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10. Stellar Populations and Chemical Evolution

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10. Stellar Populations and Chemical Evolution 10. Stellar Populations and Chemical Evolution How individual stars evolve with time? Most stars, during most of their evolution, can be considered as: spherically symmetric with constant mass and in hydrostatic equilibrium. Under these conditions, the evolution of a star is almost completely determined by its mass and chemical composition through a set of ordinary differential equations that describe the structure of the star. Spherically Symetric? (i) stars are in hydrostatic state (ii) stars are spherically symmetric. Structure of a star described by a set of equations in which all quantities depend only on the distance to the center. Of course stars are not completely spherical; as a gaseous body a star can be flattened by rotation. Consider an element of mass m near the equator of a star. The centrifugal force on the mass element due to rotation is mω2R, where ω is the angular velocity of the rotation and R is the equatorial radius of the star. This force will not cause significant departure from spherical symmetry if it is much smaller than the gravitational force, GMm/R2, if ω2R3/GM ≪ 1. For the Sun, ω ≈ 2.5 × 10−6 s−1, ω2R3/GM ≈ 2 × 10−16, and so flattening due to rotation is completely negligible. In general, normal stars can be considered to be spherical to very high accuracy. Hydrostatic Equilibrium? To check hydrostatic equilibrium, consider the acceleration of a mass element at a radius r in a spherical star: partial derivative of P is used because the pressure is a function of both radius r and time t. Two time scales based on the ratio between the radius of the star, R, and the acceleration |r ̈|. −1/2 1) gravitational dynamical time scale, tdyn = (Gρ) , which corresponds to setting the pressure gradient to zero. 2) hydrodynamical time scale, thydro = R/cs (with cs the sound speed of the stellar gas), which corresponds to setting the gravitational term to zero. −3 10 For the Sun with ρ ∼ 1 g cm , R ∼ 7 × 10 cm and T ∼ 7000K, these two time scales are tdyn ∼ 50min and thydro ∼ 1day, respectively. These two time scales characterize how fast the gravity and sound waves can adjust a perturbed stellar structure to a new equilibrium configuration - should be compared to any perturbator. Hydrostatic Equilibrium? Thermal time scale of a star, ratio between - thermal energy of a star, Eth - energy loss rate at its surface, L: tth = Eth/L. Gives the rate at which a star changes its structure by radiating its thermal energy. 7 For the Sun, tth ∼ 10 yr. Since this is much longer than both tdyn and thydro, the structure of a normal star can adjust quickly to a new configuration as it radiates. Main energy source in a normal star is nuclear reactions and, for a star in equilibrium, the energy generation rate is equal to its luminosity. Nuclear energy generation time-scale, ratio between - energy resource in a star 2 −3 - current luminosity, L: tnuc ∼ ηMc /L, η ∼ 10 is the efficiency of nuclear reactions converting rest-mass to radiation For the Sun, this time scale is too long, about 1010 yr. Thus, for a normal star like the Sun, tnuc ≫ tth ≫ tdyn. Implies that the evolution of a star is determined by nuclear reactions, with thermal and mechanical equilibrium achieved during the evolution. Exceptions are variable stars and stars in their late evolutionary stages, where the luminosity of a star can change significantly over a time scale comparable to the dynamical time scale. 10.1.1 Basic Equations of Stellar Structure Under the assumption of spherical symmetry and hydrostatic equilibrium, the equation governing the structure of a star is the hydrostatic equation, Two differential equations, three quantities, P, M and ρ. Another independent relation among these quantities is needed. The equation of state of the stellar material can provide such a relation. In general, the equation of state relates the local pressure P to the local density ρ, temperature T, and mass fractions {Xi} of elements i = 1,2,···,n, and can formally be written as where the exact form of P(ρ,T,{Xi}) has yet to be specified. Temperature Two new quantities, T and {Xi}, we need two new equations to complete the description. To get an equation for the temperature, we use the fact that a temperature gradient in a star causes energy transport. Three ways for the energy transport: (i) heat conduction, transfers energy as electrons from hotter regions collide and exchange energy with electrons from cooler regions; (ii) radiation, transfers energy as photons propagate from hotter regions to cooler regions; (iii) convection, transfers energy as material is convected between hot and cold regions. Consider radiative stars where photon diffusion dominates. The energy flux F, in terms of the