A Multi-observing-technique Study of the Dynamical Evolution of the Viscous Disk around the Be Star ω CMa
Mohammad R. Ghoreyshi, Alex C. Carciofi, Carol E. Jones, Daniel M. Faes, Dietrich Baade, Thomas Rivinius
DDraft version February 9, 2021
Typeset using L A TEX twocolumn style in AASTeX62
A Multi-observing-technique Study of the Dynamical Evolution of theViscous Disk around the Be Star ω CMa.
Mohammad R. Ghoreyshi,
1, 2
Alex C. Carciofi, Carol E. Jones, Daniel M. Faes, Dietrich Baade, andThomas Rivinius Department of Physics and Astronomy, Western University, London, ON N6A 3K7, Canada Instituto de Astronomia, Geof´ısica e Ciˆencias Atmosf´ericas, Universidade de S˜ao Paulo, Rua do Matao 1226, SP 05508-900, Brazil Gemini Observatory/NSF’s NOIRlab, 670 N. A’ohoku Place, Hilo, Hawai’i, 96720, USA European Organisation for Astronomical Research in the Southern Hemisphere (ESO), Karl-Schwarzschild-Str. 2, 85748 Garching beiM¨unchen, Germany European Organisation for Astronomical Research in the Southern Hemisphere (ESO), Casilla 19001, Santiago 19, Chile (Received February 9, 2021; Revised February 9, 2021; Accepted 00-00-0000)
Submitted to ApJABSTRACTThe observed emission lines of Be stars originate from a circumstellar Keplerian disk that are gen-erally well explained by the Viscous Decretion Disk model. In an earlier work we performed themodeling of the full light curve of the bright Be star ω CMa (Ghoreyshi et al. 2018) with the 1-Dtime-dependent hydrodynamics code
SINGLEBE and the Monte Carlo radiative-transfer code
HDUST .We used the V -band light curve that probes the inner disk through four disk formation and dissipationcycles. This new study compares predictions of the same set of model parameters with time-resolvedphotometry from the near UV through the mid-infrared, comprehensive series of optical spectra, andoptical broad-band polarimetry, that overall represent a larger volume of the disk. Qualitatively, themodels reproduce the trends in the observed data due to the growth and decay of the disk. However,quantitative differences exist, e.g., an overprediction of the flux increasing with wavelength, too slowdecreases in Balmer emission-line strength that are too slow during disk dissipation, and the discrep-ancy between the range of polarimetric data and the model. We find that a larger value of the viscosityparameter alone, or a truncated disk by a companion star, reduces these discrepancies by increasingthe dissipation rate in the outer regions of the disk. Keywords:
Line: profiles; Polarization; Techniques: photometric, polarimetric, spectroscopic; Stars:massive, rotation, circumstellar matter, variables: Be, individual star: ω CMa INTRODUCTIONBe stars are a specific subclass of main sequence B-type stars (Jaschek et al. 1981; Collins 1987) that arecharacterized by the presence of one or more hydrogenemission lines in their spectrum. The emission includesmainly the first members of the Balmer line series. Theyoriginate in a circumstellar environment that is in theform of an equatorial, dust-free disk that rotates in a(nearly) Keplerian fashion. The Be stars have initialmasses from ≈ (cid:12) to ≈
17 M (cid:12) . In a statistical anal-
Corresponding author: Mohammad Reza [email protected] ysis, Cranmer (2005) found that they rotate moderatelyfast ( ≈ . v crit , usually seen in the early types; e.g., κ CMa; Meilland et al. 2007) to close to the critical rota-tional speed ( ≈ . v crit , usually seen in the late types;e.g., α Eri; Domiciano de Souza et al. 2012). Due to fastrotation, the stellar equatorial material is loosely boundbut an additional mechanism is required to launch thematerial with sufficient angular momentum (AM) to re-main in orbit.Non-radial pulsations (NRP; e.g., Rivinius et al. 2003;Kee et al. 2016b) are a mechanism that may facilitatethe release of material and, in turn, may play a role inthe variability of Be stars (Rivinius et al. 2013b; Baadeet al. 2017, 2018a; Semaan et al. 2018, see Section1.1). a r X i v : . [ a s t r o - ph . S R ] F e b Ghoreyshi et al.
Changes in both brightness and spectral line profilesare typical variations in Be stars. They are knownto be variable on a range of timescales from hours toyears (Peters 1986; Hanuschik et al. 1993; Baade et al.2016). Associated with the photosphere, NRPs are thecause of both short- (Baade 2000; Huat et al. 2009) andintermediate-period variability, the latter through thenon-linear coupling of several NRP modes (Baade et al.2018a). Disk processes, on the other hand, cause vari-ability on all timescales. For instance, one-armed den-sity oscillations (Okazaki 1997; ˇStefl et al. 2009) usuallyresult in variations from months to several years. Themost frequent cause of disk variability is changes in therate of AM injection from the central star into the disk,˙ J , which manifests itself on all timescales from days toweeks (e.g., Carciofi et al. 2007; Levenhagen et al. 2011)to years and decades (Haubois et al. 2012; R´ımulo et al.2018). Finally, binarity effects are also an importantsource of intermediate-period disk variations (Panoglouet al. 2018, and references therein).Outbursts and quiescence are routinely observedstates in Be stars (Rivinius et al. 1998) and they areattributed to long term, secular variations in the disk.When the variations are both of lower amplitude andduration, outbursts are commonly referred to as flick-ers (e.g., Keller et al. 2002; R´ımulo 2017; R´ımulo et al.2018). For a (nearly) pole-on system, an outburst istypically exhibited by a rapid rise in the visible andinfrared emission. Outbursts are commonly associatedto disk formation due to mass being ejected by the star,and the excess is caused by a larger light emitting andscattering area (see Haubois et al. 2012). Conversely,if the system is seen edge-on the outburst will appearas a quick decline in brightness because the cooler diskobscures part of the hotter surface of the star. Usually,an outburst is followed by a more gradual decay (or rise,in the edge-on case) back to quiescence. A quiescencephase is associated with either the cessation or reduc-tion of AM loss and the ensuing dissipation of the disk(Haubois et al. 2012; Ghoreyshi et al. 2018).The Viscous Decretion Disk (VDD) model has beensuccessful in reproducing the observed variations ofthese disks (e.g., Carciofi et al. 2009, 2010, 2012; Kle-ment et al. 2015, 2017, 2019; Faes et al. 2016; Baadeet al. 2018a; R´ımulo et al. 2018; Ghoreyshi et al. 2018;de Almeida et al. 2020; Suffak et al. 2020). In the VDDmodel, the material ejected by the star carries AM whichis redistributed within the disk facilitated by viscosity.Some material remains in orbit and slowly diffuses out-ward to form the disk, while most of it falls back ontothe star (Okazaki et al. 2002). For approximately steady state disks (i.e., a disk fedat a constant rate for an extended period of time)the VDD model has a straightforward solution if oneassumes the disk is isothermal (e.g., Bjorkman 1997;Okazaki 2001; Bjorkman & Carciofi 2005). The firstattempt to understand the dynamical evolution of cir-cumstellar disks around isolated Be stars was done byJones et al. (2008). This was later followed by a sys-tematic study by Haubois et al. (2012) who coupledthe 1-D time-dependent hydrodynamics code SINGLEBE (Okazaki 2007, see Section 3.1) and the
HDUST radia-tive transfer code (Carciofi & Bjorkman 2006, 2008, seeSection 3.2).Shakura & Sunyaev (1973) introduced the α -viscosityprescription that links the scale of the turbulence to the(vertical) scale of the disk by a constant called viscosityparameter, α , with the following formula ν = 23 αc s H, (1)where ν represents the viscosity, c s is the isothermalsound speed and H is the disk scale-height. The α pa-rameter is usually assumed to be constant and it controlsthe timescale of disk evolution. A large α speeds up thediffusion process and vice versa.1.1. ω Canis Majoris ω (28) CMa (HD 56139, HR 2749; HIP 35037; B2 IV-Ve) is one of the brightest Be stars in the sky (with m v ≈ i ≈ ◦ ) so the measured projected rotationalvelocity of 80 km s − (Slettebak et al. 1975) is only afraction of the true equatorial velocity, estimated to be350 km s − (Maintz et al. 2003). A 1.37-day line-profilevariability has been observed in ω CMa suggesting thatit is a non-radial pulsator (Baade 1982). Later, thiswas confirmed by ˇStefl et al. (1999) and Maintz et al.(2003) by studying the line-profile variations caused byNRP for various photospheric absorption lines of dif-ferent species including Balmer lines, He i , Mg ii , andFe ii . Recently, from space photometry with BRITE-Constellation (Weiss et al. 2014), Baade et al. (2017)found that the 0.73 d − frequency (corresponding to the1.37-d period) is part of a NRP frequency group be-tween ∼ − and ∼ − . Another frequencygroup between ∼ − and ∼ − seemed to ex-hibit a much increased amplitude at a time when themean brightness also increased, i.e., matter was ejectedinto the disk when the NRP amplitude was high. Obser-vations with the TESS satellite of hundreds of Be starshave established such a correlation in dozens of other Be ulti-observing-technique study of the Be star ω CMa ω CMa used in this study are sum-marized in Table 1 and were obtained by Maintz et al.(2003).Carciofi et al. (2012) used
