The Spectral Energy Distribution of Fermi Blazars

# The Spectral Energy Distributions of Fermi Blazars

## Abstract

In this paper, multi-wavelength data are compiled for a sample of 1425 Fermi blazars to calculate their spectral energy distributions (SEDs). A parabolic function, is used for SED fitting. Synchrotron peak frequency (), spectral curvature (), peak flux (), and integrated flux () are successfully obtained for 1392 blazars (461 flat spectrum radio quasars-FSRQs, 620 BL Lacs-BLs and 311 blazars of uncertain type-BCUs, 999 sources have known redshifts). Monochromatic luminosity at radio 1.4 GHz, optical R band, X-ray at 1 keV and -ray at 1 GeV, peak luminosity, integrated luminosity and effective spectral indexes of radio to optical (), and optical to X-ray () are calculated. The ”Bayesian classification” is employed to log in the rest frame for 999 blazars with available redshift and the results show that 3 components are enough to fit the log distribution, there is no ultra high peaked subclass. Based on the 3 components, the subclasses of blazars using the acronyms of Abdo et al. (2010a) are classified, and some mutual correlations are also studied. Conclusions are finally drawn as follows: (1) SEDs are successfully obtained for 1392 blazars. The fitted peak frequencies are compared with common sources from samples available ( Sambruna et al. 1996, Nieppola et al. 2006, 2008, Abdo et al. 2010a). (2) Blazars are classified as low synchrotron peak sources (LSPs) if log 14.0, intermediate synchrotron peak sources (ISPs) if , and high synchrotron peak sources (HSPs) if . (3) -ray emissions are strongly correlated with radio emissions. -ray luminosity is also correlated with synchrotron peak luminosity and integrated luminosity. (4) There is an anti-correlation between peak frequency and peak luminosity within the whole blazar sample. However, there is a marginally positive correlation for HBLs, and no correlations for FSRQs or LBLs. (5) There are anti-correlations between the monochromatic luminosities (-ray and radio bands) and the peak frequency within the whole sample and BL Lacs. (6) The optical to X-ray () and radio to optical () spectral indexes are strongly anti-correlated with peak frequency (log ) within the whole sample, but the correlations for subclasses of FSRQs, LBLs, and HBLs are different.

BL Lacertae objects: general, galaxies: active, galaxies: jets, galaxies: nuclei
1

## 1 Introduction

The most powerful active galactic nuclei (AGNs) are the sources referred to as blazars, which show rapid variability and high luminosity, high and variable polarization, superluminal motions, core-dominated non-thermal continuum and strong -ray emissions, etc. ( Abdo et al. 2009a, 2010b,c; Acero et al. 2015; Ackermann et al. 2011a,b, 2015; Aller et al. 2011; Bai, et al. 1998; Bastieri et al. 2011; Chen et al., 2012; Fan & Xie 1996; Fan et al. 2011; Ghisellini et al. 2010; Gu 2014; Gu & Li 2013; Gupta 2011; Gupta et al. 2012; Hu, et al. 2006; Marscher et al. 2011; Nolan et al. 2012; Romero et al. 2002; Urry 2011; Wills et al. 1992; Wu, et al. 2007; Yang et al. 2010a,b, 2012a,b, 2014; You & Cao, 2014). Quite recently, Massaro et al. (2015) published the largest blazar sample (The 5th edition of the Roma-BZCAT, see the BZCAT 5.0 (http://www.asdc.asi.it/bzcat/). Blazars consist of two subclasses, namely BL Lacertae objects (BL Lacs) and flat spectrum radio quasars (FSRQs), both subclasses have the common continuum properties while their emission line features are quite different, namely FSRQs have strong emission lines while BL Lacs have no emission lines or very weak emission lines. The strong radio continuum is believed to be produced via synchrotron process. This synchrotron radiation is reflected in the blazar spectral energy distribution (SED) of the vs as a bump in radio to X-ray frequencies. Another bump followed is often attributed to the inverse Compton process.