temperature gradient and an energy transport coefficient λ is: Temperature In practice, we work in terms of a quantity called opacity, which is related to λ by 2 −1 where ar is the radiation constant, c is the speed of light, and [κ] = cm g . In terms of κ, the temperature gradient can be written as where L ≡ 4πr2F is the luminosity ([L] = ergs−1) at radius r. To complete this equation, the form of κ has to be specified (then we have the two additional needed). In general κ is a function of ρ, T and {Xi}, and at the moment we denote this function as Luminosity Now we need an equation for L as well. Suppose that the energy release rate at r is ε ([ε] = ergg−1), then To complete this equation we need to know ε as a function of ρ, T and {Xi}. At the moment we assume that this function can be obtained and write it as Basic Equations of Stellar Structure Eqs. (10.2), (10.3), (10.7), and (10.9), are the four structure equations for the five quantities ρ, M, P, T , L. Together with the equation of state, they provide a complete description of a radiative star with given composition {Xi}, provided that P, κ and ε are known functions of (ρ,T,{Xi}). Boundary Conditions: Luminosity The boundary conditions for the structure of a star can be set as follows: where rs is the radius of the star. Since by definition the luminosity of a star is related to its radius and effective temperature by where σSB = arc/4 is the Stefan-Boltzmann constant, we can choose Ts = Teff so that L(r) = L at r = rs. Boundary Conditions: Pressure The value of Ps can be set by the condition of hydrostatic equilibrium for the stellar atmosphere: where g is the gravitational acceleration. Since mass in the atmosphere << mass within it, pressure on the stellar surface (bottom of atmosphere) can be written as: We relate this pressure to the optical depth of the atmosphere, Boundary Conditions: Pressure In general, ρ(r) falls rapidly as r increases and the main contribution to P(rs) and τ(rs) comes from layers near rs. We can therefore write P(rs) and τ(rs) in the following approximate forms: Boundary Conditions: Temperature Under the ‘gray-atmosphere’ approximation, i.e. the opacity is independent of photon frequency, the temperature is related to the optical depth as and so τ (rs ) = 2/3 [because T (rs ) = Teff ]. With all these relations, we can finally write the boundary conditions on the surface of a star as Since T and P on the surface of a star << stellar interior, the outer boundary conditions may be replaced by the so called ‘zero boundary conditions’: For stars whose outermost layers are in radiative equilibrium, these zero boundary conditions provide a good approximation to the actual boundary conditions. Boundary Conditions: Temperature Theoretically, the Mass of a star is a constant, while the radius is determined only after the model calculation. It is therefore more convenient to write the structure equations using M(r) (instead of r) as the independent variable, and set the boundary conditions at Ms. Normally the relation between r and M(r) is almost always one-to-one. In terms of M(r), the four stellar structure equations, for a star of mass Ms, can be written as 10.1.2 Stellar Structure Evolution Stars do evolve with time. As nuclear reactions proceed in the centre and energy is radiated away from its surface, both the structure and chemical composition of the star can change with time. Need dynamical equations in order to describe such evolution? NO Nuclear reactions drive the evolution of a star. Since in a normal star tdyn ≪ tth ≪ tnuc, any deviation from dynamical and thermal equilibrium caused by nuclear reactions can be adjusted very quickly: At any time the star can be considered to be in an equilibrium state governed by the static structure equations. The evolution comes in only because the star must adjust its structure and thermal state to a new chemical composition determined by the on-going nuclear reactions. If there is no bulk motion of material in a star (i.e., convection is negligible), the change of chemical composition is localized and is given by the chemical evolution equation, where the form on the right-hand side takes into account the fact that the rates of nuclear reactions in general depend on ρ, T and chemical composition. It is this change in chemical composition that drives stellar evolution. Solving the evolution of a star: start with initial assumption for composition as a function of r or M(r) at some time t0, solve the structure equations under the boundary conditions. With the newly obtained ρ, T and the old {Xi} calculate, for each radius in the star, the new composition at a slightly later time t = t0 + δ t : Solve again 10.1.3 Equation of State, Opacity, and Energy We must specify the equation of state, P(ρ,T,{Xi}), the opacity, κ(ρ,T,{Xi}), and the energy production rate, ε(ρ,T,{Xi}).
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