SINGLEBE and
HDUST tostudy the disk dissipation of ω CMa between 2003 and2008. With time-dependent models of the dissipatingdisk they determined the α parameter. Moreover, theyshowed that the stellar wind is not a probable mecha-nism for AM injection into the disk. Later, their workwas followed up by Ghoreyshi et al. (2018, hereafterG18) who presented a model of the full V -band lightcurve, spanning more than 30 years of data. The modeladdressed any photometric variability larger than 0.05mag and longer than 2 months. Any shorter variabilitywas excluded because the contribution to the total gascontent of the disk is small and these are usually poorlysampled.G18 showed that the VDD model could reproduce thedata well. It was determined that α changes during thedifferent epochs of the disk life as ˙ J varies over time.They also found that the light curve could only be re-produced if quiescence phases be interpreted as a reduc-tion of ˙ J , rather than a complete cessation of it, as it iscommonly assumed (see Carciofi et al. 2012).In G18, the V -band photometric data were investi-gated in detail and the results relevant to this studyare summarized here. Since 1982, ω CMa exhibitedquasi-regular cycles, each one lasting between about 7.0to 10.5 years. Each cycle consists of two main parts:1) an outburst phase represented by a fast increase inthe brightness and a subsequent plateau. This increaseis not always smooth, and the peak brightness plateaulasts about 2.5 to 4.0 years. 2) a quiescence phase last-ing about 4.5 to 6.5 years that is characterized by a fastdecline in brightness and a subsequent slow fading. Dur-ing these phases the brightness of the system in the V band changes from about 0 . m . m
5. Throughout thistext we refer to the cycles by C i and to the phases byO i and Q i for outburst and quiescence , respectively,where i is the cycle number. All four cycles are labeledin Figure 1.Because the V -band excess comes from the innermostpart of the disk (Carciofi 2011), the numerical solutionsused in G18 were not constrained past a few stellar radiifrom the star. In this paper, the same parameters as the Note the difference between regular font Q for quiescence anditalic font Q for the Stokes parameter, and also Q Q model of G18 are used to study other observables of ω CMa including polarimetric, spectroscopic, and photo-metric data at other wavelengths. Our goal is to com-pare the predictions of the G18 model for a much largervolume of the disk out to 30 stellar radii. For thiswork, and following G18, only variability longer thantwo months is investigated.Different observables emanate in different parts of thedisk (Carciofi 2011). For example, continuum polar-ization originates near the star and spectral lines formin various locations based on the wavelength-dependentopacity and the source function for the line. Therefore,probing Be star disks with a variety of observationaltechniques and in a variety of wavelength regimes allowsus to perform comprehensive studies of these systems.The main goal of this paper is to use the VDD modelto study the temporal variations seen in ω CMa’s databy a variety of observational techniques. Reproducingthe observed data in a wide range of wavelengths is a newchallenge for the VDD model that was not previouslyperformed.Section 2 describes the observational data available for ω CMa. Section 3 presents the theoretical concepts thatwere used in this work. In Section 4 the modeling of theavailable multi-observing-technique data are presented.Finally, in Section 5 possible solutions to solve the dis-crepancies between data and models are discussed. InSection 6 our conclusions and plans for future work arepresented. OBSERVATIONSA wealth of data from different observational tech-niques has been collected since 1963. The rich datasetcovering the most recent outburst with several differenttechniques is an important addition for this paper. Inthe following we present a summary of the observed databy each technique. The epochs of all available observa-tions are shown in Figure 2. We note that the data thatwere observed prior to C1 (December 1981) or after C4(December 2015) were not included in our analysis butare shown in Figure 2 for completeness.2.1.
Photometry
At the end of 2008, Sebastian Otero alerted the com-munity (in a private communication to our deceasedcolleague Stanislav ˇStefl) that a new outburst had be-gun, thus a broad suite of observations was undertaken.In addition to visual photometry,
JHKL photometrywere obtained with the Mk II photometer (Glass 1973)of SAAO (South African Astronomical Observatory)and the CAIN-II (CAmara INfrarroja) Tenerife/TCS(Telescopio Carlos S´anchez) camera (Cabrera-Lavers
Ghoreyshi et al.
Table 1.
Stellar, disk, and geometrical parameters of ω CMa. m ∗ v is the magnitude of the star during a diskless phase. Parameter Value reference i npu t p a r a m e t e r s f o r m o d e li n g s t a r M (cid:12) Maintz et al. 2003 L (cid:12) Maintz et al. 2003 T pole R pole (cid:12) Maintz et al. 2003 R eq (cid:12) Maintz et al. 2003log g pole v rot
350 km s − Maintz et al. 2003 W d i s k ˙ M inj (min) 2.0 × − M (cid:12) yr − G18˙ M inj (max) 3.7 × − M (cid:12) yr − G18 − ˙ J ∗ , std (min) 2.9 × g cm s − G18 − ˙ J ∗ , std (max) 5.4 × g cm s − G18 T R out R eq G18 o t h e r p a r a m e t e r s i ◦ this work v crit
436 km s − Maintz et al. 2003 m ∗ v ± d +13 − pc Gaia Collaboration et al. 2016, 2020 et al. 2006), Q
1- and Q UBV
Johnson and uvby
Str¨omgren (Str¨omgren 1956; Craw-ford 1958) filters collected in the Long-Term Photome-try of Variables (LTPV) project (Manfroid et al. 1995)during C1. 2.2.
Spectroscopy
The above campaign also produced optical echellespectra from UVES (Ultraviolet and Visual EchelleSpectrograph; Dekker et al. 2000) and VLT (Oct 2008-Mar 2009), feros (Fiber-fed Extended Range OpticalSpectrograph)/La Silla (Kaufer et al. 1999) and the1.6m telescope at Observat´orio Pico dos Dias (OPD; Jan2009 - 20164ri) initially using the ECASS spectrograph and, since 2012, the MUSICOS spectrograph (Baudrand& B¨ohm 1992).In addition to the observational effort described abovefor C4 (see Figure 1), we obtained other data for someof the previous cycles in the literature. For C1 and C2,we acquired spectroscopy from the Short-Wavelength This Cassegrain spectrograph consists of a 600 groove mm − grating blazed at 6563 ˚A at the first order, resulting in a reciprocaldispersion of 1.0 ˚A pixel − . Prime (SWP) camera of IUE (International Ultravi-olet Explorer) and heros (Heidelberg Extended RangeOptical Spectrograph )/ feros , respectively. For C3, wefound spectroscopy from the CES (Coud´e Echelle Spec-trometer ), feros , Lhires spectroscope in ObservatoirePaysages du Pilat , and Ondrejov Observatory . Addi-tional spectroscopic data for C4 came from ESPaDOnS(Echelle SpectroPolarimetric Device for the Observationof Stars; Donati 2003), OPD, PHOENIX (Hinkle et al.1998), Ritter Observatory , and UVES.Figure 3 provides an example of the observed hydro-gen lines of ω CMa. The top panel shows the V -bandphotometric data with the date the spectroscopic datawere observed, indicated by the colored vertical solidline. The bottom panels show the flux relative to thelocal continuum for the four main hydrogen lines. Usu-ally the peak emission to the continuum ratio (E/C)of the H α and H β lines is largest at the end of quies-cence, and lower during the outburst. This seeminglycontradictory behavior is well explained by the models https://archive.stsci.edu/iue/ stelweb.asu.cas.cz/web/index.php?pg=2mtelescope ulti-observing-technique study of the Be star ω CMa Figure 1.
