Lagrange Points - More Than You Ever Wanted to Know
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- Опубліковано 9 лип 2024
- This video covers the fundamental physics of Lagrange Points, and also dives deep into the stability of each Lagrange point in detail. Optional (but recommended!) refreshers are provided on some basic orbital mechanics, the centrifugal force, and the Coriolis force.
Time Stamps:
0:00 - Introduction
2:46 - Lagrange point basics
10:41 - Fictitious forces introduction
12:07 - Centrifugal force
13:30 - Coriolis force
16:45 - Lagrange point stability
21:16 - Potential surface derivation
25:55 - Lagrange point stability continued
#lagrangepoints #orbitalmechanics #3bodyproblem #gravity #coriolis #centrifugal #inertial #referenceframe
Great informative video, but at 5:07, the formula you use is T^2/2pi, where I believe I should have been (T/2pi)^2 or T^2/4pi^2. can anyone confirm this? thank you
Oops! Yes you are correct. Good catch!
Thank you, well done. I fell about 50 years younger. All the best to you:))
Thanks very much for the kind words. All the best to you too
Very informative and nicely animated. I would love to see the same but with the moonearth instead, obviously focusing on tidal effects
Nicely structured, beautifully animated, very well presented. Great work.
Thank you very much!
@@TheGravityAssistant couldn't agree more!
amazing video. best one i found on lagrange points. thanks. FYI, adding "0:00" before your other timestamps in your descriptions is how the youtube "chapter feature" gets enabled for easy chapter access from the scroll bar.
I'm glad you liked it :). Thanks for watching and the useful tip! I'll fix it now.
amazing video!!
i totally thank myself for picking aerospace!!
Thanks for watching, glad it was helpful
Very clear explanation! Next video should be about tidal forces and tidal locking 😉
Thanks for watching! I'll add it to the list 😀
Orbits have been stated most accurately as the "secret of flying", from the Hitchhiker's Guide to the Galaxy. Its basically, "throwing yourself at the ground, and missing".
Well done mate. I had naively thought the Lagrange points were points where the gravity was equal, although if I ever thought about it, that won't make sense.
Thanks for watching and the kind words!
Thank you… just thank you so much. I finally understand why Lagrange points work
Very glad to hear that :). Thanks for watching!
Thank you for your time and effort in making this video!
Both enjoyable and informative!
Thanks for watching and the kind words
I'm out of mind! Accidentally watched this video, now I'm forced to subscribe this channel. ❤❤❤ More info expected on DSN communication. Best wishes, keep educating.
That's for the kind words and thanks for watching!
Outstanding video!!!!!
Thank you! Thanks for watching
I am halfway trhough the video. This video does a great job explaining. Finally I understand the L4 and L4 points.
Thanks for watching, very glad to hear that it was helpful!
Trust is faith graduated to perfect healing. ❤
Amazing. Thank you!
Thanks for watching!
Amazing explanation. I'm sure your channel will grow bigger and bigger
Thanks for watching!
Someone has explained it to me, but I still only have a rough grasp of: why spacecraft at a L point dong just go there and stay there. It's seemed to me that this would minimize station-keeping fuel use. But, apparently, orbiting the L points instead is actually more efficient, station-keeping wise. I suppose it's to do with the, in real life, changeable nature of the L points. I wonder if you would perhaps include a visualization of the changing location a L points over long timespans? This video came very close to doing that, but it was mostly about what happens if you perturb a spacecraft at the L point, not so much about the dynamic nature of that point itself over long time frames. Obviously, aside from station-keeping concerns there are other good reasons for orbiting that point instead of going right there, like there might be several spacecraft who want to be there, and they can't all inhabit the same point in space!
Good video, enjoyed this.
Thanks for watching, and apologies for the very slow reply. This one slipped through the cracks.
Orbiting of un-stable Lagrange points is yet another thing on my list of things to cover when I have time to make more videos.
I think a useful (if not rigorously correct) way to think about it is like the difference between an orbiting and non-orbiting spacecraft around Earth. A sub-orbital vehicle that wants to stay at a fixed point in space will have to constantly fire its thrusters to stop from falling back down and to counteract any perturbing forces. Look up some videos of multiple kill vehicles for a visualisation of this. On the other hand, an orbiting spacecraft is expending no energy to remain in orbit. It just has a velocity and Earth's gravity is constantly bending its trajectory in a way that it just keeps rotating around Earth. All it has to do is make small station keeping / correction burns from time to time to balance out disturbances.
