Combining Computer Models to Account for Mass Loss in Stellar Evolution Nathan M. Stein1, David A. van Dyk2∗, Ted von Hippel3, Steven DeGennaro4, Elizabeth J. Jeffery5 and William H. Jefferys6 1Statistics Department, Harvard University, Cambridge, MA 02138-2901, USA 2Statistics Section, Department of Mathematics, Imperial College London, SW7 2AZ, UK 3Physical Sciences Department, Embry-Riddle Aeronautical University, Daytona Beach, FL, USA 4Department of Astronomy, University of Texas at Austin, Austin, TX, USA 5James Madison University, Harrisonburg, VA, USA 6Department of Astronomy, University of Texas at Austin, Austin, TX, USA; Department of Mathematics and Statistics, The University of Vermont, Burlington, VT, USA Received 23 August 2011; revised 11 October 2012; accepted 17 October 2012 DOI:10.1002/sam.11172 Published online in Wiley Online Library (wileyonlinelibrary.com). Abstract: Intricate computer models can be used to describe complex physical processes in astronomy such as the evolution of stars. Like a sampling distribution, these models typically predict observed quantities as a function of a number of unknown parameters. Including them as components of a statistical model, however, leads to significant modeling, inferential, and computational challenges. In this article, we tackle these challenges in the study of the mass loss that stars experience as they age. We have developed a new Bayesian technique for inferring the so-called initial–final mass relation (IFMR), the relationship between the initial mass of a Sun-like star and its final mass as a white dwarf. Our model incorporates several separate computer models for various phases of stellar evolution. We bridge these computer models with a parameterized IFMR in order to embed them into a statistical model. This strategy allows us to apply the full force of powerful statistical tools to build, fit, check, and improve the statistical models and their computer model components. In contrast to traditional techniques for inferring the IFMR, which tend to be quite ad hoc, we can estimate the uncertainty in our fit and ensure that our model components are internally coherent. We analyze data from three star clusters: NGC 2477, the Hyades, and M35 (NGC 2168). The results from NGC 2477 and M35 suggest different conclusions about the IFMR in the mid- to high-mass range, raising questions for further astronomical work. We also compare the results from two different models for the primary hydrogen- burning stage of stellar evolution. We show through simulations that misspecification at this stage of modeling can sometimes have a severe effect on inferred white dwarf masses. Nonetheless, when working with observed data, our inferences are not particularly sensitive to the choice of model for this stage of evolution. © 2013 Wiley Periodicals, Inc. Statistical Analysis and Data Mining 6: 34–52, 2013 Keywords: computer models; Bayes; astronomy; multi-level model; stellar mass loss; initial final mass relation 1. INTRODUCTION the models are the implicit solutions to nonlinear simulta- neous equations that can only be solved using sophisticated Complex models that cannot be expressed using analyt- ical forms are becoming more and more common in the computational algorithms. Such computer models are used, physical, engineering, and social sciences. In many cases, for example, to model weather and climate, the Earth’s inte- rior and seismology, genetic drift, and many other specific Correspondence to: David A. van Dyk (dvandyk@imperial. scientific phenomena. Computer models are particularly ac.uk) useful to describe complex astronomical processes, for © 2013 Wiley Periodicals, Inc. Stein et al: Combining Computer Models for Stellar Evolution 35 example, to model stellar evolution, describe the properties physical processes that power white dwarf stars on the one of planetary and stellar atmospheres, simulate chemical hand and main sequence and red giant stars on the other reactions in interstellar clouds, calculate the emergence of are completely different (i.e., latent heat and thermonuclear clusters and superclusters of galaxies in the early Universe, fusion, respectively), separate computer models are used for and determine the yield of different elements during the Big the white dwarf stars and for main sequence and red giant Bang. In this article, we explore how such computer models stars. Since white dwarfs are evolved main sequence stars, can be embedded into a principled statistical analysis and predicting the colors of white dwarfs involves first running how we can use the resulting statistical methods for model a computer model for the main sequence and red giant stars building, fitting, checking, and improvement. Our ultimate (MS/RG computer model) and then running a white dwarf goal is to enhance both the computer models and our under- model (WD computer model), using a transformation of the standing of the underlying astronomical