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Differential polarization of Hβ in γ CAS Jiang, Yiman 1993

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DIFFERENTIAL POLARIZATION OF IP IN 7 CAS By Jiang Yiman B. Sc. (Spacephysics) Beijing University  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE  in THE FACULTY OF GRADUATE STUDIES GEOPHYSICS AND ASTRONOMY We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA September 1993 © Jiang Yiman, 1993  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  Geophysics and Astronomy The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1Z1  Date:  3 2993 5 p, 2 3^  Abstract  High resolution ( 0.15 Alpixel ) spectropolarimetry across the Hfl emission line of the Be star 7  Cas  is presented. The differential polarization and polarization angle vary  both in profile and depth. It was found that the polarization profile fits the relation Pi,  = (Iwukviving fitotai)Pc, instead of the generally accepted  PL  =  PC  fiemission  (Coyne  1975). Assuming the absolute polarization in the adjoining continuum is constant, the polarization profile is used to determine Teff, geff, and v sin i for the underlying star by fitting the Hubeny (1988) atmosphere model. The Be star model developed by Poeckert and Marlborough (1978a) for .7 Cas is used as a basis to explain our polarization observations. We have made two types of modifications to the model: the first being to revise the intrinsic properties of the underlying star; the other is to make more accurate treatment of the radiation transfer equation in the disk. A search of the parameter space was made to find the best fit. We find an adequate fit to the emission line profile, but only a poor fit to the polarization profile from Poeckert and Marlborough model.  11  Table of Contents  Abstract List of Tables^  vi  List of Figures^  vii  Acknowledgement^  xi  1  Introduction  1  1.1 The Be phenomenon ^  1  1.2 1.3  Modeling Be stars ^ Past Polarization Studies of Be stars ^  1.4 -y Cas ^ 1.5 2  3  An Outline of this Thesis ^  Data Source and Reduction  3 8 10 11 12  2.1  UBC/DAO Polarization Analyzer ^  12  2.2  Observations ^  13  2.3 Data Reduction ^  14  2.4 Results ^  15  The Underlying Star  18  3.1  Problem ^  18  3.2  Basic relation of linear polarization across an emission line ^  18  3.3  Primitive analysis of the polarization data ^  3.4 Hubeny synthetic spectra ^  20 23  3.5  A possible way of determining fundamental parameters ^  25  3.6  Discussion ^  25  3.7 Pure emission ^  29  4 The Poeckert and Marlborough Model 4.1  4.2  34  Construction of the Model ^  34  4.1.1^Central Star ^  35  4.1.2^Envelope ^  35  Model for 7 Cas ^  43  5 Hi3 emission line profile  45  5.1  Principal level 6th of hydrogen atom ^  47  5.2  Hubeny atmosphere models ^  48  5.2.1^Continuum spectra ^  50  5.2.2^Line profiles ^  50  5.2.3^Kurucz model replaced by Hubeny model ^  52  5.3  Pure emission from PM model ^  52  5.4  Electron scattering from the disk ^  56  5.5  Parameter study ^  56  6 Polarization across Hfl emission line  61  6.1  Correct geometry for polarization ^  6.2  Correction of the scattered radiation  '7 Conclusions  ^  64 66 72  iv  Appendices^  73  A Figures for Chapter 5^  73  Bibliography^  82  V  List of Tables  1.1  Basic -y (Jas parameters ^  11  2.2  Observations of 7 Cas during 1990-1993 ^  13  3.3  Parameters of models used to find the best fitting model ^  25  3.4  Best-fitting model of polarization data for Sep. 23 and Sep. 24 ^  27  3.5  Most likely models representing the central star of -y Cas ^  28  3.6  Suggested best model for central star of 7 Cas ^  29  4.7  Original PM model parameters of 7 Cas ^  44  5.8  Parameters define the model for 7 Cas of Sep 23 ^  59  List of Figures  2.1 Line profile, polarization and polarization angle of 7 Cas on Sep. 23,1990 ^16 2.2 Line profile, polarization and polarization angle of -y Cas on Sep. 24,1990^17 3.3 Polarization across H13 emission line on Sep. 23,1990 (solid line). The dashed line shows Pc/how, where Pc is taken to be constant in the considered region (Poeckert 1979). We set Pc = 1.1% to roughly match the line center polarization  21  3.4 Polarization across Hfl emission line on Sep. 24,1990 (solid line). The dashed line shows Pc/how, where Pc = 1.1%. ^  22  3.5 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 ^ 24 3.6 The dashed line represents the predicted linear polarization from Eq.3.8, the underlying absorption profile is taken from the Hubeny NLTE synthetic 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 across 11$ of 7 Cas on Sep. 23,1990.  26  3.7 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 %. . . . 30 3.8 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 %. . . . 31  vii  3.9 The dotted line gives the best-fitting model for the underlying stellar absorption feature of Sep. 23. The solid line shows the total intensity of observation, with the difference between the two giving the pure emission feature from the disk, shown as the dashed line.  32  3.10 The dotted line gives the best-fitting model for the underlying stellar absorption feature of Sep. 24. The solid line shows the total intensity of observation, with the difference between the two giving the pure emission feature from the disk, shown as the dashed line.  33  4.11 A schematic picture of the envelope adopted in the PM model ^ 37 5.12 (a) PM model with their original parameter set compared to our data on Hfl of 7 Gas, 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, 1990 ^  46  5.13 Predicted H/3 emission line after adding the 6th principal level of hydrogen atoms (dashed line), compared to the original model (solid line) ^ 49 5.14 Continuum Spectrum at Tea. = 25000 K, log g =3.5 predicted by Kurucz model (dashed line) and Hubeny model (solid line) ^ 51 5.15 Intrinsic Hig line profile at Ter = 25000 K, log g =3.5 calculated by Kurucz model (dashed line) and Hubeny model (solid line) ^ 53 5.16 The dashed line gives a model prediction, in which the underlying spectrum is represented by the Hubeny atmosphere model, instead of the Kurucz atmosphere model as in the original PM model (solid line) ^ 54 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 to Sep. 23 Hfl emission line (solid line) ^ 55 VIM  5.18 Adding an gaussian profile to represent the electron scattering of the emission line photon from the disk (solid line), comparing to the Sep. 23 11)3 line  profile.  57  5.19 Line profile predicted by the final model (dashed line), overplotted on Sep ^ 23 data (solid line) ^  60  6.20 (a) Polarization profile across 11,3 emission line, predicted from original PM model (dashed line) compared to Sep. 23 data (solid line). (b) The same as (a), overlapped on Sep. 24 data ^  62  6.21 The solid line gives the predicted polarization from the PM model assuming a finite size to the central star, while the dashed line gives the predicted result when the star is treated as a point source. We see that the predicted polarization structure is the same in both cases. ^  63  6.22 Geometry of polarization in the circumstellar envelope ^ 65 6.23 The dashed line gives the predicted polarization after adopting the new geometry of polarization, comparing to that from the PM model (solid line). We find the central peak has been constrained. 67 6.24 The Dashed line gives the model prediction after the changes made according to Eq.3.8 to include the underlying feature in the scattered radiation, comparing to the prediction without such changes (solid line) 69 6.25 Polarization profile predicted by the final model (dashed line), overplotted on Sep. 23 data (solid line) ^  70  6.26 The dashed line gives the polarization predicted by applying the modified PM model predicted line profile to Eq.3.8, Pc = 1.6%, overplotted on the data of Sep. 23 (solid line) A.27 ^  71 74  A.28 ^  75  A.29 ^  76  A.30 ^  77  A.31 ^  78  A.32 ^  79  A.33 ^  80  A.34 ^  81  Acknowledgement  Thanks first to Gordon Walker, my supervisor, for his guidance and patience throughout. I am also grateful to Jaymie Matthews for his help and support, Ted Kennelly for teaching me the essentials of astronomical research, and Jason Auman for providing many useful 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 is based and responded to several of my queries • Nadine Dinshaw who started the polarimetry project for her M.Sc. thesis and provided many observations of which I made use • Katharine Wright who helped me take the observations gathered during the course of my thesis I would also like to thanks my parents for loving me so much, my sister for being there when I needed her, Angela for giving me a lot of fun, and special thanks to Andrew for everything he did.  xi  Chapter 1 Introduction  1.1 The Be phenomenon A 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 or more Balmer lines in emission.  