Problem $3$

On October $14$th$,2024,$ the Europa$-$Clipper space probe was launched from Earth on a SpaceX Falcon$-$Heavy Expendable. On its way to Jupiter's moon Europa, the spacecraft will first perform a fly$-$by at Mars on Mars on March $1$st$,2025.$

$($a$)$ Determine the orbital elements of the first leg of its interplanetary trajectory $($including the true anomaly at launch and at the fly$-$by at Mars$)$ at the C$3$ at departure.

$($b$)$ Knowing that the spacecraft has a mass of $6,065$ kg$,$ use the graph below to assess the correctness of the C$3$ found in part $($a$).$

$($c$)$ If the spacecraft had weighted only $4,000$ kg$,$ find the earliest date for the Mars fly by$.$

Explain your reasoning!