AN ACCURATE FLUX DENSITY SCALE FROM 1 TO 50 GHz

To monitor system gains, a small, known amount of noise power is added to the receivers, square-wave modulated at a frequency of 10 Hz. This additional power is synchronously detected and measured at the correlator. Changes in the detected calibration power reflect changes in electronic gains, provided the injected power is itself stable. To demonstrate the effectiveness of this system, we show in Figure 1 the correlation amplitudes (visibilities) for a point source when the gain corrections have, and have not been applied. These data were taken in clear weather over a 40 hr period at a frequency of 4885 MHz, at which the elevation-dependent antenna gain corrections and atmospheric opacity effects are negligible. The application of the switched power correction has reduced the apparent flux density variations from ∼50% to less than 1%.

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Observations made in the early years of this program clearly indicated that the primary source of error in the gain calibration process at frequencies above ∼10 GHz is due to offsets in antenna pointing. The accuracy of blind VLA pointing at night under calm conditions is about 10''. On sunny days, pointing offsets induced by differential solar heating of the antenna structure can exceed 1'—larger than the FWHM of the VLA primary beam at the highest frequency band.

To remove such errors, the method of "referenced pointing" was implemented on the VLA in the mid-1990s (Kestevan 1994). In this method, a pointing determination is made on a nearby calibrator (or on the target source itself, if sufficiently compact), and the offset correction applied to subsequent observations. Utilization of this technique can reduce pointing error to as low as 3'', providing the calibrator object is close enough—ideally within 5° in azimuth and elevation, and preferably less than 15°—and the weather calm and clear. The offset determination is normally made at a frequency within the 3 cm band, as this has the best combination of sensitivity and resolution, and the results applied to observations made at other frequency bands. This methodology was first employed in our program in 1995, and was utilized for all following observations.

The VLA antennas lose forward gain at high frequencies from a combination of deformation of the antenna surface and a bending of the quadrupod feed leg structure. The loss of gain at high frequencies is significant, but is repeatable and can be corrected for with good accuracy, as illustrated by the fits shown in Figure 2, taken from 43 GHz observations in 2011 October. Referenced pointing was applied to these observations—the scatter about the best-fit gain represents the current limit for VLA pointing—approximately 4'', rms.

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The major goals of this program included both an accurate determination of the variability of the sources, and an accurate determination of the flux density ratios between the sources over as wide a frequency range as possible. As both the VLA's original, and the new WIDAR correlator provide correlation products for two different frequency tunings within each frequency band, we strove to fix one of these tunings to a value which remained unchanged throughout the duration of the project, and to place the other at the maximum separation allowed by the electronics at the time of observing. Due to changes in the VLA's electronics over thirty years, evolution of the goals of the project, and changes in our understanding of the location and impact of RFI, the specific frequencies utilized changed over the observation period, most notably at the lower frequencies. The frequencies chosen for all sessions up to 2006 are given in Table 3.

Major changes in VLA electronics due to the VLA's upgrade permitted a much wider range in frequency selection for sessions after 2006. Observations after this date added new frequencies, but always retained those listed in the last line of Table 3. The 90 cm band observing system was disabled in 2009, so no observations were possible at this band following the 2008 session. Due to the implementation of new receivers, no 2 cm band data were taken in the 2010 session.

The general methodology was to observe each source approximately hourly at each band. To prevent holes in coverage, the durations of most sessions were at least 24 hr. Given the restrictions in antenna slew rate, source separation, and source elevation, we typically obtained 5–10 individual "snapshot" observations of each object at each band, with each observation's duration being 30–60 s. This methodology permits accurate estimation of and correction for elevation-dependent telescope characteristics, and allows a good estimation of the errors in the flux density. This multiple observation strategy also ensures minimal impact on the project goals in case of short-duration telescope malfunctions, bad weather, and sporadic interference. To permit careful editing of discrepant data, the visibility integration time was kept short, typically 3.3 s or less.

All editing, calibration, imaging, and image analysis were done within the AIPS software package (Greisen 2004). The same editing and calibration methodology was employed for every source at each band for every session, to maximize uniformity and reliability of the results. The sequence of operations was as follows.

