Stellar Structure and Evolution
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PHYS 633: Introduction to Stellar Astrophysics Spring Semester 2006 Rich Townsend ([email protected])
PHYS 633: Introduction to Stellar Astrophysics Spring Semester 2006 Rich Townsend ([email protected]) Governing Equations In the foregoing analysis, we have seen how a star usually maintains a state of hydrostatic equilibrium. We have seen how the energy created in the star through nuclear reactions, or released through contraction, makes its way out in the form of the star’s luminosity. We have seen how the physical processes responsible for transporting this energy can either be radiation or convection. And, finally, we have seen how nuclear reactions within the star, and transport processes such as convective mixing, can alter the stars chemical composition. These four key pieces of physics lead to 3 + I of the differential equations governing stellar structure (here, I is the number of elements whose evolution we are following). Let’s quickly review these equations. First, we have the most general (spherically-symmetric) form for the equation governing momen- tum conservation, ∂P Gm 1 ∂2r = − − . (1) ∂m 4πr2 4πr2 ∂t2 If the second term on the right-hand side vanishes, then we recover the equation of hydrostatic equilibrium, expressed in Lagrangian form (i.e., derivatives with respect to the mass coordinate m, rather than the radial coordinate r). If this term does not vanish at some point in the star, then the mass shells at that point will being to expand or contract, with an acceleration given by ∂2r/∂t2. Second, we have the most general form for the energy conservation equation, ∂l ∂T δ ∂P = − − c + . (2) ∂m ν P ∂t ρ ∂t The first and second terms on the right-hand side come from nuclear energy gen- eration and neutrino losses, respectively. -
Used Time Scales GEORGE E
PROCEEDINGS OF THE IEEE, VOL. 55, NO. 6, JIJNE 1967 815 Reprinted from the PROCEEDINGS OF THE IEEI< VOL. 55, NO. 6, JUNE, 1967 pp. 815-821 COPYRIGHT @ 1967-THE INSTITCJTE OF kECTRITAT. AND ELECTRONICSEN~INEERS. INr I’KINTED IN THE lT.S.A. Some Characteristics of Commonly Used Time Scales GEORGE E. HUDSON A bstract-Various examples of ideally defined time scales are given. Bureau of Standards to realize one international unit of Realizations of these scales occur with the construction and maintenance of time [2]. As noted in the next section, it realizes the atomic various clocks, and in the broadcast dissemination of the scale information. Atomic and universal time scales disseminated via standard frequency and time scale, AT (or A), with a definite initial epoch. This clock time-signal broadcasts are compared. There is a discussion of some studies is based on the NBS frequency standard, a cesium beam of the associated problems suggested by the International Radio Consultative device [3]. This is the atomic standard to which the non- Committee (CCIR). offset carrier frequency signals and time intervals emitted from NBS radio station WWVB are referenced ; neverthe- I. INTRODUCTION less, the time scale SA (stepped atomic), used in these emis- PECIFIC PROBLEMS noted in this paper range from sions, is only piecewise uniform with respect to AT, and mathematical investigations of the properties of in- piecewise continuous in order that it may approximate to dividual time scales and the formation of a composite s the slightly variable scale known as UT2. SA is described scale from many independent ones, through statistical in Section 11-A-2). -
14 Timescales in Stellar Interiors
