Doppler factors, Lorentz factors and viewing angles for quasars, BL Lacertae objects and radio galaxies


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We have calculated variability Doppler boosting factors, Lorentz factors, and viewing angles for a large sample of sources by using total flux density observations at 22 and 37 GHz and VLBI data. We decomposed the flux curves into exponential flares
    a  r   X   i  v  :   0   8   1   1 .   4   2   7   8  v   1   [  a  s   t  r  o  -  p   h   ]   2   6   N  o  v   2   0   0   8 Astronomy & Astrophysics  manuscript no. 1150 c  ESO 2013February 19, 2013 Doppler factors, Lorentz factors and viewing angles for quasars,BL Lacertae objects and radio galaxies T. Hovatta 1 , E. Valtaoja 2 , 3 , M. Tornikoski 1 , and A. L¨ahteenm¨aki 1 1 Mets¨ahovi Radio Observatory, TKK, Helsinki University of Technology, Mets¨ahovintie 114, 02540 Kylm¨al¨a, Finland e-mail: 2 Tuorla Observatory, University of Turku, V¨ais¨al¨antie 20, 21500 Piikki¨o, Finland 3 Department of Physics and Astronomy, University of Turku, Vesilinnantie 5, 20100 Turku, FinlandReceived  /  Accepted ABSTRACT Aims.  We have calculated variability Doppler boosting factors, Lorentz factors, and viewing angles for a large sample of sources byusing total flux density observations at 22 and 37GHz and VLBI data. Methods.  We decomposed the flux curves into exponential flares and determined the variability brightness temperatures of the fastestflares. By assuming the same intrinsic brightness temperature for each source, we calculated the Doppler boosting factors for 87sources. In addition we used new apparent jet speed data to calculate the Lorentz factors and viewing angles for 67 sources. Results.  We find that all quasars in our sample are Doppler-boosted and that the Doppler boosting factors of BL Lacertae objects arelower than of quasars. The new Lorentz factors are about twice as high as in earlier studies, which is mainly due to higher apparentspeeds in our analyses. The jets of BL Lacertae objects are slower than of quasars. There are some extreme sources with very highderived Lorentz factors of the order of a hundred. These high Lorentz factors could be real. It is also possible that the sources exhibitsuch rapid flares that the fast variations have remained undetected in monitoring programmes, or else the sources have a complicated jet structure that is not amenable to our simple analysis. Almost all the sources are seen in a small viewing angle of less than 20degrees. Our results follow the predictions of basic unification schemes for AGN. Key words.  galaxies: active – galaxies: jets – radio continuum: galaxies – radiation mechanisms: non-thermal – quasars: general 1. Introduction All radio-bright active galactic nuclei (AGN) have relativistic jets emitting synchrotron radiation. The jets can at the sim-plest level be modelled by using two intrinsic parameters, theLorentz factor ( Γ ), which describes the speed of the jet flow,and the viewing angle ( θ  ), which is the angle between the jetaxis and the line of sight to the observer. These parameters canbe calculated if the Doppler boosting factor (  D ) and the appar-ent speed  β app  =  v / c  are known. We can find out the  β app  fromVery Long Baseline Interferometry (VLBI) observations, and inthe past few years major progress has been made in this area(e.g. Jorstad et al. 2001; Homan et al. 2001; Kellermann et al. 2004; Jorstad et al. 2005; Piner et al. 2007; Britzen et al. 2008). The Doppler boosting factors can be calculated in various ways,and di ff  erent methods are compared in L¨ahteenm¨aki & Valtaoja (1999) (hereafter LV99).A common way to calculate the Doppler boosting fac-tors is to combine X-ray observations with VLBI compo-nent fluxes (e.g. Ghisellini et al. 1993; Guijosa & Daly 1996; Guerra & Daly 1997; Britzen et al. 2007). This method assumes inverse Compton (IC) srcin of the X-ray emission, and that thesame synchrotron photons forming the lower frequency radia-tion are responsible also for the IC emission. Assuming that theVLBI observations are done at the spectral turnoverfrequency,apredicted X-ray flux can be calculated. By