skip to main content


Title: Skye: A Differentiable Equation of State
Abstract

Stellar evolution and numerical hydrodynamics simulations depend critically on access to fast, accurate, thermodynamically consistent equations of state. We present Skye, a new equation of state for fully ionized matter. Skye includes the effects of positrons, relativity, electron degeneracy, Coulomb interactions, nonlinear mixing effects, and quantum corrections. Skye determines the point of Coulomb crystallization in a self-consistent manner, accounting for mixing and composition effects automatically. A defining feature of this equation of state is that it uses analytic free energy terms and provides thermodynamic quantities using automatic differentiation machinery. Because of this, Skye is easily extended to include new effects by simply writing new terms in the free energy. We also introduce a novelthermodynamic extrapolationscheme for extending analytic fits to the free energy beyond the range of the fitting data while preserving desirable properties like positive entropy and sound speed. We demonstrate Skye in action in theMESAstellar evolution software instrument by computing white dwarf cooling curves.

 
more » « less
NSF-PAR ID:
10362581
Author(s) / Creator(s):
; ; ; ;
Publisher / Repository:
DOI PREFIX: 10.3847
Date Published:
Journal Name:
The Astrophysical Journal
Volume:
913
Issue:
1
ISSN:
0004-637X
Format(s):
Medium: X Size: Article No. 72
Size(s):
["Article No. 72"]
Sponsoring Org:
National Science Foundation
More Like this
  1. Abstract

    We update the capabilities of the open-knowledge software instrument Modules for Experiments in Stellar Astrophysics (MESA). The newauto_diffmodule implements automatic differentiation inMESA, an enabling capability that alleviates the need for hard-coded analytic expressions or finite-difference approximations. We significantly enhance the treatment of the growth and decay of convection inMESAwith a new model for time-dependent convection, which is particularly important during late-stage nuclear burning in massive stars and electron-degenerate ignition events. We strengthenMESA’s implementation of the equation of state, and we quantify continued improvements to energy accounting and solver accuracy through a discussion of different energy equation features and enhancements. To improve the modeling of stars inMESA, we describe key updates to the treatment of stellar atmospheres, molecular opacities, Compton opacities, conductive opacities, element diffusion coefficients, and nuclear reaction rates. We introduce treatments of starspots, an important consideration for low-mass stars, and modifications for superadiabatic convection in radiation-dominated regions. We describe new approaches for increasing the efficiency of calculating monochromatic opacities and radiative levitation, and for increasing the efficiency of evolving the late stages of massive stars with a new operator-split nuclear burning mode. We close by discussing major updates toMESA’s software infrastructure that enhance source code development and community engagement.

     
    more » « less
  2. Abstract

    We explore the three-dimensional properties of convective, luminous (L≈ 104.5–105L), hydrogen-rich envelopes of red supergiants (RSGs) based on radiation hydrodynamic simulations in spherical geometry usingAthena++. These computations comprise ≈30% of the stellar volume, include gas and radiation pressure, and self-consistently track the gravitational potential for the outer ≈3Mof the simulatedM≈ 15Mstars. This work reveals a radius,Rcorr, around which the nature of the convection changes. Forr>Rcorr, though still optically thick, diffusion of photons dominates the energy transport. Such a regime is well studied in less luminous stars, but in RSGs, the near- (or above-)Eddington luminosity (due to opacity enhancements at ionization transitions) leads to the unusual outcome of denser regions moving outward rather than inward. This region of the star also has a large amount of turbulent pressure, yielding a density structure much more extended than 1D stellar evolution predicts. This “halo” of material will impact predictions for both shock breakout and early lightcurves of Type IIP supernovae. Inside ofRcorr, we find a nearly flat entropy profile as expected in the efficient regime of mixing-length theory (MLT). Radiation pressure provides ≈1/3 of the support against gravity in this region. Our comparisons to MLT suggest a mixing length ofα= 3–4, consistent with the sizes of convective plumes seen in the simulations. The temporal variability of these 3D models is mostly on the timescale of the convective plume lifetimes (≈300 days), with amplitudes consistent with those observed photometrically.

     
    more » « less
  3. ABSTRACT

    We recently derived, using the density-of-states approximation, analytic distribution functions for the outcomes of direct single-binary scatterings. Using these outcome distribution functions, we present in this paper a self-consistent statistical mechanics-based analytic model obtained using the Fokker–Planck limit of the Boltzmann equation. Our model quantifies the dominant gravitational physics, combining both strong and weak single–binary interactions, which drives the time evolution of binary orbital parameter distributions in dense stellar environments. We focus in particular the distributions of binary orbital energies and eccentricities. We find a novel steady-state distribution of binary eccentricities, featuring strong depletions of both the highest and the lowest eccentricity binaries. In energy space, we compare the predictions of our analytic model to the results of numerical N-body simulations, and find that the agreement is good for the initial conditions considered here. This work is a first step towards the development of a fully self-consistent semi-analytic model for dynamically evolving binary star populations in dense stellar environments due to direct few-body interactions.

     
    more » « less
  4. Abstract

    Chemically homogeneous evolution (CHE) is a promising channel for forming massive binary black holes. The enigmatic, massive Wolf–Rayet binary HD 5980 A&B has been proposed to have formed through this channel. We investigate this claim by comparing its observed parameters with CHE models. UsingMESA, we simulate grids of close massive binaries, then use a Bayesian approach to compare them with the stars’ observed orbital period, masses, luminosities, and hydrogen surface abundances. The most probable models, given the observational data, have initial periods ∼3 days, widening to the present-day ∼20 days orbit as a result of mass loss—correspondingly, they have very high initial stellar masses (≳150M). We explore variations in stellar-wind mass loss and internal mixing efficiency, and find that models assuming enhanced mass loss are greatly favored to explain HD 5980, while enhanced mixing is only slightly favored over our fiducial assumptions. Our most probable models slightly underpredict the hydrogen surface abundances. Regardless of its prior history, this system is a likely binary black hole progenitor. We model its further evolution under our fiducial and enhanced wind assumptions, finding that both stars produce black holes with masses ∼19–37M. The projected final orbit is too wide to merge within a Hubble time through gravitational waves alone. However, the system is thought to be part of a 2+2 hierarchical multiple. We speculate that secular effects with the (possible) third and fourth companions may drive the system to promptly become a gravitational-wave source.

     
    more » « less
  5. Abstract

    Three control problems for the system of two coupled differential equations governing the dynamics of an energy harvesting model are studied. The system consists of the equation of an Euler–Bernoulli beam model and the equation representing the Kirchhoff's electric circuit law. Both equations contain coupling terms representing the inverse and direct piezoelectric effects. The system is reformulated as a single evolution equation in the state space of 3‐component functions. The control is introduced as a separable forcing term on the right‐hand side of the operator equation. The first control problem deals with an explicit construction of that steers an initial state to zero on a time interval [0,T]. The second control problem deals with the construction of such that the voltage output is equal to some given function (with being given as well). The third control problem deals with an explicit construction of both the force profile, , and the control, , which generate the desired voltage output . Interpolation theory in the Hardy space of analytic functions is used in the solution of the second and third problems.

     
    more » « less