Astrophysics 1 PDF Lecture Notes

Astrophysics 1 Przemyslaw Walczak1 1 Instytut Astronomiczny Uniwersytet Wroclawski, Wroclaw, Poland SZ 2024/25 Walczak (UWr) Stellar Structure and evolution SZ 2024/25 1 / 134 ...

Astrophysics 1 Przemyslaw Walczak1 1 Instytut Astronomiczny Uniwersytet Wroclawski, Wroclaw, Poland SZ 2024/25 Walczak (UWr) Stellar Structure and evolution SZ 2024/25 1 / 134  Literature Literature Kippenhahn, R., Weigert, A., Weiss, A. (2012). Stellar Structure and Evolution. Springer-Verlag, Berlin, Heidelberg 2012. Jørgen Christensen-Dalsgaard, Lecture Notes on Stellar Structure and Evolution, Institut for Fysik og Astronomi, Aarhus Universitet Walczak (UWr) Stellar Structure and evolution SZ 2024/25 2 / 134  Introduction Figure: Schematic illustration of the relation between physics, stellar models and observations. Lecture Notes on Stellar Structure and Evolution (Christensen-Dalsgaard, 2023, LNSSE) Walczak (UWr) Stellar Structure and evolution SZ 2024/25 3 / 134  Hertzsprung–Russell diagram Figure: An observational Hertzsprung-Russell diagram with 22 000 stars plotted from the Hipparcos Catalogue and 1 000 from the Gliese Catalogue of nearby stars. Source: Richard Powell - The Hertzsprung Russell Diagram, CC BY-SA 2.5 Walczak (UWr) Stellar Structure and evolution SZ 2024/25 4 / 134  Hertzsprung–Russell diagram Figure: HR Diagram showing the evolutionary tracks of stars with masses between 1M⊙ and 40M⊙ (full lines, Schaller et al. 1992). The dashed line is the zero-age main sequence and the dotted line symbolizes the transition phase from the Asymptotic Giant Branch to the white-dwarf cooling track (Christensen-Dalsgaard, 2023, LNSSE) Walczak (UWr) Stellar Structure and evolution SZ 2024/25 5 / 134  Hertzsprung–Russell diagram Figure: Schematic illustration of the evolution of a moderate-mass star. The effective temperature Tef f is in K, and the luminosity Ls is measured in units of the solar luminosity L⊙. The dashed line indicates the zero-age main sequence, corresponding to the onset of hydrogen burning (Christensen-Dalsgaard, 2023, LNSSE) Walczak (UWr) Stellar Structure and evolution SZ 2024/25 6 / 134  Time scales Time scales Time scales Walczak (UWr) Stellar Structure and evolution SZ 2024/25 7 / 134  Time scales Dynamical time scale Dynamical time scale The shortest relevant time scale is the dynamical timescale tdyn. Consider a star of mass M and radius R. The gravitational acceleration at the surface of the star is GM g= , R2 where G is the gravitational constant. Hence the time required for a particle to fall the distance ℓ = 21 gt2 in the gravitational field of the star is 1/2 1/2 2ℓR2   2ℓ t= =. g GM If we take ℓ to be R/2 we obtain a time scale that is characteristic for motions over stellar scales in the gravitational field:  3 1/2 R tdyn =. GM Using the solar values M and R for M and R, we may write the equation as R 3/2 M −1/2     tdyn = 30 min. R⊙ M⊙ Stellar radii vary over a range from roughly 0.01R⊙ to roughly 1000R⊙ , whereas the mass ranges from 0.1M⊙ to 100M⊙. Hence the dynamical time scale ranges from seconds to years. Walczak (UWr) Stellar Structure and evolution SZ 2024/25 8 / 134  Time scales Dynamical time scale Thermal time scale L - the surface luminosity of the star; the amount of energy it radiates per unit time The rough estimate of the gravitational energy is GM 2 Ω=−. R Then the thermal time scale, known as the Kelvin-Helmholtz scale is GM 2 tKH = , RL  2  −1  −1 M R L tKH = 30 million years M⊙ R⊙ L⊙ Walczak (UWr) Stellar Structure and evolution SZ 2024/25 9 / 134  Time scales Dynamical time scale Nuclear time scales During most of the life of a star the energy it radiates comes from the fusion of hydrogen into helium. The total amount of energy that is available from this reaction may be estimated as ∆E = ∆mc2 , where ∆m is the difference in mass between the original hydrogen and the resulting helium, and c is the speed of light. In the fusion of hydrogen to helium about 0.7% of the mass is lost. The reaction occurs only in the inner part of the star, let’s say in about 10 per cent of the mass. Hence the total amount of energy available is approximately 7 × 10−4 M c2 , and the corresponding timescale is M c2 tnuc = 7 × 10−4 , L or   −1 M L⊙ tnuc = 1010 years. M⊙ L Since a star spends by far the largest part of its life in the hydrogen burning phase, tnuc provides a measure of the total lifetime of a star. For a 30M⊙ star the entire evolution, from birth to death, only lasts about 5 million years, whereas a star of 0.5M has barely had time to begin evolving over the age of the Universe. Walczak (UWr) Stellar Structure and evolution SZ 2024/25 10 / 134  Evolution of the star Nuclear reactions 41 H →4 He + 2e+ + 2νe. 34 He →12 C. 4 He +12 C →16 O. Walczak (UWr) Stellar Structure and evolution SZ 2024/25 11 / 134  Ideal gas Ideal gas Ideal gas Walczak (UWr) Stellar Structure and evolution SZ 2024/25 12 / 134  Ideal gas Ideal gas An ideal gas is a theoretical gas composed of many randomly moving non-relativistic point particles (classical particles). The ideal-gas law for is written as P = nkT, or P V = N kT where n is the number of particles per unit volume, N is the number of particles in a volume V and k is Boltzmann constant. Lets introduce the (dimensionless) atomic mass µ of the particles. ρ µ= , nmu where mu is the atomic mass unit, equals, by definition, as one-twelfth the mass of an atom of carbon-12. ρ is the mass density. With this we can write: ρkT P =. µmu This equation is, basically, the most commonly used form of the ideal gas law. Walczak (UWr) Stellar Structure and evolution SZ 2024/25 13 / 134  Ideal gas Ideal gas The mean internal energy in a unit volume is 3 3 ρkT U = nkT =. (1) 2 2 µmu The energetics of the gas as the star evolves or the gas moves plays a very important role for stellar evolution. The basic equation describing the changes in the properties of the gas is the first law of thermodynamics, dQ = dUV + P dV, (2) valid for a fixed amount of matter, where UV is the internal energy of the matter, and V is the volume it occupies. Here dQ is an amount of heat added to the matter. Let V be the volume corresponding to unit mass, so that 1 1 V = , and dV = − 2 dρ. ρ ρ Then UV = U · V = U/ρ is the specific internal energy, i.e., the internal energy per unit mass. Therefore we have P dQ = dUV − dρ. (3) ρ2 Walczak (UWr) Stellar Structure and evolution SZ 2024/25 14 / 134  Ideal gas Ideal gas We now consider a process where the volume does not change. We introduce the specific heat at constant volume, cV , as the amount of heat that has to be added, per unit mass, to raise the temperature one degree. It follows from eq. 2 that dQ 3 k cV ≡ =. (4) dT V 2 µmu 3 k Note also, that dUV = 2 µmu dT = cV dT. It is also of interest to consider a process where the pressure is constant. To do so we write dQ = dUV + P dV = dUV + P dV + V dP − V dP. dQ = dUV − V dP + P dV + V dP = dUV − V dP + d(P V ) = dUV − V dP + d(P/ρ). ρkT Now we use the ideal gas law P = µmu. Divide it by ρ: kT P/ρ = µmu and differentiate k d(P/ρ) = dT. µmu Walczak (UWr) Stellar