T(tau) Atmospheres

When the atm_option control is set to 'T_tau', MESA defines the temperature structure of the atmosphere using a \(T(\tau)\) relation. The atm_T_tau_relation control determines the choice of relation, as follows:

The atm_T_tau_relation control

Value	Meaning
'Eddington'	Use the Eddington grey relation \(T^{4}(\tau) = 3/4 T_{\rm eff}^{4} (\tau + 2/3)\).
'solar_Hopf'	Use an approximate Hopf function calibrated against the Sun (see section A.5 of Paxton et al. 2013, MESA II). This is equivalent to the fit given by Sonoi et al. (2019, A&A, 621, 84) to the Vernazza et al. (1981) VAL-C model.
'Krishna_Swamy'	Use the approximate Hopf function given by Krishna-Swamy (1966, ApJ, 145, 174).

For the selected \(T(\tau)\) relation, the pressure structure of the atmosphere is obtained by integrating the equation of hydrostatic equilibrium

\[\frac{{\rm d}P}{{\rm d}\tau} = \frac{g}{\kappa}\]

under the assumption that the gravity \(g\) is spatially constant. The atm_T_tau_opacity control determines how the opacity \(\kappa\) is evaluated, as follows:

The atm_T_tau_opacity control

Value	Meaning
'varying'	At each optical depth, evaluate the opacity consistent with the local thermodynamic state (typically, density and temperature); this is done via calls to MESA’s equation-of-state and opacity modules. The equation of hydrostatic equilibrium is integrated numerically. Controls that affect this integration include atm_T_tau_errtol (sets the error tolerance) and atm_T_tau_max_steps (sets the maximum number of steps).
'fixed'	Assume the same opacity at each optical depth (i.e. uniform opacity), equal to the the opacity in the outermost cell of the interior model. The equation of hydrostatic equilibrium is integrated analytically.
'iterated'	Assume the same opacity at each optical depth (again, uniform opacity), but use iteration to find an opacity consistent with the thermodynamic state at the base of the atmosphere. The equation of hydrostatic equilibrium is integrated analytically. Controls that affect the opacity iteration include atm_T_tau_errtol (sets the error tolerance) and atm_T_tau_max_iters (sets the maximum number of iterations).

Once the temperature and pressure structure of the atmosphere is known, MESA evaluates \(T(\tau)\) and \(P(\tau)\) at \(\tau=\tau_{\rm surf}\), providing the necessary \(T_{\rm surf}\) and \(P_{\rm surf}\) values for the boundary conditions.

The nominal surface optical depth \(\tau_{\rm surf}\) is calculated via

\[\tau_{\rm surf} = \tau_{\rm base} \times {\tt tau\_factor}\]

Here, \(\tau_{\rm base}\) is the optical depth where \(T(\tau) = T_{\rm eff}\) (the stellar effective temperature) for the chosen \(T(\tau)\) relation, while tau_factor is a parameter determined from various controls in the &star_job namelist (e.g., set_tau_factor).