Imaging the dynamical atmosphere of the red supergiant Betelgeuse in the CO first overtone lines with VLTI/AMBER
K. Ohnaka, G. Weigelt, F. Millour, K.-H. Hofmann, T. Driebe, D. Schertl, A. Chelli. F. Massi, R. Petrov, Ph. Stee
aa r X i v : . [ a s t r o - ph . S R ] A p r Astronomy&Astrophysicsmanuscript no. 16279 c (cid:13)
ESO 2018March 17, 2018
Imaging the dynamical atmosphere of the red supergiantBetelgeuse in the CO first overtone lines with VLTI/AMBER ⋆ K. Ohnaka , G. Weigelt , F. Millour , , K.-H. Hofmann , T. Driebe , , D. Schertl , A. Chelli , F. Massi , R. Petrov ,and Ph. Stee Max-Planck-Institut f¨ur Radioastronomie, Auf dem H¨ugel 69, 53121 Bonn, Germanye-mail: [email protected] Observatoire de la Cˆote d’Azur, Departement FIZEAU, Boulevard de l’Observatoire, B.P. 4229, 06304 Nice Cedex 4, France Deutsches Zentrum f¨ur Luft- und Raumfahrt e.V., K¨onigswinterer Str. 522-524, 53227 Bonn, Germany Institut de Plan´etologie et d’Astrophysique de Grenoble, BP 53, 38041 Grenoble, C´edex 9, France INAF-Osservatorio Astrofisico di Arcetri, Instituto Nazionale di Astrofisica, Largo E. Fermi 5, 50125 Firenze, ItalyReceived / Accepted
ABSTRACT
Aims.
We present one-dimensional aperture synthesis imaging of the red supergiant Betelgeuse ( α Ori) with VLTI / AMBER. Wereconstructed for the first time one-dimensional images in the individual CO first overtone lines. Our aim is to probe the dynamics ofthe inhomogeneous atmosphere and its time variation.
Methods.
Betelgeuse was observed between 2.28 and 2.31 µ m with VLTI / AMBER using the 16-32-48 m telescope configurationwith a spectral resolution up to 12000 and an angular resolution of 9.8 mas. The good nearly one-dimensional uv coverage allows usto reconstruct one-dimensional projection images (i.e., one-dimensional projections of the object’s two-dimensional intensity distri-butions). Results.
The reconstructed one-dimensional projection images reveal that the star appears di ff erently in the blue wing, line center,and red wing of the individual CO lines. The one-dimensional projection images in the blue wing and line center show a pronounced, asymmetrically extended component up to ∼ R ⋆ , while those in the red wing do not show such a component. The observed one-dimensional projection images in the lines can be reasonably explained by a model in which the CO gas within a region more thanhalf as large as the stellar size is moving slightly outward with 0–5 km s − , while the gas in the remaining region is infalling fastwith 20–30 km s − . A comparison between the CO line AMBER data taken in 2008 and 2009 shows a significant time variation in thedynamics of the CO line-forming region in the photosphere and the outer atmosphere. In contrast to the line data, the reconstructedone-dimensional projection images in the continuum show only a slight deviation from a uniform disk or limb-darkened disk. Wederive a uniform-disk diameter of 42 . ± .
05 mas and a power-law-type limb-darkened disk diameter of 42 . ± .
06 mas and alimb-darkening parameter of (9 . ± . × − . This latter angular diameter leads to an e ff ective temperature of 3690 ±
54 K for thecontinuum-forming layer. These diameters confirm that the near-IR size of Betelgeuse was nearly constant over the last 18 years, inmarked contrast to the recently reported noticeable decrease in the mid-IR size. The continuum data taken in 2008 and 2009 revealno or only marginal time variations, much smaller than the maximum variation predicted by the current three-dimensional convectionsimulations.
Conclusions.
Our two-epoch AMBER observations show that the outer atmosphere extending to ∼ R ⋆ is asymmetric and itsdynamics is dominated by vigorous, inhomogeneous large-scale motions, whose overall nature changes drastically within one year.This is likely linked to the wind-driving mechanism in red supergiants. Key words. infrared: stars – techniques: interferometric – stars: supergiants – stars: late-type – stars: atmospheres – stars: individual:Betelgeuse
1. Introduction
Red supergiants (RSGs) experience slow, intensive mass loss upto 10 − M ⊙ yr − , which is very important for understanding thefinal fate of massive stars. For example, our poor understandingof the RSG mass loss makes it di ffi cult to estimate the main-sequence mass range of the progenitors of the Type IIP super-novae, which are the most common type of core-collapse super-novae. The mass loss also plays a significant role in the chem-ical enrichment of galaxies. Despite this importance, there are Send o ff print requests to : K. Ohnaka ⋆ Based on AMBER observations made with the Very LargeTelescope Interferometer of the European Southern Observatory.Program ID: 082.D-0280 (AMBER Guaranteed Time Observation) no satisfactory theories for the RSG mass loss at the moment, asstressed by Harper (2010).Studies of the outer atmosphere, where the winds are accel-erated, are a key to tackling this problem. The outer atmosphereof RSGs has complicated structures. The UV observations ofthe well-studied RSG Betelgeuse ( α Ori, M1-2Ia-Ibe) with theHubble Space Telescope reveal that the hot ( ∼ ∼ O gas in
Ohnaka et al.: Imaging the dynamical outer atmosphere of Betelgeuse in the CO first overtone lines the outer atmosphere, the so-called “MOLsphere”, extending to ∼ R ⋆ with column densities on the order of 10 cm − andtemperatures of 1500–2000 K (e.g., Tsuji 2000a, 2000b, 2006;Ohnaka 2004; Perrin et al. 2004, 2007). The nature of the coolgas in the outer atmosphere of Betelgeuse has also recently beenprobed with mid-IR [Fe II] emission lines (Harper et al. 2009a).Possibly the chromospheric plasma with a small filling factor isembedded in more abundant, cooler gas. This inhomogeneous,multi-component nature of the outer atmosphere is consideredto play a crucial role in driving mass outflows in RSGs. There isobservational evidence for the asymmetric, inhomogeneous na-ture of the circumstellar material. For Betelgeuse, Kervella et al.(2009) found a very faint plume extending to ∼ R ⋆ in the near-IR, while a millimeter CO map shows a blob at ∼ ′′ ( ∼ R ⋆ )away from the star (Harper et al. 2009b). The emission of theCO fundamental lines near 4.6 µ m shows that the circumstellarenvelope is approximately spherical within 3 ′′ (140 R ⋆ ) but withsignatures of mildly clumpy structures (Smith et al. 2009).High-spectral and high-spatial resolution observations ofstrong IR molecular lines are ideal for probing the physical prop-erties of the outer atmosphere. The near-IR interferometric in-strument AMBER (Astronomical Multi-BEam combineR) at theVery Large Telescope Interferometer (VLTI) is well suited forthis goal with its high spectral resolution up to 12000 and highspatial resolution down to 1–2 mas with the current maximumbaseline of 130 m. In 2008, we observed Betelgeuse in the COfirst overtone lines near 2.3 µ m with AMBER (Ohnaka et al.2009, hereafter Paper I). The high spectral resolution of AMBERallowed us to detect salient signatures of inhomogeneities in theindividual CO lines and to spatially resolve the gas motions in astellar photosphere (and also MOLsphere) for the first time otherthan the Sun.However, in 2008, we obtained data only at six uv points,which are insu ffi cient for image reconstruction. In order to ob-tain a more complete picture of the dynamics of the inhomoge-neous outer atmosphere, we carried out new AMBER observa-tions of Betelgeuse with a better uv coverage in 2009. In thissecond paper, we report on the first one-dimensional aperturesynthesis imaging of Betelgeuse in the CO first overtone lines,as well as on time variation in the dynamics of the stellar at-mosphere in an interval of one year. The paper is structured asfollows. The AMBER observations, data reduction, and imagereconstruction are outlined in Sect. 2. We describe the resultsabout the time variations as well as the one-dimensional imagereconstruction in Sect. 3. The modeling of the velocity field pre-sented in Sect. 4 is followed by the discussion on the dynamicsin the extended outer atmosphere (Sect. 5). Conclusions are pre-sented in Sect. 6.
