationship among FSRQs, BL Lacs, and NLSy1s, we show in Figure 3a the PDF of the spectroscopic redshift of our sample. From the figure, the ranges of redshift (z) of FSRQs, BL Lacs, and NLSy1s are −1.72 < (z) < 0.79, −2.53 < (z) < 0.63 and −1.84 < (z) < 0.45, respectively. A K–S test was performed on the redshift data. Generally, our results indicate that BL Lacs do not appear to be significantly different from NLSy1s (p ∼ 0.05), indicative of their cosmological distance. The CDF is shown in Figure 3(b).

Figure 3

(a) PDF and (b) CDF of the redshift of FSRQs, BL Lacs, and NLSy1s.

3.3 γ-ray emission properties of jetted AGNs

The hypothesis of γ-ray emissions in AGNs predicts that γ-ray emission properties of jetted AGNs are strongly beamed, as can be inferred from Eq. (6). To investigate the effect of relativistic beaming on the γ-ray emission of blazars and NLSy1s, we test the consistency of this prediction using the Pearson correlation theory. We computed the Pearson product-moment correlation coefficient (r) for the combined samples using the following expression (Pavlidou et al. 2012):

(7) r = ∑ ( x i − x ¯ ) ( y i − y ¯ ) ∑ ( x i − x ¯ ) ∑ y i − y ¯ .

Here, x ¯ and y ¯ are the average values of x _i and y _i . Figure 4a shows the scatter plots of R – L _γ-ray on logarithmic scales for the current sample. There is a tendency for the samples to show a positive trend on the R – L _γ-ray plane. The trend suggests that γ-ray emissions in FSRQs, BL Lacs, and NLSy1s are relativistically beamed and suggests the existence of some intrinsic similarities between the blazars and NLSy1s. A linear regression analysis was performed on the data of our sample. The results are shown in Table 2. The slope k, intersection k ₀ , correlation coefficient r, and chance probability p and their errors are all listed in the table. However, the observed R – L _γ-ray correlation could be an artifact if there is any significant cosmological evolution or luminosity selection effects in the sample (in flux density-limited samples, luminosity, and redshift are strongly correlated due to Malmquist bias). To check for these effects in our sample, we carried out a simple statistical investigation of a possible dependence of R _r on redshift (z). Basically, our analyses involve a linear regression of R – z data. The results reveal no significant R – z correlation on the log R – log z plot (Table 2). Although the statistics of NLSy1s seem poor, the results nonetheless suggest that the difference between the blazars and NLSy1 populations may not be only due to different orientation angles but also could as well be intrinsic.

Figure 4

Relations between radio core-dominance parameter, (a) γ-ray luminosity, and (b) redshift for FSRQs, BL Lacs, and NLSy1s.

Table 2

Results of regression analysis for the whole sample taken together

Plots	Sample	k	Δk	k 0	Δk 0	r	Chance probability
log R − log L γ	Combined sample	1.23	0.02	−1.07	0.01	0.51	2.05 × 10−6
log R − log z	Combined sample	1.68	0.03	−0.98	0.02	0.32	5.08 × 10−4

Furthermore, it is known that considerable uncertainties characterize the statistical analyses of data samples of extragalactic sources. Thus, in view of the extent of uncertainties in the radio core dominance and γ-ray luminosity, as well as a redshift of blazars and NLSy1s, we theorize that a more unbiased investigation of the hypothesis γ-ray emissions in AGNs requires the use of the average values of the parameters obtained from carefully chosen bins. The binning was done over the γ-ray luminosity and redshift as follows: L _γ-ray ≤ 44.00, 44.00 < L _γ-ray ≤ 45.00, 45.00 < L _γ-ray ≤ 46.00, 46.00 < L _γ-ray ≤ 47.00, 47.00 < L _γ-ray ≤ 48.00, 48.00 < L _γ-ray ≤ 49.00 and z ≤ −1.50, −1.50 < z ≤ −1.00, −1.00 < z ≤ −0.50, −0.50 < z ≤ 0.00, and 0.00 < z ≤ 0.50, respectively. The average values of R – L _γ and R – z were calculated for each bin. The standard errors of the average values of these parameters were also calculated. The plot of the average values of R – L _γ-ray and R – z are shown in Figure 5. Linear regression analysis of R – L _γ and R – z data gives R = ( 0.71 ± 0.30 ) L γ-ray + ( 0.65 ± 0.20 ) with a correlation coefficient r ∼ 0.29 and R = ( 0.81 ± 0.20 ) z + ( 0.48 ± 0.10 ) with a correlation coefficient r ∼ 0.25. The correlation is found to be statistically significant at a 98% confidence level and suggests the level of relativistic beaming effect in FSRQs, BL Lacs, and NLSy1s.

