Design an interplanetary Hohmann transfer to return a spacecraft from Mars orbit to Earth orbit. The initial Mars

orbit is a circular parking orbit at an altitude of $3500$ km above the surface of Mars. The final Earth orbit is an

elliptical orbit with a perigee of $450$ km and an eccentricity of $0\mathrm{.}25.$ Assume that the orbits of Mars and Earth are coplanar.

Assume ideal relative positions of Earth and Mars.

$($a$)$ Calculate the heliocentric velocities at the departure from Mars and at the arrival near Earth.

$($b$)$ Calculate the $\backslash$Delta $$ required to depart from the parking orbit around Mars to enter the departure hyperbola

$($c$)$ Calculate the $\backslash$Delta $$ required to capture the spacecraft from its arrival hyperbola near Earth to enter the

elliptical parking orbit around Earth.

$($d$)$ What is the total $\backslash$Delta $$ of the interplanetary$-$transfer mission?

$($e$)$ After placed in the elliptical parking orbit around Earth, the spacecraft will undergo aero$-$braking to reduce its

speed. Aero$-$braking happens due to the atmospheric drag, which slows down the aircraft, without having to

burn any fuel. In the planned mission, the aero$-$braking is expected to reduce the velocity of the spacecraft at

the perigee by $100$ m$/$s after every revolution, which will result in a change of orbit shape. Assuming that the

aerobraking occurs instantaneously at the perigee, what will be the eccentricity of the orbit after $4$ revolutions

of the spacecraft around Earth?

Assume:

$$M $=3390$ km

$$M $=4\mathrm{.}281\backslash$times $1013$ m$3$

s

$2$

$$ Mars distance from Sun: $227\mathrm{.}9\backslash$times $109$ m

$$E $=6378$ km

$$E $=3\mathrm{.}986\backslash$times $1014$ m$3$

s