Spectral Energy Distribution for Some Fermi Blazarstwo

CHINESE ASTRONOMY AND ASTROPHYSICS Chinese Astronomy and Astrophysics 43 (2019) 199–216

Spectral Energy Distribution for Some Fermi Blazars†  ZHANG Yu-tao

FAN Jun-hui

Center for Astrophysics, Guangzhou University, Guangzhou 510006

Abstract The observational data from radio to X-ray wavebands were collected from the SSDC (Italian Space Agency Science Data Center) for 68 Fermi blazars, and their spectral energy distributions (SEDs) were calculated by means of the least square fitting with a log-parabolic function. Based on the SED fitting parameters, the correlations of the synchrotron peak frequency, curvature and effective spectral index were discussed, and an empirical formula was also proposed to estimate the synchrotron peak frequencies by using the effective spectral indexes. The main results are as follows: (1) From the linear correlation between the synchrotron peak frequency (lg νp ) and the curvature (k), we find that the result supports the energy-dependent acceleration probability model for all BL Lac objects, while the result for the BL Lac objects of lg νp > 15.3 is consistent with the model of fluctuation of fractional acceleration gain. (2) For the sources of nearly same lg νp , a significant correlation between the effective spectral index αro of the radio-optical waveband and the curvature is detected, while the effective spectral index αox of the optical-x-ray waveband is not correlated with the curvature. According to the effective spectral index αro , a relation between the synchrotron peak frequency and the curvature can be defined. Key words statistical

galaxies: active—radiation mechanisms: non-thermal—methods:

1.

INTRODUCTION

Blazars belong to a special subgroup of radio-loud active galactic nuclei (AGNs). Owing to that their jet axis is very close to the line-of-sight direction, they exhibit some extreme observational properties, such as the violent light variation, high polarization, non-thermal †

Supported by National Natural Science Foundation (11733001, U1531245)



A translation of Acta Astron. Sin. Vol. 60, No. 1, pp. 7.1–7.15, 2019

Received 2018–11–12; revised version 2018–12–03 

[email protected]

0275-1062/01/$-see front matter © 2019 B. V. AllScience rights reserved. c Elsevier 0275-1062/01/$-see front matter  2019 Elsevier B. V. All rights reserved. doi:10.1016/j.chinastron.2019.04.002 PII:

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continuum, strong γ-ray radiation, and superluminal phenomenon, etc.[1−3] . According to the observational property of blazar emission lines, they are generally classified into the Flat Spectrum Radio Quasars (FSRQs) and BL Lac objects, the main difference is that FSRQs have strong emission lines, while BL Lac objects have no or only have very weak emission lines[4,5] . The Spectral Energy Distribution (SED) of a BL Lac exhibits a double-peaked structure, the radiation of the low-energy peak generally covers the radio to X-ray waveband ( in the case of low peak-frequency the radiation covers the radio to ultraviolet waveband), and originates from the synchrotron radiation of extremely relativistic electrons in the jet. In the lepton model, the radiation of the high-energy peak is commonly believed to be the Inverse Compton (IC) scattering, in which the Synchrotron Self-Compton (SSC) model suggested that the seed photons are originated from the synchrotron radiation of relativistic electrons in the jet[6−8] , while the External Compton (EC) model suggested that the seed photons are originated beyond the jet, for example the accretion disk, the reflection of broad-line region, etc.[9,10] . About the SEDs of blazars, there have been some studies[11−18] . In the study made by Abdo et al.[13] , the SEDs of 48 blazars were calculated by using the quasi-simultaneous data, and according to the logarithmic value of the calculated radiation peak frequency (lg (νp /Hz)) of each blazar (in the following lg (νp /Hz) is denoted as lg νp ), they divided these blazars into the LSP (Low Synchrotron Peaked: lg νp  14), ISP (Intermediate Synchrotron Peaked: 14 < lg νp  15), and HSP (High Synchrotron Peaked: lg νp > 15), and finally according to their sample obtained an empirical relation to estimate the synchrotron peak frequency by the spectral index (the effective spectral index αro of the radio-optical waveband, and the effective spectral index αox of the optical–X-ray waveband): ⎧ ⎨2.3X + 13.85 X1 < 0 and Y1 < 0.3 1 lg νp = , (1) ⎩6.58Y1 + 13.15 otherwise in which, X1 = 0.565−1.433αro +0.155αox , Y1 = 1−0.661αro −0.339αox . In the study made by Fan et al.[16] (briefly called as FAN in the following), in a similar way they also made a classification on blazars (LSP: lg νp  14; ISP: 14 < lg νp  15.3; HSP: lg νp > 15.3), and obtained as well an empirical relation: ⎧ ⎨4.238X + 16 X2  0 2 lg νp = , (2) ⎩4.005Y2 + 16 X2 > 0 in which, X2 = 1 − 1.262αro − 0.623αox , Y2 = 1 + 0.034αro − 0.978αox . The relation of the spectral index and synchrotron peak frequency makes us able to estimate the synchrotron peak frequency with less data information. However, these empirical relations have a rather large error in the estimated values of HSP blazars[16] , indicating that probably some unconsidered factors have influenced the estimation on the peak frequency by the spectral index.

ZHANG Yu-tao et al. / Chinese Astronomy and Astrophysics 43 (2019) 199–216

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In this paper, we will discuss further the relation between the spectral index and synchrotron peak frequency. In the work of FAN, they collected a sample consisted of 1425 Fermi-blazars, and obtained the synchrotron peak frequencies of 1392 sources in which by fittings. After checking the synchrotron peak frequency fittings of these sources, it is found that because of the rather few observational data, the synchrotron peak frequency errors of some sources are relatively large, this phenomenon is especially significant for the sources with a higher synchrotron peak frequency, and for the sources with a synchrotron peak frequency of lg νp > 15, the relation of the spectral index and synchrotron peak frequency becomes weak, which makes the error of the empirical formula become large for the HSP blazars. Hence, in this paper we have selected the sources of lg νp > 15, or the sources with a synchrotron peak frequency error of σ lg νp > 1 from the FAN’s sample, then we have collected the total observational data of theses sources from radio to X-ray wavebands from the SSDC1 , recalculated their SEDs, and analyzed the relations of the synchrotron peak frequency, curvature, and effective spectral index for this part of sample. 2.

