This Orbit is the WORST

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.