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Differential polarization of Hβ in γ CAS Jiang, Yiman 1993

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DIFFERENTIAL POLARIZATION OF IP IN 7 CASByJiang YimanB. Sc. (Spacephysics) Beijing UniversityA THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinTHE FACULTY OF GRADUATE STUDIESGEOPHYSICS AND ASTRONOMYWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIASeptember 1993© Jiang Yiman, 1993In presenting this thesis in partial fulfilment of the requirements for an advanced degree atthe University of British Columbia, I agree that the Library shall make it freely availablefor reference and study. I further agree that permission for extensive copying of thisthesis for scholarly purposes may be granted by the head of my department or by hisor her representatives. It is understood that copying or publication of this thesis forfinancial gain shall not be allowed without my written permission.Geophysics and AstronomyThe University of British Columbia2075 Wesbrook PlaceVancouver, CanadaV6T 1Z1Date:5 p, 2 3  ^2993AbstractHigh resolution ( 0.15 Alpixel ) spectropolarimetry across the Hfl emission line ofthe Be star 7 Cas is presented. The differential polarization and polarization angle varyboth in profile and depth. It was found that the polarization profile fits the relationPi, = (Iwukviving fitotai)Pc, instead of the generally accepted PL = PC fiemission (Coyne1975). Assuming the absolute polarization in the adjoining continuum is constant, thepolarization profile is used to determine Teff, geff, and v sin i for the underlying star byfitting the Hubeny (1988) atmosphere model.The Be star model developed by Poeckert and Marlborough (1978a) for .7 Cas is usedas a basis to explain our polarization observations. We have made two types of modifi-cations to the model: the first being to revise the intrinsic properties of the underlyingstar; the other is to make more accurate treatment of the radiation transfer equation inthe disk. A search of the parameter space was made to find the best fit. We find anadequate fit to the emission line profile, but only a poor fit to the polarization profilefrom Poeckert and Marlborough model.11Table of ContentsAbstractList of Tables^ viList of Figures viiAcknowledgement^ xi1 Introduction 11.1 The Be phenomenon ^ 11.2 Modeling Be stars 31.3 Past Polarization Studies of Be stars ^ 81.4 -y Cas ^ 101.5 An Outline of this Thesis ^ 112 Data Source and Reduction 122.1 UBC/DAO Polarization Analyzer ^ 122.2 Observations ^ 132.3 Data Reduction 142.4 Results ^ 153 The Underlying Star 183.1 Problem ^ 183.2 Basic relation of linear polarization across an emission line ^ 183.3 Primitive analysis of the polarization data ^ 203.4 Hubeny synthetic spectra^ 233.5 A possible way of determining fundamental parameters ^ 253.6 Discussion ^ 253.7 Pure emission 294 The Poeckert and Marlborough Model 344.1 Construction of the Model ^ 344.1.1^Central Star 354.1.2^Envelope^ 354.2 Model for 7 Cas 435 Hi3 emission line profile 455.1 Principal level 6th of hydrogen atom ^ 475.2 Hubeny atmosphere models ^ 485.2.1^Continuum spectra 505.2.2^Line profiles ^ 505.2.3^Kurucz model replaced by Hubeny model^ 525.3 Pure emission from PM model ^ 525.4 Electron scattering from the disk 565.5 Parameter study ^ 566 Polarization across Hfl emission line 616.1 Correct geometry for polarization ^ 646.2 Correction of the scattered radiation 66'7 Conclusions 72ivAppendices^ 73A Figures for Chapter 5^ 73Bibliography^ 82VList of Tables1.1 Basic -y (Jas parameters ^ 112.2 Observations of 7 Cas during 1990-1993 ^ 133.3 Parameters of models used to find the best fitting model ^ 253.4 Best-fitting model of polarization data for Sep. 23 and Sep. 24 ^ 273.5 Most likely models representing the central star of -y Cas ^ 283.6 Suggested best model for central star of 7 Cas ^ 294.7 Original PM model parameters of 7 Cas 445.8 Parameters define the model for 7 Cas of Sep 23^ 59List of Figures2.1 Line profile, polarization and polarization angle of 7 Cas on Sep. 23,1990^162.2 Line profile, polarization and polarization angle of -y Cas on Sep. 24,1990^173.3 Polarization across H13 emission line on Sep. 23,1990 (solid line). Thedashed line shows Pc/how, where Pc is taken to be constant in the con-sidered region (Poeckert 1979). We set Pc = 1.1% to roughly match theline center polarization    213.4 Polarization across Hfl emission line on Sep. 24,1990 (solid line). Thedashed line shows Pc/how, where Pc = 1.1%. ^  223.5 Hubeny NLTE synthetic spectrum of WI with Teir = 25000K, log g = 3.5,v sin i = 400 km/s, and microturbulent velocity= 10 km/s^ 243.6 The dashed line represents the predicted linear polarization from Eq.3.8,the underlying absorption profile is taken from the Hubeny NLTE syn-thetic spectrum of HP with Teff = 25000K, log g = 3.5, v sin i = 400 km/s,and Pc set to be 1.7%. The solid line is the observed polarization across11$ of 7 Cas on Sep. 23,1990.   263.7 Best fitting model of the linear polarization across lif3 on Sep. 23, 1990,with Teff = 30000 K, log g = 4.0, v sin i = 340.0 km/s, Pc = 1.6 %. . . . 303.8 Best-fitting model of the linear polarization across Hi3 on Sep. 24, 1990,with Ter = 30000 K, log g = 3.5, v sin i = 310.0 km/s, Pc = 1.5 %. . . . 31vii3.9 The dotted line gives the best-fitting model for the underlying stellar ab-sorption feature of Sep. 23. The solid line shows the total intensity ofobservation, with the difference between the two giving the pure emissionfeature from the disk, shown as the dashed line.   323.10 The dotted line gives the best-fitting model for the underlying stellar ab-sorption feature of Sep. 24. The solid line shows the total intensity ofobservation, with the difference between the two giving the pure emissionfeature from the disk, shown as the dashed line.   334.11 A schematic picture of the envelope adopted in the PM model ^ 375.12 (a) PM model with their original parameter set compared to our data onHfl of 7 Gas, Sep. 23, 1990. Dashed line is the model prediction. Solidline is the observation. (b) The same as (a), except overplotted on thedata of Sep. 24, 1990 ^  465.13 Predicted H/3 emission line after adding the 6th principal level of hydrogenatoms (dashed line), compared to the original model (solid line)^ 495.14 Continuum Spectrum at Tea. = 25000 K, log g =3.5 predicted by Kuruczmodel (dashed line) and Hubeny model (solid line) ^ 515.15 Intrinsic Hig line profile at Ter = 25000 K, log g =3.5 calculated by Kuruczmodel (dashed line) and Hubeny model (solid line) ^ 535.16 The dashed line gives a model prediction, in which the underlying spec-trum is represented by the Hubeny atmosphere model, instead of the Kur-ucz atmosphere model as in the original PM model (solid line) ^ 545.17 After replacing the Kurucz model by the Hubeny model, we apply thebest-fitting parameter set to represent the central star (dashed line), andcompare to Sep. 23 Hfl emission line (solid line) ^ 55VIM5.18 Adding an gaussian profile to represent the electron scattering of the emis-sion line photon from the disk (solid line), comparing to the Sep. 23 11)3line profile.   575.19 Line profile predicted by the final model (dashed line), overplotted on Sep ^23 data (solid line) ^  606.20 (a) Polarization profile across 11,3 emission line, predicted from originalPM model (dashed line) compared to Sep. 23 data (solid line). (b) Thesame as (a), overlapped on Sep. 24 data ^  626.21 The solid line gives the predicted polarization from the PM model assum-ing a finite size to the central star, while the dashed line gives the predictedresult when the star is treated as a point source. We see that the predictedpolarization structure is the same in both cases. ^  636.22 Geometry of polarization in the circumstellar envelope ^ 656.23 The dashed line gives the predicted polarization after adopting the newgeometry of polarization, comparing to that from the PM model (solidline). We find the central peak has been constrained.   676.24 The Dashed line gives the model prediction after the changes made accord-ing to Eq.3.8 to include the underlying feature in the scattered radiation,comparing to the prediction without such changes (solid line)   696.25 Polarization profile predicted by the final model (dashed line), overplottedon Sep. 23 data (solid line) ^  706.26 The dashed line gives the polarization predicted by applying the modifiedPM model predicted line profile to Eq.3.8, Pc = 1.6%, overplotted on thedata of Sep. 23 (solid line)   71A.27 ^  74A.28 ^ 75A.29 76A.30 ^ 77A.31 78A.32 ^ 79A.33 80A.34 ^ 81AcknowledgementThanks first to Gordon Walker, my supervisor, for his guidance and patience through-out. I am also grateful to Jaymie Matthews for his help and support, Ted Kennelly forteaching me the essentials of astronomical research, and Jason Auman for providing manyuseful comments as the second reader.My thesis was made easier with the help of• J.M. Marlborough, who co-wrote the code, with R. Poeckert, on which this thesis isbased and responded to several of my queries• Nadine Dinshaw who started the polarimetry project for her M.Sc. thesis and providedmany observations of which I made use• Katharine Wright who helped me take the observations gathered during the course ofmy thesisI would also like to thanks my parents for loving me so much, my sister for beingthere when I needed her, Angela for giving me a lot of fun, and special thanks to Andrewfor everything he did.xiChapter 1Introduction1.1 The Be phenomenonA classical Be star is, following Jaschek et al (1981), defined as:A non-supergiant B-type star whose spectrum has, or had at sometime one ormore Balmer lines in emission.Far from being exceptional, the presence of emission lines is pervasive throughout theHR diagram. Among normal stars of every spectral type (X), one can find emission-linestars of the same spectral type (Xe). However, Be stars are the best-studied type amongall the emission-line stars.Be stars were first discovered in 1866 by A. Secchi (Underhill 1982). Ever sincethen, Be stars have been extensively observed from the radio to the X-ray regions of thespectrum, and new candidates are still added to the list from normal B stars, of which theymake up a large subset (about 20 %). Besides the most distinguishing feature of emissionin hydrogen lines (which is often accompanied by emission from singly ionized metals),Be stars also exhibit excess IR emission with respect to normal B-type stars, intrinsiclinear polarization, high speed stellar wind (seen in the UV region) and superionizationin the wind.It is now generally accepted that a Be star is surrounded by an extended envelope,where the gas is ionized by the ultraviolet radiation of the star. Observational evidencefrom the visible and infrared shows that this outer envelope is relatively cool — with1Chapter 1. Introduction^ 2temperatures of order 104 K, so that the matter in the envelope is subionized comparedto the photosphere, quiet — without large mass-ejection and high expansion velocities.The hydrogen emission lines are generated in the envelope, which are expected to resultfrom recombinations of the ionized atoms, as is the infrared excess, which is attributedto free-free radiation emitted by the ionized gas in the outer envelope. However, theobservations in the visible, infrared, and radio regions show that there is a great diversityin