ASTROMETRICALLY REGISTERED SIMULTANEOUS OBSERVATIONS OF THE 22 GHz H₂O AND 43 GHz SiO MASERS TOWARD R LEONIS MINORIS USING KVN AND SOURCE/FREQUENCY PHASE REFERENCING

The full details of SFPR are provided in two VLBA memos (Dodson & Rioja 2009; Rioja & Dodson 2009) and in two journal papers (Rioja & Dodson 2011; Rioja et al. 2011). The method is generally applicable, but has particular relevance in cases where the frequency is so high that conventional phase referencing (PR) cannot, or struggles to, succeed. In PR, the antennas nod rapidly between a calibrator (assumed to be achromatic with no core shift of its own, in both PR and SFPR) and the target (Alef 1988; Beasley & Conway 1995). For frequencies above 22 GHz, the coherence time becomes so short that it becomes increasingly difficult to slew from one source to the other and back within this period. Additionally, the number of calibrators strong enough to be detected, yet within the same tropospheric "patch" as the target, rapidly approaches zero. SFPR avoids this limit by observing at multiple frequencies and assuming that the atmosphere is dominated by non-dispersive contributions. At millimeter wavelengths this is certainly the case, as the troposphere is the limiting source of error, and the troposphere is non-dispersive. By self-calibrating the observations of the target source at the low (reference) frequency and applying those solutions to the higher frequency data, we correct for the dominant non-dispersive terms. We call the resultant calibrated high-frequency data set the "frequency phase transferred" (hereafter FPT), or troposphere-free, target data set. However, the remaining dispersive errors prevent the recovery of the position of the target source using a Fourier inversion at this stage. The residual dispersive contributions arising from the ionospheric propagation and other effects need to be removed using the alternating observations of the second source. This source can be visited less frequently and can lie significantly farther from the target than in conventional PR, as the residual terms these observations correct for are much weaker and vary much more slowly than the dominant tropospheric and other non-dispersive effects. After this second calibration step, the resultant sfpr-calibrated data sets can be Fourier inverted to provide the astrometrically registered target image between the two observing frequencies.

As stated, the first step does not provide astrometry-ready data, as shown in Figure 1. Following Equation (2) in Rioja & Dodson (2011), the FPT data set has residual phases ϕ^FPT that can be expressed as the sum of the contributions:

$$\begin{eqnarray} \phi ^{{\rm FPT}} &=& \phi ^{{\rm high}} - R\,. \phi _{{\rm self-cal}}^{{\rm low}} \nonumber \\ & =& \phi _{{\rm str}}^{{\rm high}} + \left(\phi _{{\rm geo}}^{{\rm high}} - R\,. \phi _{{\rm geo}}^{{\rm low}}\right) + \left(\phi _{{\rm tro}}^{{\rm high}} - R\,. \phi _{{\rm tro}}^{{\rm low}}\right) \nonumber \\ &&+ \left(\phi _{{\rm ion}}^{{\rm high}} - R\,. \phi _{{\rm ion}}^{{\rm low}}\right) + \left(\phi _{{\rm inst}}^{{\rm high}} - R\,. \phi _{{\rm inst}}^{{\rm low}}\right) \nonumber \\ &&+ 2\pi (n^{{\rm high}}-R\,.n^{{\rm low}}), \end{eqnarray} \tag{ 1 }$$

