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Stellar Astrophysics: Introduction Q. Daniel Wang Astronomy Department University of Massachusetts Why should we care about stellar astrophysics? Why should we care about stellar astrophysics? I stars are a major constituent of the visible universe I understanding how stars work is probably the earliest major triumph of astrophysics I stars are responsible for the chemical composition of the universe I mass and energy feedback (supernovae) stellar astrophysics is the foundation for the astronomy and astrophysics in general I I I I cosmic yardsticks to estimate the distances (e.g., Cepheid variables) kinematic status of large scale structure (redshifts) time scales and mechanisms for the formation of stellar populations and galaxies (stellar ages) Outline Review of the course syllabus Basic equations Mass conservation Momentum Conservation and Hydrostatic Equilibrium Application example: constant-density model Molecular Weights Energy Generation Energy Transportation The HR diagram Homology and Scaling Law Virial theorem Dimensional Analysis Summary and Conclusions Approaches A lot to cover! Emphasize on the physical principles, methodologies, and connections to what you might be interested in doing in your research. I lectures are mainly to provide a coherent thread of the course and to cover essential and/or difficult parts. I reading materials will be discussed and quizzed; your questions can be part of the class discussion I homework is to consolidate and expand your understanding I exams to review systematically what has been learned and to check the mastering of the materials I Research and writing assignments to further expand and integrate your knowledge. Scopes of this chapter I introduce some basic concepts and physical processes. I estimate various time and physical scales of stars. I paint a preliminary picture of the stellar interior. To be expanded later to the modern development of stellar structure and evolution. See next chapter for a narrative description of the development. Assumptions: single isolated, spherical symmetric, and mechanically steady. Outline Review of the course syllabus Basic equations Mass conservation Momentum Conservation and Hydrostatic Equilibrium Application example: constant-density model Molecular Weights Energy Generation Energy Transportation The HR diagram Homology and Scaling Law Virial theorem Dimensional Analysis Summary and Conclusions Mass conservation What is a star? Mass conservation What is a star? A steady shining body, self-sustained, getting most energy from nuclear burning. Mass conservation What is a star? A steady shining body, self-sustained, getting most energy from nuclear burning. How is the balance achieved in a star? Mass conservation What is a star? A steady shining body, self-sustained, getting most energy from nuclear burning. How is the balance achieved in a star? Consider a shell with a thickness of dr , the mass included is dMr = ρ4πr 2 dr . (1) The integration of this equation gives the mass Mr within the radius r . Since the radius of a star can change greatly during its lifetime, sometimes quickly, while the change in mass is relatively small and slow, it is convenient to use the Mr as the coordinate; i.e., expressing various equations in the Lagrangian form. Momentum Conservation and Hydrostatic Equilibrium Stars are mostly steady, spending most of their lifetime converting H into He. Mostly forces are balanced between gravity - only the mass within r matters and acts as if it is at the center of a star (the gravitational potential is different). I pressure - most importantly its differences between the inner and outer boundaries of a shell considered here. I The equation of motion: GMr ρ dP − (2) dr r2 For a steady star, the balance of forces leads to the hydrostatic equilibrium: GMr ρ dP =− 2 , (3) dr r which in the Lagrangian form is ρr̈ = − dP GMr =− . (4) dMr 4πr 4 This form is especially useful when the stellar evolution (hence the structure change) is discussed. Application example: constant-density model Working out a simple model of a star with a constant density, which satisfies the above hydrostatic equilibrium equation and the boundary