2. Observations
AMBER (Petrov et al. 2007) is the near-IR (1.3—2.4 µ m)spectro-interferometric instrument at VLTI, which combinesthree 8.2 m Unit Telescopes (UTs) or 1.8 m AuxiliaryTelescopes (ATs). AMBER measures the amplitude of theFourier transform—the so-called visibility or visibility ampli-tude and two observables that contain information about thephase of the object’s Fourier transform: di ff erential phase (DP)and closure phase (CP). The DP roughly represents how the ob-ject’s phase in a spectral feature deviates from that in the con-tinuum. Non-zero DP represents information about the photo-center shift in a spectral feature with respect to the continuum. The CP is the sum of the measured Fourier phases around aclosed triangle of baselines (i.e., ϕ + ϕ + ϕ ), not a ff ectedby the atmospheric turbulence. The CP is always zero or π forpoint-symmetric objects, and non-zero and non- π CPs indicatean asymmetry in the object. Moreover the CP is important foraperture synthesis imaging in optical / IR interferometry.Betelgeuse was observed on 2009 January 5 and 6 withAMBER using three ATs in the E0-G0-H0 linear array config-uration with 16–32–48 m baselines (AMBER Guaranteed TimeObservation, Program ID: 082.D-0280, P.I.: K. Ohnaka). As inPaper I, we used the K -band high-resolution mode (HR K) witha spectral resolution of 12000 covering wavelengths from 2.28to 2.31 µ m to observe the strong CO first overtone lines nearthe (2,0) band head. Fringes could be detected on all three base-lines without the VLTI fringe tracker FINITO. We obtained atotal of 54 data sets on two half nights. The data sets taken morethan ∼ uv points align ap-proximately linearly at position angles of 73 ± ◦ . This linear uv coverage allows us to sample the visibility function densely atthis particular position angle and to reconstruct one-dimensionalprojection images as described in Sect. 2.2. Each data set con-sists of 500 frames (NDIT) with each frame taken with a detec-tor integration time (DIT) of 120 ms. Sirius ( α CMa, A1V, K = − .
4) was observed for the calibration of the interferometricdata of Betelgeuse. We adopted the same angular diameter of5 . ± .
15 mas from Richichi & Percheron (2005) as adoptedin Paper I. We only used the calibrator data sets obtained justbefore and after each data set on Betelgeuse. A summary of theobservations is given in Table A.1.We reduced our AMBER data with amdlib ver.2.2 , whichis based on the P2VM algorithm (Tatulli et al. 2007). Some datasets, particularly those measured on the longest baseline and / ornear the CO band head, are too noisy for the analysis. Therefore,we improved the SNR by binning the entire raw data (object,dark, sky, and P2VM calibration data) in the spectral directionwith a running box car function as described in Paper I. We useddi ff erent binnings with these spectral resolutions:1. Spectral resolution = = individual, isolatedCO lines (Sect. 3.4 and 3.5, Fig. 5).3. Spectral resolution = CO band head at 2.294 µ m (Sect. 3.6,Fig. 6).We checked for a systematic di ff erence in the calibrated vis-ibilities and di ff erential / closure phases by taking the best 20%,50%, and 80% of all frames in terms of the fringe SNR (Tatulliet al. 2007). The di ff erence between the results obtained with thebest 20% and 80% frames is typically ∼ ff eren-tial / closure phases do not show this systematic dependence onthe frame selection criterion. Therefore, we included the best80% of all frames for the final DPs and CPs. The errors of the Available at http: // / data processing amber.htmhnaka et al.: Imaging the dynamical outer atmosphere of Betelgeuse in the CO first overtone lines 3 resulting visibilities, DPs, and CPs were estimated in the samemanner as in Paper I.We removed telluric lines from the observed spectra ofBetelgeuse as best as possible by using Sirius as a spectroscopicstandard star. The telluric lines identified in the spectrum ofSirius were also used for wavelength calibration. As a templateof the telluric lines, we convolved the atmospheric transmis-sion spectra measured at the Kitt Peak National Observatory to match the spectral resolutions of the data. The uncertaintyin wavelength calibration is 2 . × − µ m (2.6 km s − ). Wenote that the uncertainties in wavelength calibration for the 2008data and 2006 data in Paper I were mistakenly overestimated.The correct uncertainties in the 2008 data and 2006 data are2 . × − µ m (3.0 km s − ) and 1 . × − µ m (25.1 km s − ),respectively. The good linear uv coverage along the position angle of 73 ◦ shown in Fig. 1 provides an opportunity to reconstruct the so-called one-dimensional projection image, which is obtained byintegrating the object’s two-dimensional intensity distributionalong the direction perpendicular to the linear uv coverage onthe sky (central slice theorem or Fourier slice theorem). In otherwords, this one-dimensional projection image represents thetwo-dimensional intensity distribution compressed or squashedonto the linear uv coverage on the sky. For example, the one-dimensional projection image of a uniform disk is a semi-circle(see also the two-dimensional image of a limb-darkened disk andits one-dimensional projection image shown in Figs. B.1a andB.1c). The reconstruction of one-dimensional projection imageswas first proposed for radio interferometry by Bracewell (1956).Whereas the information in the direction perpendicular to thebaseline vector is lost in one-dimensional projection images,they still provide model-independent information about the geo-metrical extent and asymmetry of the object. The reconstructionof one-dimensional projection images from IR interferometricdata or lunar occultation data has been carried out (e.g., Navarroet al. 1990; Leinert et al. 1991; Tatebe et al. 2006; Chandler etal. 2007).We used the MiRA package ver. 0.9.9 (Thi´ebaut et al. 2008)to reconstruct one-dimensional projection images at each spec-tral channel (details of our image reconstruction procedure aredescribed in Appendix B). We first carried out the image recon-struction using computer-simulated data to examine e ff ects ofthe uv coverage and reconstruction parameters such as the ini-tial model, prior, and regularization scheme on the reconstructedimages. These tests with simulated data are crucial for examin-ing the credibility of aperture synthesis imaging particularly forobjects with complex structures.With appropriate reconstruction parameters determined fromthese tests, we attempted to reconstruct one-dimensional projec-tion images from the observed 162 visibility amplitudes and 54CPs. While this worked well for the continuum, the reconstruc-tion in the CO lines turned out to be very sensitive to the recon-struction parameters. For example, depending on the size of theuniform disk used as the initial model, the reconstructed one-dimensional projection image in the CO lines shows a faint re-gion on the eastern or western side. Therefore, we used the self-calibration technique, which has recently been successfully ap- http: // / sci / facilities / paranal / instruments / isaac / tools / spectra / atmos S K.fits http: // / labo / perso / eric.thiebaut / mira.html plied to AMBER data for the first time by Millour et al. (2011).We added modifications to their technique to deal with some is-sue specific to our data of Betelgeuse as described in AppendixC. This technique allows us to restore the phase of the complexFourier transform of the object’s intensity distribution from theDP measurements. Image reconstruction with the complex visi-bility (i.e., visibility amplitude and phase) removes the ambigu-ity of the solution derived with the visibility amplitude and CPalone.
