Constructing A Star From The Inside Out (Part 2)

Section of Table for Lane-Emden function as used by Marshall Wrubel for his stellar interiors integration procedure. Upper arrow denotes values for core, and lower (partial- extending downward) for envelope.

We left off with the generic forms for opacity and energy generation which now need to be adapted to the specific modeling of a 2 solar mass star with convective core.

We saw:

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)

These will now be adjusted so that (for our specific modeling problem):

e  _o  = 10^-142 ( X X(CN))

K _o =  4.34 x 10²⁵  Z(1 + X)

For which:

K =  K _o r  T^--3.5

And:  e =   e _o  r  T ^u

The foregoing are used as factors in two essential equations employed in the construction of consistent stellar models. These relate the star's parameters in a general way - and permit us to proceed with the theoretical model. The equations, after incorporating the two factors, are:

C = {3/4ac (k / HG)^7.5  (1/ 4p)^3.  [ K _o  / (m)^7.5  ][LR^.5 / M^5.5  ] }

= {3/4ac (k / HG)^7.5  (1/ 4p)^3.  [ 4.34 x 10²⁵  Z(1 + X)   / (0.64)^7.5  ][LR^.5 / M^5.5  ] }

And:

D = {[(HG/ k)²⁰  1/ 4p ]   [e _o  m ²⁰] [M ²²/ LR²³]}

=   {[(HG/ k)²⁰  1/ 4p ]   [10^-142 ( X X(CN))    0.64 ²⁰] [M ²²/ LR²³]}

Where:  H  = 1.672  x 10^-24

  G = 6.67 x  10^-8    dyne-cm ²/ g^.2

k =  1.379  x 10 ^-16   cm² g s^-2 K^-1

M =  3.970  x 10³³   g


From the above we first determine p _c  ,and   t _c   using Wrubel's interior integrations for the convective core with Schwarzschild's and Harm's envelop solutions. The latter are tabulated as dimensionless variables for a specific C -value - in thei case of log C =  -6.0.

The core temperature and pressure meanwhile can be obtained from D  by an integral equation.

1  =  D  (2.5)^3/2 / b  [ p _c ] ^1/2 [ t  ] ²¹ ò _o ^R  t* ²³  x*  ² dx


Where:  x*   =   b  x Ör _c  /   2.5 t _c

t*  =  t/  t _c


The preceding formulations pave the way for construction of the model from the core outward and then joining the envelope solution - assuming the Cowling model for a star with radiative envelope and convective core.  The latter integrations are done in terms of the Emden variable, delineated as a tabulated function by Wrubel and shown at the very top.  If we go from    x = 0.0  to 1.1  we get the core interval corresponding to associated U. V values. The (double) barred level at   1.2  is the  interface between the core and the envelope.  The latter proceeds from 1.3 onwards.

The right -sided red lined border of the table then, gives the convective core solution used in the Lane-Emden function. The interface marks where the U, V plane is found to closely intersect the envelope. In other words, the interface is here used as a reference level to determine the magnirudes of  r _c   ,  p _c  ,  T _c   etc.

As noted in Part 1, at the interface one finds:  q  = 0. 7839768

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

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

If the reader will refer back to the table in Part 1, and locate the same U, V pair shown at the interface in the table above (e.g. U = 2.578, V = 1.236) above he will be able to track back the convective core from radius fraction r/R = 0.172 to 0.0.  In other words, the U,V values in the Lane -Emden function of Wrubel's table 6a enables us to complete the convective solution for the actual model.

He will also be able to see how each non-dimensional parameter value has been obtained. For example, we see at r/R = 0.172 that q  = 0. 7839768.  But, for example,  t  = t_c   q    .  Not shown is the envelope solution for the same interface which yields (for the same U.V. pair): log t = -0.15695 and log p = 1.7194. 

Take antilog ( -0.15695 ) so t =  0.6969  then:

0.6969 =    t_cc   q   =     t_cc    (0.7839768)  so that:   t_cc   =  0.8876

In a similar manner we compute  p _c =   96.32   i.e. from log p = 1.7194 in the envelope solution table

Since:  q    =  (p/ p _c ) ^{(1+ 1/n)}     and n = 1.5

Then:   p _c   =     p / q ^5/2      and   q ^5/2 =  0.5441

 

These core values can then be used to solve for D from the earlier multiple parameter equation:

1  =  D  (2.5)^3/2 / b  [ 96.32 ] ^1/2 [ 0.8876  ] ²¹ ò _o ^R  t* ²³  x*  ² dx

where the integral at right side can be evaluated using Simpson's rule:

ò _o ^R  t* ²³  x*  ² dx  =   ò _o ^R   (0.785) ²³  Ör _c  /   2.219  ² dx 

And with D obtained, e.g. D = 0.2253,  one can now solve for the radius R by making it the subject of the equation:

0.2253  = {[(HG/ k)²⁰  1/ 4p ]   [10^-142 ( 0.65)  (0.01)    0.64 ²⁰] [M ²²/ LR²³]}

leading to:   R  =   9.80  x 10¹⁰   cm

With R determined, the dimensional reference values can now be obtained which immediately enable the computation of all the core values: T _c  ,  r _c ,  P_c  , and the mass of the convective core,  m _c .


At the interface, r/R  = 0.172  so r =   (0.172) R 

= (0.172) 9.80  x 10¹⁰   cm  = 1.68  x 10¹⁰   cm 


If the total mass of the star is given as M =  3.970  x 10³³  g  then the core mass (which extends to the radial fraction r/R = 0.172) is:  6.85  x 10³²  g.  However, consultation with the envelope solution shows the mass fraction at the interface is given by (- log q) = 0.8293 so taking the antilog, q = 0.148 and the actual core mass at that level is 5.88  x 10³²  g.  What gives? Well, only that the core radius is not precise and subject to some uncertainty on account of blending the two differing solutions - for convective core and radiative envelope.

By using the appropriate U.V pairs for the remaining core and radiative envelope solutions, the model star can be completed out to the photosphere.


Suggested Comprehension Exercises:

1) Using the appropriate stellar model parameter relations calculate the core values: T _c  ,  r _c ,  P_c  .

2) Do the first two radiative envelope computations for temperature, density and pressure, i.e. at radial fractions r/R = 0.183 and r/R = 0.200.

3) Estimate the uncertainty in the core radius as a consequence of the 'fuzziness' of the interface parameters.