condition: P = 0 at the star surface. Application example: constant-density model Working out a simple model of a star with a constant density, which satisfies the above hydrostatic equilibrium equation and the boundary condition: P = 0 at the star surface. First, we need to eliminate r . The constant density implies Mr = r3 M R3 (5) Now, we have GM 4/3 −1/3 dP M =− dMr 4πR 4 r Integrating this equation from the center to Mr gives P − Pc = − 3 GM 4/3 2/3 M 2 4πR 4 r (6) (7) Application example: constant-density model Working out a simple model of a star with a constant density, which satisfies the above hydrostatic equilibrium equation and the boundary condition: P = 0 at the star surface. First, we need to eliminate r . The constant density implies Mr = r3 M R3 (5) Now, we have GM 4/3 −1/3 dP M =− dMr 4πR 4 r Integrating this equation from the center to Mr gives 3 GM 4/3 2/3 M 2 4πR 4 r Using the boundary condition: P = 0 at Mr = M, we have P − Pc = − Pc = 3 GM 2 2 4πR 4 (6) (7) (8) We then have Mr 2/3 P = Pc 1 − (9) M This central pressure is a lower limit if density always decreases outward. By using the equation of state (EoS) for a monatomic ideal gas, P = nkT , one can get the temperature structure. But what is an ideal gas? We then have Mr 2/3 P = Pc 1 − (9) M This central pressure is a lower limit if density always decreases outward. By using the equation of state (EoS) for a monatomic ideal gas, P = nkT , one can get the temperature structure. But what is an ideal gas? It is composed of a set of randomly-moving, non-interacting point particles except when they collide elastically. We also need to know how to calculate n. Molecular Weights The thermodynamic relations between P, ρ, and T , as well as the calculation of stellar opacity requires knowledge of the mean molecular weight µ (the mean mass of a particle in Atomic Mass Units). ρ = ni + ne (10) n≡ µmA 1 1 1 ≡ + µ µi µe If the mass fraction of species i is xi , then we have 1 x = Σi i µi Ai (11) (12) 1 x = Σi i Zi fi (13) µe Ai where Ai is the atomic weight of the species, Zi be the atomic number of species i, and fi be the species ionization fraction, i.e., the fraction of electrons of i that are free. Note that in the case of total ionization (fi = 1), this equation simplifies greatly. Since Zi /Ai = 1 for hydrogen, and ∼ 1/2 for everything else, µe = −1 2 1 = X + (Y + Z ) 2 1+X (14) 1 . (X + Y /4) (15) µi ≈ For a “zero-age main sequence” star (ZAMS), X ≈ 0.7, Y = 0.3, and Z ≈ 0.03: µi = 1.3, µe = 1.2 and µ = 0.6. Using this µ value in the EoS, we can now get the temperature structure. In particular, the central temperature for the Sun can be estimated as Tc = 1 GM µmA ≈ 6 × 106 K 2 R k (16) To balance the gravity, a star must have a high pressure, which is realized with both high temperature and density. How to maintain such a high temperature? Using this µ value in the EoS, we can now get the temperature structure. In particular, the central temperature for the Sun can be estimated as Tc = 1 GM µmA ≈ 6 × 106 K 2 R k (16) To balance the gravity, a star must have a high pressure, which is realized with both high temperature and density. How to maintain such a high temperature? We need to consider both the energy generation and transportation. Energy Generation Neglecting terms due to the energy input or loss to gravitational expansion or contraction and due to neutrinos, we can express the energy equation as dLr = . dMr (17) Over some sufficiently restricted range of T , ρ, and composition, one may approximate in a power law form = 0 ρλ T µ , (18) where λ = 1, 1, and 2 and µ ≈ 4, 15, and 40 for pp-chain, CNO-cycles, and triple-α modes, respectively. For the conversion of hydrogen to Helium, about 0.7% of the rest mass energy is released, which is 6 × 1018 ergs for energy per gram of hydrogen consumed. Energy Generation Neglecting terms due to the energy input or loss to gravitational expansion or contraction and due to neutrinos, we can express the energy equation as dLr = . dMr (17) Over some sufficiently restricted range of T , ρ, and composition, one may approximate in a power law form = 0 ρλ T µ , (18) where λ = 1, 1, and 2 and µ ≈ 4, 15, and 40 for pp-chain, CNO-cycles, and triple-α modes, respectively. For