Summary of stellar equations

Transcription

1 Chapter 8 Summary of stellar equations Two equations governing hydrostatic equilibrium, dm dr = 4πr2 ρ(r) Mass conservation dp dr = Gm(r) r 2 ρ Hydrostatic equilibrium, three equations for luminosity and temperature gradients, dl dr = 4πr2 ε(r) dt dr = 3ρ(r)κ(r) L(r) 4acT 3 (r) 4πr ( ) 2 dt γ 1 T dr = dp γ P dr equations governing nucleosynthesis, dn dt = 1 τ n and an equation of state, Luminosity Radiative T gradient Convective T gradient, Nucleosynthesis, P=P(T,ρ,X i,...) Equation of state. 289

2 290 CHAPTER 8. SUMMARY OF STELLAR EQUATIONS The two temperature-gradient equations are to be employed as follows: The radiative gradient ( ) dt dr rad rad = 3ρ(r)κ(r) 4acT 3 (r) L(r) 4πr 2 should be used unless the condition ( ) ( dt < 1 1 ) T dp dr γ P dr for convective instability is satisfied, in which case the adiabatic gradient should be used: ( ) ( ) dt γ 1 T dp = dr γ P dr Some equations in this set, like the last two, dn dt = 1 τ n ad P=P(T,ρ,X i,...) are to be understood schematically. Nucleosynthesis, Equation of state, Nucleosynthesis will in general involve a complex coupled network of differential equations The equation of state will depend on the physics of the problem and may take a variety of forms.

3 291 These equations represent a considerably simplified description of a star. Even in this simplified form their solution for realistic cases presents formidable numerical problems. Relatively specialized techniques must be used for some aspects of the solution 1. because of the boundary conditions required for a star, and 2. because these equations couple processes having characteristic time and length scales that may differ by many orders of magnitude.

4 292 CHAPTER 8. SUMMARY OF STELLAR EQUATIONS Let us now consider solution of the stellar equations. We shall address three aspects Using timescale analysis to avoid solving the equations directly. Obtaining solutions by approximating the stellar equations. Full numerical solution of the stellar equations.

5 8.1. SUMMARY OF IMPORTANT STELLAR TIMESCALES 293 Table 8.1: Some important stellar timescales Timescale Characteristic value Value for Sun R 3 Dynamical τ dyn 55 min 2GM Thermal adjustment τ therm GM2 RL yr Nuclear burning τ nuc ε Mc2 L yr 8.1 Summary of important stellar timescales A timescale τ s characteristic of some important physical process represented by a quantity s may be defined as τ s = s/ṡ. This is just a generalization of the standard example from introductory physics of estimating a time to travel some distance x as t = x/ẋ=x/v, where v is the average velocity. At several points in the preceding discussion, three important timescales have been discussed. These are summarized in Table 8.1 and discussed further below.

6 294 CHAPTER 8. SUMMARY OF STELLAR EQUATIONS 1. Dynamical timescale: A dynamical timescale is defined by a characteristic time to restore hydrostatic equilibrium: τ dyn = R R 3 = v esc 2GM 1 G ρ where v esc =(2GM/R) 1/2 ρ = 3M/4πR 3 were used. For the Sun τ dyn 55 minutes. 2. Thermal adjustment timescale: The thermal adjustment (or Kelvin Helmholtz) timescale is associated with time for a star to shed thermal energy, so τ therm = U L = GM2 LR, where U is the internal energy and L the luminosity, and U GM 2 /R by the virial theorem. The Sun has a thermal adjustment timescale of about yr. 3. Nuclear burning timescale: The time to burn the star s nuclear fuel may be approximated by τ nuc = εm 0c 2 L, where ε is the efficiency for conversion of mass into energy in hydrogen fusion, M 0 is the mass of hydrogen available to burn in the star. For the Sun this gives τ nuc yr.

7 8.2. AN APPROXIMATE SOLUTION: THE LANE EMDEN EQUATIONS An Approximate Solution: The Lane Emden Equations The equations of hydrostatic equilibrium may be combined to give the differential equation dm=4πr 2 ρ(r)dr ( r 2 ) dp = 4πGρ. dr dp dr = Gm(r) r 2 ρ 1 d r 2 dr ρ We then approximate the equation of state in polytropic form, P=Kρ γ = Kρ 1+1/n γ 1+ 1 n. Introducing dimensionless variables ξ and θ through ρ = ρ c θ n (n+1)kρ c (1 n)/n r=aξ a=, 4πG where ρ c ρ(r= 0) is the central density, the differential equation embodying hydrostatic equilibrium for a polytropic equation of state may be expressed in terms of the new dependent variable θ(ξ) and independent variable ξ as, 1 d ξ 2 dξ ( ξ 2dθ dξ ) = θ n.

