The Intrinsic $\gamma$-ray Emissions of $Fermi$ Blazars

TL;DR

This study investigates the intrinsic gamma-ray emissions of Fermi blazars by estimating de-beamed fluxes and luminosities, revealing correlations with redshift, variability timescales, and synchrotron peak frequencies, and comparing Doppler factors.

Contribution

It introduces a method to estimate intrinsic gamma-ray properties of blazars and analyzes their correlations, providing insights into jet physics and emission mechanisms.

Findings

Intrinsic gamma-ray flux density follows a redshift relation: log f = -2.0 log z + const.

Strong positive correlation between intrinsic gamma-ray flux and synchrotron peak frequency.

Intrinsic gamma-ray luminosity correlates with variability timescales, consistent with theoretical relations.

Abstract

Beaming effect is important for the observational properties of blazars. In this work, we collect 91 $Fermi$ blazars with available radio Doppler factors. $\gamma$-ray Doppler factors are estimated and compared with radio Doppler factors for some sources. The intrinsic (de-beamed) $\gamma$-ray flux density ($f^{\rm in}_{\gamma}$), intrinsic $\gamma$-ray luminosity ($L^{\rm in}_{\gamma}$), and intrinsic synchrotron peak frequency ($\nu_{\rm p}^{\rm in}$) are calculated. Then we study the correlations between $f^{\rm in}_{\gamma}$ and redshift and find that they follow the theoretical relation: $\log f = -2.0 \log z + {\rm const}$. When the subclasses are considered, we find that stationary jets are perhaps dominant in low synchrotron peaked blazars. 63 $Fermi$ blazars with both available short variability time scales ($\Delta T$) and Doppler factors are also collected. We find that the intrinsic relationship between $L ^{\rm in}_{\gamma}$ and $\Delta T^{\rm in}$ obeys the Elliot \& Shapiro and the Abramowicz \& Nobili relations. Strong positive correlation between $f_{\gamma}^{\rm in}$ and $\nu_{\rm p}^{\rm in}$ is found, suggesting that synchrotron emissions are highly correlated with $\gamma$-ray emissions.

Tables

Table 1. Table 1: The Sample of 91 Fermi Blazars.

3FGL name	Other name	z	log${\nu }_p^s$	Class	$f_{2GeV}$	${\alpha }_{\gamma }^{\mathrm{ph}}$	${\delta }_R$	Ref
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)
J0050.6-0929	PKS 0048-09	0.300	14.60	IPB	12.67	2.09	9.6	H09
J0102.8+5825	TXS 0059+581	0.643	12.73∗	LPQ	24.44	2.25	10.91	F09
J0108.7+0134	PKS 0106+01	2.099	13.53	IPQ	28.83	2.39	18.2	S10
J0112.1+2245	RX J0112.0+2244	0.265	14.39	IPB	29.55	2.03	9.1	S10
J0137.0+4752	S4 0133+47	0.859	12.69	LPQ	21.08	2.27	20.5	S10
J0151.6+2205	PKS 0149+21	1.320	13.14	LPQ	2.21	2.65	4.72	LV99
J0205.0+1510	4C +15.05	0.405	12.10	LPQ	2.82	2.53	15.0	S10
J0217.5+7349	S5 0212+73	2.367	13.35	LPQ	7.72	2.91	8.4	S10
J0217.8+0143	PKS 0215+015	1.721	14.66	IPQ	17.03	2.19	5.61	F09
J0222.6+4301	3C 66A	0.444	14.76	IPB	62.03	1.94	2.6	H09
J0237.9+2848	B2 0234+28	1.207	13.59	LPQ	45.20	2.35	16.0	S10
J0238.6+1636	PKS 0235+164	0.940	13.24	LPB	39.92	2.17	23.8	S10
J0303.6+4716	4C +47.08	0.475	14.10	IPB	8.89	2.28	4.33	F09
J0309.0+1029	PKS 0306+102	0.863	14.04	IPQ	11.18	2.23	2.79	F09
J0336.5+3210	B2 0333+32	1.259	13.55	LPQ	5.26	2.89	22.0	S10
J0339.5-0146	PKS 0336-01	0.852	13.40	LPQ	15.32	2.42	17.2	S10
J0423.2-0119	PKS 0420-01	0.915	12.88	LPQ	23.83	2.30	19.7	S10
J0424.7+0035	PKS 0422+00	1.025	14.22	IPB	7.45	2.20	6.11	F09
J0442.6-0017	PKS 0440-00	0.844	13.04∗	LPQ	16.65	2.50	12.9	H09
J0449.0+1121	PKS 0446+11	1.207	13.09	LPQ	15.07	2.55	4.90	LV99
J0501.2-0157	PKS 0458-02	2.286	13.50	IPQ	13.33	2.41	15.7	S10
J0522.9-3628	PKS 0521-36	0.055	13.75	LPQ	17.06	2.44	1.83	F09
J0530.8+1330	PKS 0528+134	2.070	12.53	LPQ	22.96	2.51	30.9	S10
J0608.0-0835	PKS 0605-08	0.872	13.88	IPQ	9.27	2.37	7.5	S10
J0721.9+7120	1H 0717+714	0.310	14.96	IPB	75.14	2.04	10.8	S10
J0725.8-0054	PKS 0723-008	0.127	14.00	IPB	3.93	2.19	2.50	LV99