• 1.  
  Initial editing. Most sources are strong enough to allow a simple initial editing to purge obviously invalid data—typically showing up as abnormally low amplitudes. On occasion, RFI or solar interference was seen, with the affected data removed in the same way.
• 2.  
  Estimation of atmospheric opacity. For the observations made between 1995 and 2007, antenna dips were done every six hours at 1.3 cm and 0.7 cm bands to permit estimation of atmospheric opacity (Butler 1996). These values, or the values from an atmospheric model using surface and seasonal weather data for the years where no dip data were available (Butler 2002), were applied to the visibility data.
• 3.  
  Removal of atmospheric and instrumental phase perturbations. Phase fluctuations were removed by application of phase self-calibration for each source, using a point-source model for the unresolved sources, or a "clean-component" model for resolved sources. In the latter case, the model was derived from the data, using an imaging/self-calibration loop. For both, the phase solutions were determined for each time integration, and the results applied to the corresponding visibility data. The signal-to-noise ratio for the low-frequency planetary observations was generally too low to permit this procedure; however, at these frequencies, the phase stability was always good enough that such detailed phase fluctuation removal was not necessary.
• 4.  
  Removal of data taken with bad pointing. The most difficult error to detect—and the error which ultimately limits the accuracy of high-frequency observing—was that due to residual pointing errors. We did this by first making an amplitude calibration solution for each source, using either a point-source model, or a "clean-component" model. We then examined these solutions, flagging those data for which the corresponding power gains deviated by more than a fixed fraction, varying from 1% for the low-frequency bands to 25% for the highest frequency band. In nearly all cases, the flagging is on data whose amplitudes are too low—consistent with the cause being pointing offsets. To prevent bias, all sources at any one band were flagged with the same threshold. We emphasize that these amplitude gain solutions were never applied to the data—they were utilized only to identify bad data.
• 5.  
  Estimation and removal of elevation gain dependency. The elevation gain dependency was estimated by jointly fitting the calibration gain solutions for the source flux densities and an antenna-based elevation dependency of the form G = G₀ +  G₁sec(z) +  G₂sec²(z). The resulting gain function was then applied to all the visibility data. For all but the last two sessions, the objects utilized were the four sources which are nearly unresolved at all frequencies in our observations: 3C48, 3C138, 3C147, and 3C286. For the final two sessions, approximately 30 additional point sources were added to the source list and were included in this calibration step. Separation of the elevation gain dependency from individual pointing offsets required two or three iterations of this and the previous step.
• 6.  
  Final gain calibration. After all discrepant data were removed, and the elevation dependency measured and corrected for, a final amplitude calibration solution was made using a single source, either 3C147 or 3C286. The average gain from these solutions was then applied, without any temporal trend, to all sources in the data set. Because the primary data products from these observations are the flux density ratios between the sources, the value of the flux density assigned to this source is not important. The only rationale for this final step is one of convenience—to put the source visibilities on a scale close to the correct one.

We made no effort to remove any temporal changes in the amplitude gains, except in the final observing session, which included an additional 30 point sources. These were added for the purpose of investigating the VLA's gain stability over wide ambient temperature changes. A very small effect—less than 2%—was observed at some frequencies, and removed with a 6 hr smoothing window.

For each source at each session and frequency, the flux density—on the arbitrary scale as noted above—and a likely error were determined with the following procedure—utilized for both unresolved and resolved objects.

• 1.  
  The source flux density was determined from an image utilizing all the data for one session. The image provides two estimates, one from the sum of the clean components, the other from integration over the source brightness. For the latter, the integration was over a window sufficient to cover the source solid angle. No base level adjustment is needed, as these interferometric images have zero mean in regions absent detectable brightness. For sources which were unresolved or marginally resolved, a third estimate from a fit to a two-dimensional Gaussian profile to the image was made. In all cases, these estimates agreed to within the 1σ image noise errors, generated as described below.
• 2.  
  For the four planets, an additional estimation was made from fitting the complex visibility to an appropriate limb-darkened disk model (Butler & Bastian 1999; Butler et al. 2001). A comparison of this estimate to that from the image total flux generated by integration of the brightness over the planet's disk showed excellent agreement to within the 1σ error estimate derived as described below.
• 3.  
  An estimate of the error in the derived mean flux density was made from the dispersion of the N individual "snapshot" observations of each source at each band. For each snapshot observation, an estimate of the flux density was made by summing the clean components generated by the deconvolution. The dispersion, σ, was determined from these N estimates, and the error estimate of the mean, σ_u, was then determined from $\sigma _\mu = \sigma /\sqrt{N-1}$. Typical 1σ errors for the planet Mars for a single epoch are ∼4% at the 10 and 20 cm bands, 1%–2% for the 6, 3, 1.5, and 1 cm bands, rising to 7% at the high-frequency end of the 0.6 cm band.

