© ESO, 2015

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) or RadioAstron (Kardashev et al. 2013). In these situations, more basic measurements of flux density, S_ν, and emitting area, Ω, 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 (hν ≪ kT), the brightness, I_ν, is approximated by I_ν = 2ν²kT/c², and the respective brightness temperature is T_b = I_νc²/ (2 kν²), 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_ν and Ω, the resulting brightness is I_ν = S_ν/ Ω = S_ν/ [2π (1 − cosρ_d)], if the emitting region is a uniformly bright circle of angular radius ρ_d. For small ρ_d, the term 1 − cosρ_d is approximated by ${\mathit{\rho }}_{\mathrm{d}}^{\mathrm{2}}\mathit{/}\mathrm{2}$, which yields ${\mathit{I}}_{\mathit{\nu }}\mathrm{\approx }{\mathit{S}}_{\mathit{\nu }}\mathit{/}\mathrm{(}\mathit{\pi }\hspace{0.17em}{\mathit{\rho }}_{\mathrm{d}}^{\mathrm{2}}\mathrm{)}$. If the emitting region is unresolved, ρ_d can be constrained by the resolution limit, θ_lim, of the measurement, providing lower limits on the brightness, ${\mathit{I}}_{\mathit{\nu }}\mathrm{\ge }\mathrm{4}{\mathit{S}}_{\mathit{\nu }}\mathit{/}\mathrm{(}\mathit{\pi }\hspace{0.17em}{\mathit{\theta }}_{\lim }^{\mathrm{2}}\mathrm{)}$, and brightness temperature, ${\mathit{T}}_{\mathrm{b}}\mathrm{\ge }\mathrm{2}{\mathit{S}}_{\mathit{\nu }}{\mathit{c}}^{\mathrm{2}}\mathit{/}\mathrm{(}\mathit{\pi }\hspace{0.17em}\mathit{k}\hspace{0.17em}{\mathit{\nu }}^{\mathrm{2}}{\mathit{\theta }}_{\lim }^{\mathrm{2}}\mathrm{)}$.

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 θ_maj and θ_min. This translates into I_ν = (4 ln 2 /π) S_g/ (θ_majθ_min) and T_b = [2 ln 2/(πk)] S_gc²/ (ν²θ_majθ_min). These expressions are used for the bulk of brightness temperature estimates based on decomposition of the observed structure into one or more two-dimensional 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 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. Brightness temperature limits from interferometric visibilities

We consider an emitting region with a brightness distribution, I_r, observed instantaneously at a wavelength, λ, by an interferometer consisting of two receiving elements (telescopes) separated by a baseline distance, B. This observation corresponds to measuring the Fourier transform of I_r at a single spatial (Fourier) frequency q = B/λ (also called uv distance or uv radius). It yields an interferometric visibility, V = V_qe^{− iφ_q}, described by its amplitude V_q and phase φ_q, and their respective errors σ_q and σ_φ. Generally speaking, V_q depends on the shape and angular extent of the brightness distribution, and φ_q = φ_p + φ_o is a function of its position and geometry. The position-dependent term of the phase, φ_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, φ_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, φ_o ≡ 0, independently of the baseline orientation.

2.1. Minimum brightness temperature

Without a priori information about the specific structural shape of I_r, the symmetry assumption can be employed and the angular extent of the emission can be estimated from V_q 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 I_r. For instance, for a circular Gaussian distribution, the respective expression for V_q is

${\mathit{V}}_{\mathit{q}}\mathrm{=}{\mathit{V}}_{\mathrm{0}}\exp \left(\mathrm{-}\frac{{\mathit{\pi }}^{\mathrm{2}}{\mathit{\theta }}_{\mathit{r}}^{\mathrm{2}}{\mathit{q}}^{\mathrm{2}}}{\mathrm{4}\ln \mathrm{2}}\right)$and it can be used for obtaining an estimate of the size, θ_r, of the emitting region: ${\mathit{\theta }}_{\mathit{r}}\mathrm{=}\frac{\mathrm{2}\sqrt{\ln \mathrm{2}}}{\mathit{\pi }}\frac{\mathit{\lambda }}{\mathit{B}}\sqrt{\ln \mathrm{(}{\mathit{V}}_{\mathrm{0}}\mathit{/}{\mathit{V}}_{\mathit{q}}\mathrm{)}}\mathit{.}$(1)With this expression, the brightness temperature can be estimated from ${\mathit{T}}_{\mathrm{b}}\mathrm{=}\frac{\mathit{\pi }}{\mathrm{2}\mathit{k}}\frac{{\mathit{B}}^{\mathrm{2}}\hspace{0.17em}{\mathit{V}}_{\mathrm{0}}}{\ln \mathrm{(}{\mathit{V}}_{\mathrm{0}}\mathit{/}{\mathit{V}}_{\mathit{q}}\mathrm{)}}\mathrm{·}$(2)Appendix A lists the respective expressions derived for several other patterns of brightness distributions commonly used in the analysis of astronomical data.

