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A&A 574, A84 (2015) DOI: 10.1051/0004-6361/201425084
c ESO 2015

Astronomy & Astrophysics

Brightness temperature constraints from interferometric visibilities
Andrei Lobanov1,
1 2

2

Max-Planck-Institut fЭr Radioastronomie, Auf dem HЭgel 69, 53121 Bonn, Germany e-mail: alobanov@mpifr-bonn.mpg.de Institut fЭr Experimentalphysik, UniversitДt Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany

Received 30 September 2014 / Accepted 4 December 2014
ABSTRACT

Context. The brightness temperature is an effective parameter that describes the physical properties of emitting material in astrophysical objects. It is commonly determined by imaging and modeling the structure of the emitting region and estimating its flux density and angular size. Aims. Reliable approaches for visibility-based estimates of brightness temperature are needed for interferometric experiments in which poor coverage of spatial frequencies prevents successful imaging of the source structure, for example, in interferometric measurements made at millimeter wavelengths or with orbiting antennas. Methods. Such approaches can be developed by analyzing the relations between brightness temperature and visibility amplitude and its rms error. Results. A method is introduced for directly calculating the lower and upper limits of the brightness temperature from visibility measurements. The visibility-based brightness temperature estimates are shown to agree well with the image-based estimates obtained in the 2 cm MOJAVE survey and the 3 mm CMVA survey, with good agreement achieved for interferometric measurements at spatial frequencies exceeding 2 в 108 . Conclusions. The method provides an essential tool for constraining brightness temperature in all interferometric experiments with poor imaging capability.
Key words. galaxies: jets ­ techniques: interferometric ­ methods: analytical

1. Introduction
Interferometric measurements offer a powerful tool for probing the finest structures of emitting objects by extending the effective instrumental diameter to the maximum distance (baseline length) between individual elements of an interferometer. However, for interferometric measurements made at extreme baseline lengths, imaging the structure of the target object becomes increasingly limited, owing to incomplete sampling of the Fourier plane. This is often the case in radio interferometric measurements made with very long baseline interferometry (VLBI) at millimeter wavelengths (cf. Doeleman et al. 2012) and with space-ground interferometers such as VSOP (Horiuchi et al. 2004)orRadioAstron (Kardashev et al. 2013). In these situations, more basic measurements of flux density, S , and emitting area, , of the structure can still be obtained (i.e., from model fitting of the visibility distribution) and can be combined to yield a brightness temperature estimate. The latter can then be used as a generic indicator of the physical conditions of the emitting material (cf. Lobanov et al. 2000; Kovalev et al. 2005; Homan et al. 2006; Lee et al. 2008). For the black-body spectrum in the Rayleigh-Jeans limit (h kT ), the brightness, I , is approximated by I = 22 kT /c2 , and the respective brightness temperature is T b = I c2 /(2 k 2 ), where h are k are the Planck and Boltzmann constants, respectively, and c is the speed of light. In terms of the measured quantities, S and , the resulting brightness is I = S / = S /[2 (1 - cos d )], if the emitting region is a uniformly bright circle of angular radius d . For small d , the term 1 - cos d is approximated by 2 /2, which yields I S /(2 ). d d