This method for determining the measurement error is valid provided the errors in the N observations are statistically independent. We believe this is the case, since the primary source of error in the flux determination at the higher frequencies is from residual pointing errors, which change over a timescale of an hour or less. Furthermore, any residual errors in the elevation gain dependency correction are likely due to variable atmospheric opacities, which we expect are also variable on timescales of an hour or less. Since the observations of any individual source are typically separated by an hour or more, we can consider each observation independent of the others.

At low frequencies (90, 20, and 10 cm bands), the errors are dominated by residuals in the deconvolution procedure since the background sources are never perfectly removed, and by errors in the corrections for gain variations. Again, we believe that the characteristic timescale for changes in these errors is typically one hour or less, so that the individual observations are expected to have independent errors.

The primary data products of this program are the flux density ratios between all of the sources. These were determined utilizing the flux densities derived as described above. The errors of these ratios were determined using standard propagation of error methodology, assuming the errors in the flux density of one source are independent of those for the others.

The flux density of Mars is variable as a function of time, due to both geometrical factors (distance, and sub-Earth latitude and longitude) and to seasonal factors (because of the waxing and waning of the polar caps). It is thus necessary to use a thermophysical model to calculate the expected emission from the planet. Such thermophysical models exist, for example those of Wright (1976) and Rudy et al. (1987). We choose to use the latter model (Rudy et al. 1987; Muhleman & Berge 1991; Sidher et al. 2000) because it is based on observations at centimeter wavelengths which overlap those of the absolutely calibrated WMAP measurements of Weiland et al. (2011).

WMAP measured the brightness temperature of Mars at five frequencies (22.85 GHz (K band), 33.11 GHz (Ka band), 40.82 GHz (Q band), 60.85 GHz (V band), and 93.32 GHz (W band)) during seven "seasons" of observing (Weiland et al. 2011). Three of these frequency bands overlap those available at the VLA, thus providing a good way to transfer the WMAP flux density scale to the higher frequencies of the VLA. By utilizing the thermophysical emission model described above, and incorporating lower-frequency observations described in Baars et al. (1977), the entire radio frequency spectrum from ∼1 GHz to 50 GHz can be placed on an absolute calibration scale.

The spectral flux density measured by a total power telescope for any particular pointing is the antenna beam-weighted integral of the sky brightness over the full sky:

$$\begin{equation} S = \int BPd\Omega , \end{equation} \tag{ 2 }$$

where B is the sky brightness, P is the normalized antenna power pattern, and the integration is over 4π steradians. The increment in spectral flux density, S_p, provided by a uniform, discrete source of solid angle Ω_p (assumed much smaller than the antenna main beam), brightness B_p, and opacity τ—hence the value actually measured by the telescope—is

$$\begin{equation} S_p = [B_p-B_{{\rm bg}}(1-e^{-\tau })]\Omega _p, \end{equation} \tag{ 3 }$$

where B_bg is the sky brightness behind the foreground object. The second term on the right-hand side accounts for the attenuation of the background sky by the source. For an optically thick source, such as a planet, this becomes

$$\begin{equation} S_p = (B_p - B_{{\rm bg}})\Omega _p. \end{equation} \tag{ 4 }$$

WMAP employs a differencing system with a pair of feed horns, one for the target, the other as a reference. A straightforward argument shows that the difference power obeys the same relationships given in Equations (3) and (4).

Thus, flux density measured by the telescope is proportional to the brightness difference between the object and the background sky. To this value must be added that amount occulted by the object in order to determine the actual flux density. These considerations apply equally to interferometers, as the uniform background sky is resolved out by the interferometric coherence pattern.

