Solar calibration of $\alpha_{\rm MLT}$ using {\AE}SOPUS opacities in MESA

The simplistic but ubiquitous Mixing Length Theory (MLT) formalism is used to model convective energy transport within 1D stellar evolution calculations. The formalism relies on the free parameter $\alpha_{\rm MLT}$, which must be independently calibrated within each stellar evolution program and for any given set of physical assumptions. We present a solar calibration of $\alpha_{\text{MLT}}$ appropriate for use with the AESOPUS opacities, which have recently been made available for use with the MESA stellar evolution software. We report a calibrated value of $\alpha_{\rm MLT}=1.931$ and demonstrate the impact of using an appropriately calibrated value in simulations of a $3 M_{\odot}$ asymptotic giant branch star.


INTRODUCTION
The Mixing Length Theory of convection (MLT) was first applied to stellar interiors by Erika Boehm-Vitense (Vitense 1953) and remains one of few we have for approximating convection in 1D stellar models on evolutionary timescales (along with, e.g., the Full Spectrum of Turbulence model of Canuto & Mazzitelli 1991;Canuto et al. 1996). The MLT formalism contains a free parameter, α MLT , that can be thought of as characterizing the "efficiency of convection," or the "mean-free path" of a fluid parcel by analogy with molecular heat transfer (Prandtl 1925). As α MLT does not have a physical meaning in 3D fluid dynamics, it must be calibrated empirically. Most often, and for obvious reasons, this calibration is performed to the Sun (though exceptions include Joyce & Chaboyer 2018a,b). However, as α MLT may absorb modeling inconsistencies or inherit other artifacts from the environment in which it is used, α MLT must be calibrated anew for each stellar evolution code and for each choice of input physics-especially when the physical assumptions heavily impact the outer layers of stars: regions where the temperature gradient becomes superadiabatic (see, e.g., Jermyn et al. 2022 for detailed discussion). Different choices of opacities, for example, will therefore require different solar mixing length calibrations.

METHODS
This note is a tangential result from the work of Cinquegrana, Joyce and Karakas (2022, in prep), wherein MESA is used to model high-(i.e. super-solar) metallicity intermediate mass and massive stars. We discuss here the relevant physics for this science case as well as the variations applied for the current demonstration.

Software tools
The AESOPUS: Accurate Equation of State and OPacity Utility Software (Marigo & Aringer 2009) tool allows users to create low-temperature Rosseland mean opacity tables customized to the situation they are modelling by, e.g., choosing different CNO abundances or alpha-element enhancement levels. The necessity of using low-temperature opacity tables that can follow composition changes along the evolution of a star, rather than relying on the star's initial metal content, has been reiterated by numerous groups over the past two decades (e.g., Marigo 2002;Cristallo et al. 2007;Ventura & Marigo 2009;Constantino et al. 2014;Fishlock et al. 2014). The importance of this feature is especially apparent when modeling thermally pulsing asymptotic giant branch (AGB) stars, which undergo various processes such as third dredge up and hot bottom burning that significantly change the chemical composition of the envelope from its initial chemical makeup (for further details on AGB stars, see: Busso et al. 1999;Herwig 2005;Nomoto et al. 2013;Karakas & Lattanzio 2014;Karakas et al. 2022;Ventura et al. 2022). Such opacity tables have also proved useful in modelling R Coronae Borealis stars, which have envelopes deficient of hydrogen but enhanced in CNO elements (Schwab 2019). The impacts of using appropriate opacities in Figure 1. Two 3M , solar metallicity models are compared: one with αMLT=1.931 (continuous line) and one with αMLT = 2.0 (dashed line). The left panel shows the stellar tracks of the two models. The right panels shows the variance of the helium burning luminosity and radius as the models approach the thermally pulsing AGB.
these situations further extend to stellar yield calculations as well as predictions for envelope expansion, mass loss rates and stellar lifetimes.
The AESOPUS tool has been available since 2009, however the integration of AESOPUS with the Modules for Experiments in Stellar Astrophysics (MESA; Paxton et al. 2010Paxton et al. , 2013Paxton et al. , 2015Paxton et al. , 2018Paxton et al. , 2019 software is recent (MESA Instrument Paper VI, in prep). Schwab (2019) was the first to publish science utilising AESOPUS within MESA, however, there is currently no commentary on the appropriate mixing length to use with these opacities published in the literature.

