From Low Earth Orbit to Mars
The most important part of the journey involves transferring from Earth to Mars. The propulsion requirements for this part of the journey are extremely severe and probably determinant in the choice off the propulsion system.
The most energy efficient way of getting from Earth to Mars is by means of an elliptical, quasi planetary orbit that is tangential to both the Earth and Mars orbit. This type of orbit is called a Hohmann transfer orbit.
In this particular trajectory, the spacecraft will use the terrestrial orbit's velocity around the Sun to launch it to Mars. An initial rocket burn is required and then in theory the spacecraft can coast gravitationally to Mars. As it can be seen in the diagram, the total angle relative to the Sun covered by the spacecraft will be 180° or pi radians.
The point where the spacecraft leaves Earth orbit will be the periapsis of the orbit (point in the orbit that is closest to the center, in this case, the Sun.) In this case, as the spacecraft will be traveling in a heliocentric orbit it can directly be called perihelion. When it intercepts the Martian orbit, the spacecraft will be at apoapsis, in this case aphelion, the point in its orbit when it will be furthest away from the Sun.
In order to transfer from terrestrial orbit to an orbit that will take the spacecraft further out from the Sun, as we know, we will need extra speed to attain this higher orbit. Our goal then is to calculate the increase in speed required for the transfer orbit.