Correlation between Brightness Variability and Spectral Index Variability for Fermi Blazars

In this work, we obtained 851 common sources with V. I. > 18.48 by cross-checking the 4FGL catalog (Abdollahi et al. 2020; Ajello et al. 2020) and the work by Fan et al. (2016) (whose sample is from 3FGL), the sources are listed in Table 1. There are 383 BL Lacs, 396 FSRQs, and 72 BCUs. Their redshifts are from NEDs (https://ned.ipac.caltech.edu/classic/), the redshift is in a range of z = 0.00034 for 4FGL J1436.9+5638 to z = 6.395 for 4FGL J1233-0144 for the whole sample. There are three objects with z ≥ 4.0, z = 6.39520 for 4FGL J1233.7-0144, z = 4.26396 for 4FGL J1104.4+0730, and z = 4.16248 for 4FGL J0929.3+5014, they are all BL Lacs. For all sources, 708 have available redshift giving an averaged redshift of 〈z〉 = 0.937.

Table 1. A Sample of Common Fermi Blazars in 3FGL and 4FGL

4FGL Name	Class3	Class4	Redshift	log νp	References	${f}_{1-100\ \mathrm{GeV}}^{3}$	${\sigma }_{{f}_{1-100\ \mathrm{GeV}}^{3}}$	${\alpha }_{\mathrm{ph}}^{3}$	${\sigma }_{{\alpha }_{\mathrm{ph}}^{3}}$	${f}_{1-100\ \mathrm{GeV}}^{4}$	${\sigma }_{{f}_{1-100\ \mathrm{GeV}}^{4}}$	${\alpha }_{\mathrm{ph}}^{4}$	${\sigma }_{{\alpha }_{\mathrm{ph}}^{4}}$	V. I. 4
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)	(12)	(13)	(14)	(15)
J0001.2-0747	BLLac	BLLac		14.372	F16	6.95E-10	8.96E-11	2.150	0.090	8.2321E-10	5.6777E-11	2.105	0.050	33.23
J0001.5+2113	FSRQ	FSRQ	1.106	16.786	F16	2.94E-10	7.58E-11	2.310	0.180	1.3591E-09	6.8599E-11	2.659	0.023	1564.42
J0004.4-4737	FSRQ	FSRQ	0.88	14.144	F16	6.93E-10	8.13E-11	2.400	0.080	4.3596E-10	3.7467E-11	2.366	0.059	139.12
J0152.2+3714	bcuI	bcu	0.5	13.629	F16	3.01E-10	6.54E-11	2.410	0.140	2.1998E-10	3.3958E-11	2.343	0.109	22.39
J0005.9+3824	FSRQ	FSRQ	0.234	14.034	F16	6.06E-10	8.90E-11	2.620	0.080	4.2515E-10	4.8877E-11	2.672	0.057	26.77
J0016.2-0016	FSRQ	FSRQ	1.577	13.584	F16	6.41E-10	8.50E-11	2.620	0.070	4.1684E-10	4.4171E-11	2.741	0.049	82.14
J0017.0-0649	bcuII	bcu		14.639	F16	3.88E-10	7.60E-11	2.120	0.140	3.8784E-10	4.1868E-11	2.298	0.080	26.48
J0017.5-0514	FSRQ	FSRQ	0.227	14.475	F16	7.95E-10	9.66E-11	2.510	0.080	7.582E-10	5.5543E-11	2.563	0.044	230.16
J0019.2-5640	bcuII	bcu		13.348	F16	2.86E-10	5.81E-11	2.390	0.170	2.9021E-10	3.1761E-11	2.205	0.084	91.02
J0019.3-8152	BLLac	BLLac		14.690	ZF19	6.53E-10	8.74E-11	2.080	0.100	6.1294E-10	4.6995E-11	2.112	0.060	19.44
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Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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For the subclasses BL Lacs, FSRQs, and BCUs, we have following results: For the 383 BL Lacs, only have 279 sources redshifts, z ranges from z = 0.00034 to z = 6.395. For the 396 FSRQs, it is found that z is in a range from z = 0.0031 for 4FGL J1434.2+4204 to z = 3.104 for 4FGL J0539.9-2839. For the 72 BCUs, 33 sources have redshift, which ranges from z = 0.04638 for 4FGL J0602.0+5315 to z = 2.08500 for 4FGL J1418.4+3543.