14Timescales inStellarInteriors Having dealt with the stellar photosphere and the radiation transport so rel- evant to our observations of this region, we’re now ready to journey deeper into the inner layers of our stellar onion. Fundamentally, the aim we will de- velop in the coming chapters is to develop a connection betweenM,R,L, and T in stars (see Table 14 for some relevant scales). More specifically, our goal will be to develop equilibrium models that describe stellar structure:P (r),ρ (r), andT (r). We will have to model grav- ity, pressure balance, energy transport, and energy generation to get every- thing right. We will follow a fairly simple path, assuming spherical symmetric throughout and ignoring effects due to rotation, magneticfields, etc. Before laying out the equations, let’sfirst think about some key timescales. By quantifying these timescales and assuming stars are in at least short-term equilibrium, we will be better-equipped to understand the relevant processes and to identify just what stellar equilibrium means. 14.1 Photon collisions with matter This sets the timescale for radiation and matter to reach equilibrium. It de- pends on the mean free path of photons through the gas, 1 (227) �= nσ So by dimensional analysis, � (228)τ γ ≈ c If we use numbers roughly appropriate for the average Sun (assuming full Table3: Relevant stellar quantities. Quantity Value in Sun Range in other stars M2 1033 g 0.08 �( M/M )� 100 × � R7 1010 cm 0.08 �( R/R )� 1000 × 33 1 3 � 6 L4 10 erg s− 10− �( L/L )� 10 × � Teff 5777K 3000K �( Teff/mathrmK)� 50,000K 3 3 ρc 150 g cm− 10 �( ρc/g cm− )� 1000 T 1.5 107 K 106 ( T /K) 108 c × � c � 83 14.Timescales inStellarInteriors P dA dr ρ r P+dP g Mr Figure 28: The state of hydrostatic equilibrium in an object like a star occurs when the inward force of gravity is balanced by an outward pressure gradient. -
X-Ray Sun SDO 4500 Angstroms: Photosphere
ASTR 8030/3600 Stellar Astrophysics X-ray Sun SDO 4500 Angstroms: photosphere T~5000K SDO 1600 Angstroms: upper photosphere T~5x104K SDO 304 Angstroms: chromosphere T~105K SDO 171 Angstroms: quiet corona T~6x105K SDO 211 Angstroms: active corona T~2x106K SDO 94 Angstroms: flaring regions T~6x106K SDO: dark plasma (3/27/2012) SDO: solar flare (4/16/2012) SDO: coronal mass ejection (7/2/2012) Aims of the course • Introduce the equations needed to model the internal structure of stars. • Overview of how basic stellar properties are observationally measured. • Study the microphysics relevant for stars: the equation of state, the opacity, nuclear reactions. • Examine the properties of simple models for stars and consider how real models are computed. • Survey (mostly qualitatively) how stars evolve, and the endpoints of stellar evolution. Stars are relatively simple physical systems Sound speed in the sun Problem of Stellar Structure We want to determine the structure (density, temperature, energy output, pressure as a function of radius) of an isolated mass M of gas with a given composition (e.g., H, He, etc.) Known: r Unknown: Mass Density + Temperature Composition Energy Pressure Simplifying assumptions 1. No rotation à spherical symmetry ✔ For sun: rotation period at surface ~ 1 month orbital period at surface ~ few hours 2. No magnetic fields ✔ For sun: magnetic field ~ 5G, ~ 1KG in sunspots equipartition field ~ 100 MG Some neutron stars have a large fraction of their energy in B fields 3. Static ✔ For sun: convection, but no large scale variability Not valid for forming stars, pulsating stars and dying stars. 4. -
Stars IV Stellar Evolution Attendance Quiz
Stars IV Stellar Evolution Attendance Quiz Are you here today? Here! (a) yes (b) no (c) my views are evolving on the subject Today’s Topics Stellar Evolution • An alien visits Earth for a day • A star’s mass controls its fate • Low-mass stellar evolution (M < 2 M) • Intermediate and high-mass stellar evolution (2 M < M < 8 M; M > 8 M) • Novae, Type I Supernovae, Type II Supernovae An Alien Visits for a Day • Suppose an alien visited the Earth for a day • What would it make of humans? • It might think that there were 4 separate species • A small creature that makes a lot of noise and leaks liquids • A somewhat larger, very energetic creature • A large, slow-witted creature • A smaller, wrinkled creature • Or, it might decide that there is one species and that these different creatures form an evolutionary sequence (baby, child, adult, old person) Stellar Evolution • Astronomers study stars in much the same way • Stars come in many varieties, and change over times much longer than a human lifetime (with some spectacular exceptions!) • How do we know they evolve? • We study stellar properties, and use our knowledge of physics to construct models and draw conclusions about stars that lead to an evolutionary sequence • As with stellar structure, the mass of a star determines its evolution and eventual fate A Star’s Mass Determines its Fate • How does mass control a star’s evolution and fate? • A main sequence star with higher mass has • Higher central pressure • Higher fusion rate • Higher luminosity • Shorter main sequence lifetime • Larger -