the dimensions of the regions emitting at these wavelengths, from several stellar radii,which is necessary for weak Ha emission, to very extended envelopes, where the densitymust still be significant at hundreds or thousands of stellar radii, in order for radioemission to be detectable.On the other hand, ultraviolet spectra observed over the last 20 years show a strik-ingly different picture from those of visible to infrared. There is no specific distinctionbetween Be and normal B stars in the UV spectrum. One important phenomenon ap-pearing in both B and Be stars is the presence of superionized lines in the UV region.Resonance lines arising from ionization states as high as 0 VI and N V indicates theexistence of superionized regions with temperatures of order 105 K — much higher thanpredicted from the effective temperatures of these same stars under conditions of radia-tion equilibrium. A nonradiative energy flux is required to explain their formation. Thefact that resonance lines have large displacement or asymmetries, which may sometimesexceed their escape velocity at the surface of star, suggests fairly active regions in theouter envelope with violent ejection of matter taking place.A large body of observations show Be stars to be variable. Their variability presentsitself in a variety of forms. The Balmer line spectra may undergo, in their life, threedifferent phases, in any order : Be, shell, and normal B phase. In a Be spectrum, emissionlines show either no reversal or a more or less central reversal. In a shell spectrum, Balmerlines and singly ionized metal lines exhibit narrow and deep absorption cores, which mayChapter 1. Introduction^ 3or may not be bordered by emission wings. Generally there is a gradual transition fromB phase to Be or shell phase, but no clear break between Be and shell. Furthermore,within a given phase, intensities, profiles and line widths vary with time, such as V/Rvariations—the ratio of the violet to red peak in emission lines, and E/C variation—theratio of the emission in the lines to the adjacent continuum. A similar effect is seenin the continuum, which is mostly observed as the changing in stars' magnitudes bybroadband photometry in UBV. Even greater variations are observed in the ultraviolet,which mainly involve the shifts and profiles of the superionized lines; in another words,these lines are used to determine the mass loss of these stars, thus indicating that themass flux is strongly variable. All the variations are generally irregular, their time scalesranging from a few hours to several decades.1.2 Modeling Be starsThe first suggestion to explain the origin of the emission lines in Be stars was putforth by Struve (1931). His picture of the circumstellar envelope has since become thefundamental picture of Be stars.As most Be stars are rapid rotators, Struve suggested that their rotational velocitieshad reached their critical value, so that matter could be ejected at the equator due tothe rotational instability, forming an equatorial disk or ring. Extrapolated from themechanism applied to the gaseous envelopes of planetary nebulae, dilution of the stellarradiation makes it possible to convert high-energy radiation from the star to low-energyradiation regenerated from the envelope. Atoms in the envelope are ionized by theultraviolet flux from the central star. Recombinations to higher levels and then cascadesto lower levels produce the observed emission in hydrogen lines. But what distinguishesthis situation from the planetary nebula is the dilution factor, which is about 10-i toChapter 1. Introduction^ 410-2, compared to 10-14 in the planetary nebula case. So in Be stars, the envelope isbelieved to be subionized relative to its photosphere, and the envelope may be opaque inthe lines of the subordinate series.Struve's model explains the diversity of the emission line profiles by geometric effectsalone. Whether a Be star exhibits a normal Be phase or a shell phase, depends on if itis viewed closer to pole-on or equator-on.Struve's model was a great success in first trying to explain the overall Be phenomena.But it encountered severe observational difficulties, even at its earliest stages. First ofall, although most Be stars are fast rotators (statistically speaking, the Be stars exhibitlarger values of v sin i than normal B stars), no Be star has been found to be rotating atits critical velocity. Secondly, critical velocity alone cannot explain the ejection of matter,because it can only balance the gravitational force with the centrifugal force; it would notallow matter leaving too far from the stellar surface, which means that rotation alonecan't be responsible for the formation of outer envelope. Besides , some Ha emissionlines in Be stars have great extended wings, which could exceed 1000 kms-1. This is twoor three times larger than their v sin i, which implies an unrealistic conclusion that theirenvelope rotates faster than the stars. Also the large observed variabilities could not beexplained solely by the geometrical orientation effects.During the next 50 years after Struve's rotation model was first unveiled, quite a fewBe star models have been suggested based on his picture. They are basically of two types:1. Be star as a single star.• Elliptical ring model, such as Huang (1972,1973)• Disk-like models, such as Kogure (1969), Hirata and Kogure (1977,1978)• Stellar wind models, such as Doazan (1965), Hutchings (1968), Marlborough(1969), Poeckert and Marlborough (1978a)Chapter 1. Introduction^ 52. Be star as member of binary system, in which the envelope is produced by masstransfer in a close binary system, where the companion fills its Roche-lobe, such asKriz and Harmanec(1975)Good reviews for those models can be found in Marlborough (1975), Poeckert (1981) andreferences therein.Without exception, all those models faced two critical problems:1. Hydrodynamics of the formation of the circumstellar envelope.2. Solution of the transfer equations, in order to compare the model spectrum withthe observed spectrum.It is the first problem which fundamentally defines the ad hoc property of all existingBe star models. Far from completely understanding the construction of the envelopes,one generally guesses the solution to the hydrodynamic problem, so that the densitydistribution in the envelope could be obtained (cf. Chapter 4). The fact that eachmodel uses a different approach to determine the excitation and ionization state of theenvelope and to solve the radiative transfer problem and can simultaneously satisfactorilyreproduce the same spectral feature, indicates there is no unique solution. One importantreason for this is that almost all the models have a large number of ad hoc assumptions,which corresponds to a large number of free parameters that can be arbitrarily set. Itis impossible to decide which of the above models best represents the Be stars. Onemotivation behind this thesis is to put as many constraints on Be stars or their envelopesas possible from our polarization observations, based on the Poeckert and Marlboroughmodel (cf. Chapter 4,5,6).Nevertheless, the real trouble comes from the ultraviolet observations. They raiseseveral major contradictions with the classical idea of Be stars. As mentioned before,Chapter /. Introduction^ 6instead of cold, quiet regions around Be stars, analysis of ultraviolet spectra provideevidence of a large mass-loss flux, high temperature and superionized region. It was thelarge mass-loss detected in the ultraviolet which convinced people of the ability to formthe extended envelopes observed in the visible spectra. However, the problem is how thetwo strictly different regions around Be stars are related to each other if both of themactually exist; how they are formed; why only Be stars have a cold envelope dense orlarge enough to have emission lines in the visible region, although both normal B andBe stars have high-temperature, mass-outflow regions. Those problems are still unsolvedmysteries for Be stars.Only recently have theoretical models begun to take into account the dynamic evi-dence of the kinematical structure assumed in empirical models. Snow (1982) noticedthat mass-loss rates of Be stars determined from UV lines seem to be what are expectedif one extrapolates mass-loss rates of 0 and early B stars to late B types. This suggestedthat radiation-driven stellar winds might have something to do with the Be phenomenon.Actually, as early as 1934, Gerasimovic (1934) had first suggested that radiation pres-sure, both line and continuum, is critical in the formation of the circumstellar envelopeof Be stars (cf. Marlborough 1986 ). Consequently, recent work on the rotation modelhas concentrated on self-consistent radiation-driven stellar winds.It was clear from the ultraviolet observation that stellar winds are widespread amongearly-type OB stars. Castor, Abbott and Klein (1975, hereafter CAK) first introduced acomplete theory of radiation-driven stellar winds for Of stars. The basic idea behind theoriginal CAK model is that a spherically symmetric wind is raised by radiation pressureforce through the scattering of continuum photons in a very large number of spectrallines, from resonance to subordinate lines.When applied to Be stars, however, it was found that the luminosities of B-type starslater then B2 are insufficient to initialize the stellar wind by radiation pressure (AbbottChapter 1. Introduction^ 71979,1985). Therefore some other mechanism must operate in Be stars later than B2 toprovide initial acceleration of the wind. Several other factors which might be importantto OB stars are considered, such as rotation, magnetic fields, and non-radial pulsation.Marlborough and Zamir (1984) discussed the effect of rotation on a radiation-drivenwind from a hot star. They came to the conclusion that by departing from the sphericallysymmetric CAK model, a two-component stellar wind could be formed naturally underrapid rotation: one past is from the equator, with low expansion velocity, low mass lossrate, high density, and relatively low temperature (i.e. subionized), closely resembling theclassical ad hoc picture of Be stars; the other is from the polar region, with much higherexpansion velocity and high mass-loss rate, but low density, fairly high temperature andsuperionized region, to explain the UV observations. This provided the first dynamicaljustification for these earlier empirical stellar wind models.Various authors have proposed models in which magnetic fields may play an importantdynamical role in the circumstellar envelopes of Be stars (Weber and Davis 1967, Barker1979, Nerney 1980, Underhill 1983). One significant effect of a magnetic field is toincrease the Lorentz force at large radii (Friend and MacGregor 1984), thereby raisingthe terminal velocity and mass loss rates. But this effect is considerably reduced afterallowing for the finite size of the star (Poe and Friend 1986).Time variability is a fundamental aspect of the Be phenomenon, but the majority ofstellar wind models try only to represent a time-invariant star in its Be phase. Littleconsideration has been given to either the transition from B to Be phase and vice versa,or to variability within the Be phase itself. Evidence (Harmanec 1983, Percy 1986, Vogtand Penrod 1983, Penrod 1986, Baade 1986, Kambe 1993) has been found to relate non-radial