Structure and evolution SZ 2024/25 15 / 134  Ideal gas Ideal gas 3 kT Remember that UV = U · V = U/ρ = 2 µmu. Therefore 3 k k dQ = dUV − V dP + d(P/ρ) = dT − V dP + dT. 2 µmu µmu After simplification 5 k dQ = dT − V dP. 2 µmu The specific heat at constant pressure (dP = 0) is then dQ 5 k cp ≡ =. (5) dT P 2 µmu dQ 3 k Since cV ≡ dT V = 2 µmu we can notice that: 5 k 3 k k cP − cV = − =. (6) 2 µmu 2 µmu µmu Walczak (UWr) Stellar Structure and evolution SZ 2024/25 16 / 134  Ideal gas Ideal gas A particularly important type of process are the adiabatic processes, which occur without any exchange of heat, i.e., with dQ = 0. If we put this into the first law of thermodynamics: dQ = dUV + P dV = cV dT + P dV, we get P kT dρ dT dρ cV dT = −P dV or cV dT = dρ or cV dT = or cV = (cP − cV ). (7) ρ2 µmu ρ T ρ To obtain a relation between the changes in P and V (or ρ) for an adiabatic process, we use again the ideal gas law P = ρkT /µmu. If we differentiate it we get kT ρk dP = dρ + dT. µmu µmu Lets divide it by pressure P dP dρ dT = +. P ρ T Walczak (UWr) Stellar Structure and evolution SZ 2024/25 17 / 134  Ideal gas Ideal gas We have:   dP dρ dρ cV − = (cP − cV ). P ρ ρ dP dρ (cP − cV ) dρ − =. P ρ cV ρ and finally dP cP dρ dρ dV = =γ = −γ , P cV ρ ρ V where we introduced γ = cP /cV. For the ideal gas which we are considering it follows from eq. 4 and 5 that 5 γ=. 3 Walczak (UWr) Stellar Structure and evolution SZ 2024/25 18 / 134  Ideal gas Ideal gas We may also write above equation as   ∂ ln P = γ, (8) ∂ ln ρ s where the subscript “s” indicates that the partial derivative is taken at constant s From this equation, and using the ideal gas law, one can derive relations between the changes in other thermodynamic variables under adiabatic changes. In particular, one finds that   ∂ ln P γ = , (9) ∂ ln T s γ−1 and   ∂ ln T = γ − 1. (10) ∂ ln ρ s Walczak (UWr) Stellar Structure and evolution SZ 2024/25 19 / 134  Ideal gas Ideal gas he above derivation assumes that particle motion is the sole contributor to internal energy. This holds true for a fully ionized ideal gas or for a gas of neutral atoms. Under more complex conditions, however, the relationships derived – specifically eq. 8 - 10 - -do not apply. Nevertheless, it remains useful to describe adiabatic changes using analogous equations. This is the motivation for defining the adiabatic exponents   ∂ ln P Γ1 = , (11) ∂ ln ρ s   Γ2 ∂ ln P = , (12) Γ2 − 1 ∂ ln T s   ∂ ln T Γ3 − 1 =. (13) ∂ ln ρ s Walczak (UWr) Stellar Structure and evolution SZ 2024/25 20 / 134  Ideal gas Ideal gas These three quantities are not independent. In fact, from the chain rule of differentiation it follows that, for example       Γ2 ∂ ln P ∂ ln P ∂ ln ρ Γ1 = = = Γ2 − 1 ∂ ln T s ∂ ln ρ s ∂ ln T s Γ3 − 1 without any assumptions about the equation of state. For ideal gas we have Γ1 = Γ2 = Γ3 = 5/3. Walczak (UWr) Stellar Structure and evolution SZ 2024/25 21 / 134  Ideal gas Fully ionized gas of different elements Ideal gas: mixture of elements Real stellar matter consists of a mixture of different elements. Most of the atoms are largely or fully ionized. If the gas consists of different types of particles each behaving like an ideal gas, the total pressure in the gas is obtained as the sum of the partial pressures Pi = ni kT , i.e., X X P = Pi = ni kT. i i The same is applied for internal energy X 3X U = Ui = ni kT. i 2 i Walczak (UWr) Stellar Structure and evolution SZ 2024/25 22 / 134  Ideal gas Fully ionized gas of different elements Ideal gas: mixture of elements We denote the atomic number and atomic mass of element j by (Zj , Aj ), and its mass fraction by Xj. When fully ionized, each atom contributes Zj + 1 particles (Zj electrons and one nucleus). The number of atoms of element j per unit volume is ρXj /(Aj mu ), and hence the total number of particles per unit volume from element j is ni = ρXj (Zj + 1)/(Aj mu ). Thus the total pressure is X Zj + 1 ρkT P = ρXj kT = j Aj mu µmu where X Zj + 1 µ−1 = Xj j Aj is the mean molecular weight. Walczak (UWr) Stellar Structure and evolution SZ 2024/25 23 / 134  Ideal gas Fully ionized gas of different elements Ideal gas: mixture of elements The mass fractions of H and He are usually denote by X and Y , respectively, and the mass fraction of the remaining, so-called heavy, elements by Z. X + Y + Z = 1. So Z1 + 1 Z2 + 1 X Zj + 1 µ−1 = X +Y + Xj. A1 A2 j=3 Aj To obtain an approximate expression for µ we take A1 = 1 (for hydrogen), A2 = 4 (for helium), and we approximate (Zj + 1)/Aj by 1/2 for the heavy elements. Then 3 1 µ−1 ≈ 2X + Y + Z 4 2 or 4 µ≈ 3 + 5X − Z Let’s also introduce the mean molecular weight per electron, defined as:  −1 ρ X Zj µe = = Xj  ne mu j Aj In the condition of the fully ionized matter, we have 2 µe ≈ 1+X Walczak (UWr) Stellar Structure and evolution SZ 2024/25 24 / 134  Ideal gas Fully ionized gas of different elements Abundances Figure: Typical abundances of chemical elements (Iliadis, 2015, Nuclear Physics of Stars) Walczak (UWr) Stellar Structure and evolution SZ 2024/25 25 / 134  Ideal gas Fully ionized gas of different elements Ideal gas: mixture of elements + radiation Radiation pressure and energy: 1 4 Pr = aT , Ur = aT 4 , 3 where a is radiation constant. In the system consisting of a mixture of particles behaving like an ideal gas and radiation, the total pressure and internal energy are the sum of the particle and radiation: ρkT 1 P = Pg + Pr = + aT 4. µmu 3 And the internal energy: 3 ρkT U = Ug + Ur = + aT 4. 2 µmu Let’s define Pg β= P Walczak (UWr) Stellar Structure and evolution SZ 2024/25 26 / 134  Ideal gas Fully ionized gas of different elements Ideal gas: mixture of elements + radiation The first law of thermodynamics for photons dQ = dUV + P dV. For radiation we have Ur = 3P , UV = Ur V so dQ = 3d(P V ) + P dV = 3V dP + 3P dV + P dV = 3V dP + 4P dV. For adiabatic changes, dQ = 0: 0 = 3V dP + 4P dV, and dP 4 dV 4 dρ =− =. P 3 V 3 ρ The equation above can be written as   ∂ ln P 4 = = Γ1. ∂ ln ρ s 3 Walczak (UWr) Stellar Structure and evolution SZ 2024/25 27 / 134  Ideal gas Fully ionized gas of different elements Ideal gas: mixture of elements + radiation 1 4 Since P = 3 aT 4 , dP = 3 aT 3 dT and 1 1 4 dρ 4aT 3 dT = P 3 3 ρ dT 1 dρ =. T 3 ρ This equation was derived under adiabatic assumption, so   d log T 1 = = Γ3 − 1, d log ρ s 3 and 4 Γ3 =. 3 Similarly 4 Γ2 =. 3 Walczak (UWr) Stellar Structure and evolution SZ 2024/25 28 / 134  Ideal gas Fully ionized gas of different elements Ideal gas: mixture of elements + radiation For the mixture of non-relativistic gas and ultra-relativistic photons we derive 32 − 24β − 3β 2 Γ1 =. 24 − 21β 32 − 24β − 3β 2 Γ2 =. 24 − 18β − 3β 2 32 − 27β Γ3 =. 