J0738.1+1741	PKS 0735+17	0.424	14.23	IPB	18.41	2.01	3.92	F09
J0739.4+0137	PKS 0736+01	0.191	14.43	IPQ	10.46	2.48	8.5	S10
J0757.0+0956	PKS 0754+100	0.266	14.05	IPB	6.71	2.18	5.5	S10
J0807.9+4946	S4 0804+49	1.436	13.28	LPQ	1.60	2.57	35.2	S10
J0811.3+0146	OJ 014	0.407	13.28	LPB	8.97	2.16	5.39	F09
J0818.2+4223	B3 0814+425	0.245	13.52	LPB	22.22	2.11	4.6	S10
J0820.9-1258	PKS 0818-128	0.407	14.77	IPB	1.03	2.27	3.18	F09
J0830.7+2408	B2 0827+24	0.941	13.50	LPQ	5.79	2.63	13.0	S10
J0831.9+0430	PKS 0829+046	0.230	13.84	LPB	12.43	2.24	3.80	F09
J0841.4+7053	RBS 0717	2.218	14.44	IPQ	11.43	2.84	16.1	S10
J0849.9+5108	SBS 0846+513	1.860	13.36∗	LPB	8.79	2.28	6.40	LV99
J0850.2-1214	PMN J0850-1213	0.566	13.10	LPQ	0.27	0.11	16.5	H09
J0854.8+2006	PKS 0851+202	0.306	14.21	IPB	21.46	2.18	16.8	S10
J0948.6+4041	B3 0945+408	1.249	13.86	IPQ	1.81	2.67	6.3	S10
J0956.6+2515	OK 290	0.712	13.98	IPQ	3.95	2.44	4.3	H09
J0957.6+5523	4C +55.17	0.901	14.74	IPQ	34.97	2.00	4.63	LV99
J0958.6+6534	S4 0954+65	0.367	14.02	IPB	5.47	2.38	5.93	F09
J1037.0-2934	PKS 1034-293	0.312	13.92	IPQ	2.03	2.49	2.80	F09
J1058.5+0133	PKS 1055+01	0.888	13.79	IPQ	25.88	2.21	12.1	S10
J1129.9-1446	PKS 1127-14	1.187	13.99	IPQ	5.69	2.79	3.22	F09
J1159.5+2914	B2 1156+29	0.729	13.04	LPQ	32.64	2.21	28.2	S10
J1221.4+2814	W Comae	0.102	14.83	IPB	14.24	2.10	1.2	H09
J1224.9+2122	PG 1222+216	0.432	14.53	IPQ	99.31	2.29	5.2	S10
J1229.1+0202	PKS 1226+02	0.158	15.12	IPQ	36.15	2.66	16.8	S10
J1256.1-0547	3C 279	0.536	12.69	LPQ	83.67	2.34	23.8	S10
J1309.5+1154	PKS 1307+121	0.407	13.72	LPB	2.12	2.14	1.22	F09
J1310.6+3222	B2 1308+32	0.997	13.22	LPQ	15.17	2.25	15.3	S10
J1326.8+2211	B2 1324+22	1.400	12.97	LPQ	8.64	2.45	21.0	S10
J1337.6-1257	PKS 1335-12	0.539	13.25	LPQ	6.63	2.44	8.3	S10
J1408.8-0751	PKS B1406-076	1.494	12.86	LPQ	8.91	2.38	8.26	LV99
J1416.0+1325	PKS 1413+135	0.247	12.57	LPB	2.02	2.36	12.1	S10
J1419.9+5425	OQ 530	0.151	14.27	IPB	3.53	2.31	2.79	F09
J1504.4+1029	PKS 1502+106	1.839	13.34	LPQ	107.57	2.24	11.9	S10
J1512.8-0906	PKS 1510-089	0.360	13.97	IPQ	161.49	2.36	16.5	S10
J1540.8+1449	PKS 1538+149	0.605	13.97	IPB	1.53	2.34	4.3	S10
J1608.6+1029	PKS 1606+10	1.226	13.39	LPQ	7.41	2.62	24.8	S10
J1613.8+3410	B2 1611+34	1.397	13.44	LPQ	3.11	2.35	13.6	S10
J1635.2+3809	B3 1633+382	1.814	13.21	LPQ	60.94	2.40	21.3	S10
J1637.9+5719	S4 1637+57	0.751	14.22	IPQ	1.53	2.81	13.9	S10
J1642.9+3950	3C 345	0.593	13.46	LPQ	10.90	2.45	7.7	S10
J1719.2+1744	PKS 1717+177	0.407	13.91	IPB	5.43	2.04	1.94	F09
J1728.5+0428	PKS 1725+044	0.293	13.32	LPQ	3.65	2.59	3.8	H09
J1733.0-1305	PKS 1730-130	0.902	12.62	LPQ	24.61	2.35	10.6	S10
J1740.3+5211	S4 1739+52	1.379	13.42	LPQ	8.66	2.45	26.3	S10
J1744.3-0353	PKS 1741-03	1.054	14.06	IPQ	2.70	2.27	19.5	S10
J1748.6+7005	S4 1749+70	0.770	14.27	IPB	14.58	2.06	3.75	F09
J1751.5+0939	PKS 1749+096	0.322	12.99	LPB	15.92	2.25	11.9	S10
J1800.5+7827	S5 1803+78	0.684	13.90	IPB	19.78	2.22	12.1	S10
J1806.7+6949	3C 371	0.051	14.60	IPB	11.80	2.23	1.1	S10
J1824.2+5649	S4 1823+56	0.664	13.25	LPB	8.95	2.46	6.3	S10
J1829.6+4844	S4 1828+48	0.692	13.04∗	LPQ	6.38	2.37	5.6	S10
J1924.8-2914	PKS B1921-293	0.352	12.53	LPQ	8.84	2.50	9.51	F09
J2005.2+7752	S5 2007+77	0.342	13.55	LPB	6.65	2.22	4.68	F09
J2123.6+0533	PKS 2121+053	1.941	13.40	LPQ	2.02	2.17	15.2	S10
J2134.1-0152	PKS 2131-021	1.285	13.17	LPB	4.62	2.21	7.00	F09
J2147.2+0929	PKS 2144+092	1.113	13.87	IPQ	12.93	2.54	5.96	LV99
J2148.2+0659	PKS 2145+06	0.990	13.29	LPQ	2.30	2.77	15.5	S10
J2158.0-1501	PKS 2155-152	0.672	13.09	LPQ	4.27	2.27	2.31	F09
J2202.7+4217	B3 2200+420	0.069	15.10	IPB	58.52	2.25	7.2	S10
J2203.7+3143	S3 2201+31	0.295	14.43	IPQ	0.89	3.07	6.6	S10
J2225.8-0454	3C 446	1.404	13.24	LPQ	9.37	2.36	15.9	S10
J2229.7-0833	PKS 2227-088	1.562	13.34	LPQ	20.79	2.55	15.8	S10
J2232.5+1143	PKS 2230+11	1.037	13.65	LPQ	25.54	2.52	15.5	S10
J2236.3+2829	B2 2234+28A	0.795	12.88	LPB	17.70	2.28	6.0	H09
J2254.0+1608	PKS 2251+15	0.859	13.54	LPQ	463.39	2.35	32.9	S10