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.

Equation (2) provides the lowest value of T_b for V₀ = eV_q (see Fig. 1), which yields an estimate of the minimum brightness temperature supported by the measured visibility amplitude V_q: ${\mathit{T}}_{\mathrm{b}\mathit{,}\min }\mathrm{=}\frac{\mathit{\pi }\hspace{0.17em}\mathit{e}}{\mathrm{2}\mathit{k}}\hspace{0.17em}{\mathit{B}}^{\mathrm{2}}\hspace{0.17em}{\mathit{V}}_{\mathit{q}}\hspace{0.17em}\mathrm{\approx }\hspace{0.17em}\mathrm{3.09}{\left(\frac{\mathit{B}}{\mathrm{km}}\right)}^{\mathrm{2}}\left(\frac{{\mathit{V}}_{\mathit{q}}}{\mathrm{mJy}}\right)\hspace{0.17em}\mathrm{\left[}\mathrm{K}\mathrm{\right]}\mathit{.}$(3)This expression describes the absolute minimum of the brightness temperature that can be obtained from the measured visibility amplitude V_q 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

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/V_q. The maximum T_b can only be derived from upper limits on V₀ for V₀>eV_q and from lower limits on V₀ for V₀<eV_q. In the opposite cases, the combination of V₀ and V_q 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 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 = V_q + σ_q can be adopted. This limit effectively corresponds to requiring that V_q probes a structural detail that is marginally resolved (we recall that V_q ≡ 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.

Fig. 2 Comparison of the visibility amplitude distribution and Gaussian modelfit representation of the compact structure in a compact structure-dominated 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 Mλ, where λ is the observing wavelength). These spatial frequency ranges are best suited to directly estimate the brightness temperature from the visibility measurements.

Fig. 3 Limiting, Tb,lim (top row of datapoints) and minimum, Tb,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 Mλ in extent. Both the original and the averaged rows of the brightness temperature indicate that at q ≳ 150 Mλ, the interval [Tb,min,Tb,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 Tb,min and Tb,lim made over these ranges constrain the respective Tb,mod estimates well.

The requirement of marginal resolution of the observed structure implies that its size should be larger than ${\mathit{\theta }}_{\lim }\mathrm{=}\frac{\mathrm{2}\sqrt{\ln \mathrm{2}}}{\mathit{\pi }}\frac{\mathit{\lambda }}{\mathit{B}}\sqrt{\ln \frac{{\mathit{V}}_{\mathit{q}}\mathrm{+}{\mathit{\sigma }}_{\mathit{q}}}{{\mathit{V}}_{\mathit{q}}}}\mathrm{·}$(4)Correspondingly, the brightness temperature of this feature should not exceed the limit of $\begin{array}{ccc}{\mathit{T}}_{\mathrm{b}\mathit{,}\lim }& \mathrm{=}& \frac{\mathit{\pi }{\mathit{B}}^{\mathrm{2}}\hspace{0.17em}\mathrm{(}{\mathit{V}}_{\mathit{q}}\mathrm{+}{\mathit{\sigma }}_{\mathit{q}}\mathrm{)}}{\mathrm{2}\mathit{k}}{\left[\ln \frac{{\mathit{V}}_{\mathit{q}}\mathrm{+}{\mathit{\sigma }}_{\mathit{q}}}{{\mathit{V}}_{\mathit{q}}}\right]}^{-1}\\ & \mathrm{=}& \end{array}$(5)For visibilities with a signal-to-noise ratio V_q/σ_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 V_q and σ_q are made over extended time intervals, they correspond to averaging the visibility function over finite ranges of spatial frequencies (Δq, Δψ), with ψ describing the positional angle of V_q in the Fourier plane. In this case, the estimate provided by Eq. (5) holds for as long as the condition V_q,ψ ≈ const. is satisfied over the given measurement interval [Δq, Δψ]. This condition effectively requires that the measured V_q 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 Mλ, the interval [T_b,min,T_b,lim] represents 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. 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.