If the emitting region is unresolved, d can be constrained by the resolution limit, lim , of the measurement, providing lower limits 2 on the brightness, I 4S /(lim ), and brightness temperature, 2 22 T b 2S c /( k lim ). In absence of information about the actual brightness distribution of emission in a compact, marginally resolved region, it is often assumed that it can be represented satisfactorily by a two-dimensional Gaussian distribution described by a flux density S g and respective major and minor axes maj and min . This translates into I = (4 ln 2/) S g /(maj min ) and T b = [2 ln 2/( k)] S g c2 /(2 maj min ). These expressions are used for the bulk of brightness temperature estimates based on decomposition of the observed structure into one or more twodimensional Gaussian features (Gaussian components). Several other analytical patterns of brightness distribution patterns have been employed to analyze different astrophysical objects (cf. Berger 2003) such as resolved stars (Dyck et al. 1998; Ohnaka et al. 2013), young supernovae (Marcaide et al. 2009), recurrent novae (Chesneau et al. 2007), protoplanetary disks (Malbet et al. 2005), or active galaxies (Weigelt et al. 2012). In all of these cases, successful fitting of a given brightness distribution pattern to visibility data is a strong prerequisite for recovering structural and physical information about the target object. However, the most extreme cases of interferometric observations, such as millimeter and space VLBI measurements, often do not provide enough data to warrant reliable model fitting owing to lack of short-baseline measurements and the complexity of the fine structure in most of the targets. These observations require a different approach for estimating the brightness temperature. In this paper, such an approach is proposed, based on A84, page 1 of 9

Article published by EDP Sciences


A&A 574, A84 (2015)

individual visibility measurements and their errors (which can be reliably estimated in most of the measurements). The methodology of this approach is described in Sect. 2 and is tested with the visibility data from two VLBA1 observations of the prominent compact radio sources 3C 345 and NGC 1052. Applications of the visibility-based T b estimates are discussed in Sect. 3 and are compared with the results from the 3 mm CMVA2 and 2 cm MOJAVE3 surveys and in connection with an analysis of space VLBI and millimeter VLBI surveys of compact radio sources. Additional potential applications of the method to other types of astrophysical targets are also discussed, and expressions for brightness temperature limits for several specific brightness distribution patterns are presented in the appendix.

2 5.5 Tb arbitrary unit 5. 4.5 4. 3.5 3. 2.5 2

4

6

8

10 5.5 5. 4.5 4. 3.5 3.

4

2. Brightness temperature limits from interferometric visibilities
We consider an emitting region with a brightness distribution, Ir , observed instantaneously at a wavelength, , by an interferometer consisting of two receiving elements (telescopes) separated by a baseline distance, B. This observation corresponds to measuring the Fourier transform of Ir at a single spatial (Fourier) frequency q = B/ (also called uv distance or uv radius). It yields an interferometric visibility, V = Vq e-i q , described by its amplitude Vq and phase q , and their respective errors q and . Generally speaking, Vq depends on the shape and angular extent of the brightness distribution, and q = p + o is a function of its position and geometry. The position-dependent term of the phase, p , is relative and can always be zeroed by an appropriate shift applied to the visibility (which is analogous to re-pointing the interferometer). The geometry-dependent term of the phase, o , depends on the structure of the brightness distribution and its orientation with respect to the projection of the interferometric baseline on the picture plane. For a circularly symmetric or axially symmetric brightness distribution, o 0, independently of the baseline orientation.
2.1. Minimum brightness temperature

6 V0 Vq

8

10

2.5

Fig. 1. Brightness temperature in relative units as a function of the ratio V0 /Vq between the zero-spacing flux density and visibility flux density measured at a given spatial frequency q, assuming that the brightness distribution is Gaussian. The lowest value of the brightness temperature is realized with the ratio V0 /Vq = e.

With this expression, the brightness temperature can be estimated from Tb = B2 V0 · 2k ln(V0 /Vq ) (2)

Appendix A lists the respective expressions derived for several other patterns of brightness distributions commonly used in the analysis of astronomical data. Equation (2) provides the lowest value of T b for V0 = eVq (see Fig. 1), which yields an estimate of the minimum brightness temperature supported by the measured visibility amplitude Vq : T
b,min

=

B e 2 B Vq 3.09 2k km

2

Vq [K]. mJy

(3)

Without a priori information about the specific structural shape of Ir , the symmetry assumption can be employed and the angular extent of the emission can be estimated from Vq alone. This assumption is routinely used for size and brightness temperature estimates made from interferometric data (cf. Lobanov et al. 2000; Kovalev et al. 2005; Lee et al. 2008). Such estimates require knowledge of the zero-spacing visibility, V0 , and rely upon assumption of a specific symmetric template for Ir . For instance, for a circular Gaussian distribution, the respective expression for Vq is Vq = V0 exp -
2 2 r q2 4ln 2