The model fit of the full V -band light curve of ω CMa obtained in G18. Each colored solid line illustrates anindividual value for the α parameter, as indicated. The observed data are shown by the dark grey triangles (see the captionof Figure 1 in G18 for references for the data). The horizontal grey band represents the visual magnitude of the star when ithas no disk, determined by G18 (see Table 1). Selected epochs for modeling the multi-technique observations of ω CMa aremarked with yellow stars (see Section 4). C i , O i and Q i stand for cycle, outburst and quiescence phases, respectively, where i is the cycle number. The vertical dashed lines display the boundaries between the cycles. The vertical dotted lines show theboundaries between outburst and quiescence phases. as will be seen in Section 4.3. The observational logsof the spectroscopic and polarimetric data used in thispaper are listed in Appendix A, and in Tables 3 and 4,respectively. Also, the data are available at CDS .2.3. PolarimetryBVRI imaging polarimetry was obtained with the 0.6-m telescope at OPD (Magalh˜aes et al. 2006). Reductionof the OPD polarimetric data, observed by the IAGPOLinstrument, followed standard procedures outlined byMagalh˜aes et al. (1984, 1996) and Carciofi et al. (2007).The IAGPOL has an instrumental polarization smallerthan about 0.005% (Carciofi et al. 2007). The middlepanel of Figure 4 displays the polarimetric data of C4 of ω CMa in B , V , R , and I filters, alongside the V -bandphotometric data (top panel) and polarization angle, θ ,measured east from celestial north (bottom panel). Thepolarization level is very small, as expected by the factthat ω CMa is observed at rather small inclination an- http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/ApJ gles (Halonen & Jones 2013). This happens because atnearly pole-on orientations an axi-symmetric disk willappear as an almost circular structure that results inthe cancellation of the polarization vectors. The smallobserved polarization level indicates that the interstel-lar (IS) polarization is likely also very small. Both com-bined factors makes analysis of the polarization datauncertain, as will be discussed in Section 4.1. MODEL DESCRIPTIONThe principal properties of the VDD model are out-lined here, and the main methods and approximationsused to obtain the solutions are described in G18. Inthis model, the central star is located at the origin of acylindrical coordinate system whose vertical axis is par-allel to the rotational axis of the star ( z direction). Thestar is oblate with equatorial and polar radii R eq and R pole , respectively, polar temperature T pole , mass M ,luminosity L , and rotational velocity v rot that is a frac-tion of the critical velocity v crit . The values adopted forthese parameters in this work are presented in Table 1.The disk is assumed to lie in the equatorial plane of the Ghoreyshi et al. (a) (b) [S17](c)
Figure 2.
Epochs of all available data of ω CMa in (a) polarimetry, (b) photometry, and (c) spectroscopy. Each vertical solidline represents the epoch when the data were observed. The instruments, observatories, or the photometric bands are indicatedin the legend and defined in the text. The dark grey triangles in each panel display the observed V -band photometric data as areference. C i , O i and Q i stand for cycle, outburst and quiescence phases, respectively, where i is the cycle number. The verticaldashed lines display the boundaries between the cycles. The vertical dotted lines show the boundaries between outburst andquiescence phases. star. The rotational velocities of the star and the diskare vectors in the azimuthal direction, φ . Also, the diskhas a radial velocity component, v r , that can be nega-tive (i.e., inflow) or positive (i.e., outflow). The mostimportant parameters describing the disk are the AMflux injected into the disk from the star at the steady-state limit ( ˙ J ∗ , std ), the disk base temperature, T , andthe outer radius of the disk, R out . We note that an al-ternative way of describing the disk feeding mechanismuses the mass injection rate into the disk, ˙ M inj which is related to ˙ J ∗ , std by − ˙ J ∗ , std = Λ ( GM ∗ R eq ) ˙ M inj (cid:16) ¯ r inj − (cid:17) , (2)where Λ is a dimensionless quantity greater than 1 andis given by Λ = 1 / (1 − ¯ r − out ), with ¯ r out = R out /R eq and R out (cid:29) R eq (R´ımulo et al. 2018). One of the reasonsthat we do not investigate ˙ M inj in this paper is becauseit cannot be determined observationally. More detailsand discussion about the relationship between ˙ M inj and˙ J ∗ , std can be found in R´ımulo et al. (2018) and G18. The ulti-observing-technique study of the Be star ω CMa Figure 3.
Top panel : The V -band light curve with vertical colored solid lines marking the dates the spectra shown in thebottom panel were taken. Bottom panel : Continuum normalized hydrogen lines of ω CMa observed by feros using the samecolor scheme as the upper panel. The spectra were arbitrary shifted vertically for ease of comparison. C i , O i and Q i stand forcycle, outburst and quiescence phases, respectively, where i is the cycle number. The vertical dashed lines display the boundariesbetween the cycles. The vertical dotted lines show the boundaries between outburst and quiescence phases. magnitude of ˙ M inj alongside the values of the parametersused for modeling are listed in Table 1.The calculations presented here were mainly com-pleted by two computational codes: the 1D time-dependent hydrodynamics code SINGLEBE and
HDUST .In the following the codes are briefly introduced.3.1.
The
SINGLEBE code
SINGLEBE solves the isothermal 1D time-dependentfluid equations (Pringle 1981) in the thin disk approxi-mation, and provides the disk surface density, Σ( r, t ).The 1D grid used in the
SINGLEBE code models thedisk between R eq (the equatorial radius of the star) and R out (the outer radius of the disk). The grid is a loga- rithmic array with an optional number of cells. One cellis arbitrary selected as the location where mass fromthe central star is entirely injected, ¯ r inj R eq , where ¯ r inj is a dimensionless quantity. The viscosity parameter,as a function of time, α ( t ), and ˙ J ∗ , std ( t ) are the inputparameters of the code. SINGLEBE determines how theinjected matter spreads in the disk. More details about
SINGLEBE can be obtained in the original publication(Okazaki 2007), and a description of the boundary con-ditions adopted can be found in R´ımulo et al. (2018).3.2.
HDUST code
The Monte Carlo radiative transfer code
HDUST is afully three-dimensional (3D) code that simultaneously
Ghoreyshi et al.
Figure 4.
Top panel : The V -band photometric data of ω CMa.
Middle panel : Polarization level in the fourth cycle inthe
BVRI bands.
Bottom panel : The observed position an-gle. C4, O4 and Q4 stand for the fourth cycle, outburst andquiescence phases, respectively. The vertical dashed linesdisplay the boundaries between the cycles. The vertical dot-ted lines show the boundaries between outburst and quies-cence phases. solves the radiative equilibrium, the radiative transfer,and non-LTE statistical equilibrium equations to obtainthe ionization fraction, hydrogen level populations, andelectron temperature as a function of position in a 3Denvelope around the star (Carciofi et al. 2004; Carciofi& Bjorkman 2006, 2008). In order to convert the surfacedensity provided by
SINGLEBE to volume density,
HDUST uses a Gaussian vertical density profile with a 1.5 power-law isothermal disk scale height. With these quantities,
HDUST produces the emergent spectral energy distribu-tion (SED), including emission line profiles, as well asthe polarized spectrum and synthetic images.To date,
HDUST has been used in a variety of theo-retical studies of Be stars disks (e.g., Carciofi & Bjork-man 2006, 2008; Haubois et al. 2012, 2014; Faes et al.2013). The most relevant study for this work is Hauboiset al. (2012) who used the hydrodynamic simulationsto show the capability of the VDD model for reproduc-ing the light curves of Be stars. In addition,
HDUST wasused in several studies where the model predictions wereconstrained by observations such as visible and infraredphotometry (e.g., Carciofi et al. 2012; Baade et al. 2018a;R´ımulo et al. 2018; Ghoreyshi et al. 2018), radio pho-tometry (e.g., Klement et al. 2017, 2019), polarimetry(e.g., Carciofi et al. 2007, 2009; Faes et al. 2016), spec- troscopy (e.g., Carciofi et al. 2010; Suffak et al. 2020),and spectro-interferometry (e.g., Carciofi et al. 2009;Klement et al. 2015; Faes et al. 2016; de Almeida et al.2020).As mentioned earlier, the central star is oblate. Con-sequently, the polar regions of the star have a greatereffective gravity than the equatorial regions. Accord-ing to the von Zeipel (1924) theorem, this latitudinaldependence of the effective gravity causes a latitudinaldependence of the flux, in the sense that the poles arebrighter (hotter) and the equator darker (cooler). Thisgravity darkening effect plays a key role in determiningthe surface distribution of the flux, and therefore thelatitudinal dependence of the temperature.In its original formalism, applicable for a purely ra-diative envelope, the von Zeipel theorem can be writtenas T eff ( θ ) ∝ g β eff ( θ ) , (3)with β = 0 .
