Fly Me To The Moon

Welcome! In this essay, you will learn how to fly to the Moon. But this isn't just any old slab of text, it's a living, breathing, interactive, and playable essay, which means you won't just read about it; you'll actually fly your very own rocketship in the simulation below. The words of the essay will be up here, but rather than having to scroll, new words will appear by themselves as you play (or sometimes just by tapping 'next', like now).

Our simulation is, of course, greatly simplified, in many respects–for example:

• It is purely 2-dimensional.
• It is wildly not-to-scale (the moon is actually much further away and much smaller).
• The rocket itself has infinite fuel.
• The Earth and Moon are both completely stationary.

Nevertheless, I promise that it is physically realistic in the key respects that make it so the essential principles of piloting this simulated rocketship to the moon are identical to piloting a real one. These simplifications are so that those essential principles will be all that you need to successfully land on the moon.

By the way, you can still scroll backwards and forwards through parts of the essay that have already appeared, —and the state of the simulation associated with them will be restored.

You'll want to takeoff straight up away from the surface of the Earth.

Now, try your best at landing on the moon. I should note, that whilst this is possible, you are of course NOT expected to be able to do so successfully yet—that's what the rest of the essay is for! Still, have a go so you can get a feel for the simulation and why its tricky.

Falling in Style

I'll admit, flying to the moon is maybe a slightly ambitious goal to begin with. Lets start with a more immediate problem: not crashing after launch.

Getting up into the sky is easy, but to paraphrase Sir Isaac Newton, 'what goes up must come down'. You might think we just need to get far enough away from the earth that we no longer feel its gravity. However, although that is possible,* its not going to work for getting to the Moon, because the Earth's gravity is still strong there—that's why the Moon orbits the Earth, after all.

To make this clearer, let's make gravity itself visible. In this visualisation, Earth gravity is blue and Moon gravity is yellow. Where they overlap, we see a mixed green colour. The brighter the color, the stronger the pull—the weaker, the darker. The black area shows where the Earth's and Moon's gravity, pulling in opposite directions with equal force, exactly cancel each other out!

For now though, we'll get rid of the moon to keep things nice and simple. Don't worry, we'll bring it back later

Let's perform an experiment. If we drop a cat off of this tower, and launch another cat sideways from the tower with a slingshot, how long will it take each of them to land? Don't worry, cats always land on their feet. Have a guess, and pull back on the slingshot to launch.

They land at the same time. This is because they fall at the same rate, and nothing has been done to slow the fall of either. Of course, we've assumed spherical cats in a vacuum, but adding air resistance would slow the fall of both by the same amount.

However, what happens when the Earth's surface starts to curve? Try launching again with this upgraded slingshot.

It stayed airborne for longer!

Let's try an even faster launch.

We can see that gravity is still curving the trajectory, but the rate of altitude loss is decreasing as we move faster, because we are getting to take advantage of more and more curvature per second whilst being subject to the same amount of gravity per second. The ground is curving away beneath our flying furball at closer and closer rate to that which the cat is falling.

Let's keep upgrading the slingshot and see if we can get the curvature of our freefall trajectory to match the curvature of the Earth.

Further still! The ground is really starting to fall away beneath the cat now.

Another upgrade. How far can it get?

The cat made it all the way around to the far side of the planet before touching down!

One more upgrade, then: let's match the curvature of our freefall to the curvature of the Earth exactly.

Ta-da! We've just achieved orbit. Specifically, a circular orbit.

But what happens if we add MORE POWER? Don't worry, it'll be fine. It's not like a little curiosity ever killed a cat.

Another orbit! This time, though, we actually gained altitude—so much altitude we went to space, in fact—resulting in this egg shaped trajectory. Note that the principle of 'What goes up must come down' is still true! After it orbits halfway around the Earth to its high point, our feline daredevil returns to its initial altitude at the tower before repeating its elliptical circuit. That highest altitude point in an orbit furthest from the Earth is called apoapsis, and the lowest altitude point of closest approach is called periapsis. When talking about Earth orbits, we use the special terms apogee and perigee for these points

If we were to launch the cat faster still, we would raise the apogee of its orbit further.

Of course, the only thing separating our beautiful never-ending orbits from our previous short-lived trajectories is that they eventually hit the ground. What would happen if our cats could pass through the ground like ghosts?

Let's start by following the dropped cat.

We see that the dropped cat doesn't stop upon reaching the centre of the earth, but continues until it reaches its initial altitude but on the other side of the Earth—what goes down must come up, it would seem. This forms a a 1-dimensional but still endlessly repeating 'orbit'.

What about our previous launches that failed to complete an orbit?

We see that, like the dropped cat, they have two apogees both at the initial altitude, but thanks to their initial velocity their orbits are two dimensional (albeit eccentric, which is to say skinny about one dimension rather than circular).

We therefore see that all objects "want" to orbit and would do so were it not for the ground getting in the way. The parabolic trajectories we associate with thrown objects are in fact orbits cut short! We do not so much as need to "achieve orbit" as achieve an orbit which doesn't intersect the surface of the planet.