Far from being exceptional, the presence of emission lines is pervasive throughout the HR diagram. Among normal stars of every spectral type (X), one can find emission-line stars of the same spectral type (Xe). However, Be stars are the best-studied type among all the emission-line stars. Be stars were first discovered in 1866 by A. Secchi (Underhill 1982). Ever since then, Be stars have been extensively observed from the radio to the X-ray regions of the spectrum, and new candidates are still added to the list from normal B stars, of which they make up a large subset (about 20 %). Besides the most distinguishing feature of emission in 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, intrinsic linear polarization, high speed stellar wind (seen in the UV region) and superionization in 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 evidence from the visible and infrared shows that this outer envelope is relatively cool — with 1  Chapter 1. Introduction^  2  temperatures of order 104 K, so that the matter in the envelope is subionized compared to the photosphere, quiet — without large mass-ejection and high expansion velocities. The hydrogen emission lines are generated in the envelope, which are expected to result from recombinations of the ionized atoms, as is the infrared excess, which is attributed to free-free radiation emitted by the ionized gas in the outer envelope. However, the observations in the visible, infrared, and radio regions show that there is a great diversity in 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 density must still be significant at hundreds or thousands of stellar radii, in order for radio emission to be detectable. On the other hand, ultraviolet spectra observed over the last 20 years show a strikingly different picture from those of visible to infrared. There is no specific distinction between Be and normal B stars in the UV spectrum. One important phenomenon appearing 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 the existence of superionized regions with temperatures of order 105 K — much higher than predicted from the effective temperatures of these same stars under conditions of radiation equilibrium. A nonradiative energy flux is required to explain their formation. The fact that resonance lines have large displacement or asymmetries, which may sometimes exceed their escape velocity at the surface of star, suggests fairly active regions in the outer envelope with violent ejection of matter taking place. A large body of observations show Be stars to be variable. Their variability presents itself in a variety of forms. The Balmer line spectra may undergo, in their life, three different phases, in any order : Be, shell, and normal B phase. In a Be spectrum, emission lines show either no reversal or a more or less central reversal. In a shell spectrum, Balmer lines and singly ionized metal lines exhibit narrow and deep absorption cores, which may  Chapter 1. Introduction^  3  or may not be bordered by emission wings. Generally there is a gradual transition from B 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/R variations—the ratio of the violet to red peak in emission lines, and E/C variation—the ratio of the emission in the lines to the adjacent continuum. A similar effect is seen in the continuum, which is mostly observed as the changing in stars' magnitudes by broadband 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 the mass flux is strongly variable. All the variations are generally irregular, their time scales ranging from a few hours to several decades. 1.2 Modeling Be stars  The first suggestion to explain the origin of the emission lines in Be stars was put forth by Struve (1931). His picture of the circumstellar envelope has since become the fundamental picture of Be stars. As most Be stars are rapid rotators, Struve suggested that their rotational velocities had reached their critical value, so that matter could be ejected at the equator due to the rotational instability, forming an equatorial disk or ring. Extrapolated from the mechanism applied to the gaseous envelopes of planetary nebulae, dilution of the stellar radiation makes it possible to convert high-energy radiation from the star to low-energy radiation regenerated from the envelope. Atoms in the envelope are ionized by the ultraviolet flux from the central star. Recombinations to higher levels and then cascades to lower levels produce the observed emission in hydrogen lines. But what distinguishes this situation from the planetary nebula is the dilution factor, which is about 10-i to  Chapter 1. Introduction^  4  10-2, compared to 10-14 in the planetary nebula case. So in Be stars, the envelope is believed to be subionized relative to its photosphere, and the envelope may be opaque in the lines of the subordinate series. Struve's model explains the diversity of the emission line profiles by geometric effects alone. Whether a Be star exhibits a normal Be phase or a shell phase, depends on if it is 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 of all, although most Be stars are fast rotators (statistically speaking, the Be stars exhibit larger values of v sin i than normal B stars), no Be star has been found to be rotating at its 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 not allow matter leaving too far from the stellar surface, which means that rotation alone can't be responsible for the formation of outer envelope. Besides , some Ha emission lines in Be stars have great extended wings, which could exceed 1000 kms-1. This is two or three times larger than their v sin i, which implies an unrealistic conclusion that their envelope rotates faster than the stars. Also the large observed variabilities could not be explained solely by the geometrical orientation effects. During the next 50 years after Struve's rotation model was first unveiled, quite a few Be 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^  5  2. Be star as member of binary system, in which the envelope is produced by mass transfer in a close binary system, where the companion fills its Roche-lobe, such as Kriz and Harmanec(1975) Good reviews for those models can be found in Marlborough (1975), Poeckert (1981) and references 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 with the observed spectrum. It is the first problem which fundamentally defines the ad hoc property of all existing Be star models. Far from completely understanding the construction of the envelopes, one generally guesses the solution to the hydrodynamic problem, so that the density distribution in the envelope could be obtained (cf. Chapter 4). The fact that each model uses a different approach to determine the excitation and ionization state of the envelope and to solve the radiative transfer problem and can simultaneously satisfactorily reproduce the same spectral feature, indicates there is no unique solution. One important reason 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. It is impossible to decide which of the above models best represents the Be stars. One motivation behind this thesis is to put as many constraints on Be stars or their envelopes as possible from our polarization observations, based on the Poeckert and Marlborough model (cf. Chapter 4,5,6). Nevertheless, the real trouble comes from the ultraviolet observations. They raise several major contradictions with the classical idea of Be stars. As mentioned before,  Chapter /. Introduction^  6  instead of cold, quiet regions around Be stars, analysis of ultraviolet spectra provide evidence of a large mass-loss flux, high temperature and superionized region. It was the large mass-loss detected in the ultraviolet which convinced people of the ability to form the extended envelopes observed in the visible spectra. However, the problem is how the two strictly different regions around Be stars are related to each other if both of them actually exist; how they are formed; why only Be stars have a cold envelope dense or large enough to have emission lines in the visible region, although both normal B and Be stars have high-temperature, mass-outflow regions. Those problems are still unsolved mysteries for Be stars. Only recently have theoretical models begun to take into account the dynamic evidence of the kinematical structure assumed in empirical models. Snow (1982) noticed that mass-loss rates of Be stars determined from UV lines seem to be what are expected if one extrapolates mass-loss rates of 0 and early B stars to late B types. This suggested that 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 pressure, both line and continuum, is critical in the formation of the circumstellar envelope of Be stars (cf. Marlborough 1986 ). Consequently, recent work on the rotation model has concentrated on self-consistent radiation-driven stellar winds. It was clear from the ultraviolet observation that stellar winds are widespread among early-type OB stars. Castor, Abbott and Klein (1975, hereafter CAK) first introduced a complete theory of radiation-driven stellar winds for Of stars. The basic idea behind the original CAK model is that a spherically symmetric wind is raised by radiation pressure force through the scattering of continuum photons in a very large number of spectral lines, from resonance to subordinate lines. When applied to Be stars, however, it was found that the luminosities of B-type stars later then B2 are insufficient to initialize the stellar wind by radiation pressure (Abbott  Chapter 1. Introduction^  7  1979,1985). Therefore some other mechanism must operate in Be stars later than B2 to provide initial acceleration of the wind. Several other factors which might be important to 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-driven wind from a hot star. They came to the conclusion that by departing from the spherically symmetric CAK model, a two-component stellar wind could be formed naturally under rapid rotation: one past is from the equator, with low expansion velocity, low mass loss rate, high density, and relatively low temperature (i.e. subionized), closely resembling the classical ad hoc picture of Be stars; the other is from the polar region, with much higher expansion velocity and high mass-loss rate, but low density, fairly high temperature and superionized region, to explain the UV observations. This provided the first dynamical justification for these earlier empirical stellar wind models. Various authors have proposed models in which magnetic fields may play an important dynamical role in the circumstellar envelopes of Be stars (Weber and Davis 1967, Barker 1979, Nerney 1980, Underhill 1983). One significant effect of a magnetic field is to increase the Lorentz force at large radii (Friend and MacGregor 1984), thereby raising the terminal velocity and mass loss rates. But this effect is considerably reduced after allowing for the finite size of the star (Poe and Friend 1986). Time variability is a fundamental aspect of the Be phenomenon, but the majority of stellar wind models try only to represent a time-invariant star in its Be phase. Little consideration 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, Vogt and Penrod 1983, Penrod 1986, Baade 1986, Kambe 1993) has been found to relate nonradial pulsation with Be phenomena, especially the moving patterns detected in the line profiles. Willson (1986) suggested that the pulsations, caused by periodic isothermal shocks, may deposit energy in high-density regions of the atmosphere, where shocked  Chapter 1. Introduction^  8  gas adiabatically expands against gravity, and thereby initiates a cool dense wind. The  advantages 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 with significant vertical amplitude, and variances in line profiles. A series of papers presenting further development of the CAK model came out recently. Abbott (1982) adopted a complete and accurate set of atomic data to calculate numerical values of the radiation pressure in spectral lines in hot stellar winds. Friend and Abbott (1986) corrected the radiation- driven stellar wind for the finite size of the star and rotation. Bjorkman and Cassinelli (1993) presented an axisymmetric two-dimensional supersonic solution of a rotating radiation-driven stellar wind. Their solution predicted that the disk is formed because the supersonic wind that originates at high latitudes of the stellar surface travels along trajectories and come down to the equatorial plane, with the ram pressure confining and compressing the disk. 1.3 Past Polarization Studies of Be stars To 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 optical radiation from Be stars is observed to be intrinsically linearly polarized provides perhaps our most conclusive clue that the circumstellar shells of these stars are disk-like in nature."  