Launch Window for Mars
How often can we travel from Earth to Mars, or Mars to Earth?
Escaping Earth and Earth's orbit and climbing up the Sun's gravity well to Mar's orbit requires a lot of energy, which means a lot of launch velocity. The rocket capability needed is an exponential function of that velocity - a percent increase in velocity can mean many percent increase in rocket size, or many percent reduction in the size of the vehicle a given rocket can launch. To maximize mission results per dollar spent, we choose trajectories that minimize the velocity needed to accomplish the mission.
The minimum velocity trajectory is a Hohmann trajectory - half of an orbital ellipse that kisses the Earth's orbit at perihelion (closest to the Sun) and kisses the Martian orbit at aphelion (farthest from the Sun). Changing from Earth's 30 kilometer per second solar orbit to the Hohmann orbit requires an additional 2.8 km/s. As the vehicle climbs away from the Sun, it loses velocity, just like a ball thrown into the air. By the time the vehicle reaches Mars, it has slowed to 21.9 km/s. Changing from the Hohmann to Mar's 24.5 km/s orbit requires an additional 2.6 kilometers per second of velocity.
At that point, Mars is moving faster, and approaching the vehicle from behind. The easiest way to get that velocity is for Mars to drag its upper atmosphere past the vehicle and speed it up. The problem? Mars is small, and its atmosphere is 0.6% of Earth's, so there is not a lot of atmosphere to slow down in. Slowing down close to the surface risks unplanned lithobraking, that is, a fatal crash. Mars is Hard.
So - while mission planners can reduce payload and add extra velocity for a shorter transfer time from Earth, this means they also must add rockets for slowing down at Mars, reducing payload further. Missing the Hohmann launch time, either to reduce travel time or because of unplanned launch delays, can cancel the mission until Earth and Mars are in the proper configuration for another Hohmann.
The Earth orbits faster than Mars; Mars drops behind. The Earth must travel once around the Sun and catch up on Mars from behind again before the configuration repeats, and the launch window (the brief time when a mission is possible) opens up again.
What is the interval between launch windows?
Warning: The following approximate calculation (incorrectly) assumes that both Mars and Earth are in circular orbits around the Sun, in the same orbital plane, and a minimum energy Hohmann transfer orbit. In actuality, both orbits are elliptical; Earth's orbit has a perihelion of x and an aphelion of x, while mars has a perihelion of 1.3814 AU and an aphelion of 1.6660 AU.
f fraction of a Mars orbit between launch windows e = Earth year, 365.256 Earth days m = Mars year, 686.971 Earth days t time between launch windows
The time t is a fraction f of a Martian year m. As we will see, f is greater than one. f is also the fraction of Mars orbit that Mars has moved ahead - time is angle. While Mars moves forward, the Earth moves forward faster. The Earth will complete (1+f) orbits in the same time that Mars completes f orbits.
So: t = (1+f) e = f m . What is f ? With a little algebra, we learn f = e / (m-e) , so t = f m = m e / (m - e)
This solves to 779.946 Earth days, approximately 780 days, or 25.6 Earth months or 2.1 years. The precision in days is misleading; because Mars and Earth are in elliptical orbits, the size of the Hohmann orbit between them changes, as does the velocities and transfer times. There will be easier and harder Hohmann orbits, the intervals between launch windows will vary more or less than 780 days, and orbital inclination and Jupiter's gravity and the seasonal temperature and thickness of Mar's atmosphere will all play a role.
Since Mars turns, the longitude of reentry will be different for every mission as well. If you want the mission to land in a particular place, you must speed up the transfer orbit (and increase entry speed) so that Mars is turned the right way when you reach it. One Mars day of launch delay changes interplanetary spacings by half a million kilometers, requiring expensive trajectory adjustments, perhaps 20 m/s of interplanetary transfer velocity amd 50 m/s of launch velocity.
The time between windows is large because the difference between the Martian and Earth years ( m - e ) is relatively small. If Earth and Mars orbits were "closer", then the time between windows would be larger . If Mars was in a 1.1 year orbit, the time between launch windows would increase to 11 years - Mars would make 10 orbits, and Earth 11 orbits, to get to the same relative position.
Lunar Gravitational Assist
It is possible (but probably not practical) to launch from Earth so that the outbound trajectory goes past the Moon, "falls" towards it, deflects, then continues towards Mars with additional velocity. This is sometimes called a "slingshot" orbit. However, this can, at best, add only a few hundred meters per second to the outbound trajectory, if and only if the Moon is at the right part of its monthly orbit, approximately opposition to the Sun. This fortuitous alignment doesn't happen for every Mars launch window; perhaps 1 in 4 launch windows, or at 10 year intervals (WAG). Nothing to count on.
100 meters per second added at the Moon is like 25 m/s added to near-Earth launch velocity. Small errors in assist velocity and trejectory can magnify aiming error towards Mars. Overall, gravitational assist provides small percentage velocity advantages at the cost of long delays and increased risk.