DOCSLIB.ORG

Lecture 7 Evolution of Massive Stars on the Main Sequence and During

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Lecture 7 Evolution of Massive Stars on the Main Sequence and During Massive Stars - Key Physics and Issues • Nuclear Physics • Evolution in HR diagram • Equation of state • Nucleosynthesis Lecture 7 • Opacity • Mass loss • Surface abundances Evolution of Massive • Convection • Presupernova structure • Rotation (magnetic fields) Stars on the Main Sequence • Binary membership • Supernova properties and During Helium Burning - Basics • Explosion physics • Remnant properties • Rotation and B-field of pulsars Massive Stars For a non-degenerate gas, the entropy is given by (Clayton 2-136) For an ideal gas 3 ⎛ 2πmk ⎞ 5 st S ln integrate the 1 0 = ⎜ 2 ⎟ + Generalities: 2 ⎝ h ⎠ 2 law of thermo- ( Reif - Statistical Physics - 7.3.6) dynamics T dS = dU +P dV The electrons are included in µ and in S0 Ideal gas (convective with negligible radiation entropy): Because of the general tendency of the interior temperature of main sequence stars to increase with mass, stars of over two P = const × ρ × T ∝ρ × ρ 2/3 = ρ 5/3 = ργ For non relativistic, but possibly partly degenerate 3/2 solar mass are chiefly powered by the CNO cycle(s) rather than T electrons, the electrons are since = constant the pp cycle(s). This, plus the increasing fraction of pressure due to ρ given as a separate term see Clayton 2-145. radiation, makes their cores convective. The opacity is dominantly n+1 3 γ = ⇒ n = due to electron scattering. Despite their convective cores, the n 2 overall main sequence structure can be crudely represented as Radiation dominated gas or a gas with constant : an n = 3 polytrope. This is especially true of the outer radiative 1 P = aT4 ∝ρ 4/3 part of the star that typically includes the majority of the mass. 3 3 T P 1− β Pgas Pgas if ∝ rad = = constant β = = P P +P ρ Pideal β tot gas rad n+1 γ = ⇒ n = 3 n (S/NAk) half way through hydrogen burning 15 and 100 solar masses More massive stars have larger entropies on the Not surprisingly then, it turns out that massive stars main sequence that are more radiation - dominated are typically hybrid polytropes with their convective cores having 3 > n > 1.5 and radiative envelopes with n approximately 3. Overall n = 3 is not bad. For normal massive stars, the ionic entropy always dominates on the main sequence, but for very massive stars Selec, Srad and Sionic can become comparable. continuing farther out Convection plus entropy from ideal gas implies n = 1.5 d ln ρ 1 = d ln P γ γ ≈ 5 / 3 for ideal gas at constant entropy Most of the mass and volume. d ln ρ 1 = 40% of the mass d ln