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I just watched Neil Tyson say the brachistochrone path discovered by the Bernoulli bros is the best ascent profile for an ascending rocket. Link

I hadn't heard this before. Is this really a thing? Or is this another factoid Neil's made up?

HopDavid
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    Not sure how to put together a full answer to this, but in my limited experience in working on rocket ascent optimizers (SORT, Simulation for Optimization of Rocket Trajectories I think?) and a cursory search of NTRS, I think Neil is not correct. There's some interesting general points made in https://space.stackexchange.com/questions/23088/brachistochrone-variation-for-earth-to-mars-orbit that point that way too. https://space.stackexchange.com/a/43735 indicates that it should refer to constant-acceleration trajectories, which ascents certainly aren't. – Erin Anne Nov 22 '23 at 01:18
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    Yes, for certain criteria of "best" – JCRM Nov 22 '23 at 12:15
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    In general, I don't think you can call a specific profile the "best" because a launch takes place in the real world, and, particularly within the atmosphere, including real-world conditions in your calculation would get "messy" very quickly. For example, wind-speeds, air pressures, air temperatures, humidity levels, etc all ensure that the profile that is the "best" now, will almost certainly not a couple minutes later from a physics/mathematics perspective. – Dragongeek Nov 22 '23 at 17:32
  • It's worth considering what we're optimizing for when we look at "the best" launch profile. Fuel efficiency? Raw speed? Mechanical complexity? Launch window? A profile that's theoretically faster or more efficient may be less appealing in the real world than one that's simple, or flexible, or less demanding on the rocket hardware. – Cadence Nov 23 '23 at 04:22

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While I greatly appreciate Tyson's dedication to educating the public and outreach to young people to use their heads and think scientifically, I believe that his "mike-drop" moment:

...so that's how to get into orbit most efficiently.

is most likely just plain wrong, or at least misleading scientifically.

Granted the curves have somewhat similar shapes (when one is flipped and stretched to match the other), but I am pretty sure the similarity is coincidental. Certainly the velocity and acceleration curves look very different.

(Of course, I would be extremely happy to see someone do the math and show that he's right in some slightly toned-down way and I'm wrong!)

If the Earth's gravity were negative - i.e. it repelled the rocket and constantly pushed on it in upwards vertically, then if you built a brachistochrone-shaped rail that terminates at a given height and downrange distance, then when you let go of the rocket and let it get pushed up the rail, it would reach the specified endpoint in the least time.

I don't know what Tyson's connection is between least time and most efficient.

There are no spaceflight-relevant tidbits to learn here. Neil had too much coffee that day I think.

As an aside, so far as we know, neither antiprotons nor anything else falls up.

uhoh
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    His lack of rigor and accuracy angers me. Some of his stuff is outright fiction.

    Here is Neil telling Chuck the James Webb Space Telescope is parked at the L2 point in earth's shadow so as to keep the sun's rays off the telescope.

    Here is Neil saying rocket propellant goes exponentially with payload mass.

    I suspect this brachistochrone ascent profile is another example.

     – HopDavid Nov 22 '23 at 12:37
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    @HopDavid yikes! maybe there should be a repository for Tyson's mistakes somewhere. He took such great pride in telling James Cameron he got the stars in Titanic wrong. – uhoh Nov 22 '23 at 15:33
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    Neil certainly demonstrates at times that he doesn't know everything. But I have learned a lot of things listening to him. I consider him sort of like ChatGPT, it often times gives you interesting things to think about and research on your own, just don't rely on it (or Neil) as being authoritative. – Steve Pemberton Nov 22 '23 at 16:24
  • @StevePemberton making me laugh out loud before I've had my morning coffee is quite a feat; yes that sums it up quite nicely :-) – uhoh Nov 22 '23 at 22:51
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    @uhoh I've made such a repository: Link – HopDavid Nov 24 '23 at 13:07
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    He is wrong about the shape as well. The force varying with height isn't negligible in this case. Check the DEQ you get. – C. Towne Springer Nov 26 '23 at 06:42
  • @HopDavid Niel's biggest issue is his insistence to try to insert himself in the same place as his idol, Carl Sagan, had in society. The issue is that Niel is half as smart as Carl and either doesn't know it/wont admit it and tries to put in his comment on too many things. Not to say Niel isn't smart, he is excellent on topics he's well versed in. He plays fast and loose with details or nuance, but typically his main points are correct. – David S Nov 27 '23 at 23:30
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This makes some sense on a logical basis. The Brachistochrone problem is a bead sliding on a frictionless wire that is lower at the end than at the start. Question, what wire shape minimizes the time to slide from start to stop? This is interesting for rockets because time = fuel. Note the speed of the bead at the end is the same regardless of the path taken since the path is frictionless and the speed depends only on the kinetic energy gained in going down hill. The solution is 1/4 of a cycloid.

How does this differ from a rocket launch? First, there are forces other than gravity. The acceleration is governed by the rocket equation since the mass is decreasing as fuel is burned and acceleration is increasing. Second, the gravitation field is weakening with altitude. Third, launching in an atmosphere means air friction. Forth, the the planets and moons are spherical, not infinite flat planes with uniform gravitation..

Intuition tells me these are not going combine to produce a cycloid of revolution. But maybe Tyson is talking about a reformulation of the problem to include these factors? Maybe today's rocketeers call the result a small b "brachistochrone"? And yes, they do. They are using the name in it's Greek meaning of least time. So any space flight that uses it's engines and direction to achieve a journey in the least time is using a "brachistochrone course". Scott Manley looks at it in that sense here.

(FYI, this problem had the math brains of Europe sweating for quite a while. When Newton heard of it he solved it in an afternoon and using something he had been fiddling around with that today we call the calculus of variations.)

C. Towne Springer
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  • "small b 'brachistochrone'" I love it! Looking forward to seeing what Manley has to say; thanks for your answer! (update: Oh, "torch ship" trajectories!) See also aswers to Brachistochrone variation for Earth-to-Mars Orbit – uhoh Nov 25 '23 at 03:53
  • so you think Neil wasn't wrong about modern rocket ascent, he was right about certain sci-fi torchship trajectories? – Erin Anne Nov 25 '23 at 07:18
  • @ErinAnne I think he WAS wrong about surface to orbit within the original meaning of the problem. He was right as far as using the jargon of trajectories in space. I don't know if it is common use or comes from The Expanse. The problem with using the name of the famous problem comes from how badly the jargon version can not be made to fit the original. Ex: there must be a least time for any given fuel amount to reach "Earth II" on the other side of the Sun. But the potential energy is the same at each end of the trip. There is no down hill. – C. Towne Springer Nov 26 '23 at 06:36
  • In my OP I link to Neil's video. He makes it clear he's talking about a specific curve that the Bernoulli brothers looked at -- the usual curve we think about with the brachistochrone. – HopDavid Dec 04 '23 at 16:13