24 − 21β Walczak (UWr) Stellar Structure and evolution SZ 2024/25 29 / 134  Basic equations Hydrostatic equilibrium Basic equations Basic equations Walczak (UWr) Stellar Structure and evolution SZ 2024/25 30 / 134  Basic equations Hydrostatic equilibrium Basic equations: Hydrostatic equilibrium Figure: LNSSE, J. Ch-D. Walczak (UWr) Stellar Structure and evolution SZ 2024/25 31 / 134  Basic equations Hydrostatic equilibrium Basic equations: Hydrostatic equilibrium The gravitational acceleration at the distance r is GMr g=−. r2 The volume of the shell is dA · dr, where dA is a surface area of the element. The mass of the element is dMr = ρ · dAdr. The gravitational force is RMr dFg = gdMr = ρ drdA. r2 The second force arises from the net pressure acting on the slab. Since pressure P is defined as force per unit area, this force is dP dFP = P (r)dA − P (r + dr)dA → − drdA. dr Now, the general equation of motion is (Newton’s second law of motion): d2 x m = F. dt2 Walczak (UWr) Stellar Structure and evolution SZ 2024/25 32 / 134  Basic equations Hydrostatic equilibrium Basic equations: Hydrostatic equilibrium In the case os the star we have: d2 r ρdrdA = dFg + dFP. dt2 After simplification: d2 r GMr dP ρ = −ρ 2 −. dt2 r dr In the case of strict equilibrium, the time derivatives disappear and dP GMr = −ρ 2. (14) dr r This is the first of the equations of stellar structure. We can easily write the second equation of stellar structure. This is an equation relating Mr to the other properties of the star. The mass in a shell between r and r + dr is 4πr2 ρdr. Therefore dMr = 4πr2 ρdr, or dMr = 4πr2 ρ dr Walczak (UWr) Stellar Structure and evolution SZ 2024/25 33 / 134  Basic equations Hydrostatic equilibrium Basic equations: Hydrostatic equilibrium: central values From equation of hydrostatic equilibrium we cat obtain an estimate of the central pressure Pc of a star with mass M and radius R. We make the following approximations: Replace dP/dr by −Pc /R. Replace Mr by M. Replace r by R. Replace ρ by the mean density, approximated as M/R3. We get 2  −4 GM 2  M R Pc ≈ = 1.1 × 1016 dyn cm−2. R4 M⊙ R⊙ ρkT The central temperature we estimate with the help of the ideal gas law (P = µmu ):   −1  µc mu Pc Gµc mu M M R µc  Tc = ≈ = 1.9 × 107 K kρc kR M⊙ R⊙ 0.85 From realistic solar models we have: Pc = 2.4 × 1017 dyn cm−2 , and Tc = 1.5 × 107 K. Walczak (UWr) Stellar Structure and evolution SZ 2024/25 34 / 134  Basic equations Hydrostatic equilibrium HR and central values Figure: Paxton et al. (2011) Walczak (UWr) Stellar Structure and evolution SZ 2024/25 35 / 134  Basic equations Hydrostatic equilibrium HR and central values Figure: Paxton et al. (2011) Walczak (UWr) Stellar Structure and evolution SZ 2024/25 36 / 134  The virial theorem The Virial Theorem The virial theorem is of great significance as it provides a framework for linking gravitational energy to internal energy. The gravitational potential on the surface of a spherically symmetric object with mass Mr and radius r is given by: VG = GM r r. For a mass element m, the gravitational energy is: GMr m dΩ = VG · m = r. Therefore, the total gravitational energy of a star can be expressed as: ZM ZR ZR GMr (m) GMr Ω=− dm = − 4πr2 ρdr = −4π GMr rρdr, r(m) r 0 0 0 where dm = 4πρr2 dr. This equation can be integrated by parts (hydrostatic equilibrium, dP dr = −ρ GMr2 r ) ZR ZR ZR GMr 3 dP (r) 3 R Ω = −4π r ρdr = 4π r dr = 4πr3 P (r) 0 −3 P (r)4πr2 dr. r2 dr 0 0 0 For a typical situation P (R) = 0 and ZR Ω = −3 P (r)4πr2 dr. 0 Walczak (UWr) Stellar Structure and evolution SZ 2024/25 37 / 134  The virial theorem The Virial Theorem Lets consider ideal gas in an non-relativistic approximation. In such a case we have ρkT 3 ρkT P = , U =. µmu 2 µmu 3 2 From this we can notice that U = 2 P or P = 3 U. So ZR ZR Z Ω = −3 P 4πr2 dr = −2 U 4πr2 dr = −2 U dV = −2Utot. V 0 0 Ω = −2Utot This relation is called the Virial Theorem. Walczak (UWr) Stellar Structure and evolution SZ 2024/25 38 / 134  The virial theorem The Virial Theorem The total energy of a star (without any additional energy sources) is then given by: 1 E = Ω + Utot = −Utot = Ω. 2 The fact that the total energy is negative indicates that the star is stable: its thermal energy is insufficient to trigger an explosion. Walczak (UWr) Stellar Structure and evolution SZ 2024/25 39 / 134  The virial theorem The Virial Theorem Let’s consider the evolution of a star that lacks nuclear energy sources: 1 E = Ω + Utot = −Utot = Ω. 2 Stars shine. They are not adiabatic systems; they continuously lose energy. Let us examine the total differential of the energy equation: 1 dE = dΩ + dUtot = −dUtot = dΩ. 2 Energy loss requires the total energy change, dE, to be negative. Consequently, −dUtot must also be negative, implying that dUtot is positive - it increases (as well as the average temperature in the gas). Simultaneously 21 dΩ must be negative, indicating that the gravitational potential energy decreases. This results in the star becoming more strongly gravitationally bound. The release of gravitational energy serves as an energy source. Half of this energy is used to increase the star’s total thermal energy, while the other half is radiated away from the surface. Walczak (UWr) Stellar Structure and evolution SZ 2024/25 40 / 134  The virial theorem The Virial Theorem The rate of change of a star’s total energy over time is commonly referred to as its stellar luminosity: dE L=−. dt We can now define the luminosity of a contracting star 1 dΩ LG = −. 2 dt Walczak (UWr) Stellar Structure and evolution SZ 2024/25 41 / 134  The virial theorem The Virial Theorem Let’s consider an extremely relativistic case. We have already considered a radiation pressure and internal energy: 1 4 Pr = aT , Ur = aT 4. 3 1 Here we have U = 3P or P = 3 U. With this we derive ZR ZR Z Ω = −3 P 4πr2 dr = − U 4πr2 dr = − U dV = −Utot. V 0 0 Ω = −Utot. Walczak (UWr) Stellar Structure and evolution SZ 2024/25 42 / 134  The virial theorem The Virial Theorem As a consequence E = Ω + Utot = Ω − Ω = 0. An extremely relativistic system is marginally stable: it may expand or contract indefinitely without any change in the total energy. Hence a small change to the system may be sufficient to push it into instability. This situation arises when the majority of the pressure within a star is contributed by radiation. Consequently, our findings suggest that very massive stars may be inherently unstable. Walczak (UWr) Stellar Structure and evolution SZ 2024/25 43 / 134  Transport of energy Transport of energy Transport of energy Walczak (UWr) Stellar Structure and evolution SZ 2024/25 44 / 134  Transport of energy Transport of energy: radiative transport The mean free path of a photon: 1 ℓ= κρ where κ is a mean absorption coefficient, i.e. a radiative cross section per unit mass (averaged over frequency). Typical values for stellar material are of order κ ∼ 1 cm2 /g. Typical mean free path are of the order of 1 cm (stellar matter is very opaque). Since the mean free path is very small (as compared to the characteristic length of stellar radius - ℓ/R ∼ 10−11 ), the transport can be treated as a diffusion process. Walczak (UWr) Stellar Structure and evolution SZ 2024/25 45 / 134  Transport of energy Transport of energy: radiative transport The diffusive flux j of particles between regions of different number density n is given by Fick’s first law: ⃗j = −D dn. dr Here, D is a diffusion coefficient and it is defined as 1 D= vℓ. 3 For photons we need to replace n by the energy density of radiation U. We then derive diffusive