P γ γ ≈ 4 / 3 for standard model (with β = const) in radiative regions Eddington’s standard model (n=3) Consider a star in which radiation pressure is important P P (though not necessarily dominant) and energy transport gas rad Define β = = 1- where P= Pgas +Prad , is by radiative diffusion P P dPrad d ⎛ 1 4 ⎞ 4 3 dT = ⎜ aT ⎟ = aT dr dr ⎝ 3 ⎠ 3 dr then Prad = P −Pgas =(1− β) P and dT 3κρ L(r) But for radiative diffusion, = so dPrad κL(r) L(r) dr 16πacT 3 r 2 = (1− β)= = dP 4πGmc L dP κρ L(r) Edd rad = − dr 4πc r 2 but hydrostatic equilibrium requires If, and it is a big IF, β (or 1-β) were a constant throughout dP Gmρ = − the star, then one could write everywhere, including the surface dr r 2 Divide the 2 eqns L(r) = 1 L dP κL(r) L(r) ( − β ) Ed (Main sequence only) rad = = dP 4πGmc LEdd 4πGMc where L = Ed κ is nearly constant 3/2 3/2 n +1 dθ ⎞ ⎛ K ⎞ 3−n Fom polytropes M ( ) 2 2n = − ξ1 ⎟ ⎜ ⎟ ρc 1− β = fraction of the pressure 4π dξ ⎠ ⎝ G⎠ (Clayton 155- 165) ξ1 from radiation 1/3 4 ⎡3(NAk) (1− β)⎤ K = ⎢ 4 ⎥ ⎣⎢ a(µβ) ⎦⎥ For n = 3 (β= constant), ρc drops out and this becomes 3/2 3/2 2.01824 Near the surface P 4 2 ⎛ dθ ⎞ ⎛ K ⎞ ⎛ K ⎞ gas M 4.56 the density declines β = = − ξ1 ⎜ ⎟ ⎜ ⎟ = ⎜ ⎟ P dξ G G precipitously making total π ⎝ ⎠ ξ ⎝ ⎠ ⎝ ⎠ 1 radiation pressure 1/2 ⎡3(N k)4(1− β)⎤ inner ~5 Msun is more important. M = 4.56 A lim M →0 convective ⎢ 4 3 ⎥ 1 ⎣⎢ a(µβ) G ⎦⎥ β→ 1/2 lim M →∞ Eddington’s quartic 18.1M ⎛ 1− β ⎞ β→0 M equation = 2 4 µ ⎝⎜ β ⎠⎟ −1 ⎡ ⎤ µ =⎣∑(1+Zi ) Yi ⎦ = 0.73 for 50% H, 50% He 0.64 for 75% H, 25% He For 20 M β ≈0.80− 0.85 µ2 M ≈11 κ L ∝(1− β) decreases as M(r) ↑ M because L is centrally concentrated, so β within a given star increases with M(r) (for radiative regions) from Clayton p. 163 inner ~8 Msun convective 1/2 18.1M ⎛ 1− β ⎞ M = µ2 ⎝⎜ β 4 ⎠⎟ 2 4 ⎛ M ⎞ ⎛ µ ⎞ 1− β = 4.13×10−4 β 4 and since ⎜ M ⎟ ⎜ 0.61⎟ ⎝ ⎠ ⎝ ⎠ For M not too far from M β is close to 1 and L ∝M3. L(r) = (1- β ) LEdd ⎛ aG3 ⎞ π 4πGc L 4 4M 2 M =⎜ 4 ⎟ 2 µ β At higher masses however the mass dependence of β 3(N k) 4 ⎝ A ⎠ 16 d / d κ 4 −2 ξ1 ( θ ξ) becomes important. Eventually M so that L M. ξ1 β ∝ ∝ 2 4 4 4 In fact, the luminosity of very massive stars approaches π ⎛ acG ⎞ ⎛ µ β ⎞ 3 = M the Eddington limit as 0 ( L(r) = 1 L ) 4 2 ⎜ 4 ⎟ ⎜ ⎟ β → ( − β ) Edd 12 d / d ⎝ (NAk) ⎠ ⎝ κ ⎠ ξ1 ( θ ξ) ξ1 4 3 2 −1 ⎛ ⎞ Mass luminosty 4πGMc 38 -1 ⎛ M ⎞ ⎛ 0.34⎞ 4 ⎛ µ ⎞ ⎛ 1 cm g ⎞ M L 1.47 10 erg s 5.5 L Edd = = × ⎜ ⎟ ⎜ ⎟ = β ⎜ ⎟ ⎜ ⎟ Relation κ M ⎝ κ ⎠ ⎜ 0.61⎟ M ⎝ ⎠ ⎝ ⎠ ⎝ κ surf ⎠ ⎝ ⎠ where κ surf is the value of the opacity near the surface. This was obtained with no mention of nuclear reactions. For n = 3 one can also derive useful equations for the central conditions