8 296 CHAPTER 8. SUMMARY OF STELLAR EQUATIONS In terms of these new variables the boundary conditions are θ(0)=1 θ (0) dθ dξ = 0, ξ=0 The first follows from the requirement that the correct central density ρ c = ρ(0) be reproduced. The second follows from requiring that the pressure gradient dp/dr vanish at the origin (necessary condition for hydrostatic equilibrium). Then we may integrate ( ) 1 d ξ 2 ξ 2dθ = θ n. dξ dξ outward from the origin (ξ = 0) until the point ξ = ξ 1 where θ first vanishes, to define the surface of the star, since at this point ρ = P=0because ρ = ρ c θ n P=Kρ γ. Solutions having this property generally exist for n<5.

9 8.2. AN APPROXIMATE SOLUTION: THE LANE EMDEN EQUATIONS 297 Table 8.2: Lane Emden constants n γ ξ 1 ξ1 2 θ (ξ 1 ) / / / / The equation ( 1 d ξ 2 ξ 2dθ ) = θ n. dξ dξ has analytical solutions for the special cases n = 0,1, and 5, but In the physically most interesting cases the equations must be integrated numerically to define the Lane Emden constants ξ 1 and ξ 2 1 θ (ξ 1 ) for given n. These are tabulated for various values of n and γ in Table 8.2.

10 298 CHAPTER 8. SUMMARY OF STELLAR EQUATIONS Corresponding solutions are plotted in the following figure 1.0 θ(ξ) Solutions of the Lane-Emden equation n = 4 n = ξ -0.2 n = 0 n = 1 n = 2 n = 3 and pressure profiles computed for polytropic equations of state with several values of n are shown in the following figure _ P P c Sun n = 3/2 0.2 n = 3 n = R/R sun The n = 3 polytrope approximates relatively well the actual pressure profile of the Sun (Standard Solar Model).

11 8.2. AN APPROXIMATE SOLUTION: THE LANE EMDEN EQUATIONS 299 The transformation equations ρ = ρ c θ n r=aξ a= (n+1)kρ c (1 n)/n, 4πG may then be used to express quantities of physical interest in terms of these constants for definite values of the polytropic index n. For example, the radius R is R=aξ 1 = (n+1)k 4πG ρ(1 n)/2n c ξ 1, and the mass M is given by (Exercise) [ M 4πa 3 ρ c ξ 2dθ ] dξ ξ=ξ 1 [ ] (n+1)k 3/2 = 4π ρ c (3 n)/2n ξ1 2 4πG θ (ξ 1 ), Eliminating ρ c between these two equations gives a general relationship between the mass and the radius, ( ) M= 4πR (3 n)/(1 n) (n+1)k n/(n 1) ξ (3 n)/(n 1) 4πG 1 ξ1 2 θ (ξ 1 ). for a solution with polytropic index n.

12 300 CHAPTER 8. SUMMARY OF STELLAR EQUATIONS Limitations of the Lane Emden approximation The Lane Emden equation has elegant solutions with a direct physical interpretation, but it has serious limitations: It reflects only the property of hydrostatic equilibrium, and then only for a polytropic equation of state. Thus it describes only the mechanical part of stellar structure and has nothing to say about temperature gradients and energy transport, and their coupling to the full problem. There are two general situations where a polytropic equation of state may be reasonable. The realistic equation of state depends on T as well as ρ, but additional physical constraints between T and P lead to a polytropic equation of state. Example: The adiabatic constraint applied to an ideal gas leads to a polytropic equation of state PV Γ 1 Pρ Γ 1 = constant. Then the temperature is effectively fixed by a constraint T = T(P) and not by coupling to the full set of equations. The realistic equation of state actually is approximately polytropic. Often true in very dense matter such as white dwarfs and neutron stars.

13 8.3. NUMERICAL SOLUTION OF THE STELLAR EQUATIONS Numerical solution of the stellar equations The stellar structure and evolution problem has some specific features that complicate obtaining numerical solutions. These issues fall primarily into two categories: Boundary conditions. Some boundary conditions must be imposed at the center and some at the surface. This requires specialized techniques to ensure compatibility of the solutions. Extreme space and time scale differences. Example: Equations governing isotopic composition and energy release for the PP chains involve timescales that can differ by orders of magnitude. They can be solved only with custom numerical methods. Numerical solution of the full set of equations describing stellar structure and stellar evolution is a specialized topic that would take us too far afield for the present discussion.