The term is the “PdV”-work, which is responsible for adiabatic heating and cooling when gas is compressed or decompressed. This is more clearly brought out by the Lagrangian form of the energy equation,
The radiative heating or cooling may be written as the negative divergence of the radiative flux,
The radiation intensity may be obtained along straight lines of sight by solving the radiative transfer equationoptical depth over the geometric interval , given the monochromatic absorption coefficient . Combining Equations 25–28 one also finds that
Because of the interplay of the and factors, the effect is largest in a thin layer near optical depth unity; at larger depths the difference tends to zero exponentially with optical depth, while at smaller depths the factor diminishes the effect as well.
The discussion above applies frequency by frequency, and because of variations in the opacity with frequency, in particular across spectral lines, the net heating or cooling can only be found by evaluating the integral over frequency that occurs in Equation 29. An approximate way to estimate that integral using a binning method was developed by Nordlund (1982) and applied in subsequent works by Stein, Nordlund and collaborators (e.g., Stein and Nordlund, 1989, 1998; Asplund et al., 2000b; Carlsson et al., 2004). The method has been tested and used by Ludwig and co-workers (Ludwig et al., 1994; Steffen et al., 1995) and by Vögler and co-workers (Vögler et al., 2004; Vögler, 2004; Vögler et al., 2005). Improvements are being developed by Trampedach and Asplund (2003).
Whenever the constituent atoms or molecules of a gas undergo ionization or dissociation, extra energy is required to heat or cool the gas; i.e., the internal energy varies more with temperature than for an ideal gas. Near the surface of the Sun, both ionization and dissociation are important; ionization of hydrogen and helium influence the equation of state greatly just below the surface, and dissociation of hydrogen molecules is important in the cooler layers of the photosphere.
In general then, one needs to know the relationset al., 1975; Mihalas et al., 1988; Däppen et al., 1988; Mihalas et al., 1990; Rogers, 1990; Rogers et al., 1996; Rogers and Iglesias, 1998; Rogers and Nayfonov, 2002; Trampedach, 2004b,a; Trampedach et al., 2006). Since computing these relations from first principles in general is quite expensive the best approach for practical computational work is to tabulate and interpolate in these relations.
By a few algebraic manipulations of the equations of motion and the energy equation one can show that
At the solar surface there is a very rapid transition between primarily convective and primarily radiative energy transport. In the atmosphere it is more relevant to consider changes of the radiative flux, through Equation 29.
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