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

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73 and their visible counterparts. The ISI measured diameter for α Ori (at 11.15 µm) is about 54 mas. This large (∼30%) increase in apparent size is not accounted for by the static, spherically symmetric, continuum models considered up to this point. Although there are many fewer observations of α Her, the same roughly 30% increase in apparent size is seen between the near-IR and 11 µm. The simplest explanation for the lower near-IR sizes might be spectral line contamination of intermediate optical depth producing very extremely limb-darkened profiles and smaller fit uniform disk diameters. The relatively wide bandwidths used in the near-IR size measurements often make it impossible not to average in some spectral lines (as opposed to the narrow ISI bandwidths and availability of 11 µm spectra. See Chapter 5.1.) However, to reduce the apparent size by 30%, the molecular gas would need to be very close to the photosphere and very cool. A 3250 K star would need a gas shell with temperature less than 1050 K just beyond the photosphere in order to decrease the 2.12 µm apparent diameter by 30%, or an even cooler gas shell farther out (based on estimate described in Section 5.2). Theoretically, a shell of gas or dust at any radius is capable of lowering the apparent size indefinitely if the opacity is high enough and the temperature sufficiently low. However, the cool temperatures would be evident in the fluxes at these wavelengths. α Ori has a flux at 2.5 µm of about 20,000 Jy. Assuming a diameter of 42 mas, this implies a temperature of the emitting region of 3516 K. We can conclude that a reduction of the apparent diameter by 30%, solely due to limb darkening, is not consistent with observations. A more plausible explanation might be the presence of surface non-uniformities. Stellar hotspots, possibly the result of giant convective cells within the star, have been predicted and observed to exist on α Ori and α Her. Schwarzschild (1975) [91] argues that convective granules, observed on the solar surface to have a typical width around 3×10 −3 R ⊙ , should exist on the surface of red giants and supergiants having a much larger width. For the largest stars, the convective granules might have a typical width on the order of the stellar radii. Variations in surface temperature due to these granules are predicted to be on the order of 1000 K. Asymmetric structures on or near the surface of α Ori and α Her were observed in the visible by Tuthill et al. (1997) [103] and the possibility of these “hotspots” originating as convective cells was deemed likely. The effect a hotspot would have on measured visibilities can be estimated by considering a stellar blackbody of temperature 2500 K having a single large hotspot with a blackbody temperature of 3500 K. The hotspot was chosen to have a width of 0.6R ∗ , and
74 Figure 3.21: 2.1 µm Image of 2500 K Star with 3500 K Hotspot of Width 0.6R ∗ for generality, was offset from the line of sight by 30 ◦ . The image of this star at 2.1 µm is shown in Figure 3.21. In the 11 µm image of the same star, the hotspot would be much less pronounced due to the more linear relationship between temperature and intensity at 11 µm (in this temperature range.) The visibility calculated from the 2.1 µm image is plotted against the magnitude of the spatial frequency in Figure 3.22. For reference, the visibility corresponding to the uniform stellar disk without the hotspot is plotted also. The image with the hotspot is asymmetric and the visibility is a function of both spatial frequency magnitude and position angle. The assorted points are plotted to show the range of values the visibility would take for various position angles. The uniform disk is symmetric and can be represented by a single curve. The visibility of the star with hotspot is very similar to that of a uniform disk with a 15% smaller radius for spatial frequencies within the first null. Thus, the presence of hotspots on a stellar disk, which may be present from large convective cells, lowers the apparent size of the star. However, in order for hotspots to account for the observed difference in apparent size between the near-IR and 11 µm, they would need to be even more extreme than is predicted by Schwarzschild for convective cells. The collection of measurements of the diameters of o Cet and R Leo from Tables 2.3 and 2.4 show similar, but much larger variations with wavelength as do those for α Ori.
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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
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 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: 72 3.5 Explanation of Variations in
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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