flux of radiative energy F. We already have dU dT U = aT 4 and = 4aT 3. dr dr Walczak (UWr) Stellar Structure and evolution SZ 2024/25 46 / 134  Transport of energy Transport of energy: radiative transport This gives (flux of radiative energy) 4ac T 3 dT F =−. (15) 3 κρ dr The flux F given above is defined as energy flowing through a unit area per unit time. For symmetrically symmetric star, however, it is more convenient to use local luminosity, Lr , which is a an energy flowing through the whole sphere of a radius r. Therefore Lr = 4πr2 F. Taking this into account we get dT 3 κρLr =−. (16) dr 16πac r2 T 3 Walczak (UWr) Stellar Structure and evolution SZ 2024/25 47 / 134  Energy equation Energy equation: nuclear energy During the majority of a star’s evolution, its energy predominantly originates from nuclear reactions. Let ϵ represent the energy production rate per unit mass. In a thin shell of the star located between radii r and r + dr the energy produced per unit time is given by 4πr2 ρϵdrThis energy production corresponds to the increase in luminosity, dLr. Thus, we can write: dLr = 4πr2 ρϵ. (17) dr Walczak (UWr) Stellar Structure and evolution SZ 2024/25 48 / 134  Energy equation Energy equation: gravitational contraction The first law of thermodynamics dQ = dUV + P dV. The energy change per unit volume and unit time is 1 dQ 1 dUV 1 dV = + P. V dt V dt V dt 1 Let’s take a volume containing unit mass V = ρ   dQ dUV dV d(U V ) dV d U P dρ ρ =ρ + ρP =ρ + ρP =ρ −ρ. dt dt dt dt dt dt ρ ρ2 dt The energy change in a whole volume of the sphere and in a unit of the time is     dQ d U P dρ 4πr2 drρ = 4πr2 dr ρ −. dt dt ρ ρ dt Walczak (UWr) Stellar Structure and evolution SZ 2024/25 49 / 134  Energy equation Energy equation The general form of the energy equation is then dQ dLr = 4πr2 ρϵdr − 4πr2 drρ dt or     dLr d U P dρ = 4πr2 ρϵ − ρ − (18) dr dt ρ ρ dt Walczak (UWr) Stellar Structure and evolution SZ 2024/25 50 / 134  Energy equation Basic equations dP GMr = −ρ 2 (19) dr r dMr = 4πr2 ρ (20) dr dT 3 κρLr =− (21) dr 16πac r2 T 3     dLr d U P dρ = 4πr2 ρϵ − ρ −. (22) dr dt ρ ρ dt Walczak (UWr) Stellar Structure and evolution SZ 2024/25 51 / 134  Energy equation Convection s Figure: Image of small part of the solar surface, showing the effect of convection in the form of granulation. The bright areas, up to 1000 km across, are warmer rising gas and the dark lanes are cooler sinking gas. (Courtesy DKIST, National Solar Observatory, USA, JCD LNSS) Walczak (UWr) Stellar Structure and evolution SZ 2024/25 52 / 134  Energy equation Convection A rough estimate of the relationship between the temperature gradient and the luminosity is to use adiabatic approximation. dT dT Γ2 − 1 T dP Γ2 − 1 T GMr = = =−. dr dr S Γ2 P dr Γ2 P r 2 Walczak (UWr) Stellar Structure and evolution SZ 2024/25 53 / 134  Energy equation Basic equations of stellar structure dP GMr = −ρ 2 (23) dr r dMr = 4πr2 ρ (24) dr in a radiative region in a convective region dT 3 κρLr dT Γ2 − 1 T GMr =− or =− (25) dr 16πac r2 T 3 dr Γ2 P r 2     dLr d U P dρ = 4πr2 ρϵ − ρ −. (26) dr dt ρ ρ dt Walczak (UWr) Stellar Structure and evolution SZ 2024/25 54 / 134  Energy equation Basic equations of stellar structure How can we determine whether a specific layer in a stellar model is radiative or convective? Walczak (UWr) Stellar Structure and evolution SZ 2024/25 55 / 134  Energy equation Stability condition for convective motion Stability condition for convective motion Figure: Christensen-Dalsgaard (2023) Motion of the bubble is determined by buoyancy. The force per unit volume at the point 2 is GMr fbuoy = −g(ρ∗2 − ρ2 ) ≡ −g∆ρ, g= r2 At the position 1, the pressure and density of a bubble is in equilibrium with surroundings, i.e. ρ∗1 = ρ1 , P1∗ = P1. Walczak (UWr) Stellar Structure and evolution SZ 2024/25 56 / 134  Energy equation Stability condition for convective motion Stability condition for convective motion To determine ∆ρ, and hence decide between stability and instability, we make the following assumptions: 1) The motion is so slow that there is pressure balance between the element and the surroundings. 2) The motion is so fast that there is no heat loss to the surroundings. The pressure is established on the dynamical time scale, time scale for the heat loss is the Kelvin-Helmholtz. The dynamical time scale is of the order of hours, while thermal time scale is of the order of thousand of years. Therefore, the assumptions should be satisfied. From assumption 1) we have that P2∗ = P2. Walczak (UWr) Stellar Structure and evolution SZ 2024/25 57 / 134  Energy equation Stability condition for convective motion Stability condition for convective motion Let’s express the density difference using Taylor expansion ∆ρ = ρ∗2 − ρ2 = ρ∗2 − ρ1 − (ρ2 − ρ1 ) = ρ∗2 − ρ∗1 − (ρ2 − ρ1 ) ∆ρ = ρ(r + ∆r)∗ − ρ(r)∗ − [ρ(r + ∆r) − ρ(r)] In the limit ∆r → 0 we have dρ∗ dρ ∆ρ ≈ ∆r − ∆r, dr dr dρ dρ ∆ρ ≈ ∆r − ∆r. dr e dr   dρ dρ ∆ρ ≈ − ∆r. dr e dr Walczak (UWr) Stellar Structure and evolution SZ 2024/25 58 / 134  Energy equation Stability condition for convective motion Stability condition for convective motion The condition for instability is that ∆ρ < 0, so the convection occurs if dρ dρ < dr e dr The instability occurs when the density does not decrease sufficiently rapidly. For stellar model calculations it is more convenient to use slightly different criterion. We will use the general equation of state. First we will note that the density is a function of pressure, temperature and chemical composition (represented by a mean atomic weight) ρ = ρ(P, T, µ). Then the total differential is ∂ρ ∂ρ ∂ρ dρ = dP + dT + dµ. ∂P T,µ ∂T P,µ ∂µ T,P Let’s rewrite it in a different way dρ ∂ log ρ dP ∂ log ρ dT ∂ log ρ dµ = + + ρ ∂ log P T,µ P ∂ log T P,µ T ∂ log µ T,P µ or in a simpler form dρ dP dT dµ =α −δ +φ ρ P T µ For ideal gas there are α = δ = φ = 1. dµ represent the change of chemical composition, i.e. the variation of the concentrations of different nuclei. Thus, dµ = 0 for the moving element that carries its composition along. However, for surroundings dµ ̸= 0. Walczak (UWr) Stellar Structure and evolution SZ 2024/25 59 / 134  Energy equation Stability condition for convective motion Stability condition for convective motion With this the instability condition can be written as α dP δ dT α dP δ dT φ dµ − < − + P dr e T dr e P P T dr µ dr First and third factors cancel out, so δ dT δ dT φ dµ − −P δ + Pφ d log P e d log P d log P and reduce P d log T d log T d log µ −δ > −δ +φ d log P e d log P d log P Now we will define some variables d log T d log T d log µ ∇≡ , ∇e ≡ , ∇µ ≡. d ln P d ln P d ln P Walczak (UWr) Stellar Structure and evolution SZ 2024/25 61 / 134  Energy equation Stability condition for convective motion Stability condition for convective motion With this we can write φ −∇e > −∇ + ∇µ δ or φ ∇ > ∇e + ∇µ. δ The quantity ∇e describes the variation of T in the element during its motion, where the position of the element is measured by P. On the other hand, ∇ and ∇µ describe the spatial variation of T and µ in the surroundings. Walczak (UWr) Stellar Structure and evolution