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 ϵ. 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 ϵ taken in steps of 0.1 between ϵ = 0.1 and ϵ = 0.9.

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 ψ 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, ϵ = θ_min/θ_maj) of 0.4 ± 0.2. For visibility measurements made at a random position angle ψ (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, ϵ, and a position angle of the major axis, ψ_jet. The angular size estimate, θ_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, $\begin{array}{ccc}{\mathit{\theta }}_{\mathrm{maj}}& \mathrm{=}& {\mathit{\theta }}_{\mathit{r}}\mathit{/}\sqrt{{\sin }^{\mathrm{2}}\mathit{\zeta }\mathrm{+}{\mathit{ϵ}}^{\mathrm{2}}{\cos }^{\mathrm{2}}\mathit{\zeta }}\mathit{,}\\ {\mathit{\theta }}_{\min }& \mathrm{=}& {\mathit{\theta }}_{\mathit{r}}\mathit{/}\sqrt{{\cos }^{\mathrm{2}}\mathit{\zeta }\mathrm{+}{\mathit{ϵ}}^{-2}{\sin }^{\mathrm{2}}\mathit{\zeta }}\mathrm{\equiv }{\mathit{\theta }}_{\mathrm{maj}}\hspace{0.17em}\mathit{ϵ,}\end{array}$where ζ = ψ − ψ_jet describes the difference between the visibility position angle and that of the major axis of the elliptical Gaussian component (hence θ_r = θ_min for ζ = 0° and θ_r = θ_maj for ζ = 90°). For the brightness temperature estimates, this results in a multiplicative correction factor ${\mathit{\varphi }}_{\mathit{ϵ}}\mathrm{=}{\mathit{\theta }}_{\mathit{r}}^{\mathrm{2}}\mathit{/}\mathrm{(}{\mathit{\theta }}_{\mathrm{maj}}\hspace{0.17em}{\mathit{\theta }}_{\min }\mathrm{)}\mathrm{=}\mathit{ϵ}\hspace{0.17em}{\cos }^{\mathrm{2}}\mathit{\zeta }\mathrm{+}{\mathit{ϵ}}^{-1}\hspace{0.17em}{\sin }^{\mathrm{2}}\mathit{\zeta }$(6)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 /ϵ² over the full range of values of ζ (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 Δψ ≲ 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).

Fig. 5 Limiting, Tb,lim (top row of datapoints) and minimum, Tb,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 ϵ = 0.5 (based on MOJAVE model fitting reported in Kovalev et al.2005). and ψjet = −104°. The limiting brightness temperature of 4.8 × 1011 K, obtained from the visibilities at the longest baselines (B> 600 Mλ), agrees well with the estimate based on model fitting the source structure.

The effect of correcting for the core elongation is illustrated in Fig. 5, where the correcting factor ϕ_ϵ 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 ϵ = 0.5 (weighted average of the model-fitting results in Kovalev et al.2005) and a jet position angle ψ_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 φ_jet and, in particular, ϵ. 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 ϵ = 0.4 and ψ_jet = −110°) results in corrected T_b,min = 1.1 × 10¹⁰ K and T_b,lim = 2.4 × 10¹⁰ K, with the latter value falling very close to T_b,mod = 2.8 × 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.

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 (λ_obs = 2 cm) MOJAVE sample (Kovalev et al. 2005) and the results of brightness temperature measurements from the 86 GHz (λ_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.

Fig. 6 Comparison of Tb,lim (circular Gaussian approximation) and Tb,mod estimates obtained from the MOJAVE data (open circles). Gray circles illustrate the effect of correcting Tb,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 Tb,lim is estimated from MOJAVE data at Ruv ≥ 0.99 Bmax to restrict the visibility information to the most compact structures. The resulting Tb,lim are on average 4.6 times higher than Tb,mod. The residual logarithmic distribution of the Tb,lim/Tb,mod ratio (inset) can be approximated by the Gaussian PDF with μ = 0.66 and σ = 0.57. At Tb,mod ≲ 1010 K the Tb,lim may be biased by large-scale structure contributions in strongly jet-dominated objects.