where $\phi ^{{\rm low}}_{{\rm geo}}, \phi ^{{\rm low}}_{{\rm tro}}, \phi ^{{\rm low}}_{{\rm ion}}$, $\phi ^{{\rm low}}_{{\rm inst}}$, $\phi ^{{\rm high}}_{{\rm geo}}, \phi ^{{\rm high}}_{{\rm tro}}, \phi ^{{\rm high}}_{{\rm ion}}$, and $\phi ^{{\rm high}}_{{\rm inst}}$ are the contributions arising from geometric, tropospheric, ionospheric, and instrumental inadequacies in the delay model for, respectively, either the low or high frequency. n^low and n^high are the integer number of full phase rotations, or ambiguities in the phase, for each of the frequencies and R is the frequency ratio between the high and low frequencies. At millimeter wavelengths, the geometric and tropospheric phase residual terms will scale with frequency and therefore the terms in parentheses will cancel; instrumental terms are assumed to have been dealt with using observations of a primary calibrator, whereas the ionospheric terms, although non-zero, are slow-changing in time and equal over a large angular separation. When R is an integer, the phases from the ambiguity terms n are always integer multiples of 2π, so are not relevant to the analysis. However, when R is not an integer phase, jumps are introduced every time n^low changes. That is, at every introduction of an ambiguity in phase, there will be a step in the phase of ϕ^FPT, which cannot be corrected from the observations of the second source; as for the calibrator, the number of ambiguities is independent along the different lines of sight. Clearly, the solution for non-integer frequency ratio SFPR is equivalent to that required to avoid introducing any untracked lobe rotations in the visibility data. Below, we describe our data reduction procedure to retain the astrometric information, even when the ratio between the observing frequencies is not an integer number.

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We used the NRAO AIPS software package for the data reduction (Griesen 2003). The information on measured system temperatures, telescope gains, and estimated bandpass corrections for the individual antennae were used to calibrate the raw correlation coefficients of the continuum calibrators and of the target spectral line source. Figure 2 shows a schematic of the steps after this initial calibration for each source and for each frequency, indicating the transfer of the calibration tables. Each of the four columns in that figure represents a different stage in the analysis.

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Column 1 is for the reference source at the reference frequency, i.e., 4C39.25 at 22 GHz. The first step is to use the AIPS task FRING, which is a global self-calibration algorithm (Cotton 1995), to estimate residual antenna-based VLBI phases and its partial derivatives with respect to frequency (group delay, τ), and time (phase delay rate), on the calibrator (4C39.25) data set. These terms result from unaccounted contributions from the atmospheric propagation and from errors in the array geometry during the data correlation. The performance of the digital filters in the KVN optics (Han et al. 2013) and backend (Oh et al. 2011) is extremely good, particularly in the absence of instrumental phase offsets introduced by the electronics at each IF and band. This enables the straightforward use of all the IFs as an effective single bandwidth for the estimation of a more precise group delay (since σ_τ∝1/bandwidth) (Schwab & Cotton 1983). With a reasonable signal-to-noise ratio and 432 MHz spanned bandwidth, we expect delay accuracies of tens of picoseconds. We used the delays derived from this calibrator and frequency, solving for a single delay across all the IFs in all the subsequent analyses of the other sources and frequencies. Finally, for this source and frequency, we applied the small (<10°) phase corrections that were constant across the whole experiment to align the IFs. These were derived with the phase self-calibration task CALIB; normally, this would be done with FRING on a single scan of a strong calibrator, but we applied this as a refinement after the application of the initial calibration across all the observations, as the single-band delay corrections required for the KVN receiver were essentially zero.

With these calibration steps completed, we (Column 2, Figure 2) calibrated and separated the IF with the spectral-line R LMi H₂O maser data and applied the Doppler corrections using the LSR standard of rest and radio velocity definitions (with SPLAT and CVEL; Reid 1995). Additional corrections to the amplitudes could be derived with ACFIT at this point; these were of the order of 10% and are possibly from the uncorrected atmospheric opacity. The 22 GHz phase-referenced spectral-line data could then be imaged and a channel with a compact, strong, point-source-dominated emission was selected for further analysis. It was immediately obvious where the peak of the brightness fell in the image, and a single model component at the site of this peak was used for a phase refinement of the data, generated by CALIB, followed by re-imaging and deconvolution. The latter step, which is mainly correcting for the differential static atmospheric contributions, would lose the astrometrical registration if we were not able to find a reasonable a priori position for the conventionally phase-referenced data and use that for the calibration model. Therefore, one of the requirements for successful non-integer SFPR is that an adequate initial phase-referenced image can be formed. As the accuracy we require for this position is a few milliarcseconds, this is not an overly demanding requirement. The outcome is that the first step produces a conventional PR image of the H₂O masers, and the second provides phase refinements to that data set while retaining the location of the maximum emission. Therefore, the H₂O maser cube is registered to the 4C39.25 calibrator. Figure 3 plots the spectra of the recovered emission (i.e., the flux density in the clean components) from the H₂O image cube with a red dotted line.