Table 2. Table 2: The Short Variability Time Scale and γ 𝛾 \gamma -ray Doppler Factor for Fermi Blazars.

3FGL name	Other name	z	Class	log$\Delta T$	Band	Ref	$F_X$	Ref	${\alpha }_X$	Ref	$F_{\gamma }$	${\alpha }_{\gamma }^{\mathrm{ph}}$	${\delta }_{\gamma }$
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)	(12)	(13)	(14)
J0141.4-0929	1Jy 0138-097	1.034	B	6.03	$\gamma$	V13	0.70	LAC	1.15	F14	20.65	2.12	4.45
J0205.0+1510	4C +15.05	0.405	Q	5.78	$\gamma$	V13	0.02	BZC	0.37	E14	6.71	2.53	0.99
J0210.7-5101	PKS 0208-512	0.999	Q	5.61	$\gamma$	V13	1.62	LAC	1.06	B97	47.40	2.30	5.61
J0222.6+4301	3C 66A	0.444	B	5.10	$\gamma$	V13	6.39	LAC	1.60	F14	192.78	1.94	4.46
J0238.6+1636	PKS 0235+164	0.940	B	5.94	$\gamma$	V13	1.24	BZC	1.59	F14	103.05	2.17	4.41
J0339.5-0146	PKS 0336-01	0.852	Q	6.02	$\gamma$	V13	0.75	BZC	0.62	E14	33.58	2.42	3.67
J0423.2-0119	PKS 0420-01	0.915	Q	5.37	$\gamma$	V13	3.87	LAC	0.86	F14	55.92	2.30	6.84
J0442.6-0017	PKS 0440-00	0.844	Q	4.95	$\gamma$	V13	4.06	LAC	0.59	E14	34.94	2.50	8.05
J0457.0-2324	PKS 0454-234	1.003	Q	4.83	$\gamma$	V13	0.60	BZC	0.48	E14	180.35	2.21	7.31
J0501.2-0157	PKS 0458-02	2.286	Q	5.73	$\gamma$	V13	0.92	BZC	0.60	E14	23.24	2.41	12.30
J0510.0+1802	PKS 0507+17	0.416	Q	4.03	$\gamma$	L15	0.38	BZC	0.50	E14	13.34	2.41	4.34
J0522.9-3628	PKS 0521-36	0.055	Q	4.57	$\gamma$	V13	22.50	LAC	0.92	A09a	47.34	2.44	2.27
J0530.8+1330	PKS 0528+134	2.070	Q	5.24	$\gamma$	D95	3.75	LAC	0.58	F14	36.86	2.51	17.83
J0538.8-4405	PKS 0537-441	0.894	B	6.04	$\gamma$	V13	4.53	LAC	1.12	F14	329.60	2.04	5.36
J0540.0-2837	1Jy 0537-286	3.104	Q	6.21	$\gamma$	V13	1.46	LAC	0.32	F14	7.91	2.78	16.66
J0721.9+7120	1H 0717+714	0.310	B	4.80	$\gamma$	V13	4.91	LAC	1.77	F14	219.99	2.04	3.62
J0738.1+1741	PKS 0735+17	0.424	B	6.05	$\gamma$	V13	2.09	LAC	1.34	F14	54.82	2.01	2.69
J0739.4+0137	PKS 0736+01	0.191	Q	4.86	Opt	B83	6.36	LAC	0.76	F14	27.34	2.48	2.94
J0831.9+0430	PKS 0829+046	0.230	B	6.06	$\gamma$	V13	0.60	BZC	1.00	L16	33.88	2.24	1.40
J0841.4+7053	RBS 0717	2.218	Q	4.41	$\gamma$	V13	10.70	LAC	0.42	F14	12.58	2.84	37.76
J0854.8+2006	PKS 0851+202	0.306	B	5.11	$\gamma$	V13	1.79	BZC	1.50	F14	59.03	2.18	2.81
J0920.9+4442	S4 0917+44	2.189	Q	5.08	$\gamma$	V13	1.83	LAC	0.39	F14	57.84	2.29	19.74
J0957.6+5523	4C +55.17	0.901	Q	5.50	$\gamma$	V13	0.77	LAC	0.84	F14	104.50	2.00	5.10
J0958.6+6534	S4 0954+65	0.367	B	5.67	$\gamma$	V13	1.12	LAC	0.24	F14	13.81	2.38	2.26
J1104.4+3812	Mkn 421	0.031	B	3.84	X	D95	678.00	LAC	1.82	F14	302.58	1.77	4.13
J1159.5+2914	B2 1156+29	0.729	Q	5.59	$\gamma$	V13	1.49	LAC	0.86	F14	83.65	2.21	4.40
J1217.8+3007	1ES 1215+303	0.130	B	4.18	Opt	G12	86.40	LAC	1.47	B00	60.51	1.97	4.66
J1221.4+2814	W Comae	0.102	B	3.79	Opt	F99b	2.29	LAC	1.24	F14	41.23	2.10	2.80
J1224.9+2122	PG 1222+216	0.432	Q	3.64	$\gamma$	V13	3.82	LAC	1.19	F14	255.37	2.29	6.68
J1229.1+0202	PKS 1226+02	0.158	Q	4.70	$\gamma$	V13	111.00	LAC	1.11	F14	94.24	2.66	4.21
J1256.1-0547	3C 279	0.536	Q	5.48	$\gamma$	V13	40.50	LAC	0.84	F14	205.75	2.34	6.29
J1310.6+3222	B2 1308+32	0.997	Q	2.65	Opt	B83	0.85	LAC	0.86	B97a	36.53	2.25	17.12
J1408.8-0751	PKS B1406-076	1.494	Q	4.76	$\gamma$	F99a	0.53	BZC	0.07	F13	17.90	2.38	12.84
J1439.2+3931	PG 1437+398	0.344	B	6.25	$\gamma$	V13	17.90	LAC	1.33	F14	4.44	1.77	3.25
J1457.4-3539	PKS 1454-354	1.424	Q	4.64	$\gamma$	A09b	0.51	BZC	0.68	E14	66.18	2.29	10.41
J1504.4+1029	PKS 1502+106	1.839	Q	4.16	$\gamma$	A10	0.16	BZC	0.84	F14	239.96	2.24	12.98
J1512.8-0906	PKS 1510-089	0.360	Q	3.84	$\gamma$	V13	1.15	BZC	0.98	F14	411.05	2.36	4.73
J1517.6-2422	AP Librae	0.049	B	2.95	Opt	B83	2.92	LAC	1.36	F14	52.34	2.11	2.89
J1535.0+3721	RGB J1534+372	0.143	B	6.42	$\gamma$	V13	0.37	LAC	1.84	F14	4.10	2.11	1.07
J1540.8+1449	PKS 1538+149	0.605	B	3.44	Opt	F96	1.82	LAC	0.66	F14	3.70	2.34	9.88
J1626.0-2951	PKS 1622-297	0.815	Q	4.14	$\gamma$	M97	2.28	LAC	0.45	E14	25.02	2.45	10.66
J1635.2+3809	B3 1633+382	1.814	Q	4.81	$\gamma$	V13	0.17	BZC	0.62	F14	114.44	2.40	10.22
J1642.9+3950	3C 345	0.593	Q	5.05	Opt	D95	4.07	LAC	0.81	F14	25.12	2.45	5.37
J1653.9+3945	Mkn 501	0.034	B	5.40	Inf	B83	65.10	LAC	1.36	F14	97.38	1.72	1.96
J1728.3+5013	I Zw 187	0.055	B	5.61	X	B83	39.60	LAC	1.39	F14	10.81	1.96	1.86
J1733.0-1305	PKS 1730-130	0.902	Q	4.90	$\gamma$	V13	6.32	LAC	0.50	F14	55.89	2.35	10.02
J1740.3+5211	S4 1739+52	1.379	Q	5.70	$\gamma$	V13	1.25	LAC	1.08	F14	16.66	2.45	6.85
J1748.6+7005	S4 1749+70	0.770	B	4.68	$\gamma$	V13	1.55	LAC	1.44	F14	41.38	2.06	6.14
J1751.5+0939	OT 081	0.322	B	5.47	$\gamma$	V13	1.18	BZC	0.74	L15	42.51	2.25	2.39
J1800.5+7827	S5 1803+78	0.684	B	4.95	$\gamma$	V13	1.71	LAC	0.45	F14	50.65	2.22	5.97
J1806.7+6949	3C 371	0.051	B	4.92	Opt	B83	4.79	LAC	0.75	F14	33.48	2.23	1.49
J1813.6+3143	B2 1811+31	0.117	B	6.26	$\gamma$	V13	1.44	LAC	2.60	L15	15.44	2.12	1.27
J1824.2+5649	S4 1823+56	0.664	B	6.60	$\gamma$	V13	2.52	LAC	0.96	F14	20.11	2.46	2.86
J1833.6-2103	PKS 1830-210	2.507	Q	4.44	$\gamma$	V13	3.25	LAC	0.13	F14	15.44	2.12	43.49
J2009.3-4849	1Jy 2005-489	0.071	B	6.48	$\gamma$	V13	80.80	LAC	1.32	F14	35.54	1.77	1.78
J2134.1-0152	PKS 2131-021	1.285	B	5.88	$\gamma$	V13	0.67	LAC	1.05	F14	11.18	2.21	5.66
J2143.5+1744	S3 2141+17	0.211	Q	5.45	$\gamma$	V13	1.76	LAC	1.44	F14	44.18	2.52	1.92
J2158.8-3013	PKS 2155-304	0.117	B	6.69	$\gamma$	V13	572.00	LAC	1.62	F14	216.84	1.83	2.64
J2202.7+4217	B3 2200+420	0.069	B	4.96	$\gamma$	V13	7.42	LAC	0.83	F14	164.77	2.25	1.81
J2225.8-0454	3C 446	1.404	Q	3.48	Opt	B83	2.12	LAC	0.59	F14	19.43	2.36	22.99
J2232.5+1143	PKS 2230+11	1.037	Q	4.63	Opt	B83	3.06	LAC	0.51	F14	50.19	2.52	11.05
J2250.1+3825	B3 2247+381	0.119	B	6.15	$\gamma$	V13	7.93	LAC	1.51	F14	11.02	1.91	1.69
J2254.0+1608	PKS 2251+15	0.859	Q	2.94	$\gamma$	V13	19.00	LAC	0.62	F14	1060.92	2.35	26.71

