================================================================ DESCRIPTION OF THE HONORS OPTION FOR ASTRONOMY 485 - NIEL BRANDT ================================================================ ---------------------------------------------------------------- The honors option for this course will involve a computational investigation of the structure of white dwarf stars. You will learn about the following: * Observations of white dwarf stars * The equation of state for degenerate white dwarf matter * The differential equations describing white dwarf stars * Polytropes and solutions to the Lane-Emden equation * The mass-radius relation for white dwarf stars * The Chandrasekhar mass limit * The effects of chemical composition on white dwarf properties * Efficient numerical integration of ordinary differential equations ---------------------------------------------------------------- The basic work that will be done for the honors option (roughly in order) is the following: * Review the basic facts about white dwarfs. For example, you could read the following: "Compact Stars for Undergraduates" I. Sagert et al. (arXiv:astro-ph/0506417) "White Dwarf Stars" D. Koester (http://adsabs.harvard.edu/abs/2013pss4.book..559K) Chapter 15 of "Modern Astrophysics" book by B.W. Carroll and D.A. Ostlie Chapter 15 of "Modern Stellar Astrophysics" book by D.A. Ostlie and B.W. Carroll Chapter 15 of "High Energy Astrophysics: Volume 2" book by M.S. Longair Chapter 3 of "Black Holes, White Dwarfs, and Neutron Stars" book by S.L. Shapiro and S.A. Teukolsky Slides from a 2017 conference on white dwarf stars: http://www.cnls.lanl.gov/External/whitedwarf/ http://www.cvent.com/events/current-challenges-in-the-physics-of-white-dwarf-stars/event-summary-06cf2ff9feef497ebb6dba446bec9b71.aspx * Learn about observations of white dwarf masses and radii. Here you will read a few relevant papers from the scientific literature and talk with white dwarf experts in the Department. Relevant papers include the following: "The Mass and Radius of 40 Eridani B from Hipparcos: An Accurate Test of Stellar Interior Theory" H.L. Shipman et al. (1997, The Astrophysical Journal, 488, L43) "Testing the White Dwarf Mass-Radius Relation with Hipparcos" J.L. Provencal et al. (1998, The Astrophysical Journal, 494, 759) "PG 2131+066: A Test of Pre-White Dwarf Asteroseismology" M.D. Reed et al. (2000, The Astrophysical Journal, 545, 429) "A Redetermination of the Mass of Procyon" T.M. Girard et al. (2000, The Astronomical Journal, 119, 2428) "Procyon B: Outside the Iron Box" J.L. Provencal et al. (2002, The Astrophysical Journal, 568, 324) "Hubble Space Telescope Astrometry of the Procyon System" H.E. Bond et al. (2015, ApJ, 813, 106) "The Sirius System and Its Astrophysical Puzzles: Hubble Space Telescope and Ground-based Astrometry" H.E. Bond et al. (2017, ApJ, 840, 70) "Relativistic Deflection of Background Starlight Measures the Mass of a Nearby White Dwarf Star" K.C. Sahu et al. (2017, Science, 356, 1046) "Astrophysical Implications of a New Dynamical Mass for the Nearby White Dwarf 40 Eridani B" H.E. Bond et al. (2017, ApJ, 848, 16) "The Gravitational Redshift of Sirius B" S.R.G. Joyce et al. (2018, MNRAS, 481, 2361) "The White Dwarf Mass-Radius Relation and Its Dependence on the Hydrogen Envelope" A.D. Romero et al. (2019, MNRAS, 484, 2711) Howard Bond and Richard Wade in the Department are white dwarf experts. They can provide further references if you'd like to learn more about a specific issue. * Learn about efficient numerical integration of ordinary differential equations, especially the Runge-Kutta method. Solve some simple differential equations with this method, and understand the concept of coupled differential equations. You will work through Chapter 16 of the book "Numerical Recipes in C" by W.H. Press et al. See http://www.nr.com/ for details. * Derive and appropriately scale the differential equations describing white dwarf stars. Here you will follow Chapter 2 (pages 42-48) of the book "Computational Physics" by S.E. Koonin. You will also learn about polytropes and solutions to the Lane-Emden equation. * Write a computer program that numerically integrates the differential equations describing white dwarf stars. This program should be as elegant and generalizable as possible. Please use standard good programming techniques. This includes using good overall coding architecture, commenting your code, using good formatting, using consistent and meaningful naming conventions, using consistent indentation, and avoiding repetition. See https://en.wikipedia.org/wiki/Best_coding_practices and similar. * Verify the correctness of your computer program by comparisons with observations of white dwarf masses and radii. You will also recover the Chandrasekhar mass limit and make comparisons with analytic solutions of the Lane-Emden