25. However, interferometric studies of stel-lar spectral classes of F to B suggested that the β pa-rameter is typically in the range of 0.18–0.25 (van Belleet al. 2006; van Belle 2012; Monnier et al. 2007; Cheet al. 2011; Hadjara et al. 2018; Domiciano de Souzaet al. 2018) with the most likely value of 0.21 (van Belle2012). The theoretical study of Espinosa Lara & Rieu-tord (2011) suggested that the value of β is a function ofthe rotational rate of the star. Following these authors,G18 adopted a β of 0.19 for ω CMa. The same valuewas used in this paper. Using the β parameter, the polarradius of the star ( R pole ), and the critical fraction of therotational velocity ( W , see Rivinius et al. 2003, for howthis parameter is defined) as input parameters, HDUST calculates the geometrical oblateness and gravitationaldarkening of the star.Although
HDUST has the ability to take into accountthe opacity of dust grains (e.g., for B[e] stars, Carciofiet al. 2010), our models were calculated for dust-freegaseous disks consistent with our current understandingof Be stars. MULTI-OBSERVING-TECHNIQUE MODELINGRecall, in G18 our model was limited to the V -bandwhich in turn is sensitive only to variations in the diskregions very close to the star (see Figure 1 of Car-ciofi 2011). An important next step consists of extend-ing the analysis to the other observables (photometryat longer wavelengths, polarimetry and spectroscopy),which probe different disk regions.Here, various line profiles and the entire emergent po-larized spectrum from the UV to the mid-infrared forabout 80 selected epochs covering different phases of thedisk evolution were computed using HDUST . The selected ulti-observing-technique study of the Be star ω CMa
Polarimetry
The linear polarization level can be expressed in termsof the Stokes parameters, Q and U (Clarke 2010) as P = (cid:112) Q + U . (4)The polarization position angle is θ = 12 arctan (cid:18) UQ (cid:19) . (5)One common issue regarding interpretation of polari-metric data is the removal of the IS contribution to theobserved signal. The observed polarization, decomposedin its Stokes Q and U parameters, can be written as: Q obs = Q IS + Q int , (6)and U obs = U IS + U int , (7)i.e., without knowing the IS components ( Q IS and U IS )of the observed polarization, the components ( Q int and U int ) of intrinsic polarization are unknown. This isshown schematically in the top panel of Figure 5. Mea-suring the IS component of the polarization can be achallenging task (e.g., Wisniewski et al. 2010). As thereis no reliable information about this quantity for ω CMain the literature, we employ below three different meth-ods, to determine the position angle of the intrinsic po-larization and to estimate the IS polarization ( θ IS and P IS ). 4.1.1. Q − U method The first method explores the fact that the intrinsicpolarization is variable, while on the same timescale andat a given wavelength, the P IS is not. We begin exam-ining the bottom panel of Figure 5, that shows, in aschematic way, how the process of formation and dissi-pation of a Be disk appears in the Q − U diagram. Theintrinsic polarization angle on the sky is θ int . When P int is zero (no disk), the observed polarization willbe due solely to the IS component ( P obs = P IS ). Asthe disk grows (and dissipates), the magnitude of P int changes, but not the angle (assuming that the disk isaxi-symmetric and lies along equatorial plane). Thisis shown in the bottom panel of Figure 5 as the track (a)(b) Figure 5.
Schematic diagrams showing the components andthe temporal variability of the observed polarization. Notethat no actual data for ω CMa have been presented here. (a)The components of the observed polarization vector ( P obs ) inthe Q − U diagram: P IS and P int . θ obs , θ IS and θ int representthe observed, IS and intrinsic polarization angle, respectively.(b) The variations of the polarization vector and its compo-nents with the disk growth and decay. The red stars andblack circles show the schematic observed polarization levelduring disk growth (toward up and left) and decay (towarddown and right), respectively. of points along the 2 θ int direction, which indicates thatthe angle of the track is a measure of θ int (Draper et al.2014). More specifically, θ int should be parallel to theminor elongation axis of the Be disk (we note that thedisk may not be elliptic but appears as an ellipse in theplane of the sky in the line of sight of observer, if it isnot seen pole-on). In the case of disks confined to theequatorial plane, θ int also describes the position angleof the spin axis of the star, measured east from celestialnorth.Figure 6 shows the Q − U diagram of the polariza-tion data of C4. The original data have some individualpoints with significant variations and large error. There-fore, we binned the data in time intervals of 100 days.For all four bands, the measurements form a straightpath in the Q − U diagram as explained above. A simplelinear least squares regression fit (solid red lines in Fig-ure 6) indicates that the angle of this path is 117 ◦ ± . ◦ ,102 ◦ ± . ◦ , 110 ◦ ± . ◦ , and 104 ◦ ± . ◦ for B , V , R ,and I , respectively, which means that θ int should be half0 Ghoreyshi et al.
Figure 6. Q − U diagram of polarization data of the fourthcycle of ω CMa. The date of observations are indicated inthe legend on the right. The red solid lines are linear fits tothe data. The cyan bands show the range of uncertainties inthe fit. The data are binned in 100-day time intervals. of this value. The errors were estimated using ± σ un-certainty. These numbers are listed in Table 2 with the QU superscript, to indicate that they were obtained us-ing the Q − U diagram method. The average value of theintrinsic position angle for the four filters is 54.2 ◦ ± . ◦ ,where the quoted uncertainty is the standard deviationof the mean.The Q − U method requires that the position angle ofthe disk remains relatively constant over time. We canestimate the validity of this assumption by measuringthe correlation between the Stokes Q and U . ρ ≈ − ρ ≈
0, means the disk behavior is complexand the validity of the method is compromised. Weobtained ρ = -0.34, -0.51, -0.35, and -0.60 for B , V , R , and I filters, respectively, which infer that the er-rors derived by the linear least squares regression fit areunderestimated. However, our results suggest that theintrinsic polarization angle of the star might be close to52 ◦ corresponding to the correlation coefficient closestto -1 ( I -band). 4.1.2. Light curve method
The second method uses the light curve itself. Theresults of G18 indicate that at the end of C4, the V -band excess is very small. According to the best G18model fits, the inner disk at that phase is very tenuous(Figure 1, see also second panel of Figure 11 of G18).Therefore, if one assumes that P int at that phase is verysmall, the observed polarization should be very close to P IS . According to the relatively nearby distance of ω Figure 7.
Top : The V -band observed photometric data for ω CMa. Two vertical red lines indicate the epoch duringwhich the star was assumed to be diskless.
Bottom : The ob-served (grey circles) and intrinsic polarization (blue circles)of the fourth cycle of ω CMa in the V filter, measured bythe field star method. The estimate of P fieldIS is shown bythe horizontal orange line. C i , O i and Q i stand for cycle,outburst and quiescence phases, respectively, where i is thecycle number. The vertical dashed lines display the bound-aries between the cycles. The vertical dotted lines show theboundaries between outburst and quiescence phases. CMa, small values for P IS are expected (e.g., Serkowskiet al. 1975; Yudin 2001).Figure 7 presents the photometric and polarimetricdata of ω CMa in the V band. The observed polariza-tion is shown as grey circles. Two vertical red lines inFigure 7 indicate the boundaries of the phase assumedfor the star to be almost diskless. An average of the dataat this phase (a single point for the B , V , and R filters,and a few points for the I filter; see Figure 18), givesus Q LCIS and U LCIS , which in turn provide the values for P LCIS and θ LCIS , listed in Table 2 for each filter. Here thesuperscript LC indicates that the estimates were madeusing the light curve itself. Using Eqs. 4 to 7 and theestimated value of interstellar polarization, the intrin-sic polarization of ω CMa, P int and θ int , was calculated.The average value of ¯ θ LCint is 60.2 ◦ ± . ◦ , in good agree-ment with the value estimated using the Q − U method.We defer for later a discussion on the intrinsic polariza-tion levels. 4.1.3. Field star method
Finally, we estimated the P IS using the field starmethod, by which one or more stars that are physicallyclose to the target star and that are known to have no in-trinsic polarization are used as a proxy of the interstellar ulti-observing-technique study of the Be star ω CMa Table 2.