the conversion of hydrogen to Helium, about 0.7% of the rest mass energy is released, which is 6 × 1018 ergs for energy per gram of hydrogen consumed. With this energy efficiency, how long will sun last? (assuming that about 10% of the Sun’s mass will be fused and 70% of it is hydrogen). Energy Generation Neglecting terms due to the energy input or loss to gravitational expansion or contraction and due to neutrinos, we can express the energy equation as dLr = . dMr (17) Over some sufficiently restricted range of T , ρ, and composition, one may approximate in a power law form = 0 ρλ T µ , (18) where λ = 1, 1, and 2 and µ ≈ 4, 15, and 40 for pp-chain, CNO-cycles, and triple-α modes, respectively. For the conversion of hydrogen to Helium, about 0.7% of the rest mass energy is released, which is 6 × 1018 ergs for energy per gram of hydrogen consumed. With this energy efficiency, how long will sun last? (assuming that about 10% of the Sun’s mass will be fused and 70% of it is hydrogen). t ∼ 1010 yrs Energy Transportation modes Now, what determines the luminosity of a star? Energy Transportation modes Now, what determines the luminosity of a star? The energy leak makes a star shine. But this leakage must be slow to maintain a star steady. A feedback mechanism is needed, like a thermometer. Energy Transportation modes Now, what determines the luminosity of a star? The energy leak makes a star shine. But this leakage must be slow to maintain a star steady. A feedback mechanism is needed, like a thermometer. The energy generation also needs to be balanced by energy removal, but not too fast. Then we call the material is in “thermal balance”. Energy Transportation modes Now, what determines the luminosity of a star? The energy leak makes a star shine. But this leakage must be slow to maintain a star steady. A feedback mechanism is needed, like a thermometer. The energy generation also needs to be balanced by energy removal, but not too fast. Then we call the material is in “thermal balance”. Three major modes of energy transportation: 1 radiation (photon) transfer 2 convection of hotter and cooler materials 3 heat conduction (only important under degenerate condition, i.e., in white dwarfs). Energy Transportation modes Now, what determines the luminosity of a star? The energy leak makes a star shine. But this leakage must be slow to maintain a star steady. A feedback mechanism is needed, like a thermometer. The energy generation also needs to be balanced by energy removal, but not too fast. Then we call the material is in “thermal balance”. Three major modes of energy transportation: 1 radiation (photon) transfer 2 convection of hotter and cooler materials 3 heat conduction (only important under degenerate condition, i.e., in white dwarfs). For the time being we consider the radiation transfer or diffusion. Radiation transfer Consider a system of particles diffusing across a boundary in the z direction. The net number flux of particles diffuse across the boundary is F ≈ 1 1 dn v̄ [nz−l − nz+l ] = − v̄ l , 6 3 dz (19) where l ≡ 1/nσ is the particle mean free path, while σ is the cross section of the collision. We have the Fick’s law of diffusion: dn F = −D , (20) a dz a v̄ Diffuse of particles where D = is the diffusion across a boundary in z 3nσ direction. coefficient. Radiation transfer Similarly, we can compute the energy flux of radiative energy across a boundary. Treat the photons as particles and recall the energy density U = aT 4 and lph = 1/κρ, where κ is the absorption coefficient. Thus, the energy transportation equation is clph dU 4caT 3 dT =− , (21) F =− 3 dz 3κρ dz or (4πr 2 )2 4caT 3 dT , (22) Lr = − 3κ dMr where a = 4σs /c and σs = 5.7 × 10−5 cgs is the Stefan-Boltzmann’s constant. Opacity The calculation of the opacity is a whole industry. Here we use a generic opacity form κ = κ0 ρn T −s cm2 g−1 . (23) For the Thompson scattering in an ionized medium, n = 0 and s = 0, where n = 1 and s = 3.5 for Kramers’ opacity, characteristic of radiative processes involving atoms. Summary In summary, under the steady and spherical assumptions, we have described the basic equations (mass, momentum, energy, and heat transfer): dr 1 = , (24) dMr 4πr 2 ρ GMr dP =− , dMr 4πr 4 