RADIO ASTROMETRY OF THE TRIPLE SYSTEMS ALGOL AND UX ARIETIS

VLBI visibilities from all epochs were calibrated and imaged using the NRAO AIPS software package (Greisen 2003), except for the 1983.5-1994.4 UX Arietis positions that were analyzed using the SPRINT software package (cf. Lestrade et al. 1999). We followed standard VLBI amplitude and delay/rate calibration procedures. Geometric delays introduced by small errors in the predicted values of the Earth Orientation Parameters (EOP) were corrected by downloading tables of the measured EOP for the time of each observation. We also corrected ionospheric and tropospheric delays—these are discussed in Section 3.3 below.

The phase calibrator sources used in the observations are compact extragalactic core-jet sources (e.g., Mutel et al. 1998). We made self-calibrated images of the calibrators (S/N ∼ 5000:1) which we used to recompute the fringe solution. In principle, the core position of a calibrator could shift with frequency. However, Fomalont et al. (2011) observed four ICRF calibrators from 8 to 43 GHz and found the core positions are coincident to within 0.02 mas. Hence, we have assumed that the core centroids of the calibrators have a negligible shift from 8 to 15 GHz.

For the Algol observations, we also corrected for source motion within the observing period. Since the radio emission in Algol is associated with the K sub-giant (Lestrade et al. 1993), it may move nearly 2 mas during a 12 hr observation, especially if the observation is centered near one of the eclipses. Images made without correcting for this motion would result in a smeared-out source along the direction of motion. Using the already well-determined orbital elements of the inner binary in Algol, we introduced a time-varying phase-tracking correction to the visibilities (using AIPS task CLCOR) equal to the K star's position offset in the inner binary orbit.

In order to determine radio centroid positions, we made Stokes I images using target visibilities that had been corrected from linearly interpolated delay-rate and phase solutions of calibrator scans bracketing each target scan. We then self-calibrated these images in order to improve image fidelity and better discern the source structure. If the source was unresolved, we fitted a Gaussian to the centroid of the emission (AIPS task JMFIT) and used the Gaussian FWHM/2 as the position uncertainty estimate. For epochs with a resolved double structure we found the centroid of each radio lobe and used the midpoint between the lobes as the radio position for that epoch. In both cases, the position uncertainties were increased if the calculated position uncertainty due to tropospheric and ionospheric delays (Section 3.3) exceeded the image-based estimate.

We combined our own VLBI observations, as well as those from the NRAO data archive, with radial velocity data (Duemmler & Aarum 2001; Hill et al. 1971, 1993; Güdel et al. 1999) and differential positions from the Fourth Catalog of Interferometric Positions of Binary Stars (Hartkopf et al. 2010) to determine the best-fit values for the orbital parameters designated "variable" in Tables 3 and 4. We used the Nelder–Mead simplex (a.k.a. "amoeba") algorithm (Press et al. 1992; Nelder & Mead 1965) to minimize the sum-squared difference between the measured positions and radial velocities compared to those predicted by the trial parameters. This algorithm has the advantage that it does not need to calculate partial derivatives to determine the direction of steepest descent, which is problematic when dealing with the transcendental orbital equations. It also changes the step size based on the detected steepness of the target function in parameter space, allowing it to converge quickly at first, but also to a within a very small tolerance once it approaches a minimum.

Like many such minimization schemes, the simplex algorithm can become trapped in a local minimum in parameter space, resulting in a non-ideal solution. To mitigate this problem, we ran the algorithm multiple times with different initial values for the parameters. The orbital and astrometric parameters of the Algol system had already been measured by various means (Lestrade et al. 1993; Carlos & Popper 1971; Hill et al. 1971; Zavala et al. 2010) and produced good agreement with our observations, so our initial values spanned a small volume of parameter space centered on the accepted values to allow for a similar, but competing set of parameters based on our more accurate VLBI data. However, the orbit of the third component in UX Arietis and the constant linear proper motion of the center of mass of the system had not been previously determined. This made it necessary to search a larger volume of parameter space. Inclusion of radial velocity and separation/P.A. (i.e., optical speckle and interferometric) data, which are indifferent to proper motion, helped to refine our solutions. Our final solution, except for a few exceptions as described in Section 4, agrees with all the data within one or two standard deviations.

An ideal interferometer with projected baseline length B has a positional accuracy given by (Thompson et al. 1986)

$$\begin{equation} \sigma = \frac{1}{2\pi }\ \frac{1}{{\rm S/N}}\ \frac{\lambda }{B}, \end{equation} \tag{ 1 }$$

where S/N is the signal-to-noise ratio of the target source and λ the observing wavelength. This ideal estimate, of order several μas for centimeter-wavelength VLBI, is almost never realized in practice since instrumental and intervening atmospheric effects degrade the phase of the incoming signals.

