Total Flux Density Variations in Extragalactic Radio Sources. I. Decomposition of Variations into Exponential Flares

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Total Flux Density Variations in Extragalactic Radio Sources. I. Decomposition of Variations into Exponential Flares
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  T HE A STROPHYSICAL J OURNAL , 511:112  È  117, 1999 January 20 1999. The American Astronomical Society. All rights reserved. Printed in U.S.A. ( TOTAL FLUX DENSITY VARIATIONS IN EXTRAGALACTIC RADIO SOURCES. II. DETERMINING THELIMITING BRIGHTNESS TEMPERATURE FOR SYNCHROTRON SOURCESA. L A   HTEENMA   KI Radio Observatory, Helsinki University of Technology, FIN-02540 Finland;Metsa hovi Kylma la , alien = kurp.hut.Ð E. V ALTAOJA Tuorla Observatory, University of Turku, FIN-21500 FinlandPiikkio , AND K. W IIK Radio Observatory, Helsinki University of Technology, FIN-02540 FinlandMetsa hovi Kylma la , Received 1998 March 12; accepted 1998 August 21 ABSTRACTThe maximum intrinsic brightness temperature for powerful synchrotron-emitting radio sources T   b ,lim is usually assumed to be  B 10 12  K, limited by the inverse Compton catastrophe. A lower value of  B 5 ] 10 10  K, based on the equipartition brightness temperature, has been suggested by Readhead onthe basis of distributions derived from VLBI observations. We present two new methods for esti- T   b ,obs mating in extragalactic radio sources by using total Ñux density variations. A reasonable estimate T   b ,lim of the value of for a source can be obtained by comparing the Doppler boosting factors derived T   b ,lim from total Ñux density variations at 22 and 37 GHz with traditional estimates based on the radio andsynchrotron self-Compton (SSC) X-ray Ñuxes. Another independent estimate of is obtained by T   b ,lim comparing the brightness temperatures derived from variability data with the values calculated fromVLBI observations. Using several data sets, we Ðnd that both methods yield a value of   ¹ 10 11  K, inaccordance with the equipartition brightness temperature limit proposed by Readhead. Subject headings:  galaxies: photometry È radiation mechanisms: nonthermal È radio continuum: galaxies È X-rays: galaxies 1 .  INTRODUCTION The observed brightness temperature in a source is T   b ,obs P  S h 2 l 2  , (1)where  S  is the Ñux,  l  is the frequency, and  h  is the angulardiameter of the source. The angular size can either be mea-sured directly with VLBI or computed from the Ñux densityvariability timescale. The observed brightness temperaturecan be transformed to the source proper frame by multi-plying by (1 ] z ) in the case of VLBI, and by (1 ] z ) 3  in thecase of Ñux density variations.The upper limit for the intrinsic brightness tem- T   b ,lim perature in incoherent synchrotron sources such as T   b ,int active galactic nuclei (AGNs) is usually taken to be  B 10 12 K, independent of wavelength (Kellerman & Pauliny-Toth1969). This limit marks the beginning of the inverseCompton (IC) process, which rapidly leads to catastrophicelectron energy losses and the su†ocation of synchrotronemission. Under nonstationary conditions, the 10 12  K limitmay be exceeded, and brightness temperatures up to 10 15  Kand more may be reached during the Ðrst few days after theinjection of relativistic electrons (Slysh 1992). The ICprocess produces X-ray photons from a population of rela-tivistic synchrotron electrons via the synchrotron self-Compton (SSC) mechanism. In a number of sources, theX-ray spectra follow the radio spectra smoothly in shape,strength, and simultaneous variationsÈan importantfeature in conÐrming the presence of the SSC process. Yetwe observe sources with K and no IC- T   b ,obs [ 10 12 scattered X-ray emission, although the high photon andrelativistic electron densities imply a strong X-ray Ñux. Thecommonly accepted explanation for observed brightnesstemperatures exceeding the synchrotron limit is relativisticboosting in the source, which a†ects the observed proper-ties. The relativistic speed of the jet in a radio sourcechanges its apparent Ñux density, making it appear muchbrighter, blueshifts the radiation, and compresses the time-scales. The Doppler boosting factor in a jet making an angle /  to the line of sight and having an intrinsic Lorentz factor ! is D \ [ ! (1 [ b  cos  / )] ~1  . (2)Relativistic beaming enhances the observed variabilitybrightness temperature by a factor  D 3 , and the VLBIbrightness temperature