based upon the original polytropic equation for mass For the n=3 polytrope 2/3 3 2 dθ ⎞ 3 ⎛ M ⎞ M = − 4πα ρ ξ = 2.01824 (4πα ρ ) T = 4.6 × 106 K µβ ρ1/3 (in general for n = 3) c 1 d ⎟ c c ⎜ ⎟ c ξ ⎠ ξ ⎝ M ⎠ 1 1/2 1/2 ⎡P (n +1)⎤ ⎡ P ⎤ For stars on the main sequence and half way through and the definitions α = ⎢ c ⎥ = ⎢ c ⎥ 4 G 2 G 2 hydrogen burning, µ ≈ 0.84 and, unless the star is very massive, ⎣⎢ π ρc ⎦⎥ ⎣⎢π ρc ⎦⎥ β ≈ 0.8 - 0.9. Better values are given in Fig 2-19 of Clayton P ρ N kT ρ 4πR3 ρ and P = ideal = c A c and c = c = 54.18 replicated on the next page. c β µβ ρ 3M The density is not predicted from first principles since the actual 2 ⎛ M / M ⎞ 17 ( ) radius depends upon nuclear burning, but detailed main Pc =1.242×10 ⎜ 4 ⎟ ⎜ R / R ⎟ ⎛ 10 M ⎞ --3 ⎝ ( ) ⎠ sequence models (following page) give ρ ≈ 10 gm cm ,So c ⎝⎜ M ⎠⎟ ⎛ M / M ⎞ 6 ( ) 1/3 T =19.57×10 βµ ⎜ ⎟ K 7 ⎛ M ⎞ c R / R T ≈ 3.9 ×10 β K (main sequence only) ⎝ ( ) ⎠ c ⎜ ⎟ ⎝ 10M ⎠ 2/3 T 4.62 106 M / M 1/3 K c = × βµ ρc ( ) 7 37 M Tc/10 C L/10 Competition between the p-p 9 3.27 9.16 2.8 chain and the CNO Cycle 12 3.45 6.84 7.0 All evaluated in 15 3.58 5.58 13 L M2.5 actual models at 20 3.74 4.40 29 ∝ a core H mass 25 3.85 3.73 50 fraction of 0.30 40 4.07 2.72 140 for stars of solar 60(57) 4.24 2.17 290 metallicity. 85(78) 4.35 1.85 510 L ~ (1− β) L µ ≈0.8 120(99) 4.45 1.61 810 Ed ρc decreases with mass as a general consequence of the fact that T3 c ∝ M 2β 3µ 3 and H burning happens at a relatively constant ρc temperature. Until about 40 M, the density decreases roughly as M-1. After that it decreases more slowly. Recall β ∝ M-1/2 for very large masses The temperature dependence of the CNO cycle is given by the sensitivity of the proton capture rate of 14N. See previous lectures The Primary CNO Cycle CNO tri-cycle CN cycle (99.9%) O Extension 1 (0.1%) Ne(10) O Extension 2 F(9) O Extension 3 O(8) N(7) In a low mass star C(6) The slowest reaction is 14N(p,)15O. For temperatures near 2 x 107 K. 1/3 3 4 5 6 7 8 9 ⎛ 2 2 14 ⋅1 ⎞ 7 1 neutron number n τ -2 ⎜ 14 + 1 ⎟ ε nuc ∝T n = τ = 4.248 ⎜ ⎟ =60.0 3 ⎜ 0.020 ⎟ ⎝ ⎠ All initial abundances within a cycle serve as catalysts and accumulate at largest ! n =18 Extended cycles introduce outside material into CN cycle (Oxygen, …) (More on nucleosynthesis later) In general, the rates for these reactions proceed through known resonances whose properties are all reasonably well known. There was a major revision of the rate for 14N(p,γ )15O in 2001 by Bertone et al., Phys. Rev. Lettr., 87, 152501. The new rate is about half as large as the old one, so the main sequence lifetime of massive stars is longer (but definitely not linear in the reciprocal rate).