Phase-referenced astrometric observations using modern VLBI arrays typically have very small antenna position and timing uncertainties. Neither of these uncertainties contribute significantly to the astrometric accuracy so long as the switching time between calibrator and source is smaller than the coherence time. The two most important sources of position uncertainty result from path length fluctuations in the troposphere and the ionosphere (e.g., Fomalont 1995; Ros & Reid 2005; Pradel et al. 2006). A good phase-referencing scheme (i.e., with a short enough cycle time) should compensate for these delays, but for sub-mas astrometry they can still be a source of error and should be addressed.

The daytime ionospheric delay is dispersive (τ∝λ²) and is highly dependent on time of day. At 10 GHz, the daytime zenith ionospheric delay is quite large, typically 1 ns, corresponding to an excess path length 30 cm. Fortunately, since 1998 there have been several global databases of ionospheric delay maps available for online download. These maps are generated every 2 hr and are derived from observed Global Positioning System (GPS) delays (Ros et al. 2000). We have corrected all post-1998 VLBI data for these delays (AIPS task TECOR). Ros et al. (2000) found that the residual ionospheric uncertainty after this correction is less than ±0.15 ns (4.5 cm) at 8.4 GHz on global baselines.

The tropospheric delay is non-dispersive, and consists of a dry component well predicted by the local atmospheric pressure, and a highly variable wet component whose value can exceed 10 cm excess path length. Both GPS delay maps and water vapor radiometers have been used to measure and correct for the wet component (e.g., Snajdrova et al. 2006), reducing the uncertainty to about 3 cm. However, although all VLBA sites measure the local atmospheric pressure and hence the dry component accurately, they are not equipped with water vapor radiometers, so there is no on-site measurement of wet component delay. Rather, the correlator applies a seasonally based average value, which can be in error by several cm on a given day, especially for humid sites such as St. Croix.

For phase-referenced observations on a single baseline, these path delay uncertainties result in a position uncertainty given by (see Reid et al. 1999)

$$\begin{equation} \delta \theta = \left(\frac{c\ \delta \tau _0}{\lambda }\right) \tan Z \sec Z \cdot \delta Z\cdot \Theta, \end{equation} \tag{ 2 }$$

where Z is the mean zenith angle, δZ is the zenith angle difference between calibrator and target, δτ₀ is the zenith path length uncertainty, and Θ is the baseline angular resolution.

In Figure 1 we show a representative plot of the positional uncertainty expected from ionospheric and tropospheric delay uncertainties as a function of frequency for a global VLBI array using the nodding technique at mean zenith angle 70° and target-calibrator angular separations 1° and 4°. We have also assumed a tropospheric uncertainty ±0.1 ns (3 cm) and mean daytime ionospheric uncertainty of ±0.1 ns (3 cm). Nighttime ionospheric delay fluctuations are much smaller (∼10% daytime values) and are unimportant at frequencies above a few GHz.

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It is clear that at 15 GHz, the tropospheric contribution dominates, resulting in position uncertainties 0.12 mas at 1° calibrator-target separation and 0.5 mas at 4° separation. At 8.4 GHz, the daytime ionospheric and the tropospheric contributions are nearly equal, resulting in a total uncertainty of 0.18 mas at 1° separation and 0.68 mas uncertainty at 4° separation. Unless given otherwise, we use these uncertainties as formal uncertainties for all VLBI measurements in Table 1.

4.1.1. Outer Orbit

Figure 2 shows the position of the radio-loud secondary (Algol B) on the sky, after correction for proper motion and parallax. The model trajectory is shown as gray line, while the observed and predicted positions at each observing epoch are shown as blue error bars and red ×'s, respectively. Our outer orbital solution agrees very well with Zavala et al. (2010), including the orientation of the Algol C orbit, which differed from earlier determinations (see Zavala et al. for discussion). We note that the derived mass of the tertiary component (1.57 ± 0.01 M_☉), although determined to high accuracy, depends on the masses of A and B, which we have assumed are exactly known.