Likewise at an unstable Lagrange point. If you aim to 'hover' at exactly the point, you will frequently have to thrust against disturbances before they become run-away instabilities (like shown in this video). These kind of thrusting maneuvers directly against the direction of motion take more energy than glancing re-direction burns. Kind of like in cricket or baseball how hitting the ball directly back where it came from is much harder than lightly redirecting it.
An orbit around a Lagrange point is kind of like an orbit around Earth, but with different forces (in the rotating reference frame). Instead of gravity deflecting the velocity, it's the Coriolis and centrifugal effects, and so the orbit shape looks more like a bean than a circle. Again, like gravity at Earth, we're letting these other forces do most of the work, and just making small corrections from time to time.
Let me know if that helps!
@@TheGravityAssistant Yes, that was helpful. Thanks for taking the trouble to reply.
visualizations and animations very helpful, thank you
Thanks for watching!
That was awsome. Great explanation. Don't think it could have been better. Lots of work I bet.
Thanks for watching!
Excellent!
Thank you and thanks for watching
Very interesting thanks. Would also like to see what happens at L4 and L5 sometime in a future video. Are these depressions rather than saddles?
Thanks for watching, L4 is animated from approx 18:00 to 21:00 (when I detour into the potential surface) and continues again from 25:46. I didn't bother with L5 as well because it would be basically the same behaviour as L4. Would you like to see more detail on those points in particular?
L4 and L5 are 'peaks', so the potential surface slopes away from them in every direction. This image shows it well:
wow, it is totally NOT A CLICKBAIT - I really never wanted to know this!
I'm not sure if you're being sarcastic or serious 😅
@@TheGravityAssistant just joking, you titled video as "MORE than you ever wanted to know" though! :)
@@sergiob8501 oh sorry, my mistake, I misunderstood. Thanks for watching!
@@TheGravityAssistant just trying to be funny :)
very informative video, Sir!
Great video.
Thanks for watching!
I have an orbital mechanics exam in a few days, this video was super useful!
I'd love to see a video on planetary fly-by!
Thanks for watching! I have a couple of videos in progress related to interplanetary missions and flybys, hopefully they will be helpful. Good luck for the exam!
@@TheGravityAssistant thanks a lot!
this was awesome, loved your explanations
what did you use to make the animations?
Thanks for watching! I've used MATLAB up until now, but I'm planning to transition over to python for all future videos.
@@TheGravityAssistant Ooh, could you make a video where you show how you make the animations?
@@carljohangrevby7217 I hadn't thought of that, thanks for the suggestion. I will think about it, I would have to clean up my code as it always ends up embarrassingly messy at the end of each video 😅
Hi! Thank you so much for this wonderful explanation :). I'm a high school student who is really interested in lagrange points and i'm trying to simulate them too. Could I possibly use your simulation of orbits around l4 & l5 in a youtube video im making? it's for a contest - with credit, of course. Either way, your video has really helped me understand the concept better and learn more about it and I really appreciate that sir
Hi, thanks for watching, glad it was helpful! Yes, sure no problems.
great video, can you make a video on nrho orbits ?
Thank you! It's on my to-make wish list, but unlikely to happen any time soon unfortunately. My understanding of NRHO orbits at the moment is not deep enough for me to concisely explain them in a coherent way. The things higher up the list are topics I'm in a better position to explain!
Wish you got more attention
Baby steps, world domination later 😉
Subscriiibed! (:
Thank you! Thanks for watching
A couple questions:
1. How much gravity do LaGrange Points have?
Is there a direct relationship to the strength of that points gravity when compared to the celestial bodies that created it?
2. Are LaGrange Points taken into account when trying to predict the orbits of the planets?
3. Do Lagrange Points create gravitational lensing?
Efforts are currently underway to use the suns gravity as a telescope, thanks to the gravitational lensing the sun creates. If these LaGrange Points have the necessary gravity, could they be used the same way as, "Gravity Telescopes", so to speak?
Hi, thanks for watching! To answer your questions:
1 - Lagrange points don't 'have' any gravity of their own, as they are just points in space with no mass. The two bodies (e.g. the Sun and the Earth) have mass and so generate gravitational fields. The Lagrange points are just empty points in space where the two gravity fields of the bodies interact in a way that allows a third body with a small mass to orbit around the main body in a way that would not 'normally' be possible if there was no second body.
2 - No - similar to point 1, the Lagrange points do not attract / repel / interact with anything as they are just empty points in space that are a result of two interacting gravity fields. It is the behaviour of the gravitational fields around a Lagrange point that causes the peculiar motion of the third body. The gravity of each of the planets however, must be taken into account when predicting the orbits of the other planets, as the gravitational fields of each planet propagate infinitely and interact and change the motion of each of the planets.