This expression describes the absolute minimum of the brightness temperature that can be obtained from the measured visibility amplitude Vq under the assumption that the brightness distribution is well approximated by a circular Gaussian. The lowest brightness temperature is realized for any visibility distribution that has an inflection point. The respective expressions of T b,min for other patterns of brightness distributions are listed in Appendix A.
2.2. Maximum measurable brightness temperature

and it can be used for obtaining an estimate of the size, r ,ofthe emitting region: 2 ln 2 r = ln(V0 /Vq ). (1) B
1

Very Long Baseline Array of National Radio Astronomy Observatory, Socorro NM, USA; http://www.nrao.edu 2 Coordinated Millimeter VLBI Array, currently succeeded by the Global Millimeter VLBI Array; http://www3.mpifr- bonn.mpg. de/div/vlbi/globalmm/ 3 http://www.phyiscs.purdue.edu/astro/MOJAVE/ A84, page 2 of 9

The expression for T b,min given by Eq. (3) is independent of V0 , while estimating the maximum brightness temperature will necessarily require knowledge, or at least a reasonable assumption, of V0 . With the latter, it should be kept in mind that if only a limit on V0 can be assumed, the nature of the resulting estimate of T b depends on the ratio of V0 /Vq . The maximum T b can only be derived from upper limits on V0 for V0 > eVq and from lower limits on V0 for V0 < eVq . In the opposite cases, the combination of V0 and Vq yields an estimate of the minimum brightness temperature. The zero-spacing visibility V0 is often approximated by the total flux density, S tot , measured at a single receiving element. Generally, this is a poor approximation because S tot contains contributions from all angular scales. A better constraint on V0 is provided by the flux density of a respective Gaussian component if the data warrant reliable Gaussian decomposition. For


A. Lobanov: Brightness temperature from interferometric visibilities

extremely poor coverages of the Fourier domain, this is not the case, and no satisfactory estimate of V0 can be made. In this situation, a lower limit of V0 = Vq + q can be adopted. This limit effectively corresponds to requiring that Vq probes a structural detail that is marginally resolved (we recall that Vq const. results from the Fourier transform of a point source). This assumption is well justified for visibility measurements made at sufficiently long baselines, where the visibility amplitude is dominated by the most compact structure observed in the target object. It can furthermore be verified through the observed absence of amplitude beating at slightly shorter baselines (with the latter reflecting the presence of multiple compact emitting regions in the object). Examples of visibility (baseline) ranges dominated by contributions from the most compact structures are shown in Fig. 2 for two radio sources (NGC 1052 and 3C 345) that represent the typical cases of a compact radio source with and without a strong contribution from extended emission. The requirement of marginal resolution of the observed structure implies that its size should be larger than lim 2 ln 2 = B ln Vq + q · Vq (4)

5

4

Vu Jy

3

2

1

0 0 100 200 300 400 q M 500 600

2

Correspondingly, the brightness temperature of this feature should not exceed the limit of T
b,lim

Vu Jy 1 0 0 100 200 q M 300 400

=

B (Vq + q ) Vq + q ln 2k Vq
2

-1

= 1.14

Vq + q mJy

B km

2

ln

Vq + q Vq

-1

[K].