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In Figure 3 we compare our model Algol C orbit to optical observations of Algol tabulated in the Fourth Catalog of Interferometric Measurements of Binary Stars (FCIMBS) (Hartkopf et al. 2010). Most of these observations did not include an absolute P.A. calibration, and thus contain a 180° ambiguity. The authors attempted to settle this ambiguity when possible using the orbital elements available at the time (Balega et al. 1984; Bonneau 1979; McAlister & Fekel 1980; Pan et al. 1993), but we found that many of the catalog positions needed to be corrected by 180° in order to be consistent with the orbital solution found by Zavala et al. (2010), which did employ an absolute P.A. calibration. It also appears that a subset of the observations, interspersed in time with the rest of the data, is shifted to the northwest of their model positions by ∼10 mas. We cannot explain the offset at this time, but the uncertainty of these positions may be underestimated. With these considerations, the optical observations of Algol agree well with our model orbit.

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4.1.2. Inner Orbit

To fit the parameters of the inner binary, we used only our high-accuracy 15 GHz data, since the position uncertainty of the rest of the data is twice as large and provides little additional constraint of the inner binary orbit. Figure 4 shows the orbit of Algol B in the inner binary center of mass frame, along with the mean observed (blue error bars) and model (red ×'s) positions at the six observing epochs observed at 15 GHz (experiment BM267). The orientation of the inner binary, Ω = 48° ± 6°, agrees with the earlier VLBI result of Lestrade et al. (1993; 52° ± 5°) and the more recent optical determination on Zavala et al. (2010; 474 ± 52). We also find i = 99° ± 5°, confirming the conclusion of Zavala et al. (2010), who found that the inner orbit is retrograde.

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4.1.3. Algol's Radio Morphology: Co-rotating Coronal Loop Centered on KIV Star

Unlike Algol, the orbital elements of a third component in the UX Arietis system were not previously known, although the presence of a third component is not unexpected—companions in close binaries are quite common. Tokovinin et al. (2006) found that 63% of SBs that they surveyed had at least a tertiary companion. For SBs with a period less than 3 days, the fraction with companions is 96%, suggesting that the shorter-period systems exchanged angular momentum with their companions, shortening their orbital periods.

In the UX Arietis system, a third component had been inferred by detection of nonlinear proper motion (e.g., Boboltz et al. 2003) and spectral lines of a possible third component (Aarum Ulvås & Engvold 2003a). However, the putative third component's orbit was poorly determined. The only published orbital solution consisted of two quite different models: a 10.7 year circular orbit and a 21.5 year highly eccentric orbit (Duemmler & Aarum 2001). Therefore, it was necessary to undertake a comprehensive parameter search, constrained by the VLBI positions, third component radial velocity data (Duemmler & Aarum 2001; Aarum Ulvås & Engvold 2003a; Glazunova et al. 2008), and optical interferometric observations in the FCIMBS (Hartkopf et al. 2010). The search was complicated by the relatively high proper motion of UX Arietis (μ ∼ 100 mas yr⁻¹), which magnifies a small fractional proper motion uncertainty over decades into a large aggregate position shift.

4.2.1. Outer Orbit

Figure 5 shows the radio position of UX Arietis on the sky plane, from epoch 1980.0 (origin) to 2009.8. The blue circles are observed positions relative to 1985.0. The undulating shape of the trajectory, previously interpreted as an acceleration (Lestrade et al. 1999; Boboltz et al. 2003; Fey et al. 2006), results from the reflex motion of the inner binary in a 111 year period outer orbit. The position of the radio component in the frame of the outer binary center of mass in shown in Figure 6. The gray line is the predicted position of the inner binary center of mass, with blue error bars indicating observed positions and red ×'s the model positions after correction for proper motion and parallax.

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We next compare the model solution with FCIMBS interferometer observations. Figure 7 shows the model position of the tertiary component with respect to the inner binary center of mass (green line). The red and corresponding blue symbols are the model and observed positions of the tertiary component as listed in the FCIMBS. These observations did not use an absolute P.A. calibration, and thus also have the 180° ambiguity (see Section 4.1.1). The listed positions prior to 2002 showed Algol C curving clockwise across the sky from the northeast to the north. Our VLBI observations over the same time range, with proper motion removed, show the inner binary following a trajectory with the same direction of curvature. A different value of the proper motion would not affect the optical observations, but also cannot change the direction of curvature of the VLBI observations. Thus, we conclude that the P.A. measurements prior to 2002 are flipped by 180°.

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Finally, Figure 8 shows the predicted radial velocity of the inner binary center of mass (gray line) and tertiary component (green line), along with corresponding radial velocity observations (Duemmler & Aarum 2001; Massarotti et al. 2008; Glazunova et al. 2008). In general, the agreement with all three data sets is very good, with the exception of the outlier optical observation at epoch 1991.25 from the Hipparcos catalog (Perryman et al. 1997).