For our application, both the planet brightness B_p and the background sky brightness B_bg are described by the Planck function

$$\begin{equation} B(\nu, T) = \frac{2h\nu ^3}{c^2}\frac{1}{e^{h\nu /kT}-1}, \end{equation} \tag{ 5 }$$

where T is the brightness temperature of the source. The WMAP brightness temperatures of Mars published in Weiland et al. (2011) are given in terms of the "Rayleigh–Jeans" brightness temperature, defined as

$$\begin{equation} T_{{\rm RJ}} = \frac{\lambda ^2 S_\nu }{2k\Omega }, \end{equation} \tag{ 6 }$$

and have not been adjusted by the blockage of the sky background. Since the Rudy model provides an estimate of the total brightness of the source in true "Planck" units, we must adjust the published WMAP values for both the blockage and the difference in units.

From Equations (4), (5), and (6), we can relate the true brightness temperature of an optically thick object, T_P, to the value assigned using the Rayleigh–Jeans approximation, T_RJ, accounting for the flux blocked by the planet's disk:

$$\begin{equation} \frac{h\nu }{kT_P}=\ln \left[1+\frac{1}{(e^{h\nu /kT_{{\rm bg}}}-1)^{-1} +\frac{kT_{{\rm RJ}}}{h\nu }}\right], \end{equation} \tag{ 7 }$$

where T_bg = 2.725 K is the cosmic microwave background (CMB) brightness temperature (Mather et al. 1994, 1999; Fixsen 2009). We find that the published WMAP values for Mars must be increased by 2.78, 2.82, 2.86, 3.00, and 3.32 K for the five frequency bands used by WMAP. The WMAP observations of the brightness temperature of Mars for all frequencies and sessions are shown in Table 6, along with the values adjusted as described above.

Given the WMAP observation dates, we then calculate the Rudy model brightness temperatures for each observation, averaged over a full rotation of Mars, since the Rudy model has variations as a function of planetary longitude, based on surface thermophysical properties, and the Weiland et al. (2011) results do not include a time of day of the observations. The Rudy model values calculated in this way are also shown in Table 6. Figure 4 shows the fits along with the model calculations for all five bands.

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With the WMAP observations and the Rudy model predictions for a given frequency, we then find a correction factor to apply to the Rudy model to make the best agreement with the WMAP observations, given by

$$\begin{equation} f = \frac{\sum w_iM_iD_i}{\sum w_iM_i^2}, \end{equation} \tag{ 8 }$$

where M_i are the Rudy model values, D_i are the WMAP observed values, and w_i are the WMAP weights (w_i = 1/σ²).

Table 7 shows the derived corrections at each band, and an error estimated from the dispersion in the individual ratios—for these estimates we did not use the "season 1" WMAP observations in this fit, since they were taken during a global dust storm (Weiland et al. 2011). These storms are known to change the planet's emission, and are not included in the model.

We find little evidence of the unmodeled frequency-dependent emissivity reported by Weiland et al. (2011), what they reported may have been a byproduct of their extrapolation of the Wright model to inappropriate frequencies. Because we find no strong indication for variation of the correction factor across these five bands, we have applied the average correction factor of 0.975 from the lowest three WMAP frequencies to the spectral flux densities derived from the Rudy model.

The Rudy model calculates the total spectral flux density from the planet Mars. To determine that actually measured by the VLA requires subtracting the background flux blocked by the planet's disk, as described in Section 7.1. The background brightness temperature T_bg is

$$\begin{equation} T_{{\rm bg}} = T_{{\rm gal}} + T_{{\rm cmb}}, \end{equation} \tag{ 9 }$$

where T_cmb = 2.725 K and T_gal is the brightness temperature of the Galactic background. This is widely variable, depending on Galactic longitude and latitude. For our observations, Mars was always sufficiently far from the Galactic plane that we can utilize the expression T_gal ∼ 2.5 ν^−2.7_G K, where ν_G is the frequency in GHz, to estimate the Galactic contribution. In general, the Galactic synchrotron brightness is negligible compared to the CMB at frequencies above 4 GHz (where the galactic brightness is typically 60 mK). For observations below 2 GHz, the Galactic background can contribute a small offset (0.4 K at 2 GHz, and 1.5 K at 1 GHz), but even this offset can be neglected for our observations of Mars, as it is small compared to the planet brightness, and much smaller than the errors in the observations due to the low flux density of Mars below 2 GHz.