The first detection of intrinsic polarization in Be stars was made by Behr (1959), when he was searching for variation of polarization in 7 Cas during its shell phase. Following that, Shakhovskoi (1962) found variable polarization for x Oph and later (Shakhovskoi 1964) for some other Be stars. It is its peculiar wavelength dependence pattern, p(A), and its variation with time which makes the intrinsic polarization different from interstellar  Chapter 1. Introduction^  9  polarization, so that it is possible to separate the two. Later observations have produced evidence for polarized radiation in the majority of Be stars surveyed (Underhill 1982). Although there are significant differences from star to star, the intrinsic polarization of Be stars always has two basic characteristics: • The continuum polarization (around 1%) decreases rapidly with wavelength in the ultraviolet spectrum across the Balmer limit (Serkowski 1968, Coyne 1971). • The polarization across spectral lines is less than the continuum polarization (Zellner and Serkowski 1972, Clarke and McLean 1974). Theoretical work (Nagirner 1962, Collins 1970, Rucinski 1970 ) showed polarization from pure photospheric effects can not exceed 0.1%. As proposed by Shakhovskoi (1964) and Rucinski (1966), the simplest model of producing such a large linear polarization in most Be stars is to have the stellar radiation scattered by particles contained in a volume that is not spherically symmetric with respect to the observer. Multicolour polarimetry by Serkowski (1968, 1970) and Coyne et a/. (1967, 1969, 1974) led them to notice that hydrogen absorption played a significant role in the observed wavelength dependence pattern 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 scattering in the envelope, whose geometry is aspherical with respect to the observer, and where the wavelength dependence of the polarization is mainly due to continuous absorption by hydrogen. The critical factors in this model are the geometry of the envelope and the opacities of hydrogen and electron scattering. During the next 10 years, more effort was put into the study of polarization across the spectral lines. In the first approximation, the decrease in polarization across the lines could be explained as the dilution of additional unpolarized emission radiation from the envelope (Clarke and McLean 1974, Poeckert 1975). However, the addition of unpolarized  Chapter 1. Introduction^  10  emission flux does not appear to be an adequate explanation for all Be stars. High resolution observations showed that there is residual polarization after the unpolarized emission flux (Coyne and McLean 1975) is taken into account, which indicates some other mechanism must be contributing to the decrease. 1.4 7 Cas Gas was the first emission-line star discovered (Secchi 1867), and ever since then Cas has been carefully traced for more than a century. Table 1.1 gives the basic parameters of 7 Gas. As a particularly interesting Be star, 7 Gas has gone through strong and weak Be phases, shell phases and a normal B phase (Goraya and Tur 1988). During 1932-1942, -y Gas displayed spectacular variations in colour, brightness, line spectrum etc.(Telting et al. 1993), which finished with the star as a quasi-normal B star. From 1942 till now, 7 Gas slowly and irregularly increased its emission lines, undergoing some small fluctuations (Underhill 1982). In 1976 7 Cas was identified as the optical counterpart of the low-luminosity, variable hard 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 were made that 7 Gas is a binary system, its companion being either an accreting neutron star or a white dwarf (White et al. 1982, Mura.karni et d. 1986). Several detailed models have been developed for 7 Gas (Poeckert and Marlborough 1978, Scargle et al. 1978). Optical interferometry measurements of 7 Gas (Mourard et al. 1989, Quirrenbach et al. 1993) well resolved the target star in the Ha line. The  results clearly showed a spherically asymmetric envelope in rotation, which is consistent  with the disc model.  Chapter 1. Introduction^  11  Reference Cas RA(1900) DEC Vm  Spectral Class V sin i  HR 264 (HD 5394) 00:50:40.1 +60°10'31" 2.47 B0.5IVe 300 km/s 230 km/s  Lesh 1968 Boyarchuk 1964 Slettebak 1982  Table 1.1: Basic 7 Cas parameters  1.5 An Outline of this Thesis The goal of this thesis is to interpret spectropolarimetry data of one of the brightest Be stars — -y Cas, based on an improved Poeckert and Marlborough ad hoc stellar wind model. The data were obtained at the Dominion Astrophysical Observatory, with a polarization analyzer developed by Walker and Dinshaw (Dinshaw 1991, Walker, Dinshaw etc. 1993). A summary of the instrument and data reduction will be found in Chapter 2, and a primitive analysis of data in Chapter 3. Chapter 4 reviews the original Poeckert and Marlborough model. Improvements on this model are described in Chapter 5 and Chapter 6. In the last chapter, we discuss the successes and failures of the improved model in fitting observations, and further investigations are suggested.  Chapter 2  Data Source and Reduction  In order to study the nature of Be stars, a microcomputer-controlled polarization analyzer 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 differential spectropolarimetry at the same spectral resolution as the spectrograph-detector system.  2.1 UBC/DAO Polarization Analyzer The polarization analyzer was designed for linear polarization measurements. It consists of two components: a beamsplitter cube followed by a quarter-wave plate. The bearasplitter is constructed of two right-angle prisms separated by a multilayer dielectric. The quarter-wave plate is set with its optical axis oriented at a 45° angle to the beamsplitter axis. The plate co-rotates with the beamsplitter so as to maintain this angle. The beamsplitter cube separates the unpolarized or partially polarized starlight into two beams, with orthogonal polarizations. The beam whose plane of polarization is aligned with the transmission axis of the beamsplitter goes straight through the cube and emerges as a beam of linearly polarized light, and the other beam is reflected. The linearly polarized light is converted into circularly polarized light by the quarter-wave plate at a 45° angle. (Ideally only 50% of the star light will be lost in this process, it is about 60% in practice.) 12  Chapter 2. Data Source and Reduction^  13  The analyzer is controlled remotely by computer under an iRMX 86 environment. At the 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 Observations Table 2.2 presents the polarization observations of 7 Cas during 1990-1993, which lists 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 spectrograph, we found short exposures gave better results (Walker and Dinshaw 1991). The rotation of the analyzer was alternated between clockwise and counter-clockwise in order to minimize the effect of zero-point drifts in the analyzer. The 21181 spectrograph and a RCA 620 x 1024 CCD were used together to give a resolution of 0.15 A/pixel. CCD detectors are used because of low readout noise. Table 2.2: Observations of 7 Cas during 1990-1993 Date (UT) Sep. 23,1990  Number of Spectra 225  Exposure Time (s) 20  Sep. 24,1990  255  20  Nov. 9,1992 Mar. 8,1993 Mar. 9,1993 Mar. 10,1993  35 100  Spectral Region  Hp Hp  S/N 500 500  30 30  Ha  90  Iiii  120  114  30  Hi3  140  90  40  HP  155  Position Angles 20°, 60°, 100°, 140°, 180° 20°, 60°, 100°, 140° , 180° 00, 45°, 90°,135° 00, 300,600, 900, 120°, 150° 0°, 30°, 60°, 90°, 120°, 150° 0°, 300, 60°, 90°, 120°, 150°  Chapter 2. Data Source and Reduction^  14  2.3 Data Reduction  The CCD data were reduced using IRAF. The standard processing methods were used, with the exception of the following: • Mean spectra Two types of mean spectra are used. The total mean is the mean spectrum for a single star obtained by averaging the spectra, at all position angles, from one night. 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 should be normalized. • Residual spectra No flat field is used during this study, in order to avoid the additional noise from the flat field spectra. Instead, the residual spectra are formed by dividing each spectrum by its own cycle-mean spectrum. In this way, not only the pixel-topixel variations are cancelled, but the differential polarization across the line is also revealed (Fahlman 1975). • Polarization analysis The polarization across the lines are at about the 1% level in Be stars. In order to detect the structure of such a low signal, we need as high S/N level as possible. So we combine residual spectra at each position angle, which will finally form a set of residual spectra obtained at four to six different rotation positions of the analyzer 1The reason we are doing this, instead of taking longer exposures to get high S/N, is that, 7 Cas is a strongly variable star and we wish to avoid a significant change during a single cycle of the position angle of the analyzer.  