Load more

Recommended publications
  • Spectroscopic Analysis of Accretion/Ejection Signatures in the Herbig Ae/Be Stars HD 261941 and V590 Mon T Moura, S
    Spectroscopic analysis of accretion/ejection signatures in the Herbig Ae/Be stars HD 261941 and V590 Mon T Moura, S. Alencar, A. Sousa, E. Alecian, Y. Lebreton To cite this version: T Moura, S. Alencar, A. Sousa, E. Alecian, Y. Lebreton. Spectroscopic analysis of accretion/ejection signatures in the Herbig Ae/Be stars HD 261941 and V590 Mon. Monthly Notices of the Royal Astronomical Society, Oxford University Press (OUP): Policy P - Oxford Open Option A, 2020, 494 (3), pp.3512-3535. 10.1093/mnras/staa695. hal-02523038 HAL Id: hal-02523038 https://hal.archives-ouvertes.fr/hal-02523038 Submitted on 16 May 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. MNRAS 000,1–24 (2019) Preprint 27 February 2020 Compiled using MNRAS LATEX style file v3.0 Spectroscopic analysis of accretion/ejection signatures in the Herbig Ae/Be stars HD 261941 and V590 Mon T. Moura1?, S. H. P. Alencar1, A. P. Sousa1;2, E. Alecian2, Y. Lebreton3;4 1Universidade Federal de Minas Gerais, Departamento de Física, Av. Antônio Carlos 6627, 31270-901, Brazil 2Univ. Grenoble Alpes, IPAG, F-38000 Grenoble, France 3LESIA, Observatoire de Paris, PSL Research University, CNRS, Sorbonne Universités, UPMC Univ.
    [Show full text]
  • Plotting Variable Stars on the H-R Diagram Activity
    Pulsating Variable Stars and the Hertzsprung-Russell Diagram The Hertzsprung-Russell (H-R) Diagram: The H-R diagram is an important astronomical tool for understanding how stars evolve over time. Stellar evolution can not be studied by observing individual stars as most changes occur over millions and billions of years. Astrophysicists observe numerous stars at various stages in their evolutionary history to determine their changing properties and probable evolutionary tracks across the H-R diagram. The H-R diagram is a scatter graph of stars. When the absolute magnitude (MV) – intrinsic brightness – of stars is plotted against their surface temperature (stellar classification) the stars are not randomly distributed on the graph but are mostly restricted to a few well-defined regions. The stars within the same regions share a common set of characteristics. As the physical characteristics of a star change over its evolutionary history, its position on the H-R diagram The H-R Diagram changes also – so the H-R diagram can also be thought of as a graphical plot of stellar evolution. From the location of a star on the diagram, its luminosity, spectral type, color, temperature, mass, age, chemical composition and evolutionary history are known. Most stars are classified by surface temperature (spectral type) from hottest to coolest as follows: O B A F G K M. These categories are further subdivided into subclasses from hottest (0) to coolest (9). The hottest B stars are B0 and the coolest are B9, followed by spectral type A0. Each major spectral classification is characterized by its own unique spectra.
    [Show full text]
  • • Classifying Stars: HR Diagram • Luminosity, Radius, and Temperature • “Vogt-Russell” Theorem • Main Sequence • Evolution on the HR Diagram
    Stars • Classifying stars: HR diagram • Luminosity, radius, and temperature • “Vogt-Russell” theorem • Main sequence • Evolution on the HR diagram Classifying stars • We now have two properties of stars that we can measure: – Luminosity – Color/surface temperature • Using these two characteristics has proved extraordinarily effective in understanding the properties of stars – the Hertzsprung- Russell (HR) diagram If we plot lots of stars on the HR diagram, they fall into groups These groups indicate types of stars, or stages in the evolution of stars Luminosity classes • Class Ia,b : Supergiant • Class II: Bright giant • Class III: Giant • Class IV: Sub-giant • Class V: Dwarf The Sun is a G2 V star Luminosity versus radius and temperature A B R = R R = 2 RSun Sun T = T T = TSun Sun Which star is more luminous? Luminosity versus radius and temperature A B R = R R = 2 RSun Sun T = T T = TSun Sun • Each cm2 of each surface emits the same amount of radiation. • The larger stars emits more radiation because it has a larger surface. It emits 4 times as much radiation. Luminosity versus radius and temperature A1 B R = RSun R = RSun T = TSun T = 2TSun Which star is more luminous? The hotter star is more luminous. Luminosity varies as T4 (Stefan-Boltzmann Law) Luminosity Law 2 4 LA = RATA 2 4 LB RBTB 1 2 If star A is 2 times as hot as star B, and the same radius, then it will be 24 = 16 times as luminous. From a star's luminosity and temperature, we can calculate the radius.