SZ 2024/25 62 / 134  Energy equation Stability condition for convective motion Stability condition for convective motion Let’s come back to the assumption 2): the moving convective element changes adiabatically. Then we can write Γ2 − 1 ∇e = ∇S =. Γ2 If there is no convection, energy transport occurs through radiation. The radiative energy transport equation is (eq. 33) dT 3 κρLr =− dr 16πac r2 T 3 We will modify slightly that equation dT dP d log T T dP 3 κρLr = =− , dP dr d log P P dr 16πac r2 T 3 use hydrostatic equilibrium equation dP dr = −ρ GM r2 r and ideal gas law P = ρkT /µmu. We can then write d log T 3k κLr ρ =− ≡ ∇rad d log P 16πacGmu µMr T 3 Walczak (UWr) Stellar Structure and evolution SZ 2024/25 63 / 134  Energy equation Stability condition for convective motion Stability condition for convective motion The final condition for convective instability is expressed as: φ ∇rad > ∇S + ∇µ. (27) δ which is known as the Ledoux criterion for dynamical instability, named after Paul Ledoux. In regions with a homogeneous chemical composition, where ∇µ = 0, this reduces to the simpler Schwarzschild criterion for dynamical instability, named after Karl Schwarzschild. ∇rad > ∇S. (28) Walczak (UWr) Stellar Structure and evolution SZ 2024/25 64 / 134  Energy equation Stability condition for convective motion Stability condition for convective motion 3k κLr ρ ∇rad = − 16πacGmu µMr T 3 Γ2 − 1 ∇S =. Γ2 From the equations above, it is evident that convection may be expected under the following conditions: Large Lr /Mr : This indicates a high average rate of energy generation per unit mass within the radius r. Such a condition is typical in the interiors of massive stars, where energy generation is a steep function of temperature and becomes strongly concentrated toward the stellar core. As a result, Lr /Mr is large, leading to the presence of a convective core. Large κ (opacity): This is satisfied in the outer layers of relatively low-mass stars on the main sequence, or more generally in stars with low surface temperatures. In such stars, the temperature in the outer layers is low, leading to high opacity. The ionization of hydrogen further increases opacity in these regions. Large ρ/T 3 : This condition is typically satisfied in the outer layers of relatively cool stars. Small ∇S (entropy gradient): This condition is met in the hydrogen ionization zone, which is again located in the outer regions of cool stars. Walczak (UWr) Stellar Structure and evolution SZ 2024/25 65 / 134  Energy equation Stability condition for convective motion Stability condition for convective motion Figure: The typical occurrence of convection zones in main-sequence stars. In relatively massive stars there is a convective core, whereas in relatively light stars on the main sequence, and in general in stars with low effective temperature, there is an outer convection zone. In red giants this convection zone occupies by far the largest fraction of the stellar radius, and a substantial fraction of the stellar mass. Walczak (UWr) Stellar Structure and evolution SZ 2024/25 66 / 134  Energy equation Stability condition for convective motion Model of Sun Figure: Temperature gradients in a model of the present Sun. The solid line shows the actual temperature gradient ∇, and the dotted line shows the adiabatic gradient ∇ad the dashed line shows the radiative gradient ∇rad. In the radiative region, for r < 0.72R⊙ , ∇ = ∇rad , and the solid and dashed lines coincide. In the convection zone ∇ is so close to ∇ad that the solid and dotted lines are indistinguishable Walczak (UWr) Stellar Structure and evolution SZ 2024/25 67 / 134  Energy equation Stability condition for convective motion Low mass stars: evolution of the structure Figure: Kippenhahn diagram showing the evolution of the internal structure of the non-rotating solar metallicity models of 0.3 (top), 0.5, 1.0 and 1.5M⊙ (bottom) from the PMS up to the end of the main sequence. The upper line represents the surface radius and hatched areas refer to convective regions. The green line displays the H-burning limit. The five pink vertical lines indicate the ages of open clusters used as markers of the evolution. Amard et al. (2019) Walczak (UWr) Stellar Structure and evolution SZ 2024/25 68 / 134  Energy equation Stability condition for convective motion Massive stars: evolution of the structure Figure: Kippenhahn diagram of a stellar model, with the x-axis showing the time until CC. Light blue shaded regions are convective. Solid colored lines show helium, carbon, oxygen, silicon, and iron core masses. We have labeled convective core burning phases with the element being burned in each phase. Dashed vertical lines show the locations (from left) of our helium core burning, carbon shell burning, oxygen shell burning, and silicon shell burning. Fuller et al. (2015) Walczak (UWr) Stellar Structure and evolution SZ 2024/25 69 / 134  Nuclear energy generation Nuclear energy generation Throughout most of their lifetimes, stars generate the energy they radiate through nuclear reactions. As these reactions progress, they gradually alter the chemical composition of the star, driving its evolution. Therefore, understanding the properties of these nuclear reactions is essential for tracing the life cycle of a star. The reaction of a type A+a→Y +y will be written as A(a, y)Y. Walczak (UWr) Stellar Structure and evolution SZ 2024/25 70 / 134  Nuclear energy generation Nuclear energy generation Nuclear reactions occur due to the strong force acting between nucleons (protons and neutrons). However, the strong force is short-ranged, effectively confined to the size of the nucleus. For a reaction to take place, the nuclei must be brought sufficiently close that they almost touch, requiring the Coulomb repulsion between them to be overcome. The challenge of initiating a reaction becomes evident when considering the height of the Coulomb barrier at the surface of the nucleus. For a typical nuclear radius of r0 ∼ 1013 cm, the Coulomb energy is approximately given by: Z1 Z2 e2 ECoulomb ≈ ≈ Z1 Z2 MeV. r0 On the other hand, the typical kinetic energy of a single particle is: 3 Ekin = U/n = kT ∼ keV. 2 The average kinetic energy of a particle is approximately three orders of magnitude smaller than the energy needed to overcome the Coulomb barrier. Even when considering the distribution of particle energies, it becomes evident that, within the framework of classical mechanics, nuclear reactions would be virtually impossible. Walczak (UWr) Stellar Structure and evolution SZ 2024/25 71 / 134  Nuclear energy generation Nuclear energy generation Reactions and ultimately our existence are made possible by quantum mechanics, which allows for a finite probability that nuclei can tunnel through the Coulomb barrier and react. However, the significant height and width of the barrier make this probability small. As a result, nuclear burning in stellar interiors is typically a very slow process Walczak (UWr) Stellar Structure and evolution SZ 2024/25 72 / 134  Nuclear energy generation Nuclear energy generation The energy released