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PASJ: Publ. Astron. Soc. Japan 54, 159–169, 2002 April 25
c 2002. Astronomical Society of Japan.
The Multiwavelength Doppler Factors for a Sample of Gamma-Ray Loud Blazars
Li Z HANG,1,2 Jui-Hui FAN,3 and Kwong-Sang C HENG2
1
Department of Physics, Yunnan University, Kunming, China
[email protected]
2
Department of Physics, University of Hong Kong, Pokfulam Road, Hong Kong, China
3
Center for Astrophysics, Guangzhou University, Guangzhou 510400, China
(Received 2001 April 22; accepted 2002 January 18)
Abstract
The high luminosity and rapid variability detected in γ -ray loud blazars imply that the beaming effect plays an
important role in these sources. The Doppler factor, depending on two unobservable parameters (Lorentz factor and
viewing angle), is very important for understanding the basic properties of blazars. Although the viewing angle is
unobservable, increasing direct and indirect evidence has shown up to indicate that the jet is bent, and therefore the
viewing angle also varies. In this sense, the Doppler factor, which is dependent on the viewing angle and the Lorentz
factor, should also vary. In the present paper, using available data of blazars at different (radio, optical, X-ray, and
γ -ray) wavebands and assuming that multi-waveband radiation is produced by the radiation of accelerated particles
in a jet, we discuss the properties of the Doppler factors of blazars at different wavebands. Our results indicate that
the Doppler factor of a blazar is a function of frequency. The Doppler factor decreases with frequency from radio
to X-ray bands, but increases with frequency from X-ray to γ -ray bands. The γ -ray Doppler factors found in the
present paper are correlated with those by Ghisellini et al. (1998, AAA070.159.155), suggesting that our estimation
is reasonable. Furthermore, BL Lacertae objects are found to show lower Doppler factors and higher synchrotron
peak frequencies compared with FSRQs; the promising TeV sources are BL Lacertae objects.
Key words: galaxies: active — galaxies: nuclei — jets: radiation mechanism
1. Introduction
The observed broadband spectrum of a blazar indicates a
two-component feature: low-frequency and high-frequency
components (see, e.g., Ulrich et al. 1997 for a review;
Ghisellini et al. 1998). The low-frequency component is up
to the soft X-ray region, and its spectral energy distribution
(SED) can be roughly described by a convex parabolic νFν
spectrum, which is generally believed to be produced by the
synchrotron radiation of relativistic electrons in the jet. The
high-frequency component, which includes γ -rays, is usually
believed to be produced by Compton radiation from these same
electrons. In order to understand the spectra of blazars, there
are two popular kinds of models for γ -ray emission, i.e., the external Compton (EC) models, in which soft photons are directly
from a nearby accretion disk or from disk radiation reprocessed
in some region of AGNs, e.g. broad emission line region (e.g.
Dermer et al. 1992, 1997; Sikora et al. 1994; Zhang, Cheng
1997). The other is the synchrotron self-Compton (SSC) models, in which soft photons originate as synchrotron emission
in the jet (e.g. Maraschi et al. 1992; Ghisellini 1993; Bloom,
Marscher 1996). SSC models are responsible for some blazars,
and thus EC models would be required for some blazars. At
this moment, we would like to explain the overall spectrum
by a combination of SSC and EC models. In these models, the
Doppler factor, a key physical parameter, is assumed to be constant for all radiation in the EC model and in the SSC model.
Therefore, the beaming effect on the radiation spectrum is the
same for all bands. Because the Doppler factor depends on the
bulk Lorentz factor of the jet and the viewing angle, a constant
Doppler factor requires that the Lorentz factor of the jet and
the view angle are the same for the radiation at different bands.
Generally, this is not true; the emissions from different regions
may result in different Lorentz factors (Ghisellini, Maraschi
1989) and various viewing angles are quite possible (Ciliegi
et al. 1995). In fact, some observations in the radio to X-ray
bands require that the jets change the angles with respect to the
line of sight (see below). The various Lorentz factors and viewing angles will certainly result in various Doppler factors. For
BL Lac objects, which constitute about 25% of the detected
high-energy γ -ray sources by EGRET (Thompson et al. 1995,
1996; Hartman et al. 1999), the observed differences in the
spectral properties between radio-selected and X-ray selected
BL Lac objects can be understood if the Doppler factors in different bands are different (Fan et al. 1993, 1997; Fan, Xie 1996;
Georganopoulos, Marascher 1999). Therefore, a possible case
is that the Doppler factor is different at various bands and the
observed multi-waveband radiation from blazars is a combination of the beaming effect and intrinsic radiation.
Multi-frequency VLBI images of blazars show some interesting features. In 4C 39.25, three radio components (a, b,
and c) are observed. The interesting behaviour of component b
suggests that a shock wave travels along a bent relativistic jet
(G´omez et al. 1994; Alberdi et al. 2000, and reference therein).
For PKS 0735 + 178, to fit the kinematical properties of the superluminal components, Baath et al. (1991) suggested that a
bending of the jet is required during the inner milliarcsecond,
while direct evidence of a curved structure in the inner jet of
0735 + 178 was presented by Kellermann et al. (1998). Later,
G´omez et al. (1999) confirmed the existence of a very twisted
160
L. Zhang, J.-H. Fan, and K.-S. Cheng
structure in the inner region of the jet in 0735 + 178. The direction change of the radio components is also found in other
sources, such as 3C 345, 3C 120, and 3C 279 (von Montigny
et al. 1995). G´omez et al. (1999) also pointed out that many