Stellar Structure and Evolution
Lecture Notes on Stellar Structure and Evolution Jørgen Christensen-Dalsgaard Institut for Fysik og Astronomi, Aarhus Universitet Sixth Edition Fourth Printing March 2008 ii Preface The present notes grew out of an introductory course in stellar evolution which I have given for several years to third-year undergraduate students in physics at the University of Aarhus. The goal of the course and the notes is to show how many aspects of stellar evolution can be understood relatively simply in terms of basic physics. Apart from the intrinsic interest of the topic, the value of such a course is that it provides an illustration (within the syllabus in Aarhus, almost the first illustration) of the application of physics to “the real world” outside the laboratory. I am grateful to the students who have followed the course over the years, and to my colleague J. Madsen who has taken part in giving it, for their comments and advice; indeed, their insistent urging that I replace by a more coherent set of notes the textbook, supplemented by extensive commentary and additional notes, which was originally used in the course, is directly responsible for the existence of these notes. Additional input was provided by the students who suffered through the first edition of the notes in the Autumn of 1990. I hope that this will be a continuing process; further comments, corrections and suggestions for improvements are most welcome. I thank N. Grevesse for providing the data in Figure 14.1, and P. E. Nissen for helpful suggestions for other figures, as well as for reading and commenting on an early version of the manuscript. -
Evolution of the Sun, Stars, and Habitable Zones
Evolution of the Sun, Stars, and Habitable Zones AST 248 fs The fraction of suitable stars N = N* fs fp nh fl fi fc L/T Hertzsprung-Russell Diagram Parts of the H-R Diagram •Supergiants •Giants •Main Sequence (dwarfs) •White Dwarfs Making Sense of the H-R Diagram •The Main Sequence is a sequence in mass •Stars on the main sequence undergo stable H fusion •All other stars are evolved •Evolved stars have used up all their core H •Main Sequence ® Giants ® Supergiants •Subsequent evolution depends on mass Hertzsprung-Russell Diagram Evolutionary Timescales Pre-main sequence: Set by gravitational contraction •The gravitational potential energy E is ~GM2/R •The luminosity is L •The timescale is ~E/L We know L, M, R from observations For the Sun, L ~ 30 million years Evolutionary Timescales Main sequence: •Energy source: nuclear reactions, at ~10-5 erg/reaction •Luminosity: 4x1033 erg/s This requires 4x1038 reactions/second Each reaction converts 4 H ® He 56 The solar core contains 0.1 M¤, or ~10 H atoms 1056 atoms / 4x1038 reactions/second -> 3x1017 sec, or 1010 years. This is the nuclear timescale. Stellar Lifetimes τ ~ M/L On the main sequence, L~M3 Therefore, τ~M-2 10 τ¤ = 10 years τ~1010/M2 years Lower mass stars live longer than the Sun Post-Main Sequence Timescales Timescale τ ~ E/L L >> Lms τ << τms Habitable Zones Refer back to our discussion of the Greenhouse Effect. 2 0.25 Tp ~ (L*/D ) The habitable zone is the region where the temperature is between 0 and 100 C (273 and 373 K), where liquid water can exist. -
Useful Constants