phenomena. outputs from the MS/RG model as inputs. We focus on developing statistical methods that take Our goal is to build principled statistical models advantage of a set of computer models for stellar evolution and methods that embed these computer models into a with the aim of improving our understanding of the likelihood function as components of a statistical model composition, age, mass, and location of clusters of stars. that can be used to learn about the underlying parameters Stars are formed when the dense parts of a molecular cloud of scientific interest. The basic components of our model are collapse into a ball of hot plasma. If the mass of this ball described in refs. [1–3] and applied in ref. [4]. These articles is sufficient, its core will ignite in a thermonuclear reaction employ a Gaussian measurement error model for multiple that is powered by the fusion of hydrogen into helium, photometric magnitudes per star, allow for contaminated forming a so-called main sequence star. This reaction can data, and account for binary star systems composed of two continue for millions or billions of years, depending on the separately evolving stars that appear as a single star from original mass and composition of the star. More massive Earth. This article starts with this baseline and develops stars are denser, hotter, and burn their fuel more quickly. the linkage between the component computer models for When the hydrogen at the core has been depleted the core stellar evolution and describes how we evaluate them and collapses, heating up the inner star and igniting the same their fit to data. In particular, we propose a parametric reaction in regions surrounding the core. At the same time, model to link the MS/RG model with the WD computer the diameter of the star increases dramatically and the model and include the parameters as part of our likelihood surface temperature cools, resulting in a red giant star. This function. This enables us to evaluate and improve existing period of the star’s life is much shorter, lasting only about models that predict the mass of a white dwarf star from the one-tenth as long as the initial period. The temperature of initial mass of its progenitor star, the so-called initial–final the core continues to rise and for more massive stars will mass relationship (IFMR). This is done within the context become hot enough to fuse helium into carbon, oxygen, of a fully Bayesian model that allows us to quantify the neon, and perhaps heavier elements. During this time, stars uncertainty in the fitted IFMR. with mass less than about eight times that of the Sun The primary scientific goal of this article is to study undergo mass loss leading to the formation of a very short the total mass loss between the initial mass of a newly lived planetary nebula. The dense core of such a star formed main sequence star and the final mass of a white eventually stabilizes as the outer layers of the star blow dwarf. Mass loss occurs as a main sequence star fuses away, leaving a small hot dense ember, known as a white atoms to produce energy and more dramatically as the dwarf star. Nuclear reactions cease in a white dwarf star, outer layers of a red giant blow away. This latter process which slowly cools via radiation from its surface. is not well understood [5] but drives the relationship From Earth, main sequence, red giant, white dwarf, and between stellar ages, masses, and composition and observed other types of stars can be distinguished by the relative photometric magnitudes for the brightest stars of a cluster. intensity of the light emitted by the star at different This relationship is quantified by a MS/RG computer wavelengths. We use photometric data which records the model and is particularly important in the study of extra- luminosity or magnitude of each star under observation in a galactic clusters, especially those outside our local group particular range of wavelengths. The luminosity is a direct of galaxies. For such distant clusters, photometric data measurement of the amount of energy an astronomical can only be obtained for the brightest stars. Thus, an object radiates per unit time. A magnitude is a negative understanding of red giant mass loss may be our only logarithmic transformation of the luminosity, thus smaller clue to the ages of such distant stellar systems. Yet this magnitudes correspond to brighter objects. The computer clue is fraught with uncertainty because we lack a properly models that we employ predict the photometric magnitudes determined mass loss formula for stellar models. Our study as a function of a number of parameters that describe the is aimed directly at improving our understanding of stellar composition, size, distance, and age of the star. Because the mass loss. Statistical Analysis and Data Mining DOI:10.1002/sam 36 Statistical Analysis and Data Mining, Vol. 6 (2013) The hallmark of our approach is principled statistical fitting in a Bayesian context.