In Table 1, Col. (1) gives the 4FGL name; Col. (2) Classification in 3FGL; Col. (3) Classification in 4FGL; Col. (4) redshift; Col. (5) synchrotron peak frequency, log ν_p (HZ); Col. (6) reference for log ν_p (HZ), F16: (Fan et al. 2016), ZF19: (Zhang & Fan 2019); Col. (7) photons in 1–100 GeV given in 3FGL, ${f}_{1-100\ \mathrm{GeV}}^{3};$ Col. (8) error in photons in 1–100 GeV given in 3FGL, ${\sigma }_{{f}_{1-100\ \mathrm{GeV}}^{3}};$ Col. (9) photon spectral index in 3FGL, ${\alpha }_{\mathrm{ph}}^{3};$ Col. (10) error in photon spectral index, ${\sigma }_{{\alpha }_{\mathrm{ph}}^{3}};$ Col. (11) photons in 1–100 GeV given in 4FGL, ${f}_{1-100\ \mathrm{GeV}}^{4};$ Col. (12) error in photons in 1–100 GeV given in 4FGL, ${\sigma }_{{f}_{1-100\ \mathrm{GeV}}^{4}};$ Col. (13) photon spectral index in 4FGL, ${\alpha }_{\mathrm{ph}}^{4};$ Col. (14) error in photon spectral index, ${\sigma }_{{\alpha }_{\mathrm{ph}}^{4}};$ Col. (15) variability index in 4FGL, (V. I. )⁴; Col. (11)–(15) are from Abdollahi et al. (2020). This table is available in its entirety in machine-readable forms.

The γ-ray luminosity, L_γ , in the energy range of E_L and E_U can be calculated from the detected photons, ${N}_{({E}_{{\rm{L}}}\sim {E}_{{\rm{U}}})}$ (Fan et al.2012, 2013; Yang et al. 2017). From ${N}_{({E}_{{\rm{L}}}\sim {E}_{{\rm{U}}})}$, an integral flux, F, in the corresponding energy range of E_L and E_U, can be obtained by

$$\begin{eqnarray*}&&F={N}_{({E}_{{\rm{L}}}\unicode{x00303}{E}_{{\rm{U}}})}(\frac{{E}_{{\rm{L}}}\times {E}_{{\rm{U}}}}{{E}_{{\rm{U}}}-{E}_{{\rm{L}}}})\mathrm{ln}\frac{{E}_{{\rm{U}}}}{{E}_{{\rm{L}}}},\mathrm{if}\ {\alpha }_{\mathrm{ph}}=2,\,\mathrm{otherwise}\end{eqnarray*}$$

$$\begin{eqnarray*}&&F={N}_{({E}_{{\rm{L}}}\sim {E}_{{\rm{U}}})}\left(\displaystyle \frac{1-{\alpha }_{\mathrm{ph}}}{2-{\alpha }_{\mathrm{ph}}}\right)\left(\displaystyle \frac{{E}_{{\rm{U}}}^{2-{\alpha }_{\mathrm{ph}}}-{E}_{L}^{2-{\alpha }_{\mathrm{ph}}}}{{E}_{{\rm{U}}}^{1-{\alpha }_{\mathrm{ph}}}-{E}_{{\rm{L}}}^{1-{\alpha }_{\mathrm{ph}}}}\right).\end{eqnarray*}$$

So, one can get

$$\begin{eqnarray}&&{L}_{\gamma }=4\pi {d}_{L}^{2}{\left(1+z\right)}^{({\alpha }_{\gamma }-1)}F,\end{eqnarray} \tag{ 3 }$$

where d_L is a luminosity distance with Ω_Λ ∼ 0.692, Ω_M ∼0.308 and H₀ = 67.8 km s⁻¹ Mpc⁻¹, z is a redshift, α_γ is a γ-ray spectral index, α_γ = α_ph − 1, α_ph is a photon spectral index, and F is in units of GeV cm⁻² s⁻¹. In the work, E_L and E_U correspond to 1 GeV and 100 GeV respectively.

Redshift-z: For the 851 blazars and blazar candidates, redshift is available for 702 sources from NED. For all the known redshift, an averaged value 〈z〉 = 0.937 is obtained. If we only considered 396 FSRQs and 279 BL Lacs with available redshift, then the averaged value of redshift is 〈z〉 = 0.945. In Figure 1(a), the redshift distribution for the 396 FSRQs is shown in the upper left panel, and that for the 279 BL Lacs is shown in the lower left panel while their cumulative distribution for FSRQs and BL Lacs is shown in the right panel. The averaged values are z = 1.168 ± 0.622 for FSRQs, and z = 0.628 ± 0.724 for BL Lacs. When a K-S test is performed to the distributions, a probability p = 1.35 × 10⁻³¹ for the two distributions to be from the same parent distribution is obtained.

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Photon Spectral Index-α_ph: From the 4FGL catalog (Abdollahi et al. 2020; Ajello et al. 2020), the photon spectral index distributions are shown in the upper left panel for FSRQs, in the lower left panel for BL Lacs in Figure 1(b), a cumulative distribution is shown in the right panel of Figure 1(b). One can get the averaged values of 〈α_ph〉 = 2.45 ± 0.16 for FSRQs, and 〈α_ph〉 = 2.05 ± 0.18 for BL Lacs. The probability for the two distributions to be from the same parent distribution is zero (p = 1.17 × 10⁻¹¹⁰).