SAMPLE AND SED CALCULATION

2.1 Sample This paper is purposed for studying the relation between the effective spectral index and synchrotron peak frequency for the HSP blazars, so we have to build a sample of high synchrotron peak frequency. In the FAN’s sample there are 398 blazaes of lg νp > 15, and there are 38 blazars of lg νp < 15 but σ lg νp > 1. Among these 436 blazars, there are 277 BL Lac objects, 14 FSRQs, and 145 blazars of uncertain type (BCU). From the SSDC we have collected the observational data of 436 blazars from radio to X-ray wavebands, and selected from which the blazars with the complete data from radio to X-ray. Finally, we have obtained 68 blazars in accordance with our conditions (57 BL Lac objects, 1 FSRQ, and 10 BCU), and calculated their SEDs. 2.2 Calculations of SED and Effective Spectral Index As same as FAN, the SED calculation of this paper adopts the least square fitting with a log-parabolic function. The fitting formula is given as follows: lg (νfν ) = −k(lg ν − lg νp )2 + lg (νp fνp ) ,

(3)

in which, k is curvature, fν is the flux density at the frequency ν, lg (νfν ) is the abbreviation of lg [(νfν )/(erg · cm−2 · s−1 )], it expresses the logarithmic value of the flux at the frequency ν, similarly lg (νp fνp ) expresses the logarithmic value of the flux at the peak frequency. In order to reduce the error in the calculation of SED, and obtain the reliable parameter of synchrotron radiation peak as possible, we have made some treatments in the calculation: 1 http://www.ssdc.asi.it/

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(1) To delete the observational data without an error bar or with only an upper limit; (2) In the optical waveband some blazars may have much more observational data than other wavebands, in order to balance the fitting weights of different wavebands, we have made the weighted average on the observational data at the same frequency according to the error, the calculation formulae are given as follows: lg (νfν ) = lg ν =

n n  lg (νi fi )  1 / , err2i err2i i=1 i=1

(4)

n n  lg νi  1 / ; err2i i=1 err2i i=1

(5)

(3) In the infrared-ultraviolet waveband, some blazars have a thermal hump produced by the thermal radiations from the dust ring, accretion disk, and host galaxy[13,15,19,20] , its existence has a significant influence on the fitting of synchrotron radiation peak. We have selected artificially the sample blazars with thermal humps, then we calculate their SEDs after their thermal hump data (13.5 < lg ν < 15) are removed. Finally we have obtained a sample of 68 Fermi blazars. The effective spectral index can be calculated by the following definition formula: α12 = −

lg (f1 /f2 ) , lg (ν1 /ν2 )

(6)

in which, f is flux density, ν is frequency. After the K correction2 is made for the observational data at 5 GHz, V-band (5500˚ A), and 1 keV by though fν = fν (1 + z)α−1 , the radio-optical band and optical–X-ray band spectral indexes are calculated, in which the sample blazars without any redshift data adopt the average redshift 0.32 to make calculations[13,16,21,22] . In this paper the spectral indexes used for the K correction are as follows: for the BL Lac objects, we adopt the radio-band spectral index αr = 0, opticalband spectral index αo = 0.5, and X-ray-band spectral index αx = 1.30; for FSRQs, we adopt αr = 0, αo = 1, αx = 0.78; and for BCU, we adopt αr = 0, αo = 1, αx = 1.05[13,16,20] . The results of SED calculation and spectral indexes are listed in Table 1, in which there are 57 blazars of lg νp > 15, and 51 blazars of lg νp > 15.3. In Table 1, the first column gives the name in the 3FGL catalog, the second column is the optical classification, the third column is the redshift (z), the 4th to 7th columns are respectively the fitting parameters of synchrotron radiation peak: the energy spectrum curvature k, the logarithmic peak frequency lg νp and its error σ lg νp , the logarithmic flux at the peak frequency lg(νp fνp ) and its error σ lg(νp fνp ), and the χ2 -value of the χ2 test, the 8th and 9th columns are respectively the calculated effective spectral indexes αro and αox . Figs.1-2 are the fitting results. 2 Because

of the spectrum deformation caused by the redshift, during the luminosity calculation the

process to remove the redshift effect is called the K correction.

Table 1 Sample of 68 Blazars class[15] BL Lac bcuII BL Lac BL Lac BL Lac BL Lac bcuII FSRQ BL Lac BL Lac BL Lac BL Lac BL Lac bcuII BL Lac BL Lac BL Lac BL Lac BL Lac BL Lac BL Lac BL Lac bcuII BL Lac bcuII BL Lac BL Lac bcuII BL Lac BL Lac BL Lac BL Lac BL Lac BL Lac

z [16] – – 0.086 – 0.313 – – 0.55 0.318 – 0.443 0.287 – – 0.314 – 0.273 – 0.32 0.203 – 0.125 – 0.315 – 0.053924 1.545 – 0.212 0.361 – 0.186 0.031 0.1392