Equations

$$\frac{dN}{dE}=N_0{E}^{-{\alpha }^{\mathrm{ph}}},$$

$$f_E=\frac{N_{(E_{\mathrm{L}}\sim E_{\mathrm{U}})}(1-{\alpha }^{\mathrm{ph}})}{E_{\mathrm{U}}^{1-{\alpha }^{\mathrm{ph}}}-E_{\mathrm{L}}^{1-{\alpha }^{\mathrm{ph}}}}\cdot E_0^{1-{\alpha }^{\mathrm{ph}}},$$

$$F=\frac{N_{(E_{\mathrm{L}}\sim E_{\mathrm{U}})}{E}_{\mathrm{U}}\times E_{\mathrm{L}}}{E_{\mathrm{U}}-E_{\mathrm{L}}}\ln (\frac{E_{\mathrm{U}}}{E_{\mathrm{L}}}),\mathrm{for}{\alpha }^{\mathrm{ph}}=2$$

$$F=N_{(E_{\mathrm{L}}\sim E_{\mathrm{U}})}\frac{1-{\alpha }^{\mathrm{ph}}}{2-{\alpha }^{\mathrm{ph}}}\frac{(E_{\mathrm{U}}^{2-{\alpha }^{\mathrm{ph}}}-E_{\mathrm{L}}^{2-{\alpha }^{\mathrm{ph}}})}{(E_{\mathrm{U}}^{1-{\alpha }^{\mathrm{ph}}}-E_{\mathrm{L}}^{1-{\alpha }^{\mathrm{ph}}})},otherwise.$$

$$\log\nu_{\rm p}^{\rm s}=\left\{\begin{array}[]{ll}16+4.238X&\,\,\,\,\,\,\,X<0\\ 16+4.005Y&\,\,\,\,\,\,\,X>0,\end{array}\right.$$

$${\delta }_{\gamma }\ge {\left[1.54\times 10^{-3}(1+z{)}^{4+2\alpha }{\left(\frac{d_{\mathrm{L}}}{\mathrm{Mpc}}\right)}^2{\left(\frac{\Delta T}{\mathrm{hr}}\right)}^{-1}\left(\frac{F_{\mathrm{KeV}}}{\mu \mathrm{Jy}}\right){\left(\frac{E_{\gamma }}{\mathrm{GeV}}\right)}^{\alpha }\right]}^{\frac{1}{4+2\alpha }},$$

$$d_{\mathrm{L}}=(1+z)\frac{c}{H_0}{\int }_1^{1+z}\frac{1}{\sqrt{{\Omega }_M{x}^3+1-{\Omega }_M}}\mathrm{d}x$$

$$r_{ij,k}=(r_{ij}-r_{ik}{r}_{jk})/\sqrt{(1-r_{ik}^2)(1-r_{jk}^2)},$$

Full text

1 Introduction

Two major subclasses of active galactic nuclei (AGNs) are radio loud AGNs and radio quiet AGNs. The radio-loud AGNs are blazars that have high and variable polarization, rapid and high amplitude variability, superluminal motions, and strong $\gamma$-ray emissions, etc. Blazars have two subclasses, namely flat spectrum radio quasars (FSRQs) and BL Lacertae objects (BL Lacs). The difference between the two subclasses is mainly that BL Lacertae objects show very weak (or even no) emission line features while FSRQs display strong emission lines. However, the continuum emission properties of BL Lacs and FSRQs are quite similar (Zhang & Fan 2003; Fan et al. 2009, 2014a; Xiao et al. 2015). BL Lac objects were separately discovered through radio or X-ray surveys, and were divided into radio selected BL Lacertae objects (RBLs) and X-ray selected BL Lacertae objects (XBLs). From spectral energy distributions (SEDs), blazars can be divided into low synchrotron peaked (LSP, $\nu_{\rm{peak}}^{s}<10^{14}$ Hz), intermediate synchrotron peaked (ISP, $10^{14}$ Hz $<\nu_{\rm{peak}}^{s}<10^{15}$ Hz), and high synchrotron peaked (HSP, $\nu_{\rm{peak}}^{s}>10^{15}$ Hz) blazars. This classification was first proposed by Abdo et al. (2010a) (see also Wu et al. 2007; Yang et al. 2014; Ackermann et al. 2015; Fan et al. 2015a; Lin & Fan 2016). In our recent work (Fan et al. 2016), a sample of 1392 Fermi blazars is collected, and SEDs of them are obtained. Then we followed the acronym of Abdo et al. (2010a), proposed a classification by using the Bayesian classification method, as follows: $\nu_{\rm peak}^{\rm s}<10^{14}$ Hz for LSP, $10^{14}$ Hz $<\nu_{\rm peak}^{\rm s}<10^{15.3}$ Hz for ISP, and $\nu_{\rm peak}^{s}>10^{15.3}$ Hz for HSP. We also pointed out that there is no ultra-HSP blazars (Fan et al. 2016).

Blazars have strong $\gamma$-ray emissions, some of them even were detected in TeV energy region (Weekes T. C. 1997; Catanese & Weekes 1999; Holder, 2012; Xiong et al. 2013; Lin & Fan 2016). But the origin of high energy emissions are still unclear. The Fermi Large Area Telescope (LAT) was launched in 2008, and detected many blazars at $\gamma$-ray energy region (Abdo et al. 2010b; Nolan et al. 2012; Acero et al. 2015; Ackermann et al. 2015). Compared with the predecessor the Energetic Gamma Ray Experiment Telescope (EGRET), the Fermi/LAT satellite has unprecedented sensitivity in the $\gamma$-ray band (Abdo et al. 2010a). The 3rd Fermi Large Area Telescope source catalog (3FGL) contains 3033 sources (see Acero et al. 2015), which gives us a large sample to analyze the mechanism of $\gamma$-ray emissions and other observed properties for blazars.

Beaming effect is included in the explanations of the whole electromagnetic emissions including $\gamma$-ray emissions for blazars, and many authors found that their $\gamma$-ray emissions have strong beaming effect (eg., Fan et al. 1999a, 2013a, 2014b, c, 2015b; Fan, 2005; Ruan et al. 2014; Chen et al. 2015, 2016; Cheng et al. 1999). Correlations are found between $\gamma$-ray emissions and other bands, and gamma-ray loud blazars are found to have larger Doppler factors than non-gamma-ray loud ones (Dondi & Ghisellini 1995; Valtaoja & Teräsranta 1995; Xie et al. 1997; Fan et al. 1998; Cheng et al. 2000; Jorstad et al. 2001a, b; Lähteenmäki & Valtaoja 2003; Kellermann et al. 2004; Lister et al. 2009; Savolainen et al. 2010; Zhang et al. 2012; Li et al. 2013; Xiong et al. 2013; Wu et al. 2014; Xiao et al. 2015; Chen et al. 2016; Pei et al. 2016).

In a beaming model, the relativistic jet emissions are boosted, $f^{\rm{ob}}=\delta^{q}f^{\rm{in}}$, here $f^{\rm{in}}$ is the intrinsic (de-beamed) emissions in the source rest frame, $\delta$ is a Doppler boosting factor, $q$ depends on the shape of the jet: $q=2+\alpha$ for the stationary jet, $q=3+\alpha$ for the jet with distinct “blobs”, and $\alpha$ is an energy spectral index ($f_{\nu}\propto\nu^{-\alpha}$). The Doppler boosting factor, which is defined as $\delta=[\Gamma(1-\beta\cos\theta)]^{-1}$, is important to investigate the intrinsic properties of blazars. But it depends on two unobservable factors: the bulk Lorentz factor, $\Gamma=(1-\beta^{2})^{-1/2}$, and the viewing angle, $\theta$, here $\beta$ is the jet speed in units of the speed of light (see Fan et al. 2009; Savolainen et al. 2010).

Some methods are proposed to estimate the Doppler factors. Ghisellini et al. (1993) gave a method of estimating the Doppler factors, which was based on the synchrotron self-Compton model. This method assumes the X-ray flux originates from the self-Compton components, so a predicted X-ray flux can be calculated by using Very Long Baseline Interferometry (VLBI) data. By comparing this to the observed X-ray flux, the Doppler boosting factors can be calculated. After that, Lähteenmäki & Valtaoja (1999, hereafter LV99) proposed a more accurate and reliable method: the variability brightness temperature of the source ($T_{\rm b}^{\rm ob}$) obtained from the variability of VLBI data is used to compare with intrinsic brightness temperature of the source ($T_{\rm b}^{\rm in}$). $T_{\rm b}^{\rm in}$ is assumed to be the equipartition brightness temperature ($T_{\rm b}^{\rm{eq}}$), namely $T_{\rm b}^{\rm in}=T_{\rm b}^{\rm{eq}}=5\times 10^{10}$K. So the Doppler factor can be estimated by using $\delta=(T_{\rm b}^{\rm ob}/T_{\rm b}^{\rm in})^{1/3}$. When the variability timescales are obtained from total flux density (TFD) observation, the variability brightness temperature can be calculated by using exponential flares and the variability timescales (LV99, see also Fan et al. 2009; Hovatta et al. 2009).