equation. * Use your computer program to investigate the dependence of white dwarf properties upon factors such as chemical composition. * Examine how the accuracy of your numerical solutions depends upon computational method and program parameters (for example, integration step size). How can you solve the white dwarf problem accurately with the best computational efficiency? * Investigate other selected issues, as time allows. + How much can adaptive step size control with the Runge-Kutta algorithm improve computational efficiency? + How do predictor-corrector methods or the Bulirsch-Stoer method compare to the Runge-Kutta method in terms of solving the white dwarf problem efficiently? I might predict that Bulirsch-Stoer would perform best, but I am willing to be corrected. + Calculate the properties of an "iron white dwarf" in the center of a star about to go supernova. How do its properties compare to those of the "normal" white dwarfs you have studied above? + Include rotation and/or magnetic fields into your computer code to determine how these might alter the structural properties of white dwarfs. Some good hints on how to get started with this are given in Chapter 7 of the book by Shapiro & Teukolsky. + Could white dwarfs exist in a universe with only two spatial dimensions? If so, what would be their properties? What would white dwarfs be like in universes with four or five spatial dimensions? + Adapt your code so that it works for neutron stars. You'll need to check the book by Shapiro & Teukolsky for hints on the possible equation of state to utilize for bulk nuclear matter. Also, you should read about the Tolman-Oppenheimer-Volkoff equation. * Write a final report describing the results of your investigations. This report should be typed and well written. This report should include at least the following: + A brief review of white dwarfs and observations of their masses and radii. + A review of the equations describing white dwarf structure, demonstrating that you understand these equations physically. + A description of the code you have written to solve the white dwarf problem numerically. The clearly commented code itself should be included as an appendix to the report. + A justification of the initial conditions used to start the numerical integration. + Basic tests of your code that demonstrate that it works correctly. For example, you should test your code versus analytic solutions of the Lane-Emden equation. You should recover the Chandrasekhar mass and the white dwarf mass-radius relation. You should show that your mass-radius relation agrees with modern observations of white dwarfs. When making these observational comparisons, please use the up-to-date information from your preparatory reading to update your list of white dwarf masses and radii (some of the ones listed in Step 5 of the Koonin book are out of date). You should illustrate your test results with appropriate plots. + Examinations of the dependence of white dwarf structure upon chemical composition. You should illustrate your results with appropriate plots. + An investigation of computational efficiency in solving the white dwarf problem. + A summary of your main findings. + Suggestions for future avenues of investigation. Suggestions of how this honors project might be improved in the future for other students. + A bibliography giving proper reference information for the references used. + Written solutions to the questions and problems on pages 46-48 of the Koonin book. These do not need to be typed but should be written clearly. When you are solving the Lane-Emden equation for Gamma = 2 and Gamma = 6/5, please actually solve the equation (that is, do not just plug in the stated solutions to show that they work). You should show full analytic derivations/solutions, and not just use Mathematica (or similar programs) to produce answers in a "deus ex machina" fashion. Your final report will be due on the last weekday of Penn State classes for the relevant semester (typically this is the Friday before the start of Final Exams Week). Please plan ahead over the semester to avoid a last-minute crisis. Last-minute extensions will not be given. You will turn in your final report via email to "wnbrandt@gmail.com". Your report should be a single PDF file submitted as an attachment. If needed for inclusion in your report, you should scan your handwritten solutions to the questions and problems in the Koonin book. ---------------------------------------------------------------- ------------------------------------------------------------------------ W. Niel Brandt; Department of Astronomy & Astrophysics; The Pennsylvania State University; 525 Davey Lab; University Park, PA 16802; USA wnbrandt@gmail.com; 814-865-3509; Skype: nielbrandt; Office number: 514A ------------------------------------------------------------------------