3.1. Preparation of the visibility data

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, (B_uv,B_max) 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 B_max found in the data for each individual object. The MOJAVE datasets were clipped at B_uv = 0.99 B_max (i.e., in an annulus located within 1% of B_max). The CMVA survey data, with substantially fewer visibilities per target object, were clipped at B_uv = 0.9 B_max. The resulting average uv distances and visibility position angles in the clipped data are 435 ± 27 Mλ and −88° ± 6° for the MOJAVE data and 2270 ± 660 Mλ and −78° ± 20° for the CMVA data. In each case, the fraction of visibilities selected is ~1 /N_bas, where N_bas 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.

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.

Fig. 7 Correlation between Tb,lim and Tb,mod(θmaj/θmin) 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 Tb,lim is is only 20% higher than the respective corrected values of Tb,mod , and the residual logarithmic distribution of the Tb,lim/Tb,mod ratio (inset) can be approximated by the Gaussian PDF with μ = 0.09 and σ = 0.70.

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

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 & Königl 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 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, θ_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,modθ_maj/θ_min, which has the effect of calculating the size of the core region as $\mathit{\pi }\hspace{0.17em}{\mathit{\theta }}_{\min }^{\mathrm{2}}\mathit{/}\mathrm{4}$. This simple correction brings the two estimates to a very good agreement, as demonstrated by both the average ratio between the two estimates and the distribution of logarithmic residuals of this ratio. The only notable discrepancy between the two estimates is the persistently higher T_b,lim values observed for the low T_b,mod ≲ 3 × 10¹⁰. Such brightness temperatures are typically measured in objects in which the cores are strongly resolved along both axes of the fitted Gaussians, hence they may require a correction for the resolution along the minor axis of the Gaussian as well.

Such a correction can be performed by substituting the fitted effective size, ${\mathit{\theta }}_{\mathrm{comp}}\mathrm{=}\sqrt{{\mathit{\theta }}_{\min }{\mathit{\theta }}_{\mathrm{maj}}}$ of the Gaussian component with its respective resolution limit θ_lim derived in Kovalev et al. (2005). This operation should account for resolving the core region along and across the jet direction as marked by the position angle of the major axis of the fitted Gaussian component. Applying this correction is illustrated by Fig. 8, which shows an excellent agreement between the two estimates of the brightness temperature and indeed effectively brings the low T_b,mod outliers to the main correlation trend.

The examples set by substituting the measured core size by either θ_min or θ_lim indicate that the visibility-based estimates of T_b,lim can be related to brightness temperature estimates obtained from Gaussian model fitting and that the T_b,lim estimates present a reliable account of brightness temperatures measured at the limit of angular resolution of the respective interferometric data sets.

Fig. 9 Comparison of Tb,lim (circular Gaussian approximation) and Tb,mod estimates obtained from the 86 GHz survey data (Lee et al. 2008). Open circles indicate objects with lower limits on Tb,mod. The dashed line marks the one-to-one correspondence between the two estimates. For each object, the Tb,lim is estimated from the 86 GHz data at Ruv ≥ 0.90 Bmax to restrict the visibility information to the most compact structures. The resulting Tb,lim are essentially equal to Tb,mod. The residual logarithmic distribution of the Tb,lim/Tb,mod ratio (inset) can be approximated by the Gaussian PDF with μ = −0.06 and σ = 0.33.

3.3. Results from the CMVA survey data

If T_b,lim is indeed determined by the limiting resolution of the data, the difference between the T_b,lim and T_b,mod estimates should become progressively smaller with increasing q_max = B_max/λ_obs and for model fitting performed with circular Gaussian components (essentially forcing model fitting to be more responsive to the resolution limit). Both these conditions are fulfilled with the 86 GHz data from the CMVA survey.

An analysis of the 86 GHz survey data presented in Fig. 9 demonstrates that at spatial frequencies q ≥ 2 Gλ, the estimates T_b,lim and T_b,mod yield similar brightness temperatures. This indicates that at these spatial frequencies, the visibility distribution is strongly dominated by the most compact structure, which is reflected both in the modelfit results and in the visibility-based estimates of T_b,lim. It can therefore be expected that the visibility-based estimates provide reliable brightness temperature limits for RadioAstron space VLBI measurements (Kardashev et al. 2013) and millimeter VLBI observations (Doeleman et al. 2012), reaching frequency spacings well in excess of 1 Gλ.

