There's a common misconception among the general public that physics is boring, yet nothing could be further from the truth. As a physicist I can say with confidence physics is awesome, it's just physics teachers that are boring.
I hesitate to say all physics teachers are boring, because I've met a few really exceptional ones... but there's also been plenty of others who some how manage to suck all the colour and joy out of the incredible relationships that govern our universe. So with this in mind we're going to tackle arguably one of the most abstract, mathematically complex, and potentially boring concepts in Newtonian physics - Orbital Mechanics - and we're going to make it awesome instead.
Also I don't mean in that fake-smiling "YAY!!! ISN'T THIS FUN KIDS?!" way where you're desperately trying to convince yourself and others that your entire life's work means something, while your soul slowly crumples inside as you fight the creeping existential dread that the universe is unloving and ambivalent to your existence and everything you do... I mean in a "Holy crap the universe is ridiculous, awful and weird, and I need to know more!" equation & jargon-free kind of way to explain how we'll get to Mars.
Which I think we can all agree is a lot more fun than reading Nietzsche and embracing nihilism over a cup of tea.
Firstly some basics. If you want to go anywhere in space, you either need to a) increase your spacecraft's velocity using a rocket or other propulsion system (we'll cover propulsion in the next article) for a little to increase the size of your orbit and coasting through space as gravity to pulls you around on a curved path, b) have a ludicrously powerful propulsion system to brute force a straight line to wherever you want to go, or c) travel at 88mph and use 1.21 Gigawatts of energy to tear a hole through the fabric of space-time and pop out wherever/whenever you like.
Because we don't yet have anything even remotely powerful enough to brute force a straight line through space, and neither Doc Brown or Sam Neill have been opening any portals to hell recently, that leaves firing a rocket for a bit to increase the size of our orbit and letting gravity do the rest of the work. The most fuel-efficient way to do this is called a "Hohmann Transfer", where you increase your velocity just enough to reach where you're going. When you're trying to get from Earth to Mars that means burning your rockets when your spaceship is closest to Earth (to get the most out of the rocket thrust) and after coasting for 8.5 months you arrive at Mars at the slowest point of your new orbit.
Above: Burn your rocket when you're travelling fastest at #1 (Earth), slow down as you travel along the yellow line, arrive at #3 (Mars) when you're at the slowest point of the new orbit
But "fuel-efficient" is slow and boring - the space exploration equivalent of having sex while listening to Enya. It's fine if you don't have anything better to do with your afternoon - or if you want to launch cargo to Mars that can take 8.5 months to get there - but the longer you spend in deep space the more cosmic radiation (and Enya) you're being exposed to. Humans also need food and water and oxygen and a bunch of other nonsense robots and cargo don't, so Hohmann transfers aren't ideal for sending humans to Mars unless you really hate them.
Getting to Mars in less than the 8.5 months means we have to leave faster. Sounds simple, but this gets ridiculously complicated really quickly. The three things to remember though are the more you accelerate:
We'll talk more about propulsion systems in the next post, but right now using traditional chemical rockets the quickest we can get to Mars is about 6 months. Which looks something like this:
Interplanetary transfer for the Mars Odyssey probe in 2001
Obviously you also don't aim for where Mars is when you're launching from Earth, because it won't be in the same place you were aiming for 6 months later. Like throwing a water-bomb at a toddler you aim ahead to where your target will be in the future, letting gravity and the easily predictable path of a planet or under 5 do the work for you.
Because Earth orbits the Sun once every 365.25 days and Mars orbits the sun once every 687 Earth days*, they only line up for this kind of transfer once every 22 Earth months.
*Mars has a "day" of 24 hour and 36 minutes called a "Sol", so 1 year on Mars is 668.6 sols
Alright, enough already
There a couple of other little tricks of gravity we can also use to get to Mars quicker and with less fuel too, namely Orbital Slingshots and Ballistic Capture.
Orbital Slingshots AKA "Gravity Assists" AKA "Big Thing Make Spaceship Go Fast"
Turns out you can actually use an entire planet to speed up your spacecraft if you're willing to swing in close enough. The gravitational attraction between a planet and a spacecraft doesn't just move the spacecraft - it also moves the planet a tiny fraction too! So by flying up behind a planet as it orbits and letting gravity swing your spacecraft towards it you'll slow the planet down (increasing it's "year" by a few nanoseconds) but massively increase the velocity of your spaceship!
The last diagram, I swear
This is actually what they use in The Martian to get the Hermes back to Mars and save Mark Watney. While Donald Glover is being a mentalist with a stapler in a NASA boardroom, he's describing an especially powerful orbital slingshot. The speed boost the Hermes gets swinging around Earth is the reason they can get back to Mars so quickly, but it's also why they're going so fast at the other end.
Kristen Wiig will have none of your swingline stapler shenanigans
Recently we've discovered another way to get things from Earth to Mars that doesn't require you waiting nearly 2 years for an alignment or having Sam Neill take you through a portal to Hell... but it's even slower than the "Enya-Space-Sex" Hohmann Transfer. This "Ballistic Capture" approach involves getting just close enough to a planet or moon that it's gravity slowly pulls your spacecraft into it at low velocity without needing any extra fuel to slow down. It's just like knocking a pool ball towards a pocket and having it stop right on the edge: it'll either roll in on it's own after a few seconds, or you give the table a little bump to help it in.
Ballistic capture was used by the Japanese probe "Hiten" to orbit the Moon in 1990, but until recently it was believed that Mars was too small and too far away for ballistic capture to work. Some clever folks with a super computer recently worked out though that you can launch towards Mars anytime as long as you don't mind taking up to a year to get there. For a human crew this would be like having sex to Enya playing at half tempo, so you might prefer the trip through actual Hell with the Event Horizon instead.
Before you realise Sam Neill is playing Enya through the PA too...
For someone like Mark Watney though - slowly starving on Mars because his potato crops were suddenly freeze dried - this would have been pretty handy. Building a new probe full of food, testing it properly (rather than just glancing at it and saying "Yeah mate, she'll be right") and launching it on a 1 year trajectory using a ballistic capture would have been considerably quicker and safer than the mentalist orbital slingshot the Hermes crew do in the film. Although I guess staying put and eating potatoes for a few more years isn't as "Hollywood" as:
Me during the last 10 mins of The Martian
So there you have it: orbital mechanics that's awesome and not lame/boring. Obviously there is so much math to dig into if that's what gets you off, and I'm not one to kink-shame: go and get wild solving three-body problems or dig out on the crazy equations describing Lagrange points, gravitational keyholes, Halo orbits, Lissajous orbits and Horseshoe orbits, or Hill spheres... if that's your thing.