Different estimates of the IS polarization and the position angle of the intrinsic polarization. QU , LC, and fieldindicate the Q − U diagram, the light curve, and the field star, respectively, as the methods that have been used to estimatethe intrinsic polarization of ω CMa. The numbers in bold give the average value of the intrinsic position angle of four differentfilters measured by each particular method.Method Parameter B Filter V Filter R Filter I Filter average
QU θ QU int ( ◦ ) 58.4 ± ± ± ± ± P LCIS (%) 0.12 ± ± ± ± ± θ LCIS ( ◦ ) 56.4 ± ± ± ± ± θ LCint ( ◦ ) 56.2 ± ± ± ± ± P fieldIS (%) 0.13 ± ± ± ± ± θ fieldIS ( ◦ ) 59.7 ± ± ± ± ± θ fieldint ( ◦ ) 54.5 ± ± ± ± ± polarization. By the IAGPOL, we observed HD56876,a B5Vn star (Houk 1982) with m v ≈ +4 − pc and an angular distance of 0.67 ◦ from ω CMa. The distance of ω CMa inferred from the paral-lax measured by Gaia is 280 +13 − pc (Gaia Collaborationet al. 2016, 2020).The results of this method are listed in Table 2 withthe “field” superscript. The agreement with the previ-ous method based on the light curve is quite good, asboth results have very similar values for P IS and θ IS .Furthermore, the three estimates of θ int (shown by boldnumbers in Table 2) also agree. By modeling the Br- γ interferometric data for ω CMa, ˇStefl et al. (2011)showed that the position angle of the major axis of thestar’s disk is -29 ◦ (see the lower left panel of Figure 1in their paper). Since there is a 90 ◦ difference betweenthe elongation of the minor and major axes, this meansthe position angle for the minor axis of the disk is 61 ◦ which is in agreement with the results presented here.The estimated P int level from the field star method isillustrated with blue circles in the bottom panel of Fig-ure 7. A positive correlation between the polarizationlevel and the brightness of the star can be seen. It ap-pears that the polarization level follows the variation ofthe V -band photometric data with a lag, for instance, inthe dissipation phase the drop of polarization is slowerthan the drop in brightness. This agrees with the the-oretical studies of Haubois et al. (2014). This behavioris easily explained when we consider that the V -bandpolarization probes a slightly larger volume of the diskthan the V -band (see Figure 1 of Carciofi 2011) and,therefore, the timescales for viscous dissipation will belonger. There is also some intrinsic scatter in the datathat is likely the result of disk variability. As shownby Haubois et al. (2014), the polarization level respondsquickly to changes in the ˙ J . The flickers seen in the V -band light curve of ω CMa should, therefore, have apolarimetric counterpart. Indeed, some of the polariza- tion variability seems to be directly related to flickeringevents (e.g., epochs 56300 to 56800).Below we use the values of the field star method, be-cause the light curve method can have systematic errorsif the P int at the epochs chosen (bracketed by the redlines in Figure 7) is non-zero. In fact, our model cal-culations (see Figure 8 below) indicate that this maybe the case. The same can be said about the field starmethod, of course, as the field star may not probe thesame P IS as the target star. Our choice for the field starmethod is thus based solely on the fact that one method(the light curve one) very likely suffers from systematicerrors while for the other method this is unknown.4.1.4. Model-data comparison
The top panel of Figure 8 displays the G18 modelfor three different inclination angles. To increase thesignal-to-noise (S/N) of the data, we use an inverse-error-weighted average of all filters (i.e. grey filter) avail-able for each epoch. The model was also averaged inthe same way, to ensure a proper comparison with thedata. The agreement between the G18 best fit model(for which i = 15 ◦ and reduced χ , χ = 50) and thedata from C4 is reasonable. Recall that this model wasdeveloped based solely on the V -band photometry, sothe broad agreement for the polarization level is encour-aging. The model seems to reproduce the variations dueto the short (partial) formations and dissipations duringthe main formation phase (O4). However, it is apparentthat the model cannot reproduce the rate of polarimet-ric variations during Q4, as P int drops faster than themodel does. The implications of this result will be fur-ther discussed in Section 4.3.In the bottom panel of Figure 8 we display θ int , aver-aged for all filters (i.e., grey filter) in the same way aswas done for P . Interestingly, there is clear evidence fora trend in θ int , from about 40 ◦ at the beginning of O4to 60 ◦ towards the end of this phase. After that the po-2 Ghoreyshi et al. (a)(b)
Figure 8. (a) The averaged polarimetric data of ω CMavs. the model for the last cycle. The data are presentedwith blue circles. The colored lines represent the syntheticpolarization, with the colors representing different inclina-tions as indicated in the legend. The best fit is for i = 15 ◦ with χ = 50 compared with 107 and 215, for i = 12 ◦ and18 ◦ , respectively. For calculating χ the data were binned(green circles) in time intervals of 100 days to combine theobserved points in the vicinity of epochs for which the modelwas computed. (b) The averaged polarization position an-gle of ω CMa. In all panels, the red points indicate datataken in epochs with a clear photometric flicker. The V -band photometric data are shown with grey triangles in thebackground. C i , O i and Q i stand for cycle, outburst andquiescence phases, respectively, where i is the cycle number.The vertical dashed lines display the boundaries between thecycles. The vertical dotted lines show the boundaries be-tween outburst and quiescence phases. sition angle remains constant, with the exception of onepoint near MJD = 57300. This last point is marked inred, as well as two points at MJD = 54500, to indicatethat they coincide precisely with a photometric flickerin the light curve. The behavior of θ int might be an in-dication of matter being injected in the disk outside ofthe equatorial plane. Evidence for this comes from thefact that whenever matter is fed into a low-density disk(red points and the beginning of O4) the angle is differ-ent than when a fully formed disk is present (end of O4)or no matter is being ejected (Q4). In this scenario, the position angle of the disk would be about 60 ◦ , while theposition angle of the injected matter would be roughly20 ◦ different. It should be emphasized that the abovetrend for θ int can only be seen in the averaged data, andis not discernible in the data for each filter (Figure 18).Finally, similar to the polarization level, the polar-ization angle has a large scatter, much larger than theobservational errors. The scatter in the data may alsobe related to the disk feeding process: depending on howmatter is ejected from the star, an axial asymmetry maydevelop in the inner disk, which can cause large varia-tions in the polarization position angle. For instance,Carciofi et al. (2007) detected changes in P int of up to8 ◦ in less than one hour for the star Achernar, followingputative mass loss events. Similar variations in θ int havebeen observed for other Be stars, and some of these couldbe related to specific outburst events (see discussion insection 4 of Draper et al. 2014).The similarity of the results for each method that havebeen discussed in Sections 4.1.1 to 4.1.3 suggests thatthe results presented here would not be altered withminor changes in P IS .4.2. Magnitudes and colors
In this section our comparison of the model with pho-tometric data in wavelengths other than the visible ispresented.Figure 9 displays the synthetic light curves of ω CMa.The top panel of Figure 9 illustrates the V -band mod-eling presented in G18. The second panel shows thesynthetic UBV bands together. Interestingly, the U -band light curve displays the largest variations, whichis expected as this band should be more sensitive to theinner disk conditions, where the density varies widely(see Figure 11 in G18). The model predicts a complexbehavior. In general, we see that the longer the wave-length, the slower the rate of magnitude variations inthe model light curve. This is explained by the factthat larger disk volumes (from where the long wave-length continuum fluxes originate; see Figure 7 of Riv-inius et al. 2013b) respond more slowly to variations inthe inner disk. One interesting feature of the modelsis that each subsequent dissipation reached a lower fluxlevel when compared to the previous. This reflects thefinding in G18 that in ω CMa a true quiescence value isnever realized, but rather the star transitions betweenhigh and low mass loss rate states.The left panel of Figure 10 shows the comparison be-tween the observed
UBV data and color-indices with themodel. The model fits the data generally well. There area few outliers, which are likely explained by the fact thatwe did not model the short term flickering events (i.e., ulti-observing-technique study of the Be star ω CMa Figure 9.