pulsation with Be phenomena, especially the moving patterns detected in the lineprofiles. Willson (1986) suggested that the pulsations, caused by periodic isothermalshocks, may deposit energy in high-density regions of the atmosphere, where shockedChapter 1. Introduction^ 8gas adiabatically expands against gravity, and thereby initiates a cool dense wind. Theadvantages of NRP models are that they offer answers to classical problems of Be stars,such as why only Be stars develop disks, why other B-type stars don't pulsate withsignificant vertical amplitude, and variances in line profiles.A series of papers presenting further development of the CAK model came out re-cently. Abbott (1982) adopted a complete and accurate set of atomic data to calculatenumerical values of the radiation pressure in spectral lines in hot stellar winds. Friend andAbbott (1986) corrected the radiation- driven stellar wind for the finite size of the star androtation. Bjorkman and Cassinelli (1993) presented an axisymmetric two-dimensional su-personic solution of a rotating radiation-driven stellar wind. Their solution predicted thatthe disk is formed because the supersonic wind that originates at high latitudes of thestellar surface travels along trajectories and come down to the equatorial plane, with theram pressure confining and compressing the disk.1.3 Past Polarization Studies of Be starsTo this point there is one important aspect of Be stars which has not been mentioned:their intrinsic polarization. As Coyne (1981) commented: "The mere fact that the op-tical radiation from Be stars is observed to be intrinsically linearly polarized providesperhaps our most conclusive clue that the circumstellar shells of these stars are disk-likein nature."The first detection of intrinsic polarization in Be stars was made by Behr (1959), whenhe was searching for variation of polarization in 7 Cas during its shell phase. Followingthat, Shakhovskoi (1962) found variable polarization for x Oph and later (Shakhovskoi1964) for some other Be stars. It is its peculiar wavelength dependence pattern, p(A), andits variation with time which makes the intrinsic polarization different from interstellarChapter 1. Introduction^ 9polarization, so that it is possible to separate the two. Later observations have producedevidence for polarized radiation in the majority of Be stars surveyed (Underhill 1982).Although there are significant differences from star to star, the intrinsic polarizationof Be stars always has two basic characteristics:• The continuum polarization (around 1%) decreases rapidly with wavelength in theultraviolet spectrum across the Balmer limit (Serkowski 1968, Coyne 1971).• The polarization across spectral lines is less than the continuum polarization (Zell-ner and Serkowski 1972, Clarke and McLean 1974).Theoretical work (Nagirner 1962, Collins 1970, Rucinski 1970 ) showed polarization frompure photospheric effects can not exceed 0.1%. As proposed by Shakhovskoi (1964) andRucinski (1966), the simplest model of producing such a large linear polarization in mostBe stars is to have the stellar radiation scattered by particles contained in a volume thatis not spherically symmetric with respect to the observer. Multicolour polarimetry bySerkowski (1968, 1970) and Coyne et a/. (1967, 1969, 1974) led them to notice thathydrogen absorption played a significant role in the observed wavelength dependencepattern p(A). The current, widely accepted model, suggested by Coyne and Kruczwski(1969), argues that the intrinsic polarization of Be stars is caused by electron scatteringin the envelope, whose geometry is aspherical with respect to the observer, and wherethe wavelength dependence of the polarization is mainly due to continuous absorptionby hydrogen. The critical factors in this model are the geometry of the envelope and theopacities of hydrogen and electron scattering.During the next 10 years, more effort was put into the study of polarization acrossthe spectral lines. In the first approximation, the decrease in polarization across the linescould be explained as the dilution of additional unpolarized emission radiation from theenvelope (Clarke and McLean 1974, Poeckert 1975). However, the addition of unpolarizedChapter 1. Introduction^ 10emission flux does not appear to be an adequate explanation for all Be stars. Highresolution observations showed that there is residual polarization after the unpolarizedemission flux (Coyne and McLean 1975) is taken into account, which indicates some othermechanism must be contributing to the decrease.1.4 7 CasGas was the first emission-line star discovered (Secchi 1867), and ever since thenCas has been carefully traced for more than a century. Table 1.1 gives the basicparameters of 7 Gas. As a particularly interesting Be star, 7 Gas has gone through strongand weak Be phases, shell phases and a normal B phase (Goraya and Tur 1988). During1932-1942, -y Gas displayed spectacular variations in colour, brightness, line spectrumetc.(Telting et al. 1993), which finished with the star as a quasi-normal B star. From1942 till now, 7 Gas slowly and irregularly increased its emission lines, undergoing somesmall fluctuations (Underhill 1982).In 1976 7 Cas was identified as the optical counterpart of the low-luminosity, variablehard X-ray source MX 0053+604 (Jernigan 1976). Both in photometry and spectroscopy,Cas/MX 0053+604 is greatly similar to that of the X-ray pulsar X Per/4U 0352+30,which is a Be and neutron star binary (Moffat et al. 1973). Therefore, suggestions weremade that 7 Gas is a binary system, its companion being either an accreting neutronstar or a white dwarf (White et al. 1982, Mura.karni et d. 1986).Several detailed models have been developed for 7 Gas (Poeckert and Marlborough1978, Scargle et al. 1978). Optical interferometry measurements of 7 Gas (Mourard etal. 1989, Quirrenbach et al. 1993) well resolved the target star in the Ha line. Theresults clearly showed a spherically asymmetric envelope in rotation, which is consistentwith the disc model.Chapter 1. Introduction^ 11CasRA(1900)DECVmSpectral ClassV sin iHR 264 (HD 5394)00:50:40.1+60°10'31"2.47B0.5IVe300 km/s230 km/sReferenceLesh 1968Boyarchuk 1964Slettebak 1982Table 1.1: Basic 7 Cas parameters1.5 An Outline of this ThesisThe goal of this thesis is to interpret spectropolarimetry data of one of the brightestBe stars — -y Cas, based on an improved Poeckert and Marlborough ad hoc stellarwind model. The data were obtained at the Dominion Astrophysical Observatory, with apolarization analyzer developed by Walker and Dinshaw (Dinshaw 1991, Walker, Dinshawetc. 1993). A summary of the instrument and data reduction will be found in Chapter2, and a primitive analysis of data in Chapter 3. Chapter 4 reviews the original Poeckertand Marlborough model. Improvements on this model are described in Chapter 5 andChapter 6. In the last chapter, we discuss the successes and failures of the improvedmodel in fitting observations, and further investigations are suggested.Chapter 2Data Source and ReductionIn order to study the nature of Be stars, a microcomputer-controlled polarizationanalyzer was designed here at UBC and DAO, by Dinshaw and Walker (Dinshaw 1991,Walker, Dinshaw etc. 1993). With the DAO 1.83-m telescope, the analyzer allows dif-ferential spectropolarimetry at the same spectral resolution as the spectrograph-detectorsystem.2.1 UBC/DAO Polarization AnalyzerThe polarization analyzer was designed for linear polarization measurements. It con-sists of two components: a beamsplitter cube followed by a quarter-wave plate. Thebearasplitter is constructed of two right-angle prisms separated by a multilayer dielec-tric. The quarter-wave plate is set with its optical axis oriented at a 45° angle to thebeamsplitter axis. The plate co-rotates with the beamsplitter so as to maintain thisangle.The beamsplitter cube separates the unpolarized or partially polarized starlight intotwo beams, with orthogonal polarizations. The beam whose plane of polarization isaligned with the transmission axis of the beamsplitter goes straight through the cubeand emerges as a beam of linearly polarized light, and the other beam is reflected. Thelinearly polarized light is converted into circularly polarized light by the quarter-waveplate at a 45° angle. (Ideally only 50% of the star light will be lost in this process, it isabout 60% in practice.)12Chapter 2. Data Source and Reduction^ 13The analyzer is controlled remotely by computer under an iRMX 86 environment. Atthe start of each run, the analyzer is automatically initialized to the zero-degree position.It switches to certain position angles at the rotation rate of 12.503-i.2.2 ObservationsTable 2.2 presents the polarization observations of 7 Cas during 1990-1993, whichlists the number of spectra, typical exposure time, spectral region, average S/N level,and chosen position angles of the analyzer.To limit the effects of incomplete depolarization and systematic flexure of the spec-trograph, we found short exposures gave better results (Walker and Dinshaw 1991). Therotation of the analyzer was alternated between clockwise and counter-clockwise in orderto minimize the effect of zero-point drifts in the analyzer. The 21181 spectrograph anda RCA 620 x 1024 CCD were used together to give a resolution of 0.15 A/pixel. CCDdetectors are used because of low readout noise.Table 2.2: Observations of 7 Cas during 1990-1993Date(UT)Number ofSpectraExposureTime (s)SpectralRegion S/NPositionAnglesSep. 23,1990 225 20 Hp 500 20°, 60°, 100°, 140°, 180°Sep. 24,1990 255 20 Hp 500 20°, 60°, 100°, 140° , 180°Nov. 9,1992 35 30 Ha 90 00, 45°, 90°,135°Mar. 8,1993 100 30 Iiii 120 00, 300,600, 900, 120°, 150°Mar. 9,1993 114 30 Hi3 140 0°, 30°, 60°, 90°, 120°, 150°Mar. 10,1993 90 40 HP 155 0°, 300, 60°, 90°, 120°, 150°Chapter 2. Data Source and Reduction^ 142.3 Data ReductionThe CCD data were reduced using IRAF. The standard processing methods wereused, with the exception of the following:• Mean spectraTwo types of mean spectra are used. The total mean is the mean spectrum fora single star obtained by averaging the spectra, at all position angles, from onenight. Second is the cycle mean, which is a mean spectrum for each cycle (e.g.,position angle changing from 0° 450 —> 90° —+ 135'or135° —■ 90° —> 45° -4 00).The cycle-mean spectra are used to produce the residual spectra, so they shouldbe normalized.• Residual spectraNo flat field is used during this study, in order to avoid the additional noise fromthe flat field spectra. Instead, the residual spectra are formed by dividing eachspectrum by its own cycle-mean spectrum. In this way, not only the pixel-to-pixel variations are cancelled, but the differential polarization across the line is alsorevealed (Fahlman 1975).