Variability Index—V.I.: From the 4FGL catalog, we can get the variability in the γ-ray band, the logarithm of the γ-ray variability index is obtained and their distribution is shown in the upper left panel for FSRQs, in the lower left panel for BL Lacs in Figure 1(c), and a corresponding cumulative distribution is in the right panel of Figure 1(c). The obtained corresponding averaged values are $\langle \mathrm{log}\,V.I.\rangle =2.35\,\pm 0.68$ for FSRQs, and $\langle \mathrm{log}\,V.I.\rangle =1.81\,\pm 0.51$ for BL Lacs. The corresponding p is 2.83 × 10⁻³⁰.

γ-ray Luminosity—$\mathrm{log}\,{L}_{\gamma }$: From Equation (3), one can calculate the γ-ray luminosity, L_γ . However, it is found that some sources have no known redshift, so an averaged value of 〈z〉 = 0.628 is adopted for BL Lacs without known redshift, and 〈z〉 = 0.937 for the BCUs without redshift.

The logarithm of the γ-ray luminosity is obtained and their distribution is shown in the upper left panel for FSRQs, in the lower left panel for BL Lacs in Figure 1(d), and a corresponding cumulative distribution is in the right panel of Figure 1(d). The corresponding averaged values are $\langle \mathrm{log}\,{L}_{\gamma }\rangle =47.50\,\pm 0.98$ for FSRQs, and $\langle \mathrm{log}\,{L}_{\gamma }\rangle =46.95\,\pm 1.29$ for BL Lacs. The corresponding p is 1.06 × 10⁻¹⁰.

Variation of Photon Spectral Index—Δα_ph: From 3FGL and 4FGL catalogs, we can obtain the change of photon spectral index ${\rm{\Delta }}{\alpha }_{\mathrm{ph}}={\alpha }_{\mathrm{ph}}^{4\mathrm{FGL}}-{\alpha }_{\mathrm{ph}}^{3\mathrm{FGL}}$. We show their histograms in the upper left panel for FSRQs, in the lower left panel for BL Lacs in Figure 1(e), and a corresponding cumulative distribution is in the right panel of Figure 1(e). The corresponding averaged values are Δα_ph = 0.044 ± 0.147 for FSRQs, and Δα_ph = 0.008 ± 0.080 for BL Lacs. The corresponding probability p = 2.42 × 10⁻⁴.

Variation of Luminosity—${\rm{\Delta }}\mathrm{log}{L}_{\gamma }$: From 3FGL and 4FGL catalogs and the luminosity calculation of Equation (3), we can obtain the logarithm of luminosity, $\mathrm{log}\,{L}_{\gamma }^{3\mathrm{FGL}}$ and $\mathrm{log}\,{L}_{\gamma }^{4\mathrm{FGL}}$, and the corresponding variability ${\rm{\Delta }}\mathrm{log}\,{L}_{\gamma }=\mathrm{log}\,{L}_{\gamma }^{4\mathrm{FGL}}\,-\mathrm{log}\,{L}_{\gamma }^{3\mathrm{FGL}}$. We show their histograms of luminosity variability in the upper left panel for FSRQs, in the lower left panel for BL Lacs in Figure 1(f), and a corresponding cumulative distribution is in the right panel of Figure 1(f). The corresponding averaged values are ${\rm{\Delta }}\mathrm{log}\,{L}_{\gamma }=0.07\,\pm 0.30$ for FSRQs, and ${\rm{\Delta }}\mathrm{log}\,{L}_{\gamma }\,=0.02\,\pm 0.21$ for BL Lacs. The probability for the two distributions to be from the same parent distribution is p = 7.54 × 10⁻⁸.

We will make the mutual correlation analysis for physics parameters.

Photon Spectral Index versus Redshift (α_ph − z): In the present sample, redshift is known for 708 blazars and BUCs. When a linear regression is adopted to the available redshift and photon spectral index, it is obtained ${\alpha }_{\mathrm{ph}}=(0.799\pm 0.017)\mathrm{log}\,(1+z)+2.077\pm 0.018$ for the 708 blazar with available redshift with a correlation coefficient r = 0.459 and a chance probability of p < 10⁻⁴. When the two subclasses are considered separately, it is found that ${\alpha }_{\mathrm{ph}}=(0.248\,\pm 0.062)\mathrm{log}\,(1+z)+2.370\pm 0.021$ with r = 0.20 and p < 10⁻⁴ for the 396 FSRQs, and ${\alpha }_{\mathrm{ph}}=(0.30\pm 0.075)\mathrm{log}\,(1+z)+1.991\pm 0.018$ with r = 0.23 and p = 1.06 × 10⁻⁴ for the 279 BL Lacs. See Figure 2(a).