Chapter 2. Data Source and Reduction^  15  Generally, when a beam of partially polarized light passes through a polarizer, its intensity changes according to the position angle of polarizer as: 1 4,(0) = — 4(1 -I- cos(2(614, — 64))) 2  (2.1)  where i refers to each data point at wavelength Ai, Op is the plane of polarization, 0 is the axis of the polarizer. I  ,  It are intensities of incident light and light coming  out 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 polarization in the continuum, since the reduction process we adopted removes polarization in the continuum(Pc) as well as the instrumental polarization. From Eq. 2.2, we are able to determine the wavelength dependence of differential polarization(AP) and its polarization angle(4) across the line. A Fortran program stokes was written by Dinshaw to give a least-squares fit of the above parameters.  2.4 Results The following figures give the final line profiles, differential polarization profiles and polarization 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 angle outside the lines are meaningless because the errors there are too large, we didn't display them on the plots.  Chapter 2. Data Source and Reduction^  Figure 2.1: Line profile, polarization and polarization angle of 7 Cas on Sep. 23,1990  16  Chapter 2. Data Source and Reduction  Figure 2.2: Line profile, polarization and polarization angle of 7 Cas on Sep. 24,1990  17  Chapter 3 The Underlying Star  3.1 Problem Be stars are theoretically B-type stars, with additional emission from their circumstellar disk. However, we have very little direct information about the fundamental parameters of underlying Be stars, including mass, radius, luminosity, effective temperature, and v sin i, etc. (Slettebak 1986). The determination of all these parameters is very difficult, due to contamination from the emission disk, and so highly uncertain. 7 Cas is the brightest Be star in the northern sky. Unfortunately, no conclusive evidence for the intrinsic properties of the underlying star has been presented. The effective 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) found that the ultraviolet continuum, which is believed to originate from the photosphere, and be the region of the spectrum least affected by the envelope, could be equally well fitted by several Kurucz(1979) atmosphere models. They concluded that the photosphere of 7 Cas cannot be represented uniquely by a single Kurucz model. 3.2 Basic relation of linear polarization across an emission line The general relation for polarization across an emission line in Be stars is given by Coyne(1975), PL=  PA + PE X 1+X  18  (3.3)  Chapter 3. The Underlying Star^  19  where Pi,  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 the adjoining continuum. To make it more simple, Coyne considered that the intensity of the underlying absorption is small compared with the intensity of emission flux from the disk. It is true for strong emission lines, especially Ha, that the shallow rotationally broadened photospheric absorption feature can sometimes be neglected in comparison to the emission line strength (McLean and Clarke 1975). X could then be taken as the ratio of the emission flux to the continuum flux. Considering the continuum flux is normally set to be unity, then 1+ X = /total, the total intensity of the emission line. With all these assumptions, and further assuming there is no polarization in the emission flux, Coyne concluded that, FL =  Pc fit otai^  (3.4)  This relation is interpreted qualitatively such that the intrinsic polarization across the line is diluted by the unpolarized emission flux from the circumstellar disk. Poeckert(1975) claimed they found the same relationship between linear polarization and the total intensity of the line, from their linear polarization measurement across Ha in 12 Be stars, at a resolution of 2A. Further, using Eq. 3.4, they separated interstellar polarization and intrinsic polarization for Be stars (Poeckert and Marlborough 1976). So did McLean and Clarke (1975).  Chapter 3. The Underlying Star^  20  3.3 Primitive analysis of the polarization data We have a better resolution (0.15 A) of linear polarization across the lines than the previously cited authors. While applying Eq. 3.4 to our Sep. 23, 1990 and Sep. 24, 1990 polarization data 1, as shown in Fig. 3.3, Fig. 3.4, we found additional polarization which deviates from the above dependence. This inconsistency indicates either that PE 0 or that Eq. 3.3 is not applicable. It is time to reconsider Eq. 3.4. Notice that the polarization profiles do not have the same width as their line intensity profile. They tend to be wider than the line profile. We assume the basic, Eq. 3.3 is still applicable. Another possibility is that the intensity of emission flux from the disk is not as big as it was thought, comparing to the underlying absorption feature, which means it could not be negligible. Still adopting PA = Pc,  PE = 0, we rederive Eq. 3.3 as following, PL=  Pc 1+ X  (3.5)  where X = 'E/IA. Therefore, PC  FL-11 + -1:  PL =  IA.  I A + IE  Pc^  (3.6) (3.7)  we get,  IA n ii, — --7.—.1-c^ /total  (3.8)  Eq. 3.8 shows, not only the total intensity of the line, but also the underlying absorption feature is a fairly important factor, which affects the linear polarization across the line, especially when it is concerned with faint emission lines. lAmong all the data on -y Cas we have, the data of Sep. 23 and Sep. 24 have the highest S/N. We will only use these two nights data throughout this thesis.  Chapter 3. The Underlying Star  ^  21  _ ^  -  _ 1 4840  i^I^I  4860 Wavelength (Â)  f  4880  Figure 3.3: Polarization across HP emission line on Sep. 23,1990 (solid line). The dashed line shows Pc/how, where Pc is taken to be constant in the considered region (Poeckert 1979). We set Pc = 1.1% to roughly match the line center polarization.  Chapter 3. The Underlying Star ^  22  Figure 3.4: Polarization across Hfl emission line on Sep. 24,1990 (solid line). The dashed line shows Pc/how, where Pc = 1.1%.  Chapter 3. The Underlying Star^  23  3.4 Hubeny synthetic spectra A 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 spectra program SYNSPEC have been used to generate photosphere spectra of the underlying star. 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 selected atomic 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 departures from LTE are taken into account only in the continuum, and finally a model in which NLTE affects are considered both in the continuum and line transitions. It should be noticed that the only difference between an LTE and NLTE model in this program is the form of the statistical equilibrium equations, where the NLTE model counts both radiative and collisional transitions, but the LTE model is simply the populations from the Saha-Boltzmann equation. TLUSTY includes as many atomic data in the program as computing time allows. Therefore it gives a great flexibility in choosing chemical species, ionization degree, energy levels, etc. Generally, all the important atomic data for hydrogen and helium are already included in the program; the user doesn't need to worry about them unless more accurate or extensive data are available. In our case, we only need to provide the effective temperature Tog, surface gravity log g, and microturbulent  velocity. SYNSPEC uses the previously computed model atmosphere from TLUSTY, and calculates a synthetic spectrum in a chosen wavelength region. It uses the same input files as TLUSTY, plus an additional file specifying the wavelength region and another  24  Chapter 3. The Underlying Star^  one 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 with fluxes 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 example is given in Fig. 3.5.  _  -  0.95  ■  ■  0.9  1 1 1 1 1  cts 0.85  1 1 1 0.8  1  4840  1  ^ 4860 Wavelength (Â)  ^  4880  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/s  Chapter 3. The Underlying Star^  25  3.5 A possible way of determining fundamental parameters When we applied the above Hubeny synthetic spectrum and the observed emission  line profile to the polarization profile, using Eq. 3.8, as shown in Fig. 3.6, we got a fairly good fit. This proves that the underlying absorption feature is not completely smeared out 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 could possibly use Eq. 3.8 to determine the intrinsic parameters of Be stars, under the assumption 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 the linear polarization across the emission lines. A series of models (parameters are listed in Table 3.3) are applied to the two nights' polarization data by Eq. 3.8. I wrote a Fortran program to derive the best-fitting parameter set, using the least-square method. Table 3.3: Parameters of models used to find the best fitting model Hubeny model v sin i Teff log g Pc 20000K 4.0 25000K 3.0 50 km/s 0.5 % 25000K 3.5 4 -V 25000K 4.0 750 km/s 5.5 % every every 30000K 3.0 10 km/s 0.1 % 30000K 3.5 30000K 4.0  3.6 Discussion  The best-fitting model among the above models we used is given in Table 3.4. They are shown in Fig. 3.7, Fig. 3.8.  Chapter 3. The Underlying Star ^  26  Figure 3.6: The dashed line represents the predicted linear polarization from Eq.3.8, the underlying absorption profile is taken from the Hubeny NLTE synthetic spectrum of HP with Ten- = 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 across Hfl of -y Cas on Sep. 23,1990.  