3. Results
To compare with the K -band continuum visibilities from the2008 data that were derived from the binned data with a spectralresolution 4800, we also derived the visibilities from the 2009data binned with the same spectral resolution. As in Paper I,we selected continuum points shortward of the CO band headat 2.294 µ m. For each data set, we averaged the visibilities overthe selected continuum points. We took the simple mean of theerrors as the errors in the average continuum visibilities withoutreducing by √ N cont , where N cont is the number of the selectedcontinuum points. The reason is that the measurement errors aredominated by the systematic error in the absolute visibility cali-bration and do not become smaller by the averaging. We appliedthis averaging to the 2008 data as well. Since the di ff erent con-tinuum spectral channels correspond to slightly di ff erent spatialfrequencies, we also averaged the spatial frequencies from theselected continuum points.Figure 2 shows the K -band continuum visibilities measuredin 2008 and 2009 as a function of spatial frequency. The figurereveals that the nearly linear uv coverage shown in Fig. 1 enabledus to sample the visibility function quite densely from the firstto the fifth visibility lobe. Uniform-disk fitting to the 2009 dataresults in a diameter of 42 . ± .
05 mas with a reduced χ of3.8. Fitting with a power-law-type limb-darkend disk (Hestro ff eret al. 1997) results in a limb-darkened disk diameter of 42 . ± .
06 mas and a limb-darkening parameter of (9 . ± . × − with a better reduced χ of 2.5. While the reduced χ value isstill higher than 1, Fig. 2 shows that the deviation from the limb-darkened disk is not strong, as found for the 2008 data. Only atthe highest spatial frequency (i.e., the smallest spatial scale) isthe deviation noticeable, but the errors are also large there.The limb-darkened disk diameter derived from the K -bandcontinuum data and a bolometric flux of (111 . ± . × − W cm − (Perrin et al. 2004) lead to an e ff ective tempera-ture of 3690 ±
54 K. We propose this value as an e ff ective temper-ature of the continuum-forming layer, approximately free fromthe e ff ects of molecular lines. Perrin et al. (2004) modeled K -broadband interferometric measurements of Betelgeuse with acontinuum-forming blackbody sphere and an extended molecu-lar shell. They derived 3690 ±
50 K for the continuum-formingsphere. This value excellently agrees with our e ff ective tempera-ture of the continuum-forming layer. Our e ff ective temperature isslightly higher than the 3600 ±
66 K recently derived by Hauboiset al. (2009) from the the H -band observations with the InfraredOptical Telescope Array (IOTA), but both agree within the un-certainties.Figure 2 reveals that the continuum visibilities show noor only marginal time variations between 2008 (green dots)and 2009 (red and blue dots) within the measurement errors.We compare this observational result with the current three-dimensional convection simulation for RSGs by Chiavassa et Ohnaka et al.: Imaging the dynamical outer atmosphere of Betelgeuse in the CO first overtone lines al. (2009). The visibility predicted for 2.2 µ m (Fig. 18 ofChiavassa et al. 2009), which approximately samples the con-tinuum, shows maximum time variations of, for example, ± − ) show no timevariation within the error bars (3–5%), and these error barsare 8–13 times smaller than the predicted maximum variation.While it is not very likely that we observed Betelgeuse at twoepochs when it accidentally showed the same, weak deviationsfrom the limb-darkened disk, this possibility cannot be entirelyexcluded. However, it is also possible that the current three-dimensional convection simulation predicts too pronounced sur-face structures and time variations owing to the gray approxima-tion adopted for the radiative transfer, as Chiavassa et al. (2009)mention. The most direct test for three-dimensional convectionsimulations is to measure the amplitude of the temporal fluctua-tions in the visibility and closure phase as well as the time scaleof fluctuations by long-term monitoring observations and com-pare these with the model predictions. AMBER observations atmore epochs would be necessary to draw a definitive conclu-sion about whether or not Betelgeuse seen in the continuumindeed shows much weaker inhomogeneities and much smallertime variations than predicted by the current three-dimensionalconvection simulation.The deviation from the limb-darkened disk in the K -bandcontinuum visibilities is lower than that observed in the H broad-band by Haubois et al. (2009). Their measurements show devi-ations of the visibilities from the limb-darkened disk as high as80–120% already in the fourth lobe (converted from the squaredvisibilities plotted in their Fig. 4), where our K -band continuumdata still follow the limb-darkened disk within the measurementerrors of 5–10% except for the data points near the visibility nullat ∼
100 arcsec − . The cause of this di ff erence is not yet clear,because of a number of di ff erences between their observationsand ours (e.g., di ff erences in the observed wavelengths, spec-tral resolution, position angle coverage). H -band observationswith higher spectral resolution and / or K -band observations witha wider position angle coverage are necessary to clarify this is-sue. Figure 3 shows the K -band uniform-disk diameters ofBetelgeuse from the literature and the archival data summerizedin Paper I, together with the 11 µ m uniform-disk diameter pre-sented in Townes et al. (2009), who found a noticeable decreasein the 11 µ m size in the last 15 years, and the one-epoch mea-surement by Perrin et al. (2007). In marked contrast to the notice-able decrease in the 11 µ m size, the K -band diameter has beenquite stable for the last 18 years with only a possible, slight long-term decrease. These results can be qualitatively explained asfollows. While the size of the star itself has been stable over thelast 18 years, the temperature and densities and / or shape of theouter atmosphere have changed significantly (e.g., decrease intemperature and / or density). Because the mid-infrared apparentsize is largely a ff ected by the MOLsphere and by dust (Ohnaka2004; Verhoelst et al. 2006; Perrin et al. 2007), the changes inthe outer atmosphere lead to a noticeable change in the 11 µ msize. This interpretation has also been recently reached by Raviet al. (2010) based on the estimation of the surface temperatureseen at 11 µ m. On the other hand, the angular size measuredwith the K -broadband filter is only slightly a ff ected, because thestrong molecular bands of CO and H O are present only at the short and long wavelength edge of the K band. Detailed model-ing of the mid-IR interferometric and spectroscopic data, whichis necessary to quantitatively derive the change in the physicalproperties of the MOLsphere and dust, is beyond the scope ofthis paper and will be pursued in a forthcoming paper. Figure 4 shows a comparison between the data taken in 2008and 2009 for four representative CO lines. The results for the2009 data were obtained from the merged data of the data sets uv points very close toone of the data sets obtained in 