Constructing A Star From The Inside Out (Part 1)

Part of the table completed for the model star with a convective core, according to specifications given. From left to right, the quantities are: layer position as fraction of stellar radius (r/R), fraction of stellar mass to that radius in terms of total mass, fraction of luminosity relative to total luminosity L, the log of the pressure at that stellar layer, The log of the temperature at that layer, the log of the density at that layer, the opacity at that layer. Opac. indicates convective so for r/R for those values we are looking at the convective core.

Constructing a model star today is easily done using high speed computers and a number of models can be completed in hours. Back in 1970, there were only limited computing resources at most universities and so those of us assigned the task  - say as part of a project in a course ('Stellar Structure & Evolution') had only Wang calculators to do the work.

And so, as our AST 533 class was assigned this problem by visiting Yale Astronomy professor, James H.  Hunter, we knew what we were up against - all five of us. We had 3 weeks to do it, and the key for success was getting enough of the theoretical work done before doing the computations.

The problem that defined the stellar model construction had been posed thus:

Neglect radiation pressure and degeneracy and assume the gas is nearly totally ionized. Utilizing Wrubel's interior integrations and Schwarzschild's and Harm's envelop solutions:

a) Construct a consistent stellar model utilizing the U-V plane fitting technique, and

b) Calculate the following physical properties of the star: L, R, T_c  ,  r _c ,  P_c  , and the mass of the convective core,  m _c .

The work was arduous and entailed computing the properties using what is called the “U-V” plane fitting technique. From the Lane -Emden function for a specified polytrope, almost all model parameters can be consistently derived.  Let me note here that the "polytrope" or more exactly "polytropic gas sphere" is the abstract, theoretical template for physical star.

In the case of the 2 M_s (two solar mass) star I was constructing the main breakthrough arrived on recognizing the critical interface for the Lane-Emden function where the U-V plane was found to intersect with the stellar envelope. Hence this interface (its values) could be used as  reference point to determine the core values for  r _c   ,  p _c  ,  T _c   etc.

At the interface for this tabulated function one found:  q  = 0. 7839768

This value is related to the pressure, temperature and density at the interface in the following manner:

T  = T _c   q    ,    q    = (  r / r _c  )^1/n   =   (P/ P_c ) ^{(1+ 1/n)}


Polytropic gas spheres are basically mathematical entities used for modelling of actual stars. As usual, some basic assumptions are made (often in terms of temperatures, pressures, potential energies etc.) and these are then used to develop one or more "polytropic" models to test to see if they can work for a given star. Or, more likely, be employed as a guide to model a star.

A primary objective is to develop a basis for a self-gravitating sphere.  This means the force of attraction between M_r  e.g. the mass enclosed inside the stellar sphere of radius, r and r dr  (the mass of an element) is the same as that between a mass M_r at the center and  r dr at r.   By Newton’s law this attractive force is given by:

F = G M_r  r dr/ r²


Since the attraction due to the material outside r is zero, we should have for equilibrium:

- dP = G M_r r dr/ r²

 
Or:     dP/dr = - G M_r r / r²


 Consider now the mass of the shell between an outer  layer of a given star  and a deeper stellar layer. This is approximately, 4p r² r dr, provided that dr  (shell thickness) is small. The mass of the layer is the difference between M(r + dr) and M_r which for a thin shell is:

M(r + dr) -  M_r  = (dM/ dr) dr

The two equations, for dP/dr and dM/dr represent the basic equations of stellar structure, without which the innards of a star would be inaccessible to investigation.  The layout and form of a stellar model basically treats the star like an onion with layers of width dr, which are assessed for their respective parameters,  e.g. log P, log T,  log   r   etc. for each of the layers.


In the most desirable of cases, one first works to attain a simple relationship between the pressure P, and density ( r ) of a form: 

P  = K(1/ r ) ^{(1+ 1/n)}

where K and n are constants, and n is known as "the polytropic index" and K the "polytropic constant".

The polytropic index n can be defined:   n = 1/ (g   - 1)

where g   is the ratio of specific heats. (g  =  C _p / C _v )


In a non-relativistic limit, for example, one will have g = 5/3 and

n = 1 / (5/3 - 1)  =  1/ (2/3)  =  3/2

in which case,

P =  K (r)^{(1+ 1/3/2)}   =  K (r)^5/3

The so-called U-V plane begins with two definitions:

U = r/ m(r) = dm(r)/dr  (The polytropic definition)


V = - r/ r  (r)   dp(r)/ dr      (Schwarzschild's definition)


The values of U, V for the U,V plane fitting method are found in tabulations of the Lane-Emden function. The trick is to match these theoretical values to actual physical parameters and that's where Wrubel's interior integrations (for the convective core) and Schwarzschild's and Harm's envelope solutions come in.

One sample computation is shown below for given reference values of U, V:

U= 1.286   V = 4.843 

r/R = 0.360    r   =   3.528  x 10¹⁰  cm

M_r/M = 0.667    M_r = 2.647  x 10³³  g

Density:  r  =   UM_r/4p r³

=  [(1.286) (2.647  x 10³³  g)]/ 4p (3.528  x 10¹⁰  cm) ³

r  =   6.172  g/ cm ³        log   r  = 0.7904

As can be ascertained by comparing the above with the model table values (top) the relevant layer falls below the lowest tabled value, as we'd expect with  the fractional radius r/R = 0.360     while the lowest tabled value shown is r/R = 0.285.  But the casual reader can still get the basis, i.e. that a series of computations involving U,V pairs generate the layered parameters using similar calculations.    

In initially approaching the problem I first set out to determine the theoretical core values for two stellar parameters,  p _c  and  t _c  for pressure and temperature.  At the same time I needed to obtain the mean molecular weight  m  for the relative abundances given.  To that end one uses the Schwarzschild eqn.:

1/ m  =2X   +  3Y/4   + Z/2

Where X, Y and Z denote the hydrogen, helium and heavier elements abundances, e.g.

X = 0.65,  Y= 0. 32 and  Z = 0.03

Then:  1/ m  =  1.555   and    m  = = 0.64

To construct a stellar model-  say for a star of two solar masses  ( M_s = 1.988  x 10³³  g)  with a convective as opposed to a radiative core and with composition: X = 0.65 (i.e. 65% hydrogen), Y = 0.32 (i.e. 32% helium) and Z = 0.03 (i.e. 3 % heavy elements) is then a straightforward task once one has the appropriate theoretical framework with which to work..  Say also starting with an energy generation function:

e= 10^-142 (r  X X(CN)).· (T²⁰ )

where: X(CN) = 0.01   and :

K =  4.34 x 10²⁵  Z(1 + X)  r  T ^--3.5   (opacity)


A lot more work, of course, needs to be done. We will look at some other steps I had to take  in Part 2.

Suggested Comprehension  Exercise:

From the next to last U,V values given in the model table (i.e. at r/R = 0.262), confirm the values for: M_r/M, log P and log (rho) ,  i.e.   log   r .  Note the pressure at given M_r  is given by :

P   =      G  r M_r/ V r