A particular star is d $=28\mathrm{.}1$ light$-$years $($ly$)$ away, with a power output of P $=4\mathrm{.}401026$ W$.$ Note that one light$-$year is the distance traveled by the light through a vacuum in one year.

$($a$)$

Calculate the intensity of the emitted light at distance d $($in nW$/$m$2).$

nW$/$m$2$

$($b$)$

What is the power of the emitted light intercepted by the Earth $($in kW$)?($The radius of Earth is $6\mathrm{.}37106$ m$.)$

kW

What If$?$ Of the more than $150$ stars within $20$ light$-$years of Earth, $90$ are very dim red dwarf stars each with an average luminosity of $2\mathrm{.}001025$ W$,$ about $5\%$ the luminosity of the Sun. If the average distance of these objects from the Earth is $10\mathrm{.}0$ ly$,$ find the following.

$($c$)$

the ratio of the total intensity of starlight from these $90$ stars to the intensity of the single bright star found in part $($a$)$

Idwarf starsIsingle star

$=$

$($d$)$

the ratio of the total power the Earth intercepts from these stars to the power intercepted from the bright star in part $($b$)$

Pdwarf starsPsingle star

$=$