k ± σk 0.132 ± 0.0077 0.115 ± 0.006 0.0576 ± 0.007 0.0822 ± 0.0019 0.0573 ± 0.0047 0.0926 ± 0.0022 0.0636 ± 0.0082 0.0677 ± 0.0062 0.0783 ± 0.011 0.0639 ± 0.0088 0.0897 ± 0.0044 0.0745 ± 0.0047 0.0822 ± 0.0062 0.0932 ± 0.0058 0.0659 ± 0.0024 0.14 ± 0.0084 0.0653 ± 0.0028 0.0879 ± 0.0091 0.0616 ± 0.0063 0.0752 ± 0.0047 0.137 ± 0.0088 0.0439 ± 0.0043 0.109 ± 0.021 0.0871 ± 0.013 0.103 ± 0.0032 0.106 ± 0.0039 0.123 ± 0.0073 0.0675 ± 0.0066 0.0695 ± 0.0026 0.0743 ± 0.0029 0.166 ± 0.0071 0.049 ± 0.0048 0.0745 ± 0.0047 0.0771 ± 0.0078

lg νp ± σ lg νp 14.69 ± 0.0851 15.343 ± 0.139 17.629 ± 0.413 16.148 ± 0.0692 16.685 ± 0.276 16.28 ± 0.061 16.826 ± 0.382 15.958 ± 0.225 16.526 ± 0.408 17.083 ± 0.473 16.068 ± 0.104 16.651 ± 0.185 15.922 ± 0.179 16.095 ± 0.14 17.705 ± 0.132 14.539 ± 0.0688 17.26 ± 0.149 14.723 ± 0.149 14.523 ± 0.143 16.624 ± 0.194 14.457 ± 0.0827 18.671 ± 0.457 14.784 ± 0.287 16.073 ± 0.337 15.162 ± 0.0698 14.618 ± 0.0593 15.486 ± 0.127 16.378 ± 0.343 16.261 ± 0.144 16.958 ± 0.121 14.282 ± 0.0587 18.094 ± 0.399 16.739 ± 0.172 16.307 ± 0.258

lg(νp fνp ) ± σ lg(νp fνp ) −11.319 ± 0.101 −11.677 ± 0.0408 −10.598 ± 0.0521 −11.281 ± 0.0153 −11.828 ± 0.0422 −10.739 ± 0.0193 −11.029 ± 0.0745 −11.939 ± 0.0606 −12.043 ± 0.091 −11.174 ± 0.0698 −11.468 ± 0.0312 −11.034 ± 0.0414 −11.887 ± 0.0634 −11.31 ± 0.0452 −10.715 ± 0.0263 −11.664 ± 0.117 −11.042 ± 0.0239 −11.609 ± 0.137 −12.383 ± 0.0968 −10.911 ± 0.0397 −11.576 ± 0.123 −10.915 ± 0.0622 −11.817 ± 0.288 −11.272 ± 0.141 −11.409 ± 0.0284 −11.815 ± 0.0425 −12.09 ± 0.0552 −11.972 ± 0.0581 −10.766 ± 0.0291 −10.996 ± 0.023 −11.458 ± 0.0746 −10.771 ± 0.0476 −9.8062 ± 0.051 −11.149 ± 0.0667

χ2 0.524 0.00442 0.389 0.287 0.0196 0.111 0.101 0.0224 0.0996 0.059 0.34 0.196 0.0456 0.11 0.154 0.0118 0.0198 0.0951 0.0942 0.105 0.0071 0.14 0.085 0.128 0.0701 0.0946 0.0491 0.00275 1.26 0.0975 0.0708 0.246 10.6 0.173

αro 0.258 0.223 0.324 0.336 0.45 0.262 0.381 0.505 0.305 0.28 0.269 0.332 0.395 0.244 0.286 0.381 0.276 0.509 0.486 0.313 0.344 0.325 0.342 0.192 0.403 0.405 0.237 0.315 0.388 0.319 0.369 0.343 0.222 0.308

αox 1.43 1.28 1.06 1.09 0.934 1.05 0.972 1.03 0.856 1.01 0.992 0.832 1.03 0.95 0.84 1.56 0.91 1.35 1.61 0.95 1.46 1.01 1.16 0.978 1.26 1.34 0.887 1.16 1.13 0.67 1.75 0.782 1.23 0.986

ZHANG Yu-tao et al. / Chinese Astronomy and Astrophysics 43 (2019) 199–216

3FGL name J0018.9−8152 J0030.2−1646 J0035.9+5949 J0045.3+2126 J0109.9−4020 J0136.5+3905 J0137.8+5813 J0202.3+0851 J0208.6+3522 J0209.4−5229 J0316.1−2611 J0416.8+0104 J0505.9+6114 J0506.9−5435 J0508.0+6736 J0515.8+1526 J0543.9−5531 J0612.8+4122 J0617.6−1717 J0650.7+2503 J0707.0+7741 J0710.3+5908 J0720.0−4010 J0744.3+7434 J0746.6−4756 J0816.7+5739 J0912.4+2800 J0917.3−0344 J1015.0+4925 J1031.2+5053 J1047.6+7240 J1103.5−2329 J1104.4+3812 J1117.0+2014

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class[15] BL Lac BL Lac BL Lac bcuII BL Lac BL Lac BL Lac BL Lac BL Lac bcuII BL Lac BL Lac BL Lac BL Lac BL Lac BL Lac BL Lac BL Lac BL Lac bcuII BL Lac BL Lac BL Lac BL Lac BL Lac bcuII BL Lac BL Lac BL Lac BL Lac BL Lac BL Lac BL Lac BL Lac

z [16] 0.124 0.136 0.045 – 0.183648 1.0654 – – 0.237 – 0.129 0.343663 0.16309 0.0654 0.702 – 1.545 0.018 – – – – 0.047 0.254 0.23 – 0.449 0.429 – 0.229 – 0.274 0.044 0.331

