The Classification of Blazar Candidates of Uncertain Types

Abstract

In this work, the support vector machine (SVM) method is adopted to separate BL Lacertae objects (BL Lacs) and flat spectrum radio quasars (FSRQs) in the plots of the photon spectrum index against the photon flux, ${\alpha }_{\mathrm{ph}}\sim \log F$, those of the photon spectrum index against the variability index, ${\alpha }_{\mathrm{ph}}\sim \log \mathit{V}\mathit{I}$ and those of the variability index against the photon flux, $\log VI\sim \log F$. Then, we used the dividing lines to distinguish BL Lacs from FSRQs in the blazar candidates of uncertain types from the Fermi/LAT catalogue. Our main conclusions are: 1. We separate BL Lacs and FSRQs by ${\alpha }_{\mathrm{ph}}=-0.123\log F+1.170$ in the ${\alpha }_{\mathrm{ph}}\sim \log F$ plot, ${\alpha }_{\mathrm{ph}}=-0.161\log VI+2.594$ in the ${\alpha }_{\mathrm{ph}}\sim \log VI$ plot and $\log VI=0.792\log F+9.203$ in the $\log VI\sim \log F$ plot. 2. We obtain 932 BL Lac candidates and possible BL Lac candidates, and 585 FSRQ candidates and possible FSRQ candidates. 3. Discussion is given regarding comparisons with the literature.

1. Introduction

As a special subclass of active galactic nuclei (AGNs), blazars show certain extreme observational properties: high amplitude and rapid variability superposed on the long-term slow variation light curve, high polarization, powerful $\gamma$-ray emissions, sources emitting TeV emissions, strong broad emission line features or with no emission line at all or superluminal motions, [1,2,3,4,5,6,7,8,9,10,11,12,13,14]. Those extreme observational properties are explained by a beaming model, in which there is a central supermassive black hole surrounded by an accretion disk, and two jets that are perpendicular to the disk.

When the jet points to the observer, the observed emission, $f^{\mathrm{ob}}$, is boosted and the variability time scale, $\Delta t^{\mathrm{ob}}$, is shortened by $f^{\mathrm{ob}}={\delta }^q{f}^{\mathrm{in}}$ and $\Delta t^{\mathrm{ob}}=\Delta t^{\mathrm{in}}/\delta$, where $f^{\mathrm{in}}$ and $\Delta t^{\mathrm{in}}$ are the emission and the variability time scale in the comoving frame, $\delta ={\left[\mathsf{\Gamma }(1-\beta cos\theta )\right]}^{-1}$ is a boosting factor (or Doppler factor), $\mathsf{\Gamma }=1/\sqrt{1-{\beta }^2}$, is the Lorentz factor, $\theta$ is the viewing angle, and $\beta$ is the jet speed in units of the speed of light, $\beta =v/c$, and q is a parameter depending on the jet case: $q=2+\alpha$ is for the case of a continuous jet, while $q=3+\alpha$ is for the case of a moving sphere [15], and $\alpha$ is a spectral index defined by $f_{\nu }\propto {\nu }^{-\alpha }$. Based on the behaviour of emission lines, blazars can be classified into two subclasses—namely, BL Lacertae objects (BL Lacs) and flat spectrum radio quasars (FSRQs).

FSRQs have strong broad emission lines with the equivalent width greater than 5 ($\mathit{E}\mathit{W}>5$), while BL Lacs show only weak or no emission lines at all, or $\mathit{E}\mathit{W}<5$ [6,16,17]. BL Lacs were classified as radio-selected BL Lacs (RBLs) and X-ray-selected BL Lacs (XBLs) from surveys. The both have differently observational properties in Hubble diagrams, multiwavelength correlations, spectral index correlations, and linear optical polarizations, etc. [18] and the references therein.

The physical classification of BL Lacs was by Padovani and Giommi [19], who calculated the spectral energy distributions (SEDs) for a sample of BL Lacs objects and proposed to use the synchrotron peak frequency (${\nu }_{\mathrm{p}}$) to separate BL Lacs into highly peaked BL Lacs (HBLs) with $\log {\nu }_{\mathrm{p}}\left(\mathrm{Hz}\right)>15$ (the base of the logarithms is ten throughout this paper) and lowly peaked BL Lacs (LBLs) with $\log {\nu }_{\mathrm{p}}\left(\mathrm{Hz}\right)<15$, and we found that most RBLs belong to LBLs while XBLs belong to HBLs.