comparing this to theobserved X-ray flux, and by interpreting the excess flux as dueto Doppler boosting, the Doppler boosting factors can be calcu-lated. If the VLBI frequency is not at the turnover, large errorsare induced in the Doppler boosting factors. This method alsosu ff  ersgreatlyfromnon-simultaneousX-rayandVLBI data,andas was argued in LV99, gives much less accurate estimates forthe Doppler boosting factors.Using VLBI it is possible to directly observe the bright-ness temperature of the source ( T  b , obs ). This can be comparedto the intrinsic brightness temperature of the source ( T  b , int ),which is often assumed to be the equipartition temperature( T  eq ) (Readhead 1994; L¨ahteenm¨aki et al. 1999). The excess of   T  b , obs  is interpreted as caused by Doppler boosting. Thismethod also requires the values to be obtained at the turnoverfrequency, which enhances the errors in the Doppler boostingfactors (LV99).Another way to use VLBI observations is shown inJorstad et al. (2005) who estimated the variability Doppler boosting factors of 15 AGN using Very Long Baseline Array(VLBA) data at 43GHz. They calculated the flux decline time( τ obs  ∝  τ int   D ) of a component in the jet and compared it tothe measured size of the VLBI component (which does not de-pend on  D ). Assuming the intrinsic variability timescale corre-spondsto the light-traveltime acrossthe knot,theyestimated theDoppler boosting factors. They also estimated the Lorentz fac-tors and the viewing angles for these sources by using apparentspeed data.Variability timescales can also be obtained from total fluxdensity (TFD) observations. This is the method used in LV99and we use the same method in our analyses. We decomposeeach flux curve into exponential flares and calculate the vari-ability timescale of each flare. From this we gain the observedbrightness temperature, which is boosted by  D 3 in comparison  2 Hovatta et al.: Doppler factors, Lorentz factors and viewing angles for a sample of AGN with  T  b , int . This makes possible the extraction of the variabil-ity Doppler factor  D var  if   T  b , int  is known. In L¨ahteenm¨aki et al. (1999) it was argued, based on observations, that in every largeflare  T  b , int  reachesthe equipartitiontemperature T  eq  =  5 × 10 10 K  .In LV99 a sample of 81 sources was studied at 22 and37GHz frequencies. They calculated the Doppler boosting fac-tors based on observations from a period of over 15 years. Foreach source they combined the results of the two frequencybandsandchose thefastest flare forthe analysis. Inadditiontheydetermined the Lorentz factors and the viewing angles for 45sources. We have done similar calculations for a larger sampleof sources and using data from almost 30 years of monitoring.We will also compare our results with the results of LV99 to seeif the values have changed during the past 10 years.The paper is organised as follows: in Sect. 2 we describethe source sample and the method used. In Sect. 3 we calculatethe variability Doppler boosting factors and compare our resultswith LV99 and other related studies. The Lorentz factors andthe viewing angles are presented in Sect. 4 and the discussionfollows in Sect. 5. Finally conclusions are drawn in Sect. 6. 2. Data and the method Our sample consists of 87 bright, well-monitored AGN fromthe Mets¨ahovi Radio Observatory monitoring list. This is arather good approximation of a complete flux limited sample of the brightest compact northern sources. (Missing are a handfulof sources for which we had insu ffi cient data to calculate theDoppler factors.) The sources have been observed regularly foralmost 30 years at 22 and 37GHz with the Mets¨ahovi 14m tele-scope (Salonen et al. 1987; Ter¨asranta et al. 1992, 1998, 2004, 2005). Details of the observationmethodand data reductionpro-cess are described in Ter¨asranta et al. (1998). Our study also in- cludes unpublished data at 37GHz from December 2001 untilthe end of 2006. Data of BL Lacertae objects at 37GHz fromDecember2001 until April 2005 are publishedin Nieppola et al.