Synthetic light curves of ω CMa at different bandsfrom the UV to the far IR. The upper plot shows the fit of theVDD model to the V -band photometric data (see Figure 11of G18 for more details). The other plots display the modellight curves for different band-passes as indicated. The ver-tical dashed lines display the boundaries between the cycles.The vertical dotted lines show the boundaries between out-burst and quiescence phases. events shorter than two months, see G18) of the lightcurve as mentioned previously. A systematic mismatch,however, is seen in the color indices, most notably at B − V .The Str¨omgren LTPV data ( uvby -bands) are demon-strated in the right panel of Figure 10. The generalbehavior is similar to the UBV data. The general shape of the curve, as well as the colors, are however quite wellreproduced by the model.The rough agreement between the VDD model and the uvby -bands data is a significant result, as these band-passes probe slightly different regions of the disk. Ingeneral the flux level is related to the disk density, whilethe colors probe the density gradients. Our results in-dicate that the density scale of the inner disk is well-reproduced by the model, but the mismatch in the col-ors may point to inaccuracies in the density gradient.This is not surprising, given the simplistic nature of ourassumptions for the disk inner boundary conditions (seeG18 and R´ımulo et al. 2018, for more details).Figure 11 shows the comparison between the observed
JHKL magnitudes, colors and the model. The top panelreveals that the model is consistently brighter than thedata. This discrepancy is maximum for the L band andminimum for the J band. Thus, the longer the wave-length, the larger the discrepancy. The middle and bot-tom panels display that the color indices are also, in gen-eral, systematically shifted with respect to the models.It should be noted that it is unlikely that the discrepan-cies seen for the JHKL bands are due to uncertainties inthe inclination angle. For instance, changing the inclina-tion angle to 18 ◦ in the models would cause a magnitudeincrement of only ≈ JHKL color indicesare more or less well reproduced by the VDD model,while the actual magnitudes are not. The model is al-ways systematically brighter in the
JHKL bands thanthe observations, which means that the model may betoo dense in the outer disk regions. This might suggesta larger α in the outer disk that would drain it faster,making it less dense. This is discussed in Section 5.Figure 12 displays the model and data in the VISIR Q ∼ µ m) and Q ∼ µ m) bands.Unlike the JHKL magnitudes, the data and models agreewithin the errors. Because the observational errors arelarge and only two points of the Q Q JHKL bands.4.3.
Spectroscopy
We have a comprehensive dataset of spectra for ω CMacovering the cycles 3 and 4 (Figure 2c), as well as Q2.As examples, we show the comparison between the VDDmodel and observed equivalent width (EW), E/C, andpeak separation (PS) for H α and H β in Figure 13. Theresults for H γ and H δ are qualitatively similar, and areshown in Appendix C.4 Ghoreyshi et al. (a) (b)
Figure 10.
Comparison between the observed magnitudes and color-indices (stars) and the VDD model (lines) (a)
UBV filters (b) Str¨omgren uvby filters, both from LTPV. C1, O1 and Q1 stand for the first cycle, outburst and quiescence phases,respectively. The models shown here are limited only to the first cycle because we do not have data for the rest. The verticaldashed lines display the boundaries between the cycles. The vertical dotted lines show the boundaries between outburst andquiescence phases.
These spectra have good S/N and medium-to-high res-olution. Thus, the uncertainties of the quantities aresmall. The uncertainties for the line profile E/C andEW are dominated by their continuum determination,while for the PS the resolution is the limiting factor.It is important to emphasize that we did not adjustthe model to obtain an optimum fit. We used the samemodel and scenario described in G18 to calculate theseprofiles, in order to evalutate how this model performswhen compared to multi-technique data.The second and third panels of each plot in Figure 13display the EW and E/C of the line. The EW and theE/C ratio of a line reveal a complex interplay betweenthe line emission and that of the adjacent continuum.The H α line emission comes from a large volume of thedisk, which responds very slowly to changes in the diskfeeding rate. Conversely, the adjacent continuum re-sponds very quickly to these changes. Therefore, whenthe continuum emission rises (e.g., during an outburst),the EW initially drops in magnitude and the E/C ratiofalls, as well. However, when the continuum emissiondrops (e.g., during quiescence), the EW will increase inmagnitude and the E/C ratio will increase. These effects are more moderate for H β since at 4861 ˚A the adjacentcontinuum displays a much smaller range of magnitudevariation along a given cycle than at 6562 ˚A (Figure 9).Figure 13 can be interpreted with the above scenariosin mind. In all quiescence phases, the EW increases inmagnitude (becoming more negative) and the E/C in-creases, as a result of the quick suppression of the innerdisk, that causes the emission in the adjacent continuumto drop quickly. This initial dissipation of the inner diskdoes not affect the line emission. Only much later inthe dissipation when the entire disk empties, the lineemission drops. Then, the EW decreases in magnitudeand E/C also drops. At outburst, the converse happens:the inner disk fills up quickly, giving rise to a sudden in-crease in the continuum out to the IR. As a result, theEW decreases in magnitude and E/C decreases (recallthe apparent contradiction mentioned at the end of Sec-tion 2 and also, see Figure 3).The PS was computed by fitting a Gaussian curve toeach emission peak, in order to determine its height com-pared to the adjacent continuum (which was normalizedto one), as well as its velocity. The low inclination an-gle of ω CMa causes an almost single-peak profile in H α ulti-observing-technique study of the Be star ω CMa Figure 11.
Comparison between magnitudes and color in-dices of the VDD model and observed data in the
JHKL filters. The VDD models are demonstrated with solid linesand the data are shown with colored stars as indicated in thelegend of the plot. C4 and O4 stand for the fourth cycle andoutburst phase, respectively. The models presented here arelimited to this cycle for which we have data. The verticaldashed lines display the boundaries between the cycles. Thevertical dotted lines show the boundaries between outburstand quiescence phases. whose flux comes mainly from the larger part of the disk(in comparison to the other hydrogen lines) where theKeplerian velocities are lower. Also, some of the dataare of low resolution, which makes our analysis difficult.For this reason we show the PS in Figure 13 only for thespectra with a clear double-peaked structure. Compar-ison with the model reveals that, similarly to what wasseen for the EW and E/C, there is a better agreementduring the outburst phases than during dissipation.In general, the results for H β are similar to those forH α . The EW curve is qualitatively reproduced, buta quantitative comparison fails mainly during the qui-escence phases. Of particular significance is the closematch between the data and the model for the fast de-cline in EW of O4. The E/C is also well reproduced.It is important to recall that, since the H β opacity issmaller than H α , the formation volume of this line issmaller (Carciofi 2011). This can be seen by the PS val-ues, which lie ≈
40 km s − for H β , while for H α they are,in general, smaller than 20 km s − . The larger PS indi- Figure 12.
Comparison between magnitudes of the VDDmodel and the observed data in the Q Q cate that H β is indeed formed closer to the star, wherethe rotational velocities are larger. The fact that themodel reproduces this behavior is a significant result.The model can reproduce these variations qualita-tively, but not quantitatively. After the quick increasein magnitude of the EW at the onset of dissipation, theobserved EW decreases in magnitude at a much fasterrate than the model does (the same is observed with theE/C). Since the predictions from G18 fit the visual andinfrared band light curves well, the problem likely liesin the outer disk. It appears that while G18’s modelpredicts the correct rate of density variation in the in-ner disk, the corresponding rates in the outer part aretoo slow. In other words, the outer disk is not beingdrained of material fast enough. Further support to thiscomes from polarimetry. Recall that the observed rateof polarimetric variation is larger than the model cal-culations, also indicating faster emptying than in themodel.In the following, we provide some tentative explana-tion for this mismatch between the model and the data.One way to achieve faster dissipation rates in the outerdisk is to have larger values of the viscosity; this couldhappen either because the temperature rises with radius(which is not physically justified) or the α parameterincreases with distance from the star. Therefore, thismight be the first hint of a radially varying α in a Bestar. Another possibility is to consider that there is anunknown binary companion truncating the disk at radiismaller than the 1000 R eq assumed here (note that thiswas an arbitrary assumption for the disk size in G18).6 Ghoreyshi et al. (a) H α (b) H β Figure 13.
