On the Unification of Low Luminosity Active Galactic Nuclei

We have used the distributions of observed radio properties of a subset of AGNs called low luminosity AGNs (LLAGNs) to investigate the relationship between radio-selected BL Lacs (RBLs and XBLs) and Fanaroff-Riley Type I radio galaxies (FRI) in accordance with the unified scheme. Analysis of the core and extended luminosity show that RBLs are more core dominated than XBLs which are also more core dominated than FRI radio galaxies. Distributions of core dominance parameter (R) give R m ≈ 42.41 ± 10.77, 5.08 ± 1.57, and 0.34 ± 0.0024 for RBLs, XBLs, and FRI, corresponding to mean viewing angles (ϕ m ) in the range ϕ ≈ 7 0 − 17 0 , 13 0 − 24 0 and 20 0 − 28 0 respectively. Regression analysis of the three samples shows that the core to extended luminosity plot (L C − L E ) yields a strong correlation of r ≈ 0.67±0.52, 0.85±0.46, and 0.79±0.61 for FRI, RBLs and XBLs at 95% confidence level. Similarly, regression analysis of R and ϕ for FRI, RBLs, and XBLs all yield a strong anti-correlation of r ≈ −0.98±0.02, −0.64±40.52, and 0.75±4.45. These results are consistent with BL Lac/FRI unification and suggest that relativistic beaming of the core and extended emissions is significant in low luminosity AGNs and thus a strong support for the unified scheme.

 determine the average viewing angle of each sub-sample from the observed data.  determine the nature and strength of correlation among various beaming and orientation parameters of the three sub-samples. In light of the above, the synopsis of the present article is: In § (2), we make a brief review of the data selection process, and the constraints imposed on their parameters. In § (3), we present the theoretical relationship underlying the physics of AGNs and their applications as widely understood by applying distribution and correlation statistics for analyzing the relationship of the sub-samples. In §(4), we present the results of the application of theoretical laws and statistics to our data sets for the three sub-samples obtained from the literature and discuss its implication as applied to the unified scheme for low luminosity radio sources. Lastly, in § (5), a general discussion is presented on our expectation for the unified scheme, and conclusions are drawn respectively.

2.0
Data Description and Source BL Lac objects (RBLs and XBLs) are inherently identical to other low luminosity radio counterparts under the present relativistic beaming and orientation-based unification approach for LLAGNs The relativistic effects on the apparent luminosity of emitting matter moving at relativistic speeds are known as relativistic beaming (Iyida et (Kollgaard et al., 1996). For a better understanding of BL Lacs-FRI unification based on the premise of the unified scheme, we will quantitatively investigate the relationship between these subclasses based on already established relativistic models for which the beaming exponent (n) can be either 2 (jet model) or 3 (blob model) using statistical methods. These statistical analyses are aimed at comprehending the behaviour of the two-sub class of BL Lacs population (RBLs and XBLs) with that of FRI samples under relatively near physical conditions. This implies that we will determine if there is continuity in the observed beaming and orientation parameters of the three sub-samples (i.e., the core dominance parameter (R), viewing angle (ϕ), and projected linear size (D). Notwithstanding, we will also bear in mind that FRI is the parent population of BL Lacs (RBLs and XBLs). These analyses if properly carried out will mean that any observed properties of these parameters over a range of frequencies will be compared for the support of the unified scheme.