BL Lacertae objects can be classified as radio selected BL Lacs (RBLs) and X-ray selected BL Lacs (XBLs) from surveys. In 1995, Giommi et al. constructed radio to X-ray energy distributions ( vs ) of a sample of 121 BL Lacs to investigate the difference between XBLs and RBLs, and found that the synchrotron peak for RBLs locates at the IR/optical band while for XBLs it locates at the UV/X-ray band. BL Lacs were proposed to distinguish by the ratio of as high-energy cut-off BL Lacs (HBLs) and low-energy cut-off BL Lacs (LBLs) respectively when or (where X-ray fluxes cover 0.3 3.5 keV in units of erg/cm/s while the radio fluxes are at 1.4 GHz in units of Jy) (Padovani & Giommi 1995). In 1996, with a parabola which was used by Landau et al. (1986) to parameterize the spectral flux distribution in , Sambruna et al. parameterized the power per decade energy distribution by a logarithmic parabola form using a sample of blazars including complete samples of RBLs and XBLs. They obtained integrated luminosity, peak frequency, and the peak luminosity. The averaged peak frequency is for EMSS XBLs and for 1 Jy RBLs. In 1996, Padovani & Giommi classified BL Lacs as HBLs () and LBLs ().

Later, Fossati et al. (1998) computed the average SED from radio to -rays using three complete samples of blazars (FSRQs, RBLs, and XBLs), and found that there is a clear continuity for more luminous blazars to have lower first peak frequency and for less luminous blazars to have higher first peak frequency. However observations show high luminosity HBLs (Giommi et al. 2005) and low power LBLs (Padovani et al. 2003; Caccianiga & March, 2004).

In 2002, we calculated SEDs using a sample of -ray blazars with available radio Doppler factors and estimated the Doppler factors at -ray band (Zhang, et al. 2002). Nieppola et al. (2006) calculated SEDs in the form of for 308 blazars, classified BL lacertae objects into LBLs, IBLs, and HBLs, and set the boundaries as for LBLs, for IBLs, and for HBLs. They also calculated peak frequencies for 135 AGNs by fitting in 2008 (Nieppola, et al. 2008). Abdo et al. (2010a) calculated SEDs for 48 Fermi blazars, and proposed an empirical parametrization for synchrotron peak frequency based on effective spectral indexes of (radio-optical) and (optical-X-rays) using the available fitting peak frequencies and the effective spectral indexes. They also extended the definition to all types of non-thermal dominated AGNs using new acronyms: low synchrotron peaked blazars (LSP, Hz), intermediate synchrotron peaked blazars (ISP, Hz), and high synchrotron peaked blazars (HSP, Hz). From the above work, it is noted that the criteria of the synchrotron peak frequency boundaries used to assign a type of HSP, ISP and HSP, is not uniform.

For the 2nd bump in the plots of of blazars, the peak frequencies are in the region of GeV to TeV bands. Blazars are strong -ray emitters. EGRET/GRO has detected about 60 high confidence -ray bright blazars (Hartman et al. 1999). The second generation of -ray detector, Fermi/LAT, has detected about 1800 blazars and unidentified blazars ( see Abdo et al 2010c, Ackermann, et al. 2011a, Nolan et al. 2012, Acero et al. 2015; Ackermann, et al. 2015).

From the 3FGL (Acero et al. 2015), a sample of Fermi detected blazars (classified as FSRQs, BLs and blazars of uncertain type-BCUs) is now available. In this paper, we have compiled multiwavelength data from NED for a sample of 1425 Fermi blazars, calculate their SEDs and discuss the relationships between some relevant parameters. In Sect. 2, we present our Fermi detected blazar sample, calculate their SEDs, discuss their classification, and analyze the correlations. In Sect. 3, correlation analysis results are given. In Sect. 4, some discussions and in Sect. 5, some conclusions are both presented.

The spectral index is defined as , and all luminosities are denoted simply by .