Nieppola et al. [20] calculated the SEDs, $\log \nu F_{\nu }-\log \nu$ for 308 BL Lacs and classified them into three subclasses: LBLs, IBLs and HBLs with the boundaries being $\log {\nu }_{\mathrm{p}}\left(\mathrm{Hz}\right)=14.5$ and $\log {\nu }_{\mathrm{p}}\left(\mathrm{Hz}\right)=16.5$, respectively. Thus, it was set as LBLs if $\log {\nu }_{\mathrm{p}}\left(\mathrm{Hz}\right)<14.5$, IBLs if $14.5<\log {\nu }_{\mathrm{p}}\left(\mathrm{Hz}\right)<16.5$ and HBLs if $\log {\nu }_{\mathrm{p}}\left(\mathrm{Hz}\right)>16.5$ for HBLs.

Later on, Abdo et al. [21] calculated the SEDs using the quasi-simultaneous data of 48 Fermi blazars and extended the definition to all types of non-thermal dominated AGNs using new acronyms: low synchrotron peaked blazars (LSP) if $\log {\nu }_{\mathrm{p}}\left(\mathrm{Hz}\right)<14$, intermediate synchrotron peaked blazars (ISP) if $14<\log {\nu }_{\mathrm{p}}\left(\mathrm{Hz}\right)<15$ and high synchrotron peaked blazars (HSP), if $\log {\nu }_{\mathrm{p}}\left(\mathrm{Hz}\right)>15$. They also proposed an empirical parametrization to estimate the synchrotron peak frequency using the effective radio-optics spectral index (${\alpha }_{\mathrm{ro}}$) and the effective optics-X-ray spectral index (${\alpha }_{\mathrm{ox}}$).

Following Abdo et al. [21], and using their acronyms, Fan et al. [22] calculated the SEDs for 1392 Fermi blazars and proposed following classifications: LSPs if $\log {\nu }_{\mathrm{p}}\left(\mathrm{Hz}\right)\le 14.0$, ISPs if $14.0<\log {\nu }_{\mathrm{p}}\left(\mathrm{Hz}\right)\le 15.3$ and HSPs if $\log {\nu }_{\mathrm{p}}\left(\mathrm{Hz}\right)>15.3$. They found no component with the extreme high peak frequency of $\log {\nu }_{\mathrm{p}}\left(\mathrm{Hz}\right)>19$.

Recently, Yang et al. [9] calculated the SEDs for 2709 blazars (including BCUs) in 4FGL DR3 and obtained the following classifications: LSPs if $\log {\nu }_{\mathrm{p}}\left(\mathrm{Hz}\right)\le 13.7$, ISPs if $13.7<\log {\nu }_{\mathrm{p}}\left(\mathrm{Hz}\right)\le 14.9$ and HSPs if $\log {\nu }_{\mathrm{p}}\left(\mathrm{Hz}\right)>14.9$. These blazar classification boundaries are summarized in

Table 1. The boundaries of blazar classification.

Type	Lower	Intermediate	Higher	Ref.	N
BL Lacs	$\log {\nu }_{\mathrm{p}}\left(\mathrm{Hz}\right)<14.5$	$14.5<\log {\nu }_{\mathrm{p}}\left(\mathrm{Hz}\right)<16.5$	$\log {\nu }_{\mathrm{p}}\left(\mathrm{Hz}\right)>16.5$	Nieppola et al. [20]	308
	$\log {\nu }_{\mathrm{p}}\left(\mathrm{Hz}\right)<14.0$	$14.0<\log {\nu }_{\mathrm{p}}\left(\mathrm{Hz}\right)<15.0$	$\log {\nu }_{\mathrm{p}}\left(\mathrm{Hz}\right)>15.0$	Abdo et al. [21]	48
Blazars	$\log {\nu }_{\mathrm{p}}\left(\mathrm{Hz}\right)<14.0$	$14.0<\log {\nu }_{\mathrm{p}}\left(\mathrm{Hz}\right)<15.3$	$\log {\nu }_{\mathrm{p}}\left(\mathrm{Hz}\right)>15.3$	Fan et al. [22]	1392
	$\log {\nu }_{\mathrm{p}}\left(\mathrm{Hz}\right)<13.7$	$13.7<\log {\nu }_{\mathrm{p}}\left(\mathrm{Hz}\right)<14.9$	$\log {\nu }_{\mathrm{p}}\left(\mathrm{Hz}\right)>14.9$	Yang et al. [9]	2709

N denotes the sample size that the authors used to find the boundary.

.

Fermi/LAT missions have detected many $\gamma$-ray emitters. More than 60% of the Fermi/LAT detected sources are AGNs, and 99% of the Fermi/LAT AGNs are blazars Thus, $\gamma$-ray emission is a typically observational property of blazars, and $\gamma$-ray emission was taken as one of the observation properties of blazars. Up to now, several catalogues have been released [2,3,21,23,24].