2.1
Sources of Data Kollgaard et al. (1996) compiled x-ray-selected BL Lacs objects (XBLs) detected by First High-Energy Astrophysics Observatory Large Area Sky Survey (HEAO-1 LASS ) and compared their core and extended luminosity with radio-selected BL Lacertae objects (RBLs) from Einstein Medium Sensitivity Survey (EMSS) (Rector et al., 2000) in the range of 10 23 to 10 25 WHz −1 . Zhou et al. (2007) later used these same data to do a statistical study of XBLs, RBLs and FSRQs at 1.5GHz. In our work however, we did a careful selection of the data from Zhou et al. (2007) and Rector et al. (2000) with specific reference to the core luminosity (LC), extended luminosity (LE), core dominance parameter (R), and Redshift (z) for the RBL and XBL samples. Furthermore, we employed Fanaroff-Riley Type I (FRI) radio samples from Wang et al. (2006) to complete our sample selection. Information regarding the projected linear sizes (D) for RBLs and XBLs samples was updated using 1.5GHz data from Odo et al. (2011). Finally, after completing the sample parameters at 5GHz for the FRI sample, their luminosity information from Odo For the core luminosity in which = 0, we obtained: while for the extended luminosity for which = 1 we obtained: After tuning down the frequencies, the core dominance parameter (R) for the FRI samples was then recalculated using the expression In Equations 1 to 4, α is the spectral index (SναS ±α ) and z is the Redshift for the FRI samples. We have assumed αext = 1 for the extended luminosity because they are lobe-dominated sources with steep spectra and αcore = 0 for the core luminosity because they are core-dominated sources with flat spectra (see The ratio of the two luminosity components LC/LE is called the core dominance parameter (R). Some authors use the ratio of the flux densities while others e.g. Ubachukwu (1997) use the ratio of core-to-lobe luminosity to quantify this parameter. After our compilations and frequency reductions, we were left with a total of 24 RBLs, 17 XBLs, and 28 FRI radio galaxies with complete information on Redshift (z), core luminosity (LC), extended luminosity (LE), core dominance parameter(R), and projected linear size (D), making a total of sixty-eight (68) samples. Other parameters displayed in Table 1 (columns 8 and 9) were determined by calculation.   3.0 Theory of Relationship Various beaming and orientation parameters utilized for the testing of the unified scheme can be determined using these parameters (i) viewing Angle ( ), (ii) core dominance parameter (R), and (iii) Projected Linear Size (D). In the relativistic beaming and source orientation paradigm, these parameters are used to test for the unification of LLAGNs. Following, we shall discuss briefly these parameters and see how they are used in this work.
I. The viewing angle is defined as the angle between the jet and the line of sight of the observer(s). It is such that it gives rise to the different sub-classes of AGNs because it accounts for the orientation of the AGNs. Mathematically, the viewing angle can be computed by the expression: where Rm = the mean value of R for RBLs, XBLs, and FRIs, n = relativistic beaming model (for continuous jet n = 2 or blobs n = 3), but generally 2 ≤ n ≤ 3, α = spectral index (We have adopted α = 0 because they are core dominated sources with flat spectra dominated by synchrotron process), RT is the minimum value of R, at ϕ = 90 0 . We also adopt RT = 0.0024 because it appears to be consistent with the FRI-BL Lac's unification scheme for low luminosity surveys . II. The core dominance parameter also known as the core-to-lobe ratio of an AGN is the ratio of the core luminosity to the extended or lobe luminosity of any active galactic nuclei. This parameter is a measure of the relative orientation of the source . Mathematically, considering the relativistic property of the source for two opposite jets, the core dominance parameter was given by Fan where "β" denotes the source's relativistic velocity, "α" denotes its spectral index, and "ϕ" denotes its viewing angle. where D0 is the intrinsic linear size and ϕ is the viewing angle obtained from equation (5) of the source.

Results and Discussion
To investigate a unification scheme and comprehend the relationship between the numerous classes and subclasses of LLAGN, distributive analyses of the observed source parameters are crucial (Iyida et al., 2021). Similarly, correlations between different orientation parameters of these radio sources can be used to probe the physics of their radio source components (Ubachukwu, 1997). The core dominance parameter (R) for example is also a statistical indicator of source orientation in extragalactic radio sources (Orr & Browne, 1982). On this premise, we have statistically analysed the distribution of R, LC, LE, D and ϕ (Figures 1 to 5 under the beaming and orientation scenario (n = 2 and 3). Similarly, the correlation of R with other beaming and orientation parameters [LC, LE, D, and ϕ (n = 2,3)] as shown in Figures 6 to 10 for the three sub-classes of LLAGNs (RBLs, XBLs, and FRI) were also analysed to test for unification amongst them.