in a reaction A(a, y)Y is Q = c2 [m(A) + m(a) − m(Y ) − m(y)] The energy generated by nuclear reactions is released primarily as the kinetic energy of the resulting particles and, in some cases, as γ-ray photons. This energy is then redistributed among the other constituents of the gas through collisions and photon absorption. Under the assumption of thermodynamic equilibrium in the gas, the specifics of this redistribution process are irrelevant; only the total amount of heat added to the gas is significant. An exception to this arises when neutrinos are emitted during the reaction. Due to their extremely small interaction cross section, neutrinos typically do not interact with the other particles in the gas and instead escape directly from the star. Consequently, to accurately compute the energy release in the region where nuclear reactions occur, the energy carried away by neutrinos must be subtracted. Walczak (UWr) Stellar Structure and evolution SZ 2024/25 73 / 134  Nuclear energy generation Nuclear energy generation Throughout most of their lifetimes, stars generate energy by fusing hydrogen into helium. This process can be represented schematically as: 41 H →4 He + 2e+ + 2νe. As we can see, this reaction produces two positrons and two electron neutrinos as by-products. However, it is clear that the reaction does not occur as directly indicated in the equation above; the likelihood of four protons simultaneously coming together and reacting at a single point is exceedingly small. Instead, the reaction proceeds through a series of intermediate steps along various possible pathways. Regardless of the specific path, we can determine the total energy released in the reaction to be: Qtot = 26.73 MeV. However, neutrinos escape directly from the star without interacting with its matter and, therefore, do not contribute to the energy released within the star. To calculate the effective energy, Qeff , released per 4 He nucleus formed—and consequently, the energy generation rate, ε, we must subtract the energy carried away by neutrinos. The exact amount of this neutrino energy depends on the specific reactions in which the neutrinos are produced. Walczak (UWr) Stellar Structure and evolution SZ 2024/25 74 / 134  Nuclear energy generation Nuclear energy generation: the pp cycle Figure: The pp cycle. p stands for 1 H, d is the deuterium 2 H, α is 4 He, β + means beta decay with emission of neutrino and positron e+. Nuclear Physics of Stars, Christian Iliadis, 2007 pp pp 1 Qpp1 = 26.73 MeV −2E ν = 26.19 MeV, E ν = 265 keV, 7 7 pp Be Be 2 Qpp2 = 26.73 MeV −E ν − E ν = 25.65 MeV, E ν = 815 keV 8 8 pp B B 3 Qpp3 = 26.73 MeV −E ν − E ν = 19.75 MeV, E ν = 6.715 MeV Walczak (UWr) Stellar Structure and evolution SZ 2024/25 75 / 134  Nuclear energy generation Nuclear energy generation: the CNO cycle A second sequence of reactions is possible in stars that contain C, N, and O. Figure: The CNO cycle Nuclear Physics of Stars, Christian Iliadis, 2007. Thus the reactions proceed through a sequence of proton captures, interrupted by positron decay (with emission of neutrinos) to convert protons into neutrons; as indicated 12 C is produced at the end of the sequence of reactions, and hence acts as a catalyst. The total average neutrino loss in the cycle is 1.71 MeV, and hence the effective Q-value is QCN O1 = 25.02 MeV. Walczak (UWr) Stellar Structure and evolution SZ 2024/25 76 / 134  Nuclear energy generation Nuclear energy generation The 14 N(p, γ)15 O reaction has the smallest probability amongst the reactions in the cycle; hence, once the cycle operates in equilibrium this reaction determines the overall reaction rate. A second consequence is that in equilibrium almost all the original C and N is converted into 14 N. 13 N Eν = 0.710 MeV 15 O Eν = 1.00 MeV 17 F Eν = 0.94 MeV Walczak (UWr) Stellar Structure and evolution SZ 2024/25 77 / 134  Nuclear energy generation Nuclear energy generation The energy-generation rate from the CNO cycle is significantly more dependent on temperature compared to the energy-generation rate of the proton-proton (PP) chains. As a result, the pp chains dominate at relatively low temperatures, while the CNO cycle becomes the dominant process at higher temperatures. Figure: Ratio of energy producion rate in pp cycle to CNO cycle Nuclear Physics of Stars, Christian Iliadis, 2007 Walczak (UWr) Stellar Structure and evolution SZ 2024/25 78 / 134  Nuclear energy generation Nuclear energy generation: 3α cycle After the exhaustion of hydrogen, the next reactions involve 4 He (an α particle). However, the reaction between two 4 He nuclei produces nuclei with an atomic weight of eight. The challenge is that there are no stable nuclei with atomic weight eight. Therefore, 4 He burning must occur through the so-called triple-alpha process: 34 He →12 C + γ. In reality, the reaction proceeds through the formation of 8 Be: 4 He +4 He ↔8 Be (29) and then 8 Be(α, γ) →12 C. (30) The half-life time of beryllium-8 is only T1/2 = 6.7 × 10−17 s. The typical temperature of helium burning cycle is T = 1 − 2 × 108 K. Walczak (UWr) Stellar Structure and evolution SZ 2024/25 79 / 134  Nuclear energy generation Nuclear energy generation: 3α cycle The possible further reactions 12 C(α, γ)16 O(α, γ)20 Ne(α, γ)24 Mg. 14 N(α, γ)18 F(β + ν)18 O(α, γ)22 Ne Created 22 Ne can be consumed in two reactions: 22 Ne(α, γ)26 Mg (Q = 10.62 MeV) or 22 Ne(α, n)25 Mg (Q = −0.48 MeV). The total production of these nuclei during the helium-burning phase is rather small, though. Walczak (UWr) Stellar Structure and evolution SZ 2024/25 80 / 134  Nuclear energy generation Nuclear reaction rate The reaction rate rlm refers to the number of reactions per unit volume and time that convert nuclei of type l into nuclei of type m. In general, an element i can be affected simultaneously by multiple reactions, some of which produce it (rji ) while others consume it (rik ). These reaction rates directly indicate the rate of change of ni per second. With mi ni Xi = ρ we can write   ∂Xi mi X X = rji − rik . ∂t ρ j k Walczak (UWr) Stellar Structure and evolution SZ 2024/25 81 / 134  Nuclear energy generation Nuclear reaction rate ε - energy generation rate per unit mass. formally, it consists of sum of all reaction: XX 1 XX ε= εij = rij Qij. i j ρ i j By introducing energy generation rate of one unit mass of element i: Qij qij = , mi we can write: ∂Xi X εji X εik = −. ∂t j qji k qik Walczak (UWr) Stellar Structure and evolution SZ 2024/25 82 / 134  Nuclear energy generation Nuclear reaction rate In the simplest case we have ∂X εH =− , ∂t qH where εH is an overall generation rate of H (the sum over all single reactions) and qH is the energy release per unit mass when hydrogen is converted into helium. Walczak (UWr) Stellar Structure and evolution SZ 2024/25 83 / 134  Nuclear energy generation Basic equations of stellar structure and evolution 1 dP GMr = −ρ 2 (31) dr r 2 dMr = 4πr2 ρ (32) dr 3 in a radiative region in a convective region  φ   φ  if ∇rad < ∇S + ∇µ if ∇rad > ∇S + ∇µ δ δ dT 3 κρLr dT Γ2 − 1 T GMr =− or =− (33) dr 16πac r2 T 3 dr Γ2 P r 2 4     dLr