Comparison of the models (lines) and the (a) H α and (b) H β EW, E/C, and PS. The top plot in each panel displaysthe V -band light curve with vertical blue dashed lines marking the date the spectrum was taken. Observed EW, E/C, and PSare shown with blue circles, triangles, and pentagons, respectively, in the second to the fourth panels. C i , O i and Q i stand forcycle, outburst and quiescence phases, respectively, where i is the cycle number. Models for the first cycle are not presenteddue to lack of data. The vertical dashed lines display the boundaries between the cycles. The vertical dotted lines show theboundaries between outburst and quiescence phases. If this were the case, the mass reservoir of the outerdisk would be smaller and the whole disk would dissi-pate faster, as suggested by the observations. Finally,a third possible explanation for the mismatch is radia-tive ablation. In the absence of active feeding, ablationcould act in addition to viscosity to dissipate disk ma-terial. However, the results of Kee et al. (2016a, andsubsequent papers) indicate that ablation is more effi-cient at clearing the inner disk rather than the outer.Therefore, ablation, if included in our models, wouldlikely make the mismatch between the rates of the dis-sipation of the inner and outer disk worse. In the nextsection, we investigate the first two possibilities, namelylarger α in the outer disk and binary truncation. Also,we discuss the influence of ablation in more details. TESTING ALTERNATIVE MODELSSo far, we showed that the model presented in G18 isunable to reproduce some of the characteristics of theobserved spectra, multi-wavelength photometry and op-tical polarimetry. These discrepancies seem to indicate that the models predict an outer disk that is too mas-sive and a rate of dissipation during quiescence phasesthat is too slow. Thus, we need to adjust our modelsso that the disk dissipation rate at larger radii is larger.We investigate two possible solutions, namely: 1) largervalues of the α parameter, and 2) disk truncation by abinary companion.In the first test, we compare the new models basedon larger values of α with the polarimetric data and H α EW during the quiescence phase of the fourth cycle, C4,to probe the effectiveness of the method for two datasetsoriginating from regions within the disk at greater radiifrom the central star. (We note that C4 is the onlycycle for which we have simultaneous polarimetric andspectroscopic data.) To further support our findings,we verify this approach by using it to model also the H α EW during the second quiescence (Q2) since the EWdata for Q2 show a clearer pattern.Figure 14 displays the results of this test for the po-larimetric data and confirms that larger values of α of ulti-observing-technique study of the Be star ω CMa χ , χ = 11) enhance the fitsignificantly. Recall that the optimum value found forthe α parameter for Q4 by fitting the V -band data was0.11 (G18, with χ = 69; also see Figure 1). With agreater α , the rate of polarimetric variation of the modelis larger due to the increased dissipation rate, matchingthe lower data points that were not reproduced by theoriginal model (see Figure 8).Figure 15 confirms that an even larger value of α suchas 0.22, leading to quicker disk evolution, is requiredfor a better agreement between the model and the data.With a larger value for α , the EW rises fast enough tomatch the data and later, at the middle of the dissipa-tion phase, starts to drop simultaneously with the H α EW data.Since the EW data for the H α line in C4 are sparse,we repeat this test for Q2. Figure 16 demonstrates aresult similar to Q2: a larger value of α of about 0.25with χ ≈ . χ ≈
130 determined from V -band photometricmodeling) provides much more consistency between thedata and the model. These tests indicate a common pat-tern: larger values of α than what was obtained for the V -band lightcurve are required to match the observedrate of variations for the polarization (Q4) and H α EW(both Q2 and Q4). Therefore, recalling that the polar-ization and H α probe a radial extent of the disk of about1.5 – 5 times larger than the V -band continuum, respec-tively, this may suggest that the α parameter grows withdistance from the star.For the second test (disk truncation), we calculatedthe H α EW during Q2 for a disk with R out = 25 R eq .The result for the H α line is shown in Figure 17 with χ ≈
210 and is compared with the result of G18 thatuses R out = 1000 R eq (the same model as α = 0 .
13 inFigure 16 with χ ≈ α strength, asshown in Figure 17. Thus, although the truncated diskhypothesis seems a viable solution for increasing the rateof dissipation, it creates another problem, namely an H α emission that is too small. It is worth mentioning thatHarmanec (1998) and ˇStefl et al. (2003) did not findany evidence of binarity. However, if the orbital planeand the circumstellar disk plane are about the same, theradial velocity signature of a companion is very difficultto find owing to the small inclination angle, especiallyif it is a subluminous star. The results discussed aboveare unchanged regardless of the cycle, because the trendsseen in the EW curve of all cycles are similar. Finally, it is worth discussing an effect that was notincluded in our models but could affect the results byincreasing the disk dissipation rate. Kee et al. (2016a)showed that radiation forces, especially for the very hotO-type stars with strong winds, can ablate the entiredisk in timescales of the order of days to years. Theysuggest that this is the reason why disks are not com-monly observed surrounding these types of stars. Theyalso showed that for a B2-type star it would take a cou-ple of months to destroy an optically thin, low densitydisk. However, Kee et al. (2018a) concluded that, fora more massive optically thick disk (like ω CMa) thiseffect would decrease the ablation rate by only 30% orless. Therefore, although the ablation does not seem ad-equate for the disk dissipation timescale of ω CMa ( ≈ ω CMa, the values of α for the dissipation phases quoted inG18 and Section 5 of this paper would represent upperlimits. However, since this mechanism does not seem tobe strong enough in the outer parts of the disk, it can-not help the problem of slow dissipation of the disk seenin our models. Future work, combining viscosity andablation, is necessary to properly address this problem. CONCLUSIONSWe use the VDD model to study the observed datafrom a range of wavelengths and techniques for the Bestar, ω CMa, in a dynamical fashion. We adopt thesame model presented in G18 that was used to study the V -band continuum emission of this star. In this work,we compare model predictions with a range of differentobservations (multi-band photometry, spectroscopy, andpolarimetry). Since different wavelengths originate fromvarious parts of the disk (Figure 1 of Carciofi 2011), thismethod is a solid test for the VDD model.The results were mostly positive: qualitative and evenquantitative agreements were found, but in some casesimportant differences could be noted. We see the bestagreement for the visible photometric data, but for theIR band the differences are more significant. This makessense because the model parameters used here are thesame as G18 that were used for modeling the V -bandoriginating mostly from the inner regions of the disk( (cid:46) R eq ) while the IR photometric data (e.g., JHKL and Q Q
3) come from a much larger volume of the disk( (cid:46) R eq ). The models predicted larger IR excessesthan observed, indicating that the disk mass is likelyoverestimated.8 Ghoreyshi et al.
Figure 14.
Top panel shows comparison between the mod-els with various α values and observed polarimetric data.The blue circles display the observed data and the lines rep-resent the models corresponding to values of α as indicatedin the legend. The V -band photometric data are shown withgrey triangles in the background. The vertical dashed linesdisplay the boundaries between the cycles. The vertical dot-ted lines show the boundaries between outburst and quies-cence phases. C4, O4 and Q4 stand for the fourth cycle, for-mation and quiescence phases, respectively. Bottom paneldisplays the goodness of the fit ( χ ) for each α value. Forcalculating the χ the data were binned (green circles) tocombine the observed points in the vicinity of epochs forwhich the model was computed. The average polarization level was fitted acceptably byour models, but the model rate of dissipation during qui-escence was too slow. One important point to stress isthat the observations showed a much slower decline rateof the polarization than the V -band light curve. Thisslower decay indicates that the polarization originatesin a larger radial extent than the V band continuum,which is consistent with our model predictions.Although our model could fit the spectroscopic dataqualitatively, we find, similarly to the polarization level,that the rate of EW decay during quiescence is too slow.This seems to point to the fact that in the original G18model the rate of density variation in the outer disk istoo small, meaning that, during quiescence and at largerdistances from the star, the disk is not being drained fastenough. Two tests were conducted to look for possibleremedies to this issue.In the first test, we experimented with models withincreased values of α . These models produced an EWcurve very similar to the observed one, specially for Q4.Given that lower values of α are required to match the V -band light curve, as in G18, these results hint at thepossibility of a radially increasing α in Be star disks.As a second alternate scenario we considered the ef-fect of truncating the disk by an unresolved binary com- Figure 15.