## 2 Sample and Results

### 2.1 Sample and SED Results

In this paper, a sample of 1425 Fermi detected blazars (FSRQs, BLs, and blazars of uncertain type-BCUs) are from the 3FGL (Acero et al. 2015). The multi-frequency data (radio to X-ray bands) collected from NED are used to calculate their SEDs. In doing so, the infrared and optical data are corrected using in NED for reddening/galactic absorption. Then the corrected infrared and optical magnitudes are transferred into flux densities, and afterwards, all the flux densities are K-corrected by , where (, is the photon spectral index for X-ray and -ray bands) is the spectral index at frequency , and is the redshift. If the redshift and spectral index are not available, then we can adopt averaged values of the sub-sample to replace them. For redshift, the following averaged values from our sample are obtained: for BLs, and for BCUs. For spectral indexes, we adopt for radio band (Donato et al. 2001, Abdo et al. 2010a), while for optical band, = 0.5 for BLs and = 1 for the rest of the sources as similar to what have been done by Donato et al. (2001): for BLs, for FSRQs, and for BCUs.

The spectral energy distributions (SEDs) are calculated by fitting following relation with a least square fitting method,

 log(νFν)=P1(logν−P2)2+P3,

where , and are constants with being the spectral curvature, the peak frequency () and peak flux (). The SED fitting figures of 1425 sources are shown in Fig 1 and Appendix. From the fitting results, it can be seen that SEDs have been successfully fitted only for 1392 sources with resulting , , and . The fitting results (peak frequency, spectral curvature), peak luminosity, integrated luminosity, monochromatic luminosity, and effective spectral indexes are listed in Table 1, where

Col. (1) gives the 3FGL name;

Col. (2) redshift from NED database at IPAC;

Col. (3) gives the SED classification by our method. HF stands for HSP FSRQ, IF for ISP FSRQ, LF for LSP FSRQ, HBL for HSP BL Lac, IBL for ISP BL Lac, LBL for LSP BL Lac, HU for HSP U-Blazar, IU for ISP U-Blazar, and LU for LSP U-Blazar;

Col. (4) radio luminosity and its uncertainty at 1.4 GHz ();

Col. (5) optical R luminosity and its uncertainty () ;

Col. (6) X-ray luminosity and its uncertainty at 1 keV ();

Col. (7) -ray luminosity and its uncertainty at 1 GeV ();

Col. (8) and (9) give the effective spectral indices and the corresponding uncertainties of radio to optical () and optical to X-ray (). They are calculated by a formula (Ledden & O’Dell 1985), , where and are the flux densities in frequencies and , respectively, and . In this paper, GHz, Hz and Hz are adopted.

Col. (10) spectral curvature () and its uncertainty;

Col. (11) synchrotron peak frequency (, Hz) and its uncertainty;

Col. (12) peak luminosity (, ) and its uncertainty;

Col. (13) integrated (bolometric) luminosity (, ) and its uncertainty.

### 2.2 Classification of Blazars

In Table 1, the peak frequency, classification, luminosities, spectral indexes are listed for 1392 blazars, where 461 are FSRQs, 620 are BLs and 311 are BCUs. Amongst the 1392 blazars, 999 objects have available redshift. For the whole 1392 sources, a distribution for logarithm of fitted peak frequencies is shown in Fig. 2a. From the distribution, it can be seen that there are 4 peaks locating at about , , , and , and 3 concave points at about , , and .

For the 999 blazars (463 BLs, 461 FSRQs, and 75 BCUs) with available redshifts, a distribution for the logarithm of fitted peak frequencies is shown in Fig. 2c. From the distribution, it can be seen that there are 4 peaks locating at about , , , and , and 4 concave points at about , , , and .

When the peak frequencies are corrected to the rest frame, we have = + . A distribution for the logarithm of corrected peak frequencies is shown in Fig. 2d. From the distribution, it can be seen that there are many peaks locating at about , , , , , , and , and many concave points at about , , , , , , , , and .