Because of the short term variability, a highly compact engine exists at the center of blazars. The balance between gravitation and radiation pressure determines an upper limit of luminosity, namely Eddington luminosity, for any active galactic nuclei (Bassani et al. 1983). If the short variability time scale is assumed to be equal or greater than the time that light travels across the Schwarzschild radius of a black hole, then the observed luminosity and short variability time scale should obey to the so called Elliot & Shapiro Relation (E-S Relation): $\rm logL\leq 43.1+log\,\,\Delta T$ (Elliot & Shapiro 1974), where $L$ is luminosity in units of erg/s, and $\Delta T$ is variability time scale in units of second (s). Generally, short variability time scale is assumed to be the time scale which is shorter than one week (Fan 2005). When the anisotropy of emissions is considered, above limit was replaced by the Abramowicz & Nobili Relation (A-N Relation): $\rm logL\leq 44.3+log\,\,\Delta T$ (Abramowicz & Nobili 1982).

The intrinsic properties are required to analyze the origin of $\gamma$-ray emissions for blazars. In our recent work of Xiao et al. (2015), we considered the beaming effect of Fermi blazars in Nolan et al. (2012) (2FGL), then analysed the correlation between $\gamma$-ray flux density and redshift for 73 blazars, and the relation between $\gamma$-ray short variability time scale and $\gamma$-ray luminosity by comparing with the E-S Relation and the A-N Relation for 28 blazars. In this work, we use a larger sample to revisit the relation between $\gamma$-ray flux density and redshift, and the relation between $\gamma$-ray luminosity and short variability time scale. The subclasses of blazars, and the short variability time scale from X-ray and optical bands are also considered. Then we have 91 Fermi blazars with available radio Doppler factors and 63 Fermi blazars with both available short variability time scales and Doppler factors. $\gamma$-ray Doppler factors are estimated for the Fermi blazars with available short variability time scales at optical, X-ray or $\gamma$-ray band. In addition, the correlation between $\gamma$-ray emissions and synchrotron peaked frequency is also studied in this work. This work is arranged as follows: we will describe our sample and corresponding results in section 2, and give some discussions and conclusions in section 3.

2 Sample and Results

2.1 Sample

In this work, we collect blazars with available radio Doppler factors from the literatures: LV99, Fan et al. (2009), Hovatta et al. (2009) and Savolainen et al. (2010). These references used the same method introduced by LV99. If the Doppler factors of some sources are available in more than one reference, we choose the value from the latest paper. Based on the third catalog of active galactic nuclei detected by the Fermi Large Area Telescope (3LAC)333http://www.asdc.asi.it/fermi3lac/ (Ackermann et al. 2015), we collect a sample of 91 Fermi blazars with available radio Doppler factors, which are listed in Table 1. In Table 1, $\gamma$-ray data are from 3LAC, only that for PKS 2145+06 is from 2LAC444http://www.asdc.asi.it/fermi2lac/ (Ackermann et al. 2011).

We get the Fermi integral photon flux at 1–100 GeV, as we did in our previous works (Fan et al. 2013a, 2014b), let

[TABLE]

where ${\rm\alpha^{ph}}$ is a photon spectral index, here $\alpha^{{\rm ph}}=\alpha-1$. Then the flux density at energy $E_{0}$ in units of GeV/cm2/s/GeV can be expressed as

[TABLE]

where $N_{(E_{\rm L}\sim E_{\rm U})}$ is integral photons in units of ${\rm ph/cm^{2}/s}$ in the energy range of $E_{\rm L}\sim E_{\rm U}$. Because the integral flux in GeV/cm2/s can be obtained by $F=\int^{E_{\rm U}}_{E_{\rm L}}EdN=\int^{E_{\rm U}}_{E_{\rm L}}f_{E}dE$, therefore

[TABLE]

[TABLE]

For the Fermi sources in this work, $E_{\rm L}$ and $E_{\rm U}$ correspond to 1 GeV and 100 GeV respectively.

The synchrotron peak frequency (${\rm log\nu^{s}_{p}}$) of PKS 2145+06 is not available in Fan et al. (2016) or 2LAC. We use the empirical relationship introduced in Fan et al. (2016) to estimate it, as follows

[TABLE]

where $X=1-1.262\alpha_{\rm RO}-0.623\alpha_{\rm OX}$, and $Y=1.0+0.034\alpha_{\rm RO}-0.978\alpha_{\rm OX}$ (Fan et al. 2016), $\alpha_{\rm RO}$ and $\alpha_{\rm OX}$ are the effective spectral indexes. For PKS 2145+06, we can get $\alpha_{\rm RO}=0.666$ and $\alpha_{\rm OX}=1.283$ from 2LAC, so we obtain $\log\nu_{\rm p}^{s}=13.29$ Hz.

2.2 $\gamma$-ray Flux Density and Redshift

Our sample in Table 1 contains 32 BL Lacs and 59 FSRQs, or 40 ISP and 51 LSP based on the SED classification in Fan et al. (2016). In this work, we transform the $\gamma$-ray integral photon flux at 1–100 GeV into the flux density in units of mJy at $E_{0}=2$GeV by using the equation (1), and adopt a linear regression to the correlation between flux density and redshift for the whole sample, BL Lacs, FSRQs, ISP, and LSP respectively. All the fluxes are K-corrected by using $f=f^{\rm obs}(1+z)^{\alpha-1}$.

Whole sample: For the whole sample of 91 blazars, we have $\log f_{\gamma}=-(0.01\pm 0.15)\log z-(8.96\pm 0.06)$ with a correlation coefficient $r=-0.01$ and a chance probability $p=93.08\%$. As we introduce in the Section 1, we can calculate the intrinsic flux density, then we have $\log f_{\gamma}^{\rm in}=-(1.82\pm 0.30)\log z-(12.44\pm 0.13)$ with $r=-0.54$ and $p=2.98\times 10^{-8}$ for the case of $q=2+\alpha$, and $\log f_{\gamma}^{\rm in}=-(2.32\pm 0.37)\log z-(13.46\pm 0.16)$ with $r=-0.55$ and $p=1.66\times 10^{-8}$ for $q=3+\alpha$. The corresponding figure is shown in Fig. 1.

BL Lacs: For the 32 BL Lacs, we have $\log f_{\gamma}=-(0.13\pm 0.24)\log z-(9.05\pm 0.13)$ with $r=-0.10$ and $p=59.96\%$; $\log f_{\gamma}^{\rm in}=-(1.30\pm 0.47)\log z-(11.82\pm 0.26)$ with $r=-0.45$ and $p=1.03\%$ for $q=2+\alpha$; and $\log f_{\gamma}^{\rm in}=-(1.65\pm 0.61)\log z-(12.68\pm 0.34)$ with $r=-0.44$ and $p=1.13\%$ for $q=3+\alpha$. The corresponding figure is shown in the upper panel of Fig. 2

FSRQs: For the 59 FSRQs, we have $\log f_{\gamma}=-(0.02\pm 0.23)\log z-(8.94\pm 0.07)$ with $r=-0.01$ and $p=94.20\%$; $\log f_{\gamma}^{\rm in}=-(1.34\pm 0.45)\log z-(12.61\pm 0.14)$ with $r=-0.37$ and $p=3.90\times 10^{-3}$ for $q=2+\alpha$; and $\log f_{\gamma}^{\rm in}=-(1.73\pm 0.55)\log z-(13.68\pm 0.17)$ with $r=-0.39$ and $p=2.50\times 10^{-3}$ for $q=3+\alpha$. The corresponding figure is shown in the lower panel of Fig. 2.