Table 1 Continued k ± σk lg νp ± σ lg νp 0.077 ± 0.0061 16.542 ± 0.231 0.0477 ± 0.0068 18.214 ± 0.57 0.0651 ± 0.004 16.289 ± 0.17 0.0703 ± 0.0029 16.185 ± 0.106 0.0709 ± 0.0042 17.086 ± 0.191 0.0746 ± 0.0045 16.703 ± 0.221 0.0798 ± 0.01 15.109 ± 0.271 0.0946 ± 0.017 15.123 ± 0.326 0.0585 ± 0.0031 17.631 ± 0.179 0.0816 ± 0.0051 16.41 ± 0.182 0.0533 ± 0.0045 18.163 ± 0.337 0.0861 ± 0.004 15.986 ± 0.119 0.0575 ± 0.0038 17.528 ± 0.256 0.09 ± 0.0043 15.777 ± 0.122 0.08 ± 0.0044 16.776 ± 0.165 0.103 ± 0.0094 15.667 ± 0.266 0.119 ± 0.0042 15.163 ± 0.0701 0.0732 ± 0.0073 16.327 ± 0.291 0.0942 ± 0.0055 15.93 ± 0.135 0.0589 ± 0.011 17.111 ± 0.659 0.0715 ± 0.0052 16.383 ± 0.196 0.0957 ± 0.0057 15.959 ± 0.144 0.0646 ± 0.0061 17.013 ± 0.283 0.0763 ± 0.0088 14.798 ± 0.187 0.0837 ± 0.0097 15.3 ± 0.28 0.0876 ± 0.0045 16.639 ± 0.141 0.0712 ± 0.0029 16.555 ± 0.123 0.0919 ± 0.008 14.706 ± 0.122 0.0747 ± 0.0056 16.229 ± 0.191 0.121 ± 0.0049 14.474 ± 0.0572 0.0686 ± 0.0062 16.963 ± 0.325 0.109 ± 0.0077 15.044 ± 0.132 0.0546 ± 0.0036 17.063 ± 0.194 0.0813 ± 0.0095 15.834 ± 0.259

lg νp fνp ± σ lg νp fνp −11.273 ± 0.0491 −11.186 ± 0.0713 −11.021 ± 0.0378 −11.71 ± 0.0283 −10.841 ± 0.0359 −10.751 ± 0.0503 −11.953 ± 0.132 −12.249 ± 0.188 −11.16 ± 0.0231 −11.792 ± 0.0453 −10.874 ± 0.0449 −11.234 ± 0.0354 −11.277 ± 0.0446 −10.981 ± 0.0429 −10.995 ± 0.0376 −11.776 ± 0.0639 −11.565 ± 0.0303 −11.06 ± 0.065 −11.376 ± 0.042 −11.977 ± 0.0742 −11.889 ± 0.0416 −11.378 ± 0.0534 −10.452 ± 0.0564 −12.289 ± 0.124 −11.877 ± 0.111 −11.584 ± 0.0385 −11.466 ± 0.024 −12.145 ± 0.112 −11.447 ± 0.0527 −11.503 ± 0.068 −11.468 ± 0.0566 −11.512 ± 0.0759 −10.951 ± 0.0305 −12.004 ± 0.108

χ2 0.102 0.131 0.313 0.0116 0.471 0.476 0.0613 0.0581 0.209 0.0033 0.637 0.0656 0.0474 0.0341 0.238 0.0133 0.333 0.173 0.0577 0.0529 0.0414 0.0635 1.65 0.0282 0.00504 0.414 0.0133 0.0365 0.148 0.0188 0.0312 0.0895 0.785 0.00901

αro 0.324 0.295 0.385 0.435 0.301 0.258 0.386 0.307 0.314 0.241 0.254 0.332 0.29 0.287 0.322 0.203 0.391 0.333 0.268 0.381 0.353 0.302 0.281 0.606 0.429 0.202 0.377 0.485 0.353 0.433 0.327 0.31 0.339 0.216

αox 0.892 1.04 1.01 0.987 0.825 0.98 1.3 1.16 0.857 1.05 0.82 0.966 0.991 1.21 0.734 1.2 1.32 1.13 1.25 1.1 0.901 1 1.08 1.22 1.17 0.908 0.861 1.4 0.972 1.49 0.837 1.16 1.12 1.51

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3FGL name J1120.8+4212 J1136.6+6736 J1136.6+7009 J1141.6−1406 J1221.3+3010 J1243.1+3627 J1254.5+2210 J1330.6+7002 J1417.8+2540 J1418.9+7731 J1428.5+4240 J1439.2+3931 J1442.8+1200 J1444.0−3907 J1517.6+6524 J1546.0+0818 J1630.7+5222 J1725.0+1152 J1811.2+0340 J1820.3+3625 J1841.7+3218 J1926.8+6154 J2000.0+6509 J2001.8+7041 J2014.3−0047 J2036.6−3325 J2131.5−0915 J2143.1−3928 J2247.8+4413 J2251.9+4031 J2304.6+3704 J2340.7+8016 J2347.0+5142 J2356.0+4037

14

16

18

14

16

10 12 14 16 18

−9

12

14

16

18

−16

−15

3FGL J0508.0+6736 10 12 14 16 18

3FGL J0650.7+2503 10

12

14

16

18

−11

10 12 14 16 18

−12 −13 14

16

18

3FGL J1015.0+4925 8

−15

−14 12

−12

10

3FGL J1117.0+2014 10

12

14

16

18

3FGL J0746.6−4756 10

10 12 14 16 18

−12 3FGL J1104.4+3812 8 10 12 14 16 18 20 lg (ν /Hz)