It is hard from the distributions to set boundary by eyes for different subclasses. To classify different subclass, a ¡¯normal mixture model¡¯ of Gaussian components is applied to the peak frequencies to confidently identify the peaks present. Then the existence of 3 vs. 2 or 4 components can be quantitatively established using maximum likelihood estimation (via the expectation-maximization (EM) Algorithm) and Bayesian Information Criterion (BIC) for model selection. The CRAN package ‘mclust’ (Chris et al. 2012, Chris & Adrian 2002) within the public domain R statistical software environment is used for the analysis. ‘mclust’ provides iterative EM methods for maximum likelihood clustering with parameterized Gaussian mixture models. First, density estimation via Gaussian finite mixture modeling was conducted. The BIC values for our 999 peak frequencies in the rest frame are shown in Fig 3d. There are 2 kinds of models fitted by ‘mclust’, one is the V (univariate, unequal variance) models, the other is the E (univariate, equal variance) models. An astro-oriented tutorial could be found in Sec. 9.92 of Modern Statistical Methods for Astronomy with R Applications (Feigelson & Bau, 2012). Our analysis indicates that the better one is the V model with 3 components: For the 1st component, it has a mean value of = 13.56 with a variance of 0.129 and a clustering probability of 0.32; the 2nd one has a mean of = 14.49 with a variance of 0.294 and a clustering probability of 0.38; and the 3rd one has = 15.46 with a variance of 1.701 and a clustering probability of 0.30. The density function with the 3 components is plotted also in Fig 2d. As can be seen in Fig 2d, the crossing points of two adjacent Gaussian curves are at and . If we choose the frequencies at the jointing points as the boundaries for classification and follow the acronyms of LSP, ISP, and HSP (Abdo et al. 2010a), the following classifications can be set:

for LSPs,

for ISPs, and

for HSPs.

Based on the classifications, we obtain: 38.6% of the 999 blazar sample are LSPs, 42.9% are ISPs, and 18.4% are HSPs.

For the 999 peak frequencies at observer’s frame, the same process is performed, the following results are obtained: the 1st component has a mean value of = 13.19 with a variance of 0.090 and a clustering probability of 0.28; the 2nd one has a mean of = 14.20 with a variance of 0.310 and a clustering probability of 0.40; the 3rd one has = 15.24 with a variance of 1.724 and a clustering probability of 0.32. The BIC values and the corresponding density function are shown in Fig 3c and Fig 2c respectively.

For the 1392 peak frequencies at observer’s frame, we have: the 1st component has a mean value of = 13.19 with a variance of 0.091 and a clustering probability of 0.22; the 2nd one has a mean of = 14.23 with a variance of 0.445 and a clustering probability of 0.43; and the 3rd one has = 15.45 with a variance of 1.861 and a clustering probability of 0.36. The BIC values and the corresponding density function are shown in Fig 3a and Fig 2a respectively.

For the 1392 objects, we adopt the averaged redshift values for the unknown sources ( for BLs and for BCUs), then we obtain: the 1st component has a mean value of = 13.59 with a variance of 0.151 and a clustering probability of 0.30; the 2nd one has a mean of = 14.61 with a variance of 0.314 and a clustering probability of 0.33; and the 3rd one has = 15.62 with a variance of 1.824 and a clustering probability of 0.37. The BIC values and the corresponding density function are shown in Fig 3b and Fig 2b respectively.

The classifications for 1392 blazars are shown in Col. (3) in Table 1, where 34.77% of the whole sample are LSPs, 40.09% are ISPs, and 25.14% are HSPs. See Table 2 for details.

## 3 Correlations

### 3.1 Correlations between γ-ray Luminosity and Other Luminosities

In Table 1, monochromatic luminosities at 1.4 GHz (), optical R band (), X-ray () at 1 KeV, and -ray () at 1 GeV are given. Here, the luminosity is calculated by a formulae , where, ()(Pedro & Priyamvada 2007) is luminosity distance and is the K-corrected flux density at the corresponding frequency .