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The 43 GHz 4C39.25 data (Column 3, Figure 2) was first adjusted so that the reference frequency value was set, using the AIPS task PUTHEAD, to 43.620 GHz, which is double the reference frequency value of the lower frequency and outside the observed band. The frequency axis reference pixel was also changed. This ensured that the labeling of the data was correct while preserving an integer reference frequency ratio, and that the phases could be doubled without consideration of the ambiguities. These two steps are specifically required for non-integer frequency ratio analyses. Then, we doubled the value of the phases measured from the delay observations of 4C39.25 at 22 GHz using our own external script to produce an FPT data set. As a word of warning to users, we mention that we do not recommend the AIPS task SNCOR for this purpose, as it does not do a good job when there are multiple IFs unless the separations between the IFs also scale by the same frequency ratio factor. However, it functions correctly when there is only one IF per frequency band. We calibrated the remaining dispersive phase corrections, which were changing on ∼hour-long timescales, with CALIB.

Once again (Column 4, Figure 2), we calibrated and separated those IFs with the R LMi SiO maser data using SPLAT, using the scaled 4C39.25 delays from 22 GHz and the dispersive phase corrections from 4C39.25 at 43 GHz. The Doppler and amplitude corrections were calculated with CVEL and ACFIT as before. We transferred the phase solutions from the H₂O maser and scaled these values by the frequency ratio between the H₂O and the SiO maser v = 1 and v = 2, that is, 1.939 and 1.926, respectively. This data was then ready for imaging. However, the images were not of very good quality and, looking back at the solutions, it was clear that in the observing gap at zenith a turn of phase had been missed on the Ulsan antenna in the second half of the experiment. We added the phase jump that would have been introduced by this missed turn of phase ($2\pi \,(2-\nu _{\rm SiO}/\nu _{{\rm H}_2{\rm O}})$ for the two transitions, equal to 27° and 22°, respectively) using SNCOR. Figure 4 shows the measured phase corrections (i.e., the phase residuals) for the H₂O maser and the derived corrections for the SiO v = 1 maser before and after adding the lost phase turn. With these steps completed, we were able to directly image the SiO masers at the two transitions, which were now accurately registered to the H₂O emission and that in turn to 4C39.25. We imaged the data via the normal methods. Given the limited number of baselines, we found that the most reliable deconvolution came with fitting a small number of model components to the data, which we did in difmap (Shepherd et al. 1994). Figure 3 plots with blue and green solid and dashed lines the spectra of the recovered emission (i.e., the flux density in the clean components) from the SiO image cubes. The images are shown in Section 4. Plotting of the cubes for the figures in this paper was performed with Miriad (Sault et al. 1995).

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Our analysis followed the steps required to produce PR maps of the H₂O maser emission, with respect to 4C39.25 and SFPR maps of the two transitions of the SiO maser emission, with respect to the H₂O maser emission. We present images of the astrometrically registered velocity-averaged (moment zero) cubes of the H₂O and SiO maser emission (Figure 5) and astrometrically registered velocity cubes of the two transitions of SiO maser emission (Figure 6). The image coordinates are relative to the peak of the SiO maser emission in v = 1. On Figure 5, we have overlaid the position of the optical source R LMi, as measured by Hipparcos, for the observing epoch (2011.17) with 1σ errors (shown with a large cross; van Leeuwen 2007). This position is listed in Table 1, which summarizes all the astrometric positions we have measured. The Hipparcos position has errors that are dominated by the measurement errors in the proper motion, multiplied by the two decades since those observations were made. We measured the peak of the H₂O emission with IMFIT in the moment zero map, which is listed in Table 1 with the formal errors of that fit. The much larger systematic errors are discussed in the following section. Similarly, the peak of the SiO v = 1 emission was measured and the position and errors are in the same table. We fitted a ring to a mask of the combined v = 1 and 2 emission using a least-squares maximization of the overlap of a ring model with the pixels in the image in either transition, with emission greater than 10% of the peak in that transition. The center of this ring should be the site of the center of the optical emission of R LMi. We take the width of the ring as the uncertainty in that position. The ring is shown as the circle and the center and the uncertainty are shown as a small cross in both Figures 5 and 6. The values are listed in Table 1.