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Spectral Index versus Peak Frequency (α_ph − ν_p): Fan et al. (2016) obtained synchrotron peak frequency ($\mathrm{log}{\nu }_{{\rm{p}}}$) for 1392 3FGL blazars from their SED fittings. In this work, we only considered 851 common sources with 4FGL and V. I. > 18.48. When a linear correlation analysis is adopted to the photon spectral index (α_ph) from 4FGL catalog and the peak frequency ($\mathrm{log}{\nu }_{{\rm{p}}}$), it is found that

$$\begin{eqnarray*}&&{\alpha }_{\mathrm{ph}}=-(0.151\pm 0.007)\mathrm{log}\,{\nu }_{{\rm{p}}}+4.386\pm 0.095\end{eqnarray*}$$

with r = − 0.61 and p < 10⁻⁴ for the whole sample (851 blazars and BCUs). The corresponding best fitting result is shown in Figure 2(b). When BL Lacs and FSRQs are considered separately, one has ${\alpha }_{\mathrm{ph}}=-(0.116\pm 0.006)\mathrm{log}\,{\nu }_{{\rm{p}}}+3.750\pm 0.093$ with r = − 0.683 and p < 10⁻⁴ for 383 BL Lacs, and a marginal correlation ${\alpha }_{\mathrm{ph}}=(0.027\pm 0.109)\mathrm{log}\,{\nu }_{{\rm{p}}}+2.082\pm 0.168$ with r = 0.109 and p = 2.9% for 396 FSRQs.

Luminosity versus Photon Spectral Index (L_γ − α_ph): From the calculated γ-ray luminosity and the photon spectral index from the 4FGL catalog, we investigated their mutual correlation and obtained

$$\begin{eqnarray*}&&{\alpha }_{\mathrm{ph}}=(0.023\pm 0.009)\mathrm{log}\,{L}_{\gamma }\,+1.162\pm 0.405\end{eqnarray*}$$

with r = 0.262 and p = 5.9 × 10⁻³ for all the 708 blazars and BCUs with available redshift. The corresponding best fitting result is shown in Figure 2(c). When BL Lacs and FSRQs with available redshift are considered separately, one has ${\alpha }_{\mathrm{ph}}\,=-(0.038\pm 0.008)\mathrm{log}\,{L}_{\gamma }\,+4.255\pm 0.376$ with r = − 0.235 and p < 10⁻⁴ for 396 FSRQs, and there is no clear correlation between α_ph and $\mathrm{log}\,{L}_{\gamma }$ for BL Lacs.

Variability Index versus Peak Frequency (V. I. − ν_p): When the linear correlation analysis is adopted to the variability index ($\mathrm{log}\,V.I.$) from 4FGL catalog and $\mathrm{log}\,{\nu }_{{\rm{p}}}$ (Fan et al. 2016), it is found that

$$\begin{eqnarray*}&&\mathrm{log}\,V.I.=-(0.188\pm 0.021)\mathrm{log}\,{\nu }_{{\rm{p}}}+4.731\pm 0.301\end{eqnarray*}$$

with a correlation coefficient r = − 0.297 and a chance probability of p < 10⁻⁴ for the whole sample. The corresponding best fitting result is shown in Figure 2(d). When BL Lacs and FSRQs are considered separately, one has $\mathrm{log}\,V.I.\,=-(0.133\pm 0.053)\mathrm{log}\,{\nu }_{{\rm{p}}}+4.157\pm 0.723$ with r = − 0.125 and p = 1.2% for 396 FSRQs, and $\mathrm{log}\,V.I.=-(0.058\,\pm 0.024)\mathrm{log}\,{\nu }_{{\rm{p}}}+2.669\pm 0.356$ with r = − 0.123 and p =1.6% for 383 BL Lacs.

Luminosity versus Variability Index (L_γ − V. I. ): From the present γ-ray luminosity and variability index obtained from the 4FGL catalog, we obtained their mutual correlation

$$\begin{eqnarray*}&&\mathrm{log}\,V.I.=\,(0.266\pm 0.019)\mathrm{log}\,{L}_{\gamma }\,-10.459\pm 0.916\end{eqnarray*}$$

with r = 0.460 and p < 10⁻⁴ for all the 708 blazars and BCUs with available redshift. The corresponding best fitting result is shown in Figure 2(e). While for BL Lacs and FSRQs with available redshift, one has $\mathrm{log}\,V.I.\,=\,(0.357\pm 0.030)\mathrm{log}\,{L}_{\gamma }\,-14.621\pm 1.425$ with r = 0.512 and p < 10⁻⁴ for 396 FSRQs, and $\mathrm{log}\,V.I.\,=(0.108\pm 0.023)\mathrm{log}\,{L}_{\gamma }\,-3.286\pm 1.069$ with r = 0.275 and p < 10⁻⁴ for 279 BL Lacs.

