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

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65 the apparent size significantly. We conclude that continuum effects are likely to increase the 11 µm apparent diameter of a supergiant such as α Ori relative to near-IR apparent diameters due to increased free-free opacities in the atmosphere, but reasonable density stratification estimates put the increase at about 1%. Additionally, the 11 µm apparent diameter is predicted to be about 2.5% larger than the 1.6 µm continuum monochromatic diameter 4 which is believed to be closely related to the Rosseland diameter since most of the star’s energy is radiated in the near-IR. As discussed in Section 3.4.2, pulsationally driven stellar atmospheres can be expected to be significantly more extended than static atmospheres. Consequently, it is expected that the disparity between diameters measured at different wavelengths would be larger in variable stars. We can estimate the effect of this extension by calculating the 11 µm opacity for the density and temperature distributions modelled in Höfner et al. (1998) [47]. Then, integrating over an LTE source function, we can synthesize an 11 µm intensity profile and visibilities and compare with observations. The density and temperature distributions of the Höfner model and calculated continuum opacities are shown in Figure 3.17 at minimum and maximum phase. model is based on a 7000L ⊙ , 1M ⊙ hydrostatic initial model atmosphere with an effective temperature of 2880 K driven using a piston velocity of 4 km/s. Density and temperature were estimated from Figure 5 of Höfner et al. (1998) [47]. For the minimum phase model, and the 1.6 µm opacity calculations, it was necessary to extrapolate the given density and temperature functions to slightly lower radii. So, for these cases, there may be some errors. Opacity was calculated from the density and temperature assuming LTE. The monochromatic diameter resulting from the Höfner model is 22.08 mas at maximum and 20.06 mas at minimum at 11.15 µm and 19.48 mas at maximum and 19.42 mas at minimum at 1.6 µm. The intensity profiles resulting from this model are plotted in Figure 3.18. The presence of a shock front close to the photosphere at maximum causes the apparent size of the star to be larger. At 11.15 µm, the star appears 18.5% larger at maximum than at minimum. At 1.6 µm, this change is only 7.5% 5 . In addition, the density fall off is not as steep at maximum. Thus, the 11 µm apparent diameter is more enlarged (as compared to the 1.6 µm apparent diameter) than at minimum. At maximum, the 11 µm apparent size is 4 Once again, the monochromatic diameter refers to the radius at which the optical depth at a given wavelength equals unity. The apparent diameter refers to the width of the intensity distribution at a given wavelength. The
Figure 3.17: Density and Temperature from Höfner et al. (1998) [47] and Calculated 11 µm LTE Opacity at Maximum (solid line) and Minimum (dashed line) Phase 66
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The Size, Structure, and Variabilit
Page 3 and 4:
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:
Page 23 and 24: Figure 1.6: Stellar Evolution of a
Page 25 and 26: 16 such as Mira variables, become u
Page 27 and 28: 18 wavelengths, however, its range
Page 29 and 30: 20 Figure 2.1: ISI Data for o Cet f
Page 31 and 32: 22 Figure 2.3: Uniform Disk, Limb D
Page 33 and 34: 24 2.2 Comparison of ISI Data with
Page 35 and 36: 26 Table 2.3: Diameter Measurements
Page 37 and 38: 28 measurements. χ Cyg and R Leo a
Page 39 and 40: 30 metric flux (from the reference
Page 41 and 42: 32 their surface. Very often, mater
Page 43 and 44: 34 Mira. Finally, variations in tim
Page 45 and 46: 36
Page 47 and 48: 38 Figure 3.2: Intensity Distributi
Page 49 and 50: 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: 64 Figure 3.16: Normalized Syntheti
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 and 104: 94 H 2 O on their measured diameter
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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