2. Let's model a star. Our star will be a ball of gas compressed by its own gravity, held together by gas pressure, and is leaking out energy from its core due to radiative diffusion.
a) Write down the three equations of stellar structure that we have learned so far: for hydrostatic equilibrium, radiative diffusion, and mass conservation. Supplement this with an equation of state appropriate for matter inside a star. These four equations (plus an equation for energy generation) are the building blocks of our stellar model.
Mass conservation: \(\frac{dM(r)}{dr}=4\pi r^2\rho(r)\)
Hydrostatic equilibrium: \(\frac{dP}{dr}=\frac{-GM(r)\rho(r)}{r^2}\)
Radiative diffusion: \(\frac{dT(r)}{dr}=\frac{3L(r)\kappa(r)\rho(r)}{16\pi^2acT^3(r)r^2}\)
Equation of state (gas pressure): \(P(r)=nkT=\frac{\rho(r)}{m(r)}kT(r)\)
b) Consider the boundary conditions for the Sun and rewrite the differential equations as proportionalities.
Mass conservation: \(\frac{\Delta M}{\Delta r}=4\pi r^2\rho(r)\)
If we rearrange this to solve for density, we get \(\rho={M}{4\pi r^3}\), which simplifies to \(\rho\sim\frac{M}{r^3}\). This makes sense based on how we calculate density.
Hydrostatic equilibrium: \(\frac{\Delta P}{\Delta r}=\frac{-GM(r)\rho(r)}{r^2}\)
We can plug in the previously determined proportionality relating density and mass: \(\frac{\Delta P}{\Delta r}=\frac{-GM(r)}{r^2}\frac{M}{r^3}\)
By rearranging, getting rid of the constants, and solving for pressure, we get \(P\sim\frac{M^2}{r^4}\).
Radiative diffusion: \(\frac{\Delta T}{\Delta r}=\frac{3L(r)\kappa(r)\rho(r)}{16\pi^2acT^3(r)r^2}\)
Since opacity and luminosity depend on more than just the radius, we will leave these terms in the proportionality. However, we can substitute in the proportionality for density: \(\frac{\Delta T}{\Delta r}=\frac{3L(r)\kappa(r)}{16\pi^2acT^3(r)r^2}\frac{M}{r^3}\)
This simplifies to the proportionality \(T^4\sim\frac{M\kappa(r)L(r)}{r^4}\).
Correct
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