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\title{Are we alone in the Universe?\\ The Drake Equation}


\author[ ]{Matteo Cantiello}

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\textbf{Previous "\href{https://www.authorea.com/users/2/articles/10997/_show_article}{Habitable Planets}" -- Next "\href{https://www.authorea.com/users/2/articles/24715/_show_article}{Astrobiology}"} \\

There is on average one planet orbiting every star in the Universe \citep{2013ApJ...764..105S, 2012Natur.481..167C}. If this sounds exciting, you might wanna read the \href{https://www.authorea.com/10997/}{previous post} in this series. Our Galaxy (the Milky Way) is an immense disk of gas and stars with a diameter of about 100~000 light years, hosting about 100 billion stars and, therefore, also about 100 billion planets. Take a deep breath.
Now, it turns out the Milky Way is just one of 100 billion galaxies that populate our Universe, a colossal expanding stretch of spacetime with an age of 13.7 billion years.
The math is trivial: There are about 10~000~000~000~000~000~000~000 = $10^{22}$  planets out there. 
This number is extremely large. Apparently larger than the\href{http://www.npr.org/blogs/krulwich/2012/09/17/161096233/which-is-greater-the-number-of-sand-grains-on-earth-or-stars-in-the-sky}{ number of grains of sand} found in every beach and every desert on Earth.

But how many of these planets host life? And in particular,\textbf{ how many planets host intelligent life we might be able to communicate with}? 

In order to estimate the number of technological civilizations that might exist among the stars, in 1961 Frank Drake proposed the following simple equation: 
%\begin{equation}\nonumber
%N = R \times f_p \times n_e \times f_l \times f_i \times f_c \times L
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\caption{{\textbf{The Drake equation:} \textit{it estimates the number $N$ of civilizations in The Milky Way Galaxy whose electromagnetic emissions are detectable. \href{https://www.authorea.com/28104}{Interactive version here}.}%
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It is a product of factors giving the number $N$ of civilizations in the Milky Way Galaxy with whom we could make contact. The terms in the equation are:
\begin{itemize}
%\item $N$ is the number of civilizations in the Milky Way Galaxy with whom we could make contact.
\item $R$ is the rate of star formation, which tells how many stars are born every year in our Galaxy.
\item $n_e$ is the average number of habitable planets in any planetary system
\end{itemize}
All the $f$ terms are factors $\le 1$:
\begin{itemize}
\item $f_p$ is the fraction of stars that have planets 
\item $f_l$ is the fraction of habitable planets hosting life
\item $f_i$ is the fraction of life-bearing planets that develop an intelligent life-form
\item $f_c$ is the fraction of intelligent life-forms that decide to communicate
\end{itemize}
Finally $L$ is the longevity of a communicative civilization (in years). Humankind has been "communicative" only for a few decades and it's hard to predict what $L$ will be for us. However I think that the current lack of detection of E.T. communications could be used to put some limits on the maximum longevity of a communicative civilization. 
More on this in a \href{https://www.authorea.com/users/25815}{follow up post}.   


Scientifically speaking, a lot has happened since 1961. For example some of the terms in the Drake equation are now known, so we can re-write it in a simpler form.

$R$, the \textbf{rate of star formation} tells how many stars are born every year in our Galaxy. This number is known and is about 10. It might seem quite small, but the Galaxy is about 10 billion years old, so plenty of stars have born (and died) in the meanwhile. $R\approx 10$. 

$f_p$ Represents the \textbf{fraction of stars that form planets}. 
%We now know that, on average, there is one planet orbiting every star in the Universe. To put it in the words of W. %Borucki, Principal Investigator of the KEPLER satellite: "When you wish upon a star, you are wishing upon a star with %planets". So $f_p \approx$ 1
As already mentioned this number is now known: on average, there is one planet orbiting every star in the Universe \citep{2013ApJ...764..105S,2012Natur.481..167C}. $f_p \approx$ 1

$n_e$ Is the number of \textbf{Earth-like planets} per planetary system. 
We just got to know this factor! Earth-like planets are very common. Statistically speaking at least 1 in 5 planets around Sun-like stars could potentially support life \cite{Petigura_Howard_Marcy_2013}. $n_e \approx$ 0.2

So the product of the first 3 terms is now well established and is on the order of \textbf{2}. 

\begin{equation}\nonumber
N =\underbrace{\overbrace{R}^{\approx10} \times \overbrace{f_p}^{\approx 1} \times \overbrace{n_e}^{\approx 0.2}}_{\sim 2} \times \underbrace{f_l \times f_i \times f_c \times L}_{?}
\end{equation}\label{Drake}

We can then simplify the Drake equation and re-write it in its "2015 form\footnote{Note all the factors are dimensionless numbers and $L$ should be considered $L/yr$, so that also N is a dimensionless number}" as:

\begin{equation}\nonumber
\boxed{N \approx 2\, f_l \, f_i \, f_c \, L\,}
\end{equation}\label{Drake_simplified}

The interesting fact about this compact version of the Drake equation is that it only contains factors dealing with the emergence of life itself. Broadly speaking astrophysics has answered the question "Is the Universe likely to provide the right environment for life as we know it ?" with a big "yes!". \\
Time to talk Astrobiology in the \href{https://www.authorea.com/24715}{next post}!

\textbf{Previous "\href{https://www.authorea.com/users/2/articles/10997/_show_article}{Habitable Planets}" -- Next "\href{https://www.authorea.com/users/2/articles/24715/_show_article}{Astrobiology}"}

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