= Polytrope of index one
{title2=$P=K\rho^2$}
A <stellar polytrope> with <polytropic index> one has $\nabla\Phi=-2K\nabla\rho$ in <hydrostatic equilibrium>. Combining this with the <Poisson equation for Newtonian gravity> gives the <Helmholtz equation>
$$
\nabla^2\rho+k^2\rho=0,\qquad k^2=\frac{2\pi G}{K}.
$$
The spherical solution regular at the origin is $\rho(r)=\rho_c\sin(kr)/(kr)$. Its first zero gives $R=\pi/k=\sqrt{\pi K/(2G)}$, independent of central <mass density>, and $\bar\rho/\rho_c=3/\pi^2$.
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