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In my previous video on space navigation, we talked about some apparent paradoxes, like how to catch up with someone in the same orbit, you first have to slow down! Or how you have to speed up (twice) to switch to a higher, slower orbit. Here are three more even weirder paradoxes of space navigation, including the most surprising one I've ever come across - which I only found out about recently, and which is truly bonkers. First: There's a worst orbit to get to. It seems like the further out your destination orbit is, the more fuel would be required to get there. But
in fact, after a certain point, going out begins to require less fuel. The worst orbit to aim for is about 15 times farther out than your current orbit, which for us is between Saturn and Uranus. This fact is profoundly bizarre. Ultimately, it has to do with the interplay between how much you slow down on the way out to the new orbit verses how much speed you need to stay there. The simplest method to get to a different circular orbit - which we talked about in the last video - requires two changes of speed: the first burn puts you onto an elliptical transfer orbit,
and the more you increase your speed with that burn, the higher the high point of the ellipse. This makes intuitive sense: the more fuel you use, the faster you'll go and the further out you'll end up. Except you're not done - gravity constantly pulls to slow you down as you go out along the transfer orbit, so when you arrive at your target radius, you need to speed up in order to get into a circular orbit there (otherwise you'll keep falling back to where you started). And this second, re-circularising burn is what makes things downright weird. The amount you need to speed up to circularize
your orbit depends, of course, on the difference between your speed upon arriving at the top of the elliptical orbit and the speed you need to be in a circular orbit there. It turns out the arrival speed at the top of the ellipse falls roughly like one over r, while the target speed needed for a circular orbit falls roughly as one over the square root of r, which is bigger - comparing the two, you can see that the difference between the target speed and the arrival speed initially increases for short range transfers, then shrinks once your target radius is more than around
six times farther out than your starting point. You might think that the worst orbit is therefore around six times farther out, but this is just the worst point for the second, circularizing, burn - once we remember to add in the first burn (which is the speedup orbit necessary to get onto the transfer orbit in the first place) we find it's hardest to get into an orbit around 15 and a half times larger than your starting orbit. Beyond 15 times, it's easier to get there! A bizarre consequence of this 'worst' orbit is that it takes less fuel to escape
the solar system entirely than to go into orbit between Saturn and Uranus. Actually, it's easier to escape the solar system than to go into a circular orbit anywhere beyond the asteroid belt; between Saturn and Uranus is just the hardest possible place to get to. And the difference is pretty substantial - it takes almost 30% more fuel to transfer to the "worst" circular orbit than it does to go to infinity! This fact applies generally, whether you're orbiting the sun and trying to go out to Saturn, or orbiting earth and trying to go
to the moon. Like, it takes almost the same amount of fuel to get into a geostationary orbit 6 and a half times out from low earth orbit as it does to get to the moon, which is sixty times farther out. The general inefficiency of medium-range orbital transfers leads to - what's to me - the most surprising paradox of space navigation, and one I didn't know about until recently: it's that you can actually save fuel by going out too far, and then coming back. Basically, you do the orbital transfer with an extra step: rather than going directly out to the destination orbit and circularizing, first, you completely overshoot your destination,
then come back and circularize. It's called a bi-elliptic transfer, and it saves fuel because it does its intermediate burn out where gravity is really weak, AND because circularizing an orbit is much easier when you're arriving from above, rather than arriving from below. For bi-elliptic magic, you first boost yourself onto an elliptical orbit that overshoots 100 or 1000 times further out than you need to go - it doesn't cost much extra fuel vs going directly to your final destination because a gravitational well requires less and less
additional speed to go further and further out. Then when you're at the furthest away point, you're going so slowly and gravity is so weak it takes almost no effort to change orbits, so you can speed up just a miniscule amount to get onto a new transfer ellipse back down to your destination orbit. Then, since you're coming from above you'll be going too fast and need to slow down to circularize your orbit - but it turns out it's much easier to circularize an orbit arriving from above than below. We already mentioned that the target speed for a circular orbit is proportional to one over the square root of r,
while coming from below your arrival speed is proportional to one over r, which is much smaller, you might only have 1% or 5% of the target speed, so you need to speed up a lot to circularize from below. Coming from above, though, your arrival speed is proportional to one over the square root of r, just like your target speed - and in fact, it's just roughly 1.4 times your target speed, meaning you need to slow down only ~30% to get onto a circular orbit from above. The takeaway is that when you come from above and then circularize your orbit, you don't have to work nearly as hard as if you come from below and circularize.
So, the genius of the bi-elliptic transfer is this: you do a little bit more work to go out farther than you need, and from where it's very easy to come back, in order to save effort on circularizing the final orbit. All-in-all, overshooting is more efficient when your destination is more than around 12 times farther out, but it's not particularly big savings. If your destination is 20 times out and you overshoot to 40 times out before coming back, then you save 1.7% compared with a direct transfer. If your destination is 100 times
further out and you overshoot all the way to 1 million times out before coming back, then you save 7.6% over a direct transfer. Not very much… and there's a big cost: time. Overshooting so far takes a long time since you slow down more and more the further out you go (so you'd be traveling farther AND doing it more slowly); going 10, 100, or 1000 times further out than your target takes around 600, 20,000, or 700,000 times longer than a direct transfer. So: what's more valuable, your fuel or your time? Well, if you have a limited amount of fuel,
but all the time in the world, then you may want to hear about this last paradox: when doing a bi-elliptic transfer, the more you overshoot, the more fuel you save. Here's the total fuel needed for a bi-elliptic transfer vs how far out you overshoot, and you can see clearly that the more you overshoot, the less fuel you need. This fact seems ridiculous, because the further out you go, the more fuel is needed during the initial burn to get out all that way, and then, because you're falling back down from further away, you'll arrive at your destination going faster and also require
more fuel to slow down and recircularise. The reason overshooting farther actually does save you fuel is that you get a bigger saving from the middle transfer burn being really, really far out, than the extra fuel required to get there and return. Specifically, compared to the fuel savings for the middle burn, the extra fuel cost for the final burn is roughly half as much, and the extra fuel cost for the first burn is roughly half times one over the square root of r as much. Since a half plus a half divided by the square root of r is less than one, that means you save more fuel the more you overshoot!
The natural conclusion is that, to be as fuel-efficient as possible, your best course of action is to overshoot all the way to infinity! An infinite bi-elliptic transfer is the most efficient simple way to transfer to any destination more than twelve times further away then you're currently orbiting, saving up to 8% of your fuel. The only problem, other than the savings are not that great, is that it takes an infinite amount of time… Here's a paradox about AI: people who are more concerned about the risks of AI are less likely to work at AI companies,
so then AI products are less likely to take AI safety into account, making the risks even worse! Luckily, some people and organizations are working to push AI in the right direction, like BlueDot Impact, the sponsor of this video. BlueDot Impact is a nonprofit helping people become informed about AI and involved in shaping its future. They're specifically looking for people who feel like they're missing something about AI and want to meaningfully contribute. If that's you, BlueDot Impact has created a number of completely free courses on AI and AI safety.
The decisions being made now about how this technology should be developed are still being made by a relatively small number of people, and these decisions will set the direction for a long time to come. So it is more important than ever to get a wide range of voices educated and involved in AI development from an AI safety and security perspective to make sure it goes well for all of humanity, not just the super-rich. To help you understand AI and find your own place in shaping it, check out the free courses available over at bluedot.org/minutephysics, no technical background required.
