ner_tags: O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O
tokens: but the authors note that the resulting parameters A and B are not particularly sensitive to this, provided that the function can adequately reproduce the variation of s ad . This function comprises six parameters that define the relationship of s ad across the – plane, and the least squares minimization algorithm of Markwardt ( 2009 ) was then used to determine their values (listed in Table 1 ). The process is repeated for different convective envelope compositions (see Figure 3 ), each of which require a unique projection of the – plane. This fitting process effectively reduces the dimensionality of the initial variation of s ad by projecting the – plane onto an axis that is aligned with the convective envelope adiabats. The process used in this work is only one possible method for reducing the dimensionality of the problem, and further study using other statistical tools, such as principle component analysis, may yield additional insights into the fundamental relationship between convection zone entropy and the stellar surface parameters. Figure 3. Adiabatic entropy from sets of 3D simulations with varied chemical compositions (similar to the lowermost panel in Figure 2 ) presented along projections in the – plane. Each metallicity requires different coefficients for A and B , which correspond to different adiabatic contours in the – plane. Table 1 Parameters for the Function Fit to s ad from Magic et al. Simulations in the Projected – Plane () Since the convection zone adiabatic entropy value in a stellar model is determined by the mixing length parameter, the curves in Figure 3 basically show how we need to change α as a function of and . Of course, the first step would be to determine which numerical value of α yields a particular s ad given the rest of the physics to set the mixing length scale, and thus determine how s ad changes with α . After setting the relationship between s ad and α , all that is required is to follow this relationship (i.e., the curve in Figure 3 ) as the star evolves. Since each time step in a stellar evolution calculation changes and T eff , we will need to keep changing α as we evolve a model. The two panels of Figure 4 outline, in principle, the steps that must be taken to translate the adiabatic specific entropy derived from a 3D simulation into the corresponding entropy calibrated value of α to be used in the 1D MLT stellar model. Figure 4 (left) shows, in the same entropy calibration plane as Figure 3 , the locus of a set of 3D simulations, all with the same chemical composition, but with different values of s ad in the deep part. Also in the left panel are three 1D MLT models, all with the same metallicity and surface conditions as the 3D simulations. Presented relative to the projected and coordinates in this way, the evolution tracks begin on the zero-age main sequence with a relatively low s ad , which increases as the model approaches the terminal age. For the purpose of demonstrating our calibration