−15

18

3FGL J0744.3+7434

18

16

−15 −14 −13 −12 −11

−15

−13 16

−14 14

18

−13 14

−16 12

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3FGL J0209.4−5229 10

−15 −14 −13 −12 −11

12

3FGL J0617.6−1717 8

3FGL J0917.3−0344 10

14

−11

18

12

−15 −14 −13 −12 −11

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16

3FGL J0109.9−4020 10

12

14

16

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3FGL J1031.2+5053 10 12 14 16 18

−15 −14 −13 −12 −11

−12 −13 −14 3FGL J0720.0−4010

14

3FGL J0506.9−5435 10

−15 −14 −13 −12 −11

−11

10 12 14 16 18

10

18

3FGL J1103.5−2329 8

−11

−12 −13 −14 3FGL J0612.8+4122 8

−12 16

18

−14 18

−16 −15 −14 −13 −12

16

12

−15 −14 −13 −12 −11

14

−13 14

16

−12

−12 −14 −16 −15 −14 −13 −12 −11 −15 −14 −13 −12 −11

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−14 12

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−15 −14 −13 −12 −11 −10

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3FGL J0912.4+2800 10

14

−14

−13 −15

−14

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18

10 12 14 16 18

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3FGL J1047.6+7240 10

16

10

10

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3FGL J0208.6+3522

18

−10

−16

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14

−12

12

12

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3FGL J0505.9+6114

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3FGL J0710.3+5908

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3FGL J0816.7+5739 10

10

14

−12

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16

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−13 −15

−14

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3FGL J0543.9−5531

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3FGL J0707.0+7741 10

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3FGL J0202.3+0851 10

−15

−13

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−12

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10

3FGL J0045.3+2126

10 12 14 16 18

3FGL J0416.8+0104

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3FGL J0515.8+1526 10

3FGL J0035.9+5949

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−16

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−12

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−14 −15

lg [νfν /(erg·cm−2·s−1)]

12

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10 12 14 16 18 −15 −14 −13 −12 −11

3FGL J0316.1−2611 10

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3FGL J0137.8+5813

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−15 −14 −13 −12 −11

16

−15 −14 −13 −12 −11

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−15 −14 −13 −12

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−16 −15 −14 −13 −12

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3FGL J0030.2−1646 10

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3FGL J0018.9−8152 10 12 14 16 18

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ZHANG Yu-tao et al. / Chinese Astronomy and Astrophysics 43 (2019) 199–216

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Fig. 1 Spectral energy distributions of part sample. The black points are the fitted observational data. The gray points are the thermal hump data or the observational data without any error information

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−13 −14 3FGL J2001.8+7041 10

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3FGL J2014.3−0047 10

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12 14 16 lg (ν /Hz)

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3FGL J1841.7+3218 10

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−12 3FGL J1820.3+3625

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3FGL J1546.0+0818 10

−10

18

−15 12

−14 3FGL J2000.0+6509 10 12 14 16 18

16

12

14

16

18

−12

18

−16 −15 −14 −13 −12 −11

16

10

−13

−12 −14 14

−12

12

−16

3FGL J1926.8+6154 10

3FGL J1811.2+0340

18

14

−13

16

12

−14

14

−12

12

10

−13

−15 −14 −13 −12 3FGL J1725.0+1152 10

−15

18

−15 −14 −13 −12 −11

16

−16

−14 14

−10

−15 −14 −13 −12 −11

12

3FGL J1517.6+6524

18

−16

16

−16 −15 −14 −13 −12

14

−10

12

−16 −15 −14 −13 −12

−12

−9 −11 −15

3FGL J1444.0−3907 10

−12

−12 −13 −14 −15

3FGL J1630.7+5222 10

−15

lg [νfν /(erg·cm−2·s−1)]

3FGL J1442.8+1200 10 12 14 16 18

18

Fig. 2 Same as Fig.1, the SEDs of the remaining sample

18

−15

18

−15

16

−15 −14 −13 −12 −11

14

−11

12

−16

3FGL J1439.2+3931 10

−15

−16

−13

−14

−14

−12

−12

−10

10 12 14 16 18

12

−16

3FGL J1254.5+2210 8

3FGL J1141.6−1406 10

−15 −14 −13 −12 −11

−15

−14

−13

−16 −15 −14 −13 −12

−12

10 12 14 16 18

−16

3FGL J1136.6+7009 8

−15

−14

−14 −16

3FGL J1136.6+6736 10 12 14 16 18

−15 −14 −13 −12 −11

−12

−10

−15 −14 −13 −12 −11

−10

ZHANG Yu-tao et al. / Chinese Astronomy and Astrophysics 43 (2019) 199–216

3FGL J2304.6+3704 10

12

14

16

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ZHANG Yu-tao et al. / Chinese Astronomy and Astrophysics 43 (2019) 199–216

3.

207

RESULTS AND DISCUSSIONS

20

20

19

19

18

18

lg (vp3LAC /Hz)

lg (vpFAN /Hz)

In 2015, Ackermann et al. published the 3LAC catalog (The third catalog of active galactic nuclei detected by the Fermi-LAT) containing 1559 blazars, and made fitting of their synchrotron radiation SEDs by using cubic polynomials[15] . The cubic polynomial fitting is more reliable when the two synchrotron peaks are asymmetrical. We have compared the synchrotron peak frequencies of this paper with those of FAN and the fitting results in the 3Lac catalog. In our sample, there are 67 sources with the fitting results in FAN, and 68 sources with the fitting results in 3Lac. The compared result is shown in Fig.3, the left panel is the comparison with FAN, and the right panel is the comparison with 3LAC. The slopes of linear fitting are respectively 0.610 and 0.681, the correlation coefficients are respectively 0.621 and 0.819, and the p-values of the Pearson test are respectively 2.003 × 10−8 and 1.41 × 10−17 , the black solid line indicates the case when the two fitting results of synchrotron peak frequency are identical. Compared to the fitted synchrotron peak frequencies in FAN, the peak frequencies obtained by fitting SEDs in this paper are more close to the results given by 3LAC.