• Polarization analysisThe polarization across the lines are at about the 1% level in Be stars. In order todetect the structure of such a low signal, we need as high S/N level as possible. Sowe combine residual spectra at each position angle, which will finally form a set ofresidual spectra obtained at four to six different rotation positions of the analyzer1The reason we are doing this, instead of taking longer exposures to get high S/N, is that, 7 Cas isa strongly variable star and we wish to avoid a significant change during a single cycle of the positionangle of the analyzer.Chapter 2. Data Source and Reduction^ 15Generally, when a beam of partially polarized light passes through a polarizer, itsintensity changes according to the position angle of polarizer as:14,(0) = —2 4(1 -I- cos(2(614, — 64))) (2.1)where i refers to each data point at wavelength Ai, Op is the plane of polarization, 0is the axis of the polarizer. I , It are intensities of incident light and light comingout of polarizer, respectively.When applied to the residual spectra, it has the form,Li.e,(0) = APicos(2(44, — 0i)) (2.2)where AP denotes the differential polarization across the line relative to the po-larization in the continuum, since the reduction process we adopted removes polar-ization in the continuum(Pc) as well as the instrumental polarization.From Eq. 2.2, we are able to determine the wavelength dependence of differentialpolarization(AP) and its polarization angle(4) across the line. A Fortran programstokes was written by Dinshaw to give a least-squares fit of the above parameters.2.4 ResultsThe following figures give the final line profiles, differential polarization profiles andpolarization angle profiles for each night of observations. The data of Sep. 23 and 24,1990 were reduced by Dinshaw for her M.Sc. thesis. Notice that the polarization angleoutside the lines are meaningless because the errors there are too large, we didn't displaythem on the plots.Chapter 2. Data Source and Reduction^ 16Figure 2.1: Line profile, polarization and polarization angle of 7 Cas on Sep. 23,1990Chapter 2. Data Source and Reduction 17Figure 2.2: Line profile, polarization and polarization angle of 7 Cas on Sep. 24,1990Chapter 3The Underlying Star3.1 ProblemBe stars are theoretically B-type stars, with additional emission from their circum-stellar disk. However, we have very little direct information about the fundamentalparameters of underlying Be stars, including mass, radius, luminosity, effective tempera-ture, and v sin i, etc. (Slettebak 1986). The determination of all these parameters is verydifficult, due to contamination from the emission disk, and so highly uncertain.7 Cas is the brightest Be star in the northern sky. Unfortunately, no conclusiveevidence for the intrinsic properties of the underlying star has been presented. Theeffective temperature, Telt, and surface gravity, log g, which define the model atmosphere,are very critical for this thesis. The most recent studies on 7 Cas by Telting (1993) foundthat the ultraviolet continuum, which is believed to originate from the photosphere, andbe the region of the spectrum least affected by the envelope, could be equally well fittedby several Kurucz(1979) atmosphere models. They concluded that the photosphere of 7Cas cannot be represented uniquely by a single Kurucz model.3.2 Basic relation of linear polarization across an emission lineThe general relation for polarization across an emission line in Be stars is given byCoyne(1975),= PA + PE XPL 1 + X (3.3)18Chapter 3. The Underlying Star^ 19wherePi, is the observed polarization across the line.PA is the polarization at the same wavelength in the underlying absorption.PE is the polarization of the emission flux.X is the intensity ratio of the emission flux to that in the underlying absorption.In practice, PA is normally assumed to be the same as Pc, the polarization in theadjoining continuum. To make it more simple, Coyne considered that the intensity ofthe underlying absorption is small compared with the intensity of emission flux from thedisk. It is true for strong emission lines, especially Ha, that the shallow rotationallybroadened photospheric absorption feature can sometimes be neglected in comparison tothe emission line strength (McLean and Clarke 1975). X could then be taken as the ratioof the emission flux to the continuum flux. Considering the continuum flux is normallyset to be unity, then 1+ X = /total, the total intensity of the emission line. With all theseassumptions, and further assuming there is no polarization in the emission flux, Coyneconcluded that,FL = Pc fitotai^ (3.4)This relation is interpreted qualitatively such that the intrinsic polarization across theline is diluted by the unpolarized emission flux from the circumstellar disk.Poeckert(1975) claimed they found the same relationship between linear polarizationand the total intensity of the line, from their linear polarization measurement across Hain 12 Be stars, at a resolution of 2A. Further, using Eq. 3.4, they separated interstellarpolarization and intrinsic polarization for Be stars (Poeckert and Marlborough 1976). Sodid McLean and Clarke (1975).Chapter 3. The Underlying Star^ 203.3 Primitive analysis of the polarization dataWe have a better resolution (0.15 A) of linear polarization across the lines than thepreviously cited authors. While applying Eq. 3.4 to our Sep. 23, 1990 and Sep. 24, 1990polarization data 1, as shown in Fig. 3.3, Fig. 3.4, we found additional polarization whichdeviates from the above dependence. This inconsistency indicates either that PE 0 orthat Eq. 3.3 is not applicable.It is time to reconsider Eq. 3.4. Notice that the polarization profiles do not have thesame width as their line intensity profile. They tend to be wider than the line profile. Weassume the basic, Eq. 3.3 is still applicable. Another possibility is that the intensity ofemission flux from the disk is not as big as it was thought, comparing to the underlyingabsorption feature, which means it could not be negligible. Still adopting PA = Pc,PE = 0, we rederive Eq. 3.3 as following,PL= 1+ Xwhere X = 'E/IA. Therefore,PC FL-11 (3.6)+ -1:IA. PL = I A + IE Pc^ (3.7)we get,I A nii, — --7.—.1-c^ (3.8)/totalEq. 3.8 shows, not only the total intensity of the line, but also the underlying absorp-tion feature is a fairly important factor, which affects the linear polarization across theline, especially when it is concerned with faint emission lines.Pc (3.5)lAmong all the data on -y Cas we have, the data of Sep. 23 and Sep. 24 have the highest S/N. Wewill only use these two nights data throughout this thesis._-^-_i^I^I 4860Wavelength (Â)1 f4840 4880Chapter 3. The Underlying Star^ 21Figure 3.3: Polarization across HP emission line on Sep. 23,1990 (solid line). The dashedline shows Pc/how, where Pc is taken to be constant in the considered region (Poeckert1979). We set Pc = 1.1% to roughly match the line center polarization.Chapter 3. The Underlying Star^ 22Figure 3.4: Polarization across Hfl emission line on Sep. 24,1990 (solid line). The dashedline shows Pc/how, where Pc = 1.1%.Chapter 3. The Underlying Star^ 233.4 Hubeny synthetic spectraA theoretical prediction of the intrinsic line profile is needed as input to Eq. 3.8.The Hubeny(1988) stellar atmosphere model (program TLUSTY) and synthetic spectraprogram SYNSPEC have been used to generate photosphere spectra of the underlyingstar.TLUSTY calculates non-LTE model stellar atmospheres, assuming "classical" plane-parallel, horizontally homogeneous atmospheres in radiative and hydrostatic equilibrium,while allowing for departures from LTE for a set of occupation numbers of selectedatomic and ionic energy levels. The NLTE model is obtained in a sequence of steps:the program starts with a full LTE model, then an NLTE model in which departuresfrom LTE are taken into account only in the continuum, and finally a model in whichNLTE affects are considered both in the continuum and line transitions. It should benoticed that the only difference between an LTE and NLTE model in this program isthe form of the statistical equilibrium equations, where the NLTE model counts bothradiative and collisional transitions, but the LTE model is simply the populations fromthe Saha-Boltzmann equation. TLUSTY includes as many atomic data in the programas computing time allows. Therefore it gives a great flexibility in choosing chemicalspecies, ionization degree, energy levels, etc. Generally, all the important atomic data forhydrogen and helium are already included in the program; the user doesn't need to worryabout them unless more accurate or extensive data are available. In our case, we onlyneed to provide the effective temperature Tog, surface gravity log g, and microturbulentvelocity.SYNSPEC uses the previously computed model atmosphere from TLUSTY, andcalculates a synthetic spectrum in a chosen wavelength region. It uses the same inputfiles as TLUSTY, plus an additional file specifying the wavelength region and another0.950.9cts0.850.8■■1111111111_ -Chapter 3. The Underlying Star^ 24one of line list (typically Kurucz-Peytremann line list). It gives an output file (for007),which contains a condensed synthetic spectrum, and tables of wavelengths together withfluxes at the stellar surface.Finally, the output file from SYNSPEC (for007) serves as input to program ROTIN,which produces a rotationally and/or instrumentally broadened spectrum. An exampleis given in Fig. 3.5.4840^4860^4880Wavelength (Â)Figure 3.5: Hubeny NLTE synthetic spectrum of HP with Tea- = 25000K, log y = 3.5,v sin i = 400 km/s, and microturbulent velocity= 10 km/sChapter 3. The Underlying Star^ 253.5 A possible way of determining fundamental parametersWhen we applied the above Hubeny synthetic spectrum and the observed emissionline profile to the polarization profile, using Eq. 3.8, as shown in Fig. 3.6, we got a fairlygood fit. This proves that the underlying absorption feature is not completely smearedout by emission. It has fairly strong effects on the linear polarization across the line.Further, related to the problem mentioned in the beginning of this chapter, we couldpossibly use Eq. 3.8 to determine the intrinsic parameters of Be stars, under the as-sumption that Eq. 3.8 is correct. According to Eq. 3.8 and Hubeny synthetic spectrum,only four parameters (Ta-, log g, v sin i, and Pc) are needed to completely describe thelinear polarization across the emission lines. A series of models (parameters are listed inTable 3.3) are applied to the two nights' polarization data by Eq. 3.8. I wrote a Fortranprogram to derive the best-fitting parameter set, using the least-square method.Table 3.3: Parameters of models used to find the best fitting modelHubeny modelTeff log g v sin i Pc20000K 4.025000K 3.0 50 km/s 0.5 %25000K 3.5 4 -V25000K 4.0 750 km/s 5.5 %30000K 3.0 every every30000K 3.5 10 km/s 0.1 %30000K 4.03.6 DiscussionThe best-fitting model among the above models we used is given in Table 3.4. Theyare shown in Fig. 3.7, Fig. 3.8.Chapter 3. The Underlying Star^ 26Figure 3.6: The dashed line represents the predicted linear polarization from Eq.3.8, theunderlying absorption profile is taken from the Hubeny NLTE synthetic spectrum of HPwith Ten- = 25000K, log g = 3.5, v sin i = 400 km/s, and Pc set to be 1.7%. The solidline is the observed polarization across Hfl of -y Cas on Sep. 23,1990.Chapter 3. The Underlying Star^ 27Table 3.4: Best-fitting model of polarization data for Sep. 23 and Sep. 24Sep. 23, 1990 Sep. 24, 1990Telt- 30000 K 30000 Klog g 4.0 3.5sin i 340.0 km/s 310.0Pc 1.6% 1.5%X2 131.78 72.051.0 1.0The next question is how reliable this method is for determining the underlying stellarparameters of 7 Cas. The two nights' data gives fairly consistent values of the intrinsicparameters of the underlying star, although not exactly the same. It still indicates thissimple model is realistic. However, there are about 600 data points across the polarizationprofiles