This Orbit is the WORST — Transcript

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1. 0:01In my previous video on space navigation, we  talked about some apparent paradoxes, like how
2. 0:05to catch up with someone in the same orbit, you  first have to slow down! Or how you have to speed
3. 0:09up (twice) to switch to a higher, slower orbit. Here are three more even weirder paradoxes of
4. 0:14space navigation, including the most surprising  one I’ve ever come across - which I only found
5. 0:18out about recently, and which is truly bonkers. First: There’s a worst orbit to get to. It seems
6. 0:24like the further out your destination orbit is,  the more fuel would be required to get there. But
7. 0:27in fact, after a certain point, going out begins  to require less fuel. The worst orbit to aim for
8. 0:32is about 15 times farther out than your current  orbit, which for us is between Saturn and Uranus.
9. 0:37This fact is profoundly bizarre. Ultimately, it  has to do with the interplay between how much
10. 0:41you slow down on the way out to the new orbit  verses how much speed you need to stay there.
11. 0:44The simplest method to get to a different  circular orbit – which we talked about in the last
12. 0:48video – requires two changes of speed: the first  burn puts you onto an elliptical transfer orbit,
13. 0:53and the more you increase your speed with  that burn, the higher the high point of the
14. 0:56ellipse. This makes intuitive sense: the  more fuel you use, the faster you’ll go
15. 0:59and the further out you’ll end up. Except you’re not done - gravity
16. 1:03constantly pulls to slow you down as  you go out along the transfer orbit,
17. 1:06so when you arrive at your target radius, you need  to speed up in order to get into a circular orbit
18. 1:10there (otherwise you’ll keep falling back to where  you started). And this second, re-circularising
19. 1:14burn is what makes things downright weird. The amount you need to speed up to circularize
20. 1:19your orbit depends, of course, on the difference  between your speed upon arriving at the top of the
21. 1:23elliptical orbit and the speed you need to be in  a circular orbit there. It turns out the arrival
22. 1:27speed at the top of the ellipse falls roughly  like one over r, while the target speed needed
23. 1:31for a circular orbit falls roughly as one over  the square root of r, which is bigger - comparing
24. 1:35the two, you can see that the difference between  the target speed and the arrival speed initially
25. 1:39increases for short range transfers, then shrinks  once your target radius is more than around
26. 1:43six times farther out than your starting point. You might think that the worst orbit is therefore
27. 1:47around six times farther out, but this is just  the worst point for the second, circularizing,
28. 1:51burn – once we remember to add in the first  burn (which is the speedup orbit necessary to
29. 1:55get onto the transfer orbit in the first place)  we find it’s hardest to get into an orbit around
30. 1:5915 and a half times larger than your starting  orbit. Beyond 15 times, it’s easier to get there!
31. 2:04A bizarre consequence of this ‘worst’  orbit is that it takes less fuel to escape
32. 2:07the solar system entirely than to go into  orbit between Saturn and Uranus. Actually,
33. 2:12it’s easier to escape the solar system than  to go into a circular orbit anywhere beyond
34. 2:15the asteroid belt; between Saturn and Uranus  is just the hardest possible place to get to.
35. 2:20And the difference is pretty substantial - it  takes almost 30% more fuel to transfer to the
36. 2:24“worst” circular orbit than it does to go  to infinity! This fact applies generally,
37. 2:28whether you’re orbiting the sun and trying to go  out to Saturn, or orbiting earth and trying to go
38. 2:31to the moon. Like, it takes almost the same amount  of fuel to get into a geostationary orbit 6 and a
39. 2:36half times out from low earth orbit as it does to  get to the moon, which is sixty times farther out.
40. 2:40The general inefficiency of medium-range  orbital transfers leads to - what’s to me – the
41. 2:44most surprising paradox of space navigation,  and one I didn’t know about until recently:
42. 2:48it’s that you can actually save fuel by  going out too far, and then coming back.
43. 2:52Basically, you do the orbital transfer with  an extra step: rather than going directly
44. 2:56out to the destination orbit and circularizing,  first, you completely overshoot your destination,
45. 3:01then come back and circularize. It’s called  a bi-elliptic transfer, and it saves fuel
46. 3:06because it does its intermediate burn out where  gravity is really weak, AND because circularizing
47. 3:10an orbit is much easier when you’re arriving  from above, rather than arriving from below.