(2007). In our sample we have 30 high polarisation quasars(HPQs), which have optical polarisation exceeding 3 percent atsome point in the past. In addition we have 22 low polarisa-tion quasars (LPQs), 8 quasars (QSOs) for whichno polarisationdata were available, 22 BL Lacertae objects (BLOs), and 5 ra-dio galaxies (GALs). We have used the classification used in theMOJAVE 1 programmewheneverpossible. All the quasars in oursample can be classified as flat spectrum radio quasars (FSRQs),which show blazar-like properties. Our sample does not containordinary quasars that have larger viewing angles and which donot show the rapid variability common for FSRQs. In our anal-yses we have usually combined QSOs with LPQs, following thechoice of LV99.We decomposedthe flux curves into exponentialflares of theform ∆ S  ( t  )  =  ∆ S  max e ( t  − t  max ) /τ ,  t   <  t  max , ∆ S  max e ( t  − t  max ) / 1 . 3 τ ,  t   >  t  max ,  (1)where  ∆ S  max  is the maximum amplitude of the flare in janskys, t  max  is the epoch of the flare maximum and  τ  is the rise timeof the flare. The method is described in detail in Valtaoja et al.(1999). An example of a fit is presented in Fig. 1, where the flux curve of a HPQ source 1156 + 295 is decomposed into ex-ponential flares. The solid line represents the fitted sum curveof the individual components and the points are actual observed 1 http:  //  /  MOJAVE  /  Fig.1.  Flux curve of the HPQ source 1156 + 295(points) decom-posed into exponential flares (solid line) at 22GHzdata. We can see that the overall correspondence between thetwo is very good. In LV99 and Savolainen et al. (2002) it was also shown that the individualexponentialflare componentscor-respond very well to the emergence of new VLBI components,indicating that these flares obtained from the fits are indeed re-lated to the actual jet physics.From the fits we obtain the necessary parameters to calcu-late the observedvariabilitybrightnesstemperatureof the source T  b , var  (in the source proper frame) T  b , var  =  1 . 548 × 10 − 32  ∆ S  max d  2L ν 2 τ 2 (1 +  z ) ,  (2)where  ν  is the observed frequencyin GHz,  z  is the redshift,  d  L  istheluminositydistanceinmetres,and ∆ S  max  and τ  aredefinedinEq. 1. The numerical factor in Eq. 2 corresponds to using  H  0  = 72kms − 1 Mpc − 1 ,  Ω m  =  0 . 27 and  Ω Λ  =  0 . 73, and to assumingthatthesourceis a homogeneoussphere.Inthecalculationoftheluminosity distances we have made use of the python version of the cosmologycalculatorcreated by Edward L. Wright 2 (Wright2006).The variability Doppler factor can then be calculated as  D var  =  T  b , var T  b , int  1 / 3 .  (3)We used the value  T  b , int  =  5  ×  10 10 K. This value wassuggested by Readhead (1994) and the use of it was jus- tified in L¨ahteenm¨aki et al. (1999). Based on simulations, Kellermann et al. (2004) also find the  T  b , int  to be of the orderof 10 11 K.BycombiningtheDopplerboostingfactorswithapparentsu-perluminal component velocities  β app , obtained using VLBI, wecan calculate the variability Lorentz factors Γ var  and the viewingangles  θ  var  by using Eqs. 4 and 5. Γ var  =  β 2app  +  D 2var  + 12  D var (4) θ  var  =  arctan2  β app  β 2app  +  D 2var  − 1(5) 2 http:  //  /   wright  /  CosmoCalc.html  Hovatta et al.: Doppler factors, Lorentz factors and viewing angles for a sample of AGN 3 We obtained 67  β app  values from the MOJAVE sample, ob-served with the VLBA at 15GHz. The values are taken fromthe website on September 9, 2008 and some of them may stillbe preliminary and change slightly in the final results (Lister etal. in preparation). This is the most homogeneous and largestsample of   β app  available at higher radio frequencies. These val-ues representthe fastest reliablespeedin eachjet as measuredbythe MOJAVE programmeat 15 GHz, usingVLBA data spanningbetween 5 to 13 years, depending on the individual source. 