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4.2.2. Inner Orbit

Since both inner binary components of UX Arietis exhibit chromospheric activity (Aarum Ulvås & Engvold 2003a), it is unclear whether the radio emission originates from a single component, both components or perhaps an accretion region between the two stars. The orbital elements are well characterized by spectroscopic observations except for the orientation (Ω) and the inclination (i ∼60°; Duemmler & Aarum 2001), which is degenerate about 90° reflection with respect to radial velocity curves. Hence, we fixed the other orbital parameters and, as with Algol, used only the 15 GHz observations to solve for Ω and i.

We confirm that the K0IV primary is the source of the radio emission in UX Arietis by testing the quality of the fit assuming the alternatives: that the emission is either from the secondary or from a location between the two components. The alternative assumptions both yielded unacceptable fits (χ²_ν = 7.6 and χ²_ν = 2.8, respectively), while the solution for radio emission from the primary was very good, yielding a good fit at the 99.7% confidence level.

Figure 9 shows the best-fit orbital solution overlaid with contour maps of the radio emission at each epoch. We obtain a tentative solution for Ω (82° ± 14°). Our data cannot constrain the solution in i, as can be seen by the fact that the timing of our observations were all very close to the nodal points in the inner binary orbit. Fixing the inclination at 592 (Duemmler & Aarum 2001) produces a slightly better fit than the radial velocity-degenerate value (i = 1208), thus we use the former value in our final solution.

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4.2.3. Source Structure

The UX Arietis outer orbit solution can be used to derive the mass of the tertiary component, m_c = 0.75 ± 0.01 M_☉, where the formal uncertainty is contingent on fixed values for the inner binary masses, as given in Table 4. This mass corresponds to a spectral type K1 main-sequence star (Zombeck 1990). Duemmler & Aarum (2001) estimated the mass of UX Arietis' tertiary component between 0.30 < m₃ < 0.46 M_☉, for their circular and elliptical orbit solutions, respectively, although they reject the smaller mass, since it would be an M star too faint to be responsible for the spectral lines ascribed to the tertiary. Aarum Ulvås & Engvold (2003a) fit the continuum spectrum of the UX Arietis system and found that after correction for the primary and secondary spectra, the derived color index is best fit to a K5 main-sequence star, although they cast doubt on this identification, since the implied radius (0.83 solar radii) is in the range of a K1-type star. Our derived mass agrees with the hotter type, so perhaps starspots on the tertiary are influencing its color index as is suspected with the primary (Aarum Ulvås & Engvold 2003b).

Peterson et al. (2010) recently described a polar loop model for Algol in which the double-lobed radio components are associated with footpoints of an active region of meridional magnetic field lines originating at the star's magnetic poles. A similar geometry was already suggested for this system by Franciosini et al. (1999). We now consider whether the single radio component seen in the UX Arietis system can be interpreted within this model.

A large, stable polar spot has been detected on the K primary of UX Arietis (Vogt & Hatzes 1991; Elias et al. 1995). These polar spots are a common phenomenon: starspots on magnetically active cool stars preferentially appear near the poles (e.g., Hatzes et al. 1996; Bruls et al. 1998). This preference for high latitudes may be due to rapid rotation, which leads to the Coriolis force dominating over buoyancy forces in the dynamics of magnetic flux tubes. As a consequence, flux tubes in the stellar convection zone migrate nearly parallel to the axis of rotation and thus surface at high latitudes (Schuessler & Solanki 1992). Hence, magnetically active stars with rapid rotation exhibit magnetic flux eruption at high latitudes and polar starspots.

The VLBI images of UX Arietis indicate that the radio centroid lies in the hemisphere facing away from the G star. The same structure was seen in single-dish observations of UX Arietis during a strong flare (Trigilio et al. 1998). This is also consistent with observations of a similar late-type binary, HR1099, in which Doppler imaging (Donati et al. 1992) and X-ray monitoring indicate that the active regions lie in the hemisphere facing away from the inactive companion. Audard et al. (2001) suggest that this is because tidal locking of the rotation of the K star alters the internal dynamo in such a way that strong activity on the hemisphere facing the inactive star is suppressed.

Since the inner binary orbital plane is inclined at 30° to the observer's line of sight (i ∼ 60°), if the magnetic axis is close to perpendicular to the inner binary's orbital plane, only one pole will be visible. The polar loop model posits that the radio emission arises above both polar regions, but in this case, the far-side emission will be largely occulted by the K star. This geometry may explain the absence of two radio lobes.