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

It's not about getting high, its about going fast

My hero and god, Randall Munroe makes this point infinitely more entertaining than I ever could, but essentially getting into space is not about how high you can shoot yourself up, its about how fast you can encourage yourself to go sideways. You can spend all day firing things up in the air with whatever legal (or illegal) means you can acquire. You might even get above the 150km-above-sea-level mark and be in 'space' (or not depending on which circles you travel). But if you don't build up enough sideways speed, you're always going to fall right back down.

Sounding rockets and even balloons can get you to the edge of space

To get to space you need to first get in orbit, and to get in orbit you need to travel in a circle. Once your spacecraft starts moving in circle fast enough, the force of gravity keeps you going in a circle instead of forcing you back down to the ground.

A diagram showing orbital velocity, via aerospaceweb.com

How much speed? Well at 600km above sea level, about where the Hubble Space Telescope sits, you need to be going at 7.56 km per second, or about 27 000 km per hour. And so to accelerate the 11 tonne hubble space telescope to that kind of speed, we pretty much have to rely on rockets.

Rockets. Goddamn Rockets. 

Rockets manage to be awesome and awful all at the same time.

Rockets are cool because all controlled explosions are cool and all in all they are pretty amazing feats of engineering. Rockets suck because they're huge, they weigh more than they should and shake a crap ton.

Let's deal with size and weight, To launch a heavy spacecraft or any lump of mass, you need to burn fuel. Fuel adds mass - mass that you will need to launch with more fuel. More fuel means more mass which needs more fuel. Hopefully you see where I'm going with this.

Eventually after a little bit of algebra you come up with the Tsiolkovsky rocket equation. The equation maps delta v (a fancy term rocket scientists use to indicate how much faster you want to go) against the ratio of payload mass (that we need to work in space) and fuel mass. The graph ends up looking like something below

The rocket equation, mapping mass ratio against delta V / exhaust velocity

There are two scary things about that above graph.

Firstly - look at that curve! The more delta v you need the steeper the graph gets (trust me you need more delta V to get further into space). That means that the amount of fuel you'll need gets insane unless you're launching something really really tiny.

Secondly - the x axis maps mass ratio, not absolute mass. So if you need to send up a payload twice as large, you'll need twice as much fuel. Imagine having to double the size of your rockets to send up a 10 tonne space telescope vs what is already huge to launch a 5 tonne communications satellite.

An example

No beginner's blog would be complete without having some horrendously inaccurate back of the envelope calculations. So why not dredge up one of my old university assignments and simulate ourselves an asteroid capture.

Let's say that we want to rendezvous with an asteroid that has roughly the same orbit as Earth but is some distance away from us. Capturing asteroids is all the rage lately, so lets say we want to hit it with NASA's Asteroid Redirect Vehicle - which will weigh aproximately 16 tonnes if you include all the fuel it's carrying.

Now any rocket carrying this vehicle will need to

• Launch into Earth Orbit
• Escape from Earth's Orbit
• Perform a manoeuvre to meet with the asteroid

All of these steps need (you guessed it) delta v. And as we have learned above, delta V needs lots of fuel. Now, accounting for ideal launch conditions, getting the Earth's spin and orbit around the sun to help us gain some speed, we eventually figure out that we need roughy 8km/s of delta V. And when you plug all the maths into the rocket equation and use the specs from the Atlas V user's guide (which is super easy to find btw), you end up with a little over 500 000kg of fuel.

Atlas V rocket launch

Five hundred thousand kilograms of fuel to launch something that is only 16 thousand kilograms. Fully 97 percent of the total mass of this mission is just fuel to burn to get it where it needs to be. That's not even accounting for the mass of the rocket parts like empty tanks or flight electronics. Is that not insane!?

To drive the point home, to get to space, you need to go fast. To go fast you need rockets. To go faster you need an unholy amount of fuel. Getting to space is hard, and gravity wells really suck. Aim small, slim down and conserve your weight.

Cool Links

• The Atlas V rocket by United Launch Alliance - go shopping online for a heavy rocket launch!
• NASA's Asteroid Redirect Mission Concept - they actually put up plans to grab an asteroid

PS: Some pedantic details

Ok so the rocket equation graph above doesn't strictly show the relationship between mass ratio and delta v. It maps mass ratio and delta v/v_e. v_e is what is known as exhaust velocity and essentially describes how efficiently your rocket turns mass of propellant into thrust. This varies - liquid and solid propellants having relatively 'bad' v_e and newer electric propulsion doing much better (I'll get into that in a later blog post). That detail aside - the graph is still pretty crazy if you consider v_e to be a constant. Getting to the point where your delta v/v_e ratio is around 3-4 is not unheard of for longer missions. So what I said above isn't completely wrong...

As for my actual rocket calculations.... they're pretty long winded and I took some liberties with estimation of exhaust velocities and getting a 'prefect' flight'. Send me a message if you want to see the full working out

Space Blog? Why?

If you're reading this particular blog post then you probably know me personally - in which case you realise that I have an irrational need to send something (or indeed many things) into space. You may also then know that I have a slightly unusual amount of practical experience in the matter, given that Australia is one of the only developed countries to not have a space agency. 

But let's not make any assumptions. Let me introduce myself. 

I spent all of my undergraduate time at University being involved in the BLUEsat project. During which time I realised that my lifelong goal hadn't ever changed from when I was 5 years old - I wanted to build spaceships. So I put myself to work, eventually becoming the technical lead of BLUEsat, publishing research on CubeSats and spending a stint in France studying satellite attitude control. 

So why the blog? Because I have some of my own ideas about the way Space Design should be taught. To be honest, you could read Wertz' Space Mission Analysis and Design and get a much more thorough understanding of how to launch things into space - but hopefully this blog will be more fun. To start off with I'm going to post some starter guides on the basic stuff. I'll try not to pitch it too high - high school physics/science should do. Hopefully you'll like what I'm writing enough to start doing your own research, or even head to your local university and start a major.

So, here we go.