(5)

For visibilities with a signal-to-noise ratio Vq /q > e - 1, the estimate of T b,lim provides the highest brightness temperature that can be obtained from the measured visibility amplitude and its error while requiring that the respective brightness distribution is a) circularly Gaussian; and b) marginally resolved by the measured visibility. Expressions for T b,lim derived for several other brightness distribution patterns are given in Appendix A. As measurements of Vq and q are made over extended time intervals, they correspond to averaging the visibility function over finite ranges of spatial frequencies (q, ), with describing the positional angle of Vq in the Fourier plane. In this case, the estimate provided by Eq. (5) holds for as long as the condition Vq, const. is satisfied over the given measurement interval [q, ]. This condition effectively requires that the measured Vq is dominated by a contribution from a single emitting region that is marginally resolved at the spatial frequency q. Equations (3) and (5) provide a robust bracketing for the brightness temperature obtained from interferometric measurements that have a limited sampling of the visibility distribution of the target. This can be demonstrated by applying these equations to every visibility of the VLBI datasets used in the examples shown in Fig. 2. Results of this application are presented in Fig. 3, where the visibility-based estimates of T b,min and T b,lim are plotted against the spatial frequency (uv radius) of the respective visibilities. These estimates can be compared with the brightness temperature, T b,mod estimated from the model fit parameters of the "core" component. This comparison indicates that at uv radii >150 M, the interval [T b,min , T b,lim ] repre sents a reasonably good bracketing for the expected maximum brightness temperature. This substantially exceeds the conservative expectations for the uv ranges (indicated by braces in

Fig. 2. Comparison of the visibility amplitude distribution and Gaussian modelfit representation of the compact structure in a compact structuredominated object (top, a 22 GHz VLBA observation of the quasar 3C 345 made on 23/08/1999) and an extended structure-dominated object (bottom, a 15 GHz VLBA observation of the radio galaxy NGC 1052 from MOJAVE Survey, made on 16/12/1995; Kellermann et al. 2004). The visibility amplitude distributions are projected onto a PA of -104 (3C 345) and -110 (NGC 1052) for illustration purposes, enabling a representation by a single, composite modelfit rather than by fits to individual baselines. The insets show Gaussian modelfit images of the source structure, with component locations and sizes indicated. The resulting fits of the visibility amplitudes are shown by the solid curves. The dotted curves show visibility representations of the individual modelfit components. The thick dashed lines correspond to the respective visibility responses from the most compact "core" component that has the highest brightness temperature. Braces indicate the spatial frequency (uv distance) ranges in which the contribution from the core component dominates the measured visibility amplitude distribution (the uv ranges are measured in units of M, where is the observing wavelength). These spatial frequency ranges are best suited to directly estimate the brightness temperature from the visibility measurements.

Fig. 3) suitable for estimating the brightness temperature. Within these ranges, the average T b,lim is only marginally (factors of 1.5 and 2.2) higher than the T b,mod obtained from modelfits. Hence, T b,lim estimates made at long baselines can constrain the maximum brightness temperature well in both core-dominated and jet-dominated compact radio sources.
A84, page 3 of 9


A&A 574, A84 (2015)
13 Tb
,lim

9.3 1011 K

1. 0.75 0.5

0.1 0.2

12 Tb log10 Tb K Tb
,mod

10



4.2 1011 K 3.5 1011 K

0.25 0. 0.25 0.5 0.75 1. 0 10 20 30 40 50 jet ° 60 70 80 90 0.9

11

,min

10

9 0 100 200 300 400 q M 500 600

T 11

b,lim

4.3 1010 K

Fig. 4. Brightness temperature correction to account for the elongation of the emitting region. The elongated region is described by an elliptical Gaussian with an axial ratio . The corrections are plotted against the difference between the position angle of a visibility measurement and the position angle of the major axis of the Gaussian. Different curves correspond to the corrections for values of taken in steps of 0.1 between = 0.1and = 0.9.

