<p>Consider the following problem:</p>
<blockquote>
<p>Sun has a convection zone from <span class="math-container">$0.7 R_\odot$</span> to the solar surface, when <span class="math-container">$R_\odot$</span> is the radius of Sun. If the matter is marginally convectively unstable and the convection zone contains a very small fraction of Sun's mass so that the gravitation field in the convection zone can be taken to be <span class="math-container">$-GM_\odot /r^2$</span> when <span class="math-container">$M_\odot$</span> is mass of Sun, obtain the density, pressure and temperature profiles as functions of the radial coordinate. Assume the convection zone consists of <span class="math-container">$75\%$</span> <span class="math-container">${}^{1}$</span>H, <span class="math-container">$20\%$</span> <span class="math-container">${}^{4}$</span>He, <span class="math-container">$3\%$</span> <span class="math-container">${}^{12}$</span>C and <span class="math-container">$2\%$</span> <span class="math-container">${}^{56}$</span>Fe and is hot enough such that inter-atomic potential energies of the matter therein are negligible compared to the kinetic energies of the particles.</p>
</blockquote>
<hr />
<p>For convection region, We know that
<span class="math-container">$$\frac{dT}{dr}=\left(1-\frac{1}{\gamma}\right)\frac{T}{P}\frac{dP}{dr}$$</span></p>
<p>We also have the equation of state given by
<span class="math-container">$$P=\frac{k_B}{\mu m_H}\rho T=\alpha \rho T\rightarrow \frac{dP}{dr}=\alpha\left(T\frac{d\rho}{dr}+\rho\frac{dT}{dr}\right)$$</span>
putting the value of <span class="math-container">$dT/dr$</span>, We get
<span class="math-container">$$(\gamma-1)T=\rho\frac{dT}{d\rho}$$</span>
<span class="math-container">$$\int_{T_\text{eff}}^T\frac{dT}{(\gamma-1)T}=\int_0^\rho\frac{d\rho}{\rho}$$</span></p>
<p>You can see that the second term have a problem with boundary condition. Is something wrong with my approach? Why this boundary condition is not fitting?</p>

CC BY-SA 4.0
2021-09-16T21:00:26.390Z
Hint:  Where on the sun is the mass density zero?
honeste_vivere
CC BY-SA 4.0
2021-09-17T03:03:27.493Z
@honeste_vivere On the surface, I guess.
Himanshu
CC BY-SA 4.0
2021-09-17T14:05:25.197Z
No, it's not zero there or anywhere.  At the photosphere, the number density is still something like $10^{15} \ cm^{-3}$ or on that order (it's around $10^{9} \ cm^{-3}$ in the corona).
honeste_vivere
<the-sun><stellar-structure>
Solution of stellar structure equation for Sun in Convective Zone
Himanshu
# Solution of stellar structure equation for Sun in Convective Zone

(Asked by: username_0 on 2021-09-16)

Consider the following problem:

> Sun has a convection zone from $0.7 R\_\odot$ to the solar surface, when $R\_\odot$ is the radius of Sun. If the matter is marginally convectively unstable and the convection zone contains a very small fraction of Sun's mass so that the gravitation field in the convection zone can be taken to be $-GM\_\odot /r^2$ when $M\_\odot$ is mass of Sun, obtain the density, pressure and temperature profiles as functions of the radial coordinate. Assume the convection zone consists of $75\%$ ${}^{1}$H, $20\%$ ${}^{4}$He, $3\%$ ${}^{12}$C and $2\%$ ${}^{56}$Fe and is hot enough such that inter-atomic potential energies of the matter therein are negligible compared to the kinetic energies of the particles.

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For convection region, We know that
$$\frac{dT}{dr}=\left(1-\frac{1}{\gamma}\right)\frac{T}{P}\frac{dP}{dr}$$

We also have the equation of state given by
$$P=\frac{k\_B}{\mu m\_H}\rho T=\alpha \rho T\rightarrow \frac{dP}{dr}=\alpha\left(T\frac{d\rho}{dr}+\rho\frac{dT}{dr}\right)$$
putting the value of $dT/dr$, We get
$$(\gamma-1)T=\rho\frac{dT}{d\rho}$$
$$\int\_{T\_\text{eff}}^T\frac{dT}{(\gamma-1)T}=\int\_0^\rho\frac{d\rho}{\rho}$$

You can see that the second term have a problem with boundary condition. Is something wrong with my approach? Why this boundary condition is not fitting?

**Tags:** the-sun, stellar-structure
> **Comments:**
> **username_1**: Hint:  Where on the sun is the mass density zero?  
> **username_0**: @honeste_vivere On the surface, I guess.  
> **username_1**: No, it's not zero there or anywhere.  At the photosphere, the number density is still something like $10^{15} \ cm^{-3}$ or on that order (it's around $10^{9} \ cm^{-3}$ in the corona).  

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## 0 answers