2008 (data set uv points very closeto the same 2008 data set, and the results from these data setsagree well with those shown in Fig. 4. We used the binning withthe same spectral resolution as applied to the data from 2008:spectral resolution of 12000 (no binning) for the 16 m baselinedata, 8000 for the 32 m baseline data, and 4800 for the 48 mbaseline data and CP, respectively.In marked contrast to the continuum data, Fig. 4 reveals sig-nificant time variations in the CO line visibilities. The visibilitywithin each CO line obtained on the 16 m baseline in 2008 wascharacterized by the maxima in the blue wing and minima in thered wing (black line in Fig. 4b). In the 2009 data, the visibil-ity on the 16m baseline does not show the maxima in the bluewing anymore (red line in Fig. 4b) and is characterized only bythe minima in the red wing. The visibilities on the 32 m and48 m baselines also show time variations, although the data onthe 48 m baseline are noisy.Time variations are even clearer in the DPs and CPs. Non-zero DPs were not detected in the CO lines on the 16m baselinein 2008, but the 2009 data show clear non-zero DPs in the COlines. On the other hand, the non-zero DPs on the 32m baselineobtained in 2009 are much weaker than those in the 2008 data.The DPs on the 48m baseline as well as the CPs measured in2009 also show significant time variations. The non-zero DPsand non-zero / non- π CPs indicate the asymmetry of the CO-line-forming region in 2009, as found in 2008.The observed spectra also reveal changes in the line profiles.The lines observed in 2009 are redshifted by ∼ − com-pared to those observed in 2008, as shown in Fig. 4a. The spectrataken on 2009 Jan 5 and 6 agree very well, although they werecalibrated independently. This confirms that the redshift of theCO lines in the 2009 data is real. All these results suggest thatthe dynamics in the atmosphere of Betelgeuse has changed in aninterval of one year. The reconstructed one-dimensional projection image in the con-tinuum at 2.30662 µ m is shown by the black line in Fig. 5 (com-parison between the observed interferometric data and thosefrom the reconstructed image is shown in Fig. D.1). Also plottedis the one-dimensional projection image of the limb-darkeneddisk with the angular diameter of 42.49 mas and the limb-darkening parameter of 0.097 (gray line, overlapping with thegreen line at angular distances between +
15 and −
15 mas).These one-dimensional projection images are already convolvedwith the Gaussian beam with a FWHM of 9.8 mas as describedin Appendix B. Figure 5 shows that the stellar surface is wellresolved with the beam size of 1 / hnaka et al.: Imaging the dynamical outer atmosphere of Betelgeuse in the CO first overtone lines 5 continuum spectral channels agree well within the uncertainty ofthe reconstruction ( ∼ ∼
5% on the eastern side (position angle = ◦ ). Because inho-mogeneities in the direction perpendicular to the baseline vec-tor on the sky are smeared out in the one-dimensional projec-tion image, we estimated the upper limit on the strength of in-homogeneities using a uniform disk with one Gaussian-shapeddark spot. The H -band image of Betelgeuse shows spots withFWHMs up to 10–11 mas (Haubois et al. 2009), which we adoptfor our model. For a given amplitude of the spot, we generated10000 models with random positions of the spot and countedthe number of models whose one-dimensional projection im-age shows deviation from the uniform disk smaller than 5%.For spot amplitudes lower than 20% of the stellar disk inten-sity, more than 70% of the models show deviations compatiblewith the observations. However, the fraction of these models is13%, 3%, and < Figure 5 shows the one-dimensional projection images recon-structed in the blue wing, line center, and red wing within the COline centered at 2.3061 µ m because of the two transitions (2,0) R (26) and R (75) with a spectral resolution of 6000 (compari-son between the observed interferometric quantities and thosefrom the reconstructed images is shown in Fig. D.1). The one-dimensional projection images are normalized with the peak in-tensities but are not artificially registered with one another, be-cause the relative astrometry is preserved thanks to the restoredvisibility phase. The round or blunt shape of the images primar-ily results from the projection of the two-dimensional imagesonto the baseline vector on the sky. The level of the image re-construction noise is estimated to be ∼ ∼ µ m agree well with those shownin Fig. 5, which adds fidelity to the image reconstruction. Theseone-dimensional projection images represent the imaging of thephotosphere and MOLsphere of an RSG, for the first time, in theindividual CO first overtone lines.
Clearly, the one-dimensional projection images in the bluewing and line center are more extended than those in the con-tinuum, with the extension of 15% and 30% on the eastern andwestern side, respectively, when measured at the noise level ofthe image reconstruction ( ∼ O bands (e.g., Tsuji 2000a, 2006; Ohnaka 2004; Perrin etal. 2004, 2007). On the other hand, the red wing one-dimensionalprojection image shows only a slight deviation from the contin-uum without a trace of the extended component. Furthermore,the di ff erent appearance of the extended component in the blueand red wing suggests that the vigorous, inhomogeneous gas mo-tions are present not only in the photosphere extending to ∼ R ⋆ (see Sect. 4 in Paper I) but also in the layers extending to ∼ R ⋆ . This is because if the inhomogeneous gas motions werepresent only in the photosphere, the extended component wouldappear the same in the blue and red wing. Figure 6 shows the one-dimensional projection images recon-structed at four di ff erent wavelengths in the CO (2,0) band headwith a spectral resolution of 1600 (see Fig. D.2 for a comparisonbetween the observed interferometric quantities and those fromthe reconstructed images). The peak of the one-dimensional pro-jection image at the band head is shifted to the east with respectto that in the continuum and has geometrical extensions of 18%and 13% on the eastern and western side, respectively, whenmeasured at the noise level of the image reconstruction.In none of the reconstructed images in the continuum, COlines, and CO band head did we detect a feature correspondingto the faint plume reported by Kervella et al. (2009), althoughtheir (single-dish) observations were carried out coincidentallyalmost simultaneously with our AMBER measurements. This ispresumably because of the very faint nature of the plume. Itsintensity is below 1% of the center of the stellar disk even at < ∼ µ m, where it appears the most prominent, and the plume iseven less pronounced at 2.12 and 2.17 µ m. Such a faint structureis below the noise of the image reconstruction from the presentdata (1.5% and 3% in Figs. 5 and 6, respectively). However, fu-ture observations with better accuracies could reveal the pres-ence of the plume in the CO lines, which would be useful forunderstanding the nature of the plume. Furthermore, the recon-structed images in the CO band head with a spectral resolution1600 can be compared with future medium-spectral resolutionAMBER observations.