jets in BL Lacertae objects and quasars appear to be curved.
It has been proposed that a binary black hole model can
explain the orbital periods of a few tens of years in quasars,
such as the BL Lacertae object OJ 287 (Begelman et al. 1980;
Sillanp¨aa¨ et al. 1988). Such a model is also used to explain the
observational properties in both the radio and optical bands of
3C 273 (Abraham, Romero 1999; Romero et al. 2000a) and the
observational properties of Mkn 501 (Villata, Raiteri 1999).
To explain the symmetric optical light curve in 3C 345, a
jet which is rotating about its axis due to the conservation of
angular momentum carried by the accreted plasma is proposed
(Schramm et al. 1993). In such a scenario, density perturbations within the jet travel on helical paths. If the fluid moves
relativistically, the emission is beamed into the forward direction. Because the direction of the boosted emission changes,
its inclination to the line of sight of the observer also changes
(e.g. Camenzind, Krockenberger 1992; Wagner et al. 1995;
Chiaberge, Ghisellini 1999).
In addition, it is also possible that relativistic particles move
along the radius in the jet cone. Although the particles on the
shock front should be identically accelerated, the moving direction is different to the observer. Thus, the synchrotron emissions from the same shock front are the same in speed, but different in viewing angle. Therefore, the emissions at different
frequencies will have different Doppler factors.
In principle, the Doppler factor can be obtained if the
Lorentz factor and the viewing angle are known. Unfortunately, the two parameters are unobservable. Therefore, it can only
be obtained by other methods. There are some methods to
estimate the Doppler factor: (i) the Doppler factor (δssc ) can
be derived by using VLBI observations combined with X-ray
flux density in the SSC model (see Ghisellini et al. 1993); (ii)
the Doppler factor (δeq ) is estimated using single-epoch radio
data by assuming that the sources are in equipartition of energy
between radiating particles and the magnetic field (Readhead
1994; Guerra, Daly 1997) and (iii) the Doppler factor (δvar )
is estimated using radio flux density variations (L¨ahteenm¨aki,
Valtaoja 1999). Furthermore, the lower limits of the Doppler
factor have been estimated for the γ -ray-loud blazars (Mattox
et al. 1993; Dondi, Ghisellini, 1995; Cheng et al. 1999a; Fan
et al. 1999; Ghisellini et al. 1998). Since the Doppler factors
were estimated by L¨ahteenm¨aki and Valtaoja (1999), we describe their method in a little more detail. That method is based
on the total flux density flares associated with new VLBI components emerging from the AGN core (Valtaoja et al. 1999),
which gives an observed variability brightness temperature in
the source frame
λ2 Smax √
Tb,var = 5.87 × 1021 h−2 2 ( 1 + z − 1)2 ,
τobs
where λ is the observed wavelength in meters, z is the redshift, Smax is the maximum amplitude of the outburst in Jy,
and τobs = dt/d(ln S) is the observed variability timescale in
days (e.g. Valtaoja et al. 1999). The variability Doppler factors
can be obtained by comparing Tb,var and the intrinsic brightness
[Vol. 54,
temperature, Tb,int , which can be replaced by the equipartition
brightness temperature, Teq = 5 × 1010 K during the flare state.
Therefore,
1/3
Tb,var
δvar =
.
5 × 1010 K
In this paper, we consider whether the Doppler factors at different frequencies are different. To do so, we use the Doppler
factors for active galactic nuclei given by L¨ahteenm¨aki and
Valtaoja (1999). Then we analyze the properties of the Doppler
factors at different wavebands by using the data (e.g. Cheng
et al. 2000; Fossati et al. 1998; Ter¨asranta et al. 1998) and assuming that the multi-waveband intrinsic radiation (S int ) from
a blazar is non-thermal and that the observed data (S obs ) are
boosted. They are associated with each other, S obs = δ β S int (1 +
z )α−1 (here Sν ∝ ν −α ). The value of β depends on the shape of
the emitted spectrum and the detailed physics of the jet (Lind,
Blandford 1985); β = 3 + α is for a moving sphere and β = 2 + α
is for the case of a continuous jet.
2.
Observation Data and Analysis Results
The observed data used here are as follows. For the Doppler
factors, L¨ahteenm¨aki and Valtaoja (1999) have estimated the
Doppler factors of 81 AGNs using radio flux variation monitoring data at 22 and 37 GHz, in which 31 AGNs are γ -ray
loud blazars, 20 of which are included in the paper by Fossati
et al. (1998), who obtained the average flux densities in the radio and optical bands. In the present paper, the averaged radio
flux densities of these blazars are taken from Ter¨asranta et al.
(1998). The optical data are from the paper of Fossati et al.
(1998) for the 20 objects. As for the optical data of the remaining 11 objects and the X-ray data, we use the observed
multi-waveband data of the blazars complied by Cheng et al.
(2000), in which the flux densities in both low and high states
are given. The average flux density is obtained by averaging
the values in the low and high states. For the average flux densities and spectral index of these blazars in γ -ray band, we use
the data given by Hartman et al. (1999). Therefore, we have a
sample including 31 blazars shown in table 1. To analyze them,
we need to convert the gamma-ray photon flux into flux density
at a given energy. The γ -ray flux density at average energy E
is
αγ − 1
Sγ (E) ≈ 66
Fγ (> 100 MeV)
1 − 100αγ −1
−(αγ −1)
E
pJy,
(1)
×
100 MeV
where αγ is the differential spectral index of γ -rays and
Fγ (> 100 MeV) is the γ -ray flux above 100 MeV in units of
10−7 cm−2 s−1 . In deriving the above equation, we have assumed that the γ -ray energy range is from 100 MeV to 10 GeV
and that the average energy E is in units of 100 MeV.
In the beaming model, the observed flux density can be written in the form
Siobs = δiβ Siint (1 + z )αi −1 ,
(2)