Chapter 3. The Underlying Star ^  27  Table 3.4: Best-fitting model of polarization data for Sep. 23 and Sep. 24 Sep. 23, 1990 Telt30000 K log g 4.0 sin i 340.0 km/s 1.6% Pc 2 131.78 X 1.0  Sep. 24, 1990 30000 K 3.5 310.0 1.5% 72.05 1.0  The next question is how reliable this method is for determining the underlying stellar parameters of 7 Cas. The two nights' data gives fairly consistent values of the intrinsic parameters of the underlying star, although not exactly the same. It still indicates this simple model is realistic. However, there are about 600 data points across the polarization profiles of each night. From the last two lines of Table 3.4, we could see that the errors in the polarization data were somehow overestimated. On the other hand, although this best-fitting model is defined as the one with the least-chi-square value, the best-fitting model 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 for both nights would be the most likely models of 7 Cas, which are listed in Table 3.5 (Pc is not listed in Table 3.5, as it is not an intrinsic parameters of the star. Pc is around 1.2 % to 1.7 %). It is still not possible at this stage to definitely determine the intrinsic parameters of Cas. To properly estimate the likelihood of the above models, and further pin down the intrinsic parameters of underlying star, we need data with higher signal-to-noise, and a good estimate of the error in the data.  28  Chapter 3. The Underlying Star^  Table 3.5: Most likely models representing the central star of 7 Cas v sin i (km/s) (km/s) 50-200 210-230 240-270 280-300 310-330 340-360 370-400 410-750  20 4.0  25 3.0  Teff(103K) & log g 25 3.5 25 4.0 30 3.0  30 3.5  * *  * * * * *  * * * *  * * * * *  30 4.0  * * * *  Chapter 3. The Underlying Star^  29  Here 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 in the later chapters. Table 3.6: Suggested best model for central star of 7 Cas Teff 30000 K log g 4.0 v sin i 325.0 km/s 1.6% Pc  3.7 Pure emission  Assuming the underlying absorption line has been determined, we have a direct approach 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 using the suggested best underlying stellar model. In this way it would be easy to model the emission flux and tell whether the line profile variations originate from the star itself or from its circumstellar disk.  Chapter 3. The Underlying Star^  30  Figure 3.7: Best fitting model of the linear polarization across lig on Sep. 23, 1990, with Teff = 30000 K, log g = 4.0, v sin i = 340.0 km/s, Pc = 1.6 %  Chapter 3. The Underlying Star  4840  ^  ^  4860  ^  31  4880  Figure 3.8: Best-fitting model of the linear polarization across HP on Sep. 24, 1990, with Teff = 30000 K, log g = 3.5, v sini = 310.0 km/s, Pc = 1.5 %  Chapter 3. The Underlying Star^  32  Figure 3.9: The dotted line gives the best-fitting model for the underlying stellar absorption feature of Sep. 23. The solid line shows the total intensity of observation, with the difference between the two giving the pure emission feature from the disk, shown as the dashed line.  33  Chapter 3. The Underlying Star ^  0.5  1 4840  i  I^,  4860 Wavelength (A)  1 4880  Figure 3.10: The dotted line gives the best-fitting model for the underlying stellar absorption feature of Sep. 24. The solid line shows the total intensity of observation, with the difference between the two giving the pure emission feature from the disk, shown as the dashed line.  Chapter 4 The Poeckert and Marlborough Model  Poeckert and Marlborough's model (hereafter PM model) was first developed in 1969 by Marlborough, and later improved by Poeckert. Like all the Be star models, because of insufficient knowledge of relevant physical effects in the circumstellar disc around Be stars, their model is also ad hoc. Its main advantage lies in the large number of observable features it manages to compute and reproduce nicely, which gives a powerful tool to compare with the observations and improve the model. 4.1 Construction of the Model The stellar wind model, based upon the original ideas of Struve (1931 cf. Chapter 1), postulates an asymmetric circumste,llar envelope strongly concentrated to the equatorial plane of the star, which is formed through rotationally forced ejection resulting from the breakup rotation of the central star. Generally one must guess the velocity, fT(i:), and temperature, T(7), distributions in the disk as functions of position (and time if appropriate) and then obtain the density distribution, p(r), from the equation of continuity. Approximate solutions of the radiative transfer equations then give the ionization-excitation conditions as a function of position. The PM model and its applications are described in detail by Limber, Poeckert, Marlborough (Limber and Marlborough 1968, Marlborough 1969, Poeckert and Marlborough 1977, Poeckert and Marlborough 1978a, Poeckert and Marlborough 1978b, Poeckert and Marlborough 1979) and a good review is given by Underhill (1982). Here we outline the 34  ^  Chapter 4. The Poeckert and Marlborough Model  ^  35  model. 4.1.1 Central Star  The central star is assumed to rotate as a solid body at its critical velocity at the equator. The physical parameters describing the star are assumed to be the same as a normal spherically symmetric star, of the same spectral type as the Be star, whose continuum energy distribution is taken from Kurucz et al (1974). 4.1.2 Envelope  The envelope is assumed to be in a steady state, and symmetric about both the rotation axis and the equatorial plane. The envelope consists only of hydrogen, and is treated as a perfect gas. The envelope is assumed to be isothermal, with a temperature Te throughout, which is lower than the effective temperature,  Teff,, of  the central star.  Density distribution  Consider a cylindrical coordinate system (r,0,z) with the origin at the center of the star, and the i-axis along the stellar rotation axis. In a meridian plane, it is assumed that steady flow takes place along streamlines, which are taken to be straight lines converging at one point. The density distribution is set by hydrostatic equilibrium in the z-direction. Using these assumptions plus conservation of mass and steady state, Limber and Marlborough (1968) derived the total number density distribution of atoms and ions, N(r,z), in the envelope as: N(r,  where  Z)^  N(1, °)  exp^[;-!7-^r/:+zi2 V,.(r) r(r—p) V,(1) 1—p  (4.9)  Chapter 4. The Poeckert and Marlborough Model ^  36  N(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 the surface of the star. r and z where the envelope is assumed to be in hydrostatic equilibrium in the z p  direction.  distance of the outflow converging point from the rotation axis, p < 1, in units of stellar radius.  It is clear from Eq. 4.9 that the density distribution in the disc is completely determined by, 1. N(1,0) ;  2. p; 3. Vr(r). All these factors are chosen in an ad hoc manner  in the PM model. The disc is chosen to be wedge shaped and bounded by 1/7* < r < 5014 and straight lines diverging by a certain open angle. Outside this region the density is assumed to be negligible, and the source function and opacity are set equal to zero. A schematic picture of the disc can be found in Fig. 4.11. Velocity distribution  The velocity distribution in the envelope results from a combination of rotation and expansion. In the original model, it was assumed that the envelope is supported by centrifugal force plus some viscous force. Assuming partial conservation of angular momentum, the rotational velocity of the envelope is given by:  got = V[/3r  2+  (1  ^ fi)r - 111/2  where Vrot is in the same direction as the stellar rotation K.  (4.10)  0 is an ad hoc function of  distance from the star which, depending on its assigned value, allows a transition between  Chapter 4. The Poeckert and Marlborough Model ^  Figure 4.11: A schematic picture of the envelope adopted in the PM model  37  Chapter 4. The Poeckert and Marlborough Model^  38  two extreme cases: the envelope rotating at critical velocity (fl = 0) or conservation of angular momentum (fl = 1). The expansion velocity adopts the canonical form (Castor 1970, Castor, Abbott and Klein 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 surface of the star, K. the terminal velocity derived from the displacement of UV resonance lines. Excitation-ionization structure The envelope is considered to be composed of hydrogen atoms only. Each hydrogen atom is assumed to consist of 6 bound levels: the ground level, 2S and 2P treated separately 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 envelope by assuming a steady state, i.e. at each point in the envelope, the radiation field and the population of all levels are independent of time. Therefore, the statistical equilibrium equations for the population of bound level n could be written as: Na  + En' >. Nn' [An'n Bn'nu(vn' 4:^Nni,  n u (Pe n  En,,, on Nni" Ne Q n(u', 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 of bound level n: 'Angular momentum substates / for a given n with n > 3 were not treated individually because collisional transitions between substates of one principle level are much faster than the radiative transitions between 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 B coefficients However, 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 2S In the rate equations, the radiative recombination coefficients, an, are from Burgess (1964). Einstein A and B coefficients are taken from Capriotti (1964). Photoionization cross-sections at the series limits are also taken from Burgess (1964). The collisional excitation and deexcitation rate coefficients for transitions between bound levels, except 2S 4-+ 2P, are derived from the cross-sections given by Saraph (1967), while 2S 4-+ 2P is taken from Seaton (1955). Radiation field In general, the radiation density in the continuum is the combined result of the radiation emitted by the star and radiation produced in the envelope, the so-called diffuse field. The computer code of the PM model is written in such a way that the contribution  Chapter 4. The Poeckert and Marlborough Model^  40  to ionization and excitation by the envelope is completely neglected; in other words, at each point in the envelope, only radiation contained in the right circular cone subtended at (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 along a given direction from (r, z) to a point on the surface of the star. Only bound-free absorption processes are included in rm, which assumes photo-ionization is the dominant source of continuous opacity. It is known that the actual state of material in the envelope is significantly different from the prediction of the LTE model. However, the non-LTE treatment requires a completely self-consistent simultaneous solution of both the radiative transfer and statistical equilibrium equations, which is difficult to achieve. It is reasonable to believe that the true radiation field and population distribution could be reached by deviation from the LTE case. Poeckert and Marlborough, claiming that a simpler approach can be used without too great a sacrifice in accuracy, adopt the following procedure to solve for the radiation field and line profiles: 1. The initial radiation field in the envelope only includes the radiation emitted from the central star diluted by the distance factor. 