Chapter $8$: $2$ Body Central Force Problems

The height of a satellite at perigee is $300$ km above the earth's surface and it is $3000$ km above the earth's surface at apogee.

a$)$ Find the orbit's eccentricity. $(10$ points$)$

b$)$ If we take the orbit to define the $xy$ plane and the major axis in the $x$ direction with the earth at the origin, what is the satellite's height when it crosses the $y$ axis? $(10$ points$)$ Hint: the earth's radius is $6400$ km $.$ The mass of the earth is $5\mathrm{.}97{10}^{24}kg,$ and $G=6\mathrm{.}67{10}^{-11}N\frac{m^2}{k}{g}^2.$

c$)$ If we want to transfer the satellite to a circular orbit when it is at apogee, and give it a constant height of $3000$ km above the earth's surface, what thrust factor $$ should we use? $($Assume the thrust factor is the fraction by which the speed of the satellite changes, as defined in the text in chapter $8.)$