Photon Spectral Index versus Variability Index (α_ph − V. I.): The photon spectral index and variability are obtained from the 4FGL catalog for the 851 sources, and a linear mutual correlation analysis gives

$$\begin{eqnarray*}&&\mathrm{log}(V.I.)=(0.537\pm 0.083){\alpha }_{\mathrm{ph}}+0.857\pm 0.189\end{eqnarray*}$$

with r = 0.216 and p < 10⁻⁴ for the whole sample, the corresponding best fitting result is shown in Figure 2(f). However, when BL Lacs and FSRQs are considered separately, it is found that $\mathrm{log}\,V.I.\;=\;-(1.400\pm 0.206){\alpha }_{\gamma }+5.780\pm 0.504$ with r = − 0.325 and p < 10⁻⁴ for 396 FSRQs, and almost no correlation for BL Lacs.

Luminosity Variation versus Spectral Index Variation $({\rm{\Delta }}\mathrm{log}\,{L}_{\gamma }-{\rm{\Delta }}{\alpha }_{\mathrm{ph}})$: From above calculations, we can get the γ-ray luminosity variation (${\rm{\Delta }}\mathrm{log}\,{L}_{\gamma }=\mathrm{log}\,{L}_{\gamma }^{4\mathrm{FGL}}-\mathrm{log}\,{L}_{\gamma }^{3\mathrm{FGL}}$), and the spectral index variation (${\rm{\Delta }}{\alpha }_{\mathrm{ph}}={\alpha }_{\mathrm{ph}}^{4\mathrm{FGL}}-{\alpha }_{\mathrm{ph}}^{3\mathrm{FGL}}$), which give a linear correlation

$$\begin{eqnarray*}&&{\rm{\Delta }}\mathrm{log}\,{L}_{\gamma }=-(1.352\pm 0.050){\rm{\Delta }}{\alpha }_{\mathrm{ph}}-0.006\pm 0.007\end{eqnarray*}$$

with r = − 0.715 and p < 10⁻⁴ for the whole sample. The corresponding best fitting result for the whole sample is shown in Figure 3. When BL Lacs and FSRQs are considered separately, it is found that ${\rm{\Delta }}\mathrm{log}\,{L}_{\gamma }=-(1.448\,\pm 0.073){\rm{\Delta }}{\alpha }_{\mathrm{ph}}-0.002\pm 0.011$ with r = − 0.707 and p < 10⁻⁴ for 396 FSRQs, and ${\rm{\Delta }}\mathrm{log}\,{L}_{\gamma }=-(1.206\pm 0.068){\rm{\Delta }}{\alpha }_{\mathrm{ph}}\,-0.006\pm 0.009$ with r = − 0.731 and p < 10⁻⁴ for 279 BL Lacs.

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The present analysis based on our sample shows that, on average, the photon spectral index (α_ph), the variability index ($\mathrm{log}\,V.I.$), the γ-ray luminosity ($\mathrm{log}\,{L}_{\gamma }$), and its variability (${\rm{\Delta }}\mathrm{log}\,{L}_{\gamma }$) in FSRQs are all clearly greater than those in BL Lacs. The K-S test indicates that the probabilities for the distribution for FSRQs and that for BL Lacs to be from the same parent distribution is less than 7.54 × 10⁻⁸. For the average variation in spectral index (〈Δα_ph〉), we can also obtain that 〈Δα_ph〉 of FSRQs is greater than that of BL Lacs, the K-S test indicates that the probability for the distribution in FSRQs and that in BL Lacs to be from the same parent distribution is p = 2.42 × 10⁻⁴.

The fact that FSRQs show greater γ-ray luminosity and the photon spectral index than the BL Lacs was mentioned in previous studies (Ghisellini et al. 2009; Ackermann et al. 2015; Chen 2018; Abdollahi et al. 2020). It is found that the photon spectral index decreases with synchrotron peak frequency as shown in Figure 2(b), which is consistent with the result that 〈α_ph〉 in FSRQs is averagely greater than that in BL Lacs (Figure 1(b)) because FSRQs have, on average, lower synchrotron peak frequency ($\mathrm{log}\,{\nu }_{\mathrm{ph}}$) than do BL Lacs. Our result is also consistent with that in Ackermann et al. (2015), who displayed that the photon spectral index decreases with synchrotron peak frequency. The γ-ray emission is originated from the inverse Compton (IC) process in a leptonic model, the FSRQs are believed to have richer seed photons from the external field than the BL Lacs (Fossai et al. 1998; Ghisellini et al. 1998) that have been proved by the emission line observations (Shaw et al. 2012; Paliya et al. 2021). From the view of the relativistic beaming model, blazars display a strong beaming effect (Hovatta et al. 2009; Liodakis et al. 2018) and the FSRQs show a larger Doppler factor than the BL Lacs in γ-ray band (Zhang et al. 2020; Xiao et al. 2022). Thus, it is natural that the FSRQs are showing brighter γ-ray luminosity than the BL Lacs. A rich soft photon field yields a lower synchrotron and IC peaks, which is mostly observed in FSRQs, due to the efficient relativistic electrons cooling. In this case, those electrons can hardly up-scatter soft photons to extremely high energy regime. Consequently, FSRQs have lower IC peak frequency and steeper γ-ray spectrum than the BL Lacs at GeV γ-ray band.