    [Show full text]
  • Structure and Energy Transport of the Solar Convection Zone A
    Structure and Energy Transport of the Solar Convection Zone A DISSERTATION SUBMITTED TO THE GRADUATE DIVISION OF THE UNIVERSITY OF HAWAI'I IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN ASTRONOMY December 2004 By James D. Armstrong Dissertation Committee: Jeffery R. Kuhn, Chairperson Joshua E. Barnes Rolf-Peter Kudritzki Jing Li Haosheng Lin Michelle Teng © Copyright December 2004 by James Armstrong All Rights Reserved iii Acknowledgements The Ph.D. process is not a path that is taken alone. I greatly appreciate the support of my committee. In particular, Jeff Kuhn has been a friend as well as a mentor during this time. The author would also like to thank Frank Moss of the University of Missouri St. Louis. His advice has been quite helpful in making difficult decisions. Mark Rast, Haosheng Lin, and others at the HAO have assisted in obtaining data for this work. Jesper Schou provided the helioseismic rotation data. Jorgen Christiensen-Salsgaard provided the solar model. This work has been supported by NASA and the SOHOjMDI project (grant number NAG5-3077). Finally, the author would like to thank Makani for many interesting discussions. iv Abstract The solar irradiance cycle has been observed for over 30 years. This cycle has been shown to correlate with the solar magnetic cycle. Understanding the solar irradiance cycle can have broad impact on our society. The measured change in solar irradiance over the solar cycle, on order of0.1%is small, but a decrease of this size, ifmaintained over several solar cycles, would be sufficient to cause a global ice age on the earth.
    [Show full text]
  • The Hr Diagram for Late-Type Nearby Stars
    379 THE H-R DIAGRAM FOR LATE-TYPE NEARBY STARS AS A FUNCTION OF HELIUM CONTENT AND METALLICITY 1 2 3 2 1 1 Y. Lebreton , M.-N. Perrin ,J.Fernandes ,R.Cayrel ,G.Cayrel de Strob el , A. Baglin 1 Observatoire de Paris, Place J. Janssen - 92195 Meudon Cedex, France 2 Observatoire de Paris, 61 Avenue de l'Observatoire - 75014 Paris, France 3 Observat orio Astron omico da Universidade de Coimbra, 3040 Coimbra, Portugal Key words: Galaxy: solar neighb ourho o d; stars: ABSTRACT abundances; stars: low-mass; stars: HR diagram; Galaxy: abundances. Recent theoretical stellar mo dels are used to discuss the helium abundance of a numberoflow-mass stars for which the p osition in the Hertzsprung-Russell di- 1. INTRODUCTION agram and the metallicity are known with high accu- racy. The knowledge of the initial helium abundance of Hipparcos has provided very high quality parallaxes stars b orn in di erent sites with di erent metallicities of a sample of a hundred disk stars, of typeFtoK,lo- is of great imp ortance for many astrophysical stud- cated in the solar neighb ourho o d. Among these stars ies. The lifetime of a star and its internal structure we have carefully selected those for which detailed very much dep end on its initial helium content and sp ectroscopic analysis has provided e ective temp er- this has imp ortant consequences not only for stellar ature and [Fe/H] ratio with a high accuracy. astrophysics but also in cosmology or in studies of the chemical evolution of galaxies. Wehave calculated evolved stellar mo dels and their Direct measurement of the helium abundance in the asso ciated iso chrones in a large range of mass, for photosphere of a low mass star cannot b e made since several values of the metallicity and of the helium there are no helium lines in the sp ectra.
    [Show full text]
  • Evolution, Mass Loss and Variability of Low and Intermediate-Mass Stars What Are Low and Intermediate Mass Stars?