Comparison between the models with a rangeof α values and the observed H α EW. The upper panel showsthe observed V -band photometric data, and the middle paneldisplays the EW for each model with the individual α param-eter. The blue circles display the observed data and the linesrepresent the models with different α parameter as indicatedin the legend. C4 and Q4 stand for the fourth cycle andquiescence phase, respectively. The vertical dashed lines dis-play the boundaries between the cycles. The vertical dottedlines show the boundaries between outburst and quiescencephases. The lower panel demonstrates the goodness of thefit ( χ ) for each α value. Figure 16.
Same as Figure 15 for C2. ulti-observing-technique study of the Be star ω CMa Figure 17.
Comparison between the model presented inG18 and the truncated disk model. The top panel illus-trates the observed V -band photometric data, and the bot-tom panel displays the EW for each model. The blue circlesdemonstrate the observed data and the lines represent themodels with different disk size as shown in the legend. C i ,O i and Q i stand for cycle, outburst and quiescence phases,respectively, where i is the cycle number. The vertical dashedlines display the boundaries between the cycles. The verti-cal dotted lines show the boundaries between outburst andquiescence phases. panion that could potentially decrease the density inthe outer part of the disk. This effect was consideredby changing the outer radius of the simulation from1000 R eq to 25 R eq . While the rate of line strength varia-tion approached the observed data, truncating the diskcreated the undesired effect of reducing the line emis-sion.It is important to note that the above results could beinterpreted in a different way. The discrepancies abovemight simply be the result that the V -band photometryalone cannot fully constrain the disk at all radii. As theG18 model likely suffers from degeneracies, it is possiblethat different model parameters (e.g., different ˙ J andvalues of the α parameter) might be able to explain thefull set of observations, without the need to resort to thevariable α or truncated disk scenarios. This possibilitywill be explored in future models.Also, it is worth mentioning that line-driven ablationmay play a role helping the disk to dissipate faster (Keeet al. 2016a, 2018a,b). This means that if this effectis important, the values we find for the α parameterare upper limits. However, ablation affects mostly theinner disk. Therefore, its role, if any, should be morenoticeable in the observables that are more sensitive tothe inner disk variations, e.g., V -band photometry andpolarimetry. In this regard, including ablation in themodel might exacerbate the mismatch between observa- tions and model concerning the rate of polarimetric andspectroscopic variations during quiescence.Finally, it is worth discussing the instability of the diskof ω CMa. As an early-type Be star, it is more likelythat ω CMa possesses an unstable disk in comparisonto the late-type Be stars (Labadie-Bartz et al. 2018)and all our multi-technique data confirm this statement,showing variations on timescales of few days to severalyears. The origins of this instability may be caused byone or more of the following mechanisms: discrete mass-loss events caused by the nonlinear coupling of multipleNRP modes (e.g., Baade et al. 2016), fast rotation (e.g.,Rivinius et al. 2013a reports variations of the width ofphotospheric lines likely linked to changes in the rotationrate of the surface layers), and ablation (e.g., Kee et al.2016a).In Section 5 we discussed the possible existence of anundetected binary companion and its effect on disk size.Investigating the long-baseline interferometric data of ω CMa may help us to have a better understanding ofthe morphology of the disk and of the possible existenceof a companion object. In the future, we also plan toextend our analysis to include interferometric data andlong wavelength (radio) photometry and Balmer decre-ment variations in order to continue to explore the limitsof the VDD model.ACKNOWLEDGEMENTSWe would like to thank the anonymous referee forher/his constructive criticism, careful reading, thought-ful and insightful comments and time and efforts towardsimproving our manuscript.This work made use of the computing facilitiesof the Laboratory of Astroinformatics (IAG/USP,NAT/Unicsul), whose purchase was made possible bythe Brazilian agency FAPESP (grant 2009/54006-4)and the INCT-A. M.R.G. acknowledges the supportfrom CAPES PROEX Programa Astronomia and thegrant awarded by the Western University PostdoctoralFellowship Program (WPFP). A.C.C acknowledges sup-port from CNPq (grant 311446/2019-1) and FAPESP(grant 2018/04055-8). C.E.J. acknowledges the NaturalSciences and Engineering Research Council of Canadafor the financial support. D.M.F acknowledges Supportsby the international Gemini Observatory, a program ofNSF’s NOIRLab, which is managed by the Associationof Universities for Research in Astronomy (AURA) un-der a cooperative agreement with the National ScienceFoundation, on behalf of the Gemini partnership of Ar-gentina, Brazil, Canada, Chile, the Republic of Korea,and the United States of America.0
Ghoreyshi et al.
This research used the observations collected at theEuropean Organization for Astronomical Research inthe Southern Hemisphere under ESO programs 68.D-0095(A), 68.D-0280(A), 69.D-0381(A), 74.D-0240(A),75.D-0507(A), 82.A-9202(A), 82.A-9208(A), 82.A-9209(A), 268.D-5751(A), and 282.D-5014(B).Some of the data presented in this paper were ob-tained from the Mikulski Archive for Space Telescopes(MAST). STScI is operated by the Association of Uni-versities for Research in Astronomy, Inc., under NASAcontract NAS5-26555. Support for MAST for non-HSTdata is provided by the NASA Office of Space Sciencevia grant NNX13AC07G and by other grants and con-tracts.This work has made use of data from the Euro-pean Space Agency (ESA) mission
Gaia
Gaia
Gaia
Multilateral Agreement.The IR-band data were provided by Juan FabregatLlueca and they were obtained at the South African As-tronomical Observatory (SAAO), and at the Teide Ob-servatory (Tenerife, Spain).We thank Wagner J.B. Corradi from UniversidadeFederal de Minas Gerais for providing part of the po-larimetric data and Daniel Bednarski from Universi-dade de S˜ao Paulo for providing the polarimetric dataof HD56876.Also, this research has made use of the SIMBADdatabase and VizieR catalogue access tool, operated atCDS, Strasbourg, France.REFERENCES
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Table 3.
Spectroscopic Data LogsReference Number of Points Time Coverage (MJD) Wavelength (˚A)CES 2 52659 – 52660 4184 – 7314ESPaDOnS 16 55971 – 55971 3696 – 8868 feros
444 52277 – 54822 3527 – 9215 heros
435 50102 – 51301 3438 – 8629IUE 12 43833 – 44975 1000 – 3200Lhires 25 54083 – 57465 6511 – 6610Ondrejov 7 53060 – 56737 6258 – 6770OPD 8 56636 – 57645 4118 – 9183PHOENIX 10 54776 – 55311 21604 – 21700Ritter 20 57329 – 57496 6471 – 6634UVES 141 54784 – 54913 3055 – 10426
Table 4.
Polarimetric Data Logs, observed by OPDReference Number of Points Time Coverage (MJD) U Filter 3 55497 – 56050 B Filter 39 54765 – 57624 V Filter 81 54505 – 57626 R Filter 37 54975 – 57624 I Filter 39 54975 – 57624 Ghoreyshi et al. B. ADDITIONAL POLARIMETRIC DATA (a) (b)(c) (d)
Figure 18.
Same as Figure 7 for (a) B , (b) V , (c) R and (d) I filters. Also, the position angle for each filter is shown. Theobserved (grey pentagons) and intrinsic polarization angles (blue pentagons) of ω CMa. The θ fieldIS and the average value of theintrinsic polarization angle are shown with orange and purple horizontal lines, respectively. The vertical dashed lines displaythe boundaries between the cycles. The vertical dotted lines show the boundaries between outburst and quiescence phases. ulti-observing-technique study of the Be star ω CMa C. MODEL FITS FOR H γ AND H δ LINES (a) H γ (b) H δ Figure 19.
Same as Figure 13 for (a) H γ and (b) H δδ