From the perspective of blazar γ-ray emission, a more variable γ-ray emission and spectral is expected for FSRQs. Because γ-ray emission is believed to be a combination of synchrotron self-Compton (SSC) and of IC process for FSRQs, the SSC is considered to be the only γ-ray emission mechanism in BL Lacs (Ghisellini et al. 1998). The strength of the external photons field, which is provided by the accretion disk, board emission-line region (BLR), and dusty torus (DT), could vary with the black hole and disk activities. Thus, we should expect larger γ-ray luminosity variability (${\rm{\Delta }}\mathrm{log}{L}_{\gamma }$), spectral index variability (〈Δα_ph〉), and variability index ($\mathrm{log}\,V.I.$). One can also find that the variability index ($\mathrm{log}\,V.I.$) decreases with synchrotron peak frequency ($\mathrm{log}\,{\nu }_{{\rm{p}}}$) as shown in Figure 2(d), which is consistent with that in Ackermann et al. (2015).

For the γ-ray luminosity and the photon spectral index, it was found that the spectral index increases with the γ-ray luminosity in Ghisellini et al. (2009), who proposed that FSRQs occupy the region with α_γ > 1.2 and $\mathrm{log}\,{L}_{\gamma }\ (\mathrm{erg}\,{{\rm{s}}}^{-1})\gt 47,$ and BL Lacs in the region with α_γ < 1.2 and $\mathrm{log}\,{L}_{\gamma }\ (\mathrm{erg}\,{{\rm{s}}}^{-1})\lt 47.$ In this work, a positive correlation, ${\alpha }_{\mathrm{ph}}=(0.024\pm 0.009)\mathrm{log}\,{L}_{\gamma }\,+1.162\pm 0.405$ shows up for the 708 sources with available redshift as shown in Figure 2(c), which is consistent with the results in Ghisellini et al. (2009), Ackermann et al. (2015).

However, when BL Lacs and FSRQs are considered separately, they both display different correlations between the spectral index and the γ-ray luminosity. ${\alpha }_{\mathrm{ph}}=-(0.038\,\pm 0.008)\mathrm{log}\,{L}_{\gamma }\,+4.255\pm 0.376$ with r = 0.235 and p < 10⁻⁴ is obtained for 396 FSRQs showing an anti-correlation, and there is also a marginal anti-correlation, ${\alpha }_{\mathrm{ph}}=-(0.011\,\pm 0.008)\mathrm{log}\,{L}_{\gamma }\,+2.562\pm 0.392$ for BL Lacs. It is found that the more luminous the source is the harder the γ-ray spectrum, suggesting brighter source have harder spectrum. So, the observed positive correlation for the whole blazars between the γ-ray luminosity and the photon spectral index is an apparent correlation.

Our analysis shows that BL Lacs and FSRQs occupy different regions in the plot of spectral index against the γ-ray luminosity (α_ph versus $\mathrm{log}\,{L}_{\gamma }$) as in Ghisellini et al. (2009), in which the separation between FSRQs and BL Lacs results from two independent populations with different physical conditions as to the origin, propagation and radiative properties of their jets. We employ a support vector machine (SVM), a kind of machine learning (ML) method, to find a dividing line for separating the two blazar subclasses in the diagram of α_ph versus $\mathrm{log}\,{L}_{\gamma }$. The result gives an accuracy of 91.0% for the separation and predicts a dividing line of ${\alpha }_{\mathrm{ph}}=-0.048\,\mathrm{log}\,{L}_{\gamma }+4.498$ as shown in Figure 4. In 2018, Chen (2018) obtained that ${\alpha }_{\mathrm{ph}}\,=-0.127\,\mathrm{log}\,{L}_{\gamma }+8.18$ with an accuracy of 88.6%. It is found in Figure 4 that majority of FSRQs occupy the region above the line while majority of BL Lacs below the line. Therefore, a source is classified as an FSRQ candidate if ${\alpha }_{\mathrm{ph}}\,\gt -0.048\,\mathrm{log}\,{L}_{\gamma }+4.498$, otherwise, it is classified as a BL Lac candidate. When the 72 BCUs are considered, 50 BCUs are in the region above the dividing line, so those BCUs can be taken as FSRQ candidates and 22 BCUs are in the region below the line, which can be taken as BL Lac candidates.

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Figure 2(f) shows a positive correlation between variability index $\mathrm{log}\,V.I.$ and the photon spectral index (α_ph). FSRQs, which lie in the upper-right region, have higher variability and larger power index than BL Lacs, which occupy the lower-left region. So, the positive correlation is apparent. When BL Lacs and FSRQs are considered separately, there is an anti-correlation for FSRQs but no correlation for BL Lacs as shown in Figure 2(f).