by a factor  D . Consequently, if   T   b ,lim is known, an observed brightness temperature in excess of the limiting value can be used to calculate the Dopplerboosting factor of the source and further to estimate itsLorentz factor and viewing angle using the VLBI expansionspeeds (e.g., & Valtaoja 1994; &Tera sranta La hteenma kiValtaoja 1997).Readhead (1994) has argued that instead of the inverseCompton catastrophe limit, a more reasonable upper valuefor the intrinsic brightness temperature is ¹ 10 11  K, basedon the assumption that the sources are near equipartition of energy between the radiating particles and the magneticÐeld. Readhead has pointed out that the inverse Comptoncatastrophe occurs only in conditions requiring enormousdepartures from the equipartition and minimum-energyconditions, and further suggested that some as yet unknownmechanism may instead maintain the sources close to equi-partition. For such a source, the corresponding intrinsicbrightness temperature, called the equipartition brightnesstemperature by Readhead, is weakly dependent on the T   eq redshift, the observing frequency, the observed Ñux, and thespectrum of the source. For reasonable values of these112  EXTRAGALACTIC RADIO FLUX DENSITY VARIATIONS. II. 113parameters, is within a factor of 2 of 5 ] 10 10  K. In T   eq support of this equipartition limit, Readhead has analyzedVLBI data for samples of compact powerful radio sources,which appear to show an upper cuto† in the observedbrightness temperature distribution consistent with if  T   eq ,modest Doppler boosting is also assumed to be present.Such a value for an order of magnitude lower than the T   b ,lim ,commonly accepted limit of   B 10 12  K, would among otherthings mean that the amounts of Doppler boosting incompact radio sources have hitherto been systematicallyunderestimated, with important implications for sourcephysics and uniÐcation scenarios.In principle, can be estimated from simultaneous T   b ,int total Ñux density and VLBI observations by calculatingand and then eliminating the T   b ,obs (VLBI)  T   b ,obs (var),common unknown factor  D . Alternatively, if the value of   D is known, then can be calculated from either value of  T   b ,int The upper limit can then be estimated from the T   b ,obs .  T   b ,lim distribution of the values. & Daly (1996) have T   b ,int  Guijosacompared the SSC Doppler boosting factors calculatedfrom radio and X-ray data with equipartition Dopplerboosting factors calculated from and T   b ,obs (VLBI)  T   eq .Using a sample of 105 sources, they found that and D SSC  D eq are, in general, comparable, supporting the equipartitionlimit hypothesis. However, the calculated SSC and equi-partition boosting factors are not independent, since thesame observed quantities (frequency, Ñux, apparent size) areused in the calculation of both.In this paper we instead attempt to solve the crucial ques-tion of the true value of by using the total Ñux density T   b ,lim variation data at 22 and 37 GHz obtained in the Metsa hoviquasar monitoring program et al. 1992, 1998),(Tera srantapresented in  °   2. In  °   3 we compare the brightness tem-peratures of sources with simultaneous VLBI and total Ñuxdensity observations in order to obtain estimates for  T   b ,int .In °  4 we use traditional Doppler factors based on observedSSC X-ray Ñuxes to calculate the intrinsic brightness tem-peratures for a number of sources. We estimate the T   b ,int observed variability brightness temperatures by T   b ,obs (var)modeling the total Ñux density variations with exponen-tially growing and decaying Ñares. This method, presentedin Valtaoja et al. (1998, hereafter Paper I), provides a gooddescription of major total Ñux density variations. We haveadopted the operative deÐnition that major radio variationsexceed 10% of the total Ñux density and have timescalesexceeding 10 days. This is sufficient to exclude intraday andother rapid small-scale variations at 22 and 37 GHz thatmay have di†erent srcins (Wagner & Witzel 1995). Byexcluding smaller variations occurring in timescales of days,we also avoid possible strongly nonstationary situations(Slysh 1992). As °°  3 and 4 show, our results strongly indi-cate that the equipartition brightness temperature proposedby Readhead does provide a good estimate for the limitingintrinsic brightness temperature in strong, compact extra-galactic radio sources. Preliminary results of this study haveearlier been presented in & Valtaoja (1998). InLa hteenma kiet al. (1998, hereafter Paper III) we will con-Lahteenmakisider the implications of this result for the Doppler boostingfactors, the viewing angles, and the uniÐed models of AGNs. 2 .  