d U P dρ = 4πr2 ρϵ − ρ −. (34) dr dt ρ ρ dt 5 ∂Xi X εji X εik = −. (35) ∂t j qji k qik Walczak (UWr) Stellar Structure and evolution SZ 2024/25 84 / 134  Nuclear energy generation Nuclear energy generation Figure: HR diagram with 1M⊙ star evolution path. Nuclear Physics of Stars, Iliadis (2015) Walczak (UWr) Stellar Structure and evolution SZ 2024/25 85 / 134  Nuclear energy generation Nuclear energy generation Figure: HR diagram with massive star evolution paths. Nuclear Physics of Stars, Iliadis (2015) Walczak (UWr) Stellar Structure and evolution SZ 2024/25 86 / 134  Stellar Evolution Stellar Evolution Stellar Evolution Walczak (UWr) Stellar Structure and evolution SZ 2024/25 87 / 134  Stellar Evolution Pre Main Sequence Stellar Evolution: Pre Main Sequence Stars are formed out of interstellar gas clouds, which become unstable to gravitational collapse. During the collapse the cloud is heated up by the release of gravitational potential energy. When the temperature gets sufficiently high in the core of the star, nuclear burning starts, and the star settles down on the main sequence. Star formation is still taking place in the Milky Way Galaxy. Perhaps the most spectacular example of a star-forming region is the Orion Nebula. Figure: By NASA, ESA, M. Robberto (Space Telescope Science Institute/ESA) and the Hubble Space Telescope Orion Treasury Project Team Walczak (UWr) Stellar Structure and evolution SZ 2024/25 88 / 134  Stellar Evolution Pre Main Sequence Stellar Evolution: The Jeans instability The condition that the cloud is gravitationally bound, and hence potentially unstable towards collapse, is that Etot < 0. This leads to a simple condition for a minimal mass of a cloud that can collapse due to gravitation: 1/2 T3  MJ = 5 × 10−10 M⊙. (36) ρ where MJ is the Jeans mass. Typical values for the temperature and density in the gas clouds are T = 100 K, ρ = 10−22 gcm−3 and MJ ∼ 5 × 104 M⊙. The estimate is reasonably consistent with the masses of open clusters of stars. The initial gravitational instability leads to the collapse of a cloud of gas which eventually breaks up into the stars in a cluster. Indeed this breakup may be caused by several mechanisms. In the initial phases of the collapse the gas is so thin that radiation escapes immediately from the cloud. Consequently, the temperature of the cloud is roughly constant. It follows that the critical mass MJ decreases as the cloud contracts and the density increases. Thus as long as the cloud is roughly isothermal, it becomes unstable to gravitational collapse on smaller and smaller scales as it contracts. This may lead to the break-up into masses that are closer to stellar mass. An additional contribution to the break-up may come from rotational instability. The rotational velocity increases rapidly as the cloud contracts. As a result, the ratio between the centrifugal acceleration and the gravitational acceleration increases during the contraction, possibly to the point where the cloud becomes unstable to rotational breakup. This process may then repeat itself, again leading to the point where clouds of stellar mass are formed. Walczak (UWr) Stellar Structure and evolution SZ 2024/25 89 / 134  Stellar Evolution Pre Main Sequence Stellar Evolution: Pre Main Sequence The initial collapse of an interstellar cloud results in the formation of a core which subsequently accretes matter from the surrounding cloud. The angular momentum of the infalling material probably leads to the formation of a disk around the star It is likely that the early contraction, and the accretion of matter in the disk, leads to an initial rapid rotation of the star. In fact, it is observed that young stars generally Figure: Pre Main Sequence evolution. LNSSE, J. Ch-D. Walczak (UWr) Stellar Structure and evolution SZ 2024/25 90 / 134  Stellar Evolution Pre Main Sequence Stellar Evolution: Pre Main Sequence This is an illustration of planet-forming processes that have been obtained with the Atacama Large Millimeter/submillimeter Array (ALMA) high-resolution observations. Modelling has shown that the observed gaps are indeed consistent with the presence of newly formed planets. The planet-forming processes probably happen on a time scale comparable with, or shorter than, the gravitational contraction of the star. Walczak (UWr) Stellar Structure and evolution SZ 2024/25 91 / 134  Stellar Evolution Pre Main Sequence Stellar Evolution: Pre Main Sequence In the initial collapse, while the temperature is held roughly constant, the pressure does not increase sufficiently rapidly to balance the increasing gravitational force. However, when the density becomes so high that the gas becomes opaque to radiation, the contraction changes to being effectively adiabatic. Then pressure increases with decreasing radius more rapidly than does the gravitational force. Pressure may increase sufficiently to balance gravity, so that hydrostatic equilibrium can be achieved. The gravitational energy released in the contraction of the protostar partly goes to heating it up and is partly released as radiation from the star; the radiation finally stops the accretion and blows away the surrounding material, such that the star becomes directly observable: the star has reached the so-called ‘birthline’. Walczak (UWr) Stellar Structure and evolution SZ 2024/25 92 / 134  Stellar Evolution Pre Main Sequence Stellar Evolution: Pre Main Sequence Figure: Evolutionary tracks in the Hertzsprung-Russell diagram for pre-main sequence evolution of models of mass M/M⊙ = 0.5, 1.0, 1.25, 1.5, 2.25, 3, 5, 9 and 15. The luminosity Ls is in units of 3.86 × 1033 erg sec−1 , and Te is the effective temperature in K. The composition of the models is X = 0.708, Z = 0.02. (From Iben 1965.). LNSSE, J. Ch-D. Walczak (UWr) Stellar Structure and evolution SZ 2024/25 93 / 134  Stellar Evolution Pre Main Sequence Stellar Evolution: Pre Main Sequence Figure: The variation with time t (in years) of important quantities during the contraction of a 1M⊙ star towards the main sequence. The figure illustrates the effective temperature Tef f (in units of K), luminosity Ls (in units of 3.86 × 1033 erg sec−1 ), stellar radius R (in units of 6.96 × 1010 cm), central over mean density ρc /ρ, mass fraction qRC in the radiative interior, total energy generation Lnuc in units of Ls and the central 12 C abundance X12 C,c. Results from a MESA calculation (e.g., Paxton et al., 2019) by Johannes Holm Jorgensen. (Inspired by Iben 1965.). LNSSE, J. Ch-D. Walczak (UWr) Stellar Structure and evolution SZ 2024/25 94 / 134  Stellar Evolution Main Sequence Stellar Evolution: Main Sequence The zero-age main sequence (or ZAMS) is defined as the location of stars that have just settled down to hydrogen burning. Walczak (UWr) Stellar Structure and evolution SZ 2024/25 95 / 134  Stellar Evolution Main Sequence Stellar Evolution: Main Sequence Figure: Evolutionary tracks in the Hertzsprung-Russell diagram for the evolution following the main sequence. The unit of luminosity Ls is 3.86 × 1033 ergs−1 , and the effective temperature Te is in K. The models were computed with