where Siobs and Siint are the observed and intrinsic flux densities
No. 2]
Doppler Factors of Gamma-Ray Loud Blazars
161
Table 1. Sample of 31 blazars.
Source
Class
z
SR
SO
SX
Fγ
αγ
log νp
0202 + 149
0219 + 428
0234 + 285
0235 + 164
0336−019
0440−003
0446 + 112
0458−020
0528 + 134
0735 + 178
0804 + 499
0827 + 243
0836 + 710
0851 + 202
0954 + 556
0954 + 658
1156 + 295
1219 + 285
1222 + 216
1226 + 023
1253−055
1406−076
1510−089
1606 + 106
1611 + 343
1633 + 382
1739 + 522
1741−038
2200 + 420
2230 + 114
2251 + 158
Q
B
Q
B
Q
Q
Q
Q
Q
B
Q
Q
Q
B
Q
B
Q
B
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
B
Q
Q
1.202
0.444
1.213
0.940
0.852
0.844
1.207
2.286
2.070
0.424
1.433
2.050
2.172
0.306
0.901
0.368
0.729
0.102
0.435
0.158
0.538
1.494
0.361
1.227
1.404
1.814
1.375
1.054
0.069
1.037
0.859
2.61
0.97
2.63
2.51
2.56
1.33
1.60
2.73
5.24
2.21
1.44
1.17
1.51
2.20
1.05
0.72
1.65
0.56
1.88
35.12
16.35
1.17
2.88
1.28
3.44
1.99
1.61
3.68
2.39
2.59
10.31
0.01
4.034
0.14
2.969
0.245
0.108
0.04
0.11
0.375
1.783
0.38
0.33
1.938
2.639
0.405
1.835
9.10
3.047
0.19
29.834
2.073
0.09
1.276
0.15
0.553
0.401
0.15
0.753
5.527
0.701
1.407
0.05
1.558
0.14
1.70
0.253
0.109
···
0.07
0.951
0.248
0.20
0.22
0.819
1.063
0.112
0.158
0.65
0.409
0.20
12.074
1.246
···
0.718
0.04
0.194
0.258
0.13
0.213
0.936
0.486
1.082
0.87
1.87
1.38
2.59
1.51
1.63
1.49
1.12
9.35
1.64
1.07
2.49
1.02
1.06
0.91
0.60
0.75
1.15
1.39
1.54
7.42
2.74
1.80
2.50
2.65
5.84
1.82
1.17
1.11
1.92
5.37
2.23
2.01
2.53
1.85
1.84
2.37
2.27
2.45
2.46
2.60
2.15
2.42
2.62
2.03
2.12
2.08
1.98
1.73
2.28
2.58
1.96
2.29
2.47
2.63
2.42
2.15
2.23
2.42
2.60
2.48
2.21
···
15.01
···
15.21
13.06
13.29
···
···
13.55
14.18
···
···
15.27
13.40
14.52
14.18
···
14.50
···
14.51
13.19
···
13.36
···
15.01
14.87
···
13.85
14.17
13.51
13.52
Note. SR , SO and SX are flux densities at radio, optical and X-ray bands in units of Jy, mJy and µJy
respectively. Fγ is the γ -ray flux above 100 MeV with a differential spectral index, αγ , in units of
10−7 cm−2 s−1 .
at the ith waveband, respectively. δi is the Doppler factor at the
ith waveband. The factor (1 + z )αi −1 represents a K-correction,
where αi is the spectral index (Si ∝ νi−αi ).
In principle, the intrinsic flux densities at different wavebands can be estimated if the Doppler factor of a blazar is
assumed to be constant. However, a constant Doppler factor requires the same region of the broadband emission and
the same view angle. Although simultaneous observations in
some cases (Wagner et al. 1995; McHardy 1996; Romero et al.
2000b; and references therein) suggest that the emissions are
from the same regions, this is not at all the case for most objects; one possible case is that the broadband emissions of the
blazar is produced in different emission regions and high energy emission is produced in the region nearer to the central
engine (Blandford, Levinson 1995; Levinson, Blandford 1995;
Levinson 1996). If the Doppler factors are the same in different
frequencies, then equation (2) gives
int
,
Siobs νiαi (1 + z )1−αi = δiβ S0,i
(3)
int −αi
ν . In this sense, we know that the ratio of
where Siint = S0,i
S obs ν α (1 + z )1−α at any two frequencies, say νi and νj , should
be only a function of δ, αi , and αj , namely
Ri,j =
Siobs νiαi (1 + z )1−αi
= δ (αi −αj ) ,
α
Sjobs νj j (1 + z )1−αj
which suggests that the ratio between the radio and X-ray
bands should be less than unity, while that between the γ -ray
and X-ray bands should be unity. This can be tested by observations. When the data listed in table 1 were adopted to
this calculation, we found that the ratios are RRadio,X-ray =
9.29 ± 3.21, ROptical,X-ray = 3.33 ± 0.59, and RGamma,X-ray =
173.4 ± 45.40, respectively. Therefore, the Doppler factors
are likely not to be constant at different frequencies. This implies that either the viewing angle is changeable, as observed
in the radio observations and shown in the theoretical discussions (Kellermann et al. 1998; G´omez et al. 1999; Wagner et al.
1995), or that the Lorentz factor is changeable, as proposed
by Ghisellini and Maraschi (1989), or the spectral index, α,
varies because of further acceleration and the radiation loss of
162
L. Zhang, J.-H. Fan, and K.-S. Cheng
[Vol. 54,
Fig. 1. Spectral energy distributions of 20 blazars. The filled circles represent the observed data and the solid curves are the fit results using equation (14).
the relativistic particles.
We now relax the limitation of the constant Doppler factor.
Using equation (2), the ratio of two different waveband flux
densities gives
1/βi
Sjint
Siobs
βj /βi
η
δi = δj (1 + z )
,
(4)
Sjobs
Siint
where η = (αj − αi )/βi , βi = 3 + αi or βi = 2 + αi , depending
on the physics of the jet. From equation (4), in order to estimate the Doppler factor, we need to know the ratio of the
intrinsic energy densities between different wavebands, which
is unknown. Therefore, we make the following assumptions.
The relativistic particles satisfy a power-law distribution with
a spectral index, p. The radiation from a blazar in the radio to
X-ray regions is produced by the synchrotron radiation of these
No. 2]
Doppler Factors of Gamma-Ray Loud Blazars
particles in the jet, which gives S(ν) ∝ ν −(p−1)/2 for ν ≤ νp and
S(ν) ∝ ν −p/2 for ν > νp , where νp is the peak frequency (see
Ghisellini et al. 1993). The γ -rays are produced by Compton
scattering of the same particles and S(ν) ∝ ν −p/2 . Under above
assumptions, we have
10(2.87p−3) νR(1−p)/2 νp−0.5 if νp < νO
SRint
(5)
=
SOint
10(2.87p−2.87) ν (1−p)/2
if νp > νO ,
SRint
SXint
= 10(4.19p−3)
and
SRint
= 10(6.69p−3)
Sγint
R
νX
νR
p/2 E
νR
νp
νR
p/2 −0.5
νp
νR
,
(6)
−0.5
,
(7)
where SRint , SOint , SXint , and Sγint are the intrinsic flux densities in
the radio (νR in GHz), optical (V band), X-ray (νX in KeV)
and γ -ray (E in units of 100 MeV) bands respectively, νp (in
units of 1015 Hz) is the peak frequency. Therefore, we have the
following relations for the optical Doppler factor:
 (5 + p)/(6 + p)