ISP: For the 40 ISP blazars, we have $\log f_{\gamma}=-(0.12\pm 0.22)\log z-(9.02\pm 0.11)$ with $r=-0.09$ and $p=58.81\%$; $\log f_{\gamma}^{\rm in}=-(1.33\pm 0.44)\log z-(12.01\pm 0.22)$ with $r=-0.44$ and $p=4.12\times 10^{-3}$ for $q=2+\alpha$; and $\log f_{\gamma}^{\rm in}=-(1.67\pm 0.54)\log z-(12.89\pm 0.27)$ with $r=-0.45$ and $p=3.66\times 10^{-3}$ for $q=3+\alpha$. The corresponding figure is shown in the upper panel of Fig. 3.

LSP: For the 51 LSP blazars, we have $\log f_{\gamma}=(0.09\pm 0.22)\log z-(8.94\pm 0.07)$ with $r=0.06$ and $p=68.03\%$; $\log f_{\gamma}^{\rm in}=-(1.95\pm 0.44)\log z-(12.66\pm 0.14)$ with $r=-0.54$ and $p=5.20\times 10^{-5}$ for $q=2+\alpha$; and $\log f_{\gamma}^{\rm in}=-(2.51\pm 0.55)\log z-(13.75\pm 0.18)$ with $r=-0.55$ and $p=3.40\times 10^{-5}$ for $q=3+\alpha$. The corresponding figure is shown in the lower panel of Fig. 3.

2.3 Short Variability Time Scale and Luminosity

Observations suggest that $\gamma$-ray loud blazars are variable on time scales of hours although there is no preferred scale for the variation time of any source (Fan et al. 2014a). For example, Fermi LAT detected a variability time scale of $\sim$12 hour for PKS $1454-354$ (Abdo et al. 2009), and a doubling time of roughly 4 hours for PKS 1502+105 (Abdo et al. 2010c). In the literatures, available short variability time scales are collected, eg., Bassani et al. (1983), Dondi & Ghisellini (1995), Fan et al. (1999a), Gupta et al. (2012), Vovk & Neronov et al. (2013).

For the sources with available short variability time scales, X-ray, and $\gamma$-ray emissions, we can estimate their $\gamma$-ray Doppler factors. Follow our recent work (Fan et al. 2013b, 2014a), a Doppler factor can be estimated using:

[TABLE]

here $\Delta T$ is the variability time scale in units of hrs, $\alpha$ is the X-ray spectral index, $F_{\mathrm{KeV}}$ is the flux density at 1 KeV in units of $\mu\mathrm{Jy}$, $E_{\gamma}$ is the energy in units of GeV at which the $\gamma$-rays are detected, and $d_{\mathrm{L}}$ is the luminosity distance in units of Mpc. The average energy $E_{\gamma}$ can be calculated by $E_{\gamma}=\int EdN/\int dN$, and the luminosity distance can be expressed in the form

[TABLE]

from the $\Lambda$-CDM model (Capelo & Natarajan 2007) with $\Omega_{\Lambda}\simeq 0.7$, $\Omega_{\rm M}\simeq 0.3$ and $\Omega_{\rm K}\simeq 0.0$. We adopt $H_{0}=73$ Km/s/Mpc throughout the paper. Then, we estimate the $\gamma$-ray Doppler factors ($\delta_{\gamma}$) of 63 blzazrs, and show them in Table 2.

To compare $\gamma$-ray Doppler factors estimated in this work with radio Doppler factors, we show the plot of $\gamma$-ray Doppler factors against radio Doppler factors in Fig. 4. We find that the $\gamma$-ray Doppler factors that we estimated are on average lower than the radio Doppler factors. The reason maybe from the fact that the derived value in this work is a lower value as discussed in Fan et al. (2014a).

The luminosity can be calculated using $L_{\gamma}=4\pi d_{\rm L}^{2}(1+z)^{\alpha-1}F_{\gamma}$, where $F_{\gamma}$ is the integral flux calculated by equation (2), $(1+z)^{\alpha-1}$ stands for a K-correction into the source rest frame (Fan et al. 2013a; Kapanadze 2013). In a beaming model, the observed photon energy are also beamed, $E^{\rm ob}=\delta E^{\rm in}$, here $E^{\rm in}$ is the intrinsic energy. Because $F=\int fdE$, so we have $L^{\rm ob}=\delta^{3+\alpha}L^{\rm in}$ for the case of $q=2+\alpha$, $L^{\rm ob}=\delta^{4+\alpha}L^{\rm in}$ for the case of $q=3+\alpha$, and $\Delta T^{\rm in}=\delta\Delta T^{\rm ob}$. Here $L^{\rm in}$ and $\Delta T^{\rm in}$ are the intrinsic luminosity and the intrinsic variability time scale respectively.

For calculating $L_{\gamma}^{\rm in}$ and $\Delta T^{\rm in}$, we use radio Doppler factors in Table 1, but if there is no available radio Doppler factor, we used $\gamma$-ray Doppler factors in Table 2 instead. When the Doppler boosting effect is considered, the plot of short variability time scale against the $\gamma$-ray luminosity is shown in Fig. 5, where the observed properties and intrinsic properties are shown respectively. Then we find that 9 blazars violate the E-S or A-N relation in $L_{\gamma}^{\rm ob}$ versus $\Delta T^{\rm ob}$. However, the whole sample follows the E-S and A-N relations in $L_{\gamma}^{\rm in}$ versus $\Delta T^{\rm in}$, see Fig. 5. When the subclasses of blazars are considered, 9 FSRQs violate the E-S relation and 3 FSRQs violate the A-N relation in observed data, while all blazars follow those relations in intrinsic data, see Fig. 6.

2.4 $\gamma$-ray Emissions and Synchrotron Peaked Frequency

For the whole 91 blazars, the linear regression analysis is adopted to the correlations between $\gamma$-ray flux density ($\log f_{\gamma}$) and synchrotron peak frequency ($\log\nu^{s}_{\rm p}$). The synchrotron peak frequencies are corrected to the rest-frame by $\nu_{\rm p}^{\rm res}=(1+z)\nu_{\rm p}^{\rm obs}$ before analysis.

For $\log f_{\gamma}$–$\log\nu^{s}_{\rm p}$ correlation, we have $\log f_{\gamma}=(0.06\pm 0.09)\log\nu^{s}_{\rm p}-(9.75\pm 1.21)$ with $r=0.07$ and $p=51.51\%$, $\log f_{\gamma}^{\rm in}=(1.00\pm 0.12)\log\nu^{\rm in}_{\rm p}-(25.03\pm 1.60)$ with $r=0.65$ and $p=2.54\times 10^{-12}$ for $q=2+\alpha$, and $\log f_{\gamma}^{\rm in}=(1.30\pm 0.15)\log\nu^{\rm in}_{\rm p}-(29.84\pm 1.95)$ with $r=0.68$ and $p=1.97\times 10^{-13}$ for $q=3+\alpha$, here $\nu_{\rm p}^{\rm in}=\nu_{\rm p}^{\rm res}/\delta$ is an intrinsic peak frequency. The corresponding figure is shown in Fig. 7.