4.1
Distributions of observed radio source parameters for RBLs, XBLs, and FRI The plots shown in Figures 1 to 4 give the distribution of the core luminosity (LC), extended Luminosity (LE), core-dominance parameter (R), Projected linear size (D), and viewing angle (ϕ) for the BL Lacs (RBLs, XBLs,) and FRI all in log scale for simplicity. We adopted the blue colour for FRI, brown for RBLs, and green for XBLs for simplicity. Similarly, Table 6 shows a summary of the mean distributions we obtain alongside the standard error for the three sub-samples.
In Figure 1(a) and 1(b), Rm for the three sub-samples yielded 0.34±0.0024, 42.41±10.77, and 5.08±1.57 for FRIs, RBLs, and XBLs respectively. The distribution shows that RBLs are larger than XBLs and FRI which implies that the RBLs are more core beamed than XBL and FRI as expected (Alhassan et al., 2013;Bai & Lee, 2001). Similarly, The average projected linear sizes (Dm) as shown in Table 6 for the three sub-samples yields 108.65 ± 22.31 for XBLs, 197.95 ± 30.23 for RBLs and 136.86 ± 23.62 FRI respectively. From Table 6, it is evident that the RBLs and FRI have larger linear sizes (D) than those of XBLs. If BL Lacs are of the same population as FRI then their linear sizes should be smaller. Our result shows that this may not be the case thus suggesting that the linear size (D) is again not a consistent parameter and thus not good for testing of unification for LLAGNs.  Table 6 shows that the mean value for the core luminosity (LC − mean) for the FRI, RBLs, and XBLs are: 23.95 ± 0.13, 26.73 ± 0.18 and 24.97 ± 0.23 respectively. The core luminosity for the three sub-samples can be compared for unification since they all fall within the same range in Figure 2 (a) and (b). Furthermore, we noticed that the core luminosity of FRI is slightly lower for the three samples. This should mean that FRI is not as core beamed as RBLs and XBLs, thus strong support for the unified scheme. In the context of the unified scheme as shown by Fan et al. (2011), isotropic properties such as the extended luminosity are expected to be shared equally by both the RBLs and XBLs with their parent population FRI. This is not the case for our samples, as the distribution of the LE from Figure 2(b) shows otherwise. One obvious explanation for this is that the LE for FRI are more isotropic than RBLs and XBLs which is also a strong support for the unified scheme.  Table 6, using RT = 0.0024 given in Equation 5, we obtained the mean viewing angles for RBLs as ϕm ≈ 7 0 − 17 0 , XBL as 13 0 -24 0 and FRI as 20 0 -28 0 for n = 2 and 3 respectively. In the simplest beaming and radio source orientation scenario, Odo et al. (2012); and Zhou et al. (2007) have shown that large values of R are attributed to Doppler boosting of the core emission at small viewing angles. Our result confirms this statement as the Rm value for the three sub-samples showed that the least of the samples FRI with Rm of 0.34 ± 0.0024 has a corresponding viewing angle ϕ ≈ 20 0 and 28 0 , followed by XBLs with Rm of 5.08 ± 1.57 corresponding to ϕm ≈ 13 0 and 24 0 and RBLs with Rm of 42.41 ± 10.77 corresponding to ϕm ≈ 7 0 and 17 0 for n = 2 and 3 respectively. Thus, there exists an inverse relationship between the core dominance parameter (R) and the viewing angle (ϕ). This according to the unified scheme implies that the smaller the viewing angle, the more the source seems to align with the line of sight (Odo, 2012;Ubachukwu, 1997), thus supporting the unification of BL Lacs and FRI radio galaxies. On the other hand, the linear size is supposed to reduce progressively as the beaming angle reduces. This was not observed in the case of three samples, showing a strong indication that the linear size is not consistent with the BL Lac-FRI unification.