δ
(1 + z )−1/(6 + p)

 R



2/(6 + p)



SOobs (mJy)
× a1 obs
, if νp ≤ νO
(8)
δO ≈
SR (Jy)



2/(5 + p)



S obs (mJy)

 δR a2 O
, if νp > νO

SRobs (Jy)
with the X-ray Doppler factor being
δX ≈ δR(5 + p)/(6 + p) (1 + z )−1/(6 + p)
p/2 −1/2
νp
(4.19p−9) νX
× 10
νR
νR
2/(6 + p)
SXobs (µJy)
× obs
SR (Jy)
(9)
and the γ -ray Doppler factor being
δγ ≈ δR(5 + p)/(6 + p) (1 + z )−1/(6 + p)
p/2 −0.5
νp
(6.69p−15) E
× 10
νR
νR
2/(6 + p)
Sγobs (pJy)
× obs
SR (Jy)
(10)
for β = 3 + α, where a1 = 10(2.87p−6) νR(1−p)/2 νp−0.5 and a2 =
10(2.87p−5.87) νR(1−p)/2 . Further
 (3 + p)/(4 + p)

δR
(1 + z )−1/(4 + p)






 S obs (mJy) 2/(4 + p)

× b1 Oobs
, if νp ≤ νO ,
δO ≈
(11)
SR (Jy)



2/(3 + p)



SOobs (mJy)