3. The variability Doppler boosting factors 3.1. Estimation of the Doppler boosting factors  We were able to calculate the Doppler boosting factor (  D var ) for86 sources at 22GHz, and for 72 sources at 37GHz. For manysources we were able to determine the  D var  for more than oneflare. The medians of   D var  are slightly larger at 22GHz than at37GHz,whichcouldbeduetodi ff  erentintrinsicbrightnesstem-peratures at the two frequency bands. In our analysis we chosethefastestflareofeachsource(eitherat22or37GHz,whicheverhad the fastest flare) to calculate the  D var , as was done in LV99.The argument for using the fastest flare to determine  D var  is thatthey are most likely to reach the limiting brightness temperatureand least likely to su ff  er from blending of flares which tends toincrease the fitted timescale. This way we were able to calculatethe  D var  for all the 87 sources in our sample. By visual examina-tion, we divided the fits into three categories based on the good-ness of the fit. No single numerical value, such as the  χ 2 test, isalonesuitable fordescribingthe goodness,becausetheseusuallycharacterise the entire flux curve, while we have only used one,fastest flare, to determine the  D var . In addition, relatively largeerror bars in some fainter sources cause the  χ 2 value to be small,while we consider a fit to be better when the scatter among thedatapoints is small.We classified the  D var  as excellent (E, 21 sources), good (G,24 sources) or acceptable (A, 42 sources). In the fits classified asexcellent, the exponential decomposition follows the datapointsquite precisely, as in the flares after the year 1997 in Fig. 1. Thefit is also unambiguous and other functions do not describe thebehaviour as well. Larger flares in Fig. 1 before 1995 wouldmainly be defined as good, because in these flares the fit followsthe flux curve well, but there is also some scatter, and in somecases there are not as many datapoints to define the fit as in theones defined as excellent. In the fits classified as acceptable, thescatteraroundthefit is still larger.Thisis oftenthe casewhentheflux level of the source is modest and the errorbars consequentlylarge. However, none of our results change significantly if weexclude the acceptable sources. All Doppler boosting factors areshown in Table 1 where the B1950-name, other commonly usedname, type of the object, frequency of the  D var  determination,redshift, quality of the  D var , log T  b , var ,  D var ,  β app ,  Γ var ,  θ  var , coredominanceparameter  R , andmaximumoptical polarisation  P max and its reference are listed.It is di ffi cult to determine exact error estimates for the  D var of each source. Some indications can be obtained from the stan-dard deviation of   D var  calculated from the various flares in onesource. We calculated the deviations for all the sources classi-fied as excellent, which had more than one flare to determinethe  D var , including also flares determined as good. There were45 such cases (including all the fits at both 22 and 37GHz) and,on average, each source had 5.7 well-defined flares. The medianstandard deviation for these is  ∼  27%, which can be thought of as an indication of the upper limit for the error estimate since in Table 2.  Median values of log( T  b , var ) and  D var . Type N log( T  b , var )[ K  ]  D var HPQ 30 14.31 15.98LPQ 30 13.92 11.90FSRQ 60 14.19 14.61BLO 22 13.09 6.25GAL 5 11.65 2.07ALL 87 13.94 12.02 manycases the changein the  D var  of individualflares can also bedue to di ff  erences in the source behaviour. It is also more likelyto see a very fast flare in each source the longer they are moni-tored.More insight into the errorscan be obtainedwhen ournew  D var  are compared to other studies (cf. Chapt. 3.2).In Table 2 we show the median values of log( T  b , var ) and  D var for each source class separately and for HPQs and LPQs com-bined together (FSRQ). The distributions are shown in Figs. 2and 3 and we can see that the distributions of quasars and BLOshave considerable overlap. HPQs have a tail extending to higher  D var  andLPQs seem tobe inbetweentheHPQs andBLOs. Also,it is interesting to note that all the quasars are clearly Doppler-boosted, the smallest estimated  T  b , var  being 3 . 5  ×  10 11 K for1928 + 738. We ran the Kruskal-Wallis analysis to examine thedi ff  erences between source classes. (All Kruskal-Wallis analy-ses in this paper have been performed with the Unistat statisticalpackage for Windows 3 (version 5.0).) The results confirm thatall the source classes di ff  er from the other classes significantlywith a 95% confidence limit. 