The $\gamma$-ray loud blazars are variable on different time scales [9]. Fermi/LAT detected a great deal of $\gamma$-ray emitters. There are five catalogues of Fermi/LAT mission, which provide us with a nice opportunity to investigate the variability properties in the $\gamma$-ray band. The variability level in the $\gamma$-ray was introduced by a so-called variability index ($VI$) as defined by Abdollahi et al. [1]:

$$VI=2\sum _i\log \left[\frac{{\mathcal{L}}_i\left(S_i\right)}{{\mathcal{L}}_i\left(S_{\mathrm{glob}}\right)}\right]-\max ({Ø}^2\left({\mathrm{S}}_{\mathrm{glob}}\right)-{Ø}^2\left({\mathrm{S}}_{\mathrm{av}}\right),0),$$

(1)

$${\chi }^2\left(S\right)=\sum _i\frac{{(S_i-S)}^2}{{\sigma }_i^2},$$

(2)

where $S_i$ are the individual flux values, ${\mathcal{L}}_i\left(S\right)$ is the likelihood in the interval i assuming flux S ${\sigma }_i$ are the errors on $S_i$, $S_{\mathrm{av}}$ is the average flux, and $S_{\mathrm{glob}}$ is the globe flux.

The latest fourth Fermi/LAT catalogue (4FGL) with 5099 sources was published [1,3]. Out of them 1432 are BL Lacs, 795 are FSRQs and 1518 blazar candidates of uncertain type (BCUs). The identification of the BCUs is interesting, and it can provide more sources for us to investigate the different physics in BL Lacs and FSRQs. The identification of BCUs has been performed in many works [9,25,26,27,28,29,30,31,32,33,34].

In this work, we apply the support vector machine (SVM) learning method to separate BL Lacs and FSRQs and then use the dividing line to distinguish BL Lac candidates from FSRQ candidates from the BCUs. The work is arranged as follows: In Section 2, a sample, from 4FGL_DR3 as used in the work, is described. Moreover, the distributions of the physical parameters are given for BL Lacs and FSRQs, and the SVM method is used to separate BL Lacs and FSRQs and to divide BL Lac candidates and FSRQ candidates. Our discussions and conclusions are given in Section 3 and Section 4.

2. Sample and Classifications

2.1. Samples

In this work, we obtained 3743 blazars from the 4FGL catalogue [1,3], which include 1432 BL Lacs, 794 FSRQs and 1517 BCUs. We only list 1517 BCUs in

Table 2. Classification for the BCU sources in this work.

4FGL Name	$\mathbf{log}\mathit{F}$	$\mathbf{log}\mathit{V}\mathit{I}$	${\mathit{\alpha }}_{\mathbf{ph}}$	Class${}^{{\mathit{\alpha }}_{\mathbf{ph}}-\mathit{F}}$	Class${}^{{\mathit{\alpha }}_{\mathbf{ph}}-\mathit{V}\mathit{I}}$	Class${}^{\mathit{F}-\mathit{V}\mathit{I}}$	Class-TW	Class(K19)
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)
4FGL J0001.2+4741	−9.900	1.403	2.272	BL Lac	BL Lac	FSRQ	P-B	BL Lac
4FGL J0001.6-4156	−9.549	1.421	1.775	BL Lac	BL Lac	BL Lac	BL Lac	BL Lac
4FGL J0001.8-2153	−10.043	1.390	1.877	BL Lac	BL Lac	FSRQ	P-B	NN
4FGL J0002.1-6728	−9.587	1.098	1.848	BL Lac	BL Lac	BL Lac	BL Lac	BL Lac
4FGL J0002.3-0815	−9.924	1.114	2.092	BL Lac	BL Lac	BL Lac	BL Lac	NN
4FGL J0002.4-5156	−10.108	1.248	1.914	BL Lac	BL Lac	FSRQ	P-B	NN
4FGL J0003.1-5248	−9.463	0.903	1.916	BL Lac	BL Lac	BL Lac	BL Lac	BL Lac
4FGL J0003.3-1928	−9.372	1.698	2.282	BL Lac	BL Lac	BL Lac	BL Lac	P-F
4FGL J0003.3-5905	−9.916	1.006	2.274	BL Lac	BL Lac	BL Lac	BL Lac	P-B
4FGL J0003.5+0717	−9.814	1.039	2.217	BL Lac	BL Lac	BL Lac	BL Lac	NN
4FGL J0007.7+4008	−9.351	1.552	2.140	BL Lac	BL Lac	BL Lac	BL Lac	BL Lac
4FGL J0008.0-3937	−9.920	1.220	2.626	FSRQ	FSRQ	BL Lac	P-F	FSRQ
4FGL J0008.4+1455	−9.286	1.715	2.079	BL Lac	BL Lac	BL Lac	BL Lac	BL Lac
...	...	...	...	...	...	...	...	...
...	...	...	...	...	...	...	...	...
...	...	...	...	...	...	...	...	...
...	...	...	...	...	...	...	...	...