δ
, if νp > νO
b
 R 2 obs
SR (Jy)
with the X-ray Doppler factor being
δX ≈ δR(3 + p)/(4 + p) (1 + z )−1/(4 + p)
p/2 −1/2
νp
νX
× 10(4.19p−9)
νR
νR
2/(4 + p)
SXobs (µJy)
× obs
SR (Jy)
163
(12)
and the γ -ray Doppler factor being
δγ ≈ δR(3 + p)/(4 + p) (1 + z )−1/(4 + p)
p/2 −0.5
νp
E
× 10(6.69p−15)
νR
νR
2/(4 + p)
Sγobs (pJy)
× obs
SR (Jy)
(13)
for β = 2 + α, where b1 = 10(2.87p−6) νR(1−p)/2 νp−0.5 and b2 =
10(2.87p−5.87) νR(1−p)/2 .
Using the Doppler factor at the radio band given by
L¨ahteenm¨aki and Valtaoja (1999), the flux densities at different wavebands given in table 1, and equations (8)–(13), we can
estimate the Doppler factors at the optical, X-ray and γ -ray
bands, respectively, if the frequency, νp , is known. Now we
estimate the νp .
2.1. Synchrotron Peak Frequency
In order to determine the position of the peak of the synchrotron emission in a γ -ray loud blazar, we fitted the data
for each source in a log ν − log νSν plot using the following
parabolic fits as did Landau et al. (1986), Sambruna et al.
(1996), and Fossati et al. (1998):
log(νSν ) = A(log ν)2 + B log ν + C
(14)
where A, B, and C were obtained from the fitting (cf. figure 1).
The data adopted for the fitting were from the paper of Fossati
et al. (1998). When the parabolic fitting was employed to the
relevant data of the sources, the peak frequencies, which can
be expressed as log νp = −B/2A, were obtained, as shown
in table 1 for the 20 sources with available data. The average peak frequency is log νp = 14.11 ± 0.16 for all 20 sources,
log νp = 14.37 ± 0.23 for the lower synchrotron peak frequency
BL Lacertae objects and log νp = 13.96 ± 0.21 for flat spectral
radio sources (FSRQs), respectively. In a following analysis,
we will adopt the average peak frequency of FSRQs for the
remaining 11 objects, since they are all FSRQs.
2.2. Doppler Factors
With the relevant peak frequencies and the data in the radio, optical, X-ray, γ -ray bands, and the radio Doppler factors,
we could estimate the Doppler factors in other bands. The observed data are K-corrected using
Si = Siobs (1 + z )(αi −1) .
Adopting p = 2.2, the corresponding Doppler factors were
obtained with the averaged values being as follows: δO =
5.74 ± 0.65, δX = 4.69 ± 0.54 and δγ = 11.22 ± 1.04 for
β = 3 + α case, and δO = 4.78 ± 0.59, δX = 3.58 ± 0.45 and
δγ = 11.16 ± 1.06 for β = 2 + α case, respectively, where
164
L. Zhang, J.-H. Fan, and K.-S. Cheng
[Vol. 54,
Table 2. Doppler factors of 31 blazars at different wavebands and the SSC frequencies.
Source
δR
δO
δX
δγ
log νSSC
0202 + 149
0219 + 428
0234 + 285
0235 + 164
0336−019
0440−003
0446 + 112
0458−020
0528 + 134
0735 + 178
0804 + 499
0827 + 243
0836 + 710
0851 + 202
0954 + 556
0954 + 658
1156 + 295
1219 + 285
1222 + 216
1226 + 023
1253−055
1406−076
1510−089
1606 + 106
1611 + 343
1633 + 382
1739 + 522
1741−038
2200 + 420
2230 + 114
2251 + 158
5.93
1.99
7.29
16.32
19.01
11.46
4.90
17.80
14.22
3.17
26.21
15.46
10.67
18.03
4.63
6.62
9.42
1.56
8.16
5.71
16.77
8.26
13.18
9.32
5.04
8.83
12.12
8.92
3.91
14.23
21.84
0.67
1.54
1.85
8.10
7.66
4.36
1.04
3.35
4.43
2.27
8.92
5.64
5.82
14.88
2.13
6.11
10.65
2.09
2.68
3.38
7.27
2.27
8.40
2.93
1.19
2.33
3.35
3.85
3.88
7.07
8.13
1.04
0.97
1.72
3.96
4.64
2.96
···
2.69
5.34
1.10
6.73
4.60
3.63
10.68
0.94
3.16
4.22
1.05
2.53
2.53
5.08
···
6.07
1.77
0.77
1.99
2.97
3.20
2.79
5.53
9.08
7.07
2.18
8.36
8.90
18.86
16.50
8.36
17.62
19.15
3.83
19.64
18.47
6.06
16.44
4.04
6.37
12.50
2.17
12.75
2.29
15.71
17.08
15.73
18.06
3.66
8.56
18.87
7.00
4.12
15.75
18.81
23.33
22.17
22.50
23.44
20.20
21.27
22.90
22.83
20.95
21.08
22.95
22.90
22.89
20.65
22.08
21.61
20.90
22.16
23.02
20.85
20.46
23.25
20.87
23.15
23.03
23.40
23.05
20.78
21.61
20.86
21.06
the average γ -ray energy of 31 blazars is 360 ± 20 MeV. It is
clear that the Doppler factors are not the same at different frequencies for the both cases (β = 2 + α and β = 3 + α) with the
Doppler factor at the X-ray band being minimum on average.
The averaged radio Doppler factor is δR = 10.7 ± 1.1 for the
sample considered in this paper. The Doppler factors obtained
under the condition of p = 2.2 and β = 2 + α are shown in table 2
and the average Doppler factors are plotted against frequency
in figure 2, which shows that the Doppler factors of most
blazars decrease with frequency from radio to X-ray bands, but
increase from X-ray to γ -ray bands. This frequency-dependent
tendency is not caused by the selection effect, because (1) the
sample was compiled based on their γ -ray detection and radio Doppler factors, (2) the frequency-dependent tendency obtained in the present paper from X-ray to radio band is consistent with the proposal by Ghisellini and Maraschi (1989), and
it is therefore reasonable that the X-ray Doppler factor is lower
than the radio Doppler factor, and (3) γ -ray observations indicated that the γ -ray emissions are strongly beamed, which
implies that the γ -ray Doppler factor is higher. In this sense,
there should be a tendency that the Doppler factor increases
from the X-ray to γ -ray bands.
In deriving equations (8) to (13), we used two assumptions:
(i) that the emission from radio to γ -rays is produced from the
accelerated electrons with a single distribution through SSC
mechanism and (ii) that the X-ray emission is due to the synchrotron radiation above the synchrotron peak frequency. In
such assumptions, the ratio of the intrinsic energy densities between different wavebands are estimated [see equations (5) to