3.2. Comparison with previous analyses  Our sample has 71 sources in commonwith the sample of LV99.We have re-calculated the Doppler boosting factors of LV99, us-ing the current cosmological model. Figure 4 shows the corre-lation between the Doppler boosting factors of LV99 and thenew values. The results are very similar and the values correlatewith a coe ffi cient r  =  0.77 (p = 0.0000). Kruskal-Wallis analysisalso shows that the values come from the same population. Thisconfirms that the method is reliable because the results have notchanged even though we have now ten more years of data. Thedi ff  erences are mainly due to poor fits in LV99 which are dueto poor sampling or low flux density (large scatter) in the data.Thescatter betweenthe old andthe newvalues is consistent withthe error analysis in Sect. 3.1. In some cases (e.g. 0430 + 052and1156 + 295) the source has clearly changed its behaviour and ex-hibits a much faster flare in our new dataset. The new estimateswhich are calculated using almost 30 years of data should there-fore be more representative of the source behaviour.We also compared our  D var  values with the Jorstad et al.(2005) values for 15 AGN, obtained at 43GHz. Figure 5 shows thecorrelationbetweenthetwovalues,andaSpearmanrankcor-relation gives a coe ffi cient r = 0.56 (p = 0.0123). We can see thatthevaluesin Jorstad et al. (2005)arein generalsomewhathigher than ours. This can be due to their higher observing frequency(cf. Discussion). Also, their analysis method gives only an up-per limit to some sources. The distribution of source classes issimilar to ours, with quasars having the highest Doppler boost-ing factors, GALs the lowest and BLOs being in between them. 3 http:  //  /   4 Hovatta et al.: Doppler factors, Lorentz factors and viewing angles for a sample of AGN Fig.2.  Distribution of   T  b , var  of the fastest flare in each sourceThereforewe concludethattheDopplerboostingfactorsofthesetwo analyses correspond well with each other.Homan et al. (2006) argued that during the most active state T  b , int  should be closer to 2 × 10 11 K and therefore the  D var  of theLV99 are overestimated. This would make our  D var  values evensmaller,andthecorrespondencetoJorstad et al.(2005)wouldbe worse. Higher  T  b , int  would also increase our Lorentz factors forthe fastest sources, and as is shown later (cf. Chapters 4 and 5), our new values are already twice as high as in LV99 and in somesources even extremely high. We also note that a di ff  erent valuefor  T  b , int  does not change the distributions themselves, only thenumerical values.We also compared our Doppler boosting factors to a re-cent study at a lower frequency of 5GHz (Britzen et al. 2007).They calculated the IC Doppler boosting factors by using VLBIdatafromtheCaltech-JodrellBankFlat-Spectrumsourcesample(Taylor et al. 1996) and non-simultaneous ROSAT X-ray data.The Spearmanrank correlationbetween our24 commonsources(excluding one outlier, 0836 + 710, with  D IC  =  88) is still quitegood (r = 0.63, p = 0.0004), and the slope of the linear fit between  D IC  and  D var  is almost exactly one. We believe that most of thescatter is due to the errors in the IC Doppler boosting factors,since LV99 showed that for several reasons these are likely to bemuch less accurate than the variability Doppler boosting factors. Fig.3.  Distribution of   D var  of the fastest flare in each source 3.3. Core dominance  Standard beaming models expect that more core-dominated ob- jects should be more beamed and thus have higher Doppler Fig.4. Correlationbetweenthe  D var  fromLV99andournew  D var values  Hovatta et al.: Doppler factors, Lorentz factors and viewing angles for a sample of AGN 5 Fig.5.  Correlation between the  D var  and  δ var  from Jorstad et al.(2005).boosting factors. We studied this by calculating the core-dominance parameter R from VLBA data of  Kovalev et al.