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%L_c \sim \left[\frac{V}{2\,f_l\, f_i\, f_c}\right]^{1/4}.  %\end{equation}  Since we are discussing the second hypothesis, we are assuming  that $f_i$  is $f_i \nll 1$ not a small number. Let's assume, as Frank Drake did in his estimates, that $f_1=1$.  On the other hand our discussion of the Drake equation led to the conclusion that, roughly speaking, $N \sim L$ (meaning the denominator in previous equation is $\sim$ 1) (but see the \href{http://en.wikipedia.org/wiki/Rare_Earth_hypothesis}{rare Earth hypothesis} for a different conclusion). Therefore we can further simplify and obtain   \begin{equation}