17 16 15 14 13 13 14

16 15 14

15 lg

Fig. 3

17

16

17

(vpTW

/Hz)

18

19

20

13 13 14

15

16

17

18

19

20

lg (vpTW /Hz)

The comparison of the fitted peak frequency (lg (νpTW /Hz)) and that obtained from FAN

(lg (νpFAN /Hz)) and 3LAC (lg (νp3LAC /Hz))[15,16] . The dashed line represents the linear regression result, and the solid line indicates the identity

Compared with the SEDs in FAN, the SEDs fitted by this paper have used more data, and therefore the errors of the synchrotron peak frequencies obtained by this paper are significantly reduced (Fig.4). The mean synchrotron peak frequency error (σ lg νp ) of the 68 SEDs obtained by this paper is 0.212, in which there are 59 sources with the synchrotron peak frequency errors given by FAN, their mean value is 0.705. The accuracy of the synchrotron peak frequency fitted by this paper has a significant improvement.

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ZHANG Yu-tao et al. / Chinese Astronomy and Astrophysics 43 (2019) 199–216

32 28

this paper

24

FAN

number

20 16 12 8 4 0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 σ lg (vp /Hz) Fig. 4 The distribution of synchrotron peak frequency errors

3.1 Relation of the Spectral Index and Synchrotron Peak Frequency The synchrotron peak frequencies (lg νp ) of different sources tend to distribute in different regions in the αro -αox space, and they can be estimated and classified by trough αro and αox [13,23,24] . The αro -αox distribution of the sample in this paper is shown as Fig.5. With different samples Ref.[13] and Ref.[16] analyzed respectively the relations of the effective spectral indexes αro and αox with the synchrotron peak frequency lg νp , and successively obtained two empirical formulae (Eq.(1) and Eq.(2)). Similarly, based on the effective spectral indexes and synchrotron peak frequencies of the 57 blazars of lg νp > 15, we have obtained an empirical formula by fitting as follows: lg νp = −2.813αox + 19.358 .

(7)

We have compared the synchrotron peak frequencies estimated by Eq.(7) and those calculated by this paper (see Fig.6). Similar with Ref.[13] and Ref.[16], the empirical formula obtained by this paper has a rather large fitting error in the high synchrotron peak frequency region (i.e. the part of X > 0). Besides, since the relation between αro and lg νp weakens with the increase of lg νp (see Fig.7), it makes the term relevant to αro disappear from the empirical formula. In Fig.5, there are 41 sample sources concentrated in the regions of αro = 0.3 ± 0.1 and αox = 1 ± 0.2, while the synchrotron peak frequencies (lg νp ) of these sample sources are distributed in a very wide region (as shown by Fig.7). In this region, the correlation of the spectral index and synchrotron peak frequency is very weak even vanished, leading to that the synchrotron peak frequency cannot be estimated only by the spectral index. Hence, there exist some unconsidered factors which have a certain influence on the relation between the spectral index and synchrotron peak frequency.

ZHANG Yu-tao et al. / Chinese Astronomy and Astrophysics 43 (2019) 199–216

2.0 lg (vp /Hz)˚15 lg (vp /Hz)İ15

αox

1.5

1.0

0.5

0.2

0.6

0.4 αro

Fig. 5 The scatter diagram of αro versus αox . The black points are the sources with lg νp > 15

3

lg (vpTW /Hz)-lg (vpEs /Hz)

2 1 0 −1 −2 −3

15

16

17

18

19

lg (vpTW /Hz) Fig. 6 The fitting residuals of Eq.(7). lg (νpEs /Hz) is estimated by Eq.(7)

209

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ZHANG Yu-tao et al. / Chinese Astronomy and Astrophysics 43 (2019) 199–216

20

lg (vp /Hz)

18

16

14 0.2

0.4 αro

0.6

Fig. 7 The scatter diagram of αro versus lg (νp /Hz)

3.2

Relation of the Synchrotron Peak Frequency and Curvature

From a statistical model[25] and the models of stochastic acceleration[26,27] , we can find that the electron energy spectrum in the jet is a logarithmic parabola, the electron energy spectrum of logarithmic parabola emits approximately a SED of logarithmic parabola, the two kinds of models can explain also the correlation between the synchrotron peak frequency and curvature[14] . Some scholars have studied the relation between the synchrotron peak frequency and curvature[14,27−32] . Chen[14] suggested that the value of the slope in the linear relation between 1/k and synchrotron peak frequency (1/k = A + B lg νp ) differs with the acceleration model, and obtained a theoretical estimate of the slope B under the model of stochastic acceleration, the model of energy-dependent acceleration probability, and the model of fluctuation of fractional acceleration gain to be respective 2, 2.5, and 10/3. For the 68 SEDs of the sample in this paper, the correlation coefficient between 1/k and lg νp is 0.839 (p = 4.13 × 10−19 ), and the result of linear regression is 1/k = (2.57 ± 0.20) lg νp − (29.30 ± 3.12) (see the dashed line in Fig.8); in which the correlation coefficient between 1/k and lg νp for the 57 BL Lac objects is 0.844 (p = 1.76 × 10−16 ), the result of linear regression is: 1/k = (2.57 ± 0.21) lg νp − 29.26 ± 3.31 (see the dotdashed line in Fig.8). The linear regression result of BL Lac objects obtained by Xue et al.[32] and the result of linear regression obtained by Chen[14] are respectively 1/k = (1.87±0.19) lg νp −19.21±2.76 and 1/k = (2.04±0.03) lg νp −22.08±0.43, they all approach to the theoretical slope value (B=2) predicted by the stochastic acceleration model, while the slope B of the BL Lac objects in the sample of Luo et al.[33] is 2.48, the slope of BL Lac objects obtained by this paper is 2.57±0.21, consistent with the result of Luo et al.[33] , also consistent with the theoretical estimate 5/2 of the model of energy-dependent acceleration probability, supporting the model of energy-dependent acceleration probability. The reason