Correlation/Regression Analysis
The strength of the relationship between two or more variables is determined by the correlation coefficient (r) and the coefficient of determination (r 2 ) gotten from the scatter plot of the variables involved. In this section of our work, we have carried out a regression analysis between core luminosity (LC) and extended luminosity (LE), as well as a check for the dependence of the core dominance parameter (R) with the projected linear size (D) and viewing angle (ϕ), core luminosity (LC), and extended luminosity (LE) for the three sub-samples. Table 5 shows a summary of the correlation analysis we obtain from our analysis. The scatter plots of the observed parameters for the three samples are shown in Figures 4 to 8. We used a blue circle to represent FRI, a red circle for XBLs, and a black circle for RBLs respectively.
In the context of the unified scheme, as shown by Fan et al. (2011), isotropic properties such as the extended luminosity are expected to be shared by both BL Lacs and their parent population FRI. Furthermore, the relativistic beaming and radio source orientation model proposes that the core luminosity is expected to correlate with the extended luminosity for samples believed to obey BL Lacs-FRI unification. The LE − LC scatter plot in Figure 4 shows that the three sub-sample all strongly correlate with r ≈ 0.67, 0.78, and 0.79 for FRI, RBLs, and XBLs respectively. Regression analysis in Table 5 shows that the correlations are significant at a 95% confidence level, thus indicating strong support for the unified model. Similarly, the plot suggests that XBLs and RBLs can be classified as the same objects since they the occupy same position on the plot and also have almost the same value. FRI scatter data indicate that they are less core beamed since their LC −LE values differ slightly from that of XBLs and RBLs. The implication is that RBLs and XBLs are more core beamed than FRI as expected, thus a strong support for the unification of LLAGNs. On the other hand, our R − D relations for FRI, RBLs, and XBLs, in Figure 5 showed no significant correlation with r = −0.07, 0.06 and −0.04 respectively. This may be a result of the projection effect when observed close to the line of sight which is still in the premise of relativistic beaming (Alhassan et al., 2013). An obvious implication of our result is that, since BL Lacs are believed to form the beamed counterparts of FRI sources, R−D anti-correlation should be expected . This is also consistent with our result . In Figure 6, our R−ϕ log scatter plot for XBLs and RBLs produces a strong negative correlation with r = −0.75 and −0.64 respectively, signifying a close relationship between the two samples as predicted by the unified model for BL Lacs. Similarly, the R − ϕ relationship for FRI yields an almost perfect negative correlation with r = −0.98. This according to the beaming and orientation paradigm, signifies that FRI is the parent population of BL Lacs. Our result further suggests that RBLs are the more likely beamed counterpart of FRI than XBLs. Thus, according to the unified scheme, BL Lacs (RBLs and XBLs) with FRI radio galaxies will form a homogeneous class of AGNs but will appear different due to their relative orientation with respect to the line of sight  and thus responsible for the beaming and orientation effect observed by these sub-classes of AGNs.
In the context of the current unified scheme, Wang et al. (2006) confirmed that the R−LC plot should also correlate for generally non-isotropic sources. Our result for R−LC data gives a correlation coefficient of r = 0.78, 0.31, and -0.59, for RBLs, XBLs and FRI respectively, thus in agreement with this premise for non-isotropic sources. This result implies that amongst the three samples, FRI seems to be more isotropic than RBLs and XBLs due to the negative correlation we obtained. This is generally the case for FRI which is mostly isotropic, thus in support of the unification scheme for LLAGNs.   Table 7 presents a summary that compares the mean ratios of the BL Lacs (RBLs and XBLs) parameters with their parent population FRI radio galaxy in other to investigate if these sub-classes of BL Lacs with their parent population FRI agree with the BL Lacs/FRI the unification scheme.  In the context of the unified scheme as shown by Fan et al. (2011) and Stickel et al. (1991) isotropic properties such as the extended luminosity are expected to be shared equally by both the RBLs and XBLs with their parent population FRI. The ratio of the extended luminosity (LE) as shown in Table 5 is in disagreement with this premise. The comparison showed that this statement may not entirely be correct, as the ratio of RBLs: FRI, RBLs:XBL, and XBL:FRI are > 1. The implication will be that the extended luminosity may not be entirely isotropic as previously assumed for our sample irrespective of the combination of the sub-classes they still give values > 1. On the other hand, the core luminosity (LC) ratio for the three sub-samples all yield values > 1. This implies also that the core luminosity is certainly not isotropic as predicted by the unified scheme. Thus, in agreement with the unification scheme for BL Lac-FRI unification.