This table is available in its entirety in machine-readable forms.

since we want to classify them in this work.

Redshift, which is from NEDs (https://ned.ipac.caltech.edu/classic/, accessed on 6 July 2022), is available for 2094 sources (1010 BL Lacs, 781 FSRQs and 303 BCUs). The redshift is in a range of $z=$0.0005 for 4FGL J1124.0 + 2045 to $z=$ 5.540 for 4FGL J1113.9 + 5523 for the 2094 sources. The averaged values are $\langle z\rangle =0.507\pm 0.512$ for 1010 BL Lacs, $\langle z\rangle =1.200\pm 0.662$ for 781 FSRQs and $\langle z\rangle =0.777\pm 0.710$ for 303 BCUs.

2.2. Average Values

$\gamma$-Ray Photon Flux-$\log F$: Based on the photon flux intensity from the 4FGL catalogue [1,3], we obtained the logarithm of the $\gamma$-ray photon flux ($\log F$) and show their distributions for FSRQs and BL Lacs in the upper-left panel of Figure 1, and their cumulative distributions are in the upper-right panel of Figure 1. Their averaged values are $\langle \log F\rangle =-9.294\pm 0.520$ for FSRQs and $\langle \log F\rangle =-9.434\pm 0.482$ for BL Lacs. When a K-S test is performed to the distributions, a probability $p=6.708\times {10}^{-7}$ for the two distributions to be from the same parent distribution is obtained.

Photon Spectral Index-${\alpha }_{\mathrm{ph}}$: We show the distributions of ${\alpha }_{\mathrm{ph}}$ for FSRQs and BL Lacs in the middle-left panel in Figure 1, and their cumulative distributions are shown in the middle-right panel of Figure 1. The average photon spectral indexes are $\langle {\alpha }_{\mathrm{ph}}\rangle =2.470\pm 0.201$ for 795 FSRQs and $\langle {\alpha }_{\mathrm{ph}}\rangle =2.032\pm 0.212$ for 1432 BL Lacs. The K-S test gives $p=7.77\times {10}^{-16}$.

Variability Index-VI: For the variability index, we calculated the corresponding logarithm and show their distributions for FSRQs and BL Lacs in the lower-left panel of Figure 1, and their cumulative distributions are in the lower-right panel of Figure 1. For the averaged values, we have $\langle \log VI\rangle =2.025\pm 0.777$ for FSRQs and $\langle \log VI\rangle =1.393\pm 0.481$ for BL Lacs. The probability for the two distributions to be from the same parent distribution is $p=7.77\times {10}^{-16}$.

2.3. Correlations

From the available data: photon flux $\left(F\right)$, photon Spectral Index (${\alpha }_{\mathrm{ph}}$) and variability spectral index ($VI$), we can make mutual correlations.

Photon Flux versus Photon Spectral Index$(F-{\alpha }_{\mathrm{ph}})$: From the given $\gamma$-ray photon flux and the photon spectral index from the 4FGL catalogue, we investigated their mutual correlation and obtained

$${\alpha }_{\mathrm{ph}}=(0.022\pm 0.012)\log F+2.395\pm 0.116$$

with $r=0.038$ and $p=7.5\%$ for BL Lac and FSRQs. The corresponding best fitting result is shown in the upper panel of Figure 2. When BL Lacs and FSRQs are considered separately, one has ${\alpha }_{\mathrm{ph}}=-(0.138\pm 0.012)\log F+1.189\pm 0.115$ with $r=-0.369$ and $p=4.42\times {10}^{-27}$ for FSRQs, and ${\alpha }_{\mathrm{ph}}=(0.040\pm 0.012)\log F+2.407\pm 0.110$ with $r=0.09$ and $p=6.3\times {10}^{-4}$ for BL Lacs.