10 log10 Tb K

T

b,mod

2.8 1010 K 2.1 1010 K

9

T

b,min

8

7 0 100 200 q M 300 400

Fig. 3. Limiting, T b,lim (top row of datapoints) and minimum, T b,min (bottom row of datapoints), brightness temperature estimated directly from visibility data for 3C 345 (top panel) and NGC 1052 (bottom panel), plotted against the spatial frequency q (uv radius). The estimated values are compared with the brightness temperatures of the Gaussian model fit components (dotted lines for the jet components, thick dashed line for the core component) used to describe the source structure as shown in Fig. 2. Running stairs show the respective rows of the brightness temperature averaged within radial bins of 10 M in extent. Both the original and the averaged rows of the brightness temperature indicate that at q 150 M, the interval [T b,min , T b,lim ] provides a good bracketing for the maximum brightness temperature in each of the two objects. Braces indicate the conservative ranges of spatial frequency q identified as ranges dominated by the most compact part of the source structure. Averages of T b,min and T b,lim made over these ranges constrain the respective T b,mod estimates well.

For visibility measurements made at a random position angle (hence oriented randomly with respect to the source elongation), the resulting scatter of T b estimates can approach one order of magnitude. This adverse effect of the source elongation can be taken into account in objects with a known elongation and position angle of the compact structure. The compact core region can be approximated by an elliptical Gaussian with an axial ratio, , and a position angle of the major axis, jet . The angular size estimate, r , obtained under the assumption of a circular Gaussian brightness distribution (e.g., with Eq. (1)) can then be related to the major and minor axes of the elliptical Gaussian, maj = r / sin2 + min = r / cos2 +
2

log

cos2 , sin2 maj ,

-2

where = - jet describes the difference between the visibility position angle and that of the major axis of the elliptical Gaussian component (hence r = min for = 0 and r = maj for = 90 ). For the brightness temperature estimates, this results in a multiplicative correction factor
2 = r /(maj min ) =

cos2 +

-1

sin2

(6)

2.3. Corrections for elongation of the emitting region

The estimates of T b,min and T b,lim can furthermore be refined if the potential elongation of the emitting region is considered. If visibility measurements are made over a narrow range of position angle in the Fourier plane, this elongation may bias individual estimates of the brightness temperature and introduce scatter in the statistics obtained from object samples. Gaussian modeling of fine structure in the objects from the MOJAVE sample (Kovalev et al. 2005) indicates that the core components are well described by elliptical Gaussian patterns, with an average elongation (minor to major axis ratio, = min /maj )of 0.4 ± 0.2.
A84, page 4 of 9

that should be applied to T b,min and T b,lim given by Eqs. (3) and (5). The magnitude of this correction is on the order of 1/ 2 over the full range of values of (see Fig. 4). Applying this correction may be particularly useful for analysis of space VLBI measurements made with RadioAstron at baselines in excess of ten Earth diameters (hence falling within a range of < 6 ). Other potential applications include snapshot VLBI measurements made at millimeter wavelengths (e.g., Doeleman et al. 2012; Petrov et al. 2012; Lee et al. 2013). The effect of correcting for the core elongation is illustrated in Fig. 5, where the correcting factor is applied to brightness temperature estimates made from the visibility data on 3C 345 presented in Figs. 2, 3. The corrections are derived for an axial ratio = 0.5 (weighted average of the model-fitting results


A. Lobanov: Brightness temperature from interferometric visibilities
13 Tb
,lim

4.8 1011 K

Buv

0.99 Bmax

14
12 Tb log10 Tb K
,mod

4.2 1011 K

13
11 Tb
,min

1.8 1011 K

K log10 Tb
,lim

12 11
40 B
uv

10

0.99 B 0.66 0.57

max

10
0 100 200 300 400 q M 500 600

Number of objects

9

30

20

Fig. 5. Limiting, T b,lim (top row of datapoints) and minimum, T b,min (bottom row of datapoints), brightness temperature estimated directly from visibility data for 3C 345 and corrected for an assumed ellipticity of the emitting region, with = 0.5 (based on MOJAVE model fitting reported in Kovalev et al. 2005). and jet = -104 . The limiting brightness temperature of 4.8 в 1011 K, obtained from the visibilities at the longest baselines ( B > 600 M), agrees well with the estimate based on model fitting the source structure.