(7)], and the variation of Doppler factor with frequency shown
in figure 2 is obtained. It should be pointed out, however,
that modeling based on the standard SSC model of the single electron distribution often fails to reproduce the spectrum
in the radio waveband, where a constant Doppler factor is assumed (Fossati et al. 1998; Ghisellini et al. 1998). Therefore,
there is a possibility that the emissions, especially in the radio waveband, are the integrated emission from different region in the jet. Furthermore, the intrinsic energy density ratio between X-rays and γ -rays for each blazar is estimated
based on assumption (ii). However, this assumption may not
be true for some blazars with high radio luminosities. In fact,
Fossati et al. (1998) derived the average spectral energy distributions (SEDs) of a blazar sample binned according to the
radio luminosity. The SEDs define a spectral sequence from
No. 2]
Doppler Factors of Gamma-Ray Loud Blazars
165
Doppler factor decreases with frequency from radio to X-ray
bands, but increases from X-ray to γ -ray bands (see figure 2).
Our result suggests that the emissions from a blazar at different
wavebands are produced in different emission regions and/or in
the same emission region with a different viewing angle. Fan
et al. (1993) found a relation between the Doppler factor and
the frequency based on an analysis of observed X-ray, optical
and radio flux densities from BL Lac objects, and expressed it
1 + 1 log(ν /ν)
Fig. 2. Variation of the Doppler factor with frequency. The filled circles with error bars represent the average Doppler factors at the radio, optical, X-ray and γ -ray bands and the solid curve is given by
equation (15).
red to blue. For most red blazars with their radio luminosities > 1045 erg s−1 , the X-ray emissions are thought to be due
to inverse Compton emission. In this case, the spectral index of X-rays is ∼ (p − 1)/2 and the Doppler factor, (δX )IC ,
at X-ray band should be changed. In order to account for
this, we use (δX )syn to represent the Doppler factor at the Xray band given by equation (9) or equation (12) and consider
(δX )IC /(δX )syn . This ratio is {νX /[δR (1 + z )νp ]}1/(6 + p) for β =
3 + α and {νX /[δR (1 + z )νp ]}1/(4 + p) for β = 2 + α. Therefore,
(δX )IC > (δX )syn . For 0336−019, whose break frequency is
minimum in our sample (see table 1), (δX )IC /(δX )syn ∼ 2.2 [or
(δX )IC ∼ 10.20] for β = 3 + α and ∼ 2.9 [or (δX )IC ∼ 13.46] for
β = 2 + α. If we take this effect into our estimate of the Doppler
factors at the X-ray band for the most luminous blazars, the average Doppler factor at the X-ray band will increase, though
δX is also less than δR and δγ .
3. Discussion
In the EGRET detected sources, the detected short time scale
variability and the high flux suggested that the γ -ray emission
is beamed and that the relativistic jets are involved in the emission. The beaming effect has also been supported by the fact
that some γ -ray loud AGNs show superluminal radio components. In the 3rd catalogue, about a quarter of (14 out of 60) the
confirmed γ -ray loud sources show superluminal radio components (Hartman et al. 1999; Fan et al. 1997). However, not all
of the superluminal radio sources were detected with EGRET,
nor did all of the EGRET detected sources show superluminal radio components, although the superluminal radio sources
detected with EGRET are not different from those undetected
with EGRET (von Montigny et al. 1995). Why is this the situation?
3.1. Superluminal Motion
According to our analysis, the Doppler factor of a blazar is
a function of the frequency (see table 1). On the average, the
O
, where δO and νO are the optical Doppler
as δ(ν) ∼ δO 8
factor and the optical frequency, respectively. This relation is
valid for the radiation from radio to X-ray bands of Seyfert
galaxies as well as FSRQs and BL Lacertae objects (Fan 1997;
Mei et al. 1999). Thus, this relation should be satisfied for
the radiation from radio to X-ray emissions of blazars. The
relation suggests that the Doppler factors in the radio (δR ), optical (δO ), and X-ray (δX ) bands satisfy δR > δO > δX and the
relation in Fan et al. (1993) can be expressed in the form of
δ(ν) ∼ δX1 + 0.2 log(νX /ν) , from the X-ray to γ -ray region. Cheng
et al. (1999b), who found that the relation for the Doppler factor with frequency can be expressed by δ(ν) ∼ δX1−b log(νX /ν)
for the X-ray to γ -ray bands, used b = 0.28 to fit the X-ray
and γ -ray spectra of 3C 279. The relation suggests that the
Doppler factor increases with frequency in the X-ray and γ ray regions. Similar results have been obtained by other authors. To avoid the photon–photon collisions and to let the
γ -rays escape, Protheroe et al. (1998) found that the Doppler
factors in the γ -ray region are frequency-dependent with the
Doppler factor increasing with frequency in the γ -ray region.
That the X-ray Doppler factors are lower in our analysis is consistent with the proposal by Ghisellini and Maraschi (1989). In
some analysis, no Doppler correction is requested for X-ray
data, suggesting that the X-ray Doppler factors is small (see
Maccagni et al. 1989; Fan et al. 1994; Fan, Xie 1996).
From our estimation, we also can say that there is such a
tendency of the Doppler factor decreasing with frequency from
radio to X-ray, but the γ -ray Doppler factor is higher than the
X-ray one. The averaged values of the estimated Doppler factors satisfy following relation:

 δX1 + 0.15 log(νX /ν) if ν < νX
δν =
(15)
 δ 1−0.18 log(νX /ν) if ν > ν
X
X
for the β = 2 + α case and

 δX1 + 0.09 log(νX /ν) if
δν =
 δ 1−0.11 log(νX /ν) if
X
ν < νX
ν > νX
for the β = 3 + α case.
Comparing the Doppler-factor tendency and the discussions
in Fan et al. (1993) and Cheng et al. (1999b), we can see that
the Doppler-frequency relation in the β = 2 + α case is near to
those presented in our previous papers (Fan et al. 1993; Cheng
et al. 1999b). This suggests that a continuous jet is more reasonable than a moving sphere, as we concluded based on the
Doppler factors and a polarization analysis in 1997 (Fan et al.
1997). Recently, Jose-Luis G´omez (private communication)
pointed out that a continuous jet should be the real situation.
From the definition, the Doppler factor at the ith waveband
166
is
−1
δi = Γi − (Γ2i − 1)1/2 cos θi
,
L. Zhang, J.-H. Fan, and K.-S. Cheng
[Vol. 54,
(16)
where Γi is the Lorentz factor corresponding to the bulk velocity of the jet and θi is the angle between the line of sight
and the bulk velocity (viewing angle). Generally, a frequencydependent Doppler factor indicates that the Lorentz factors
and/or viewing angles at different wavebands are different.
In the relativistic beaming model, for small θ , δi ∝ Γi , the
frequency-dependent Doppler factor requires that the Lorentz
factor of the jet is a function of the distance to the central engine, as proposed by Ghisellini and Maraschi (1989). If the
emission region in some frequency range (e.g. X-ray and γ -ray
bands) is the same (i.e. co-spatial), then different Doppler factors at different wavebands are due to various viewing angles.
It should also be pointed out that other reasonable values of the
spectral index of the relativistic particles (p) can produce similar results, which show that the Doppler factor decreases from
radio to X-ray, but increases from the X-ray to γ -ray bands.
The result presented in figure 2 clearly shows that Doppler
factor is changing with frequency, with the minimum being in
the X-ray region and that the γ -ray and radio Doppler factors are comparable on the average. However, for individual
source this is not the total situation. A source, whose minimum Doppler factor is at a frequency lower than the X-ray
region, would tend to show that the γ -ray Doppler factor is
higher than the radio Doppler factor; this source would possibly be detected in the γ -ray region, but would show no superluminal radio components. On the contrary, for a source, whose
minimum Doppler factor is at a frequency higher than the Xray region, would tend to show that the γ -ray Doppler factor
is lower than the radio Doppler factor; therefore, it would possibly show a superluminal radio component but could not be
detected in the γ -ray region. In this sense, the superluminal
radio components and the γ -ray detection do not necessarily
correspond to each other.
3.2. Doppler Factor Comparison
The Doppler factors in the γ -ray region are estimated for
many γ -ray loud blazars (see Dondi, Ghisellini 1995; Mattox
et al. 1993; Cheng et al. 1999a; Fan et al. 1999; Ghisellini et al.
1998). In Ghisellini et al. (1998), the γ -ray Doppler factors
are as high as 20, which is comparable to the radio Doppler
factors. In our analysis, the derived γ -ray Doppler factors are
in the range of 3 to 21. When we plotted a diagram of our
results against those by Ghisellini et al. (1998), we found that
these two sets of values are very closely correlated with a correlation coefficient, r = 0.77, for the left-hand part of sources
with δOurs < 11. The value of the Doppler factors of the points
in the right-hand part are perhaps underestimated in the paper
by Ghiselliini et al. (1998), since the Doppler factors are constrained so as not to exceed a value of 20–25 in their paper, and
thus the Doppler factor values are consistent with the observed
superluminal speeds. It is quite possible that the variability
time scale, 1 day for those sources, has been overestimated.
For example, for 2251 + 158, a 0.08 d variability time scale (see
Dondi, Ghisellini 1995) was detected, which would result in
the Doppler factor being 1.5-times as high as the Doppler factor estimated from a 1 day time scale; namely, the Doppler
Fig. 3. Comparison of our Doppler factors (δOurs ) with those (δG ) given
by Ghisellini et al. (1998). For 1253−055, 1156 + 295, and 2251 + 158,
the Doppler factors estimated by Ghisellini et al. (1998) become larger
if their observed shorter time scales are used (we label their increases
as the arrows; also see text.).
factor would be 15, as we show in figure 3 by the arrow. For
3C 279, if the doubling time of 6 h (see Cheng et al. 1999a; Fan
et al. 1999) is taken into account, the Doppler factor would be
1.2-times as high as that estimated from the 1 day time scale,
which is also shown in figure 3 by an arrow. For 1156 + 295,
the 0.46 d time scale also results in a larger Doppler factor, as
shown in figure 3 by an arrow. In fact, it is understandable for
the sources in the right-hand part of figure 3 to have shorter
variability time scales, because the derived γ -ray Doppler factors are comparable with the radio ones, and to some extent the
higher radio Doppler factors are associated with shorter time
scales; therefore, the higher Doppler factors suggest a shorter
time scale. In this sense, if a shorter time scale is used, the
points in the right-hand part of the figure would move up and
make the linear correlation be closer. This correlation also implies that our estimation of the γ -ray Doppler factors is reasonable. We can also expect a shorter-than-one-day time scale for
the sources located in the right-hand part.
3.3. Correlation between the Doppler Factor and the Peak
Frequency
From an observational point of view, there are some differences between BL Lacertae objects and flat spectra radio
quasars (FSRQs), although they both show many common