the composition X = 0.708, Z = 0.02. Tracks are shown for models of mass M/M⊙ = 0.5, 1.0, 1.25, 1.5, 2.25, 3, 5, 9 and 15. Dashed portions of the evolutionary tracks are estimates, Iben 1967, LNSSE, J. Ch-D. Walczak (UWr) Stellar Structure and evolution SZ 2024/25 96 / 134  Stellar Evolution Main Sequence Stellar Evolution: Main Sequence A general tendency at all masses is that the luminosity increases during the central hydrogen burning. This may be understood in terms of the changes that take place in the core of the star. As hydrogen is converted into helium, the hydrogen abundance X decreases. Then the mean molecular weight µ increases. Increasing µ causes the decrease in pressure. The core contracts, which increases ρ and T. This tends to increase the nuclear energy generation rate. This results in the increase of the luminosity of the star. In all cases the surface radius of the star expands as the star evolves, but at very different rates. For low masses the expansion is relatively modest; as a result the increase in luminosity leads to an increase in effective temperature. For higher mass the expansion is more rapid, and the effective temperature decreases. Walczak (UWr) Stellar Structure and evolution SZ 2024/25 97 / 134  Stellar Evolution Main Sequence Stellar Evolution: Main Sequence The details of the way in which hydrogen is used up during core hydrogen burning have substantial effect on the subsequent evolution of the star. Because of the temperature sensitivity of the nuclear reactions, hydrogen is consumed most rapidly at the centre. In a star with radiative energy transport there is no core mixing. It follows that helium produced by hydrogen burning remains where it is created, and hence the hydrogen abundance decreases most rapidly at the centre. The result is a variation of the hydrogen abundance with time and with location in the star. In evolved stars X increases gradually from the centre, and complete exhaustion of hydrogen first occurs precisely at the centre. Figure: Hydrogen profiles showing the gradual exhaustion of hydrogen in a star of 1M⊙. The homogeneous initial model consists of a mixture with a hydrogen abundance by mass of 0.699. X as a function of the mass fraction m/M is plotted for nine models which correspond to ages of 0, 2.0, 3.6, 5.0, 6.2, 7.5, 9.6, 11.0 and 11.6 ×109 years, after the onset of hydrogen burning, LNSSE, J. Ch-D. Walczak (UWr) Stellar Structure and evolution SZ 2024/25 98 / 134  Stellar Evolution Main Sequence Stellar Evolution: Main Sequence Massive stars have convective cores, which are mixed on a time scale very short compared with the evolutionary time scale. Hence within the convective cores the chemical composition is uniform at all times. As a result, the fusion of hydrogen to helium, which still predominantly takes place near the centre where the temperature is highest, leads to a uniform reduction in the hydrogen abundance in the convective core. The computations also show that the extent of the core generally shrinks as the star evolves; Figure: Hydrogen depletion in a 2.5M⊙ star with a shrinking convective core. The homogeneous initial model has X = 0.699. The lines show the hydrogen profiles for models of age 0, 1.5, 3.1, 4.0, 4.4, 4.6, and 4.8 ×108 years. LNSSE, J. Ch-D. Walczak (UWr) Stellar Structure and evolution SZ 2024/25 99 / 134  Stellar Evolution Evolution after the Main sequence Stellar Evolution: after the Main sequence When hydrogen is exhausted near the centre, the star is left with a core consisting of helium and a small amount of heavy elements. Initially the temperature of the core is far below the 108 K required for helium ignition, and hence there is no nuclear energy generation in the core. Surrounding the core is a region containing hydrogen where the temperature is still high enough for hydrogen burning to proceed. This region, which is known as a hydrogen shell source, provides the energy from which the luminosity of the star is derived. As the hydrogen is converted into helium in the shell source, the mass of the inert helium core increases. This leads to contraction of the core. As usual the contraction releases gravitational potential energy, part of which goes towards increasing the thermal energy of the core. As long as the core is not degenerate the increase in thermal energy leads to an increase in temperature, up to the point where the temperature of the core is sufficiently high for helium burning to begin. Very roughly the process is then repeated: the star burns helium in the core (while still maintaining a hydrogen shell source) until helium is exhausted. The star then has a contracting core consisting of 12 C and 16 O, surrounded by a helium shell source and a hydrogen shell source. The contraction of the core may proceed up to the point where the temperature is high enough for carbon ignition. Walczak (UWr) Stellar Structure and evolution SZ 2024/25 100 / 134  Stellar Evolution Evolution after the Main sequence Stellar Evolution: after the Main sequence Figure: Evolutionary tracks in the Hertzsprung-Russell diagram for the evolution following the main sequence. The unit of luminosity Ls is 3.86 × 1033 ergs−1 , and the effective temperature Te is in K. The models were computed with the composition X = 0.708, Z = 0.02. Tracks are shown for models of mass M/M⊙ = 0.5, 1.0, 1.25, 1.5, 2.25, 3, 5, 9 and 15. Dashed portions of the evolutionary tracks are estimates, Iben 1967, LNSSE, J. Ch-D. Walczak (UWr) Stellar Structure and evolution SZ 2024/25 101 / 134  Stellar Evolution Evolution after the Main sequence Stellar Evolution: after the Main sequence The shell-burning law: When the region within a burning shell contracts, the region outside the shell expands; and when the region inside the shell expands, the region outside the shell contracts. Walczak (UWr) Stellar Structure and evolution SZ 2024/25 102 / 134  Stellar Evolution Evolution after the Main sequence Stellar Evolution: after the Main sequence As the star expands, it reverts the evolution before the main sequence: the effective temperature decreases and the luminosity either decreases somewhat or stays constant. Once the effective temperature gets low enough, the star develops a very deep outer convection zone. The star continues to expand and increase the luminosity. The evolution after the Main Sequence is strongly sensitive to the stellar mass. We will first consider a moderate-mass star. Walczak (UWr) Stellar Structure and evolution SZ 2024/25 103 / 134  Stellar Evolution Evolution after the Main sequence Stellar Evolution: Moderate mass stars A notable feature as the star leaves the main sequence is the “hook” in the evolutionary track, towards higher effective temperature. This is a result of the convective cores in moderate- and high-mass stars. The hydrogen is exhausted throughout the core at the same time. This would tend to decrease the energy production, since, at least initially, the temperature outside the core is too low for significant hydrogen shell burning to take place. When the hydrogen abundance gets very low, the star attempts to ma