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This figure shows the relative error in the derivative :math:`\partial \kappa / \partial T` (:math:`X` = 0.625, :math:`Z` = 0.015),
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This figure shows the logarithmic relative error in the derivative :math:`\partial \kappa / \partial T` (:math:`X` = 0.625, :math:`Z` = 0.015),
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for an OPAL opacity table grid using Grevesse & Sauval (1998) abundances, generated from MESA’s kap module, using cubic interpolation.
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The OPLIB log(:math:`R`) = −8, 1.5 table boundaries are marked with a solid black line and the OPAL/OP log(:math:`R`) = 1.0 boundary is shown with a dashed line.
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The approximate location of the Z-dependent transition to an electron conduction dominated opacity is marked with dot-dash blue curve. Regions for Atomic, molecular,
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and compton scattering opacity are labeled and presented with their associated blending regions.
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While the opacity derivatives do not directly appear in the canonical equations of stellar structure, they do appear in the Jacobian matrix for the MESA implicit solver.
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The opacity derivatives are used to implicitly solve the stellar structure equations and reach a converged solution. Numerically unstable opacity derivatives can
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halt the progress of the solver and ultimately crash a calculation.
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While the opacity derivatives do not directly appear in the canonical equations of stellar structure, they do appear in the Jacobian matrix for MESA's implicit solver.
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Numerically unstable opacity derivatives can halt the progress of the solver and ultimately crash a calculation.
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To improve the numerical stability of MESA's cubic opacity interpolation routines, we have implemented
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automatic differentiation into the opacity interpolating functions. Now, when using cubic interpolation, the opacity derivatives for an arbitrary mixture
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in the :math:`X–Z` plane are computed by taking the derivative of the interpolating function as opposed to the interpolant of the derivatives. This improvement
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has resulted in a significant reduction in the relative derivative error and results in an increase in the numerical accuracy of opacity derivatives computed with cubic interpolation.
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has lead to a significant reduction in the relative derivative error and an increase in the numerical accuracy of opacity derivatives computed with cubic interpolation.
@@ -114,7 +113,7 @@ option (shown below), while also providing more accurate opacity physics between
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For this MESA release, linear interpolation remains the default method for interpolating in composition between opacity tables
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while we continue to investigate the residual areas where cubic interpolation appears to occasionally produce lower quality derivatives.
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However, we recommend users experiment with cubic interpolation, as it has been shown to consistently increase the overall
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However, adopting cubic interpolation has been shown to consistently increase the overall
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opacity of a model, and can directly effect the structure of solar models, see Appendix B & C in Farag et al. 2024.
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We anticipate making cubic interpolation the default in a future MESA release version.
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We encourage users to experiment with these different opacity interpolation routines and be mindful of the effect they can have on their stellar models.
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