The results of this procedure are shown in Table 8, giving the apparent flux density of Mars for the dates, times, and frequencies shown. Also shown are the approximate coordinates, in equatorial and Galactic coordinates, of Mars during our observations.

For each observing session and frequency, the ratio between our observed spectral flux density of Mars, based on the arbitrary calibration scale described earlier, and the calculated apparent spectral flux density of Mars shown in Table 8 was used to adjust the derived flux densities of the four non-varying sources 3C123, 3C196, 3C286, and 3C295 to the Mars-based scale. For the 90 cm band ratios, where the VLA cannot detect Mars, we have used the Scaife & Heald (2012) value for 3C196 as the reference source.

3C48 is a steep-spectrum quasar at redshift z = 0.367. Its radio structure is quite compact with a maximum extent of about 12, as shown by the 01 resolution VLA image shown in Figure 6. The brightest regions of this source have been imaged by VLBA, MERLIN, and EVN observations (An et al. 2010). The source is a commonly used flux density calibration source for the VLA. The plots in the right panel of Figure 6 show that the source has undergone significant changes in flux density over the past 30 years. A period of slow decline or quiescence was followed by a small flare, more prominent at the higher frequencies, beginning in 2009, which is now declining. As this source is a commonly utilized flux density calibration source, we have fitted its changing spectrum with a cubic polynomial fit for each of the 18 sessions. The results of this are given in Table 11.

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3C123 is a radio galaxy at redshift z = 0.218. With an angular size of 44'', it is the largest of the objects in this study, and is of limited use for the calibration of high-resolution interferometers. We included it in this program primarily because it is a well-known calibrator source at low frequencies. The structure of the source, with 3'' resolution at 22.5 GHz, utilizing the data taken in this program, is shown in Figure 7.

3C123 is one of the non-variable objects that we have identified. The right panel of Figure 7 shows the source's secular flux density evolution, based on our polynomial expression for 3C286. Notable is a small (∼1%) but significant drop at 1465 MHz, beginning in 2008. This drop is also seen at 1275 MHz in 2008, but at no other frequency utilized between 2008 and 2011. We believe the drop is real, but have no explanation for its origin.

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3C138 is a compact steep-spectrum quasar at redshift z = 0.759. VLBI images show that it has structure on a maximum scale of 062, with a highly polarized, one-sided jet leading from the nucleus, and some weak structure on the other side (Cotton et al. 1997a). These structures are shown in a 60 mas resolution VLA image at 43 GHz shown in Figure 8. Due to its high flux density and small angular size, this source is commonly utilized by the VLA for amplitude gain calibration.

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3C138 is not one of the Baars et al. (1977) sources, probably because it was known to be significantly variable. Our data show that this is indeed the case, with the slow decline noted after 1983 abruptly terminated with a large increase beginning in 2003, as shown in the right-hand side of Figure 8. The flare, with the proportional increase being the greatest at the higher frequencies (nearly doubling the flux density at 43 GHz), reached a peak in 2010, and is now quickly subsiding. We have fitted this variable spectrum with a cubic polynomial function for each of the 18 session, whose coefficients are given in Table 11.

3C147 is a compact steep-spectrum quasar at redshift z = 0.545. VLBI imaging shows the structure of the maximum extent of 200 mas (Rossetti et al. 2009). A VLA high-resolution image at 43 GHz, made with high-resolution data, is shown in the left panel of Figure 9. The small angular size and high spectral flux density of 3C147 have resulted in it being commonly used as a flux density scale calibrator by the VLA. This source has undergone significant changes in flux density over time, as shown in the right panel of Figure 9. For all bands other than 20 cm, a rise of 5%–10% was seen between 1990 and 1998, again with the larger increases being seen at higher frequencies. This increase was followed by a sudden drop of similar magnitude lasting until 2004, then another short and rapid rise, of similar magnitude, to 2006. Since then, the flux density has been dropping steadily. We have fitted the variable flux densities of this source with cubic polynomials for each of the 18 observing session, the results of which are shown in Table 11.

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3C196 is a quasar at redshift z = 0.871. The source comprises a pair of compact lobes symmetrically placed about a weak core, with the maximum angular extent of about 7'', as shown in the 22 GHz, 1'' resolution image in the left panel of Figure 10, made from the data taken for this program. In the right panel is shown the secular variation. This source is one of the non-variable objects we have identified. Because of this, its small angular size, and remarkably straight spectrum (Scaife & Heald 2012), 3C196 is an excellent flux density calibration source for low-resolution observations for frequencies up to ∼10 GHz.

