Can a star with a constant density profile be possible?

Question

The equation of states for a star is given by a polytropic equation, where density depends on the $n$th power of $\theta$. Please refer to the literature

First, what is this $\theta$? It can't be a constant for sure.

Second, for $n=0$, we get a solution for such a star, for which the density is constant throughout. Is this a practical solution? If we move radially outwards, the pressure at a point is proportional to the mass of the cylinder of unit cross section above it. So Pressure will always decrease with increasing radius, $r$. Then how is it possible that we can find a solution for a star with a constant density profile?

Answer 1

Not all values of $n$ are realistic models of an equation of state. As stated in Wikipedia (http://en.wikipedia.org/wiki/Polytrope), realistic constants for neutron stars are around 0.5 and 1 (this is probably because the pressure is generated by degeneracy, not temperature). All other star models mentioned by Wikipedia have larger $n$.

Considering the case, where $n \approx 0$ (which is unrealistic for a star):

$$P = K \rho^{(n + 1)/n} \approx K\rho^{1/n}$$

Clearly, as $n \rightarrow 0$, $P$ is negligable up to some density and is huge for bigger densities. Thus, the material is incompressible. This could be a rather good model for a uniform solid (or liquid) planet.

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According to the paper, $\theta(\xi)$ is not a constant. $\theta(\xi)$ depends on $\xi$, which depends on $r$ (Original text in the paper: The radius variable r is multipled by a constant which depends on n, K and other constants to be rescaled into the variable ξ.).