As they gain more launch velocity, though, the less eccentric their orbits become and their 'perigees' inside the Earth rise closer and closer to the surface, culminating in our perfectly circular orbit with constant altitude where apogee and perigee lose all meaning. As we've seen, adding even more initial velocity makes the orbit eccentric again but on a perpendicular axis.

Step 1 of getting to the Moon is reaching orbit around Earth. In the real-world, a stable orbit cannot be achieved at such a low altitude, since within the atmosphere air resistance slows us down causing us to lose the little altitude we have and eventually crash. In our simplified simulation, however, there is no atmosphere so we don't need to worry about that, but since we don't have a tower or a slingshot for our rocketship, we will still want to boost upwards until we have reached an altitude where we can safely point the rocketship 'forward' in the direction we'd like to orbit—this direction is called 'prograde'.

Then, we can perform a prograde burn just like how we launched the cats. We need to keep firing prograde until our collision trajectory becomes a proper orbit. To help you out, we're going to add a projection of your trajectory so you can see how it changes live.

Congratulations, your rocketship has achieved orbit! Remember, you are still in freefall—the only reason you aren't crashing is because of your rocketship is moving so fast towards the horizon— there is therefore no way to stay still in space, you must always be in an orbit (well there is one way, but we'll get to that). From now on, then, we're not so much moving the rocketship directly as we are changing our orbit.

Impulse Control

Like most rocketships, ours does not have the ability to adjust its thrust—it's either on or off, there's no equivalent to lightly pressing the accelerator pedal. Instead, you've been controlling impulse—change in velocity—by timing the duration of your burns. This is indeed how most rocketships work.

However, since our simulation is effectively sped up by hundreds of times, this becomes very sensitive—especially when we're further away from the Earth and only need to perform low-impulse maneouvres. So we're going to upgrade your controls to adjust the exact timing of burns in advance—and to pause time so the rocketship remains in the same place for the duration of the burn, approximating real burns. In effect, you'll recieve the full impulse of the burn in one instanteous kick—just like the slingshot from before.

Spoiler alert, but we'll need to move to a higher altitude orbit if we're gonna to reach the moon. What direction should we fire the rocket to achieve this?

It is intuitive to fire 'up', since we want to gain altitude. But remember, we didn't achieve orbit in the first place by firing up, because what goes up must come down. Indeed, you'll see that you gain altitude initially from this manouvre, but you fall right back down.

Well done! Your logic is correct; if firing prograde is what got us into orbit, then surely we'd fire prograde to increase our altitude. The issue is, as you can see, whilst this will increase the peak altitude or apoapsis, one half orbit ahead of where we fired, it won't lift the whole orbit. The minimum altitude, or peripasis, will remain the same. Don't fret though; it turns out you can't achieve the desired higher altitude circular orbit with just one firing of the rocket (one 'burn'), but this burn is the necessary first step.

So you don't have to wait to reach a certain position in the orbit, we'll give you direct control over time.

The timeline visual is composed of vertical slices taken out of each frame—past, present and future—showing the altitude of your rocketship at that frame.

You can drag on the timeline to scrub through time, letting go to resume.

Alternatively, you can click on a point on the timeline, and it will automatically fast forward or rewind to that point.

Finally, you can 'zoom' in or out on the timeline by scrolling in or out with your cursor over the timeline.

Okay, now you can control time, what do you think is the next step to 'circularise' this orbit?

Ding ding ding! Your logic is correct again. If firing raises the altitude a half orbit ahead of where we are, then if we want to raise the periapsis to circularise the orbit, we must do another prograde burn at the apoapsis. You now have the manouvre to raise your orbital altitude as much as you like. What if we want to lower the altitude? You should be able to figure this one out; what's the opposite of what we did before?

Bingo! That's called a 'retro-grade' burn. Periapsis lowered, now just need to bring in the apoapsis.

Perfect. Now, you have the power to orbit at any altitude you like! Lets test your skills. See if you can get your orbit within the green band.

Nicely flown — your orbit fit right inside the target band.

Let's go further out.

Excellent.

Now bring it all the way back down to a low, tight orbit.

Perfect. High orbits, low orbits — you can now put your rocketship wherever you like.

Ok, clearly you've got the hang of orbital manouvreing. Its time to bring back the Moon. Before we can land on the moon, we first need to get into orbit around it. This can be achieved with what you already know, we just need to put our apoapsis in the close vicinity of the moon—in both time and space—so that when we reach it we are 'captured' by its gravity. Have a go.

Incredible! You're in lunar orbit. Now all we need to do is land. This, admittedly, is a bit more complex than just lowering your orbit. You essentially have to launch in reverse. That means firing retrograde until you have a collision trajectory, then firing 'up' in short bursts in order to slow your fall so that your landing velocity is almost zero.

Absolutely brilliant, you've done it! Just for fun, I'll let you hop out and do a moon walk. When you're ready, hop back in the rocketship and see if you can use the same principles to get back home.