Appendix A Useful Constants A.1 Physical Constants Table A.1 Physical constants in SI units Symbol Constant Value c Speed of light 2.997925 × 108 m/s −19 e Elementary charge 1.602191 × 10 C −12 2 2 3 ε0 Permittivity 8.854 × 10 C s / kgm −7 2 μ0 Permeability 4π × 10 kgm/C −27 mH Atomic mass unit 1.660531 × 10 kg −31 me Electron mass 9.109558 × 10 kg −27 mp Proton mass 1.672614 × 10 kg −27 mn Neutron mass 1.674920 × 10 kg h Planck constant 6.626196 × 10−34 Js h¯ Planck constant 1.054591 × 10−34 Js R Gas constant 8.314510 × 103 J/(kgK) −23 k Boltzmann constant 1.380622 × 10 J/K −8 2 4 σ Stefan–Boltzmann constant 5.66961 × 10 W/ m K G Gravitational constant 6.6732 × 10−11 m3/ kgs2 M. Benacquista, An Introduction to the Evolution of Single and Binary Stars, 223 Undergraduate Lecture Notes in Physics, DOI 10.1007/978-1-4419-9991-7, © Springer Science+Business Media New York 2013 224 A Useful Constants Table A.2 Useful combinations and alternate units Symbol Constant Value 2 mHc Atomic mass unit 931.50MeV 2 mec Electron rest mass energy 511.00keV 2 mpc Proton rest mass energy 938.28MeV 2 mnc Neutron rest mass energy 939.57MeV h Planck constant 4.136 × 10−15 eVs h¯ Planck constant 6.582 × 10−16 eVs k Boltzmann constant 8.617 × 10−5 eV/K hc 1,240eVnm hc¯ 197.3eVnm 2 e /(4πε0) 1.440eVnm A.2 Astronomical Constants Table A.3 Astronomical units Symbol Constant Value AU Astronomical unit 1.4959787066 × 1011 m ly Light year 9.460730472 × 1015 m pc Parsec 2.0624806 × 105 AU 3.2615638ly 3.0856776 × 1016 m d Sidereal day 23h 56m 04.0905309s 8.61640905309 -
Stellar Coronal Astronomy
Stellar coronal astronomy Fabio Favata ([email protected]) Astrophysics Division of ESA’s Research and Science Support Department – P.O. Box 299, 2200 AG Noordwijk, The Netherlands Giuseppina Micela ([email protected]) INAF – Osservatorio Astronomico G. S. Vaiana – Piazza Parlamento 1, 90134 Palermo, Italy Abstract. Coronal astronomy is by now a fairly mature discipline, with a quarter century having gone by since the detection of the first stellar X-ray coronal source (Capella), and having benefitted from a series of major orbiting observing facilities. Several observational charac- teristics of coronal X-ray and EUV emission have been solidly established through extensive observations, and are by now common, almost text-book, knowledge. At the same time the implications of coronal astronomy for broader astrophysical questions (e.g. Galactic structure, stellar formation, stellar structure, etc.) have become appreciated. The interpretation of stellar coronal properties is however still often open to debate, and will need qualitatively new ob- servational data to book further progress. In the present review we try to recapitulate our view on the status of the field at the beginning of a new era, in which the high sensitivity and the high spectral resolution provided by Chandra and XMM-Newton will address new questions which were not accessible before. Keywords: Stars; X-ray; EUV; Coronae Full paper available at ftp://astro.esa.int/pub/ffavata/Papers/ssr-preprint.pdf 1 Foreword 3 2 Historical overview 4 3 Relevance of coronal -
White Dwarfs - Degenerate Stellar Configurations
White Dwarfs - Degenerate Stellar Configurations Austen Groener Department of Physics - Drexel University, Philadelphia, Pennsylvania 19104, USA Quantum Mechanics II May 17, 2010 Abstract The end product of low-medium mass stars is the degenerate stellar configuration called a white dwarf. Here we discuss the transition into this state thermodynamically as well as developing some intuition regarding the role of quantum mechanics in this process. I Introduction and Stellar Classification present at such high temperatures are small com- pared with the kinetic (thermal) energy of the parti- It is widely believed that the end stage of the low or cles. This assumption implies a mixture of free non- intermediate mass star is an extremely dense, highly interacting particles. One may also note that the pres- underluminous object called a white dwarf. Obser- sure of a mixture of different species of particles will vationally, these white dwarfs are abundant (∼ 6%) be the sum of the pressures exerted by each (this is in the Milky Way due to a large birthrate of their pro- where photon pressure (PRad) will come into play). genitor stars coupled with a very slow rate of cooling. Following this logic we can express the total stellar Roughly 97% of all stars will meet this fate. Given pressure as: no additional mass, the white dwarf will evolve into a cold black dwarf. PT ot = PGas + PRad = PIon + Pe− + PRad (1) In this paper I hope to introduce a qualitative de- scription of the transition from a main-sequence star At Hydrostatic Equilibrium: Pressure Gradient = (classification which aligns hydrogen fusing stars Gravitational Pressure. -
