Download The Internal Structure of Stars Computational Mechanics Project Fall

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 4r 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 4r
(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)
GT   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  0.21  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.