3.4. Comparisons with T_b,min

Statistical comparisons of T_b,mod and T_b,lim with the minimum brightness temperature T_b,min can also be made, although the latter estimate may be affected by a general flux density bias in the surveys. We recall that T_b,min only depends on the absolute value of the visibility amplitude, while T_b,lim effectively represents the signal-to-noise ratio (SNR) of the visibility.

The average ratios of T_b,lim/T_b,mod are $\mathrm{2.}{\mathrm{0}}_{-1.1}^{\mathrm{+}\mathrm{2.3}}$ and $\mathrm{1.}{\mathrm{6}}_{-0.5}^{\mathrm{+}\mathrm{0.7}}$ for the MOJAVE and CMVA samples, respectively. These values are similar to the ratios of 2.6 and 2.1 obtained in Sect. 2 from the full-track VLBI data on 3C 345 and NGC 1052. The rather large spread of these average ratios is indicative of random errors in the flux density scales of both surveys (which can result from self-calibration applied during imaging the survey data). The lower bounds of both averages are ≈1, which can either reflect a relatively low SNR of the long-baseline visibilities or suggest that a systematic bias may also affect the flux density scales.

The average T_b,lim/T_b,mod ratios imply SNR_MOJAVE = 3.9 and SNR_CMVA = 2.8 on the baselines at which the brightness temperature limits are estimated. This is indeed similar to the measured SNR on the longest baselines in the survey data.

The average SNR can also be used to calculate the amount of bias in the flux density scale that is needed to bring the survey results closer to the results from the two full-track cases considered. In both cases, a positive flux density bias of about 20% on the longest baselines would reconcile the survey averages with those of the full-track data. Hence, this can also be a viable possibility, in particular for the CMVA data that have been collected in three long observing runs.

The comparison of T_b,min and T_b,mod obtained from the MOJAVE data yields an average ratio T_b,min/T_b,mod = 2.3, while a ratio of ≤1 is expected. This discrepancy probably results from the same potential problems as discussed in Sect. 3.2 in connection with using elliptical modelfits to describe the core region. Indeed, applying either the θ_min or the θ_lim correction described in Sect. 3.2 reduces this ratio to 0.7. This lends further support to the conclusion that elliptical modelfits may not be optimal for estimating the brightness temperature in compact jets because it is effectively determined by the transverse dimension of the flow. Following this line of argument, the CMVA data (fitted by circular Gaussian and providing substantially longer baselines for estimating T_b,min) should yield a better value of the average T_b,min/T_b,mod ratio. This is indeed the case, with the average T_b,min/T_b,mod = 0.5 measured from the CMVA data.

3.5. General implications for T_b measurements

Overall, the comparisons of T_b,lim and T_b,mod made using the MOJAVE and the CMVA survey data indicate that brightness temperature estimates obtained from the visibility flux density at longest baselines (highest spatial frequencies) provide suitable limits on the brightness temperature of the most compact emitting regions in radio sources. These comparisons also suggest that visibility-based estimates may even be more realistic in case of complex structure of the emitting region (i.e., a marginally transversely resolved flow, such as a compact conical jet, or a flow with thread-like patterns embedded).

The same situation may be realized for other types of objects studied with interferometric measurements, for instance, in radio interferometry observations of young supernovae (Marcaide et al. 2009), radio (Dyck et al. 1998) and optical (Cusano et al. 2012; Ohnaka et al. 2013; Arroyo-Torres et al. 2014) interferometry studies of resolved stars, and optical interferometry observations of protoplanetary disks (Malbet et al. 2005), recurrent novae (Chesneau et al. 2007), and active galaxies (Weigelt et al. 2012). In optical interferometry experiments that measure the normalized visibility spectrum V_q/V₀, the zero-spacing flux, V₀, may need to be determined in a separate measurement (cf. Cusano et al. 2012). In all of these cases, brightness temperature estimates can be made with a visibility response calculated for a specific pattern of brightness distribution (see Appendix A). For objects with a complex internal structure, this would correspond to estimating a pattern-averaged brightness temperature, while the brightness temperature of the most compact regions can still be estimated by using the data from the longest baselines and applying the generic Gaussian case described in Sect. 2.

The arguments presented above demonstrate that, for a given interferometric measurement, a range of plausible brightness temperatures of the most compact emitting structure can be determined from the visibility amplitude and its error measured at the longest baselines contributing to the measurement. This procedure effectively addresses the brightness temperature of emitting regions detected at the limiting resolution of the measurement.