THE DATA AND THE CALCULATION OF VARIABILITYBRIGHTNESS TEMPERATURES The almost 20 yr of continuum monitoring of active gal-axies with the 13.7 m radio telescope hasMetsahoviresulted in an extensive database at 22 and 37 GHz. Itconsists of   B 130 sources, many of which have beenobserved since 1980. Monitoring data prior to 1990.5 havebeen published by Salonen et al. (1987) and et al.Tera sranta(1992).By using this unique high-frequency total Ñux densityvariation data, we have calculated the intrinsic variabilitybrightness temperature and the Doppler boosting factor foreach source in the database. The method is described thor-oughly in Paper I. First we subtracted the quiescent Ñuxlevel of the source from the total Ñux density, to account forthe contribution of the constant Ñux from the nonvariablecomponents. For each source, we then identiÐed the well-deÐned outbursts and Ðtted each of them with the com-mercial PeakFit program, using a function with anexponential rise, a sharp turnover, and an exponentialdecay with a timescale 1.3 times longer than the rise time-scale. For such an exponential outburst, the logarithmicvariability timescale q obs \  dtd  (ln S ) (3)remains constant during the growth (or decay) stage. Nextwe calculated the observed variability brightness tem-perature (in the source proper frame), T   b ,obs (var) \ 5.87 ] 10 21  h ~2  j 2 S max q obs2  ( J  1 ] z [ 1) 2  , (4)where  j  is the observed wavelength in meters, is the S max maximum amplitude of the outburst in janskys,  z  is theredshift, and is the observed variability timescale in q obs days. (The numerical factor in eq. [4] corresponds to using h  km s ~1  Mpc ~1  and and to assuming H 0 \ 100  q 0 \ 0.5,that the source is a homogeneous sphere.) The observedvariability brightness temperature is related to the Dopplerboosting factor and to the intrinsic brightness tem- D var perature by T   b ,int D var \ C T   b ,obs (var) T   b ,int D 1@3 . (5)To get optimal results, we chose only the sources that hadat least one strong radio outburst with good data coverageand that could be Ðtted reliably, giving us 76 sources forwhich could be calculated for one or more out- T   b ,obs (var)bursts.For comparison with VLBI-derived values of we did a similar Ðtting at the epoch of each T   b ,obs (VLBI),VLBI observation. Instead of in equation (4), we used S max * S , the amplitude of the Ñare at the time of the VLBI obser-vation, thus obtaining the associated for com- T   b ,obs (var)parison with simultaneous Altogether, we T   b ,obs (VLBI).used three di†erent VLBI data sets, obtaining a total of 48individual pairs. 3 .  ESTIMATING FROM SIMULTANEOUS VLBI AND T   b ,lim VARIABILITY DATA We have compared the total Ñux density variation bright-ness temperatures at 22 GHz with brightness T   b ,obs (var)temperatures from simultaneous 22 GHz VLBI obser-vations using three di†erent VLBI data sets: the VSOPPre-Launch Survey (hereafter PLS; Moellenbrock et al.  114 LAHTEENMAKI, VALTAOJA, & WIIK Vol. 5111996), and those of Bloom et al. (1998) and Wiik et al. (Wiiket al. 1998; Wiik & Valtaoja 1998). The PLS sample isconstructed of 140 compact Ñat- or inverted-spectrumextragalactic radio sources observed at 22 GHz. It wasmotivated by the VLBI space satellite mission VSOP (VLBISpace Observatory Program), launched in 1997 Februarywith the satellite  HAL CA , and was intended to provideguidelines for the VSOP scientiÐc observing program at 22GHz as well as for future space VLBI missions and formillimeter VLBI. However, for almost half of the 140sources, only the lower limit of is available. T   b ,obs (VLBI)Furthermore, simultaneous data from sufficientMetsa hovifor estimating exists for only a fraction of the T   b ,obs (var)PLS sources. Altogether, this leaves a total of 33 sources forcomparisons with the PLS sample. The Bloom et al. (1998)sample consists of multiwaveband observations of 30 Ñat-spectrum quasars and radio galaxies with strong millimeter-wave emission, 17 of which were observed with VLBI at 22GHz. Finally, there are 15 bright AGN in the Wiik et al.