The Size, Structure, and Variability of Late-Type Stars Measured ...

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Figure 5.6: Stellar Spectra of P20 Bandpass: The vertical lines denote the edges of the bandpass as they would appear at different times in the year. 95
96 5.3 Modelling Spectral Features in Mira The H 2 O lines apparent in the spectrum of o Cet have a characteristic absorption trough of depth up to 20% and emission peak up to 35%. The absorption occurs at a wavenumber 0.03 cm −1 lower than the emission corresponding to a relative redshift of about 10 km/s. In addition, the absorption lines are deepest for H 2 O lines which are strongest around 1700 K whereas the emission is strongest for spectral lines having temperature near 1100 K. (Such a temperature can be associated with a given spectral line since the populations of the states involved in the transition are highest at some temperature.) To explain these features, it is necessary for there to be an infalling region in front of the star having a temperature lower than the stellar photosphere which produces absorption and an even cooler region at larger radii producing the emission. This structure is predicited by the models of Höfner et al. (1998) [47] shown in Figure 3.12. We may attempt to model the observed spectral features by adding H 2 O line opacity to the models discussed in Section 3.4.3. A molecular transition will increase the coupling of the radiation field to the local temperature. Thus, the line adds opacity at frequencies near the transition frequency. The natural (quantum) width of the transition is much narrower than the thermally doppler broadened width, and the latter will be assumed here. A line is roughly characterized by two parameters. For the 11 µm H 2 O transitions considered here, the line opacity increases quite sharply with temperature up to a peak value at some temperature then slowly tapers back down as temperature increases more and the population shifts to still higher energy states. The shape of the line opacity as a function of temperature is nearly the same for all transitions and the two parameters giving the maximum value for the line strength and the temperature at which the line strength reaches it maximum value are taken as describing the line completely. Optical depth is related to line strength by the following: τ(ν, T ) = S(T ) · f(ν, T ) · n · l (5.1) where τ is the dimensionless optical depth, ν is the wavenumber (in cm −1 ), T is the temperature, S(T ) is the line strength (in cm), f(ν, T ) is the line profile as a function of wavenumber normalized to unity (in cm), n is the number density of H 2 O (in cm −3 ), and l is the optical path length (in cm). The HITRAN/HITEMP database was used to obtain the line strength and line temperature for the H 2 O transitions modelled and f(ν, T ) was taken to be a Gaussian with a width ν 0 (2kT/m) 1/2 /c characteristic of doppler broadening
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The Size, Structure, and Variabilit
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1 Abstract The Size, Structure, and
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iv 5 Spectroscopy and ISI Measureme
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vi 3.15 Density, Temperature, and C
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viii Acknowledgements Over the cour
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2 to achieve resolution characteris
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4 sources are not coaligned perfect
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6 Intensity Profile Hankel Transfor
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8 Figure 1.3: Possible Effective Ba
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10 6LJQDO'HWHFWRU 6LJQDO'HWHFWRU /D
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1995A&A...304...69P 12 Figure 1.5:
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Figure 1.6: Stellar Evolution of a
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16 such as Mira variables, become u
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18 wavelengths, however, its range
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20 Figure 2.1: ISI Data for o Cet f
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22 Figure 2.3: Uniform Disk, Limb D
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24 2.2 Comparison of ISI Data with
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26 Table 2.3: Diameter Measurements
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28 measurements. χ Cyg and R Leo a
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30 metric flux (from the reference
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32 their surface. Very often, mater
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34 Mira. Finally, variations in tim
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36
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38 Figure 3.2: Intensity Distributi
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40 Figure 3.4: Ratio of the Best-Fi
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Figure 3.5: Synthetic Near-Infrared
Page 53 and 54: 44 Angular Diameter (mas) 60 50 40
Page 55 and 56: 46 wavelengths which would cause th
Page 57 and 58: 48 3.4.1 Static Radiative Transfer
Page 59 and 60: 50 Figure 3.8: Ratio of Photospheri
Page 61 and 62: 52 3.4.2 The Effect of Dynamic Phen
Page 63 and 64: 54 Figure 3.10: “Standard” Bowe
Page 65 and 66: 56 extension is reflected at wavele
Page 67 and 68: 58 photosphere. These results suppo
Page 69 and 70: 60 Figure 3.13: Continuum Opacity o
Page 71 and 72: 62 assuming only continuum sources.
Page 73 and 74: 64 Figure 3.16: Normalized Syntheti
Page 75 and 76: Figure 3.17: Density and Temperatur
Page 77 and 78: 68 Figure 3.19: Apparent Size of H
Page 79 and 80: 70 of Fox and Wood (1985) [31] are
Page 81 and 82: 72 3.5 Explanation of Variations in
Page 83 and 84: 74 Figure 3.21: 2.1 µm Image of 25
Page 85 and 86: 76 at visible wavelengths either by
Page 87 and 88: 78 4.1 Implications of Diameter Mea
Page 89 and 90: 80 11 µm Diameter of o Cet vs. Dat
Page 91 and 92: 82 Figure 4.3: A uniform intensity
Page 93 and 94: 84 phase observed in the continuum
Page 95 and 96: 86 5.1 General Results of 11 µm Sp
Page 97 and 98: 88 depth are the contours plotted a
Page 99 and 100: 90 Figure 5.3: Best Fitting Calcula
Page 101 and 102: 92 of its three parameters. The con
Page 103: 94 H 2 O on their measured diameter
Page 107 and 108: 98 Figure 5.7: Stellar Intensity Pr
Page 109 and 110: Figure 5.9: Observed H 2 O Spectral
Page 111 and 112: Figure 5.10: Höfner Model Predicte
Page 113 and 114: 104 30 km/s causes the spectrum to
Page 115 and 116: Figure 5.13: o Cet Spectra in Obser
Page 117 and 118: 108 density prediction matches the
Page 119 and 120: 110 Dixon. The infrared angular dia
Page 121 and 122: 112 [29] M. U. Feuchtinger, E. A. D
Page 123 and 124: 114 [50] A. P. Jacob, T. R. Bedding
Page 125 and 126: 116 [70] A. A. Michelson and F. G.
Page 127 and 128: 118 [92] Martin Schwarzschild. Over
Page 129: 120 [114] P. Woitke, D. Krüger, an
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The Size, Structure, and Variability of Late-Type Stars Measured ...

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