**.**

*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)).· (T

^{20})

where: X(CN) = 0.01 and :

K = 4.34 x 10

^{25}Z(1 + X) r T

^{--3.5}(opacity)

_{}

^{}

e

_{o }= 10

^{-142}( X X(CN))

K

_{o}= 4.34 x 10^{25}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 10

^{25}Z(1 + X) / (0.64)

^{7.5}][LR

^{.5}

**/**M

^{5.5}] }

And:

D = {[(HG/ k)

^{20}1

**/**4p ] [e

**m**

_{o }^{20}] [M

^{22}/ LR

^{23}]}

= {[(HG/ k)

^{20}1**/**4p ] [10^{-142}( X X(CN))**0.64**_{ }^{20}] [M^{22}/ LR^{23}]}Where: H = 1.672 x 10

^{-24}

G = 6.67 x 10

^{-8}dyne-cm

^{2}/ g

^{.2}

k = 1.379 x 10

^{ -16}

^{ }cm

^{2}g s

^{-}

^{2}K

^{-}

^{1}

M = 3.970 x 10

^{33}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

**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.**

*core interval*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:

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

T = T

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. **q = 0. 7839768**This value is related to the pressure, temperature and density at the interface in the following manner:

T = T

_{c}_{c }q , q = ( r / r_{c})^{1/n }= (P/ P_{c})^{(1+ 1/n)}^{}^{}_{}^{}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

_{c}_{c }q = t_{c}_{c}(0.7839768) so that: t_{c}_{c}= 0.8876
In a similar manner we compute p

Since: q = (p/ p

Then: p

_{c}= 96.32 i.e. from log p = 1.7194 in the envelope solution tableSince: q = (p/ p

_{c})^{(1+ 1/n) }and n = 1.5Then: 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)

where the integral at right side can be evaluated using Simpson's rule:^{3/2}**/**b [ 96.32**]**^{1/2}[ 0.8876_{}]^{21}ò_{o}^{R }t*^{23}x*^{2}dxò

_{o}

^{R }t*

^{23}x*

^{2}dx = ò

_{o}

^{R }(0.785)

^{23}Ör

_{c}

**/**2.219

^{2}dx

_{}

^{}

0.2253 = {[(HG/ k)

^{20}1

**/**4p ] [10

^{-142}( 0.65) (0.01)

**0.64**

_{ }^{20}] [M

^{22}/ LR

^{23}]}

leading to: R = 9.80 x 10

^{10}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 10

^{10}cm = 1.68 x 10

^{10}cm

If the total mass of the star is given as M = 3.970 x 10

^{33}g then the core mass (which extends to the radial fraction r/R = 0.172) is: 6.85 x 10

^{32}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 10

^{32}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.

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