As results shown in Section 2.2 and in Lin & Fan (2016), $\gamma$-ray flux density is strongly correlated with redshift, therefore the correlations between $\gamma$-ray emissions and peak frequency maybe caused by a redshift effect. In our recent works (Fan et al. 2013a, 2015a; Lin & Fan 2016), we have removed the redshift effect from luminosity-luminosity correlation by using the partial correlation introduced by Padovani (1992). If variables $i$ and $j$ are correlated with a third one $k$, then the correlation between $i$ and $j$ can remove the $k$ effect, as follows:

[TABLE]

here $r_{ij}$, $r_{ik}$, $r_{jk}$ are the correlation coefficients between any two variables of $i$, $j$ and $k$ respectively. When the method is performed to the $\log f_{\gamma}^{\rm in}$–$\log\nu_{\rm p}^{\rm in}$ correlation, then the correlation coefficients excluding the redshift effect for the whole sample are as follows: $r_{f\nu,z}$ = 0.56 with $p_{f\nu,z}=8.09\times 10^{-9}$ for $q=2+\alpha$, and $r_{f\nu,z}$ = 0.59 with $p_{f\nu,z}=7.52\times 10^{-10}$ for $q=3+\alpha$.

3 discussion and conclusion

In this work, we collect 91 blazars with available radio Doppler factors, and calculate their intrinsic $\gamma$-ray flux density ($\log f_{\gamma}^{\rm in}$) at 2 GeV, and intrinsic synchrotron peak frequency ($\log\nu_{\rm p}^{\rm in}$). Then the correlations between between $\log f_{\gamma}^{\rm in}$ and redshift, and between $\log f_{\gamma}^{\rm in}$ and $\log\nu_{\rm p}^{\rm in}$ are investigated. We also study the intrinsic relation between short variability time scale ($\Delta T^{\rm in}$) and $\gamma$-ray luminosity ($\log L_{\gamma}^{\rm in}$) for 63 blazars by comparing to the E-S and A-N Relations.

In our recent work, we compared the $\gamma$-ray Doppler factors ($\delta_{\gamma}$) with the radio Doppler factors ($\delta_{R}$), and found they are associated with each other, see Fan et al. (2014a). So that we can investigate the beaming effect in $\gamma$-ray band by using the radio Doppler factors. We also found that the radio Doppler factor well correlates with $\gamma$-ray luminosity for the Fermi detected sources, see Fan et al. (2009). In this work, we find that $\log f_{\gamma}^{\rm in}$ of FSRQs are smaller than that of BL Lacs with a probability for the two groups to come from the same distribution (Kolmogorov-Smirnov test) being $p<10^{-5}$ for $q=2+\alpha$ and $q=3+\alpha$. The averaged values of $\log f_{\gamma}^{\rm in}$ are $<\log f_{\gamma}^{\rm in}>=-12.52\pm 1.12$ ($q=2+\alpha$), $<\log f_{\gamma}^{\rm in}>=-13.56\pm 1.38$ ($q=3+\alpha$) for FSRQs; and $<\log f_{\gamma}^{\rm in}>=-11.25\pm 1.00$ ($q=2+\alpha$), $<\log f_{\gamma}^{\rm in}>=-11.95\pm 1.29$ ($q=3+\alpha$) for BL Lacs. And a t-test indicates that the averaged difference of $\log f_{\gamma}^{\rm in}$ between BL Lacs and FSRQs is $\Delta(\log f^{\rm in}_{\gamma})=1.27\pm 0.24$ with a significant level of $p<10^{-6}$ for $q=2+\alpha$, and $\Delta(\log f^{\rm in}_{\gamma})=1.61\pm 0.30$ with $p<10^{-6}$ for $q=3+\alpha$, while the averaged difference of the observed flux density ($\log f_{\gamma}$) is $\Delta(\log f_{\gamma})=0.05\pm 0.11$ with $p=64.23\%$.

3.1 $\gamma$-ray Flux Density and Redshift

We analyse the whole 91 Fermi blazars, and find that $\log f_{\gamma}$ is weakly correlated with the redshift ($r=-0.01$, and slope is $-0.01\pm 0.15$). The result is different from our expectation: $\log f=-2.0\log z+{\rm const}$, if blazars belong to a group. But when we consider the strong beaming effect of $\gamma$-ray emissions for Fermi blazars, we find that $\log f_{\gamma}^{\rm in}$ is strongly correlated with redshift as follows: $\log f_{\gamma}^{\rm in}=-(1.82\pm 0.30)\log z-(12.44\pm 0.13)$ with $r=-0.54$ and $p=2.98\times 10^{-8}$ for $q=2+\alpha$, and $\log f_{\gamma}^{\rm in}=-(2.32\pm 0.37)\log z-(13.46\pm 0.16)$ with $r=-0.55$ and $p=1.66\times 10^{-8}$ for $q=3+\alpha$. In our recent work (Xiao et al. 2015), we found that $\log f_{\gamma}^{\rm in}$ and redshift have strong correlation ($p<10^{-4}$) with the slopes being $-2.05\pm 0.32$ ($q=2+\alpha$) and $-2.55\pm 0.34$ ($q=3+\alpha$). The results in present work suggest that $\log f_{\gamma}^{\rm in}$ and the redshift follow the theoretical relation, which is consistent with that in our previous work (Xiao et al. 2015).

For subclasses of blazars, some similar correlation results are found, but their slopes are different slightly. The slopes of correlations between $\log f_{\gamma}^{\rm in}$ and redshift are $-1.30\pm 0.47$ (BL Lacs), $-1.34\pm 0.45$ (FSRQs), $-1.33\pm 0.44$ (ISP), and $-1.95\pm 0.44$ (LSP) for the case of $q=2+\alpha$; and those are $-1.65\pm 0.61$ (BL Lacs), $-1.73\pm 0.55$ (FSRQs), $-1.67\pm 0.54$ (ISP), and $-2.51\pm 0.55$ (LSP) for $q=3+\alpha$. In our results, slopes of BL Lacs are very close to those of FSRQs, and they favor the jet case of $q=3+\alpha$. However, for the whole sample, the results of slopes indicate that two jet cases exist in blazars, since slopes being $-1.82\pm 0.30$ for $q=2+\alpha$, $-2.32\pm 0.37$ for $q=3+\alpha$, and $\sim-2.0$ for theoretical relation. For the SED classification, results of slopes suggest that ISP blazars favor the jet case of $q=3+\alpha$, while LSP blazars favor the jet case of $q=2+\alpha$. Those results indicate stationary jets ($q=2+\alpha$) are perhaps dominant in LSP blazars. A possible explanation of those results is that differences of synchrotron peaked frequency caused by the physical differences of blazars, such as the different forms of relativistic jets. In Xiao et al. (2015), we suggested that the continues case ($q=2+\alpha$) of jet is perhaps real for Fermi blazars, however, we did not discuss the subclasses at that work. In the present work, we can not discuss this further since there is no Doppler factor for HSP blazars.

3.2 Short Variability Time Scale and Luminosity

For the whole sample of 63 blazars, our results show that 9 blazars violate the E-S or A-N relation in $L_{\gamma}^{\rm ob}$ versus $\Delta T^{\rm ob}$, while the whole sample follows the E-S and A-N relations in $L_{\gamma}^{\rm in}$ versus $\Delta T^{\rm in}$, see Figs. 5 and 6. In our recent work (Xiao et al. 2015), some similar results are found for a sample of 28 blazars. Different subclasses of blazars have different properties, for instance, FSRQs have strong emission lines, so that can be determined the redshift easier and more accurate than BL Lacs, FSRQs have higher redshift and lower synchrotron peaked frequency than BL Lacs statistically. Therefore, for further analysis, the subclasses of blazars are considered. We find that 9 FSRQs violate the E-S relation and 3 FSRQs violate the A-N relation in $L_{\gamma}^{\rm ob}$ versus $\Delta T^{\rm ob}$, but all blazars follow those relations in their intrinsic properties. In addition, the averaged values of radio Doppler factors is $<\delta_{\rm R}>\,=\,6.50\pm 4.87$ for 32 BL Lacs, and $<\delta_{\rm R}>\,=\,13.47\pm 8.04$ for 59 FSRQs. From a K-S test, the probability that the distributions of $\delta_{R}$ for BL Lacs and FSRQs to be drawn from the same parent distribution is $p=6.87\times 10^{-5}$. Thus, the Doppler factors of FSRQs are larger than that of BL Lacs, which is consistent with our previous result (Fan et al. 2004). From above analysis, we find that beaming effect is an important reason that causes the blazars to violate the E-S and A-N relations, and FSRQs have stronger beaming effect than BL Lacs.