dLr = 0 ρλ T µ , dMr (25) (26) (4πr 2 )2 4caT 3 dT (27) 3κ dMr These equations, together with an E.O.S., assumed to be P ∝ ρχρ T χT , and appropriate boundary conditions, allow one to solve for r , ρ, Lr , and T as functions of Mr . Lr = − Summary In summary, under the steady and spherical assumptions, we have described the basic equations (mass, momentum, energy, and heat transfer): dr 1 = , (24) dMr 4πr 2 ρ GMr dP =− , dMr 4πr 4 dLr = 0 ρλ T µ , dMr (25) (26) (4πr 2 )2 4caT 3 dT (27) 3κ dMr These equations, together with an E.O.S., assumed to be P ∝ ρχρ T χT , and appropriate boundary conditions, allow one to solve for r , ρ, Lr , and T as functions of Mr . Based on these equations, we may also get some ideas about various scales and dependencies among various stellar properties. Lr = − Outline Review of the course syllabus Basic equations Mass conservation Momentum Conservation and Hydrostatic Equilibrium Application example: constant-density model Molecular Weights Energy Generation Energy Transportation The HR diagram Homology and Scaling Law Virial theorem Dimensional Analysis Summary and Conclusions The HR diagram I Please draw a Hertzsprung-Russell (HR) diagram, both theorist’s and observer’s versions, and explain the axes, as well as directions of stellar radius and mass). I Show an example of the diagram and point out the main sequence, giants, white dwarf etc. I How to get R, M, and age of a star? In the HR diagram? The HR diagram a a a Theorist’s version of HR diagram. a HR (or CMD) diagram of ”nearby” stars measured by Hipparcos. Colors indicate multiple stars at that position. An important way to characterize the properties of stars: power output vs. temperature or equivalent. The exact units of the axes depend on the context and who present them. For historic reason, the temperature axis has the highest value on the left. The HR diagram As the radiation from a star photosphere is close to a black-body, one can define Teff as 1/4 L Teff ≡ , (28) 4πσs R 2 a Density of stars depends on the lifetime of individual processes that govern the evolution at different stages. a Theorist’s version of HR diagram. where R is the radius of the photosphere. The HR diagram As the radiation from a star photosphere is close to a black-body, one can define Teff as 1/4 L Teff ≡ , (28) 4πσs R 2 a Density of stars depends on the lifetime of individual processes that govern the evolution at different stages. a Theorist’s version of HR diagram. where R is the radius of the photosphere. The L − Teff diagram itself gives no further information than L, Teff , and R and says nothing about mass, composition, or state of evolution. But these may be inferred from from the distribution of stars in the diagram, plus the modeling. Outline Review of the course syllabus Basic equations Mass conservation Momentum Conservation and Hydrostatic Equilibrium Application example: constant-density model Molecular Weights Energy Generation Energy Transportation The HR diagram Homology and Scaling Law Virial theorem Dimensional Analysis Summary and Conclusions Homology and Scaling Law A very useful technique, not only for the study of stars, but for other astrophysical problems. Without getting all the solutions, such analysis can provide insights into how physical quantities depend on each other and scaled. Homology and Scaling Law A very useful technique, not only for the study of stars, but for other astrophysical problems. Without getting all the solutions, such analysis can provide insights into how physical quantities depend on each other and scaled. Consider the four basic equations and remove all the constants: dr 1 ∝ 2 dMr r ρ dP Mr ∝− 4 dMr r dLr ∝ ρλ T µ dMr r 4 T 3 dT Lr ∝ − n −s , ρ T dMr together with the E.O.S., P ∝ ρ χρ T χT . a a assuming the radiation transfer and implicitly a uniform chemical When all the constants and exponents are assumed to be the same, solutions from one star to another with different masses are scalable. This family of stars is said to be in a homologous sequence. Homology and Scaling Law To show this, we do the following variable transformation in the above equations: 1 dr ∝ 2 dMr r ρ dP Mr ∝− 4 dMr r dLr ∝ ρλ T µ dMr r 4 T 3 dT Lr ∝ − n −s ρ T dMr r = r 0 M αr , ρ = ρ 0 M αρ , Lr = Lr ,0 M αL , T = T0 M αT , Mr = Mr ,0 M, where M is any constant. We then obtain, for example, M αr −1 1 dr0 ∝ M −2αr −αρ 2 . dMr ,0 r0 ρ0 If we have αr − 1 = −2αr − αρ or 3αr + αρ = 1, the above two mass equations are the exactly the same. Homology and Scaling Law 1 dr ∝ 2 dMr r ρ Mr dP ∝− 4 dMr r dLr ∝ ρλ T µ dMr r 4 T 3 dT Lr ∝ − n −s ρ T dMr All together, we have 3αr + αρ = 1 4αr + χρ αρ + χT αT = 2 λαρ − αL + µαT = −1 4αr − nαρ − αL + (4 + s)αT = 1 This set of linear equations can be solved (if the determinant is not zero) to get the α∗ values. Homology and Scaling Law For example, consider upper MS stars as a homologous sequence. For these stars, the opacity is mostly due to electron scattering (i.e., n = s = 0) and the nuclear reaction is due to CNO cycles (i.e., λ = 1 and µ = 15. The pressure is still dominated by the ideal gas law (i.e., χρ = χT = 1). Homology and Scaling Law For example, consider upper MS stars as a homologous sequence. For these stars, the opacity is mostly due to electron scattering (i.e., n = s = 0) and the nuclear reaction is due to CNO cycles (i.e., λ = 1 and µ = 15. The pressure is still dominated by the ideal gas law (i.e., χρ = χT = 1). The inferred αr = 0.78 and αL = 3.0 may be compared with the empirical fitted values of 0.75 and 3.5, respectively. In addition, we get αT = 0.22 and αρ = −1.3 so that T should increase with the mass whereas ρ decreases. This is indeed what happens! Homology and Scaling Law For example, consider upper MS stars as a homologous sequence. For these stars, the opacity is mostly due to electron scattering (i.e., n = s = 0) and the nuclear reaction is due to CNO cycles (i.e., λ = 1 and µ = 15. The pressure is still dominated by the ideal gas law (i.e., χρ = χT = 1). The inferred αr = 0.78 and αL = 3.0 may be compared with the empirical fitted values of 0.75 and 3.5, respectively. In addition, we get αT = 0.22 and αρ = −1.3 so that T should increase with the mass whereas ρ decreases. This is indeed what happens! For lower MS stars, the physics are different (different λ, µ, χρ , and χT ). The convection is also more important. The homology does not work as well. A more general discussion of the scalability of the hydrodynamic solution or simulation and application examples can be found in Tang & Wang (2009, MNRAS, 397, 2106). Outline Review of the course syllabus Basic equations Mass conservation Momentum Conservation and Hydrostatic Equilibrium Application example: constant-density model Molecular Weights Energy Generation Energy Transportation The HR diagram Homology and Scaling Law Virial theorem Dimensional Analysis Summary and Conclusions Virial theorem 1 d 2I = 2K + Ω, 2 dt 2 (29) where the momentum of inertia I = Σi mi ri2 , the kinetic energy Gmi mj 1 . K = Σi mi vi2 , and the potential energy Ω = −Σ 2 ri,j Virial theorem 1 d 2I = 2K + Ω, 2 dt 2 (29) where the momentum of inertia I = Σi mi ri2 , the kinetic energy Gmi mj 1 . K = Σi mi vi2 , and the potential energy Ω = −Σ 2 ri,j For a gaseous system (e.g., a star), Z 3P dMr (30) K = M 2ρ and Z Ω=− M GMr dMr . r (31) The system is called to be virialized (in kind of dynamic equilibrium), if the r.h.s of Eq. 29, averaged over time, is equal to zero (i.e., statistically does not change with time). dI i 1 h dI − /(t2 − t1 ) = 2K + Ω = 0, 2 dt t1 dt t2 (32) where t2 − t1 is sufficiently large. Note that the above expression of the theorem applies to the entire system (e.g., the entire star). For a portion of a star, one needs to account for external force (e.g., the pressure at the boundary). Consider a so-called “γ−law” equation of state (EoS): P = (γ − 1)ρE, (33) where E is the mass specific internal energy density. Eq. 30 R 3 then can be written as K = (γ − 1)U, where U = EdMr . 