Following our pervious papers (Fan et al. 2013, 2014; Lin & Fan 2016; Nie et al. 2014; Yang, et al. 2002b, 2014), the calculation of -ray luminosity is further conducted. The luminosity () at synchrotron peak frequency and integrated luminosity () are then obtained by , where () is from SED fittings.

The correlations between -ray luminosity and lower energy bands at radio, optical and X-ray are shown in Fig. 4, and the correlations between -ray luminosity and the peak and integrated luminosities are shown in Fig. 5. The linear regression fitting results are shown in Table 3, where the linear regression fitting relation is expressed as , is a correlation coefficient, is the number of sources in the corresponding sample (sub-sample), is a chance probability, and are correlation coefficient and the corresponding chance probability after removing redshift effect respectively.

### 3.2 Correlations between Peak Frequency and Other Parameters

Now, we investigate correlations between synchrotron peak frequency () and other parameters including monochromatic luminosity (-ray, X-ray, optical, and radio band), integrated luminosity, peak luminosity, spectral curvature (), effective spectral indexes. The spectral curvature () is from SED fitting, and the effective spectral indexes are calculated as (Ledden & O’Dell 1985).

The relations between peak frequency (log ) and monochromatic luminosities at 1.4 GHz (), optical R band (), X-ray () at 1 KeV, and -ray () are shown in Fig. 6. The relations between the peak frequency (log ) and the peak luminosity (log )/the integrated luminosity (log ) are shown in Fig. 7. The relations between the spectral curvature () and peak frequency (log ) and those between the spectral curvature () and the integrated luminosity(log ) are shown in Fig. 8. The relations between peak frequency and effective spectral indexes (, ) are shown in Fig. 9. The corresponding linear regression analysis results are listed in Table 4.

### 3.3 Effective spectral index correlation

From the calculated effective spectral indexes, the scattering diagram between and is plotted in Fig. 10, and the linear regression analysis results are listed in Table 4.

## 4 Discussions

As a special subclass of AGNs, blazars show many extreme observational properties, which are associated with a beaming effect. Blazars can be divided into BL Lacertae objects (BL Lacs) and flat spectrum radio quasars (FSRQs) by their emission line features. They are the major population of detected sources in the Fermi missions ( Abdo et al 2010b,c; Ackermann, et al. 2011a,b; Nolan et al. 2012; Acero et al. 2015; Ackermann, et al. 2015). The Fermi detected blazars provide us with a good opportunity to study the emission mechanism and beaming effects in -rays. The spectral energy distributions (SEDs) are available for some blazars and studied in the literatures (see Sambruana et al. 1996; Zhang, et al. 2002; Nieppola et al. 2006, 2008; Abdo et al. 2010a). At the present work, the multiwavelength data is compiled for a sample of 1425 Fermi blazars from the 3FGL (Acero et al. 2015) and their SEDs are calculated. SEDs for 1392 blazars have successfully been achieved, and their monochromatic luminosities at radio, optical, X-ray and -ray, and effective spectral indexes are also calculated.

BL Lacertae objects can be divided into radio selected BL Lac objects (RBLs) and X-ray selected BL Lacs objects (XBLs) from surveys, or low frequency peaked BL Lacertae objects (LBLs, log Hz) and high frequency peaked BL Lacertae objects (HBLs, log Hz) from SEDs. Generally RBLs correspond to LBLs while XBLs to HBLs ( Padovani & Giommi, 1995, 1996; Urry & Padovani, 1995). Nieppola, et al. (2006) calculated SEDs for a sample of BL Lacertae objects and set the boundaries for different subclasses: Hz for LBLs, 14.5 Hz Hz for IBLs, and Hz for HBLs. The classification was extended to all non-thermal dominated AGNs as low synchrotron peaked blazars-LSP ( Hz), intermediate synchrotron peaked blazars-ISP ( Hz), and high synchrotron peaked blazars-HSP ( Hz) by Abdo et al. (2010a).