In summary, our analysis leads to the following basic conclusions about applying visibility data to constrain the brightness temperature in astrophysical objects:

• 1.

  A measurement of visibility amplitude, V_q, alone is sufficient to constrain the minimum brightness temperature, T_b,min, that can be derived from this visibility under the assumption that the brightness distribution of the target object can be described by a Gaussian function. This holds for both circular and elliptical Gaussians and indeed for any visibility distribution that has an inflection point (for instance, for any visibility distribution that can be approximated by a Bessel function of the first kind).

• 2.

  If the brightness temperature of an emitting region with a putative Gaussian brightness distribution is constrained using an assumed limit on the zero-spacing flux density, V₀, the nature of the resulting constraint depends on the ratio V₀/V_q. For V₀>eV_q, lower/upper limits on V₀ provide lower/upper limits on T_b. For V₀<eV_q, lower/upper limits on V₀ provide upper/lower limits on T_b.

• 3.

  A measurement of visibility amplitude, V_q, and its rms error, σ_q, can be used to calculate the maximum brightness temperature, T_b,lim that can be obtained under assumption that the emitting region is marginally resolved.

• 4.

  At spatial frequencies higher than ~200 Mλ, the interval [T_b,min,T_b,lim] provides a reliable bracketing for the brightness temperature of the most compact structure in the target object.

• 5.

  Applying the analysis to samples of radio sources with brightness temperatures derived from model fitting the compact structure indicates that the brightness temperature in the relativistic jets may be largely determined by the transverse dimension of the flow.

• 6.

  The visibility-based estimates of brightness temperature can offer a suitable tool for constraining this physical parameter in a wide range of experiments in which a full reconstruction of the source structure is not feasible.

Acknowledgments

This research has made use of data from the MOJAVE database that is maintained by the MOJAVE team (Lister et al. 2009, AJ, 137, 3718) and of data obtained with the Global Millimeter VLBI Array, which consists of telescopes operated by the MPIfR, IRAM, Onsala, Metsahovi, Yebes and the VLBA. The VLBA is an instrument of the National Radio Astronomy Observatory, a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.

References

Appendix A: Visibility limits on the brightness temperature derived for specific patterns of brightness distribution

Visibility limits on brightness temperature can be obtained by combining the expression for brightness temperature, $\begin{array}{ccc}{\mathit{T}}_{\mathrm{b}}& \mathrm{=}& \frac{\mathit{h}\hspace{0.17em}\mathit{\nu }}{\mathit{k}}{\ln }^{-1}\left(\mathrm{1}\mathrm{+}\frac{\mathrm{2}\hspace{0.17em}\mathit{h}\hspace{0.17em}{\mathit{\nu }}^{\mathrm{3}}}{\mathit{I}\mathrm{\left(}\mathit{r}\mathrm{\right)}\hspace{0.17em}{\mathit{c}}^{\mathrm{2}}}\right)\mathrm{=}\frac{\mathit{h}\hspace{0.17em}\mathit{\nu }}{\mathit{k}}{\ln }^{-1}\left(\mathrm{1}\mathrm{+}{\mathrm{ℐ}}_{\mathrm{b}}\right)\hspace{0.17em}\\ & \mathrm{\to }& \end{array}$(A.1)with the visibility function, V, calculated for a specific, circularly symmetric pattern of brightness distribution, I(r), observed with an interferometer at a spatial harmonic, q, which yields a visibility measurement V(q) and its rms error σ(q). Here h is the Planck constant, k is the Boltzmann constant, c is the light speed, and ν is the frequency of measurement.

For the Planck regime, the term ℐ_r is derived for each pattern. Full expressions for T_b are given for the Rayleigh-Jeans regime. The general form of brightness distribution pattern is chosen to be I(r) = η_dS_totf(r), where S_tot ≡ V(0) is the total or zero-spacing intensity of the pattern, and f(r) and η_d are the respective pattern-specific radial dependence of brightness and its normalization factor (generally dependent on the size d of the pattern). The normalization, η_d, is chosen so that it provides a Fourier transform ℱI(r) of the form V(q) = V(0) ℱf(r). To simplify the expressions derived, I_r, V₀ and (V_q,σ_q) denote I(r), V(0) and (V(q),σ(q)) in the discussion below.