3 - No, because Lagrange points don't have any mass (and therefore gravity) of their own, they will not cause lensing.
In this video you put a lot of emphasis on the rotating coordinate system. I believe you are not doing your audience any favors with that. Please allow me to explain.
Before I get to my main point, let me first acknowledge the history of the Lagrange points problem. Today we live in the age of computers that can do high precision numerical analysis, allowing us to create visualizations such as the potential surface representation. Before the age of electronic computers the choice of way to bring the Lagrange point problem to a solution was determined by computational efficiency. Using a rotating coordinate system was more efficient, so that is what physicists used.
Choice of computation method and how to physically understand a phenomenon do not necessarily coincide. Very often they do, but not always.
Example:
there is the branch of electronic engineering that deals with the equipment for generating, or receiving, or processing electric oscillations. These circuit boards have resistors, capacitors and inductors. The electric oscillations in those circuit board can be modeled with sines and cosines, but when using sine-cosine notation it is tedious to keep track of phase. As we know, sines and cosines can be represented with exponential notation, by using complex number notation. To make the calculations more efficient electronics engineers introduce imaginary current and imaginary voltage, allowing them to move the calculation to complex number space.
The electronics engineers are aware that the imaginary current and imaginary voltage have no physical counterpart; they are purely computational tools.
To *explain* to students what is happening in the circuit board the teacher uses only the physical current and the physical voltage.
In Celestial mechanics:
The intuitive understanding of celestial mechanics is in term of the quantities momentum, angular momentum, and kinetic energy.
Example: Halley's comet:
The major axis of Halley's comet's orbit is about four times greater than the minor axis. Let's start from the aphelion of Halley's comet. Moving at its slowest velocity of all its orbit the comet starts falling to the Sun. The comet has only a bit of radial velocity, but it is enough to not hit the Sun. Moving down the gradient of the Sun's gravitational field the comet is constantly being accelerated. As the comet approaches its perihelion the gravitational acceleration becomes ever larger, and that acceleration vector is shifting to perpendicular to the instantaneous velocity. At perihelion there is a local maximum in how much the orbit curves. The ascending journey back to the aphelion is the time inverse of the descend to perihelion.
I think we will all agree that the above is the way to understand the orbit of Halley's comet intuitively. Another expression would be 'visceral understanding'.
Conversely, any attempt to present the mechanics of Halley's orbit in terms of a rotating coordinate system would be absurd.
In my opinion this generalizes to all of celestial mechanics. Why does an asteroid orbiting near L4/L5 of Jupiter never escape? Well: that is due to celestial orbital mechanics, which arises from the inverse square law of gravity and inertia. The inverse square law of gravity gives rise to a Kepler orbit, and a Kepler orbit loops back onto itself.
Let's say you are a celestial object, and you are orbiting right at the L4 point of a primary and secondary. Some perturbation (from yet another planet, far away, in that same system) makes you slide away from the L4 point. Well, there is no escape; the orbital mechanics brings you right back to where you started from; a Kepler orbit loops back onto itself.
My point is: there should not be a tacit assumption that there is a 1-on-1 relationship between efficient computation and physical understanding. They often coincide, but sometimes they don't.
Some years ago I created a physics simulation for orbiting motion along the L4/L5 points. I programmed the calculation for motion with respect to the inertial coordinate system. Using inertial coordinate system or rotating coordinate system, the computer performs the computation in real time either way, so for programming the computer there is no need to use a rotating coordinate system.
For programming the numerical analysis: using the inertial coordinate system is simpler. As to the display of that simulation: that is a two-panel display. One panel for the inertial point of view, the other for the rotating coordinate system point of view. The computer performs that coordinate transformation in just a couple of processor cycles, so of course I added that coordinate transformation.
Thanks for your thoughtful comment, and apologies for my very slow reply. I wanted to take some time to consider your post.
Firstly, I agree that in the case of an elliptical orbit such as that of Halley’s comet, it would not make sense to use a rotating reference frame, as the angular velocity of the orbiting body is constantly changing. We could have a variable angular velocity rotating frame, but then that introduces yet another abstract force…
However, Lagrange Point analysis assumes a circular orbit of the secondary body (Circular Restricted Three Body Problem). Clearly this assumption will only be valid for planets / moons with a nearly circular orbit about the primary body, and in fact the Lagrange Points become meaningless if the eccentricity of the orbit strays too far from 0. However, this assumption is good enough for many planets in our Solar system.