2. Use this radiation field to compute radiative rates in the statistical equilibrium equations, to get the solution of level populations. 3. Use these to compute the radiation field produced in the disc through radiation transfer equation. The steady-state equations are solved for five different cases for the treatment of line radiation produced in the envelope. They are:  Chapter 4. The Poeckert and Marlborough Model ^  41  1. The envelope is optically thin to all line radiation produced in it, i.e. ue(v„,n) = 0 for 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 treated by 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 in the 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 in the rest. 5. The envelope is optically thick in all lines, i.e. ue(vn,74) is given by detailed balancing for all the levels n and m. This is the LTE case. The final populations are obtained by taking a weighted average of the populations of any two consecutive cases, depending on the optical thickness of the disk. Balmer line profiles  The procedure adopted by Poeckert and Marlborough for determining the line profiles is to consider the profile from the system as a whole to be the sum of profiles from all elements 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 to a column through the envelope with its axis parallel to the observer's line of sight. The radiation transfer is integrated numerically along 2000 columns in the PM model, from the edge of the disk nearest to the observer, either to the surface of the star, or to the furthest extremity of the disk, depending on whether the column strikes the central star's  Chapter 4. The Poeckert and Marlborough Model ^  42  surface. 00  F„. -AN Svexp(—r„)dr, (4.14)  When a column intersects the stellar surface, its source function, S„, is set equal to the specific intensity at the stellar surface. Otherwise, in general, the source function for a given frequency is, (4.15) where j„L is the line volume emission coefficient, ji,c is the continuum volume emission coefficient, I, is the amount of radiation scattered into the line of sight, and k,,, is the total volume extinction coefficient, = kvc Neg.  (4.16)  where kiz is the line absorption coefficient, lex the continuum absorption coefficient, ble is 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 of continuum opacity cannot be ignored. Free-free, bound-free and recombination processes are included in the continuum emission and absorption coefficients. Unlike the procedure when the populations are determined, bound-free is considered as the only opacity source. Polarization across the Balmer lines  To predict the polarization across the lines, Poeckert and Marlborough assumed that polarization arises from single scattering of photospheric radiation within the envelope, which means that, multiple scattering of the photospheric radiation and any radiation contributed from the envelope are not included. As pointed out by Poeckert and Marlborough (1978a), while it is a drawback to treat the stellar radiation as the sole source of scattered flux, there should not be much polarized scattered flux from the radiation produced in the envelope, because it is more isotropic than the stellar radiation.  Chapter 4. The Poeckert and Marlborough Model ^  43  It is necessary to determine the second and third Stokes parameters, q and u within the 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 Iv  =E  AN f(uPs.,)exp(-7-„)dri,^ (4.18) N^°  Q„ and U„, the overall second and third Stokes parameters. And the polarization and  position angle are given by,  P„ =  (4.19)  1 = — tg-1(Uv/Q„) 2  (4.20)  Notice the F„ includes the total flux emitted per steradian, but Qv and (Iv only include the photospheric radiation. To determine q and u, Poeckert and Marlborough considered the stellar surface as a disk, which is divided into 13 sectors, instead of a point source. The geometry of the polarization calculation could be found in Poeckert and Marlborough (1977). Scattering angle (0), scattered radiation (I,) and q, u are determined for each sector at each grid point (cf. Chapter 5). 4.2 Model for 7 Cas In principle, the PM model could apply to every Be star, but this model was built up specifically for the interpretation of the observations of 7 Cas. The model calculates the populations in the envelope at 24 x 4 grid points, and the line profile along about 1500 line of sights. The basic parameter set that defines the model of 7 Cas are given in Table 4.7 (Poeckert and Marlborough 1978a). The choice of parameters in the model essentially rests on the agreement obtained between model predicted and observed continuum polarization and Ha line profile.  Chapter 4. The Poeckert and Marlborough Model^  44  The model predicts Ha, HA 117, 1115 and 1325 line profiles, and polarization and polarization angle across Ha and 11/3. It also gives the continuum energy distribution and continuum polarization. Comparison to Poeckert and Marlborough's data can be found in Poeckert and Marlborough (1978a).  Table 4.7: Original PM model parameters of 7 Cas ^ mass ^ radius ^ Tdr ^ log g Vequ  ^  Stellar Parameters 17 Mo 10 Ro 25000 K 3.5 569 kms' — 1.0 time critical speed  Envelope Parameters 20000 K T. hydrogen composition mean molecular weight 0.68 N(1,0) 3.33 x 1011cm' 0.8& P, 15.0 R,, r , z 0.0 maximum extent of the envelope 50.0 14 (line) 250.0 R..k (continuum) inclination 45° wind velocity Vo = 7.47kms" — 0.48 times the sound speed at the surface Vcc, = 253.0kms' 13 = 10.0  Chapter 5 HP emission line profile  The motivation of this thesis comes from the comparison of the original PM model with our observational data. I will only discuss the HP emission line in this chapter and leave the polarization for the next chapter. Fig. 5.12 shows the original PM model overplotted on the line profiles of Sep. 23 and Sep. 24. As can be seen , the model doesn't match the observations at all well. There could be two reasons for this: either the model is good, but because of the variations of 7 Cas, the parameters Poeckert and Marlborough chose don't fit any more; or the model simply doesn't represent 7 Cas well enough. The first thing I did was to test the PM model with a series of different parameter sets. However, no better fit to the Sep. 23 and Sep. 24, 1990 data was achieved despite an 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 the observed 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 broader than the HP line profile.  45  Chapter 5. I-113 emission line profile  46  1.4  1  4840  4860  4880  _ _ _ _  -4-  1  4840  I^, 4860 Wavelength (A)  _ 1  4880  Figure 5.12: (a) PM model with their original parameter set compared to our data on Hfl of 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, 1990  Chapter 5. Hig emission line profile^  47  In an effort to better understand Be stars, or at least 7 Cas, we decided to modify the PM code in the following respects: 1. Add the 6th principal level of hydrogen atoms, adding an additional case which assumes that the envelope is optically thick in Lyman, Balmer, Paschen and Brackett lines 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 atom  Notice in Fig. 5.12, the predicted HP line from the PM model is too wide compared with the observation. The most obvious reason for this is that 7 Cas is not rotating at a speed as high as 569 kms-1, or the rotational velocity in the envelope is much slower than was assumed. However, Marlborough (1993) claimed that simply reducing the rotational velocity either of the star or of the envelope is not correct. It is because the population of principal level 4 and 5 are overestimated. He explained this is due to the averaging process they were adopting to get the populations of each level, which gives a final population of a weighted mean of two of these five optical thickness cases (cf. Chapter 4). For example, if one grid point in the envelope is optically thick in Ha, but the optical thickness is smaller in HP, the averaging scheme will tend to weight case 3 (d. Chapter 4) heavily, which artificially increases the population of levels 4 and 5. Therefore, in this way, HP will always be too strong because from every volume element in the circumstellar envelope there will be too much Iff3 produced. Particularly those volume elements close  Chapter 5. MI emission line profile ^  48  to the central star, which have higher rotational speed, will contribute more emission than 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 include the 6th principal level and an additional optical thickness case, which considers the envelope 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. We hope that in this way, the overestimated 4th level population would pass smoothly to a situation where only the 5th and 6th level populations are overestimated, which would give a reasonably good estimate of level 4, so that HP has a better fit. Results with the same 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 level population. It has exactly the same line width. Instead, the line strength increases. As explained later by Marlborough (1993), since the averaging process leads to an overestimated 6th level population as it does to the 4th and 5th, transitions between the 4th and 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 process is found.  5.2 Hubeny atmosphere models  The underlying stellar spectrum is fairly critical in the PM model. We suspect that the reason the original PM model doesn't match our line profiles well, is partly due to the uncertainty in describing the properties of the central star. Two subroutines in the PM program deal with the underlying stellar spectrum: flux contains 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).  Chapter 5. 1113 emission line profile^  49  1.8  1.6  1.4 :14 CI)  1.2  0.8 4840  ,  4850^4860 Wavelength (A)  ,^1^,  4870  ,^1  4880  5.13: Predicted lifl emission line after adding the 6th principal level of hydrogen atoms (dashed line), compared to the original model (solid line). Figure  Chapter 5. lifi emission line profile^  50  To remain consistent with the discussion in Chapter 3, and to have better resolution of the spectrum, we replaced the Kurucz model by a Hubeny synthetic spectrum, both in the continuum and lines. 