Model Atmospheres

The stellar atmosphere consists of those layers of the star where the radiation that is transmitted directly to the observer originates. Thus in order to interpret stellar spectra, one needs to be able to compute the structure of the atmosphere and the emerging radiation.

In actual stars there are many factors, such as rotation and magnetic fields, that complicate the problem of computing the structure of the atmosphere. We shall only consider the classical problem of finding the structure, i. e. the distribution of pressure and temperature with depth, in a static, unmagnetized atmosphere. In that case a model atmosphere is completely specified by giving the chemical composition, the gravitational acceleration at the surface, g, and the energy flux from the stellar interior, or equivalently, the effective temperature Te.

The basic principles involved in computing a model stellar atmosphere are the same as for stellar interiors and will be discussed in Chap. 10. Essentially, there are two differential equations to be solved: the equation of hydrostatic equilibrium, which fixes the distribution of pressure, and an equation of energy transport, which will have a different form depending on whether the atmosphere is radiative or convective, and which determines the temperature distribution.

The values of the various physical quantities in an atmosphere are usually given as functions of some suitably defined continuum optical depth t . Thus pressure, temperature, density, ionization and the population numbers of various energy levels can all be obtained as functions of t . When these are known, the intensity of radiation emerging from the atmosphere can be computed. In *The Intensity Emerging from a Stellar Atmosphere (p. 218), it is shown that approximately the emergent spectrum originates at unit optical depth, measured along each light ray. On this basis, one can predict whether a given spectral line will be present in the spectrum.

Consider a spectral line formed when an atom (or ion) in a given energy state absorbs a photon. From the model atmosphere, the occupation number of the absorbing level is known as a function of the (continuum) optical depth t . If now there is a layer above the depth t = 1 where the absorbing level has a high occupancy, the optical depth in the line will become unity before t = 1, i.e. the radiation in the line will originate higher in the atmosphere. Because the temperature increases inwards, the intensity in the line will correspond to a lower temperature, and the line will appear dark. On the other hand, if the absorbing level is unoccupied, the optical depth at the line frequency will be the same as the continuum optical depth. The radiation at the line frequency will then come from the same depth as the adjacent continuum, and no absorption line will be formed.

The expression for the intensity derived in *The Intensity Emerging from a Stellar Atmosphere (p. 218) also explains the phenomenon of limb darkening seen in the Sun (Sect. 12.2). The radiation that reaches us from near the edge of the solar disc emerges at a very oblique angle (0 near 90°), i. e. cos 0 is small. Thus this radiation originates at small values of t, and hence at low temperatures. In consequence, the intensity coming from near the edge will be lower, and the solar disc will appear darker towards the limb. The amount of limb darkening also gives an empirical way of determining the temperature distribution in the solar atmosphere.

Our presentation of stellar atmospheres has been highly simplified. In practice, the spectrum is computed numerically for a range of parameter values. The values of Te and element abundances for various stars can then be found by comparing the observed line strengths and other spectral features with the theoretical ones. We shall not go into details on the procedures used.

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