Physical configuration
We perform a solar calibration of α MLT using MESA version 15140 (Paxton et al. 2019). Opacities are scaled according to the abundances of Lodders (2003). We determine the best-fitting mixing length by comparing a number of solar models, each adopting a different mixing length and initial helium abundance, Y initial . All models are assigned Z initial = 0.0133 per the solar metallicity determination of Lodders (2003) for consistency with the AESOPUS opacity assumptions.
Initial helium abundances from 0.2377 to 0.28 are considered (for discussion of varying Y initial as part of mixing length calibrations, see Joyce & Chaboyer (2018a), Trampedach et al. (2014), and references therein). The base metal content for the opacity tables, Zbase, is set equal to Z initial . OPAL data are used for high temperature opacities (Iglesias & Rogers 1996); low temperature opacities are built using the AESOPUS tool (Marigo & Aringer 2009). 1 All models used for the solar calibration have 1M and terminate at the solar age, but are otherwise identical to the physical prescription described above and adopted in Cinquegrana, Joyce & Karakas (2022, in prep). Using the MLT prescription of Henyey et al. (1965), values are iterated until some combination of Y initial and α MLT reproduces the solar radius and solar luminosity at the solar age to approximately 1 part in 10 4 .

RESULTS
1 The initial conditions used to create our tables can be found in the appendix § A. Instructions on how to prepare the tables for use within MESA can be found within the directory: mesa-r15140/kap/preprocessor/AESOPUS/README For each composition assumption, we run a suite of models adopting α MLT values between 1.3 and 2.2. We extract the luminosity and radius at the solar age, t . We then determine which choice of α MLT produces the best reproduction of solar parameters according to a penalty statistic that prioritizes agreement with the solar radius and solar luminosity equally. The precision on α MLT is increased until further refinement no longer reduces the residuals in luminosity and radius. We find that a combination of Y initial = 0.241 and α MLT = 1.931 performs best.

IMPLICATIONS
Composition-dependent opacity tables, such as those provided by AESOPUS, are crucial when modelling AGB stars. The reliability of AGB stellar models therefore depends on using a value of α MLT that is appropriately calibrated for use with those opacities. In particular, the mixing length is known to have a large impact on derived quantities such as the effective temperature and radius (Karakas & Lattanzio 2014;Joyce et al. 2020), the efficiency of the third dredge up (Boothroyd & Sackmann 1988), and the strength of hot bottom burning (Ventura & D'Antona 2005). Using an inappropriate α MLT can therefore lead to inaccurate predictions for fundamental stellar parameters. This is demonstrated in Fig. 1, which compares the evolutionary properties of two 3M (Z =Z ) models: one with α MLT = 1.931 and one with MESA's default value of α MLT = 2.0. In the left panel, we note little difference in their tracks along the main sequence, though variance in luminosity and temperature is apparent once the models ascend their giant branches. However, when evolved until the hydrogen envelope is less than 20% of the total stellar mass, (e.g., the post-AGB EEP as defined in Dotter (2016)), we note that the α MLT = 2.0 model has experienced 35 thermal pulses by this point, compared to the 36 thermal pulses of the model with α MLT = 1.931. As the models evolve along the thermally pulsating AGB, we also see modest variation in radius, effective temperature and the magnitude of the helium burning luminosity.
More striking, however, is the impact of decreasing α MLT on the evolutionary timescales. The model that assumes a calibrated α MLT value produces an earlier second dredge up event and shows an earlier onset of thermal pulses. Changes in timescale carry implications for stellar lifetimes and thus for nucleosynthesis and chemical evolution timescales.
In light of the clear physical changes imparted to the models by a seemingly modest absolute change in α MLT , we provide a working "best practices" value of α MLT = 1.931 for use with AESOPUS low-temperature opacities. The AESOPUS software is available online at http://stev.oapd.inaf.it/cgi-bin/aesopus. The tables used in this work were computed using the following options.

A.1. Temperature and density grid
We set logT from 3.2 to 4.5. Increments of 0.01 dex are used from 3.2 to 3.7; 0.05 dex is used above this. Similarly, logR extends from −7 to 1 in increments of 0.5 dex.

A.2. Chemical composition
We use the reference solar composition of Lodders (2003). We solve for reference metallicities of 0.014 to 0.10, given the high metallicities required for our science case. We set hydrogen abundance, X, from 0.4 to 0.8 in steps of 0.1. We do not change the abundance normalization and primordial mixture options. For the reference mixture, we choose scaled-solar, but leave the associated table uncompleted. We do not define any additional enhancement or depletion factors in the superimposed chemical pattern table.
The last option is for CNO abundance variation factors, where fc and fn are scaling factors applied to the reference abundance (see Marigo & Aringer (2009);Fishlock et al. (2014) for more detailed discussion). We note that the following values depend heavily on the mass, metallicity and evolutionary stage under consideration. In our case, none