4. Modeling of the velocity field
We used our stellar patch model presented in Paper I to char-acterize the velocity field in the photosphere and MOLsphere.This model consists of two CO layers that represent the photo-sphere and MOLsphere. An inhomogeneous velocity field is rep-resented by a patch (or clump) of CO gas moving at some veloc-ity di ff erent from the CO gas in the remaining region. Becausethe reconstructed one-dimensional projection images do not al-low us to know the actual number and shape of the patches, weassumed only one patch in our modeling to keep the number offree parameters as small as possible. Furthermore, for the tem-perature, CO column density, and the radius of the two layers, weused the same parameters as derived in Paper I for the followingreason. As discussed in Sect. 3, the CO line profiles observedin 2008 and 2009 show little time variation in the line depth andwidth except for the redshift. This implies that the physical prop-erties of the photosphere and MOLsphere, such as the densityand temperature, may not have changed significantly, althoughthere must have been temporal and spatial fluctuations. The pa-rameters adopted from Paper I are as follows: the inner CO layeris assumed to be located at 1.05 R ⋆ with a temperature of 2250 Kand a CO column density of 5 × cm − , while the outer COlayer is assumed to be located at 1.45 R ⋆ with a temperature of1800 K and a CO column density of 1 × cm − . We alsoadopted a microturbulent velocity of 5 km s − for both layersas in Paper I. This means that we attempt to explain the one-dimensional projection images observed in 2009 by changes inthe velocity field, as well as in the position and size of the patch.The wavelength scale of the model spectra was converted to theheliocentric frame assuming a heliocentric velocity of 20 km s − (Huggins 1987; Huggins et al. 1994).Figure 7 shows the two-dimensional images, one-dimensional projection images, and the spectrum predicted Ohnaka et al.: Imaging the dynamical outer atmosphere of Betelgeuse in the CO first overtone lines by the best-fit model for the same CO line as shown in Fig. 5.The model is characterized by a large, o ff -centered, circularpatch of CO gas, which dominates the upper half of the stellardisk (Figs. 7b and 7d). The CO gas within this patch is movingoutward with a velocity of 5 km s − , while the gas in theremaining region is moving inward faster with 25 km s − .Figures 7a and 7f show that the line profile and the wavelengthdependence of the observed one-dimensional projection imageswithin the CO line are reasonably reproduced, although thedi ff erence between the images in the blue wing and red wing issomewhat too pronounced, and the line profile is weaker thanthe observed data. We found out that the models with a patch ofCO gas moving slowly outward at 0–5 km s − with the gas inthe remaining region downdrafting much faster at 20–30 km s − can reproduce the observed one-dimensional projection imagesreasonably.The above model can also explain why the MOLsphere inthe blue wing is much more pronounced than in the red wing.Firstly, the velocity field with the weak upwelling and strongdowndrafting components causes the line center to be redshiftedwith respect to the stellar rest frame. This can be seen in Fig. 7a,where the positions of the two transitions responsible for the ob-served line profile, (2,0) R (26) and R (75), are marked in the stel-lar rest frame. Since the contribution of the R (26) transition tothe line profile is much larger than that of R (75) because of themuch lower excitation potential of the former transition, the lineprofile is redshifted with respect to the position of R (26) in thestellar rest frame. In other words, the wavelength of the stellarrest frame is located in the blue wing of the line. Secondly, thestrong extended CO emission is seen in the line of sight tangen-tial to the outer CO layer, because the column density along thisline of sight is the largest. The radial velocity of the CO layeralong such a line of sight is nearly zero, which means that thestrong extended CO emission appears at the stellar rest frame.Because the stellar rest frame is located in the blue wing as ex-plained above, the extended CO emission is strong in the bluewing. The emission becomes nearly absent in the red wing, be-cause the velocity di ff erence between the blue and red wing ismuch greater than the line width.The velocity field in 2009 is in contrast with that in 2008,which was characterized by the gas moving both outward andinward with velocities of 10–15 km s − . Therefore, our AMBERobservations at two epochs reveal a drastic change in the velocityfield in the photosphere and MOLsphere within one year.
5. Discussion
The drastic change in the velocity field between 2008 and 2009sets an upper limit of one year on the time scale of the change ofthe MOLsphere. This allows us to estimate the upper limit of theradial spatial scale where the inhomogeneous gas motions arepresent. We assume that the upwelling patch (or clump) with 10–15 km s − detected at 1.45 R ⋆ (radius of the MOLsphere) in 2008decelerated linearly with time over one year and corresponds tothe patch slowly moving outward with 0–5 km s − in 2009. Thenthe maximum radial distance reached by the gas patch is 0.24–0.47 R ⋆ . This means that the upwelling gas patch at 1.45 R ⋆ in2008 can reach 1.7–1.9 R ⋆ in 2009. Likewise, if we assume thatthe fast downdrafting patch with 20–30 km s − detected in 2009accelerated inward linearly with time starting from 0 km s − , itmust have traveled 0.71–1.07 R ⋆ in 1 year. This suggests that thefast downdrafting gas patch could have been located as far as at2.2–2.5 R ⋆ in 2008 and could have fallen to 1.45 R ⋆ in one year.Therefore, the vigorous gas motions can be present up to ∼ R ⋆ . These inhomogeneous gas motions in the extended atmo-sphere of Betelgeuse have also been detected by other obser-vations. Recently Harper et al. (2009a) have studied the dynam-ics of the cool extended outer atmosphere of Betelgeuse basedon high-spectral resolution mid-IR observations of the [Fe II]lines at 17.94 and 24.53 µ m. These [Fe II] lines form at ∼ R ⋆ (converted with the angular diameter of Betelgeuse derivedhere) in the cool extended outer atmosphere (see Fig. 8 of Harperet al. 2009a) with estimated excitation temperatures of 1520–1950 K. Therefore, the [Fe II] lines originate in the region sim-ilar to the MOLsphere where the CO first overtone lines form.The profiles of the [Fe II] lines indicate turbulent gas motionswithout signatures of significant outflows of > ∼
10 km s − . This isconsistent with the velocity fields derived from our two-epochAMBER observations. Harper et al. (2009a) detected no signif-icant changes in the [Fe II] line profiles at three epochs over 14months. However, this may be because the changes in the veloc-ity field are smeared out in their spatially unresolved observa-tions. As can be seen in Fig. 4a, the CO line profiles observedwith AMBER only show a low redshift, despite the remarkablechange in the velocity field.Complex gas motions have been detected in the extendedchromosphere of Betelgeuse as well. Lobel & Dupree (2001)present the modeling of the chromospheric velocity field up to ∼ R ⋆ . Moreover, the velocity field changed from overall in-ward motions to outward motions within 0.5–1 year. Therefore,both the cool and hot components in the extended outer atmo-sphere are characterized by strongly temporally variable inho-mogeneous gas motions.The physical mechanism responsible for these vigorous mo-tions and their drastic change within one year is not yet clear,although it is likely related to the unknown wind-driving mech-anism. The convective energy flux is expected to be low inthe MOLsphere, which extends to ∼ R ⋆ . This poses aproblem for the interpretation of the detected gas motions interms of convection. Other possible mechanisms include Alfv´enwaves and pulsation. The recent detection of magnetic fieldsin Betelgeuse, albeit weak ( ∼ ff ects of the Alfv´en-waves on the more dominant, cool outeratmosphere including the CO MOLsphere are not addressed.The MHD simulations of Suzuki (2007) for red giants showthat the stellar winds are highly temporally variable and “struc-tured”, in which hot ( > ∼ K) gas bubbles are embedded in cool( ∼ ∼ R ⋆ also shows sig-nificant time variations from ∼ +
40 km s − (outward motions)to ∼ −
40 km s − (inward motions). This is compatible to thechange in the velocity field detected by our AMBER observa-tions. However, the simulations of Suzuki (2007) were carriedout for red giant stars, which are much less luminous ( < ∼ L ⊙ )compared to Betelgeuse (1 . × L ⊙ , Harper et al. 2008).Extending the MHD simulations of Suzuki (2007) for more lu-minous stars, as well as the inclusion of the cool molecular com-ponent in the work of Airapetian et al. (2000), would be valuablefor a comparison with the present and future AMBER observa-tions.Lobel (2010) infers that strong shock waves generated byconvection in the photosphere that are propagating outward maycarry the energy and momentum to accelerate the wind and heatthe chromosphere. The qualitative similarity between the inho- hnaka et al.: Imaging the dynamical outer atmosphere of Betelgeuse in the CO first overtone lines 7 mogeneous velocity field in the chromosphere and in the pho-tosphere / MOLsphere may point toward this scenario. However,obviously, it is indispensable to map the dynamical structure ofthe cool outer atmosphere at various radii to clarify the drivingmechanism of mass outflows in RSGs.