5.2.1 Continuum spectra The continuum energy distribution is decided in the same way as the original PM model. About 57 wavelength points are chosen to determine the continuum level. Since in 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 the seven different ionization levels, or at the rest line wavelength between any two of the bound levels we considered to get the absolute line flux. While generating the Hubeny spectrum, only Balmer lines and several helium lines are included. 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 interpolation between the adjoining continuum. Both the continuum spectrum predicted by Kurucz atmosphere model and Hubeny atmosphere model could be found in Fig. 5.14. 5.2.2 Line profiles Basically, in the Kurucz model, 18 values of residual flux for the wavelength displacements from line center are given for each line profile. Any wavelength displacement in between any of the 18 points is interpolated linearly. The line profile is assumed to be symmetric, 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, by using different AA values in the program rotin (although it might not be meaningful in some case to get too fine a resolution). So we choose the same dispersion as the data we  Chapter 5. HP emission line profile  51  10  4 3  3.5^4 Wavelength (log A)  4.5  Figure 5.14: Continuum Spectrum at Teff = 25000 K, log g =3.5 predicted by Kurucz model (dashed line) and Hubeny model (solid line).  52  Chapter 5. HP emission line profile^  got, which is about 0.15À/pixel. The line spectra are considered 37.5  A away from the  rest line center on each side, with even space. No reflection and interpolation are needed simply because the spectrum covers the whole line profile and its wings. The predictions of rotationally broadened H(3 absorption line from Kurucz atmosphere model and Hubeny atmosphere model are shown in Fig. 5.15. 5.2.3 Kurucz model replaced by Hubeny model One model of Teff = 25000 K, log g =3.5, v sin i = 400 km/s produced by the PM model is shown in Fig. 5.16, after replacing the Kurucz model by the Hubeny model. A significant change can been seen. Since the Hubeny model predicts less UV flux than the Kurucz model does, the line profile produced by using the Hubeny profile has a lower red emission 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 represent the center star of 7 Cas. Comparing to the Sep. 23 line profile, Fig. 5.17 shows that as 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 the underlying stellar feature is applicable. However, the wings show even deeper absorption features than before, this is due to the larger value adopted for log g (4.0) than in the original PM model (log  g = 3.5).  5.3 Pure emission from PM model An equivalent or, maybe a better way of reducing the effect from the uncertainty of the underlying star is to modify the PM model to produce pure emission flux and then compare the pure emission features derived in Chapter 3. This will not be included in this thesis.  Chapter 5. Hi3 emission line profile  ^  53  0.9 -  ^  _  Z r-4 0.7 -  _  0.6 -  _ 1  4840  ,^1^,  4860^4880 Wavelength (A)  Figure 5.15: Intrinsic Hf3 line profile at Teff = 25000 K, log g =3.5 calculated by Kurucz model (dashed line) and Hubeny model (solid line)  Chapter 5. 1-1fl emission line profile ^  54  1.6  1.4  _ _ -  _. _ -  0.8 I^  I^I^I^I^_I^^ I^I_^11111 _I  4840^4850  4860^4870 Wavelength (A)  ^  4880  Figure 5.16: The dashed line gives a model prediction, in which the underlying spectrum  is represented by the Hubeny atmosphere model, instead of the Kurucz atmosphere model as in the original PM model (solid line).  55  Chapter 5. Xi emission line profile^  1.6 —  A  I j 1  1.4  1  0.8  ^  1^1^1^1^1^1^1^3^1^1^1^1^1^1^1^1^1  4840^4850^4860^4870 Wavelength (A)  •  I  4880  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 to Sep. 23 IP emission line (solid line).  Chapter 5. HO emission line profile ^  56  5.4 Electron scattering from the disk Previous studies (Marlborough 1969, Bemat and Lambert 1978) have shown that very broad weak Ha emission wings are due to electron scattering in the envelope. Poeckert and Marlborough (1979) studied the effects of electron scattering of the photons originating 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 by electrons into the wings. To investigate this, we artificially add a gaussian profile (Eq. 5.21, whose standard deviation (Eq. 5.22) is decided by the thermal electron velocity in the envelope, to represent the electron scattering of the emission line photons. The flux of the gaussian profile is about 0.5% of the total emission flux, decided by best-fitting of the line profile. A^—(V — V0)2 Ii. = ^ Lexp( ^ ) 2a2^ Viirer  (5.21)  where .Te is the total emission flux of 48 from the envelope, and 0.2 2kTdisk ^+ VI^ Me  (5.22)  It can be seen clearly from Fig. 5.18 that although the gaussian profile doesn't completely constrain the absorption features in the wings, it certainly makes the absorption less. This gives us confidence that proper calculation of the electron scattering of line photons from the radiation transfer equation would be able to explain the unmatched absorption wings. This has been done by Poeckert and Marlborough (1979), but will not be included in this thesis. 5.5 Parameter study After making all the above mentioned modifications to the PM model, we try to find the best fit to our emission line profiles for 7 Gas on Sep. 23, 1990.  Chapter 5. HO emission line profile^  57  Figure 5.18: Adding an gaussian profile to represent the electron scattering of the emission line photon from the disk (solid line), comparing to the Sep. 23 HO line profile.  Chapter 5. Hfl emission line profile^  58  First, we will discuss the influence of several major parameters on HO line profile. Similar work has been done by Poeckert and Marlborough (1978). Each parameter involved is varied with respect to the original PM model while all the other parameters are kept at their previous standard value. The central star of 7 Cas is described by the best-fitting values from Chapter 3 (71a. = 30000 K, log g =4.0, v sin i = 325 km/s), and kept as the same values throughout all the parameter tests. (Figures can be found in Appendix 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 of sight to the observer. Different inclination angles decide if the envelope of star is equation-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 the wedge; cpt, the intersection to the rotational axis) determines the thickness and open angle of the disk.  Envelope temperature, Tdisk An isothermal envelope with an electron temperature of 20000K was assumed earlier. Models are also run in which this temperature is set at 15000K and 25000K.  Chapter 5. HP emission line profile ^  59  The final choice of the parameters adopted to define the model for 7 Cas of Sep. 23 is based on numerous calculations of models of various values of each parameter, as given in 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 mentioned here, Table 5.8 does not give a unique solution to 7 Cas, several other parameters set give similar good fit to the data. Table 5.8: Parameters define the model for 7 Cas of Sep. 23. Ten.^30000 K log y^4.0 Vequ^325 kms" — 1.0 time critical speed N(1,0)^2.39 x 1011cm-1 inclination^500 Tdisk^21500 K vrot-disk^284.5 kms-1 sip^0.35 cpt^-0.25 wind velocity 170 = 17.07kms-1 = 1530.0kms' = 10.0  60  Chapter 5. .11)3 emission line profile  -  1.6  _  _ _1  1.4  -  -  _  _  -  a  0.8 -  .^,^I^,  4840  .^LI,i,i1,,,,I,  4850^4860^4870 Wavelength (Ä)  ,^,^I  4880  Figure 5.19: Line profile predicted by the final model (dashed line), overplotted on Sep.  23 data (solid line)  Chapter 6  Polarization across^emission line  The PM model also predicts the polarization across the JEW emission line. Correspondingly, Fig. 6.20 shows the original PM model overplotted on the polarization profiles of Sep. 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 profiles and polarization profiles of similar width, although the observations show that the polarization across lifl is much broader than the Etig line profile. • The predicted polarization profile across HP has a much stronger central peak than is observed. • The model predicted polarization increases in the emission-line wings, before it drops 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 absorption feature on the polarization across the emission lines (cf. Chapter 3).  61  Chapter 6. Polarization across HP emission line  ^  62  1  (a)  0 _ ,LAvvr. A P% —0.5  4840  4860  - 4880  4840  4860 Wavelength (A)  4880  0 A P% —0.5  Figure 6.20: (a) Polarization profile across up emission line, predicted from original PM model (dashed line) compared to Sep. 23 data (solid line). (b) The same as (a), overlapped on Sep. 24 data  Chapter 6. Polarization across II/3 emission line  63  0  A P% —0.5  —1 4840  4850  4860  4870  4880  Wavelength (.1k)  Figure 6.21: The solid line gives the predicted polarization from the PM model assuming  a finite size to the central star, while the dashed line gives the predicted result when the star is treated as a point source. We see that the predicted polarization structure is the same in both cases.  