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

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93 Ratio of Apparent to Actual Stellar Diameter R gas = 1.0R ∗ R gas = 1.2R ∗ R gas = 2.0R ∗ Figure 5.5: Fractional Change in Best Fit Uniform Disk Diameter Due to a Shell of Gas as a Function of the Shell Optical Depth, τ gas , and the Intensity Ratio, B ν (T gas )/B ν (T ∗ ), for Three Values of the Shell Radius, R gas . τ gas
94 H 2 O on their measured diameters greater than about 0.1% As of yet, we have not taken into account the possibility that a species other than water may be present and may contain a spectral line within the ISI P20 bandwidth. It appeared from Figure 5.1 that α Ori and α Her and possibly the Miras also had spectral features which were not attributable to H 2 O. Although unlikely, it would be possible for a spectral line from one of these species to be present in our bandpass. We will attempt to put an upper limit on their effect by examining the spectra of the P20 bandpass. The spectra from Figure 5.1 are repeated in Figure 5.6. with a higher magnification and the P20 bandpass explicitly shown. Due to the motion of the Earth, the spectra will be redshifted relative to this observation at other times in the year. The two sets of vertical lines denote the edges of the P20 bandpass as they would appear at the extrema of Earth’s motion relative to the star. The spectra of α Ori appears very flat within the P20 bandpass. The four other stars have fluctuations in their spectra which appear slightly larger than the noise. These “possible absorption lines” have a depth no greater than ∼8% for any of the stars. Since the observed absorption lines in o Cet and R Leo originated in a shell just outside the photosphere at a temperature ∼1600 K, we will calculate the effect that an 8% absorption line will have on a diameter measurement using these values. To reproduce a line depth of 0.92, a 1R ∗ /1600 K gas shell would need an optical depth of about 0.10. In this case, the uniform disk diameter for the star would be reduced by 1.31% on the line, or 0.23% after averaging over the bandpass. The magnitude of the effect is not too dependent on the choice of parameters. If the gas shell is positioned instead at a radius of 1.10R ∗ , with the same temperature, the star then appears 5.5% larger on the spectral line. The maximum radius that a 1600 K gas shell could be at and still produce an 8% absorption line is 1.30R ∗ , although the gas would have to be opaque in order to do this. If the temperature of the gas were increased to 2000 K with the shell at 1R ∗ , the diameter fit would be 1.06% too small, and if the temperature were 1200 K with the same radius, the measurement would be 1.45% too small. These changes would be lowered by a factor of ∼6, or to less than 0.25%, if a single line of width 0.03 cm −1 were present within the ISI bandpass of width 0.17 cm −1 . With a good degree of certainty, based on the spectra and theory, we can safely estimate any measured size change due to the extra opacity of a spectral line in the P20 bandpass to be less than ∼1% for all of the stars considered.
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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
Page 51 and 52: 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: 92 of its three parameters. The con
Page 105 and 106: 96 5.3 Modelling Spectral Features
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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