Photon Spectral Index versus Variability Index$({\alpha }_{\mathrm{ph}}-VI)$: The photon spectral index and variability index give the following linear mutual correlation

$${\alpha }_{\mathrm{ph}}=(0.132\pm 0.009)\log VI+1.975\pm 0.016$$

with $r=0.30$ and $p=8.3\times {10}^{-48}$ for BL Lacs and FSRQs, the corresponding best fitting result is shown in the middle panel of Figure 2. We have the following results when BL Lacs and FSRQs are considered separately,

$${\alpha }_{\mathrm{ph}}=(0.050\pm 0.012)\log VI+1.962\pm 0.017$$

with $r=0.114$ and $p=1.53\times {10}^{-5}$ for BL Lacs and

$${\alpha }_{\mathrm{ph}}=-(0.051\pm 0.009)\log VI+2.573\pm 0.020$$

with $r=-0.196$ and $p=2.65\times {10}^{-8}$ for FSRQs.

Flux versus Variability Index$(F-VI)$: From the $\gamma$-ray photon flux and the variability index, we obtained their mutual correlation

$$\log VI=(1.018\pm 0.018)\log F+11.176\pm 0.170$$

with $r=0.766$ and $p\sim 0$ for all BL Lacs and FSRQs. The corresponding best fitting result is shown in lower panel of Figure 2. While for BL Lacs and FSRQs with available redshift, one has $\log VI=(0.745\pm 0.018)\log F+8.420\pm 0.166$ with $r=0.746$ and $p\sim 0$ for FSRQs and $\log VI=(1.261\pm 0.025)\log F+13.741\pm 0.230$ with $r=0.876$ and $p\sim 0$ for BL Lacs.

2.4. Classifications

From the mutual correlation analyses, we found that BL Lacs and FSRQs show different correlation and they both occupy different regions in the plots. In this sense, we can attempt to find a dividing line to separate BL Lacs and FSRQs, and we can use this dividing line to distinguish BL Lacs from FSRQs when the BCUs are placed in the plots.

In the last version of the Fermi catalogue [1,3], there are 1517 blazar candidates unidentified type (BCUs). It is interesting to divide them into BL Lacs and FSRQs. In this work, we used a SVM, a kind of supervised machine learning (ML) method, to find a dividing line for separating the two blazar subclasses as in Yang et al. [9]. SVM is widely used for classification and regression problems in astrophysics studies [9,35,36,37,38]. Consider two linearly separable samples in the N dimensional parameter space; thus, there are infinite numbers of $N-1$ dimensional hyperplanes can be found to separate them into two sides. The SVM is, then, applied to determine the plane with the maximum margin, i.e., the maximum distance to the nearest samples.

For the case of non-linearly separable samples, SVM maps the samples to a high-dimensional space and finds the optimal separating hyperplane in the high-dimensional space. The SVM requires a training data set and a data set, that randomly takes 70% and 30% sources of each type. The training set is used to find the optimal hyperplane, the test set is used to evaluate the classification accuracy of the hyperplane. In this work, we place BL Lac and FSRQ samples in the two-dimensional parameter space, formed by either two (denote ‘A’ and ‘B’) of the three parameters $\log F$, ${\alpha }_{\mathrm{ph}}$ and $\log VI$.

Assuming the hyperplane, is a line in the two-dimensional space, is expressed as $w_{1}A+w_{2}B+m=0$. The factors $w_1$, $w_2$ and m can be determined through training SVM with the training set. The svm.LinearSVR (from sklearn package) is employed as a SVM classifier, and the hyperparameters of svm.LinearSVR need to be specified before the SVM training starts. We iterate different combinations of hyperparameters in the training process until $w_1$, $w_2$ and m converge to the the maximum margin. Finally, we find a number of different optimal dividing lines, and the one with the highest accuracy on the test set is the final optimal dividing line.

When the SVM is adopted to the $(F-{\alpha }_{\mathrm{ph}})$ data, the result gives an accuracy of 88.60% for the separation and a dividing line of ${\alpha }_{\mathrm{ph}}=-0.123\log F+1.170$ as shown in Figure 3. One can notice that FSRQs mainly occupy the region with ${\alpha }_{\mathrm{ph}}>-0.123\log F+1.170$, and the majority of BL Lacs occupy the region with ${\alpha }_{\mathrm{ph}}<-0.123\log F+1.170$.

When the 1517 BCUs are placed into the plot, we found there are 639 BCUs locate in the region above the dividing line and they can be taken as FSRQ candidates (FC) while there are 878 BL Lac candidates (BC) since they are in the region below the dividing line.

When we considered the $({\alpha }_{\mathrm{ph}}-VI)$ plot, we found that BL Lac and FSRQs can be divided by ${\alpha }_{\mathrm{ph}}=-0.161\log VI+2.594$ with an accuracy of 89.26% as shown in Figure 4. When the 1517 BCUs are placed into the plot, we found 585 FSRQ candidates (FC) and 932 BL Lac candidates (BC).

