A novel analytic atmospheric T(τ) relation for stellar models
DDraft version January 14, 2021
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A novel analytic atmospheric T ( τ ) relation for stellar models Warrick H. Ball
1, 2 School of Physics and Astronomy, University of Birmingham, Edgbaston, Birmingham B15 2TT, United Kingdom Stellar Astrophysics Centre, Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, DK-8000 Aarhus C,Denmark
ABSTRACTStellar models often use relations between the temperature T and optical depth τ to evaluate thestructure of their optically-thin outer layers. We fit a novel analytic function to the Hopf function q ( τ ) of a radiation-coupled hydrodynamics simulation of near-surface convection with solar parameters byTrampedach et al. (2014). The fit is accurate to within . per cent for the solar simulation and towithin per cent for all the simulations that are not on either the low-temperature or low-gravityedges of the grid of simulations.Standard one-dimensional stellar models require a boundary condition that summarises the transition from theiropaque depths to their observable outer layers. These are usually provided by either (a) tables of surface pressure andtemperature computed using separate codes that model the complicated physics of stellar atmospheres or (b) T ( τ ) relations that specify the stratification of the temperature T as a function of the optical depth τ . These T ( τ ) relationsfollow the general form (cid:18) T ( τ ) T eff (cid:19) = 34 ( τ + q ( τ )) (1)where T eff is the effective temperature and q ( τ ) is a Hopf function . We denote the optical depth at which T = T eff by τ eff . Hopf functions can be derived theoretically given some assumptions (e.g. the Eddington model q ( τ ) = 2 / ) orfit to data or detailed atmosphere models. Two common T ( τ ) relations, besides the Eddington relation, are those ofKrishna Swamy (1966) and analytic fits to Model C by Vernazza et al. (1981), known as VAL-C (see e.g. Paxton et al.2013; Sonoi et al. 2019).In the study of solar-like oscillations, the atmosphere’s structure can affect the mode frequencies appreciably andmust therefore be included in the equilibrium stellar models. This can be done by integrating the equations of theatmosphere’s structure and appending the integrated structure to the interior model. Alternatively, one can modifythe stellar structure equations and extend the model to smaller optical depths. Specifically, one multiplies the radiativetemperature gradient ∇ rad = ( ∂ ln T /∂ ln P ) rad by q/ d τ (see e.g. Mosumgaard et al. 2018). The first methodrequires the Hopf function; the second requires its gradient.Trampedach et al. (2014) presented Hopf functions for a set of three-dimensional radiation-coupled hydrodynamics(3D RHD) simulations of near-surface convection, as well as routines that allow stellar modellers to interpolate the Hopffunctions as a function of the surface gravity log g and effective temperature T eff . The simulated Hopf functions aremost similar to that of VAL-C but do not tend to a constant temperature at small optical depths. While we encouragemodellers to use the routines to interpolate in the full suite of Hopf functions, we present here an analytic functionthat allows relatively simple implementation of the solar Hopf function, which is itself somewhat representative of allthe Hopf functions in the grid of models.The gradient of the Hopf function for the simulation with solar parameters ( T eff = 5775 K , log g = 4 . ) can beapproximated by the function d q d x = c + e x − av e x − bw , (2)where x = log τ . This motivates fitting the Hopf function using the integral of eq. 2 (Wolfram|Alpha 2021), q ( x ) = c + c (cid:16) x − w ln (cid:16) e bw + e xw (cid:17)(cid:17) + v e x − av F (cid:16) , wv ; 1 + wv ; − e x − bw (cid:17) , (3) a r X i v : . [ a s t r o - ph . S R ] J a n where F is the hypergeometric function. We found best-fitting parameters c = 0 . , c = 0 . , a =0 . , b = 0 . , v = 0 . and w = 0 . , with which the fit reproduces the data to within . percent over the full range − . < x < . . The fit is also fairly representative of all the simulations away from the low-temperature and low-gravity edges of the grid. If we exclude simulations with T eff < or log g < T eff / − . ,the fit reproduces all the remaining Hopf functions within per cent.The hypergeometric function in eq. 3 is not always practical but the term that contains it does not contribute to thefunction for x (cid:46) . Ignoring this term is equivalent to ignoring the denominator in eq. 2, in which case the integral is q ( x ) = c + c ( x − b ) + v e x − av . (4)This is also accurate to within . per cent up to x = 0 . for the solar model and to the same accuracy upto τ eff = 0 . . Thus, if integrating an atmosphere using q ( x ) , where one usually terminates at or below τ eff ,the approximate formula in eq. 4 can be used. If including the atmosphere’s structure by modifying the structureequations, then the full equation of the gradient (eq. 2) must be used because we require d q/ d τ → for τ (cid:29) .Fig. 1 shows the Hopf functions we have discussed: the data from Trampedach et al. (2014), our fits of eqs 3 and 4to their solar simulation, and the three widely-used T ( τ ) relations. Fig. 1 and most of the preceding analysis can begenerated by a publicly available Python script (Ball 2021).ACKNOWLEDGMENTSWHB thanks the UK Science and Technology Facilities Council (STFC) for support under grant ST/R0023297/1.Funding for the Stellar Astrophysics Centre is provided by The Danish National Research Foundation (Grant agreementno.: DNRF106). Software:
NumPy (Harris et al. 2020), SciPy (Virtanen et al. 2020), Matplotlib (Hunter 2007)REFERENCES https://github.com/warrickball/atm_rnaas2021 http://matplotlib.org − − − − τ (= x )0 . . . . . . q ( τ ) Trampedach et al. (2014, all)Trampedach et al. (2014, solar)This work (eq. 3)This work (eq. 4)EddingtonKrishna Swamy (1966)VAL-C τ eff Figure 1.
All the Hopf functions q ( x ) discussed, with the corresponding values of τ effeff