Interplanetary Orbits and Hohmann Transfer to Mars

Even though Mars and Earth are relatively close when on the same side of the Sun (about 0.5 AU apart), sending spacecraft to Mars takes a long time.
Missions to Mars tend to launch every couple of years.

When launching a rocket, its trajectory follows an ellipse with the Sun at one focus, according to Kepler's and Newton's laws.
The goal is for the spacecraft to reach Mars' orbit when Mars is at the correct location to ensure a collision (or rendezvous).
Missions aim to minimize fuel consumption to reduce costs and weight.
Less fuel allows for a smaller spacecraft.
Reducing the initial amount of fuel also reduces the amount of fuel needed to launch that fuel, presenting a compounding benefit.

The most fuel-efficient method for interplanetary travel is the Hohmann transfer orbit.
After leaving Earth's orbit, the spacecraft accelerates to enter a larger, more eccentric orbit.
This elliptical orbit's perihelion (closest point to the Sun) is at Earth's orbit, and its aphelion (farthest point from the Sun) is at Mars' orbit.
Timing is critical to ensure the spacecraft arrives at Mars' orbit when Mars is in the same location.

Earth's orbital period: approximately one year.
Mars' orbital period: a little under two years.
The Hohmann transfer orbit has a semi-major axis larger than Earth's but smaller than Mars'.
The semi-major axis of the transfer orbit is the average of Earth's and Mars' semi-major axes.
- $a<em>{transfer} = (a</em>{earth} + a_{mars}) / 2$
The period of the entire transfer orbit is about 18 months.
The journey from Earth to Mars takes about half of this period, around nine months.

Earth orbits the sun faster than Mars due to its closer proximity and stronger gravitational forces.
Accelerating the spacecraft puts into a faster orbit, aligning to the cannonball thought experiment.
As the spacecraft approaches Mars, it is moving slower than Mars because it is falling back toward the Earth's orbit.
v =
sqrt{\frac{GM}{r}}, where $G$ is the gravitational constant, $M$ is the mass of the sun, and $r$ is the orbital radius.
A second acceleration is needed upon reaching Mars to match its speed for a gentle landing.
The spacecraft accelerates twice during the transfer: once to leave Earth and again to match Mars' speed.
Despite speeding up twice, the spacecraft ends up moving slower relative to the sun, matching Mars' speed.

The change in speed is due to the continuous exchange between kinetic and potential energy as per Kepler's laws.
The spacecraft loses kinetic energy and gains potential energy, resulting in a slower orbit that matches Mars' speed.
After approximately nine months, the spacecraft lands on Mars.

The principles of Hohmann transfer orbits apply to missions to other planets like Venus and Jupiter.
Understanding these orbital mechanics is crucial for planning any interplanetary voyage.