    Evolution, Mass Loss and Variability of Low and Intermediate-Mass Stars What are low and intermediate mass stars? Defined by properties of late stellar evolutionary stages Intermediate mass stars: ~1.9 < M/Msun < ~7 Develop electron-degenerate cores after core helium burning and ascending the red giant branch for the second time i.e. on the Asymptotic Giant Branch (AGB). AGB Low mass stars: M/Msun < ~1.9 Develop electron-degenerate cores on leaving RGB the main-sequence and ascending the red giant branch for the first time i.e. on the Red Giant Branch (RGB). Maeder & Meynet 1989 Stages in the evolution of low and intermediate-mass stars These spikes are real The AGB Surface enrichment Pulsation Mass loss The RGB Surface enrichment RGB Pulsation Mass loss About 108 years spent here Most time spent on the main-sequence burning H in the core (~1010 years) Low mass stars: M < ~1.9 Msun Intermediate mass stars: Wood, P. R.,2007, ASP Conference Series, 374, 47 ~1.9 < M/Msun < ~7 Stellar evolution and surface enrichment The Red giant Branch (RGB) zHydrogen burns in a shell around an electron-degenerate He core, star evolves to higher luminosity. zFirst dredge-up occurs: The convection in the envelope moves in when the stars is near the bottom of the RGB and "dredges up" material that has been through partial hydrogen burning by the CNO cycle and pp chains. From John Lattanzio But there's more: extra-mixing What's the evidence? Various abundances and isotopic ratios vary continuously up the RGB. This is not predicted by a single first dredge-up alone.
    [Show full text]
  • Spectral Classification and HR Diagram of Pre-Main Sequence Stars in NGC 6530
    A&A 546, A9 (2012) Astronomy DOI: 10.1051/0004-6361/201219853 & c ESO 2012 Astrophysics Spectral classification and HR diagram of pre-main sequence stars in NGC 6530,, L. Prisinzano, G. Micela, S. Sciortino, L. Affer, and F. Damiani INAF – Osservatorio Astronomico di Palermo, Piazza del Parlamento, Italy 1, 90134 Palermo, Italy e-mail: [email protected] Received 20 June 2012 / Accepted 3 August 2012 ABSTRACT Context. Mechanisms involved in the star formation process and in particular the duration of the different phases of the cloud contrac- tion are not yet fully understood. Photometric data alone suggest that objects coexist in the young cluster NGC 6530 with ages from ∼1 Myr up to 10 Myr. Aims. We want to derive accurate stellar parameters and, in particular, stellar ages to be able to constrain a possible age spread in the star-forming region NGC 6530. Methods. We used low-resolution spectra taken with VLT/VIMOS and literature spectra of standard stars to derive spectral types of a subsample of 94 candidate members of this cluster. Results. We assign spectral types to 86 of the 88 confirmed cluster members and derive individual reddenings. Our data are better fitted by the anomalous reddening law with RV = 5. We confirm the presence of strong differential reddening in this region. We derive fundamental stellar parameters, such as effective temperatures, photospheric colors, luminosities, masses, and ages for 78 members, while for the remaining 8 YSOs we cannot determine the interstellar absorption, since they are likely accretors, and their V − I colors are bluer than their intrinsic colors.
    [Show full text]
  • The Sun and the Solar Corona
    SPACE PHYSICS ADVANCED STUDY OPTION HANDOUT The sun and the solar corona Introduction The Sun of our solar system is a typical star of intermediate size and luminosity. Its radius is about 696000 km, and it rotates with a period that increases with latitude from 25 days at the equator to 36 days at poles. For practical reasons, the period is often taken to be 27 days. Its mass is about 2 x 1030 kg, consisting mainly of hydrogen (90%) and helium (10%). The Sun emits radio waves, X-rays, and energetic particles in addition to visible light. The total energy output, solar constant, is about 3.8 x 1033 ergs/sec. For further details (and more accurate figures), see the table below. THE SOLAR INTERIOR VISIBLE SURFACE OF SUN: PHOTOSPHERE CORE: THERMONUCLEAR ENGINE RADIATIVE ZONE CONVECTIVE ZONE SCHEMATIC CONVECTION CELLS Figure 1: Schematic representation of the regions in the interior of the Sun. Physical characteristics Photospheric composition Property Value Element % mass % number Diameter 1,392,530 km Hydrogen 73.46 92.1 Radius 696,265 km Helium 24.85 7.8 Volume 1.41 x 1018 m3 Oxygen 0.77 Mass 1.9891 x 1030 kg Carbon 0.29 Solar radiation (entire Sun) 3.83 x 1023 kW Iron 0.16 Solar radiation per unit area 6.29 x 104 kW m-2 Neon 0.12 0.1 on the photosphere Solar radiation at the top of 1,368 W m-2 Nitrogen 0.09 the Earth's atmosphere Mean distance from Earth 149.60 x 106 km Silicon 0.07 Mean distance from Earth (in 214.86 Magnesium 0.05 units of solar radii) In the interior of the Sun, at the centre, nuclear reactions provide the Sun's energy.