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The errors in Table 1 are the formal errors in the fitting to the data. For the VLBI data, these are completely dominated by the systematic effects, which we review here.

The H₂O maser peak emission position is conventionally phase referenced to 4C39.25, which has a very accurate absolute reference position with errors of about 0.1 mas (Fey et al. 2004), but an angular separation of 59 from the R LMi target. Because of this large separation, this would not normally be considered a good PR calibrator. With such an observing setup, even in good weather, one would expect of the order of 30° of phase noise from the dynamic tropospheric contributions, more than 100° of phase noise from the static tropospheric contributions, 15° of phase noise from the static ionospheric contributions, and less than 1° of phase noise from the dynamic ionospheric contributions (following the formulae in Asaki et al. 2007). However, we note that the atmospheric coherence of the KVN appears to be much better than those for typical cases (Lee et al. 2014). This may be due to better coherence between the atmosphere at the different sites, which could be from the short baselines or that the weather moves down off the Chinese landmass with little disruption. Various methods exist to reduce the dominant static tropospheric contributions, as reviewed in Honma et al. (2008), but these were not followed in this experimental setup, as we did not originally intend to attempt to achieve absolute astrometric connection. Nevertheless, on inspection of the data, we realized that it should be possible, as the peak of the brightness was clearly identifiable. To confirm that errors were minor, we compared the peak flux before and after self-calibration. This gives a measure of the scale of the random errors. If these errors are small, one can be confident that the data is largely coherent and the astrometry is preserved. The fractional flux recovery we found for the whole data set was 86%, and for the data taken in the first half of the experiment it was 96%. For our analysis, we took the position of the water maser peak from a model fit of a single component to the data, limited to regions where the phase residuals were the smallest, that is, the data between Yonsei and Tamna, plus Ulsan in the first half of the experiment, which had a fractional flux recovery of 91%. Given the large angular separation between the sources, we take a generous upper bound for the positional errors to be the resolution, 5×3 mas. In future work, we will attempt to improve on these limits.

The optical star R LMi has a relatively poor a priori position from Hipparcos. The uncertainty is in the proper motion measurement, which, given that the Hipparcos data is referenced to the observational epoch of 1991.25, results in a position error of ∼50 mas. However, as we phase reference the H₂O maser emission to 4C39.25, the positional errors in our analysis are less. Taking this as the dominant source of error in our analysis, we use it to determine the final error bars in our astrometric measurements between the maser transitions. We use the relationship of Δν/ν.Δθ, where Δθ is the astrometric accuracy of the reference (i.e., the H₂O emission) and Δν is the frequency gap between the transitions. In this case, the registration would, with Δθ of 5 mas, have an uncertainly of 2.5 mas between the H₂O and the SiO and 35 μas between the SiO v = 1 and 2 masers. In both cases, the typical separations between first the H₂O and the SiO masers and second the SiO v = 1 and 2 masers are much greater than these errors, which will allow us to have faith in any separations detected. SiO rings are close to the surface of the star and water masers maybe 10 times further out (10¹⁵ cm), which implies a typical separation of 60 mas between the sites of the H₂O and the SiO maser emission for a source at 1 kpc; that is, the astrometric errors would be of the order of 4%. An error in astrometry of 35 μas between the SiO maser transitions implies a physical scale of 5× 10¹¹ cm at 1 kpc, or 0.5% of the typical ring diameter (10¹⁴ cm).

