Figure $6$: Probability $(logP,$ color$-$bar$)$ of an ancient civilization not visiting the Earth within the last ${10}^{6}yr$ for values of $$ and settlement civilization lifetime $T_s($Carroll$-$Nellenback et al$.2019,$ ApJ, $158,117).$

$4.15$ points The "Fermi Paradox" asks why we currently don't know of any extraterrestrial intelligence $($ETI$).$ Answer the questions about the spread and detection of an ETI civilization:

$($a$)4$ points Numerical models for the spread of an ETI civilization throughout the galaxy have found that the probability $P$ of such a civilization not visiting the Earth in the last ${10}^6$ years depends on the parameter $,$ defined as:

$=f{d}_p^3$

where $f$ is the fraction of systems that are settleable, $$ is the density of systems $($unit: number per volume$),$ and $d_p$ is the probe range $($unit: length$).$ What does $$ represent?

$($b$)3$ points If $1={10}^0,$ what range of settlement civilization lifetimes $T_s$ is required for a high probability $P$ of not detecting an ETI $(P$~~$1)?$ Give your answer in yr$,$ using Figure $6.$

$($c$)3$ points If ETI civilizations last for $T_s{10}^6$ years, what values of $$ results in a $1\%$ chance of not knowing about an ETI civilization Give the range of values using Figure $6.$

$($d$)3$ points The best way of discovering such a civilization might be detecting their equivalent of radar. Currently, terrestrial airport radar have a power $($luminosity$)L={10}^{17}$erg$\frac{s}{s}.$ Assuming isotropic emission $($equal in all directions$),$ what is the flux of such a radar if located $d=15$ ly away? Give your answer in ergs $\frac{?}{?}\frac{s}{c}{m}^2,$ where $1ly=9\mathrm{.}46{10}^{17}cm.$

$($e$)2$ points A radio telescope currently under development is the Square Kilometer Array $($SKA$).$ If the SKA has a collecting area of $1k{m}^2,$ how much energy would it detect from such a radar in $t=10$min$=600s?$ Give your answer in ergs.