of each night. From the last two lines of Table 3.4, we could see that the errorsin the polarization data were somehow overestimated. On the other hand, although thisbest-fitting model is defined as the one with the least-chi-square value, the best-fittingmodel may not significantly better than models with somewhat higher chi-square value.We arbitrarily assume those models with x2 value within 5 % of its least-square value forboth nights would be the most likely models of 7 Cas, which are listed in Table 3.5 (Pcis not listed in Table 3.5, as it is not an intrinsic parameters of the star. Pc is around1.2 % to 1.7 %).It is still not possible at this stage to definitely determine the intrinsic parameters ofCas. To properly estimate the likelihood of the above models, and further pin downthe intrinsic parameters of underlying star, we need data with higher signal-to-noise, anda good estimate of the error in the data.Chapter 3. The Underlying Star^ 28Table 3.5: Most likely models representing the central star of 7 Casv sin i (km/s) Teff(103K) & log g(km/s) 20 4.0 25 3.0 25 3.5 25 4.0 30 3.0 30 3.5 30 4.050-200210-230 *240-270 * * *280-300 * * * *310-330 * * * * *340-360 * * * * *370-400 * *410-750Chapter 3. The Underlying Star^ 29Here we suggest the underlying star is represented by the model listed in Table 3.6,according to the best-fitting models to both nights' polarization data. It will be used inthe later chapters.Table 3.6: Suggested best model for central star of 7 CasTeff 30000 Klog g 4.0v sin i 325.0 km/sPc 1.6%3.7 Pure emissionAssuming the underlying absorption line has been determined, we have a direct ap-proach to reveal the pure emission flux from the circumstellar disk around the star.Stellar absorption and disk emission are separated in Fig. 3.9 and Fig. 3.10, by usingthe suggested best underlying stellar model. In this way it would be easy to model theemission flux and tell whether the line profile variations originate from the star itself orfrom its circumstellar disk.Chapter 3. The Underlying Star^ 30Figure 3.7: Best fitting model of the linear polarization across lig on Sep. 23, 1990, withTeff = 30000 K, log g = 4.0, v sin i = 340.0 km/s, Pc = 1.6 %Chapter 3. The Underlying Star^ 314840^ 4860^4880Figure 3.8: Best-fitting model of the linear polarization across HP on Sep. 24, 1990, withTeff = 30000 K, log g = 3.5, v sini = 310.0 km/s, Pc = 1.5 %Chapter 3. The Underlying Star^ 32Figure 3.9: The dotted line gives the best-fitting model for the underlying stellar absorp-tion feature of Sep. 23. The solid line shows the total intensity of observation, with thedifference between the two giving the pure emission feature from the disk, shown as thedashed line.0.5 48401 I^,i 4860Wavelength (A)48801Chapter 3. The Underlying Star^ 33Figure 3.10: The dotted line gives the best-fitting model for the underlying stellar ab-sorption feature of Sep. 24. The solid line shows the total intensity of observation, withthe difference between the two giving the pure emission feature from the disk, shown asthe dashed line.Chapter 4The Poeckert and Marlborough ModelPoeckert and Marlborough's model (hereafter PM model) was first developed in 1969by Marlborough, and later improved by Poeckert. Like all the Be star models, becauseof insufficient knowledge of relevant physical effects in the circumstellar disc aroundBe stars, their model is also ad hoc. Its main advantage lies in the large number ofobservable features it manages to compute and reproduce nicely, which gives a powerfultool to compare with the observations and improve the model.4.1 Construction of the ModelThe stellar wind model, based upon the original ideas of Struve (1931 cf. Chapter 1),postulates an asymmetric circumste,llar envelope strongly concentrated to the equatorialplane of the star, which is formed through rotationally forced ejection resulting from thebreakup rotation of the central star. Generally one must guess the velocity, fT(i:), andtemperature, T(7), distributions in the disk as functions of position (and time if appropri-ate) and then obtain the density distribution, p(r), from the equation of continuity. Ap-proximate solutions of the radiative transfer equations then give the ionization-excitationconditions as a function of position.The PM model and its applications are described in detail by Limber, Poeckert, Marl-borough (Limber and Marlborough 1968, Marlborough 1969, Poeckert and Marlborough1977, Poeckert and Marlborough 1978a, Poeckert and Marlborough 1978b, Poeckert andMarlborough 1979) and a good review is given by Underhill (1982). Here we outline the34^Chapter 4. The Poeckert and Marlborough Model^35model.4.1.1 Central StarThe central star is assumed to rotate as a solid body at its critical velocity at theequator. The physical parameters describing the star are assumed to be the same asa normal spherically symmetric star, of the same spectral type as the Be star, whosecontinuum energy distribution is taken from Kurucz et al (1974).4.1.2 EnvelopeThe envelope is assumed to be in a steady state, and symmetric about both therotation axis and the equatorial plane. The envelope consists only of hydrogen, and istreated as a perfect gas. The envelope is assumed to be isothermal, with a temperatureTe throughout, which is lower than the effective temperature, Teff, , of the central star.Density distributionConsider a cylindrical coordinate system (r,0,z) with the origin at the center of thestar, and the i-axis along the stellar rotation axis. In a meridian plane, it is assumedthat steady flow takes place along streamlines, which are taken to be straight lines con-verging at one point. The density distribution is set by hydrostatic equilibrium in thez-direction. Using these assumptions plus conservation of mass and steady state, Limberand Marlborough (1968) derived the total number density distribution of atoms and ions,N(r,z), in the envelope as:exp^[;-!7-^r/:+zi2N(r, Z)^N(1, °) V,.(r) r(r—p)V,(1) 1—p(4.9)whereChapter 4. The Poeckert and Marlborough Model^ 36N(1,0) number density at the equatorial surface of the star.Q density scale height, Q = kroTe/GmottM.V,.(r) the expansion velocity distribution, 'V,.(1) is the expansion velocity at thesurface of the star.r and z where the envelope is assumed to be in hydrostatic equilibrium in thez direction.p distance of the outflow converging point from the rotation axis, p < 1, in unitsof stellar radius.It is clear from Eq. 4.9 that the density distribution in the disc is completely deter-mined by, 1. N(1,0) ; 2. p; 3. Vr(r). All these factors are chosen in an ad hoc mannerin the PM model.The disc is chosen to be wedge shaped and bounded by 1/7* < r < 5014 and straightlines diverging by a certain open angle. Outside this region the density is assumed to benegligible, and the source function and opacity are set equal to zero. A schematic pictureof the disc can be found in Fig. 4.11.Velocity distributionThe velocity distribution in the envelope results from a combination of rotation andexpansion. In the original model, it was assumed that the envelope is supported bycentrifugal force plus some viscous force. Assuming partial conservation of angular mo-mentum, the rotational velocity of the envelope is given by:got = V[/3r 2 + (1 fi)r-111/2^(4.10)where Vrot is in the same direction as the stellar rotation K. 0 is an ad hoc function ofdistance from the star which, depending on its assigned value, allows a transition betweenChapter 4. The Poeckert and Marlborough Model^ 37Figure 4.11: A schematic picture of the envelope adopted in the PM modelChapter 4. The Poeckert and Marlborough Model^ 38two extreme cases: the envelope rotating at critical velocity (fl = 0) or conservation ofangular momentum (fl = 1).The expansion velocity adopts the canonical form (Castor 1970, Castor, Abbott andKlein 1975) of:vex, = vo + 07,0 — vo(1 — rvo)^(4.11)to be consistent with the radiation-driven wind. 170 is the expansion velocity at the surfaceof the star, K. the terminal velocity derived from the displacement of UV resonance lines.Excitation-ionization structureThe envelope is considered to be composed of hydrogen atoms only. Each hydrogenatom is assumed to consist of 6 bound levels: the ground level, 2S and 2P treated sepa-rately and the levels of principal quantum number 1 n=3, 4 and 5; plus the continuum.Populations of each of these levels are determined at specific points within the envelopeby assuming a steady state, i.e. at each point in the envelope, the radiation field and thepopulation of all levels are independent of time. Therefore, the statistical equilibriumequations for the population of bound level n could be written as:Na + En' >. Nn' [An'n Bn'nu(vn' 4:^Nni, n u (Pe nEn,,, on Nni" Ne Q (u'n , n).^Nn {Bn,c, En,/ [An„u Bnnu U(van” )} + En,>n Bnni u(unn, )&mon N.Q(n,nm))^(4.12)The following processes were included in solving the rate equations for the population ofbound level n:'Angular momentum substates / for a given n with n > 3 were not treated individually because colli-sional transitions between substates of one principle level are much faster than the radiative transitionsbetween substates of different principal levels in the case of Be stars.Chapter 4. The Poeckert and Marlborough Model^ 39• radiative recombination• photoionization• collisional transitions between level 2S^2P, n 4-t n 1• spontaneous emission^ Einstein A coefficients• photoionizations and radiative stimulated deexcitation to lower levels Einstein BcoefficientsHowever, the processes which are not included in the rate equations are:• collisional ionizations• collisional recombinations• collisional transitions between bound level n and other bound levels except n 1• two-photon emission from level 2SIn the rate equations, the radiative recombination coefficients, an, are from Burgess(1964). Einstein A and B coefficients are taken from Capriotti (1964). Photoionizationcross-sections at the series limits are also taken from Burgess (1964). The collisionalexcitation and deexcitation rate coefficients for transitions between bound levels, except2S 4-+ 2P, are derived from the cross-sections given by Saraph (1967), while 2S 4-+ 2P istaken from Seaton (1955).Radiation fieldIn general, the radiation density in the continuum is the combined result of theradiation emitted by the star and radiation produced in the envelope, the so-called diffusefield. The computer code of the PM model is written in such a way that the contributionChapter 4. The Poeckert and Marlborough Model^ 40to ionization and excitation by the envelope is completely neglected; in other words, ateach point in the envelope, only radiation contained in the right circular cone subtendedat (r, z) by the star is considered. So the radiation field could be written as= —1 dco (4.13)where is the specific intensity of the central star, and is the optical depth alonga given direction from (r, z) to a point on the surface of the star. Only bound-freeabsorption processes are included in rm, which assumes photo-ionization is the dominantsource of continuous opacity.It is known that the actual state of material in the envelope is significantly differentfrom the prediction of the LTE model. However, the non-LTE treatment requires a com-pletely self-consistent simultaneous solution of both the radiative transfer and statisticalequilibrium equations, which is difficult to achieve. It