<p>I was recently showing that an index of <span class="math-container">$n=0$</span> for the <a href="https://en.wikipedia.org/wiki/Lane%E2%80%93Emden_equation" rel="nofollow noreferrer">Lane-Emden equation</a> results in constant density throughout the star. However, in my calculations I had to use the constant <span class="math-container">$\alpha$</span> which can be defined as <span class="math-container">$$\alpha = \frac{r}{\xi}$$</span> or <span class="math-container">$$\alpha = \sqrt{\frac{(n+1)K\rho_c^{\frac{1}{n}-1}}{4\pi G}}.$$</span>
Now, using the first expression works perfectly but I just noticed that the second expression breaks down as <span class="math-container">$n=0$</span>. So how can it still be a solution to Lane-Emden? Do we use some kind of <span class="math-container">$\displaystyle \lim_{n \rightarrow 0}$</span> technique or how is this issue resolved?</p>

CC BY-SA 4.0
2023-10-25T10:25:48.387Z
Have you looked at https://en.wikipedia.org/wiki/Lane–Emden_equation ?  There appears to be a special case in the derivation for n=0
GrapefruitIsAwesome
CC BY-SA 4.0
2023-10-25T13:51:29.720Z
@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.
Kian31
CC BY-SA 4.0
2024-01-30T07:06:22.570Z
https://en.wikipedia.org/wiki/Lane%E2%80%93Emden_equation#For_n_=_0
ProfRob
<the-sun><astrophysics><mathematics>
How is an index of n=0 possible for the Lane-Emden equation?
Kian31
# How is an index of n=0 possible for the Lane-Emden equation?

(Asked by: username_0 on 2023-10-25)

I was recently showing that an index of $n=0$ for the [Lane-Emden equation](https://en.wikipedia.org/wiki/Lane%E2%80%93Emden_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?

**Tags:** the-sun, astrophysics, mathematics
> **Comments:**
> **username_1**: Have you looked at https://en.wikipedia.org/wiki/Lane–Emden_equation ?  There appears to be a special case in the derivation for n=0  
> **username_0**: @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.  
> **username_2**: https://en.wikipedia.org/wiki/Lane%E2%80%93Emden_equation#For_n_=_0  

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## 0 answers