2 Thus K = U only if γ = 5/3; that is, the total kinetic energy is the same as the total internal energy only under certain circumstances! Application of the Virial theorem From Eq. 32, we have 1 (34) K =− Ω 2 Think about the formation of a star and a galaxy from a cloud of gas. When it is virialized, only half of the potential energy loss is turned into the kinetic energy (thermal or orbit motion). Where is the rest of the energy? Application of the Virial theorem From Eq. 32, we have 1 (34) K =− Ω 2 Think about the formation of a star and a galaxy from a cloud of gas. When it is virialized, only half of the potential energy loss is turned into the kinetic energy (thermal or orbit motion). Where is the rest of the energy? Radiated away! Application of the Virial theorem From Eq. 32, we have 1 (34) K =− Ω 2 Think about the formation of a star and a galaxy from a cloud of gas. When it is virialized, only half of the potential energy loss is turned into the kinetic energy (thermal or orbit motion). Where is the rest of the energy? Radiated away! Can you think any example for the application? Outline Review of the course syllabus Basic equations Mass conservation Momentum Conservation and Hydrostatic Equilibrium Application example: constant-density model Molecular Weights Energy Generation Energy Transportation The HR diagram Homology and Scaling Law Virial theorem Dimensional Analysis Summary and Conclusions Dimensional Analysis Contraction and nuclear and their time scales. What is the dynamic scale? We can use simple dimensional analysis and central temperature estimate. Outline Review of the course syllabus Basic equations Mass conservation Momentum Conservation and Hydrostatic Equilibrium Application example: constant-density model Molecular Weights Energy Generation Energy Transportation The HR diagram Homology and Scaling Law Virial theorem Dimensional Analysis Summary and Conclusions Summary and Conclusion In summary, under the steady and spherical assumptions, we have described the basic equations (mass, momentum, energy, and heat transfer). These equations, together with an equation of the state as well as the energy generation and opacity forms, allow for the construction of a complete stellar model (e.g., solving for dependent variables: r , ρ, T , and L as function of Mr ). Summary and Conclusion In summary, under the steady and spherical assumptions, we have described the basic equations (mass, momentum, energy, and heat transfer). These equations, together with an equation of the state as well as the energy generation and opacity forms, allow for the construction of a complete stellar model (e.g., solving for dependent variables: r , ρ, T , and L as function of Mr ). We may also get some ideas about various scales and dependencies among various stellar properties, based on some scaling or dimension analyses. Summary and Conclusion In summary, under the steady and spherical assumptions, we have described the basic equations (mass, momentum, energy, and heat transfer). These equations, together with an equation of the state as well as the energy generation and opacity forms, allow for the construction of a complete stellar model (e.g., solving for dependent variables: r , ρ, T , and L as function of Mr ). We may also get some ideas about various scales and dependencies among various stellar properties, based on some scaling or dimension analyses. While the above outlines the general approach to construct stellar models, we will have in-depth look of the physics of the EoS, the heat transfer, and energy generation, before showing how the above equations can actually be solved. Review Questions 1. How is the hydrostatic balance achieved in a star? 2. What is the molecular weight (which you should be able to derive, depending on the chemical composition of the gas)? 3. Now what is the lifetime of the Sun with the nuclear power? 4. What is the virial theorem? 5. Can you have a quick estimate of the central temperature of the Sun, based on a simple dimensional analysis? How may the temperature depend on the average density and mass of a star (assuming ideal gas)? 6. Can you estimate the lifetime of the Sun if it were powered by the gravitational energy alone? 7. What is the dynamic (or free-fall) time scale of the Sun? 8. Can you derive an expression of the gravitational potential for a star (assuming spherical symmetry)? 9. What is the H-R diagram? Point out the directions of increasing temperature, luminosity, mass, and radius. Mark the locations of various types of stars you know. Trace evolutionary tracks of stars. 10. How do the central density and temperature of a main-sequence star depend on its mass, qualitatively?