(2005) at 15GHz. The core-dominance is calculated by relat-ing the flux density of the core  S  core  to the total single-dish fluxdensity observed at 15GHz  S  tot . In Kovalev et al. (2005) these are given for 250 sources observed with the VLBA at 15GHzat di ff  erent epochs. Their sample includes 80 sources for whichwe have determined  D var . We calculated the median R for eachsource from the separate epochs. Figure 6 shows the correla-tion between logR and log  D var , excluding an outlier source0923 + 392 with a very small core dominance (log  R  =  − 1 . 76).Spearman rank correlation between the parameters is  r   =  0 . 37(  p  =  0 . 0004). When the outlier source is included, the cor-relation is still significant with a coe ffi cient  r   =  0 . 39 (  p  = 0 . 0002). This shows that there is indeed some indication thatsources which are more core-dominated are also more boosted.In Kovalev et al. (2005) the core-dominance is defined to be the relation of   S  core  to the total VLBA flux density  S  VLBA . We testedthe correlation using also this parameter but the results did notchange because  S  tot  and  S  VLBA  are so similar. Using the core-dominance defined with  S  VLBA , Kovalev et al. (2005) show that quasars and BLOs are significantly di ff  erent from GALs withlower core-dominance.Similar calculations were made in Britzen et al. (2007) for their sample. They used the core flux density and total single-dish flux density at 5GHz to calculate the core-dominance pa-rameter. They found no significant correlation between theircore-dominance parameter and the IC Doppler boosting factor.We have only 25 sources in common with their sample, andwhen we compared our  D var  with their core-dominanceparame-ter, we found no correlation. 4. The Lorentz factors and viewing angles We were able to calculate Lorentz factors  Γ var  and viewing an-gles  θ  var  for 67 sources. The median values of di ff  erent sourceclasses are shown in Table 3. These are a ff  ected by two outlierswith exceptionally large  Γ var . The source 0923 + 392 has  Γ var  = 216 . 1 and 1730-130 has  Γ var  =  64 . 6. Both of these show highsuperluminal motion of   β app  >  35 c , which increases the Lorentzfactors.Thedistributionsof thesourceclasses are shownin Figs.7 and 8. In  Γ var  the distributions of quasars and BLOs over- Fig.6. Correlationbetweenlog(  R )usingdatafromKovalev et al.(2005) and log(  D var ), excluding the outlier source 0923 + 392with log(  R )  =  − 1 . 76 and log(  D var )  =  0 . 63. Table 3.  Median values of  Γ var  and  θ  var . Type N  Γ var  θ  var HPQ 26 17.41 3.28LPQ 23 13.96 3.9021 a 12.65 3.96FSRQ 49 16.24 3.37BLO 13 10.29 5.24GAL 5 1.82 15.52ALL 67 13.96 3.81 a =  excluding outliers 0923 + 392 and 1730-130 lap, but BLOs and GALs have slower jet speeds than quasars.Kruskal-Wallis analysis shows that when  Γ var  are studied with-out the outlier sources, HPQs di ff  er from other classes withhigher Lorentz factors, and GALs di ff  er with smaller Lorentzfactors. BLOs and LPQs come from the same population witha 13% confidence. There is one BLO (1823 + 568) for which Γ var  =  37 . 8. This source has also been classified as a HPQ(e.g. V´eron-Cetty & V´eron 2006) and therefore we ran the KW- analysis again by moving this source into the HPQ class. In thiscase also BLOs and LPQs di ff  er significantly from each otherwith a 96% confidence. We must note that in our samples of 13to 26 objects the significance of di ff  erences can depend on theclassification of a single extreme source. However, a clear re-sult is that the BLOs and the quasars di ff  er from each other withBLOs having slower jets (FSRQs and BLOs di ff  er significantlywith a 99% confidence if 1823 + 568 is classified as a BLO andwith a 99.9% confidence if 1823 + 568 is classified as a HPQ).This result is in accordancewith severalearlier but more indirectestimates of jet speeds. Similar results have also been obtainedwith simulations (Hughes et al. 2002).When  θ  var  is studied, the distributions overlap even moreand KW-analysis shows that GALs and BLOs di ff  er from othersourceclasseswitha95%confidence.Also,ifthedi ff  erencesbe-tween FSRQs and BLOs are studied, they di ff  er from each othersignificantly with a 99% confidence with BLOs having largerviewing angles.
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