(1998) sample, previously unobserved with 22 GHz VLBI.Here, too, the lack of simultaneous data reducesMetsa hovithe number of sources. In addition, we could not alwaysidentify a clear VLBI shock component for comparisonwith the continuum Ñare data. The total number of usefulsources in the Bloom et al. sample is 10, and in the Wiik etal. sample there are Ðve.A notable limitation to our method arises from thequality of the VLBI data. Most of the data used here (thePLS sample) provide us with the brightness temperature of the whole source rather than brightness temperatures of separate components, since no maps were made. Fortu-nately, the Bloom et al. (1998) and Wiik et al. (1998) samplesinclude 22 GHz VLBI maps, allowing us to identify theshocks and thus determine the VLBI brightness tem-perature of the component corresponding to the total Ñuxdensity Ñare and the associated variability brightness tem-perature. This provides us with much more accurate esti-mates of but the number of sources is small. However, T   b ,int ,a note of caution is necessary even here. Generally, thesource structure is completely resolved in only a few VLBImeasurements. Even if actual brightness temperatures(instead of lower limits) are derived from the data, some of them may still be underestimated, since the sources maycontain more compact unresolved components. In order tominimize such underestimates, we have used only global 22GHz data.For each source, we calculated at 22 GHz from T   b ,obs (var)our variability data at the epoch of the corresponding 22GHz VLBI observation. Because of the di†erent depen-dence of and on the Doppler factor, T   b ,obs (var)  T   b ,obs (VLBI)can be estimated. For VLBI, the formula for the T   b ,int observed brightness temperature is T   b ,obs (VLBI) \ T   b ,int (VLBI) D VLBI  . (6)For continuum observations, T   b ,obs (var) \ T   b ,int (var) D var3  . (7)Assuming that both the total Ñux density and the VLBIobservations relate to the same component, we can elimi-nate  D  and get for each source: T   b ,int T   b ,int\ S  T   b ,obs (VLBI) 3 T   b ,obs (var) . (8)We Ðnd that the observed variability and VLBI values of correlate with the probability T   b ,obs  P Spearman \ 0.00005.Figure 1 shows versus At the upper T   b ,obs (VLBI)  T   b ,obs (var).end of the distribution, the variability brightness tem- T   b ,obs peratures are much larger than the VLBI brightness tem-peratures, as expected if the sources are Doppler boosted( D [ 1; eqs. [6] and [7]). At the lower end, the VLBIbrightness temperatures are larger, again as expected if these sources are Doppler deboosted ( D \ 1). In an idealcase, the data would lie along a straight line with a slope of Even in the real world, the data points are nearly all 13 . F IG . 1.ÈObserved brightness temperatures (source frame) derived from VLBI data and from total Ñux density variations.  Open circles : VSOP Pre-Launch Survey of Moellenbrock et al. (1996).  Filled circles : Bloom et al. (1998) survey.  Filled squares : Wiik et al. (1998) survey. The straight lines correspondto intrinsic brightness temperatures of 10 12  K, 10 11  K, and 10 10  K.  No. 1, 1999 EXTRAGALACTIC RADIO FLUX DENSITY VARIATIONS. II. 115located between lines corresponding to 10 10  K \ T   b ,int \ 10 12  K. With three exceptions, the most reliable data points(from the Bloom et al. 1998 and Wiik et al. 1998 samples)are close to the 10 11  K line. The medians for the threesamples are, as listed in Table 1:1. For the PLS sample, log  T   b ,int \ 11.24 ^ 0.10( N \ 33).2. For the Bloom et al. (1998) sample, log  T   b ,int \ 10.89 ^ 0.17 ( N \ 10).3. For the Wiik et al. (1998) sample, log  T   b ,int\ 10.41 ^ 0.43 ( N \ 5).In order to estimate the errors in the brightness tem-peratures shown in Figure 1, we have made a series of tests.We have made a number of independent model Ðts to theWiik et al. (1998) sample sources using both spherical andellipsoidal components, and with Gaussian and uniformbrightness distributions. From Ðts to 19 di†erent VLBIcomponents, we Ðnd that the median uncertainty incaused by the choice of the model com-log  T   b ,obs (VLBI)ponent geometry alone is  ^ 0.20. Considering calibrationand other errors inherent in the VLBI observations, thismust be a very conservative lower limit to the true error inthe component brightness temperatures derived from VLBIdata. In the case of the PLS, where only brightness tem-peratures integrated over the whole source are available, theuncertainty is larger still.One possible source of error in estimating from T   b ,int Figure 1 is the fact that the VLBI brightness temperaturesare systematically underestimated because of insufficientresolution. This