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:
Value 
Meaning 

Use the Eddington grey relation \(T^{4}(\tau) = 3/4 T_{\rm eff}^{4} (\tau + 2/3)\). 

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) VALC model. 

Use the approximate Hopf function given by KrishnaSwamy (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
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:
Value 
Meaning 

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 equationofstate and opacity
modules. The equation of hydrostatic equilibrium is integrated
numerically. Controls that affect this integration include


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. 

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 
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
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
).