A Teaching-Learning Module on Stellar Structure and Evolution
EPJ Web of Conferences 200, 01017 (2019) https://doi.org/10.1051/epjconf/201920001017 ISE2A 2017 A teaching-learning module on stellar structure and evolution Silvia Galano1, Arturo Colantonio2, Silvio Leccia3, Emanuella Puddu4, Irene Marzoli1 and Italo Testa5 1 Physics Division, School of Science and Technology, University of Camerino, Via Madonna delle Carceri 9, I-62032 Camerino (MC), Italy 2 Liceo Scientifico e delle Scienze Umane “Cantone”, Via Savona, I-80014 Pomigliano d'Arco, Italy 3 Liceo Statale “Cartesio”, Via Selva Piccola 147, I-80014 Giugliano, Italy 4 INAF - Capodimonte Astronomical Observatory of Naples, Salita Moiariello, 16, I-80131 Napoli, Italy 5 Department of Physics “E. Pancini”, “Federico II” University of Napoli, Complesso Monte S.Angelo, Via Cintia, I-80126 Napoli, Italy Abstract. We present a teaching module focused on stellar structure, functioning and evolution. Drawing from literature in astronomy education, we identified three key ideas which are fundamental in understanding stars’ functioning: spectral analysis, mechanical and thermal equilibrium, energy and nuclear reactions. The module is divided into four phases, in which the above key ideas and the physical mechanisms involved in stars’ functioning are gradually introduced. The activities combine previously learned laws in mechanics, thermodynamics, and electromagnetism, in order to get a complete picture of processes occurring in stars. The module was piloted with two intact classes of secondary school students (N = 59 students, 17–18 years old) and its efficacy in addressing students’ misconceptions and wrong ideas was tested using a ten-question multiple choice questionnaire. Results support the effectiveness of the proposed activities. Implications for the teaching of advanced physics topics using stars as a fruitful context are briefly discussed. -
Stellar Evolution Microcosm of Astrophysics
Nature Vol. 278 26 April 1979 Spring books supplement 817 soon realised that they are neutron contrast of the type. In the first half stars. It now seems certain that neutron Stellar of the book there are only a small stars rather than white dwarfs are a number of detailed alterations but later normal end-product of a supernova there are much more substantial explosion. In addition it has been evolution changes, particularly dn chapters 6, 7 realised that sufficiently massive stellar and 8. The book is essentially un remnants will be neither white dwarfs R. J. Tayler changed in length, the addition of new nor neutron stars but will be black material being balanced by the removal holes. Rather surprisingly in view of Stellar Evolution. Second edition. By of speculative or unimportant material. the great current interest in close A. J. Meadows. Pp. 171. (Pergamon: Substantial changes in the new binary stars and mass exchange be Oxford, 1978.) Hardback £7.50; paper edition include the following. A section tween their components, the section back £2.50. has been added on the solar neutrino on this topic has been slightly reduced experiment, the results of which are at in length, although there is a brief THE subject of stellar structure and present the major question mark mention of the possible detection of a evolution is believed to be the branch against the standard view of stellar black hole in Cygnus X-1. of astrophysics which is best under structure. There is a completely revised The revised edition of this book stood.