I was recently showing that an index of $n=0$ for the Lane-Emden equation results in constant density throughout the star. However, in my calculations I had to use the constant $\alpha$ which can be defined as $$\alpha = \frac{r}{\xi}$$ or $$\alpha = \sqrt{\frac{(n+1)K\rho_c^{\frac{1}{n}-1}}{4\pi G}}.$$ Now, using the first expression works perfectly but I just noticed that the second expression breaks down as $n=0$. So how can it still be a solution to Lane-Emden? Do we use some kind of $\displaystyle \lim_{n \rightarrow 0}$ technique or how is this issue resolved?
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$\begingroup$ Have you looked at en.wikipedia.org/wiki/Lane–Emden_equation ? There appears to be a special case in the derivation for n=0 $\endgroup$– GrapefruitIsAwesomeCommented Oct 25, 2023 at 10:25
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$\begingroup$ @GrapefruitIsAwesome I have read it multiple times but cant seem to spot a remark to the porblem. It solves the Equation for $n=0$ but does not use $\alpha$ in the context of the solution. $\endgroup$– Kian31Commented Oct 25, 2023 at 13:51
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1$\begingroup$ en.wikipedia.org/wiki/Lane%E2%80%93Emden_equation#For_n_=_0 $\endgroup$– ProfRobCommented Jan 30 at 7:06
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