ZHANG Yu-tao et al. / Chinese Astronomy and Astrophysics 43 (2019) 199–216

211

causing the deviation of the slope value from the stochastic acceleration model may be that the sample selection in this paper tends to the sources of high synchrotron peak frequency, and in the sample selected by Chen[14] , the part of high synchrotron peak frequency also has a trend of enlarged slope. According to the classification of FAN, when we consider only the 42 BL Lac objects of lg νp > 15.3, the correlation coefficient between 1/k and lg νp is 0.909 (p = 9.50 × 10−17 ), the result of linear regression is: 1/k = (3.38 ± 0.34) lg νp − 42.84 ± 5.65 (see the solid line in Fig.8), the slope is very approximate to the theoretical prediction 10/3 of the model of fluctuation of fractional acceleration gain, indicating that for the BL Lac objects of high peak frequency, their jet acceleration mechanism is the model of fluctuation of fractional acceleration gain. We think that the acceleration mechanism changes with the increase of synchrotron peak frequency, HBL and LBL have different jet acceleration mechanisms, in which the jet acceleration mechanism of HBLs is the model of fluctuation of fractional acceleration gain. In the sample of Chen[14] , there are 7 BL Lac objects of lg νp > 15.3, their correlation coefficient between 1/k and lg νp is 0.8519 (p = 0.0149), the result of linear regression is: 1/k = (3.58 ± 1.07) lg νp − 46.64 ± 17.07, supporting the conclusion of this paper. In the sample of Xue et al.[32] , there are 18 BL Lac objects of lg νp > 15.3, their correlation coefficient between 1/k and lg νp is 0.0705 (p = 0.7811), compared with the samples of this paper and Chen[14] , the 1/k-values of BL Lac objects in the sample of Xue et al.[32] are apparently small (see Fig.8, the mean 1/k-values of the BL Lac objects with lg νp in the 15.3–18 range in the samples of Xue et al.[32] , Chen[14] , and this paper are respectively 9.857, 12.921, 13.492), and the synchrotron peak frequencies (lg νp ) of this sample are mainly concentrated in the 15.3–17 range.

24 20

BL Lacs (Chen[14]) BL Lacs (Xue et al.[32]) FSRQs & BCUs BL Lacs (lg (vpTW /Hz)İ15.3) BL Lacs (lg (vpTW /Hz)˚15.3)

1/k

16 12 8 4 12

14

16

18

20

lg (vpTW /Hz) Fig. 8 Synchrotron peak frequency versus curvature. The dashed line is the linear regression result of the whole sample, the triangle symbols are FSRQs and BCUs. The dot-dashed line is the linear regression result of BL Lac, and the solid line is the linear regression result of BL Lacs with lg (νp /Hz) > 15.3

212

3.3

ZHANG Yu-tao et al. / Chinese Astronomy and Astrophysics 43 (2019) 199–216

Relations among the Spectral Index, Synchrotron Peak Frequency, and Curvature

In the above we have tried to find the relation between the effective spectral index and synchrotron peak frequency, while the synchrotron peak frequency has a strong relation with the curvature. If there is no influence of spectral index in the relation between the synchrotron peak frequency and curvature, then in the sample with similar synchrotron peak frequencies, the spectral index and curvature will be uncorrelated. We have studied the relations of spectral index with k in several intervals of synchrotron peak frequency, and found the significant negative correlation between αro and k (see Figs.9(A,C) and Table 2), while the relation between αox and k is very weak (see Figs.9(B,D) and Table 2). As the sample size is rather small, it needs a bigger interval of synchrotron peak frequency, so that the reliability of the result is reduced to a certain extent. In order verify the above results, we have obtained a sample containing 994 Fermi blazars from FAN, and obtained the same result by using several smaller intervals of synchrotron peak frequency. The particular results of correlation analysis are given in Table 2. Fig.9 shows the relations between spectral index and k in several intervals of synchrotron peak frequency. From the results of correlation analysis we can find that when the intervals of synchrotron peak frequency are identical, the correlation between αro and k is stronger, while the correlation between αox and k is very weak. At the same time, the correlation between αro and k will be weakened with the increase of synchrotron peak frequency. In the sample of FAN, as the synchrotron peak frequency increases, αox and k will tend to change gradually from positive correlation to negative correlation. In the sources with similar synchrotron peak frequencies, the sources with different curvatures will have differen spectral indexes, this is natural in geometry, the above correlation analysis has verified this point as well. This indicates that the relation between the synchrotron peak frequency and curvature is not an one-by-one corresponding relation, there exists the influence of other factors (for example the spectral index). Similarly, in the relation between the spectral index and synchrotron peak frequency may exist the influence of the curvature. The spectral index cannot define very well the value of synchrotron peak frequency, but it can define a group of corresponding relations between the curvature and synchrotron peak frequency. This paper has employed a logarithmic parabola to fit the shape of synchrotron radiation peak, the frequencies ν1 and ν2 and the corresponding fluxes ought to be: ⎧ ⎨lg (ν f ) = −k(lg ν − lg ν )2 + lg ν f 1 ν1 1 p p νp ⎩lg (ν2 fν2 ) = −k(lg ν2 − lg νp )2 + lg νp fν p

.