6. Concluding remarks
We have succeeded, for the first time, in one-dimensional aper-ture synthesis imaging of Betelgeuse in the individual CO firstovertone lines, as well as in the continuum approximately freefrom molecular / atomic lines, with a spatial resolution of 9.8 masand a spectral resolution of 6000 using VLTI / AMBER.The one-dimensional projection images in the CO lines re-constructed with the self-calibration technique, which restoresthe complex visibility using di ff erential phase measurements, re-veal that the star appears di ff erent within the individual CO lines.The one-dimensional projection images in the blue wing and linecenter show a pronounced extended component up to 1.3 R ⋆ ,while the images in the red wing follow that in the continuumwithout an extended component. Our image reconstruction rep-resents the first study to image the so-called MOLsphere of anRSG in the individual CO first overtone lines. Our modeling sug-gests that the dynamics in the photosphere and MOLsphere in2009 is characterized by strong downdrafts with 20–30 km s − and slight outward motions with 0–5 km s − . This indicates adrastic change in the velocity field within one year from 2008,when the dynamics was characterized by both upwelling anddowndrafting components with 10–15 km s − .On the other hand, the reconstructed one-dimensional pro-jection images in the K -band continuum show only a small devi-ation of 5% from the limb-darkened disk with an angular diam-eter of 42 . ± .
06 mas with a power-law-type limb-darkeningparameter of (9 . ± . × − . This limb-darkened disk diam-eter results in an e ff ective temperature of 3690 ±
54 K for thecontinuum-forming layer. The deviation from the limb-darkeneddisk in the one-dimensional projection images suggests that theamplitude of stellar spots is likely smaller than 20–30% of theintensity of the stellar disk. Furthermore, we detected no or onlymarginal time variation in the continuum visibility data withinthe measurement errors, much smaller than the maximum varia-tion predicted by the current three-dimensional convection sim-ulations. It cannot be entirely excluded that Betelgeuse showedunusually weak surface structures at the times of our AMBERobservations just by chance. However, it is also possible thatthe current three-dimensional convection model for RSGs pre-dicts too strong surface structures in the continuum. A long-termmonitoring to measure the amplitude of the time variations in thevisibility and closure phase is indispensable for a definitive, sta-tistical test of three-dimensional convection simulations.The self-calibration imaging using di ff erential phase hasturned out to be very e ff ective and necessary, despite the goodlinear uv coverage from the first to fifth visibility lobe. This sug-gests that the self-calibration technique may be even more nec-essary for two-dimensional imaging, where it is di ffi cult to ob-tain a uv coverage as densely sampled as in our one-dimensionalcase. While the imaging of stellar surfaces is still challeng-ing (e.g., Creech-Eakman et al. 2010), our self-calibration one-dimensional imaging demonstrates a promising way to achievethat goal. Acknowledgements.
We thank the ESO VLTI team for supporting our AMBERobservations. We are also grateful to Eric Thi´ebaut, who kindly makes his image reconstruction software MiRA publicly available. NSO / Kitt Peak FTS data onthe Earth’s telluric features were produced by NSF / NOAO.
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Fig. 1. uv coverage of our AMBER observations of Betelgeuse.The dashed line represents the average position angle of 73 ◦ . hnaka et al.: Imaging the dynamical outer atmosphere of Betelgeuse in the CO first overtone lines 9 Fig. 2.
Continuum visibilities of Betelgeuse averaged over the continuum spectral channels between 2.28 and 2.293 µ m. The insetsshow enlarged views of the second, third, and fourth / fifth lobes. The solid and dashed lines represent the visibilities for a uniformdisk with a diameter of 42.05 mas and for a limb-darkened disk with a diameter of 42.49 mas and a limb-darkening parameter of0.097 (power-law-type limb-darkened disk of Hestro ff er 1997), respectively. The dotted lines represent the maximum range of thevariations in the 2.22 µ m visibility due to time-dependent inhomogeneous surface structures predicted by the three-dimensionalconvection simulation of Chiavassa et al. (2009), who presents the model prediction up to 70 arcsec − . Fig. 3.
Uniform-disk diameters of Betelgeuse measured in the K band and at 11 µ m as a function of time. The black filled circlesand the blue open circle represent the 11 µ m diameters mea-sured by Townes et al. (2009) and Perrin et al. (2007), respec-tively. The diamonds represent the K -band diameters from thefollowing references. IRMA: Dyck et al. (1992). IOTA: Perrin etal. (2004). VINCI: Paper I. AMBER MR K: AMBER medium-resolution ( λ/ ∆ λ = hnaka et al.: Imaging the dynamical outer atmosphere of Betelgeuse in the CO first overtone lines 11 Fig. 4.
