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The Internal Structure of Stars Computational Mechanics Project Fall 2010 In the nineteenth century it was known that the brightness (or luminosity) and the radius of a star are, nearly, unique functions of the mass of the star with small variations due to the chemical composition (i.e., the mass fraction of hydrogen, X, helium, Y, and heavier elements, Z.) In the early 1900's all of the basic relationships describing the structure of stars had been developed and by 1950 the internal energy source (i.e., nuclear fusion) was known in sufficient detail to derive the functions that had earlier been observed. The governing equations for a steady state star are: GM dP 2 r , dr r (1) dM r 4 r 2 , dr (2) dLr 4 r 2 , dr (3) 3 L r dT 4ac T 3 4r 2 dr 1 T dP 1 - P dr (radiation ) , (convection) where P is the pressure (dynes/cm2), is the density (g/cm3), G is gravitational constant (6.67x10-8dyne cm^2/g2), r is the radial position from the center (cm), M is the mass inside radius r (g), Lr is the energy emitted outward from radius r (erg/s), is the energy generated per unit mass (erg/g-s), T is the temperature (K), a is the Stefan Boltzmann constant (7.56x10-15 erg/cm3-K4, is the opacity (i.e., the fraction of energy absorbed in dr is dr (cm2/g), is the ratio of the specific heats (1/ is 0.6), and c is the speed of light (2.997x1010 cm/s). To determine which of equations (4) applies, if equation (5) is true, that isif (4) 1 T dP 3 Lr , 1 3 2 P dr 4ac T 4r (5) Then the radiation condition holds. If equation (5) is not true the convection condition holds. Three additional relationships are required. These have the following functional forms P P( , T , X , Y ) , (,T , X ,Y ) , ( ,T , X ,Y ) . where X is the mass fraction of hydrogen, and Y is the mass fraction of helium. If Z is the mass fraction of elements heavier than helium (i.e., all elements except hydrogen and helium) then X Y Z 1. (6) The pressure has two components: a mechanical which is proportional to the temperature, T, and a radiation, which is proportional to T4. Then P 1 k a T T 4 , H 3 (7) where k is Boltzmann's constant (1.379x10-16 erg/K), H is the mass of a proton (1.672x10\-24 g), is the mean molecular weight in proton masses, and is given by 1 3 1` 2X Y Z , 4 2 (8) The energy generation function, has two major components: 1) due to proton-proton fusion pp and 2) due to cn pp cn , where (9) pp T 10 6 K pp pp X 2 cn , (10) , (11) cn T cn XX cn 6 10 K and X cn 1 Z. 3 (12) The constants pp , pp , cn , cn are given as functions of temperature in Tables 1 and 2 respectively. Table 1 Constants for pp Equation T /10 K log 10 pp 5 8 11 14 20 -6.84 -6.04 -5.56 -5.02 -4.4 6 pp 6 5 4.5 4 3.5 Table 2 Constants for cn Equation T/10 K log 10 cn 6 14 20 26 30 43 -22.2 -19.8 -17.1 -15.6 -12.5 cn 20 18 16 15 13 The opacity is difficult to calculate from theory. The following algorithm gives results that are sufficiently accurate: 1) Calculate the quantum mechanics parameter, G (T ) , (1/cm3) GT 2 2 me kT 3 / 2 , (13) h3 where me is the mass of the electron (9.105x10-28 g) and h is Planck's constant (6.67x10-27 erg-s). 2) Calculate the electron density (1/cm3) Ne 1 X 2H . (14) 3) Find the guillotine factor, t, from Table 3 below. Table 3 Guillotine Factor log10t log10[G(T)] 6 T/10 1 2 3 4 5 6 8 10 12 14 16 18 20 25 ∞ 0.03 0.06 0.02 0 0.01 0.08 0.25 0.33 0.37 0.39 0.4 0.4 0.41 0.48 6 5 4 3 2 0.06 0.07 0.14 0.3 0.38 0.41 0.43 0.43 0.44 0.44 0.51 0.13 0.14 0.22 0.37 0.45 0.48 0.49 0.49 0.49 0.49 0.56 0.35 0.5 0.58 0.6 0.61 0.6 0.58 0.6 0.66 0.56 0.68 0.76 0.78 0.79 0.78 0.78 0.79 0.79 0.93 0.99 1.01 1.02 1.02 1.02 1.03 4) Calculate the opacity due to ionization (cgs units) i 130 N e T 3.5t 1 X Y . 5) Calculate the opacity due to scattering by free electrons (cgs units) (15) e 0.21 X . (16) 6) Take the opacity as the sum of the larger of the two quantities, i and e, plus 1.5 times the smaller. The sun's mass, M sun , outer radius, R sun , and luminosity, Lsun are M sun 1.985 10 33 g Rsun 6.915 1010 cm . Lsun 3.780 10 33 erg/s (17) The mass fraction of hydrogen in the sun is X 0.73 . (18) In the early 1900s the central density, c, and the central temperature, Tc, of the sun had been calculated as (cgs units) log 10 c 1.88 log 10 Tc 7.29 . (19) Take the mass fraction for the heavier elements, Z, as Z 0.02 . (20) Y 0.25 . (21) Then The boundary conditions are: 1) at the center, r 0 , M r Lr 0 , and 2) at the surface, r Rsun , 0. (22) From equation (7) either P or T can be specified at the surface. Using the formulation, find the pressure, temperature, and density as a function of the radial distance from the center of the sun. If you integrate from the center out, take equations (19) as your first guess for the central conditions. Because of the approximations made, equations (19) may not be close to the correct values. Keep track of the resulting luminosity, Lr, and mass, Mr, at the surface for all of your guesses, that result in real stars, and you will have the functions described on page 1. Note that a Taylor series in r, about r equal to zero, is required to start the integration at the center. The project will be divided into four parts: Part 1 – Write a subprogram to evaluate the energy generation in equations (11) and (12) by interpolating Tables 1 and 2. Part 2 – Write a subprogram to evaluate the total opacity. Cubic splines may be useful to interpolate Table 2.. Part 3 – Write a program to numerically integrate equations (1) through (4) using the central conditions in equations (19). Part 4 – Complete the analysis by finding central conditions that approximate the Sun.