Intrastellar: Why a trip to mars can't take less than 250 days

It turns out that the most efficient way to get from Earth to Mars is not a straight line. Even if we had the fuel to accelerate away from the gravitation pull of the sun in a straight line, we'd have some trouble trying to 'gracefully' rendezvous with Mars as it's travelling sideways fairly quickly in its own orbital path.

Straight up wastin' fuel

If you were NASA, you might plan out a mission to Mars with the following priorities:

1. Reach Mars
2. Gently touch the surface of Mars. That is, when we get to Mars we need to roughly  match it's velocity relative to the Sun (it's a little more complex but lets simplify for now).
3. Use as little fuel as possible, given how much fuel you need just to get off the surface of the Earth.

It turns out we can use mother gravity to help us here.

Gravity - why the universe goes round

Think of the universe as a giant computer simulation (philosophers, come at me). In this simulation we can set up a bunch of rules and and apply some initial conditions. In our solar system, let's focus on gravity as our defining rule and velocity as the initial condition we can apply on any of our planets (there are others, I'll talk about them more in the postscript).

Gravity is simple - two objects of non-zero mass will accelerate toward each other according to some long and boring mathematical formula. So that means if you grab two objects, say the Sun and a space ship, they will accelerate toward each other.

Now say our system has a space ship traveling at some velocity relative to the sun and lets turn on our simulation.

Not to scale. Rockets are not as big as the sun

The space ship will start at it's current velocity but the rule of gravity means that it starts to also accelerate toward the Sun. This has the effect of stopping the space ship from moving in a straight line. It in fact starts moving in a vaguely round shape (look up conic sections if you want to see exactly what shape).

Now it turns out, this 'vaguely round shape' is actually defined by the initial velocity of the space ship. If the space ship is moving slowly at the start, it looks more circular. If the space ship is moving more quickly, that circle starts to deform into an ellipse.

My art skills are super amazing

This is where we can start to have some fun.

Lets put Mars at a circular orbit below. Now if we fiddle with our initial velocity enough we can stretch out our ellipse to intersect with the path of Mars orbit.

My drawing skills continue to improve

Congratulations! You've just mapped out a trajectory to get to Mars.

Hohmann Transfer

What we've just mapped out is actually called the Hohmann transfer orbit. It's essentially an orbit (duh) that allows us to hop between planets, or even other random orbits.

The Hohmann transfer is generally considered the method of transferring orbits that consumes the least amount of fuel. It's obvious to see why - all we do is give our spacecraft a gentle push in the right direction and gravity does the rest for us. 

Carrying out a Hohmann transfer is a deceptively simple two step process:

1. Once our space-craft has escaped Earth's velocity, we burn our rocket engines just a little more. This accelerates us in the right direction and gives us the right velocity to enter into the orbital shape that we need. 
2. Once we reach Mars we then do another short burn to get captured by Mars' gravitational influence. 

The only amount of fuel that we need for this (other than launch etc) is just for the two short burns. Gravity does the rest of the work for us!

Unfortunately since we are obeying the laws of physics here, we also then have to obey the formulas that define orbital period. In this case our transfer orbit will take us about 259 days to get to Mars.

But I want to get there sooner!

Then you have to use more fuel, unfortunately. There exist other methods that require delta-v (and hence more fuel) that will get you there faster. In fact, they require so much fuel that NASA can't even be bothered to talk about them until we figure out how to better use Ion propulsion (which has more bang per kg). You could always "point and shoot" but you'd have to plan your trajectory well enough to account for the orbital velocity of Mars and the gravitational effects of Earth and the Sun.

So, pack lunch. Or pack 260 lunches. I'm not a nutritionist.

Cool Links

• Simple explanation of Hohmann Transfers - thanks to NASA. God knows why they chose comic sans
• Discussion of Hohmann Transfer alternatives - also thanks to NASA
• Trajectories and orbits - Escaping Earth's orbit
• Conic Sections - from Wolfram. AKA what orbits look like
• The two body problem - mathematical derivations that lead to the definition of orbital paths

PS: Pedantic notes

The two-body problem that I've roughly described above is actually just a teensy bit more complex than I've made it out to be. There are more rules than just gravity, and there are more initial conditions than just relative velocity (radius, position, time just to name a few). Surprisingly the amount of knowledge you need to understand the two-body problem doesn't extend far beyond high-school mathematics. If you're more interested in the equations and how everything comes out, visit Wikipedia.

Hohmann transfers are also not quite that simple. In order to get from Earth to Mars you need to align the orbits so that Mars is where you need it to be at the end of your transfer orbit. That means that you have to wait a few days in order to get your phase angle difference correct. More information in the above links on Hohmann Transfers.

Finally escaping Earth's influence and entering Mars influence are non-trivial orbital manoeuvres. Essentially once in orbit above Earth, your spacecraft needs to accelerate a certain amount so that we hit a hyperbolic escape trajectory from Earth. Then once we reach Mars we then need to 'decelerate (sort of) in order to get 'captured' into a Mars Orbit. For more information on this, read up on Trajectories and orbits