For the $(F-VI)$ plot, we found a dividing line of $\log VI=0.792\log F+9.203$ with an accuracy of 79.16% as in Figure 5. Based on which, we obtained 337 FSRQ candidates (FC) and 1180 BL Lac candidates (BC).

In our consideration, we take a BCU as a BL Lac candidate (BC) if it is below the dividing line in the three plots and as a possible BL Lac candidate (p-BC) if it is below the dividing line in any two plots; For FSRQs, we also take the same consideration. Therefore, we have 751 BL Lac candidates (BCs) and 181 possible BL Lac candidates (p-BCs)—namely, 932 BC and p-BCs in total; 210 FSRQ candidates (FCs) and 375 possible FSRQ candidates (p-FCs)—namely, 585 FCs and p-FCs in total. The ratio of the number of FCs and p-FCs versus BC and p-BC is $\sim \frac{2}{3}$(585 versus 932).

When we considered the $({\alpha }_{\mathrm{ph}}-VI)$ plot, we found that BL Lac and FSRQs can be divided by ${\alpha }_{\mathrm{ph}}=-0.161\log VI+2.594$ with an accuracy of 89.26% as shown in Figure 4. When the 1518 BCUs were placed into the plot, we found 377 FCs and 714 BCs. For the $(F-VI)$ plot, we found a dividing line of $\log VI=0.792\log F+9.203$ with an accuracy of 79.16% as in Figure 5. Based on which, we obtained 223 FCs and 888 BCs.

In our consideration, we take a BCU as a BL Lac candidate if it is below the dividing line in the three plots and as a possible BL Lac candidate if it is below the dividing line in any two plots. For FSSRQs we also take the same consideration. Therefore, we have 590 BCs and 124 p-BCs; 714 BC and p-BCs in total; as well as 145 FCs and 232 p-FCs, 377 FCs and p-FCs in total.

3. Discussions

After the launch of Fermi/LAT in 2008, a series of catalogues have been released. The latest catalogue [1] contain 3743 blazars and BCUs including 1432 BL Lacs, 794 FSRQs and 1517 BCUs. Recently, Yang et al. [9] studied the variability properties and the classification of BCUs based on a sample selected from the common blazars from [1,22] with $VI>18.48$. In the work [1], who indicated that $VI>18.48$ suggests that a source is variable. Thus, it is not necessary for one to take the restriction of $VI>18.48$ since not all sources are violently variable.

It is clear that there many blazars with $VI<18.48$, we will misclassify a great deal of BCUs if we only consider the BCUs with $VI>18.48$. This is why we considered all BCUs listed in Abdollahi et al. [1] in the present work.

3.1. The Average Values

For the observational data, the $\gamma$-ray photon flux ($\log F$), the photon spectral index (${\alpha }_{\mathrm{ph}}$) and the variability index ($\log VI$). The averaged values of three physics parameters in FSRQs are greater than those of BL Lacs. The K-S test indicates that the probability (p) for the distribution for FSRQs and that for BL Lacs to be from the same parent distribution is $p<6.708\times {10}^{-7}$.

For the subclasses of BL Lacs, we investigate the LBLs and HBLs, and we do not consider IBLs since IBLs may include some LBLs and HBLs. In this sense, we found that, for the photon spectral index, $\langle {\alpha }_{\mathrm{ph}}\rangle =2.197\pm 0.168$ for LBLs and $\langle {\alpha }_{\mathrm{ph}}\rangle =1.902\pm 0.149$ for HBLs, which show clear difference between LBLs and HBLs with $p=5.5\times {10}^{-92}$; for the photon flux, $\langle \log F\rangle =-9.301\pm 0.482$ for LBLs and $\langle \log F\rangle =-9.398\pm 0.485$ for HBLs with $p=4.25\%$; and for the variability index, $\langle \log VI\rangle =1.584\pm 0.594$ for LBLs and $\langle \log VI\rangle =1.377\pm 0.421$ for HBLs with $p=1.33\times {10}^{-6}$.

When we considered FSRQs and LBLs for comparison, we found that the probability for the two subclasses of blazars to be from the same parent population is $p=1.75\times {10}^{-75}$ for the photon spectral index, ${\alpha }_{\mathrm{ph}}$, $p=37.8\%$ for the photon flux, $\log F$ and $p=1.17\times {10}^{-20}$ for variability index, $\log VI$. The comparisons between FSRQs and HBLs give $p\sim 0$ for the photon spectral index, ${\alpha }_{\mathrm{ph}}$, $p=4.6\times {10}^{-3}$ for the photon flux, $\log F$ and $p=9.39\times {10}^{-56}$ for the variability index, $\log VI$.