would lead to a higher value of   T   b ,int .However, the Bloom et al. (1998) and Wiik et al. (1998)sample values are derived from careful model Ðts to resolv-ed shock components, so a systematic underestimation isnot likely.For the total Ñux density estimates, the largest uncer-tainty in the case of well-deÐned isolated Ñares comes fromthe subtraction of the quiescent Ñux, which cannot be accu-rately determined from total Ñux density monitoring alone(cf. Paper I). This a†ects both the estimated Ñare Ñux  S max (or  * S ) and the variability timescale (eq. [4]). We have q obs made model Ðts to 45 representative Ñares using the twoextreme alternatives: no quiescent Ñux removal before theexponential Ñare Ðt, and removal of the whole observedminimum Ñux density before the Ðtting process. The mediandi†erence between the two calculated values for  T   b ,obs (var), * T    /  T    , is 0.50, which gives a conservative  upper  limit to theerror in comparable to the  lower  limit of  T   b ,obs (var),(For some individual total Ñux density Ñares, T   b ,obs (VLBI).however, the errors may be larger because of uncertaintiesin the model Ðtting procedure.)Comparing these error estimates with the data in Figure1 and the median errors of the samples given above, we TABLE 1M  EDIAN  V ALUES OF FOR  A LL  D ATA  S  ETS log  T   b ,int Method Data Set  N  log  T   b ,int VLBI & SSC...... 3C 345 1 10.82 ^ 0.41VLBI............... Wiik et al. (1998) 5 10.41 ^ 0.43VLBI............... Bloom et al. (1998) 10 10.89 ^ 0.17VLBI............... PLS 33 11.24 ^ 0.10SSC ................ Gu ijosa & Daly (1996) 48 11.30 ^ 0.20 conclude that all our data are consistent with the assump-tion that  all   sources have intrinsic brightness temperaturesclose to the equipartition limit of   B 10 11  K. One shouldnote that this result is independent of the actual amounts of Doppler boosting in the sources, and seems to hold both forthe highly Doppler boosted sources (Fig. 1,  upper rightcorner ) and for the Doppler ““deboostedÏÏ sources with(Fig. 1,  lower left corner ). T   b ,obs > T   b ,int 4 .  ESTIMATING FROM DOPPLER BOOSTING FACTORS T   b ,int AND VARIABILITY DATA 4.1.  Synchrotron Self  - Compton Estimates We have compared our Doppler boosting factors derivedfrom variability brightness temperatures with the SSCDoppler factors from & Daly (1996), based on theGuijosadata srcinally collected by Ghisellini et al. (1993). This isthe largest existing sample of SSC Doppler factors so far,giving for 105 sources. For 48 of these sources, suffi- D SSC cient Ñux density data have been accumulated in themonitoring program to estimate the variabilityMetsa hovibrightness temperature for at least one Ñare, at either 22 or37 GHz. In the case of sources with more than one well-Ðtted Ñare, we have used the highest reliable value of in the comparisons. T   b ,obs (var)If we assume that the variability and the SSC Dopplerfactors are equal (as they should be), the intrinsic brightnesstemperature can be calculated for each source: D var \ C T   b ,obs (var) T   b ,int D 1@3\ D SSC  (9) 7 T   b ,int\ T   b ,obs (var) D SSC3  . (10)We again Ðnd a high correlation, this time between thevariability and the SSC values of   D , with probabilityFigure 2 shows the distribution of the P Spearman \ 0.0002.variability and the SSC Doppler factors for the 48 sources.Although the data points are scattered because of the uncer-tainties, one can see that K provides the best T   b ,int B 10 11 general agreement between the two totally independent esti-mates of the Doppler boosting factors. The median valuefor log is 11.30 ^ 0.20 (see Table 1), well below the T   b ,int inverse Compton limit and in agreement with the valuesderived in the previous section.One problem with the SSC Doppler factors is that theyare not very reliable. depends strongly on the Ñux and D SSC the size of the source. Even small deviations in theseobserved quantities a†ect the Ðnal result severely. Evenworse, the values used are often totally wrong. It is usuallyimpossible to identify the component in which the X-raysrcinates. In addition, instead of the observing frequencyand the corresponding observed Ñux, one should use thesynchrotron turnover Ñux and frequency in the calculations.These have been accurately determined for only a handfulof sources (e.g., Marscher & Broderick 1985), and most esti-mates of including those of & Daly (1996), are D SSC , Gu ijosabased on whatever VLBI data happens to be available. Fur-thermore, the VLBI and X-ray observations are taken atdi†erent times, often years apart. The end result is that thevalues of   l ,  S ,  h , etc., used in calculating the SSC S X ,Doppler factor usually bear little relation to the correctones.  