The retention of absolute astrometric registration in the VLBI data allows us to compare the center of the SiO maser emission, where the star in R LMi should appear, and the Hipparcos position for this star. Measuring from the astrometric image derived, we find the position offset between these to be −33 and 41 mas. The error budget on this measurement consists of the errors in the fit to the center of the circle, which is 5 mas, combined with the absolute astrometric accuracy of the H₂O emission, giving a total estimated error of 7 mas. Given the 20 years since the Hipparcos reference epoch, the errors in proper motion from the Hipparcos data dominate. The center of the circle of SiO maser emission falls within the 1σ error ($\sigma _{\rm RA_{\rm pm}}$ = 35 mas) in the Hipparcos value along μ_αcosδ and within the 2σ error ($\sigma _{\delta _{\rm pm}}$ = 25 mas) in the Hipparcos value along μ_δ (van Leeuwen 2007). Forthcoming observations from the Gaia satellite (Perryman et al. 2001) will be able to confirm this VLBI result.

Combining the VLBI position we derived in 2011.17 with uncertainties in the fit and those of the H₂O maser emission to give a total of 7 mas with that of Hipparcos from 1991.25 with uncertainties of 1 mas, we can derive the proper motion of R LMi over the twenty year baseline, and this is given in Table 1.

We subtracted from (both) the reported VLBI positions the core shift (and any other structure phases) from 4C39.25, which, because it was assumed to be achromatic in the analysis, would contaminate these results. These effects are far below the levels of accuracy achieved in this analysis, but will become significant when global baselines are included.

• 1.  
  SFPR requires that the calibrator is within the same ionospheric region (the "patch") for dispersive contributions. Fortunately, the patch size scales with frequency, so for high frequencies it will be very large. The actual scale size at millimeter frequencies is under investigation; extrapolating from measurements at lower frequencies (e.g., Thompson et al. 2001) is not possible, as the approximations are no longer valid. In Rioja & Dodson (2011), we have successfully used calibrators 10° from the target.
• 2.  
  SFPR for spectral-line sources must use the delays derived from the continuum calibrator source, as the line of sight corrections to the delays are not derivable on the line source. Therefore, the antenna position errors must be sufficiently small (less than a few centimeters) so that they do not introduce rapid phase changes, which could not be tracked on the other source.
• 3.  
  Non-integer frequency ratio SFPR requires the delay solutions to be of sufficient quality to allow extrapolation to a reference point outside of the observing band. A 0.05 ns error in the delay would introduce a 9° error in the phase for a point 500 MHz from the frequency reference pixel. We suggest that one should try to ensure that this extrapolation is no farther than the bandwidth spanned.
• 4.  
  Non-integer frequency ratio SFPR requires that the phase solutions can be tracked across wraps of phase ambiguities, so continuous observations without gaps are desirable.
• 5.  
  Non-integer frequency ratio SFPR also requires that the positions of the target source at the lower frequency are accurate. The absolute error in the target position at the reference frequency, when applied to the target frequency, is diluted by the fractional frequency span Δν/ν. For SFPR, this fractional span approaches an integer so that the errors in the registration between the frequencies can approach the absolute positional error. If needed, the positions can be derived by conventionally PR the low-frequency data. In this case, the source switching needs to be sufficiently fast to allow PR at the low frequency.
• 6.  
  For arrays that do not (yet) support simultaneous multichannel receivers (e.g., the VLBA) fast frequency switching will still work with this approach. Fast frequency switching is limited in the maximum frequencies that can be supported (as the variation in the rates cannot be higher than the cycle time of the frequency switching). In our experience, fast frequency switching between frequencies as high as 43 and 86 GHz is achievable in good conditions with a switching cycle period of 1 minute. This is about the maximum speed possible with the VLBA sub-reflector rotator.