We also employ a support vector machine (SVM) to find a dividing line for separating blazar subclasses in the diagram of $\mathrm{log}\,V.I.$ against α_ph. A dividing line of $\mathrm{log}\,V.I.\,=-10.119{\alpha }_{\mathrm{ph}}+24.855$ with an accuracy of 91% for the separation is obtained as shown in Figure 5. FSRQs occupy the region above the line while BL Lacs are below the line. Therefore, a source is classified as an FSRQ candidate if $\mathrm{log}\,V.I.\,\gt \,-10.119{\alpha }_{\mathrm{ph}}+24.855$, otherwise, it is classified as a BL Lac candidate.

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When a BCU is introduced to the diagram, it can be classified as an FSRQ candidate if it lies above the dividing line and as a BL Lac candidate if it lies below the dividing line. For the present 72 BCUs, 47 BCUs are classified as FSRQ candidates and 25 BCUs as BL Lac candidates.

We compare classification results from the above two methods, namely the photon spectral index versus γ-ray luminosity (α_ph–L_γ ) method using ${\alpha }_{\mathrm{ph}}=-0.048\,\mathrm{log}\,{L}_{\gamma }+4.498$ in Figure 4, and the photon spectral index and variability index (α_ph–V. I. ) method using $\mathrm{log}\,V.I.\,=\,-10.119{\alpha }_{\mathrm{ph}}+24.855$ in Figure 5.

When we compare the FSRQs candidates obtained from the α_ph–L_γ method (Figure 4) and the α_ph–V. I. method (Figure 5), we found 68 common classification, namely, 46 FSRQ candidates and 21 BL Lac candidates. It is found that all 47 FSRQ but one (J0644.3-6713) candidates from α_ph–V. I. method are included in the 50 FSRQ candidates from α_ph–L_γ method. For BL Lac candidates, 22 but one (J0644.3-6713) sources obtained from the α_ph–L_γ method are included in those from the α_ph–V. I. method.

There are also four sources (J2230.5-7817, J0017.2-0643, J1647.1-6438, and J1302.6+5748 ) classified as BL Lac candidates in the α_ph–V. I. method are classified as FSRQ candidates in the α_ph–L_γ method. We still keep the four BCU sources and the BCU J0644.3-6713 as BCUs in this work. We can also found that the five sources have no known redshift. It is possible that the unknown redshift results in false classification. In this case, if only are 33 BCU with available redshift considered, then it is found that the two methods give very similar classification, only one source (J1302.6+5748), which is classified as a BL Lac candidate in α_ph–V. I. method but as an FSRQ candidate in α_ph–L_γ . So, for 33 BCUs with redshift, we have 23 FSRQ candidates and nine BL Lac candidates.

When we compared our results with that by Kang et al. (2019), it is found that there are 70 common sources (J0734.3-7709 and J1532.7-1319 were not in Kang et al. 2019), out of them 61 sources in α_ph–V. I. method are the same classification as that by Kang et al. (2019) giving a 87.1% goodness of fit; and the 59 sources in the α_ph–L_γ method are the same classification as that by Kang et al. (2019) giving a 84.3% goodness of fit as shown in Table 2, which give our classification for the BCUs.

Table 2. Classification for the BCU Sources in this Work

4FGL Name	Class${}_{{\alpha }_{\mathrm{ph}}-{L}_{\gamma }}^{1}$	Class${}_{V.I.-{\alpha }_{\mathrm{ph}}}^{2}$	Class-TW	Class(K19)	Comp(1-K19)	Comp(2-K19)
(1)	(2)	(3)	(4)	(5)	(6)	(7)
J0017.2-0643	FSRQ	BL	BCU	BL	N	Y
J0019.1-5645	BL	BL	BL	BL	Y	Y
J0030.7-0209	FSRQ	FSRQ	FSRQ	FSRQ	Y	Y
J0045.2-3704	FSRQ	FSRQ	FSRQ	FSRQ	Y	Y
J0146.4-6746	BL	BL	BL	BL	Y	Y
J0152.2+3707	FSRQ	FSRQ	FSRQ	BL	N	N
J0156.3+3913	FSRQ	FSRQ	FSRQ	FSRQ	Y	Y
J0200.9-6635	FSRQ	FSRQ	FSRQ	FSRQ	Y	Y
J0205.0+1510	FSRQ	FSRQ	FSRQ	FSRQ	Y	Y
J0214.7-5823	BL	BL	BL	BL	Y	Y
J0218.9+3642	FSRQ	FSRQ	FSRQ	FSRQ	Y	Y
...	...	...	...	...	...	...
...	...	...	...	...	...	...
...	...	...	...	...	...	...
...	...	...	...	...	...	...
...	...	...	...	...	...	...