(8)

lg (f1 /f2 ) In combination with the calculation formula of the effective spectral index: α12 = − lg (ν1 /ν2 ) ,

ZHANG Yu-tao et al. / Chinese Astronomy and Astrophysics 43 (2019) 199–216

213

we can obtain the relation between the curvature k and spectral index as follows: k=

(α12 − 1) . lg ν2 + lg ν1 − 2 lg νp

(9)

According to Eq.(9), taking αro = M k lg νp + N k + P as the regression equation, and using the sample of this paper, we have obtained the following empirical relation: αro = −1.459k lg νp + 18.430k + 0.732 . 15−15.5 15.5−16 16−16.5 16.5−17 17−18

15

0.14 15.5 16

0.12

16.5

0.14 0.12

17

16

(10)

15.5 15

18

0.10

17

k

k

16.5 0.10

0.08 18

0.08

0.06

0.06 (A)

0.04 0.1

15−15.5 15.5−16 16−16.5 16.5−17 17−18

(B) 0.2

0.3

αro

0.4

0.5

0.04

0.6

0.4

0.8

0.6

1.2 αox

1.4

1.6

1.8

0.4 13

0.3

13−13.1 14−14.1 15−15.1 16−16.1

13−13.1 14−14.1 15−15.1 16−16.1

0.3

k

0.2

15 14

16

14 k

1.0

0.2

13

15 16 0.1

0.1 (C)

(D) 0.0

0.0

0.4

αro

0.8

0.7

1.4

αox

2.1

Fig. 9 αro versus k and αox versus k in several synchrotron peak frequency intervals. (A)–(B) are obtained from our sample, and (C)–(D) are obtained from the sample of FAN

From Eq.(9) we can find that for the same synchrotron peak frequency, the relation between the spectral index and curvature is related to the two frequencies used for calculating the spectral index. When the synchrotron peak frequency is far greater than (or far less than) the two frequencies ν1 and ν2 that used for calculating the spectral index (as shown by the prediction curve of lg νp =18 in Fig.9(A)), the value of the denominator in the right side of Eq.(9) is rather large, so that with the increase of lg νp the relation between αro and k becomes weak. As for the correlation between αox and k, since the magnitude of αox is

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ZHANG Yu-tao et al. / Chinese Astronomy and Astrophysics 43 (2019) 199–216

strongly related to the X-ray radiation[32] , while the inverse Compton scattering component in the X-ray radiation may influence the correlation between αox and k, this point is even more apparent for the sources with a relatively low synchrotron peak frequency. As shown by the theoretical spectrum in Fig.10, the component of inverse Compton scattering makes the X-ray radiation enhance significantly, leading to the reduction of αox . When lg νp approaches to 16 (as shown by Fig.9(B)), the denominator in Eq.(9) approaches to zero, it will also make the correlation between αox and k become very weak. In Fig.9, the dashed lines are used to denote the theoretical curves of the spectral index and curvature corresponding to the different synchrotron peak frequencies. We can find that the αro -k relation is very close to the theoretical prediction, but the αox -k relation has a very large difference from the theoretical prediction. As shown by Fig.9(D), for the sources of 13 < lg νp < 13.1, the correlation between αox and k remains to be quite strong, deviating greatly from the theoretical prediction, this deviation may be the influence of the inverse Compton component in the X-ray radiation. Table 2 Result of correlation analysis between curvature and spectral index α

Sample

lg νp interval

Number

Correlation coefficient

p-value

αro

TW

15–15.5

8

−0.604

0.1127

αro

TW

15.5–16

8

−0.779

0.0228

αro

TW

16–16.5

14

−0.791

0.00079

αro

TW

16.5–17

13

−0.805

0.00089

αro

TW

17–18

10

−0.556

0.095

αox

TW

15–15.5

8

−0.318

0.4424

αox

TW

15.5–16

8

0.102

0.8101

αox

TW

16–16.5

14

−0.279

0.3051

αox

TW

16.5–17

13

−0.147

0.6315

αox

TW

17–18

10

−0.624

0.0538

αro

FAN

13–13.1

28

−0.878

8.27 × 10−10

αro

FAN

14–14.1

38

−0.806

1.034 × 10−9

αro

FAN

15–15.1

20

−0.733

0.00024

αro

FAN

16–16.1

11

−0.529

0.094

αox

FAN

13–13.1

28

0.752

4.042 × 10−6

αox

FAN

14–14.1

38

0.272

0.098

αox

FAN

15–15.1

20

−0.0803

0.736

αox

FAN

16–16.1

11

−0.361

0.275

ZHANG Yu-tao et al. / Chinese Astronomy and Astrophysics 43 (2019) 199–216

215

−9 V band

1 keV

1g [vfv /(erg·s−1·cm−2)]

−10 −11 −12 IC −13 −14 −15

8

10

12

14 16 lg (v /Hz)

18

20

22

Fig. 10 The influence of X-ray inverse compton (IC) component on αox in the LSP blazars

4.

SUMMARY

In this paper, we have collected the data from the SSDC, obtained the SEDs of 68 Fermi blazars by fitting, and discussed the relations of the synchrotron peak frequency, spectral index, and curvature in this sample, the main conclusions are as follows: (1) We have obtained the SEDs of 68 Fermi blazars, the errors of their synchrotron peak frequencies have been significantly reduced, compared to the corresponding sources in FAN. (2) The synchrotron peak frequency and reciprocal curvature exhibit positive correlation. The result of BL Lac objects is consistent with the slope predicted by the model of energy-dependent acceleration probability. When taking only the BL Lac objects with high synchrotron peak frequencies into consideration, namely the BL Lac objects of lg νp > 15.3, the result is consistent with the model of fluctuation of fractional acceleration gain. (3) The spectral index has a certain influence on the correlation between the synchrotron peak frequency and the curvature. The spectral index cannot define very well the synchrotron peak frequency, but can define a group of relations of the synchrotron peak frequency and curvature, and according to the sample of this paper we have given an empirical relation.

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