properties. In the present work, some differences were shown
to exist between those two subgroups of active galactic nuclei. From a parabolic fitting, the average peak frequency
of BL Lacertae objects, log νp = 14.37, is higher than that of
FSRQs, log νp = 13.96, which is consistent with the results by
Fossati et al. (1998). Besides, the ratios of the γ -ray luminosity to the peak synchrotron luminosity, logνγ Sγ /νp Sp , were
calculated for BL Lacertae objects and FSRQs, respectively,
giving average values of −0.06 ± 0.07 and 0.69 ± 0.12 for the
BL Lacertae objects and FSRQs. If the γ -ray emissions are
from synchrotron self-Compton (SSC) emission, the inverse
Compton emission would dominate the synchrotron emission
in FSRQs, while the Compton emissions would be comparable to the synchrotron emission in the BL Lacertae objects. On
No. 2]
Doppler Factors of Gamma-Ray Loud Blazars
167
Fig. 5. Relation between the γ -ray flux density in pJy against the γ -ray
Doppler factor. The filled points are for BL Lacertae objects and the
open circles are for FSRQs.
Fig. 4. Relation between the synchrotron peak frequency against the
γ -ray Doppler factor. The filled points are for BL Lacertae objects and
the open circles are for FSRQs.
the other hand, it is possible that the inverse Compton emission
in the BL Lacertae objects would peak at the higher-than-GeV
regions; therefore, the γ -ray emissions used to calculate the
ratios are not the peak luminosity. If this is true, one cannot
expect TeV emissions from FSRQs, and the promising TeV
blazars should be BL Lacertae objects. Fortunately, up to now,
the known TeV γ -ray blazars are all BL Lacertae objects (see
Catanese, Weekes 1999).
From the synchrotron peak frequency and the derived γ ray Doppler factor, one can find that both parameters are anticorrelated, as plotted in figure 4. In fact, equation (6) implies that the γ -ray Doppler factor is anti-correlated with the
peak frequency, log δγ = −0.125 log νp + c, if other parameters are constant. If the three sources that deviate from the
other 17 sources are excluded, a close anti-correlation, logδγ =
−0.49logνp + 7.24 with r = 0.91 and p = 2.5 × 10−6 can be obtained. The difference between the expected correlation and the
obtained correlation is due to the fact that the obtained Doppler
factors are associated with not only the peak frequency, but
also with the radio Doppler factor, the redshift, the observed
radio and γ -ray fluxes, and the radio and γ -ray spectral indices. Figure 4 also indicates that BL Lacertae objects have a
lower Doppler factor and a higher peak frequency as compared
with FSRQs. This finding is consistent with the above results
concerning the ratio log νγ Sγ /νp Sp and suggests that the lower
ratio of BL Lacertae objects is due to the fact that BL Lacertae
objects have a higher inverse Compton frequency than FSRQs.
Equations (7) and (10) imply that there are possibly systematic effects in which the higher is the flux in γ -ray band, the
higher are the Doppler factors. To investigate this we give a
plot of the γ -ray Doppler factor and the γ -ray flux density in
figure 5, which indicates that there is no correlation between
them.
It is known that the intensity of the Compton component
with respect to the Synchrotron component is different from
objects to objects (Kubo et al. 1998) from simultaneous observations. Perhaps different Doppler factors are the reason. We
now consider the location of the Synchrotron Self-Compton
frequency. To do so, we can use formula (4) derived by Kubo
et al. (1998), namely
2
Lsync νSSC
4
12 Lsync
δ ≥ 1.6 × 10 3 2
,
4
c ∆t
LSSC νsync
where Lsync and LSSC are the synchrotron and Compton luminosities in units of erg s−1 , νsync and νSSC are the synchrotron
and the Compton frequencies in Hz, c is the speed of light in
units of cm s−1 , and ∆t is the variability time scale in units of
second. Assuming νsync = νp and Lsync = Lp , while adopting 1
day for the variability time scale in the γ -ray region, as did by
Ghisellini et al. (1998), we can estimate the νSSC by means of
the obtained γ -ray Doppler factor and the γ -ray luminosity; the
resulting νSSC are listed in the last column in table 2. The difference between νsync and νSSC falls in the range of 2.2 × 106–9 Hz
in the present sample.
168
4.
L. Zhang, J.-H. Fan, and K.-S. Cheng
Conclusion
In the present work, using the available multiwavelength
data, the synchrotron peak frequency was obtained, based on
the known radio Doppler factors and the assumption that the
intrinsic spectra of blazars are from the SSC model and the observed emissions are boosted; we thus derived a relation for
the Doppler factors, and obtained them in the optical, X-ray
and the γ -ray regions. Generally, those Doppler factors are
found to be a function of the frequency, with the Doppler factor decreasing with frequency from the radio to X-ray regions,
and then increasing from the X-ray to γ -ray regions. The γ ray Doppler factors found in the present paper are correlated
with those by Ghisellini et al. (1998), suggesting that our es-
[Vol. 54,
timation is reasonable. In addition, BL Lacertae objects are
found to show lower Doppler factors and a higher synchrotron
peak frequency as compared with the values for FSRQs, and
the promising TeV sources are BL Lacertae objects.
This work is partially supported by Outstanding Researcher
Awards of the University of Hong Kong, a Croucher
Foundation Senior Fellowship and the National 973 project of
China (NKBRAF G19990754), the National Natural Scientific
Foundation of China (19973001), and National Science Fund
for Distinguished Young Scholars (10125315). The authors
thank an anonymous referee for significant comments, and Dr.
Esko Valtaoja, Dr. G. E. Romero, and Dr. Jose-luis G´omez for
their useful suggestions and comments on the manuscript.
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