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3C286 is a compact steep-spectrum quasar at redshift z = 0.846. High-resolution VLA images show that the source has a steep-spectrum extension of length ∼25 to the SW, and a smaller knot of emission to the east, as shown in Figure 11. VLBI, EVN, and MERLIN images show a highly polarized linear structure of the maximum 50 mas extent (Cotton et al. 1997b; Jiang et al. 1996; Akujor & Garrington 1995). The milliarcsecond structure of this source has some very unusual features. The VLBA imaging shows that the compact central structure shown in the figure is steep spectrum, resolved to the VLBA, and highly polarized throughout. There appears to be no optically thick central nuclear emission. It is doubtless the lack of the nucleus, and the uniform polarization visible in the maps of Cotton et al. (1997b) that make this object an extraordinarily stable and useful calibrator.

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3C295 is a radio galaxy at redshift z = 0.464. This source is a small double of ∼5'' extent with very weak central nucleus, contributing less than 1% of the total flux density at 15 GHz (Taylor & Perley 1992). An image of the source at 23 GHz with 1'' resolution, utilizing our data, is shown in the left panel of Figure 12. In the right panel is shown the secular variation of the spectral flux density.

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The weakness of the nuclear core emission suggests that this source should have a stable flux density, as noted by Ott et al. (1994). Our observations confirm this suggestion, showing no variation in its flux density to a level below 1% over the duration of our program. 3C295 is an excellent flux density calibration source for low-resolution interferometers below ∼10 GHz.

The planetary nebula NGC 6572 was added to this program in 2000, based on its apparent similarity to NGC 7027. The source is about 13'' in angular extent, with very diffuse emission extended to the north and south, as shown in Figure 13. The lack of sharp edge emission makes the source a poor calibrator, as longer interferometer spacings provide insufficient amplitude to permit a stable gain solution for antennas located at the ends of the array. There is a possible trend in this source's flux density over a 12 year timescale, as shown in the right-hand panel of the figure, showing the data, and a linear fit. The results of the fits are shown in Table 12.

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The bright, nearby planetary nebula NGC 7027 is one of the Baars et al. (1977) sources. An image of the source at 43 GHz, made with our data and at 06 resolution is shown in Figure 14. Because of its large angular size, this source is not a good primary calibrator for high-resolution interferometers. NGC 7027 has long been known to be increasing in flux density at the frequencies where it is optically thick, and to be decreasing at frequencies where it is optically thin. Zijlstra et al. (2008), using the data taken in this program up to 2006, utilized these secular variations, combined with optical observations of the linear expansion velocity and photo-ionization models, to place tight limits on the age and distance to the planetary nebula, and on the mass of the central star. The observations taken since 2006 show the flux density continuing to change as expected, but there is now some evidence of a deceleration in the secular increase at 1465 MHz, and also a possible deceleration in the secular decrease at 4885 MHz. The results of a weighted fit are given in Table 12, and plotted in the figure. We emphasize that the statistical significance for deceleration is low—further long-term monitoring will be needed to establish the reality of this result.

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MWC 349 is a binary Be star of 24 separation. We added this source to our monitoring list, following suggestions that it may be suitable as a flux density calibrator at long-millimeter and centimeter wavelengths. Its radio structure has been extensively studied by Tafoya et al. (2004) showing 03 extent at 10 GHz, and 1'' at 3 GHz. A new, deeper image at 23 GHz with 11 resolution with our data is shown in Figure 15. The source is known to be variable in the visible, IR, and mm wavelengths (Tafoya et al. 2004, and references therein).

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Our observations show there are peculiar changes in the flux density of significant magnitude seen simultaneously over most bands. The most notable was in the 2003 February observations, when a drop in flux density of about 10% was seen at all frequencies except 1465 MHz. This drop cannot be due to pointing errors or atmospheric absorption, since these always affect high-frequency observations much more strongly. A similar, but smaller, rise was noted in the 2010 December observations. It would appear that MWC 349 has small but significant (∼10%) changes in flux density. The changes are of short duration—less than one year. Because of these, the source is not a suitable absolute flux density calibrator for cm or mm wavelengths.