is reasonable to believe that thetrue radiation field and population distribution could be reached by deviation from theLTE case. Poeckert and Marlborough, claiming that a simpler approach can be usedwithout too great a sacrifice in accuracy, adopt the following procedure to solve for theradiation field and line profiles:1. The initial radiation field in the envelope only includes the radiation emitted fromthe central star diluted by the distance factor.2. Use this radiation field to compute radiative rates in the statistical equilibriumequations, to get the solution of level populations.3. Use these to compute the radiation field produced in the disc through radiationtransfer equation.The steady-state equations are solved for five different cases for the treatment of lineradiation produced in the envelope. They are:Chapter 4. The Poeckert and Marlborough Model^ 411. The envelope is optically thin to all line radiation produced in it, i.e. ue(v„,n) = 0for all bound levels n and m.2. The envelope is optically thick in the Lyman lines and thin in all others, such that,ue(v„„,) = 0 for all lines except for Lyman series, and the Lyman lines are treatedby detailed balance of radiative transitions between levels n and m.3. The envelope is optically thick in the Lyman and Bahner lines only, and thin inthe rest, where optically thick and thin have the same meaning as above.4. The envelope is optically thick in Lyman, Balmer, and Paschen lines and thin inthe rest.5. The envelope is optically thick in all lines, i.e. ue(vn,74) is given by detailed balancingfor all the levels n and m. This is the LTE case.The final populations are obtained by taking a weighted average of the populations ofany two consecutive cases, depending on the optical thickness of the disk.Balmer line profilesThe procedure adopted by Poeckert and Marlborough for determining the line profilesis to consider the profile from the system as a whole to be the sum of profiles from allelements of area of the projected envelope as seen by the observer (Marlborough 1969).The projected area is divided into a number of sectors (N). Each sector corresponds toa column through the envelope with its axis parallel to the observer's line of sight. Theradiation transfer is integrated numerically along 2000 columns in the PM model, fromthe edge of the disk nearest to the observer, either to the surface of the star, or to thefurthest extremity of the disk, depending on whether the column strikes the central star'sChapter 4. The Poeckert and Marlborough Model^ 42surface.F„. -AN Svexp(—r„)dr, (4.14)When a column intersects the stellar surface, its source function, S„, is set equal to thespecific intensity at the stellar surface. Otherwise, in general, the source function for agiven frequency is,00(4.15)where j„L is the line volume emission coefficient, ji,c is the continuum volume emissioncoefficient, I, is the amount of radiation scattered into the line of sight, and k,,, is thetotal volume extinction coefficient,= kvc Neg. (4.16)where kiz is the line absorption coefficient, lex the continuum absorption coefficient, bleis the electron number density, and cr is the electron scattering cross-section.It should be emphasized here that in solving the radiation transfer, the effects ofcontinuum opacity cannot be ignored. Free-free, bound-free and recombination processesare included in the continuum emission and absorption coefficients. Unlike the procedurewhen the populations are determined, bound-free is considered as the only opacity source.Polarization across the Balmer linesTo predict the polarization across the lines, Poeckert and Marlborough assumed thatpolarization arises from single scattering of photospheric radiation within the envelope,which means that, multiple scattering of the photospheric radiation and any radiationcontributed from the envelope are not included. As pointed out by Poeckert and Marl-borough (1978a), while it is a drawback to treat the stellar radiation as the sole sourceof scattered flux, there should not be much polarized scattered flux from the radiationproduced in the envelope, because it is more isotropic than the stellar radiation.Chapter 4. The Poeckert and Marlborough Model^ 43It is necessary to determine the second and third Stokes parameters, q and u withinthe disk first and then integrate along the line of sights as for the radiation field.Q, = E AN 1(q/lc„)exp(—rOdr,^ (4.17)N^°t I v = E AN f(uPs.,)exp(-7-„)dri,^ (4.18)N^°Q„ and U„, the overall second and third Stokes parameters. And the polarization andposition angle are given by,P„ =1= —2 tg-1(Uv/Q„)(4.19)(4.20)Notice the F„ includes the total flux emitted per steradian, but Qv and (Iv only includethe photospheric radiation.To determine q and u, Poeckert and Marlborough considered the stellar surface as adisk, which is divided into 13 sectors, instead of a point source. The geometry of thepolarization calculation could be found in Poeckert and Marlborough (1977). Scatteringangle (0), scattered radiation (I,) and q, u are determined for each sector at each gridpoint (cf. Chapter 5).4.2 Model for 7 CasIn principle, the PM model could apply to every Be star, but this model was built upspecifically for the interpretation of the observations of 7 Cas.The model calculates the populations in the envelope at 24 x 4 grid points, and theline profile along about 1500 line of sights. The basic parameter set that defines themodel of 7 Cas are given in Table 4.7 (Poeckert and Marlborough 1978a). The choiceof parameters in the model essentially rests on the agreement obtained between modelpredicted and observed continuum polarization and Ha line profile.20000 Khydrogen0.683.33 x 1011cm'0.8&15.0 R,,0.050.0 14 (line)250.0 R..k (continuum)45°Vo = 7.47kms"— 0.48 times the sound speed at the surfaceVcc, = 253.0kms'13 = 10.0T.compositionmean molecular weightN(1,0)Pr,z,maximum extent of the envelopeinclinationwind velocityChapter 4. The Poeckert and Marlborough Model^ 44The model predicts Ha, HA 117, 1115 and 1325 line profiles, and polarization andpolarization angle across Ha and 11/3. It also gives the continuum energy distributionand continuum polarization. Comparison to Poeckert and Marlborough's data can befound in Poeckert and Marlborough (1978a).Table 4.7: Original PM model parameters of 7 CasStellar Parametersmass^ 17 Moradius 10 RoTdr 25000 Klog g^ 3.5Vequ 569 kms' — 1.0 time critical speedEnvelope ParametersChapter 5HP emission line profileThe motivation of this thesis comes from the comparison of the original PM modelwith our observational data. I will only discuss the HP emission line in this chapterand leave the polarization for the next chapter. Fig. 5.12 shows the original PM modeloverplotted on the line profiles of Sep. 23 and Sep. 24. As can be seen , the modeldoesn't match the observations at all well. There could be two reasons for this: eitherthe model is good, but because of the variations of 7 Cas, the parameters Poeckert andMarlborough chose don't fit any more; or the model simply doesn't represent 7 Cas wellenough.The first thing I did was to test the PM model with a series of different parametersets. However, no better fit to the Sep. 23 and Sep. 24, 1990 data was achieved despitean extensive search of the parameter space. The fits showed several common problems:• There is always deeper absorption in the wings of the line profile, while in theobserved line profile the absorption is almost masked by the emission.• PM model never predicts V/R ratio over 1.• PM model tends to produce broader line width than is observed.• PM model always produces line profiles and polarization profiles of similar width,although the observations show that the polarization across HP is much broaderthan the HP line profile.4546Chapter 5. I-113 emission line profile-4--_I^, 4860Wavelength (A)1_-_--_-_4840148801.414840 4860 4880Figure 5.12: (a) PM model with their original parameter set compared to our data on Hflof 7 Cas, Sep. 23, 1990. Dashed line is the model prediction. Solid line is the observation.(b) The same as (a), except overplotted on the data of Sep. 24, 1990Chapter 5. Hig emission line profile^ 47In an effort to better understand Be stars, or at least 7 Cas, we decided to modifythe PM code in the following respects:1. Add the 6th principal level of hydrogen atoms, adding an additional case which as-sumes that the envelope is optically thick in Lyman, Balmer, Paschen and Brackettlines and thin in the rest.2. Replace Kurucz (1974) atmosphere model by Hubeny (1988) model.3. Adopt the best-fitting intrinsic parameters developed from our polarization data(see Chapter 3);4. Consider the effects of electron scattering on Balmer lines from the disk.5.1 Principal level 6th of hydrogen atomNotice in Fig. 5.12, the predicted HP line from the PM model is too wide comparedwith the observation. The most obvious reason for this is that 7 Cas is not rotating at aspeed as high as 569 kms-1, or the rotational velocity in the envelope is much slower thanwas assumed. However, Marlborough (1993) claimed that simply reducing the rotationalvelocity either of the star or of the envelope is not correct. It is because the populationof principal level 4 and 5 are overestimated. He explained this is due to the averagingprocess they were adopting to get the populations of each level, which gives a finalpopulation of a weighted mean of two of these five optical thickness cases (cf. Chapter4). For example, if one grid point in the envelope is optically thick in Ha, but the opticalthickness is smaller in HP, the averaging scheme will tend to weight case 3 (d. Chapter4) heavily, which artificially increases the population of levels 4 and 5. Therefore, in thisway, HP will always be too strong because from every volume element in the circumstellarenvelope there will be too much Iff3 produced. Particularly those volume elements closeChapter 5. MI emission line profile^ 48to the central star, which have higher rotational speed, will contribute more emissionthan it should to HP, thus yielding a broader HP line (Marlborough 1993).To make a better estimate of level populations, we changed the PM code to includethe 6th principal level and an additional optical thickness case, which considers theenvelope to be optically thick in the Lyman, Balmer, Paschen and Brackett lines only.All the necessary atomic data are taken from the same source as in the PM model. Wehope that in this way, the overestimated 4th level population would pass smoothly to asituation where only the 5th and 6th level populations are overestimated, which wouldgive a reasonably good estimate of level 4, so that HP has a better fit. Results with thesame parameter set are given in Fig. 5.13, after this change.The Hig line profile does not show any obvious improvement by adding the 6th levelpopulation. It has exactly the same line width. Instead, the line strength increases. Asexplained later by Marlborough (1993), since the averaging process leads to an overes-timated 6th level population as it does to the 4th and 5th, transitions between the 4thand 6th will add additional population to level 4, thus increasing the strengths of HP.We feel there isn't any easy solution to this problem, unless a better averaging processis found.5.2 Hubeny atmosphere modelsThe underlying stellar spectrum is fairly critical in the PM model. We suspect thatthe reason the original PM model doesn't match our line profiles well, is partly due tothe uncertainty in describing the properties of the central star.Two subroutines in the PM program deal with the underlying stellar spectrum: fluxcontains the table of continuum spectra, and stelin provides the five Balmer line profiles.All these spectra are taken directly from the Kurucz model (Kurucz 1974).,4850^4860Wavelength (A),^1^,4870,^1 48804840Chapter 5. 