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

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3.4.1 Static Radiative Transfer Implied Corrections - Limb Darkening
A spherically symmetric atmosphere characterized by a balance of radiation pressure
and hydrostatic forces can be modelled through consideration of the photon-matter
interactions. Essentially, a stable solution would consist of knowing the state of the gas as
a function of radius (which may or may not be in thermodynamic equilibrium), and the
full wavelength dependent radiation field. For a steady state to occur, the radiation given
off by the stellar interior must interact with each shell of atmosphere (through absorption,
emission, or scattering by its constituents) in such a way as not to change its state (including
kinetic temperature, excitational levels, and degrees of molecular dissociation and
ionization.) Such a model involves detailed understanding of the microphysics, including a
complete list of spectral lines for all species involved. Modern computing, using millions of
atomic and molecular transitions, has refined the process a great deal.
Theoretically, the model-calculated emergent intensity distribution could be inverted
to a visibility profile and compared directly to interferometric observations. In practice,
a model will generally be fit with a uniform disk function, and this quantity, as a
function of wavelength, will be compared with observations. A static model atmosphere
has obvious limitations, but should reproduce observations for non-variable supergiants
well. A semi-regular variable, such as α Ori or α Her, may exhibit some phenomenon not
included in a static model, and the applicability of these results to Mira variables is even
more questionable.
The profiles of late-type giant stars were calculated using a plane-parallel approximation
in the visible by Manduca et al. (1977) [66] and extended to the infrared by
Manduca (1978) [65]. The model atmospheres were calculated as a function of effective
temperature, metallicity, surface gravity, and wavelength between ∼400 nm and 5 µm. The
lowest effective temperature model was 3750 K with a surface gravity of 31.6 cm/s 2 . By
comparison, a 15M ⊙ star at 131 pc with a diameter of 50 mas (similar to α Ori) would
have a smaller surface gravity of 0.83 cm/s 2 while a 1M ⊙ star at 117 pc with a diameter of
39 mas (similar to α Her) would have a surface gravity of 0.11 cm/s 2 . The Manduca model
atmosphere produces limb darkening which reduces the observable diameter monotonically
with decreasing wavelength. At 700 nm, the predicted apparent size of the star is reduced
by 9.50%, whereas, at 5 µm, it is reduced by only 2.09%. The basic cause of the reduction
in limb darkening in the infrared stems from the relative insensitivity of intensity on