One can see a clear difference in the photon spectral index (${\alpha }_{\mathrm{ph}}$) between FSRQs and LBL, between FSRQ and HBL and between LBLs and HBLs, giving $\langle {\alpha }_{\mathrm{ph}}\rangle {|}_{\mathrm{FSRQ}}>\langle {\alpha }_{\mathrm{ph}}\rangle {{|}_{\mathrm{LBL}}>\langle {\alpha }_{\mathrm{ph}}\rangle |}_{\mathrm{HBL}}$. We also found that ${\langle \log VI\rangle |}_{\mathrm{FSRQ}}>\langle \log VI\rangle {{|}_{\mathrm{LBL}}>\langle \log VI\rangle |}_{\mathrm{HBL}}$. A clear difference in photon flux ($\log F$) between FSRQ and HBL ($p=4.6\times {10}^{-3}$) and a marginal different between HBLs and LBLs with a $p=4.26\%$ are found. However, no clear difference in the photon flux between LBL and FSRQs. We can say there is a sequence from FSRQ to LBL to HBL for photons spectral index and variability index that is similar to that indicated by Fossati et al. [39]; also see in Ghisellini et al. [40].

3.2. The Correlations for FSRQs and BL Lacs

In this work, we considered the mutual correlations amongst ${\alpha }_{\mathrm{ph}}$, $\log F$ and $\log VI$ between FSRQs and BL Lacs. There is a tendency for a positive correlation between spectral index (${\alpha }_{\mathrm{ph}}$) and photon flux ($\log F$) for known blazars. When we considered BL Lacs and FSRQs separately, a clear anti-correlation was found for FSRQs with $p=4.42\times {10}^{-27}$ and a positive correlation for BL Lacs as shown in the upper panel of Figure 2. BL Lacs and FSRQs show different spectral index dependence on photon flux.

Both subclasses also show different spectral index dependence on the variability flux in the middle panel of Figure 2, which indicates that the spectral index (${\alpha }_{\mathrm{ph}}$) in FSRQs decreases with variability index $\log VI$ while that in BL Lacs increases with the variability index. While for photon flux ($\log F$) and variability index ($\log VI$), blazars and the two subclasses all show positive correlation indicating that the variability index ($\log VI$) increases with photon flux ($\log F$) as shown in the lower panel of Figure 2.

The linear correlations that we obtained amongst ${\alpha }_{\mathrm{ph}}$, $\log F$ and $\log VI$ do not mean strict mathematic linear correlation but demonstrate possible trends between two parameters. It is more reasonable to explore trends instead strict mathematical linear correlations. The theoretical relationship between the three parameters (and also for other astronomical quantities) is rarely investigated. As these observational quantities show a significant discrepancy, the discrepancy leaves enough space for various models to explain the phenomenon.

In the case of individual sources, the observational discrepancy can come for several reasons, the source’s intrinsic reason (e.g., the black hole mass and spin, the accretion ratio, etc.) and the external reason (e.g., the gas and dust density of the host galaxy, the magnetic field, the distance, etc.). In the case of many sources, the distribution of the sources may serve a selection effect of the telescope or a few sources in the universe that have very high/low values of some quantities. All the above-mentioned reasons could obstruct us from discovering the linear correlation mathematically.

For the sources in our sample, it is natural that most of the sources have a low photon flux (i.e., $\log F<-8.5$) and variability index (i.e., $\log VI<2.5$), and those sources with a high photon flux (i.e., $\log F>-7.5$) and variability index (i.e., $\log VI>4.5$) are rare in the universe; see the middle and lower panels of Figure 2.

Through the correlations study amongst ${\alpha }_{\mathrm{ph}}$, $\log F$ and $\log VI$ in this work. We conclude that the FSRQs show trends of anti-correlation for ${\alpha }_{\mathrm{ph}}$vs.$\log F$ and ${\alpha }_{\mathrm{ph}}$vs.$\log VI$, while the BL Lacs show trends of positive correlation for the two; see in the upper and middle panels of Figure 2. Furthermore, both FSRQs and BL Lacs show trends of positive correlation for $\log VI$vs.$\log F$.

3.3. The Classification for BCUs

We found that most of BL Lacs and FSRQs occupy different regions in the plots of ${\alpha }_{\mathrm{ph}}$ versus $\log F$ (Figure 3), ${\alpha }_{\mathrm{ph}}$ versus $\log VI$ (Figure 4) and $\log VI$ versus $\log F$ (Figure 5). When the support vector machine (SVM) method was adopted to the relevant data, separating lines were obtained, which can be used to give BL Lac and FSRQ candidates when BCUs are placed in the plots.