Ghisellini (1999) proposed that there is a subclass of BL Lacs with their synchrotron peak frequencies being higher than that of conventional HBLs, Hz. These objects can be called ultra-high-energy synchrotron peak BL Lacs (UHBLs) (Giommi et al. 2001). In the work presented by Nieppola et al. (2006), there are 22 objects with , of which 9 objects have . They also found the appearance of several low-radio-luminosity LBLs, which could even reach lower radio luminosities than any of the HBLs.

### 4.1 Peak Frequency

At the present work, SEDs have successfully been calculated for 1392 Fermi blazars from their multiwavelength data. The peak frequencies obtained in the range of (Hz) = 11.60 20.12 in the observer frame are listed in Col. (12) of Table 1. Out of the 1392 Fermi blazars, 999 blazars have available redshifts.

For those 999 blazars, the rest peak frequencies obtained are in the range of (Hz) = 12.15 19.39 in the rest frame (frequency in the discussions below is assumed to be in the rest frame if it is not specifically stated). There are 3 sources, whose logarithm of peak frequencies is greater than 19.0 (). When FSRQs, BLs, and BCUs are considered separately, we have: for FSRQs, their peak frequencies (Hz) are in the range of 12.15 17.11 with an averaged value of ; for BL Lacs, are in the range of 12.87 19.39 with ; while for BCUs, are in the range of 12.25 17.36 with . The distribution of peak frequencies in the rest frame for the 999 blazars is shown in Fig. 2d. It can be seen that the distribution can be fitted using 3 components when the ¡±Bayesian classification¡± analysis method is employed. When the jointing points of two adjacent Gaussian curves are used to set the boundaries for different classes and the acronyms of Abdo et al. (2010a) are used, we obtain the following classifications:

LSPs: ,

ISPs: , and

HSPs: Hz.

For the whole sample of 1392 blazars, if the averaged values of redshift, and are used to BLs and BCUs without redshifts, the rest peak frequencies can be obtained in the range of (Hz) = 12.15 20.31. In this case, there are 8 sources, whose logarithm of peak frequencies is greater than 19.0 (). Their distributions and statistical analysis results using the ”Bayesian classification” method are shown in Fig. 2b and Fig. 3b respectively. Based on the above-mentioned classification criteria, the classification (HSP, ISP, and LSP) obtained is listed in Col. 3 of Table 1. From Fig. 2b, it can be seen that the peak frequencies corresponding to the jointing points are higher than those in Fig. 2d. This implies that the averaged values of redshifts used to get the rest frequencies are over estimated for most sources without redshifts.

From the rest frame peak frequencies of the whole sample and our criteria, we have: 25.14% of them belong to HSP, 40.09% belong to ISP, and 34.77% belong to LSP. When FSRQs, BLs, and BCU are considered separately, we have: 9 are HSP FSRQs, 180 are ISP FSRQs, and 272 are LSP FSRQs for 461 FSRQs; and 235 are HSP BLs, 272 are ISP BLs, and 114 are LSP BLs for the 620 BLs. See Table 2 for details.

As shown in Table 1, there are 22 sources, whose logarithm of the rest peak frequency is greater than 18 ( ) and the other 8 sources, , however statistical analysis does not come up with an ultra-high synchrotron peak component for the 999 sources with available redshift or the whole 1392 sources (See Fig. 2d, 2b) for the rest frequencies. We do not have such an ultra-high synchrotron peak component for the observer frequency either for the 999 sources with available redshift or for the whole sample of 1392 sources (See Fig. 2c, 2a). In Fig. 27 of Abdo et al. (2010a), it is stated that there is no population of ultra high energy peaked (UHBLs) blazars. Our statistical analysis confirms the results.