1113 emission line profile^ 491.81.61.4:14CI)0.81.2Figure 5.13: Predicted lifl emission line after adding the 6th principal level of hydrogenatoms (dashed line), compared to the original model (solid line).Chapter 5. lifi emission line profile^ 50To remain consistent with the discussion in Chapter 3, and to have better resolutionof the spectrum, we replaced the Kurucz model by a Hubeny synthetic spectrum, bothin the continuum and lines.5.2.1 Continuum spectraThe continuum energy distribution is decided in the same way as the original PMmodel. About 57 wavelength points are chosen to determine the continuum level. Sincein our case the continuum level is only used for deciding the photoionization coefficients,so all the 57 wavelengths are chosen, either at the frequencies of each bound level to theseven different ionization levels, or at the rest line wavelength between any two of thebound levels we considered to get the absolute line flux.While generating the Hubeny spectrum, only Balmer lines and several helium lines areincluded. It is fairly simple to determine the continuum level of the synthetic spectrum.At the line center wavelength, the continuum level is estimated by linear interpolationbetween the adjoining continuum.Both the continuum spectrum predicted by Kurucz atmosphere model and Hubenyatmosphere model could be found in Fig. 5.14.5.2.2 Line profilesBasically, in the Kurucz model, 18 values of residual flux for the wavelength displace-ments from line center are given for each line profile. Any wavelength displacement inbetween any of the 18 points is interpolated linearly. The line profile is assumed to besymmetric, so only half of the profile is computed, the other half is obtained by reflection.For the Hubeny model, we could get as fine resolution of the spectrum as possible, byusing different AA values in the program rotin (although it might not be meaningful insome case to get too fine a resolution). So we choose the same dispersion as the data weChapter 5. HP emission line profile10513 3.5^4Wavelength (log A)44.5Figure 5.14: Continuum Spectrum at Teff = 25000 K, log g =3.5 predicted by Kuruczmodel (dashed line) and Hubeny model (solid line).Chapter 5. HP emission line profile^ 52got, which is about 0.15À/pixel. The line spectra are considered 37.5 A away from therest line center on each side, with even space. No reflection and interpolation are neededsimply because the spectrum covers the whole line profile and its wings.The predictions of rotationally broadened H(3 absorption line from Kurucz atmospheremodel and Hubeny atmosphere model are shown in Fig. 5.15.5.2.3 Kurucz model replaced by Hubeny modelOne model of Teff = 25000 K, log g =3.5, v sin i = 400 km/s produced by the PMmodel is shown in Fig. 5.16, after replacing the Kurucz model by the Hubeny model. Asignificant change can been seen. Since the Hubeny model predicts less UV flux than theKurucz model does, the line profile produced by using the Hubeny profile has a lower redemission peak than the one predicted by the Kurucz model, under the same conditions.Meanwhile, we adopted the best-fitting parameters derived in Chapter 3 to representthe center star of 7 Cas. Comparing to the Sep. 23 line profile, Fig. 5.17 shows thatas adopting the best-fitting parameter set, it gives the right width of the emission line,which in another way convinces us that using the polarization profile to determine theunderlying stellar feature is applicable. However, the wings show even deeper absorptionfeatures than before, this is due to the larger value adopted for log g (4.0) than in theoriginal PM model (log g = 3.5).5.3 Pure emission from PM modelAn equivalent or, maybe a better way of reducing the effect from the uncertainty ofthe underlying star is to modify the PM model to produce pure emission flux and thencompare the pure emission features derived in Chapter 3. This will not be included inthis thesis.-^ --Chapter 5. Hi3 emission line profile^ 530.9Zr-4 0.7-_-_0.64840,^1^, 4860 4880Wavelength (A)1--_-Figure 5.15: Intrinsic Hf3 line profile at Teff = 25000 K, log g =3.5 calculated by Kuruczmodel (dashed line) and Hubeny model (solid line)- -I^I_^11111I^I^I^I^I^_I^_I1.61.4__-_.-_-0.8Chapter 5. 1-1fl emission line profile^ 544840 4850 4860^4870^4880Wavelength (A)Figure 5.16: The dashed line gives a model prediction, in which the underlying spectrumis represented by the Hubeny atmosphere model, instead of the Kurucz atmosphere modelas in the original PM model (solid line).Chapter 5. Xi emission line profile^ 551.6 —I jA11.410.8 ^I 48801^1^1^1^1^1^1^3^1^1^1^1^1^1^1^1^14840 4850 4860 4870Wavelength (A)Figure 5.17: After replacing the Kurucz model by the Hubeny model, we apply the• best-fitting parameter set to represent the central star (dashed line), and compare toSep. 23 IP emission line (solid line).Chapter 5. HO emission line profile^ 565.4 Electron scattering from the diskPrevious studies (Marlborough 1969, Bemat and Lambert 1978) have shown that verybroad weak Ha emission wings are due to electron scattering in the envelope. Poeck-ert and Marlborough (1979) studied the effects of electron scattering of the photonsoriginating in the envelope (which is not included in the version of the code we have),and concluded the broad wings are a result of the photons in the line core scattered byelectrons into the wings.To investigate this, we artificially add a gaussian profile (Eq. 5.21, whose standarddeviation (Eq. 5.22) is decided by the thermal electron velocity in the envelope, to repre-sent the electron scattering of the emission line photons. The flux of the gaussian profileis about 0.5% of the total emission flux, decided by best-fitting of the line profile.A^—(V — V0)2Ii. = ^Lexp(^ )2a2 (5.21)Viirer where .Te is the total emission flux of 48 from the envelope, and0.2 2kTdisk^+ VI^ (5.22)MeIt can be seen clearly from Fig. 5.18 that although the gaussian profile doesn't com-pletely constrain the absorption features in the wings, it certainly makes the absorptionless. This gives us confidence that proper calculation of the electron scattering of linephotons from the radiation transfer equation would be able to explain the unmatchedabsorption wings. This has been done by Poeckert and Marlborough (1979), but will notbe included in this thesis.5.5 Parameter studyAfter making all the above mentioned modifications to the PM model, we try to findthe best fit to our emission line profiles for 7 Gas on Sep. 23, 1990.Chapter 5. HO emission line profile^ 57Figure 5.18: Adding an gaussian profile to represent the electron scattering of the emissionline photon from the disk (solid line), comparing to the Sep. 23 HO line profile.Chapter 5. Hfl emission line profile^ 58First, we will discuss the influence of several major parameters on HO line profile.Similar work has been done by Poeckert and Marlborough (1978). Each parameterinvolved is varied with respect to the original PM model while all the other parametersare kept at their previous standard value. The central star of 7 Cas is described by thebest-fitting values from Chapter 3 (71a. = 30000 K, log g =4.0, v sin i = 325 km/s), andkept as the same values throughout all the parameter tests. (Figures can be found inAppendix A, tables on each page show the condition of each model.)The parameters we considered include:Density, N(1,0) The electron number density at the base of the circumstellar disk.Inclination angle, i The angle between the rotation axis of the star and the line ofsight to the observer. Different inclination angles decide if the envelope of star isequation-on (90°) or pole-on (0°), or in between.Rotation velocity, vrot -disk The rotational velocity in the envelope.Expansion velocity The expansion velocity law is described by initial velocity (V0),terminal velocity (V00) and acceleration rate (/3), as in Eq. 4.11.Opening angle of the disk The wedge-shape of the envelope (sip, the slope of thewedge; cpt, the intersection to the rotational axis) determines the thickness andopen angle of the disk.Envelope temperature, Tdisk An isothermal envelope with an electron temperatureof 20000K was assumed earlier. Models are also run in which this temperature isset at 15000K and 25000K.Chapter 5. HP emission line profile^ 59The final choice of the parameters adopted to define the model for 7 Cas of Sep. 23 isbased on numerous calculations of models of various values of each parameter, as givenin Table 5.8. It can be seen from Fig. 5.19, this model gives a reasonable fit. However,there is no case appeared to have shown the V/R. ratio over 1. It should be mentionedhere, Table 5.8 does not give a unique solution to 7 Cas, several other parameters setgive similar good fit to the data.Table 5.8: Parameters define the model for 7 Cas of Sep. 23.Ten.^30000 Klog y 4.0Vequ^325 kms" — 1.0 time critical speedN(1,0) 2.39 x 1011cm-1inclination^500Tdisk^21500 Kvrot-disk^284.5 kms-1sip^0.35cpt -0.25wind velocity 170 = 17.07kms-1= 1530.0kms'= 10.060Chapter 5. .11)3 emission line profile_ __1-_-0.8 -.^,^I^,4840,^,^I 4880.^LI,i,i1,,,,I,4850^4860^4870Wavelength (Ä)1.4-_a-1.6 -Figure 5.19: Line profile predicted by the final model (dashed line), overplotted on Sep.23 data (solid line)Chapter 6Polarization across^emission lineThe PM model also predicts the polarization across the JEW emission line. Correspond-ingly, Fig. 6.20 shows the original PM model overplotted on the polarization profiles ofSep. 23 and Sep. 24.There exists several serious problems in interpreting our data:• As mentioned in the last chapter, the PM model always produces line profilesand polarization profiles of similar width, although the observations show that thepolarization across lifl is much broader than the Etig line profile.• The predicted polarization profile across HP has a much stronger central peak thanis observed.