Problem #$1$

$6$ points

A satellite shaped like a cube with $1$ m long sides is in Low Earth Orbit $($LEO$)$ at $400$ miles above the Earth's surface, as depicted in the figure below.

In position A$,$ the satellite is between the Sun and Earth, directly over the equator where $T_{\text{E}\text{a}\text{r}\text{t}\text{h}-A}$ is equal to $300$ K $.$ In position $B$ the satellite is over the North pole where $T_{\text{E}\text{a}\text{r}\text{t}\text{h}-B}=270K,$ and in position C it is over the equator but on the night$-$side of Earth, where $T_{\text{E}\text{a}\text{r}\text{t}\text{h}-c}=290K.$ Assuming empty space can be approximated as surroundings at $5$ K $,$ and that the solar absorptivity ${}_s,$ and emissivity $$ of the satellite surfaces are $0\mathrm{.}7$ and $0\mathrm{.}9,$ respectively, determine:

A$)$ Equilibrium temperature of the satellite in positions $A,B$ and C $(3$ points$)$

B$)$ What is the view factor between each of the $6$ faces of the satellite and the earth surface? $(3$ points$)$