Chapter 6. Polarization across 11,8 emission line^  64  6.1 Correct geometry for polarization The subroutines abint, litpat and polcon are created to calculate the polarization in the code. We tested this part of the code by replacing the star with a point source, it gave the same result as before (Fig. 6.21). In addition, the general " W " shape of the polarization across the line has never been seen to change at all. Therefore, we suspect this 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 point in the envelope is much further than the stellar radius from the center of the star, which is essentially equivalent to assuming the star is a point source. However, some of the grid points and line of sights in the PM model are chosen fairly close to the star, since it is more dense nearby the star, which could not be well represented by Eq. (14)—(17). We rederived the polarization in the envelope according to Fig. 6.22. Consider the star as a disk centered at 0, construct a spherical polar coordinate system (R, 0) at a general scattering point P in the envelope (x, y, z). Using the same notation as Poeckert and Marlborough (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 sky plane is sin 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 values of, ei = arccos(  z — vi cos Si ^ V(vi sin 602 +r2  (6.26)  Chapter 6. Polarization across 1-1# emission line  65  observer  Figure 6.22: Geometry of polarization in the circumstellar envelope  Chapter 6. Polarization across^emission line^  66  The azimuthal angle is also affected by the position along the line of sight, using Cartesian coordinate system (x, y, z), arcsin(  vi sin Si cos 00 ^ Vx2 + y2 + 2sin2 — 2 A, 42 + y2 sin S, sin  ) <ko^(6.27)  where 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 polarization, the central peak of the of polarization across the line was constrained. 6.2 Correction of the scattered radiation Until now, we still find that the problem of the width of line and polarization profile is unsolved. While Poeckert(1977) studied their polarization data across the HIS, he suggested that, because the envelope is expected to be fairly hot (between 10000 K and 30000 K), the thermal velocities of the electrons are quite large and the scattered radiation will be broadened on the order of 25A at Ha; this broadening tends to smooth out any features in the scattered radiation. Therefore, they assumed that the scattered radiation (h, q, u) is independent of wavelength across the lines and has a value equal to that in the adjoining continuum. On the other hand, as we found in the discussion in Chapter 3, the underlying spectrum has a fairly strong effect on the polarization profile, which indicates the scattered radiation is not featureless across the line. The evidence for this is hinted at the polarization profile: basically, as Poeckert proposed, the original PM model gave a polarization prediction according to Eq. 3.4. If this is the case, the polarization would always have a similar image profile as the model predicted line profile, which means the width of the polarization would be exactly the same as the line profile. We can clearly see this from earlier figures. However, the observed polarization is much wider than its corresponding line profile.  Chapter 6. Polarization across HP emission line  ^  67  —1 4840^4850^4860  4870  4880  Wavelength (A)  Figure 6.23: The dashed line gives the predicted polarization after adopting the new geometry of polarization, comparing to that from the PM model (solid line). We find the central peak has been constrained.  Chapter 6. Polarization across HP emission line ^  68  We interpret this as the scattered radiation having the same feature as the underlying stellar spectrum (Eq. 3.8). Since the emission in Ha is about 4 to 5 times stronger than the flux from the star, the feature across Ha is smeared out. While the emission across Hfl is small compared to Ha, it is about the same intensity as the underlying stellar spectrum 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 underlying feature in the PM code from the beginning. We simply modified the final part in the subroutine lines, where the original scattered radiation q, u) is weighted across II/3 by the underlying feature. It could be seen from Fig. 6.24 that the uprising wing in the polarization profile has been successfully constrained through this change, but it doesn't seem to have quite so much influence on the width. The corresponding polarization profile, adopting the parameters as in Table 5.8, is shown in Fig. 6.25. We can see that it still doesn't give a good enough fit, since the width is smaller and the depolarization deeper than is observed. We attributed this to not taking the underlying absorption feature into account properly. On the other hand, since the PM model gives a reasonably good prediction to the emission line profile, we adopt Eq. 3.8, and using the modified PM model predicted line profile (Fig. 5.19), to produce the polarization across HO. As shown in Fig. 6.26, it gives a reasonable fit.  Chapter 6. Polarization across 10 emission line^  69  A  Figure 6.24: The Dashed line gives the model prediction after the changes made according to Eq.3.8 to include the underlying feature in the scattered radiation, comparing to the prediction without such changes (solid line).  Chapter 6. Polarization across 1.113 emission line^  %.7  70  N^e  Figure 6.25: Polarization profile predicted by the final model (dashed line), overplotted on Sep. 23 data (solid line).  Chapter 6. Polarization across 11/3 emission line  ^  71  0.2  0  —0.2  —0.4  4840^4860^4880 Wavelength (A)  Figure 6.26: The dashed line gives the polarization predicted by applying the modified PM model predicted line profile to Eq.3.8, Pc = 1.6%, overplotted on the data of Sep. 23 (solid line)  Chapter 7 Conclusions  Analysis of the linear polarization data on 7 Cas reveals that the linear polarization profile 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 relationship of 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 the underlying star of 7 Cas. Two nights of polarization data show similar results for log y, and v sin i  Teff =  Teff,  30000 K, log g = 3.5 -4 4.0, v sin i = 310 340 km/s). To  further constrain the intrinsic nature of the underlying star, we need higher S/N of the data and a good estimate of the errors in the data. The Poeckert and Marlborough model is adopted as a basis to explain both the line and polarization profile. Changes have been made on the principal levels of the hydrogen atom, atmosphere model, polarization geometry in the disk, and electron scattering effect on 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 a poor fit to the polarization profile. We successfully constrained the problems of the width in the line profile, and central peak in the polarization. However, the inconsistencies in the absorption in the wings of the emission line and the widths between the line profile and 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, especially on the polarization profile.  72  Appendix A  Figures for Chapter 5  73  ^  Appendix A. Figures for Chapter 5  vrot disk  (a) 2.33 x10^cm^(b) 2.63 x10 cm^(c) 2.87 x10 cm 450 TcHA i 325 km/s sip 0.35 cpt  Vo  0.48 x V,,,„nd (8.4 km/s)  N(1,0) -  V:.^253km/s /3  1^11111^iIiI11^It^11111  -0.25 10.0  11111111111111111111-1111  _  (a)  (d) 3.33 x10 20000K  -^  _...._  1.  74  -  _  (b)  1. -L-  .-  --..--  .--,  -  -.. _._. ..._  1  -  -. . .,  -  .4-  0.8  1.6  -  ii^J  1^1^I^I^11^1111^1^  1^I^I  (c)^  1.4  1111  I _  (d)  _ _  -,-  --, _ 1  0.8  -----------  k/-------------------  ,..,I,,,,I,IIIIIIIIII, I^I 111111111^1111_11111111 4840 4850 4860 4870 4880 4840 4850 4860 4870 4880 Wavelength (A) Wavelength (1)^ Figure A.27:  —  Appendix A. Figures for Chapter 5  ^  (a) 5°^(b) 30°^(c) 45° N(1,0) 2.87 x1013cra-3 vrot-disk 325 km/s slp 0.48 x Vwund (8.4 knits) Vo l  ilt!11111111  75  (d) 60°^(e) 750 Tdisk  0.35 cpt 253km/s  (0 89° 20000K -0.25 10.0  11111111111111_11111111111111_, _._  ?'--^  (a)  _^(b) _  — (c)^_ _  Ct)  _ 0. _ _ _  _  _  _ -iliiilli I^I^I^1 I^I-1 I^I^I^I^I^I^I^I^IIIIIIIIIiiillii_ 1^I^I^I _ _ 2 — (d) (e)^_ (f)^_ —  _ _  _  _  0.5  _ _ -,^I^.^ 4850 4860 4870 Wavelength (A)  11111111111-11111111111111-filtil  4850 4860 4870 Wavelength (A) Figure A.28:  1111111*  4850 4860 4870 Wavelength (A)  76  Appendix A. Figures for Chapter 5  Tdisk  N(1,0) vrot -disk Vo  (a) 15000 K (b) 20000 K (c) 225000 K (d) 25000 K 2.87 x1013cm-3 45° 325 km/s sip 0.35 cpt -0.25 10.0 0.48 X Vsound (8.4 k111/13) Vc„, 253km/s 13  1111^111^1111  1111^1^I  (a)  1111  III/  ^II_  _ _  — -  -T— — -—  I (c) 2  MI  If  1111^1111  — — —  1  —  f  _^(b)  _  1  I  I^1^I  — _ (d) — — — — — — — — —  k....---------------■________ ______---------"N■J  -  — — -. -: u.......___________________=.  —  IIIIIIIIIIItttiolititiThilittIlliiiimililiiii  4840 4850 4860 4870 4880 4840 4850 4860 4870 4880 Wavelength (A)^ Wavelength (A) Figure A.29:  -  -  Appendix A. Figures for Chapter 5^  vrot-disk N(1,0)  (a) 325 km/s (b) 285 km/s (c) 228 km/s (d) 170 km/s 2.87 x 1013cm' 450 Tdisk 20000K sip 0.35 cpt -0.25 0.48 x V,,,,,,d (8.4 km/s) , Vo, 253km/s , f3 10.0  Figure A.30:  77  ^ ^  78  Appendix A. Figures for Chapter 5  sip, cpt^(a) 0.1, -0.08 (b) 0.2, -0 15 (c) 0.35, -0.25 (d) 0.9 -0.8 N(1,0) 2.87 x1013071-3 45° Tdisk 20000K vrot-disk 325 km/s 10.0 Vo^0.48 x Vsound (8-4 km/s) irc„, 253km/s 1111^1111  11^Il^III!^II  If^II^1111^TIII^1111^T --^  -^  _._ —^(b) —^ —^  (a)  -  -_ _ _  _ _._ —^  .. _  1  m1111111;111  5  (c)  !III  II-11111IIIIIIIIIIIIIIHM_._  _ ____ __.  (a)  _ _  4  _  03  _  2  _  lmi....„11,i,iiiiIii,_,Ii,,,I1m^I„„1,..,(11,_ 4840 4850 4860 4870 4880 4840 4850 4860 4870 4880 Wavelength (A)^ Wavelength (A) Figure A.31:  Appendix A. Figures for Chapter 5^  79  Vo (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/s 2.87 x 1013cm-3 i 450 Tdisk N(1,0) 20000K vrot-disk 0.35 cpt 325 km/s slp -0.25 253km/s p K. 10.0  Figure A.32:  Appendix A. Figures for Chapter 5  V0,3 (a) 103km/s (b) 253km/s (c) 453km/s (d) 1530km/s 2.87 x1013crn-3 i 450 Tcusk 20000K , N(1,0) vrot-disk -0.25 325 km/s sip 0.35 cpt in n -tr_ n AR ,, v.^. (la A tr,,, /.21 a  Figure A.33:  80  Appendix A. Figures for Chapter 5^  13  N(1,0) vrot-disk Vo 1.8 1.6 ^  81  (a) 1.0^(b) 5.0^(c) 10.0^(d) 20.0 2.87 x1013cm-3 i 450 Tdisk^20000K 325 km/s sip 0.35 cpt^-0.25 0.48 x Vsou„d (8.4 km/s) Voo 253km/s  1111^1111^1111^1111^1  --  (a)  I  --^  _  II  1^1111^1111^11  (b)  -  -  _  --  0.8 1.8 1.6 -  1 1^1^1  1111^I^1^1J^1^I^I^I^I^I^I1I^I^1^I1111^1111^1111  (c)  _ -  _ --------0.8 -  -  —  -^ -  (d)  _ -.. _ -  k-/----------------:,  ---  -  1,,,,lim11,11111.^1111-".11111111111111111^1,i,lii-  4840 4850 4860 4870 4880 4840 4850 4860 4870 4880 Wavelength (A)^ Wavelength (A) Figure A.34:  Bibliography  [1] Abbott,D.C. 1979, I.A.U. Symp. No. 83 Mass Loss and Evolution of 0-type Stars: 237 [2] Abbott,D.C. 1982, Ap.J. 259: 282 [3] Abbott,D.C. 1985, Progress In Stellar Spectral Line Formation Theory (Dordrecht: Reide.1): 279 [4] Baade,D. 1986, I.A.U. Colloq. 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