• The model predicted polarization increases in the emission-line wings, before itdrops sharply in the line center, which is not detected in the observation.Two major changes have been made to the polarization part of PM model:1. Adopt correct geometry of polarization.2. Correct scattered radiation. Consider the effects of underlying stellar absorptionfeature on the polarization across the emission lines (cf. Chapter 3).61Chapter 6. Polarization across HP emission line^ 621(a)0 _ ,LAvvr.A P%—0.54840 4860 - 48800A P%—0.54840 4860Wavelength (A)4880Figure 6.20: (a) Polarization profile across up emission line, predicted from originalPM model (dashed line) compared to Sep. 23 data (solid line). (b) The same as (a),overlapped on Sep. 24 data—14860Wavelength (.1k)4850 48704840 4880Chapter 6. Polarization across II/3 emission line063A P%—0.5Figure 6.21: The solid line gives the predicted polarization from the PM model assuminga finite size to the central star, while the dashed line gives the predicted result when thestar is treated as a point source. We see that the predicted polarization structure is thesame in both cases.Chapter 6. Polarization across 11,8 emission line^ 646.1 Correct geometry for polarizationThe subroutines abint, litpat and polcon are created to calculate the polarizationin the code. We tested this part of the code by replacing the star with a point source, itgave the same result as before (Fig. 6.21). In addition, the general " W " shape of thepolarization across the line has never been seen to change at all. Therefore, we suspectthis is due to incorrect calculation of the geometry concerning the polarization.Eq. (14)—(17) in Poeckert and Marlborough (1977) are only valid when the grid pointin the envelope is much further than the stellar radius from the center of the star, whichis essentially equivalent to assuming the star is a point source. However, some of the gridpoints and line of sights in the PM model are chosen fairly close to the star, since it ismore dense nearby the star, which could not be well represented by Eq. (14)—(17). Werederived the polarization in the envelope according to Fig. 6.22. Consider the star as adisk centered at 0, construct a spherical polar coordinate system (R, 0) at a generalscattering point P in the envelope (x, y, z). Using the same notation as Poeckert andMarlborough (1977), the scattering angle 8 is,cos = cos i cos 4' — sin i sin sin^ (6.23)the angle between the electric vector of the polarized scattered radiation and the skyplane issin 1.2 = (sin i cos + cos i sin sin 0) csc 8^(6.24)cos Si = sin E cos csc 8^ (6.25)where i is the inclination angle.For a given grid point P (r,z), the ith sector S^8i) on the stellar disk has the valuesof,z — vi cos Siei = arccos( ^V(vi sin 602 +r2(6.26)Chapter 6. Polarization across 1-1# emission line 65observerFigure 6.22: Geometry of polarization in the circumstellar envelopeChapter 6. Polarization across^emission line^ 66The azimuthal angle is also affected by the position along the line of sight, using Cartesiancoordinate system (x, y, z),vi sin Si cos 00arcsin( ^ ) <ko^(6.27)Vx2 + y2 + 2sin2 — 2 A, 42 + y2 sin S, sinwhere 00 is the azimuthal angle of the substellar point, dko = arctan(—y/x).It can be seen from Fig. 6.23 that after we adopted the new geometry of the polar-ization, the central peak of the of polarization across the line was constrained.6.2 Correction of the scattered radiationUntil now, we still find that the problem of the width of line and polarization profileis unsolved. While Poeckert(1977) studied their polarization data across the HIS, hesuggested that, because the envelope is expected to be fairly hot (between 10000 Kand 30000 K), the thermal velocities of the electrons are quite large and the scatteredradiation will be broadened on the order of 25A at Ha; this broadening tends to smoothout any features in the scattered radiation. Therefore, they assumed that the scatteredradiation (h, q, u) is independent of wavelength across the lines and has a value equalto that in the adjoining continuum.On the other hand, as we found in the discussion in Chapter 3, the underlying spec-trum has a fairly strong effect on the polarization profile, which indicates the scatteredradiation is not featureless across the line. The evidence for this is hinted at the polariza-tion profile: basically, as Poeckert proposed, the original PM model gave a polarizationprediction according to Eq. 3.4. If this is the case, the polarization would always have asimilar image profile as the model predicted line profile, which means the width of thepolarization would be exactly the same as the line profile. We can clearly see this fromearlier figures. However, the observed polarization is much wider than its correspondingline profile.—14840^4850^4860Wavelength (A)48804870Chapter 6. Polarization across HP emission line^ 67Figure 6.23: The dashed line gives the predicted polarization after adopting the newgeometry of polarization, comparing to that from the PM model (solid line). We find thecentral peak has been constrained.Chapter 6. Polarization across HP emission line^ 68We interpret this as the scattered radiation having the same feature as the underlyingstellar spectrum (Eq. 3.8). Since the emission in Ha is about 4 to 5 times stronger thanthe flux from the star, the feature across Ha is smeared out. While the emission acrossHfl is small compared to Ha, it is about the same intensity as the underlying stellarspectrum at Hp, so that the underlying feature might show up (cf. Chapter 3).It is too time-consuming to run a model which assumes the correct form of the under-lying feature in the PM code from the beginning. We simply modified the final part inthe subroutine lines, where the original scattered radiation q, u) is weighted acrossII/3 by the underlying feature. It could be seen from Fig. 6.24 that the uprising wingin the polarization profile has been successfully constrained through this change, but itdoesn't seem to have quite so much influence on the width.The corresponding polarization profile, adopting the parameters as in Table 5.8, isshown in Fig. 6.25. We can see that it still doesn't give a good enough fit, since thewidth is smaller and the depolarization deeper than is observed. We attributed this tonot taking the underlying absorption feature into account properly.On the other hand, since the PM model gives a reasonably good prediction to theemission line profile, we adopt Eq. 3.8, and using the modified PM model predicted lineprofile (Fig. 5.19), to produce the polarization across HO. As shown in Fig. 6.26, it givesa reasonable fit.AChapter 6. Polarization across 10 emission line^ 69Figure 6.24: The Dashed line gives the model prediction after the changes made accordingto Eq.3.8 to include the underlying feature in the scattered radiation, comparing to theprediction without such changes (solid line).Chapter 6. Polarization across 1.113 emission line^ 70N^e%.7Figure 6.25: Polarization profile predicted by the final model (dashed line), overplottedon Sep. 23 data (solid line).0.20—0.2—0.4Chapter 6. Polarization across 11/3 emission line^ 714840^4860^4880Wavelength (A)Figure 6.26: The dashed line gives the polarization predicted by applying the modifiedPM model predicted line profile to Eq.3.8, Pc = 1.6%, overplotted on the data of Sep.23 (solid line)Chapter 7ConclusionsAnalysis of the linear polarization data on 7 Cas reveals that the linear polarizationprofile across emission lines is not only affected by the emission in the circumstellar disk,but is also strongly affected by the underlying stellar absorption feature. A relationshipof PL = Pc X /A/./tota gives a good fit to our high-resolution spectropolarimetry data.This reminds us it is a possible way to determine the intrinsic parameters of theunderlying star of 7 Cas. Two nights of polarization data show similar results for Teff,log y, and v sin i Teff = 30000 K, log g = 3.5 -4 4.0, v sin i = 310 340 km/s). Tofurther constrain the intrinsic nature of the underlying star, we need higher S/N of thedata and a good estimate of the errors in the data.The Poeckert and Marlborough model is adopted as a basis to explain both the lineand polarization profile. Changes have been made on the principal levels of the hydrogenatom, atmosphere model, polarization geometry in the disk, and electron scattering effecton the Balmer lines. A fairly large parameter space has been searched to find a good fit.Several equally adequate parameter sets have been found to fit the line profile, but only apoor fit to the polarization profile. We successfully constrained the problems of the widthin the line profile, and central peak in the polarization. However, the inconsistencies inthe absorption in the wings of the emission line and the widths between the line profileand polarization profile still exist. Although we feel the PM model gives a general idea of-y Cas, it is unable to provide any detailed information without further changes, especiallyon the polarization profile.72Appendix AFigures for Chapter 5731^11111^iIiI11^It^11111_...._-^11111111111111111111-1111-_(a) _ (b)-L---..--.--.--,-.- --.._._. -...._ ...,.4- --1^1^I^I^11^1111^1^I1^I^Iii^J_(c) 1111 (d)_-,- _--,_----------- k/------------------—,..,I,,,,I,IIIIIIIIII, I^I 111111111^1111_111111111.1.11.60.80.81.41Appendix A. Figures for Chapter 5^74N(1,0) (a) 2.33 x10^cm^(b) 2.63 x10 cm^(c) 2.87 x10 cm (d) 3.33 x10i 450 TcHA 20000Kvrot-disk 325 km/s sip 0.35 cpt -0.25Vo 0.48 x V,,,„nd (8.4 km/s) V:.^253km/s /3 10.04840 4850 4860 4870 4880 4840 4850 4860 4870 4880Wavelength (1)^ Wavelength (A)Figure A.27:?'--^(a)____-iliiillil ilt!11111111 11111111111111_11111111111111_,_.__^(b)_— (c)^__-_I^I-1 1^I^I^I__I I^I^I^I^I^I^I^IIIIIIIIIiiillii__— (d)_I^I^I^1_—_(e) _ (f)^___-,^I^.^11111111111-11111111111111-filtil__1111111*Ct)0.0.52Appendix A. Figures for Chapter 5^75(a) 5°^(b) 30°^(c) 45° (d) 60°^(e) 750N(1,0) 2.87 x1013cra-3 Tdiskvrot-disk 325 km/s slp 0.35 cptVo 0.48 x Vwund (8.4 knits) 253km/s(0 89°20000K-0.2510.04850 4860 4870Wavelength (A)4850 4860 4870Wavelength (A)Figure A.28:4850 4860 4870Wavelength (A)_1111^111^1111(a)1111^1^II f 1111^1111_^(b)1111 III/ ^II___—----———-T---——I— --I^1^I— -(c) _—(d)—— -— -MI —— --———-.-:— -— u.......___________________=.— k....---------------■________ ______---------"N■J— -If IIIIIIIIIIItttiolititiThilittIlliiiimililiiii-211Appendix A. Figures for Chapter 5Tdisk (a) 15000 K (b) 20000 K (c) 225000 K (d) 25000 KN(1,0) 2.87 x1013cm-3 45°vrot -disk 325 km/s sip 0.35 cpt -0.25Vo 0.48 X Vsound (8.4 k111/13) Vc„, 253km/s 13 10.0764840 4850 4860 4870 4880 4840 4850 4860 4870 4880Wavelength (A)^ Wavelength (A)Figure A.29:Appendix A. Figures for Chapter 5^ 77vrot-disk (a) 325 km/s (b) 285 km/s (c) 228 km/s (d) 170 km/sN(1,0) 2.87 x 1013cm' 450 Tdisk 20000Ksip 0.35 cpt -0.250.48 x V,,,,,,d (8.4 km/s) , Vo, 253km/s , f3 10.0Figure A.30:78Appendix A. Figures for Chapter 52.87 x1013071-3N(1,0)vrot-disk 325 km/s1111^1111-^(a)11^Il^III!^II_._If^II^1111^TIII^1111^T^--^ -_._—^(b) --— __— __..—^ _m1111111;111(c)!III_______.II-11111IIIIIIIIIIIIIIHM-_._(a) _____lmi....„11,i,iiiiIii,_,Ii,,,I1m^I„„1,..,(11,_4840 4850 4860 4870 4880 4840 4850 4860 4870 4880Wavelength (A)^ Wavelength (A)Figure A.31:140 325Tdisk 20000K45°sip, cpt^(a) 0.1, -0.08 (b) 0.2, -0 15 (c) 0.35, -0.25 (d) 0.9 -0.8Vo^0.48 x Vsound (8-4 km/s) irc„, 253km/s 10.0Appendix A. Figures for Chapter 5^ 79Vo (in V„,„4) (a) 17 km/s (0.98) (b) 8.4 km/s (0.48) (c) 1.4 km/s (0.08) (d) 34.8 km/sN(1,0) 2.87 x 1013cm-3 i 450 Tdisk 20000Kvrot-disk 325 km/s slp 0.35 cpt -0.25K. 253km/s p 10.0Figure A.32:Appendix A. Figures for Chapter 5V0,3 (a) 103km/s (b) 253km/s (c) 453km/s (d) 1530km/s, N(1,0) 2.87 x1013crn-3 i 450 Tcusk 20000Kvrot-disk 325 km/s sip 0.35 cpt -0.25-tr_ n AR ,, v.^. (la A tr,,, /.21 a in n80Figure A.33:Appendix A. Figures for Chapter 5^ 8113 (a) 1.0^(b) 5.0^(c) 10.0^(d) 20.0N(1,0) 2.87 x1013cm-3 i 450 Tdisk^20000Kvrot-disk 325 km/s sip 0.35 cpt^-0.25Vo 0.48 x Vsou„d (8.4 km/s) Voo 253km/s1.8 -1111^1111^1111^1111^1(a)I II-----^(b)_1^1111^1111^111.6 ^ ----_-------0.8 - -1.81 1^1^1-(c)1111^I^1^1J^1^I^I^I^I^I^I1I^I^1^I1111^1111^1111-— (d)--1.6_--_ ----------_--^---.._--k-/----------------:,-------0.8 -- --- -1,,,,lim11,11111.^1111-".11111111111111111^1,i,lii-4840 4850 4860 4870 4880 4840 4850 4860 4870 4880Wavelength (A) Wavelength (A)Figure A.34:Bibliography[1] Abbott,D.C. 1979, I.A.U. Symp. 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