48. 3:14For bi-elliptic magic, you first boost yourself  onto an elliptical orbit that overshoots 100 or
49. 3:201000 times further out than you need to go  - it doesn’t cost much extra fuel vs going
50. 3:24directly to your final destination because  a gravitational well requires less and less
51. 3:27additional speed to go further and further out. Then when you’re at the furthest away point,
52. 3:31you’re going so slowly and gravity is so weak  it takes almost no effort to change orbits,
53. 3:35so you can speed up just a miniscule  amount to get onto a new transfer
54. 3:38ellipse back down to your destination orbit. Then, since you’re coming from above you’ll
55. 3:42be going too fast and need to slow down to  circularize your orbit - but it turns out it’s
56. 3:46much easier to circularize an orbit arriving  from above than below. We already mentioned
57. 3:50that the target speed for a circular orbit is  proportional to one over the square root of r,
58. 3:53while coming from below your arrival speed is  proportional to one over r, which is much smaller,
59. 3:58you might only have 1% or 5% of the target speed,  so you need to speed up a lot to circularize from
60. 4:02below. Coming from above, though, your arrival  speed is proportional to one over the square root
61. 4:06of r, just like your target speed - and in fact,  it’s just roughly 1.4 times your target speed,
62. 4:11meaning you need to slow down only ~30%  to get onto a circular orbit from above.
63. 4:14The takeaway is that when you come from  above and then circularize your orbit,
64. 4:17you don’t have to work nearly as hard as  if you come from below and circularize.
65. 4:21So, the genius of the bi-elliptic transfer  is this: you do a little bit more work to go
66. 4:25out farther than you need, and from where  it’s very easy to come back, in order to
67. 4:29save effort on circularizing the final orbit. All-in-all, overshooting is more efficient when
68. 4:33your destination is more than around 12 times  farther out, but it’s not particularly big
69. 4:37savings. If your destination is 20 times out and  you overshoot to 40 times out before coming back,
70. 4:41then you save 1.7% compared with a direct  transfer. If your destination is 100 times
71. 4:46further out and you overshoot all the way  to 1 million times out before coming back,
72. 4:50then you save 7.6% over a direct transfer.  Not very much… and there’s a big cost: time.
73. 4:57Overshooting so far takes a long time since  you slow down more and more the further out
74. 5:01you go (so you’d be traveling farther AND doing it  more slowly); going 10, 100, or 1000 times further
75. 5:06out than your target takes around 600, 20,000, or  700,000 times longer than a direct transfer. So:
76. 5:12what’s more valuable, your fuel or your time? Well, if you have a limited amount of fuel,
77. 5:16but all the time in the world, then you  may want to hear about this last paradox:
78. 5:20when doing a bi-elliptic transfer, the more  you overshoot, the more fuel you save.
79. 5:24Here’s the total fuel needed for a bi-elliptic  transfer vs how far out you overshoot,
80. 5:28and you can see clearly that the more  you overshoot, the less fuel you need.
81. 5:31This fact seems ridiculous, because the further  out you go, the more fuel is needed during the
82. 5:35initial burn to get out all that way, and  then, because you’re falling back down
83. 5:38from further away, you’ll arrive at your  destination going faster and also require
84. 5:42more fuel to slow down and recircularise.  The reason overshooting farther actually does
85. 5:46save you fuel is that you get a bigger saving  from the middle transfer burn being really,
86. 5:50really far out, than the extra fuel  required to get there and return.
87. 5:54Specifically, compared to the fuel savings  for the middle burn, the extra fuel cost
88. 5:57for the final burn is roughly half as much,  and the extra fuel cost for the first burn
89. 6:00is roughly half times one over the square root  of r as much. Since a half plus a half divided
90. 6:05by the square root of r is less than one, that  means you save more fuel the more you overshoot!
91. 6:10The natural conclusion is that, to  be as fuel-efficient as possible,
92. 6:12your best course of action is to  overshoot all the way to infinity!
93. 6:15An infinite bi-elliptic transfer is the  most efficient simple way to transfer
94. 6:19to any destination more than twelve times  further away then you’re currently orbiting,
95. 6:22saving up to 8% of your fuel. The only problem,  other than the savings are not that great,
96. 6:27is that it takes an infinite amount of time… Here’s a paradox about AI: people who are more
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