3.3 $\gamma$-ray Emissions and Synchrotron Peaked Frequency

There is no correlation between $\log f_{\gamma}$ and $\log\nu^{\rm s}_{\rm p}$ with $r=0.07$ and $p=51.51\%$. However, strong positive correlations are found between $\log f^{\rm in}_{\gamma}$ and $\log\nu_{\rm p}^{\rm in}$ for the whole sample. Those correlation coefficients and chance probabilities are $r=0.65$ and $p=2.54\times 10^{-12}$ for the case of $q=2+\alpha$, and $r=0.68$ and $p=1.97\times 10^{-13}$ for $q=3+\alpha$. When the redshift effect is removed, strong positive correlations still exist between them. In Lister et al. (2011), strong anti-correlations are found between observed radio flux density at 5GHz and synchrotron peak frequency for BL Lacs and FSRQs. From results in Lister et al. (2011) and this work, the anti-correlations (or no correlation) between observed flux densities and synchrotron peak frequency are significantly different with the positive correlations in intrinsic properties. Thus, the beaming effect can not be ignored when we investigate the physical mechanism of blazars.

The blazars sequence, which is defined by the anti-correlation between peak luminosity and peak frequency, can be explained by the cooling effect (Fossati et al. 1998; Ghisellini et al. 1998; Wu et al. 2007; Nieppola et al. 2008). However, that theoretical explanation of blazars sequence doesn’t consider the beaming effect. Therefore, the intrinsic correlation between peak luminosity and peak frequency is needed to investigate the blazars sequence. Wu et al. (2007) estimated Doppler factors ($\delta$) for a sample of 170 BL Lacs, found significant anti-correlations between $\delta$ and $\nu_{\rm p}^{\rm in}$, and between the total 408 MHz luminosity ($L_{\rm 408MHz}$) and $\nu_{\rm p}^{\rm in}$. However, the scatter of $L_{\rm 408MHz}$ versus $\nu_{\rm p}^{\rm in}$ is very large, which is contrast with much tighter relation of blazars sequence. Some similar results are found between radio power and $\nu_{\rm p}^{\rm s}$ in Nieppola et al. (2006). Recently, some high-luminosity high-$\nu_{\rm p}^{\rm s}$ and low-luminosity low-$\nu_{\rm p}^{\rm s}$ sources are detected. Those results indicate that the blazars sequence is likely to be eliminated (Wu et al. 2007).

Nieppola et al. (2008) collected a sample of 89 AGNs with available Doppler factors, found strong anti-correlation between $\delta$ and $\nu_{\rm p}^{\rm in}$, and proposed that the lower peak frequency blazars are more boosted. In Nieppola et al. (2008), a positive Spearman rank correlation between intrinsic synchrotron peak luminosity ($L_{\rm p}^{\rm in}$) and $\nu_{\rm p}^{\rm in}$ was also found with $r=0.366$ and $p=3\times 10^{-3}$ for blazars, especially for BL Lacs ($r=0.642$ and $p<10^{-3}$). They concluded that the anti-correlation between $L_{\rm p}^{\rm in}$ and $\nu_{\rm p}^{\rm in}$ which is used to determine the blazars sequence is not present, suggesting that the blazar sequence is an artefact of variable Doppler boosting across the peak frequency range. However, the scatter of the correlation between $L_{\rm p}^{\rm in}$ and $\nu_{\rm p}^{\rm in}$ is about 5 orders of magnitude for their sample. In addition, Wu et al. (2009) found a significant positive $L_{\rm p}^{\rm in}$-$\nu_{\rm p}^{\rm in}$ correlation with a Spearman correlation coefficient of $r=0.59$ at $>99.99\%$ confidence level. In this work, we find a positive correlation between $\log f^{\rm in}_{\gamma}$ and $\log\nu_{\rm p}^{\rm in}$ after correcting the redshift effect. Thus, our results and previous research indicate that there is a positive correlation between intrinsic emissions and intrinsic synchrotron peak frequency.

Interestingly, we find a strong positive correlation between $\nu_{\rm p}^{\rm in}$ and $\nu_{\rm p}^{\rm s}$: $\log\nu_{\rm p}^{\rm in}=(1.19\pm 0.06)\log\nu_{\rm p}^{\rm s}-(3.58\pm 0.79)$ with $r=0.91$ and $p=2.13\times 10^{-36}$. The strong positive $\nu_{\rm p}^{\rm in}$-$\nu_{\rm p}^{\rm s}$ correlation indicates that there is almost no difference of the order of blazars along the peak frequency between before and after considering the beaming effect. Thus the sequence of blazars would not be eliminated in intrinsic properties, although the relation between luminosity and peak frequency is changed significantly. And the observed blazars sequence is highly associated with the intrinsic one. Therefore, a new theoretical explanation is needed for the intrinsic blazars sequence. In addition, we noticed that the intrinsic blazars sequence could change what we know about blazars, such as the black hole mass difference between BL Lcas and FSRQs.

The positive correlation between $\gamma$-ray emissions and peak frequency indicates that the synchrotron emissions are highly correlated with $\gamma$-ray emissions. From the synchrotron self-Compton (SSC) process, $\gamma$-ray emissions are produced by the inverse Compton scattering process from synchrotron emissions. So that they should be associated with each other. In addition, we suppose that $f^{\rm in}_{\gamma}$–$\nu_{\rm p}^{\rm in}$ and $\nu_{\rm p}^{\rm in}$–$\nu_{\rm p}^{\rm s}$ relations can be used to estimate the Doppler boosting factors. However, larger sample is needed to find more accurate correlations.

3.4 Conclusion

In this work, we collect 91 Fermi blazars with available Doppler factors, and investigate the correlations between intrinsic flux density and redshift for the whole sample, BL Lacs, FSRQs, ISP, and LSP respectively. Then, we estimate $\gamma$-ray Doppler factors of 63 blazars, and study the relationship between $\gamma$-ray luminosity and short variability time scale for those blazars. The observed and intrinsic correlations between the $\gamma$-ray flux density and synchrotron peak frequency are also investigated for the whole blazars sample. Our main conclusions are as follows:

1. The correlation between $f^{\rm in}_{\gamma}$ and redshift follows the theoretical relation: $\log f=-2.0\log z+{\rm const}$. When the subclasses are considered, we find that the stationary jets are perhaps dominant in LSP blazars.

2. 9 FSRQs violate the Elliot & Shapiro or the Abramowicz & Nobili relation in $L^{\rm ob}_{\gamma}$ versus $\Delta T^{\rm ob}$, while the whole blazar sample obeys to the Elliot & Shapiro and the Abramowicz & Nobili relations in $L^{\rm in}_{\gamma}$ versus $\Delta T^{\rm in}$. Thus FSRQs have stronger beaming effect than BL Lacs.

3. Strong positive correlation between $f_{\gamma}^{\rm in}$ and $\nu_{\rm p}^{\rm in}$ is found, which suggests that synchrotron emissions are highly correlated with $\gamma$-ray emissions.

Acknowledgement

This work is supported by the National Natural Science Foundation of China (U1531245, U1431112, 11203007, 11403006, 10633010, 11173009, 11403006), and the Innovation Foundation of Guangzhou University (IFGZ), Guangdong Innovation Team (2014KCXTD014), Guangdong Province Universities and Colleges Pearl River Scholar Funded Scheme (GDUPS) (2009), Yangcheng Scholar Funded Scheme (10A027S), and supported for Astrophysics Key Subjects of Guangdong Province and Guangzhou City.