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\section{A lesson from the sky?}  Let's define $n=\frac{N}{V}$ the number density of communicative civilizations in the Galaxy, where $N$ is the total number of such civilizations existing at any given time in the Milky Way and $V$ is the total galactic volume. Assuming an homogeneous distribution across the Galaxy, the average separation $r$ between two neighbour communicative civilization is $r\sim n^{-1/3}$. In order for two civilization to communicate, their separation needs necessarily to be smaller than their communicative phase lifetime $L$. This is assuming communication technology that is limited by the speed of light. We can then ask what would be the minimum value for $L$ such that communication can occur (we call this value $L_c$). That is we impose $r=L$. We then obtain $L\approx (N/V)^{-1/3}$.  Now we can use equation \ref{Drake_simplified} to derive the minimum lifetime of the communicative phase in order for contact to occur  \begin{equation}  L_c \sim \left[\frac{V}{2\,f_l\, f_i\, f_c}\right]^{1/4}.  \end{equation}  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}  %L_c \sim \left(\frac{V}{L_c}\right)^{1/3} = V^{1/4}  L_c \sim V^{1/4},  \end{equation}  which states that at any given time the minimum value for the number of galactic communicative civilizations in order to achieve contact is given by the volume of the galaxy to the power 1/4. Note that at the same time this number is also the minimum communicative phase lifetime required for contact. The absenece of a contact then allows to assess that in our Galaxy $L  diff --git a/layout.md b/layout.md  index a5eafda..b03a07f 100644  --- a/layout.md  +++ b/layout.md 
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We_are_thinking_creatures_living__.tex  Astrobiology.tex  fc.tex  Derive L.tex  Last remarks.tex  figures/Drake4/Drake_Equation.jpg