    [Show full text]
  • Mass – Luminosity Relation for Massive Stars
    MASS – LUMINOSITY RELATION FOR MASSIVE STARS Within the Eddington model β ≡ Pg/P = const, and a star is an n = 3 polytrope. Large mass stars have small β, and hence are dominated by radiation pressure, and the opacity in them is dominated by electron scattering. Let us consider the outer part of such a star assuming it is in a radiative equilibrium. We have the equation of hydrostatic equilibrium: dP GM = − r ρ, (s2.1) dr r2 and the equation of radiative equilibrium dP κρL r = − r . (s2.2) dr 4πcr2 Dividing these equations side by side we obtain dP κL r = r , (s2.3) dP 4πcGMr Near the stellar surface we have Mr ≈ M and Lr ≈ L, and adopting κ ≈ κe = const, we may integrate equation (s2.3) to obtain κeLr Pr − Pr,0 = (P − P0) , (s2.4) 4πcGMr 4 where Pr,0 = P0 = aTeff /6 is the radiation pressure at the surface, i.e. at τ = 0, where the gas pressure Pg = 0. At a modest depth below stellar surface pressure is much larger than P0, we may neglect the integration constants in (s2.4) to obtain P κ L L 1+ X M⊙ L 1 − β = r = e = = , (s2.5) P 4πcGM LEdd 65300 L⊙ M where LEdd is the Eddington luminosity. Equation (s2.5) gives a relation between stellar mass, luminosity and β. The Eddington model gives a relation between stellar mass and β : 1 2 M 18.1 (1 − β) / = 2 2 , (s2.6) M⊙ µ β where µ is a mean molecular weight in units of mass of a hydrogen atom, µ−1 =2X +0.75Y +0.5Z, and X,Y,Z, are the hydrogen, helium, and heavy element abundance by mass fraction.
    [Show full text]
  • Testing Quantum Gravity with LIGO and VIRGO
    Testing Quantum Gravity with LIGO and VIRGO Marek A. Abramowicz1;2;3, Tomasz Bulik4, George F. R. Ellis5 & Maciek Wielgus1 1Nicolaus Copernicus Astronomical Center, ul. Bartycka 18, 00-716, Warszawa, Poland 2Physics Department, Gothenburg University, 412-96 Goteborg, Sweden 3Institute of Physics, Silesian Univ. in Opava, Bezrucovo nam. 13, 746-01 Opava, Czech Republic 4Astronomical Observatory Warsaw University, 00-478 Warszawa, Poland 5Mathematics Department, University of Cape Town, Rondebosch, Cape Town 7701, South Africa We argue that if particularly powerful electromagnetic afterglows of the gravitational waves bursts detected by LIGO-VIRGO will be observed in the future, this could be used as a strong observational support for some suggested quantum alternatives for black holes (e.g., firewalls and gravastars). A universal absence of such powerful afterglows should be taken as a sug- gestive argument against such hypothetical quantum-gravity objects. If there is no matter around, an inspiral-type coalescence (merger) of two uncharged black holes with masses M1, M2 into a single black hole with the mass M3 results in emission of grav- itational waves, but no electromagnetic radiation. The maximal amount of the gravitational wave 2 energy E=c = (M1 + M2) − M3 that may be radiated from a merger was estimated by Hawking [1] from the condition that the total area of all black hole horizons cannot decrease. Because the 2 2 2 horizon area is proportional to the square of the black hole mass, one has M3 > M2 + M1 . In the 1 1 case of equal initial masses M1 = M2 = M, this yields [ ], E 1 1 h p i = [(M + M ) − M ] < 2M − 2M 2 ≈ 0:59: (1) Mc2 M 1 2 3 M From advanced numerical simulations (see, e.g., [2–4]) one gets, in the case of comparable initial masses, a much more stringent energy estimate, ! EGRAV 52 M 2 ≈ 0:03; meaning that EGRAV ≈ 1:8 × 10 [erg]: (2) Mc M The estimate assumes validity of Einstein’s general relativity.