With the adopted normalization, a generic solution for the size d_q of the pattern can be determined by solving ℱ I_r = V_q/ (V₀) for d. The minimum brightness temperature supported by the visibility measurement V_q is obtained by substituting I_r with I_r(d → d_q) in Eq. (A.1) and finding the value of V₀ that minimizes the respective T_b given by this equation (e.g., deriving V₀ from the condition dT_b/ dV₀ = 0).

To derive the maximum measurable brightness temperature, the characteristic size, d_lim, of the pattern is obtained from the relation ℱI_r = V_q/ (V_q + σ_q). The maximum measurable brightness temperature T_b,lim is then calculated by substituting I_r in Eq. (A.1) with I_r(d → d_lim).

Results of the calculations are presented in Table A.1 for a circular Gaussian component, a disk of uniform brightness, and a spherical shell of finite thickness. The characteristic size d represents, respectively, the full width at half maximum of the Gaussian component, the diameter of the disk, and the outer diameter of the shell. The spherical shell is further defined by its thickness δ_r = (1 − α)d/ 2 (with 0 ≤ α ≤ 1). Thus, α = 0 describes an infinitely thin shell and α → 1 describes a filled, optically thin sphere. Calculations for the spherical shell are made separately for the average shell brightness and for the peak brightness I_r,peak in the shell, realized at r = αd/ 2, with the resulting I_r,peak = 6 S_tot/πd² (1 − α³)^1/2.

All Tables

All Figures

	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.
In the text

Fig. 2 Comparison of the visibility amplitude distribution and Gaussian modelfit representation of the compact structure in a compact structure-dominated 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 Mλ, where λ is the observing wavelength). These spatial frequency ranges are best suited to directly estimate the brightness temperature from the visibility measurements.

In the text

Fig. 3 Limiting, Tb,lim (top row of datapoints) and minimum, Tb,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 Mλ in extent. Both the original and the averaged rows of the brightness temperature indicate that at q ≳ 150 Mλ, the interval [Tb,min,Tb,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 Tb,min and Tb,lim made over these ranges constrain the respective Tb,mod estimates well.

In the text

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 ϵ. 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 ϵ taken in steps of 0.1 between ϵ = 0.1 and ϵ = 0.9.

In the text

Fig. 5 Limiting, Tb,lim (top row of datapoints) and minimum, Tb,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 ϵ = 0.5 (based on MOJAVE model fitting reported in Kovalev et al.2005). and ψjet = −104°. The limiting brightness temperature of 4.8 × 1011 K, obtained from the visibilities at the longest baselines (B> 600 Mλ), agrees well with the estimate based on model fitting the source structure.

In the text

Fig. 6 Comparison of Tb,lim (circular Gaussian approximation) and Tb,mod estimates obtained from the MOJAVE data (open circles). Gray circles illustrate the effect of correcting Tb,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 Tb,lim is estimated from MOJAVE data at Ruv ≥ 0.99 Bmax to restrict the visibility information to the most compact structures. The resulting Tb,lim are on average 4.6 times higher than Tb,mod. The residual logarithmic distribution of the Tb,lim/Tb,mod ratio (inset) can be approximated by the Gaussian PDF with μ = 0.66 and σ = 0.57. At Tb,mod ≲ 1010 K the Tb,lim may be biased by large-scale structure contributions in strongly jet-dominated objects.

In the text

Fig. 7 Correlation between Tb,lim and Tb,mod(θmaj/θmin) 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 Tb,lim is is only 20% higher than the respective corrected values of Tb,mod , and the residual logarithmic distribution of the Tb,lim/Tb,mod ratio (inset) can be approximated by the Gaussian PDF with μ = 0.09 and σ = 0.70.

In the text

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

In the text

Fig. 9 Comparison of Tb,lim (circular Gaussian approximation) and Tb,mod estimates obtained from the 86 GHz survey data (Lee et al. 2008). Open circles indicate objects with lower limits on Tb,mod. The dashed line marks the one-to-one correspondence between the two estimates. For each object, the Tb,lim is estimated from the 86 GHz data at Ruv ≥ 0.90 Bmax to restrict the visibility information to the most compact structures. The resulting Tb,lim are essentially equal to Tb,mod. The residual logarithmic distribution of the Tb,lim/Tb,mod ratio (inset) can be approximated by the Gaussian PDF with μ = −0.06 and σ = 0.33.

In the text