9

10

0 1 0 log10 T
b,lim

1 log10 T

2
b,mod

3

9

10

11
log10 Tb

12
,mod

13

14

K

in Kovalev et al. 2005) and a jet position angle jet = -104 inferred from the source structure shown in Fig. 2. The increased scatter at shorter baselines shows the effect of contributions from larger structures that are incorrectly described by the adopted values of jet and, in particular, . However, at the longest baselines, correcting for the core elongation clearly improves the T b,lim estimate and brings it well within the errors of the modelfit-based estimate. The same correction applied to NGC 1052 (with = 0.4 and jet = -110) results in corrected T b,min = 1.1 в 1010 K and T b,lim = 2.4 в 1010 K, with the latter value falling very close to T b,mod = 2.8 в 1010 K estimated from the modelfit. Both these results indicate that T b,lim can be improved by correcting for the elongation of the core region.

Fig. 6. Comparison of T b,lim (circular Gaussian approximation) and T b,mod estimates obtained from the MOJAVE data (open circles). Gray circles illustrate the effect of correcting T b,lim for the putative elongation of the core region. The dashed line marks the one-to-one correspondence between the two estimates. For each object, the T b,lim is estimated from MOJAVE data at Ruv 0.99 Bmax to restrict the visibility information to the most compact structures. The resulting T b,lim are on average 4.6 times higher than T b,mod . The residual logarithmic distribution of the T b,lim /T b,mod ratio (inset) can be approximated by the Gaussian PDF with = 0.66 and = 0.57. At T b,mod < 1010 Kthe T b,lim may be biased by large-scale structure contributions in strongly jet-dominated objects.

3.1. Preparation of the visibility data

3. Discussion
Applying the visibility-based brightness temperature estimates to VLBI data on 3C 345 and NGC 1052 has demonstrated that the method is reliable in two particular cases. A more extended testing of the method can be performed on a statistical basis by applying it to visibility data from large VLBI survey programs aimed at measuring and analyzing the brightness temperature distribution in samples of compact radio sources. The analysis of fine scale structure in the 15 GHz (obs = 2cm) MOJAVE sample (Kovalev et al. 2005) and the results of brightness temperature measurements from the 86 GHz (obs = 3 mm) CMVA survey Lee et al. (2008) offer statistically suitable samples for such tests. The MOJAVE analysis was based on elliptical Gaussian model fits of the core region. The 86 GHz data were fitted by circular Gaussian components. Resolution criteria were applied to the data from both surveys to constrain the core components with degenerate size parameters (circular diameter or one of the two axes of elliptical Gaussian) obtained from the model fitting.

To evaluate the performance of visibility-based brightness temperature limits on a self-consistent statistical basis, visibility data for each source from these two programs (244 objects in the MOJAVE sample and 123 observations of 109 individual objects in the CMVA sample) were radially and azimuthally averaged within a small annulus, ( Buv , Bmax ) in the uv plane. The purpose of clipping the data is to reduce the adverse effect of including shorter baselines that are dominated by extended structure. The upper limit of the annulus is determined by the longest baseline Bmax found in the data for each individual object. The MOJAVE datasets were clipped at Buv = 0.99 Bmax (i.e., in an annulus located within 1% of Bmax ). The CMVA survey data, with substantially fewer visibilities per target object, were clipped at Buv = 0.9 Bmax. The resulting average uv distances and visibility position angles in the clipped data are 435 ± 27 M and -88 ± 6 for the MOJAVE data and 2270 ± 660 M and -78 ± 20 for the CMVA data. In each case, the fraction of visibilities selected is 1/Nbas , where Nbas is the number of baselines in a dataset. Following the conclusions obtained from analyzing the data on 3C 345 and NGC 1052, these uv distances should be sufficiently long to reduce the visibility contamination by large-scale structures to negligible levels. The T b,lim estimates were therefore calculated for each source from all of the visibilities clipped and averaged within the respective annuli. The circular Gaussian approximation was used in these calculation.
A84, page 5 of 9