Expectations of the Unified Scheme for LLAGNs
The ratio of the mean value of the core-dominance parameter (Rm) for RBLs to that of FRI and XBLs is expected to be greater if RBLs are more beamed than XBLs and FRI. Similarly, those of the XBLs to FRI should also be greater if the XBLs are more beamed than FRI. Our comparison is in agreement with this premise for sources assumed to obey the BL Lacs-FRI unification scheme. Furthermore, Alhassan et al. (20013), confirm that large values of the core-dominance parameter (R) will yield small linear sizes (D) since the linear size is directly proportional to the viewing angle as shown in Equation 7. Thus. it is expected that the ratio of the BL Lacs (RBL/XBL) should be greater than that of (RBL/FRI and XBL/FRI). This was not the case from Table 7 as the projected linear size (D) appears to be greater for the XBL/FRI ratio followed by RBL/FRI and the smallest for the XBL/FRI ratio. The possible explanation for this difference could be that the RBL/XBL unification is more valid than other sub-combinations thus in agreement with the unified scheme for low luminosity AGNs.
According to the unified scheme, large values of the core-dominance parameter (R) will always yield a smaller viewing angle (ϕ) as shown in Figure 6's inverse relation. This according to the unified scheme implies that since RBLs have a large value of R, they should be more beamed than XBLs and FRI. Further implications of this mean that the ratio of the viewing angle (ϕ) for RBLs: FRI and RBLs: XBLs should be < 1. Similarly, the ratio of XBLs: FRI should also be < 1 by comparison. Table 7 clearly shows that this is true as the RBL/FRI, RBL/XRL, and XBL/FRI ratios are all < 1 thus in agreement with the unification scheme for low luminosity radio AGNs. Furthermore, the K-S test for the viewing angle in Table 8 identifies the largest distance d between two given cumulative distributions (FRI/RBL and FRI/XBL), then reading out the probability p that a value as large as d could have resulted by chance if the two given distributions come from the same parent distribution (FRI). A small p means that the two samples are significantly close. Our K-S text of the viewing angle for the three samples of BL Lacs and FRI show that there is a medium, positive correlation between the variables FRI/RBL with r = 0.32 and a p-value of 0.395. Thus, a medium, positive association between FRI and RBL. On the other hand, there is a high, positive correlation between sample FRI/XBL with r = 0.52 and a p-value of 0.031. Thus, high, positive association between FRI and XBLs. The implication of this is that XBLs are significantly similar to FRI as their parent population than RBLs which is also in agreement with the unified scheme for LLAGNs.

Conclusion
We have statistically tested for the unification of BL Lacs and FRI radio galaxies using distributions of the core dominance parameter (R), core luminosity (LC), extended luminosity (LE), and projected linear size (D) from a total of sixty-eight (68) LLAGN samples employed from literature. The frequencies of our data sets for BL Lacs (RBLs and XBLs) were at 1.5GHz while that of FRI radio galaxies was at 5GHz then tuned down to 1.5GHz. The mean values (see Table 4) and their correlations (see Table 5) for the three sub-samples beaming and orientation parameters were determined and discussed. Furthermore, regression analysis was carried out to test the relationship between the core dominance parameter (R) with the core luminosity (LC), extended luminosity (LE), and projected linear size (D) respectively. The results and the variations of their correlation coefficients were discussed to test for unification on the premise of the "BL Lacs-FRI Unified scheme." A comparison of the correlations amongst the different sub-classes and their parameters produced two remarkable results as a major consequence of the beaming and orientation scenario. Our results show that the scatter plot of the core dominance parameter (R) and the viewing angle (R−ϕ) are strongly anticorrelated with R ≈ −0.98±0, −0.64±40.52, and −0.75±4.45 for FRI, RBLs and XBLs respectively. In the same light, the core luminosity versus extended luminosity scatter plot (LC − LE) for the three sub-sample all show a strong correlation with R ≈ 0.67 ± 0.52, 0.78 ± 0.46 and 0.79 ± 0.61 for FRI, RBLs, and XBLs respectively, suggesting that RBLs, XBLs and FRI agree with the unified model at 95% confidence level. In both comparisons, the core dominance parameter versus the viewing angle plot (R − ϕ) and the core luminosity versus the extended luminosity (LC − LE) show that our XBLs sample seems to be an intermediate population, correlating more with RBLs and less with FRI as expected. This implies that XBLs are more similar to FRI than RBLs even though FRI is the parent population of both XBLs and RBLs, thus suggesting that the XBLs/FRI unification seems to agree more with the unified scheme than RBLs/FRI for LLAGNs. K −S test for the viewing angle also reveals that the unification of XBLs and FRI is more valid than that of RBLs and FRI, meaning RBLs are more strongly beamed counterparts of FRI than XBLs.