116 LAHTEENMAKI, VALTAOJA, & WIIK Vol. 511 F IG . 2.ÈDoppler boosting factors derived from variability brightnesstemperatures vs. the SSC-derived boosting factors from & DalyGu ijosa(1996). The values have been calculated assuming The D var  T   b ,lim \ 10 12  K.straight lines show the expected dependence between and for D var  D SSC various values of the intrinsic brightness temperature. We will discuss the reliability of SSC Doppler boostingfactors in more detail in Paper III. Here we only note thatplotting values derived by various authors from di†er- D SSC ent data sets for the same sources generally results in almostpure scatter diagrams. Comparing the sources common toMadejski & Schwartz (1983), Zhang & (1996), andBa  - a  - th& Daly (1996), we Ðnd that only in about half theGu ijosacases do the derived Doppler boosting factors agree withina factor of 2, and for a signiÐcant fraction the values di†erby an order of magnitude. This alone is sufficient to explainmost of the scatter in Figure 2. In contrast, the formal errorsin are much smaller, since it depends only on the third D var root of the observed variability brightness temperature.However, variability data clearly show that does not D var always stay constant from one outburst to another. Ideally,one should compare and calculated from simulta- D SSC  D var neous X-ray, VLBI, and total Ñux density observations. Useof noncontemporaneous data thus also increases the scatterin Figure 2 beyond the errors of the variability and SSCestimates.In view of the uncertainties associated with the SSCDoppler boosting values shown in Figure 2, we again con-clude that our data is in agreement with the hypothesis thatall sources have intrinsic brightness temperatures close tothe equipartition limit.4.2.  Doppler Boosting Factors for 3C 345 Other methods for estimating the Doppler factors inradio sources have also been suggested, all involvingvarious assumptions (see Readhead 1994). Probably thebest-studied radio source is 3C 345, for which various esti-mates of Doppler boosting factors have been presented in avery comprehensive study by Unwin et al. (1994). Theyapplied several di†erent methods for estimating the amountof Doppler boosting. They concluded that the Dopplerfactor for the brightest knot in the jet, also likely to be theorigin of the X-ray emission, lies in the range5.5 ¹ D ¹ 10.5, with  D \ 7.5 as the best value. Based oncontemporaneous X-ray multifrequency VLBI and otherdata, this is arguably the most reliable ““traditionalÏÏ esti-mate yet obtained for the amount of Doppler boosting in aradio source. From our monitoring data, we derive a valueof 2.8 ] 10 13  K for the highest observed brightness tem-perature in 3C 345, corresponding to  T   int \ 2.8 ] 10 13 /7.5 3 \ 6.6 ] 10 10  K. Once more, this is in agree-ment with the equipartition limit. (We note that if the intrin-sic brightness temperature in 3C 345 were instead  B 10 12  K,we should expect to see extremely spectacular rapid totalÑux density variations, corresponding to  T   b ,obs (var) B 4 ] 10 14  K. These would appear very di†erent from the onesactually seen in our total Ñux density monitoring of 3C 345.) 5 .  SUMMARY We have calculated intrinsic brightness temperaturesfor several samples of radio sources from total Ñux T   b ,int density variations using the 22 and 37 GHz con-Metsahovitinuum monitoring data and two di†erent methods.  T   b ,int can be estimated by comparing the observed VLBI com-ponent brightness temperatures with the correspondingobserved variability brightness temperatures (eq. [8]).Alternatively, can be calculated from if we T   b ,int  T   b ,obs (var)also have an estimate for the sourceÏs Doppler boosting,usually derived from SSC estimates (eqs. [9] and [10]).In Figure 3 we show the distribution of for each of  T   b ,int the data sets. Most of the individual values derived T   b ,int from all the di†erent data sets are below 10 12  K, clusteringaround the equipartition limit of   B 10 11  K, in accordancewith Readhead (1994). The numerical results for each dataset are compiled in Table 1. F IG . 3.ÈIntrinsic brightness temperatures calculated from the variousdata sets (see text for details).
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