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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In Table 2, Column information is as follows: Col. (1) 4FGL name, Col. (2) classification based on α_ph–L_γ method, Col. (3) classification based on V. I. –α_ph method, Col. (4) classification from the two methods, we choose the same classification from the two methods as a candidate, otherwise, we still take it as a BCU; Col. (5) classification in Kang et al. (2019), Col. (6) comparison between classification by α_ph–L_γ method and that in Kang et al. (2019). If our classification is the same as that in Kang et al. (2019), then we use "Y," otherwise, we use "N." "NN" means that the source was not included in Kang et al. (2019), Col. (7) comparison between classification by V. I. –α_ph method and that in Kang et al. (2019).

From Table 2, it is found that for the 70 common sources, we give classification for 65 sources and BCUs for five sources. For the 65 sources, our classifications for 58 sources are the same as that in Kang et al. (2019) giving a 89.3% goodness of fit. 38 out of our 44 FSRQ candidates were also classified as FSRQs candidates in Kang et al. (2019) giving a 86.4% goodness of fit. 19 out of our 21 BL Lac candidates were classified as BL Lac candidates in Kang et al. (2019) giving a 90.5% goodness of fit. So, our blazar (BL Lacs and FSRQs) candidate selection methods are available. We encourage colleagues who are interested in the classification to do spectroscopic observations from those sources to confirm the classifications. In addition, if we know the classification, then we can use the α_ph–L_γ relation to give a redshift lower limit for FSRQs candidates and an upper limit for BL Lac candidates.

It is observed from Figure 3 that there is an anti-correlation between the γ-ray luminosity variation (${\rm{\Delta }}\mathrm{log}\,{L}_{\gamma }$) and the spectral index variation (Δα_ph),

$$\begin{eqnarray*}&&{\rm{\Delta }}\mathrm{log}\,{L}_{\gamma }=-(1.046\pm 0.030){\rm{\Delta }}{\alpha }_{\mathrm{ph}}+0.028\pm 0.005\end{eqnarray*}$$

for the whole sample.

When BL Lacs and FSRQs are considered separately, such an anti-correlation is also observed for BL Lacs and FSRQs. ${\rm{\Delta }}\mathrm{log}\,{L}_{\gamma }=-(1.386\pm 0.037){\rm{\Delta }}{\alpha }_{\mathrm{ph}}+0.010\pm 0.010$ with r = − 0.71 and p ∼ 0 is found for FSRQs, and ${\rm{\Delta }}\mathrm{log}\,{L}_{\gamma }\,=-(0.890\pm 0.032){\rm{\Delta }}{\alpha }_{\mathrm{ph}}+0.028\pm 0.005$ with r = − 0.724 and p ∼ 0 for BL Lacs. The corresponding best fitting result for the whole sample is shown in Figure 3.

In the γ-ray band, the flux density can be expressed as ${f}_{\nu }={f}_{0}{\nu }^{-{\alpha }_{\nu }}$, so, the luminosity can be expressed in the form ${L}_{\nu }={L}_{0}{\nu }^{-{\alpha }_{\nu }}$. It is obtained that $\displaystyle \frac{{L}_{\nu }^{1}}{{L}_{\nu }^{2}}=\tfrac{{L}_{0}^{1}}{{L}_{0}^{2}}{\nu }^{-({\alpha }_{\nu }^{1}-{\alpha }_{\nu }^{2})},$ which results in ${\rm{\Delta }}\mathrm{log}\,{L}_{\gamma }=-A{\rm{\Delta }}{\alpha }_{\mathrm{ph}}+B$, here α_ph = α_γ + 1. The correlation suggests that when the source becomes brighter, then the spectrum becomes harder. It is common for blazars to show a blue spectrum when the sources become brighter (BWB) as discussed in the optical band (Fan & Lin 2000; Gu et al. 2006; Gupta et al. 2008; Zheng et al. 2008; Carnerero et al. 2017) and in the X-ray band (Pian et al. 1997). The BWB phenomenon could be explained in different ways (Gu et al. 2006), including a two-component (one stable component and a variable one) model for blazar optical emission or the one synchrotron model that the intense energy release is produced by the higher particles (Fiorucci et al. 2004). Some researchers suggested that the fresh electrons injection, with a harder energy distribution than the cooled one, should explain the BWB phenomenon (Kirk et al. 1998; Mastichiadis & Kirk 2002), etc. However, these models would have to involve more parameters in the blazar emission model and those models may not be able to explain the BWB phenomenon in the entire wavelength band, e.g., optical, X-ray, and γ-ray bands. We suggest that the Doppler beaming effect could be one of the main reasons for raising the BWB phenomenon, and the γ-ray band brightness and spectra change due to a variety of Doppler factors (δ), this idea was also suggested by Villata et al. (2004) to explain the optical BWB phenomenon. The blazar emission is dominated by the relativistic jet, which is strongly beamed due to a Doppler beaming effect. The luminosity becomes brighter and the spectrum becomes harder at the γ-ray band when the Doppler factor becomes greater. This is because an increase of δ can enlarge the IC peak frequency as well as its intensity (Albert et al. 2007, see Figure 21).