AMBER data in the CO first overtone lines obtained in 2008 and 2009. The black and red solid lines represent the datataken in 2008 and 2009, respectively. In the panels b–h , the scaled spectrum observed on 2009 Jan 5 is overplotted by the blue solidline to show the asymmetry of the visibilities and di ff erential / closure phases within each line profile. The spectral resolutions givenin the panels are the values of the binned data as described in Sect. 2.1. a: Normalized spectra. Two spectra derived from the datataken on 2009 Jan 5 and 6 are plotted by the red and green solid lines, respectively, while the 2008 spectrum is plotted by the blacksolid line. b–d:
Visibilities observed on the E0-G0-16m, G0-H0-32m, and E0-H0-48m baselines. e: Closure phase. f–h: Di ff erentialphases observed on the E0-G0-16m, G0-H0-32m, and E0-H0-48m baselines. Fig. 5.
One-dimensional projection images re-constructed at four di ff erent wavelengths (bluewing, line center, red wing, and continuum) within the CO line due to the two transi-tions (2,0) R (26) and R (75) (spectral resolu-tion of 6000) are shown by the blue, green,red, and black solid lines, respectively. Theone-dimensional projection image of the limb-darkened disk with the parameters derived inSect. 3.1 is also shown by the gray solid lineas a reference. The inset shows the observedline profile with the wavelengths of the im-ages marked by the filled circles with the corre-sponding colors. The one-dimensional projec-tion images are convolved with the Gaussianbeam whose FWHM (9.8 mas) is shown bythe thick solid line, and their absolute scaleis normalized to the peak intensities. The ori-entation of the one-dimensional projection im-ages is also shown. The dashed line denotes thenoise level of the image reconstruction.hnaka et al.: Imaging the dynamical outer atmosphere of Betelgeuse in the CO first overtone lines 13 Fig. 6.
One-dimensional projection images re-constructed at four di ff erent wavelengths inthe CO (2,0) band head (spectral resolution of1600) shown in the same manner as Fig. 5.4 Ohnaka et al.: Imaging the dynamical outer atmosphere of Betelgeuse in the CO first overtone lines Fig. 7.
Best-fit stellar patch model with an inhomogeneous velocity field. a: The filled diamonds represent the observed spectrum,while the black solid line represents the model spectrum. The filled circles mark the wavelengths of the two-dimensional imagesand one-dimensional projection images shown in the panels b–g . b–e: Two-dimensional model images. The orientation of the linear uv coverage is shown by the solid lines. The one-dimensional projection images are obtained by integrating the two-dimensionalimages in the direction shown by the dotted lines, which is perpendicular to the orientation of the linear uv coverage. f: The one-dimensional projection images predicted by the model at four wavelengths within the CO line. g: The observed one-dimensionalprojection images for the same CO line. hnaka et al.: Imaging the dynamical outer atmosphere of Betelgeuse in the CO first overtone lines 15
Appendix A: Summary of AMBER observations
Our AMBER observations of Betelgeuse and the calibratorSirius are summarized in Table A.1.
Appendix B: Image reconstruction of simulateddata
Aperture synthesis imaging from optical / IR interferometric datadepends on a number of parameters used in the image recon-struction process, such as the initial model, as well as the reg-ularization scheme and prior, which represent a priori informa-tion about the object’s intensity distribution. Therefore, it is im-portant to carry out the image reconstruction for simulated im-ages to derive the appropriate reconstruction parameters beforewe attempt the image reconstruction from observed data. Fromsimulated images, we generate simulated interferometric data bysampling the visibilities and CPs at the same uv points as inter-ferometric observations. With the true images known for thesesimulated data, we can examine the appropriate reconstructionparameters that allow us to reconstruct the original images cor-rectly.Because MiRA is developed for two-dimensional image re-construction, we took the following approach for the reconstruc-tion of one-dimensional projection images: two-dimensional im-age reconstruction was carried out using an appropriate ini-tial model and regularization parameters as described below.The reconstructed two-dimensional image was convolved withthe clean beam, which is represented by a two-dimensionalGaussian with a FWHM of λ/ B max = B max is themaximum baseline length of the data. The one-dimensional pro-jection image was obtained by integrating this convolved two-dimensional image in the direction perpendicular to the linear uv coverage.We generated two simulated images that represent possiblesurface patterns of Betelgeuse: a simple limb-darkened disk anda uniform disk with a bright spot, a dark spot, and an extendedhalo, as shown in Figs. B.1a and B.2a, respectively. For bothcases, the stellar angular diameter was set to be 42.5 mas, whichis the limb-darkened disk diameter derived from all continuumvisibilities measured in 2009. The visibilities and CPs were com-puted from the simulated images at the same uv points as ourAMBER observations, using the program of one of the authors(K.-H. Hofmann). Noise was also added to the simulated visibil-ities and CPs to achieve SNRs similar to the AMBER data. Wetested di ff erent initial models, priors, and regularization schemesto find out the appropriate parameter range to reconstruct theone-dimensional projection image of the simulated data cor-rectly. It turned out that uniform disks with angular diametersbetween 34 and 50 mas serve as good initial models. The priorused in the present work is a smoothed uniform disk describedas Pr ( r ) = e ( r − r p ) /ε p + , where r is the radial coordinate in mas, and r p and ε p define thesize and the smoothness of the edge ( ε p → r p and ε p were found to be 10 .. 15 (mas) and 2 .. 3 (mas), respectively.Therefore, we used six di ff erent parameter sets for the image re-construction of Betelgeuse by combining three diameters for theinitial uniform-disk model (34, 42, and 50 mas) and two di ff er-ent parameter sets for the prior (( r p , ε p = (10, 2) and (15, 3)).The final images and their uncertainties were obtained by taking the average and standard deviation, respectively, from the resultsreconstructed with these six parameter sets. The regularizationusing the maximum entropy method turned out to be appropri-ate for our reconstruction. We started the reconstruction with ahigh degree of regularization ( µ = , see Thi´ebaut 2008 forthe definition of µ ) and reduced it gradually by a factor of 10 af-ter every 500 iterations until the reduced χ reaches ∼ Appendix C: Self-calibration imaging withdifferential phase
Because the principle of the self-calibration technique using DPmeasurements is described in detail in Millour et al. (2011), wemention the actual procedure only briefly. Then we describe themodification we added to this technique to deal with an issuespecific to the AMBER data of Betelgeuse.The DP measured with AMBER at each uv point containsinformation on the phase of the complex visibility function androughly represents the di ff erence between the phase in a spec-tral feature and that in the continuum. However, two pieces ofinformation are lost because of the atmospheric turbulence: theabsolute phase o ff set and the linear phase gradient with respectto wavenumber. We can derive this lost phase o ff set and gradi-ent by a linear fit to the phase (as a function of wavenumber)from the reconstructed continuum images, if the image recon-struction in the continuum is reliable and not sensitive to thereconstruction parameters. This is indeed the case for our imagereconstruction of Betelgeuse in the continuum, as discussed inSect 3.4. Therefore, the phase in the CO lines can be restoredfrom the continuum phase interpolated at the line spectral chan-nels and the DPs measured in the lines. The image reconstructionis carried out with the measured visibilities and CPs as well asthe restored phase. This process can be iterated, but our exper-iments show that the reconstructed images do not change afterthe first iteration.We added the following modification to the technique pre-sented in Millour et al. (2011). When the phase o ff set and gra-dient are derived by a linear fit to the phase of the reconstructedimages, we only use the continuum spectral channels below2.293 µ m and those between the adjacent CO lines above 2.3 µ m,instead of using the entire spectral channels, as Millour et al.(2011) did. The reason for this selection of the spectral chan-nels is that the image reconstruction near the CO band head at2.294 µ m is so uncertain owing to the poor SNR in the databinned with a spectral resolution of 6000 that the inclusion ofthe spectral channels near the band head in the linear fit hampersthe reliable derivation of the phase o ff set and gradient.The inclusion of only the selected continuum channels hasthe following consequence. If we denote the continuum phasefrom the reconstructed continuum image at a given baseline andat the i -th spectral channel as ϕ c ( i ), the phase at the i -th spectralchannel, ϕ ( i ), is restored as ϕ ( i ) = ϕ c ( i ) + DP( i ) , where DP( i ) represents the di ff erential phase at the i -th spectralchannel measured at the same baseline. At a continuum spectralchannel denoted as i c , the restored phase ϕ ( i c ) should be equal tothe phase from the reconstructed continuum image ϕ c ( i c ). This isfulfilled if the measured DP in the continuum is zero. However, Table A.1.
