# 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 |

`'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

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 |

`'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

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`

).