Figure 5

Relation between the average values of radio core-dominance parameter, binned (a) γ-ray luminosity, and (b) redshift for FSRQs, BL Lacs, and NLSy1s.

4 Discussion

The relativistic beaming model has been proposed to explain certain observed properties of different classes of AGNs and is extensively applied in the testing emission properties of sources. A key implication of this model is that relativistic Doppler boosting and geometric projection effects are expected to be more significant in sources where the radio axes are inclined at an angle to the line of sight. Consequently, the relativistic beam from the core of an AGN is expected to be narrow and characterized for a particular class of AGNs. In this work, we compiled a sample of jetted FSRQs, BL Lacs, and NLSy1s to test the fact that gamma-ray emissions are relativistically beamed for jetted AGN sources. Hitherto, some authors have looked at the relativistic beaming effect in AGNs (see Fan et al. 2011, Pei et al. 2016, 2019, Iyida et al. 2021). The argument is based on the fact that these sources possess ultra-relativistic jets and have common properties. Generally, in relativistic beaming effects, blazars are assumed to be the beamed counterparts of the two classes of radio galaxies (see Urry and Padovani 1995).

Previously, Ghisellini et al. (1993) and Fan et al. (2011) compiled a sample consisting of different classes of AGNs. They found that the average value of log R for BL Lacs is higher than FSRQs. Meanwhile, Murphy et al. (1993) studied the distribution of R for 74 objects at 5 GHz and discovered that there is a tendency for BL Lacs to have lower values than their FQs counterparts. From the results of our analysis of the distributions of R for NLS1s, BL Lacs, and FSRQs, we observed that the R for the majority of FSRQs, BL Lacs, and NLSy1s are fairly large (>1), thus implying that γ-ray emissions in BL Lacs and FSRQs are relativistically boosted. In the same vein, values of R_r for FSRQs and BL Lacs extend by more than one magnitude, thus suggesting that relativistic beaming is more significant in blazars than NLSy1s.

Also, we showed from the distributions of γ-ray luminosity and redshift that FSRQs, BL Lacs, and NLSy1s form a sequence, which indicates the level of relativistic beaming in each class. However, one can imagine that the distributions of R, L _γ-ray, and redshift for FSRQs BL Lacs and NLS1y1s originate from the same parent distribution. We explored this further using the K–S tests. It was found that the probabilities p for both NLS1s and blazar subclasses to come from the same distribution are low (<0.005), showing that the null hypothesis cannot be rejected.

Also, very indispensable in our analysis is the investigation of the effects of relativistic beaming on the γ-ray emission of blazars and NLS1s. We found that γ-ray emissions in FSRQs, BL Lacs, and NLSy1S are relativistically beamed, and there are intrinsic similarities between the blazars and NLSy1s. The R – L _γ-ray plot is observed to be moderately correlated for the combined samples of FSRQs, BL Lac, and NLSy1s, which agrees with the relativistic beaming effects (see Orr and Browne 1982, Odo et al. 2012). It is, therefore, observed from the figure that the beaming model is more pronounced in BL Lacs and FSRQs than in NLSy1s. Thus, these results are quite consistent with several previous results on relativistic beaming and orientation effects in AGNs (see, e.g., Fan et al. 2011, 2005, Zhou et al. 2007, Resconi et al. 2017, Iyida et al. 2021). However, the fact that FSRQ, BL Lacs, and NLSy1s populate distinct and practically like regions in the log R – log L _γ-ray plot implies that they have a common history (e.g., Urry and Padovani 1995, Li et al. 2016, Pei et al. 2019). Analyses of AGN subclasses by Odo and Ubachukwu (2013) and Odo et al. (2015) suggest that sources that are projected very close to the line of sight exhibit a high beaming effect. Therefore, the correlation observed in the log R – log L _γ-ray plot is attributed to relativistic beaming and orientation effects.

5 Conclusion

Using a compiled sample from the fourth Fermi-LAT data catalog, we investigated the γ-ray properties of jetted AGNs in the context of relativistic beaming. Our analysis of the distributions of the radio core-dominance parameter (R) and γ-ray luminosity (L _γ-ray) shows that the beaming effect is stronger in FSRQs and BL Lacs compared to NLSy1s. The results indicate that FSRQs and BL Lacs are observed at smaller viewing angles and are more strongly beamed than NLSy1s, which have relatively larger viewing angles, supporting the predictions of relativistic beaming theory. Additionally, a moderate correlation (r ∼ 0.50) was found in the R–L _γ-ray plot for the combined samples. This is consistent with the relativistic beaming model, suggesting that γ-ray emissions from both blazars and NLSy1s are strongly beamed.