A&A 574, A84 (2015)

Buv

0.99 Bmax

Buv

0.99 Bmax

14 13
K K lim comp
,lim 2

14 13 12 11
Number of objects

log10 Tb

,lim

12 11
30 25 B
uv

0.99 B 0.09 0.70

Tb

35 B 30
uv

max

0.99 B 0.16 0.66

max

log

10

10 9

Number of objects

20 15 10 5 0 2 1 log10 Tb,lim 0 log
10

10 9

25 20 15 10 5 0

T

b,mod

1 2 maj min

2 log
10

1 T
b,lim

0 lim comp

2

1 2 log10 Tb,mod

9

10

11
,mod

12
maj min K

13

14

9

10

11
log10 Tb

12
,mod

13

14

log10 Tb

K

Fig. 7. Correlation between T b,lim and T b,mod (maj /min ) expressing the brightness temperature obtained assuming that the characteristic size of the brightest region is determined by the jet transverse dimension as given by the minor axis of the elliptical Gaussian fit. The average T b,lim is is only 20% higher than the respective corrected values of T b,mod ,and the residual logarithmic distribution of the T b,lim /T b,mod ratio (inset) can be approximated by the Gaussian PDF with = 0.09 and = 0.70.

Fig. 8. Correction of the distribution shown in Fig. 6 for the square of the resolution factor lim /comp . The corrected T b,lim distribution is statistically very close to the T b,mod obtained from the model-fitting analysis. The values of T b,lim are on average 40% higher than the respective values of T b,mod . This is also demonstrated by the residual logarithmic distribution of the T b,lim /T b,mod ratio (inset), which can be approximated by the Gaussian PDF with = 0.16 and = 0.66.

3.2. Results from the MOJAVE data

For the MOJAVE data, the resulting estimates of T b,lim are compared in Fig. 6 with the T b,mod estimates obtained in Kovalev et al. (2005) for each source at the same observing epoch. The two estimates agree reasonably well, with T b,lim being on average 4.6 times higher than the respective T b,mod . This is similar to the T b,lim /T b,mod ratios measured in 3C 345 and NGC 1052. Overall, the T b,lim estimates provide a reasonable upper bound on the brightness temperature, with the scatter in the estimates limited to about half a decade over nearly five orders of magnitude in brightness temperature. Correction for the putative elongation of the core region has been attempted for the MOJAVE data, based on the elliptical model fits and measured position angles of the jet. The elongation was calculated as the ratio of the minor to major axes of the elliptical Gaussian component describing the core. Two options were tried for the jet position angle: a) the position angle of the major axis of the Gaussian component; and b) the average position angle of the jet as reported in Kovalev et al. (2005). The results of applying the elongation correction with the former option are shown in Fig. 6. Neither of the two corrections has improved the correlation for either the average T b,lim /T b,mod ratio or the spread of the residuals. This result may be caused by two factors. On the one hand, it may suggest that the elongation of the core region is not accurately reflected in the parameters of the elliptical Gaussian components, for instance, if this region has a specific geometry such as a conically expanding jet (cf. Blandford & KЖnigl 1979) or a jet pervaded with thread-like instability patterns (cf. Lobanov & Zensus 2001). On the other hand, the lack of improvement achieved with the elongation correction may indicate
A84, page 6 of 9

that the highest brightness is realized in a region that is smaller than the major axis of the Gaussian component describing the core region. Again, this would be the case for a quasi-stationary conical jet or a jet dominated by the instability patterns. In each of these situations, the observed jet brightness would be largely determined by its transverse dimension. If this is the case, T b,lim could be reconciled with T b,mod derived under the assumption that only the minor axis, min , of the Gaussian component is relevant for determining the jet brightness. This hypothesis is tested in Fig. 7 by comparing T b,lim to T b,mod maj /min , which has the effect of calculating the size of 2 th