For the classifications, Abdo et al. (2010a) proposed, for LSPs, for ISPs, and for HSPs. From Fig. 2d at the present work, two jointing points at = 13.98 and = 15.30 can be obtained, and this results in that the boundaries are at = 14.0 and = 15.3. From Table 1, the uncertainties for from = 0.02 to = 1.39 with an averaged value of can be obtained. Considering the uncertainties in , we believe that our classifications are consistent with those by Abdo et al. (2010a).

For comparison, we analyze our synchrotron peak frequencies () with those () known for common sources in the literatures by Sambruna et al. (1996), Nieppola et al. (2006, 2008) and Abdo et al. (2010b). First we calculate the difference between our estimation and others’, , then investigate the relationship between the differences and our estimations.

There are 35 sources in common with Sambruna et al. (1996), and we have with a correlation coefficient 0.25 and a chance probability 14.5%. Our estimations tend to be larger than those by Sambruna et al. (1996) for some sources. It is also found that two sources with difference being larger than 2.0 are HSPs with (See Fig. 11a).

There are 129 sources in common with Nieppola et al. (2006), with -0.35 and . There is a clear anti-correlation between them, indicating that our estimations are smaller than those by Nieppola et al. (2006) for most common sources. Fig. 11b shows that our estimations of 4 source are greatly larger than those by Nieppola et al. (2006) and that the sources with large deviation are the sources with .

There are 82 sources in common with Nieppola et al. (2008), with 0.22 and . There is a marginal positive correlation between them, indicating that our estimations are larger than those by Nieppola et al. (2008) (See Fig. 11c).

There are 102 sources in common with Abdo et al. (2010a), with 0.40 and . If the right hand corner point is excluded, we have with 0.29 and . The analysis indicates that our estimations are larger than those by Abdo et al. (2010a) (See Fig. 11d). All the comparisons conducted and the best fitting results are listed in Table 5.

From the literatures (Sambruna et al. 1996; Pian et al. 1998; Nieppola et al. 2006, 2008; Abdo et al. 2010a; Giommi et al. 2000) and the present work, it is noted that the peak frequency (in observer frame) for a certain source is different from one calculation to another based on different multiwavelength observations. For some sources, the different spectral shapes may be attributed to the variability and different data sets. The comparisons suggest that there is no much difference between our estimations and those by Sambruan et al. (1996) and Nieppola et al. (2006), but ours are over estimated than those by Nieppola et al. (2008) and Abdo et al. (2010a).

Abdo et al. (2010a) and Ackermann et al. (2011a) obtained the spectral indexes, and for Fermi blazars. So comparisons can be conducted between their spectral indexes and ours.

For spectral index, , there are 724 sources in common with Abdo et al. (2010a), we have with a correlation coefficient -0.17 and a chance probability (Fig. 12a); while for , there are 498 common sources, which follow with and (Fig. 12b). The marginal positive correlation that we observe in Fig. 11d is likely due to the positive correlation we observe here in Fig. 12b.

For spectral indexes in Ackermann et al. (2011a), there are 514 for common sources, we have with a correlation coefficient -0.22 and a chance probability ; while for , there are 376 common sources, which follow with and . Results are shown in Fig. 13. As shown in Fig. 12 and Fig. 13, the differences in our estimate and those by Abdo et al. (2010a) and Ackermann et al. (2011a) scatter more or less around 0. Some of our ’s are slightly smaller than those by Abdo et al. (2010a) and Ackermann et al. (2011a) and some of our ’s are slightly larger than those by Abdo et al. (2010a) and Ackermann et al. (2011a), but no strong systematics appear. Because the flux densities used for the calculations of spectral indexes by Abdo et al. (2010a) and Ackermann et al. (2011a) are different from those at the present work, the calculation results of the spectral indexes are different as well.

In 2010, Abdo et al. (2010a) presented an empirical relation to estimate the synchrotron peak frequency, from effective spectral indexes . Following their work, we obtain an empirical relation to estimate the synchrotron peak frequency, from effective spectral indexes as

 logνEq.p={16+4.238XX<016+4.005YX>0, (1)

where , and .