    [Show full text]
  • Astronomy 112: the Physics of Stars Class 10 Notes: Applications and Extensions of Polytropes in the Last Class We Saw That Poly
    Astronomy 112: The Physics of Stars Class 10 Notes: Applications and Extensions of Polytropes In the last class we saw that polytropes are a simple approximation to a full solution to the stellar structure equations, which, despite their simplicity, yield important physical insight. This is particularly true for certain types of stars, such as white dwarfs and very low mass stars. In today’s class we will explore further extensions and applications of simple polytropic models, which we can in turn use to generate our first realistic models of typical main sequence stars. I. The Binding Energy of Polytropes We will begin by showing that any realistic model must have n < 5. Of course we already determined that solutions to the Lane-Emden equation reach Θ = 0 at finite ξ only for n < 5, which already hints that n ≥ 5 is a problem. Nonetheless, we have not shown that, as a matter of physical principle, models of this sort are unacceptable for stars. To demonstrate this, we will prove a generally useful result about the energy content of polytropic stars. As a preliminary to this, we will write down the polytropic relation (n+1)/n P = KP ρ in a slightly different form. Consider a polytropic star, and imagine moving down within it to the point where the pressure is larger by a small amount dP . The corre- sponding change in density dρ obeys n + 1 dP = K ρ1/ndρ. P n Similarly, since P = K ρ1/n, ρ P it follows that P ! K 1 dP ! d = P ρ(1−n)/ndρ = ρ n n + 1 ρ We can regard this relation as telling us how much the ratio of pressure to density changes when we move through a star by an amount such that the pressure alone changes by dP .
    [Show full text]
  • Ultraluminous X-Ray Pulsars
    Ultraluminous X-ray pulsars: the high-B interpretation Juri Poutanen (University of Turku, Finland; IKI, Moscow) Sergey Tsygankov, Alexander Mushtukov, Valery Suleimanov, Alexander Lutovinov, Anna Chashkina, Pavel Abolmasov Plan: • ULX before 2014 • A short history of ULXPs • Supercritical accretion onto a magnetized neutron star • Large or low B? Propeller? • Beamed? • Winds? Models for ULX (before 2014) • Super-Eddington accretion onto a stellar-mass black hole (e.g. King 2001, Begelman et al. 2006, Poutanen et al. 2007) • Sub-Eddington accretion onto intermediate mass black holes (Colbert & Mushotzky 2001) • Young rotation-powered pulsar (Medvedev & Poutanen 2013) Super-Eddington accretion • Slim disk models Accretion rate is large, but most of the released energy is advected towards the BH. ˙ ˙ M (r) =M 0 • Super-disks with winds Accretion rate is large, but most of the mass is blown away by radiation. Only the Eddington rate goes to the BH. ˙ ˙ r M (r) =M 0 Rsph ˙ ˙ M (rin ) =M Edd M˙ R R 0 - spherization radius sph ≈ in ˙ M Edd Shakura & Sunyaev 1973 Super-Eddington accretion • Slim disk models Accretion rate is large, but most of the released energy is advected towards the BH. ˙ ˙ M (r) =M 0 • Super-disks with winds Accretion rate is large, but most of the mass is blown away by radiation. Only the Eddington rate goes to the BH. ! ! r M (r) =M 0 Rsph ! ! M(rin ) =M Edd M! R ≈ R 0 -spherization radius sph S ! M Edd Ohsuga et al. 2005 Super-Eddington accretion Super-disks with winds and advection Accretion rate is large, a lot of mass is blown away by radiation (using fraction εW of available radiative flux), but still a significant fraction goes to the BH.
    [Show full text]