Personally, I don’t find your explanation of L4 stability satisfying. If the stability is simply a result of the inverse-square nature of gravity and a closed Keplerian orbit, why should L4 and L5 be stable while L1, L2, and L3 are not? Why should L4 and L5 become unstable if the secondary body becomes too large?
I agree that rotation frame alone is not a total solution, but IMO it can provide an intuitive explanation for something like 80% of the problem. I don’t mean to be rude, but did you watch the entire video? (I know it’s long 😊) At 20:23, 27:45, 32:24, 33:50 I have the side-by-side rotational frame and inertial frame animations that you describe from your previous work.
It’s my opinion that humans are not very good an intuiting circular motion in an inertial reference frame, as it’s not something we have day-to-day experience with (additionally, visually keeping track of a quickly rotating object and also analysing its motion at the same time gets tiring quickly). Generally, humans are IN the reference frame that is undergoing rotation. And so, even though the centrifugal force and Coriolis force are not ‘real’, we understand them and their effects in a rotating reference frame more directly. Some daily examples of the centrifugal force being a car or bus going around a corner and ‘throwing’ us outwards, or a roller coaster ‘pushing’ us downwards at the bottom of a loop. In the case of the Coriolis force, kids throwing a ball on a merry-go-round learn that it curves in funny ways, and then in high school we learn about how it causes cyclones and hurricanes due to the Earth spinning. Even things like retrograde motion of the planets are due to us observing them from a rotating reference frame, and the debate around that topic took quite some time to resolve 😉!
In any case, the Coriolis and centrifugal forces are basically IF this, THEN that rules which can be visualised quickly. Also, the rotating reference frame allows convenient calculation of the potential surface, which provides a nice graphical illustration of the points and their stability. Feel free to point me towards a reference, but I have never come across such a neat illustration of Lagrange point stability from an inertial point of view. In my mind, the inertial reference frame just provides a useful ‘anchor back to reality’ which is the last 20% of the puzzle in this analogy.
@@TheGravityAssistant I'd like to discuss the mechanics of the motion of a celestial body in a slightly eccentric orbit .
To ease the viewer into that I will start with discussing something that happens here on Earth: the physics of using the banking in track cycling.
Track cycling team pursuit
ua-cam.com/video/lP7ioj9isw0/v-deo.html
In a team pursuit event all four riders take turns riding in the lead. For each rider: when the turn is done you have to move from the front to the rear as quickly as possible, without losing energy.
As can be seen in videos of team pursuit events: this is accomplished by taking advantage of the physics of riding on a banked track. As the four riders enter the turn the rider in front allows himself to climb up the banking. This has two effects: he has to cover a longer distance, and climbing up the banking reduces his velocity.
This is intuitive; we all know that when you coast uphill gravity will decelerate you. Once the rider is halfway in allowing himself to be overtaken by the others he starts steering down again. Going downhill gravity is *accelerating* him. At the moment the rider reaches the the other three riders: gravity has assisted in accelerating him back to the (constant) velocity of the other three.
In the side-by-side animation starting at 20:22
Slow down the playback speed to 0.25, and in the righthand panel, (inertial point of view) watch the motion of the spacecraft with respect to L4.
The spacecraft is periodically pulling ahead of L4 and being overtaken by L4. The spacecraft is oscillating between moving along the inside track (inside the orbit of L4) to moving along the outside track (outside the orbit of L4)
When the orbital altitude of the spacecraft increases then gravity is slowing the spacecraft down.
So: when moving along the outside track (along highest orbital altitude) the angular velocity of the spacecraft with respect to the Sun is slower than the angular velocity of L4.
Conversely, when moving along the inside track (along lowest orbital altitude) the angular velocity of the spacecraft with respect to the Sun is quicker than the angular velocity of L4.
In the co-rotating view this oscillation in angular velocity presents itself in the form of small loops in the vicinity of L4.
More generally:
In physics what we want is to think about phenomena in a way that makes it independent from the particular perspective we happen to be using. We want to look at the quantities that are the same as seen from all perspectives. We should avoid relying on quantities that are perspective-dependent.
For orbital motion the following two quantities are what matters:
1. The distance to the center orbital motion (radial distance)
2. The rate of change of angular velocity
The radial distance is perpective-independent; it is the same in both the inertial and rotating point of view.
The rate of change of the angular velocity is perspective-independent; it is the same in both the inertial and rotating point of view.
Trust is faith graduated to perfect healing. ❤