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Orion1

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I am interested in building a three shell mathematical solar model based upon the core, the radiative zone, convective zone.

I have been researching to determine a way to build my first mathematical solar model. The best mathematical model for the core density function that I have located is from reference 1, model 4.

This density function contains three dimensionless parameters [itex]a, \delta, \gamma[/itex] and the reference does not describe any other equation solutions for model 4.reference 1 said:Note that the models in (4.1.13) and (4.1.1) are really valid only in the interior core of the Sun. The convective zone has an entirely different behavior for the density distribution. But for computing the total mass, pressure, temperature and luminosity, we have integrated out over the entire length of the solar radius [itex]R_{\odot}[/itex]. This is not appropriate. The integration should have been done only in the interior core of the Sun. Hence a more appropriate model is of the following form:

[tex]\rho(r) = \rho_0 \left( 1 - a \left( \frac{r}{R_{\odot}} \right)^{\delta} \right)^{\gamma} \; \; \; \delta > 0 \;, \gamma > 0 \;, a > 0 \; \; \; \text{(Model 4)}[/tex]

Since [itex]1 - ax^{\delta} > 0 \;, x = \frac{r}{R_{\odot}}[/itex], we have [itex]0 \leq x \leq \frac{1}{a^{\frac{1}{\delta}}}[/itex]. Hence for the total integral, the range should have been [itex]0 \leq r \leq \frac{R_{\odot}}{a^{\frac{1}{\delta}}}[/itex].

However, the impression I receive from reference 1, is that this density function is valid only in the interior core of the Sun.

I am uncertain that the other models that I have researched have addressed the issue of the different behavior for the density distribution functions.

What are the correct density functions for the core and the radiative zone and the convective zone of the Sun?

And should the dimensionless parameter [itex]a[/itex] be written as [itex]\alpha[/itex] in the formal equation?

According to reference 5, radiative transport of energy is described by the radiative temperature_gradient equation:

[tex]\frac{dT}{dr} = - \frac{3 \kappa \rho(r) L(r)}{64 \pi r^2 \sigma T(r)^3}[/tex]

According to reference 4, I should model the convection zone as a polytrope:

[tex]P(r) = K \rho(r)^{\gamma}[/tex]

Ideal gas law:

[tex]P(r) = \frac{k_B \rho(r) T(r)}{\bar{\mu} m_u}[/tex]

Polytropic convective zone pressure:

[tex]P(r) = \frac{k_B \rho(r) T(r)}{\bar{\mu} m_u} = K \rho(r)^{\gamma}[/tex]

Polytropic ideal gas convective zone density:

[tex]\boxed{\rho(r) = \left(\frac{K \bar{\mu} m_u}{k_B T(r)} \right)^{\frac{1}{1 - \gamma }}}[/tex]

Is this equation correct?

Reference 7 defines the polytropic_pressure as:

[tex]P(r) = \frac{N_A k_B T(r)}{\bar{\mu}}[/tex]

Why is this pressure_equation missing a volume dimension?

Solar core range:

[tex]r_1 = (0 \to .25) R_{\odot} = .25 R_{\odot}[/tex]

Radiative zone range:

[tex]dr_2 = (.25 \to 0.71) R_{\odot} = 0.46 R_{\odot}[/tex]

Convective zone range:

[tex]dr_3 = (0.71 \to 1) R_{\odot} = 0.29 R_{\odot}[/tex]

Reference:

http://neutrino.aquaphoenix.com/ReactionDiffusion/SERC4chap4.pdf" [Broken]

http://en.wikipedia.org/wiki/Sun" [Broken]

http://www.nasa.gov/worldbook/sun_worldbook.html" [Broken]

http://astro.berkeley.edu/~eliot/Astro252/hw2.pdf" [Broken]

http://en.wikipedia.org/wiki/Standard_Solar_Model#Equations_of_state"

http://en.wikipedia.org/wiki/Ideal_gas_law#Alternative_Forms"

http://en.wikipedia.org/wiki/Polytrope" [Broken]

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