Problem $3$: Elliptical orbit $(10$ pts$.)$

A transfer orbit is an elliptical orbit meant to move satellites between two different$-$radius circular orbits. A satellite with mass $\backslash ($ m$=3000\backslash$mathrm$\{$~kg$\}\backslash )$ is in a circular orbit with $\backslash ($ r$\_\{1\}=20,000\backslash$mathrm$\{$~km$\}\backslash ).$ It needs to be moved into a larger orbit with $\backslash ($ r$\_\{2\}=25,200\backslash$mathrm$\{$~km$\}\backslash ).$

The satellite is moved from orbit $1$ into the transfer orbit at point A $,$ and then is moved into orbit $2$ at point B $.$

A$.$ For the elliptical transfer orbit, find the following quantities:

$1.$ The semi$-$major axis $\backslash ($ a $\backslash ).$

$2.$ The distance from the center of the orbit to the satellite, $\backslash ($ c $\backslash ).($Hint: First label on the diagram the location of the $2$ foci of the ellipse, the distance $\backslash ($ a $\backslash ),$ and the distance $\backslash ($ c $\backslash ).$ Then find $\backslash ($ c $\backslash ).)$

$3.$ The eccentricity $\backslash ($ e $\backslash ).$

B$.$ What is the orbital period in the elliptical orbit?

C$.$ How much energy was added to get the satellite from orbit $1$ into the transfer orbit?