In those cases, we obtained 639 FCs and 878 BCs in the ${\alpha }_{\mathrm{ph}}$ versus $\log F$ plot (Figure 3), 582 FCs and 932 BCs in the ${\alpha }_{\mathrm{ph}}$ versus $\log VI$ plot (Figure 4) and 337 FCs and 1180 BCs in the $\log VI$ versus $\log F$ plot (Figure 5). We take a BCU as an FC if it is classified as an FC in all the three cases, and we take a BCU as a p-FC if it is classified as an FC in any two cases. We also give similar considerations for BCs and p-BCs. Our candidate classifications are shown in Table 2. Therefore, we have 932 BC and p-BCs, 585 FCs and p-FCs giving a number ratio $\sim \frac{2}{3}$.

We also made a comparison with the classification results in Kang et al. [34]. We take a BCU as an FC if it was classified as an FC in all their three considerations and as a p-FC if it was classified as an FC in any two of their considerations [34]. For BC, we also give a similar classification. If this case, we obtained that there are 302 FCs and 109 p-FCs, 556 BCs and 114 p-BCs, which are given in Col. (6) in Table 2.

When we compared our classifications with those in Kang et al. [34], we found that there were 1091 common sources, and 426 sources in the present work were not included in Kang et al. [34]. For the 1091 common sources, our analyses found that 590 BCs in this work (TW) correspond to 492 BCs, 10 FCs, 68 p-BCs and 20 p-FCs in the work Kang et al. [34]; 145 FCs in TW correspond to 4 BCs, 119 FCs, 6 p-BCs and 16 p-FCs in Kang; 124 p-BCs in TW correspond to 60 BCs, 15 FCs, 24 p-BCs and 25 p-FCs in Kang; and 231 p-FCs in TW correspond to 10 BCs, 147 FCs, 16 p-BCs and 48 p-FCs in Kang.

Therefore, our 590 BCs and 124 p-BCs (total 714) correspond to 552 BCs and 92 p-BCs (total 644) in Kang, which means that, out of 714 sources in our considerations, 644 are similar to those by Kang giving a 90.2% goodness of fit, and our 145 FCs and 231 p-FCs (total 376) correspond to 266 FCs and 64 p-FCs (total 330) in Kang giving a 87.8% goodness of fit.

4. Conclusions

In this work, 3743 blazars from the 4FGL catalogue [1,3], which includes 1432 BL Lacs, 794 FSRQs and 1517 BCUs were analysed regarding their averaged values and mutual correlation amongst the photon spectral index, variability index and the photon flux for the known blazars. The SVM method was adopted to separate BL Lacs and FSRQs. Then, we used the separating line to classify the the BCUs into BL Lac and FSRQs. We also proposed to classify a BCU as a BL Lac object if it is classified in all the three cases and as a possible BL Lac candidate if it is classified as a BL Lac in only two cases. For the FSRQ candidates, we also took similar considerations. Our classifications were compared with those of Kang et al. [34]. Our conclusions are as follows:

• The $\gamma$-ray photon flux, spectral index and variability index of FSRQs were higher than those of BL Lacs for the known blazar sample. There is a sequence from FSRQs to LBLs to HBLs that is similar to that in Fossati et al. [39].
• A positive correlation was found between the $\gamma$-ray flux and the photon spectral index for the whole sample; however, an anti-correlation was found for FSRQs and a positive correlation for BL Lacs. In addition, a positive correlation was found between the variability index ($\log VI$) and the $\gamma$-ray photon spectrum index (${\alpha }_{\mathrm{ph}}$) for the whole sample but an anti-correlation for FSRQs and a positive correlation for BL Lacs. We found that those two positive correlations for the whole sample were apparent.
• We adopted the SVM machine-learning method to classify BL Lacs and FSRQs in the ${\alpha }_{\mathrm{ph}}\mathrm{vs}.\log F$, and ${\alpha }_{\mathrm{ph}}\mathrm{vs}.VI$ plots and $\log F\mathrm{vs}.VI$. We obtained 932 BL Lac candidates and possible BL Lac candidates as well as 585 FSRQ candidates and possible FSRQ candidates.
• We compared our classifications with those in Kang et al. [34] and found that, for the common sources, there was a goodness fit of 90.2% for BL Lac and possible BL Lac candidates and a goodness of 87.8% for FSRQ and possible FSRQ candidates with those by Kang et al. [34].