Log of AMBER observations of Betelgeuse and the calibrator Sirius with the E0-G0-H0 (16-32-48m) baseline configu-ration. Seeing is in the visible. t obs B p PA Seeing τ NDIT t obs B p PA Seeing τ NDIT(UTC) (m) ( ◦ ) ( ′′ ) (ms) (UTC) (m) ( ◦ ) ( ′′ ) (ms)Betelgeuse2009 Jan 5 2009 Jan 61 01:01:44 22.24 / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / the measured DPs show noticeable non-zero values in the con-tinuum, as exemplarily shown in Figs. C.1a and C.1b. The reasonfor the non-zero DPs in the continuum is that amdlib derives dif-ferential phase by a linear fit to the instantaneous phase at all spectral channels. Owing to the strong deviation of the phase inmany CO lines from that in the continuum, this linear fit doesnot go through all the continuum points. Therefore, the non-zeroDPs in the continuum spectral channels lead to a systematic errorin the phase restored in the continuum, which a ff ects the subse-quent image reconstruction. We found out that the continuumone-dimensional projection image reconstructed using the re-stored phase shows a systematic wavelength dependence fromthe shortest to the longest wavelength of the observed spectralrange, which is not seen in the continuum images reconstructedfrom the visibilities and CPs alone.It is necessary to use the same spectral channels in the linearfit to the phase for the derivation of DP and for the derivation of the phase o ff set and gradient. Therefore, we refitted the DP fromamdlib with a linear function (with respect to wavenumber) atthe same continuum points as used for the derivation of the phaseo ff set and gradient (dashed lines in Figs. C.1a and C.1b) andsubtracted the fitted linear function from the observed DP. Thisprocedure enforces the DP in the continuum spectral channelsto zero within the measurement errors, as shown in Fig. C.1c.The phase was restored using this “refitted” DP. The continuumone-dimensional projection images reconstructed using the refit-ted DPs do not show the aforementioned systematic wavelengthdependence, which proves the validity of our procedure. hnaka et al.: Imaging the dynamical outer atmosphere of Betelgeuse in the CO first overtone lines 17 Fig. B.1.
Image reconstruction of the simu-lated data for a limb-darkened disk with theparameters derived in Sect. 3.1. a: Originaltwo-dimensional image of the simulated data.The solid line represents the orientation ofthe linear uv coverage, while the dotted linerepresents the orientation perpendicular to it. b: Two-dimensional image of the simulateddata convolved with the Gaussian beam witha FWHM of 9.8 mas. c: Comparison betweenthe original and reconstructed one-dimensionalprojection images before convolving with theGaussian beam. The one-dimensional projec-tion images are obtained by integrating thetwo-dimensional images in the direction shownby the dotted lines in the panels a and b . d: Comparison between the original and re-constructed one-dimensional projection imagesconvolved with the Gaussian beam with aFWHM of 9.8 mas. e: The filled circles and tri-angles represent the visibilities from the orig-inal simulated data and the reconstructed im-age, respectively. f: The filled circles and trian-gles represent the CPs from the original simu-lated data and the reconstructed image, respec-tively. The abscissa is the spatial frequency ofthe longest baseline of each data set.8 Ohnaka et al.: Imaging the dynamical outer atmosphere of Betelgeuse in the CO first overtone lines
Fig. B.2.
Image reconstruction of the simulateddata for a uniform disk with a bright spot, adark spot, and a halo shown in the same manneras in Fig. B.1.hnaka et al.: Imaging the dynamical outer atmosphere of Betelgeuse in the CO first overtone lines 19
Appendix D: Fit to the interferometric data with thereconstructed images
Figures D.1 and D.2 show the fit to the observed interferometricdata for the one-dimensional projection image reconstruction inthe CO line and in the CO (2,0) band head shown in Figs. 5 and6, respectively.
List of Objects ‘Betelgeuse’ on page 1‘Sirius’ on page 2‘ α CMa’ on page 2
Fig. C.1. a:
DP observed on the longest baseline in the data set B p = = b: Enlarged view of the panel a for the CO lines. Note that the DPin the continuum points between the adjacent CO lines deviatesfrom zero. c: DP after subtracting the linear fit to the continuumpoints as described in Appendix C is shown by the red solidline. The black line represents the scaled spectrum. The DP inthe continuum points between the adjacent CO lines is now zerowithin the measurement errors.
Fig. D.1.
Comparison between the observed interferometric data and those from the one-dimensional projection image reconstruc-tion for the CO line shown in Fig. 5. The first, second, third, and fourth columns show the comparison for the blue wing, line center,red wing, and continuum, respectively. The panels in the top row ( a – d ) show the observed CO line profile, and the filled circlesdenote the wavelength of the data shown in each column. In the remaining panels, the observed data are represented by the redcircles, while the values from the image reconstruction are shown by the blue triangles. The reduced χ values for the fit includingthe complex visibilities, squared visibilities, and CPs are given in the panels m – p . hnaka et al.: Imaging the dynamical outer atmosphere of Betelgeuse in the CO first overtone lines 21 Fig. D.2.
Comparison between the observed interferometric data and those from the one-dimensional projection image recon-struction near the CO (2,0) band head shown in Fig. 6. The first, second, third, and fourth columns show the comparison for thecontinuum, blue side between the continuum and the band head, bottom of the band head, and red side of the band head, respectively.The panels in the top row ( a – d ) show the observed spectrum of the CO band head, and the filled circles denote the wavelength of thedata shown in each column. In the remaining panels, the observed data are represented by the red circles, while the values from theimage reconstruction are shown by the blue triangles. The reduced χ values for the fit including the complex visibilities, squaredvisibilities, and CPs are given in the panels m – pp