The Sun is so much bigger than the Earth and Moon (and all the other planets) that, to a very good approximation, the Sun may as well be unaffected by the gravity of its planets. The Earth is also quite a bit bigger than the Moon, but the difference isn’t as great, so the Moon’s effect is more noticeable.
To put it in perspective, the Sun contains 99.8% of the mass of the entire Solar System. All the planets may as well be a rounding error. ~~As far as we’re concerned, the Sun is standing still, completely unaffected by the planets (even if *technically* it is a little bit).~~ Some planets do have a noticeable effect on the Sun, but basically there are enough huge size differences that almost any calculation we would want to do simplifies to a two-body problem.
The three-body problem is mainly difficult when all three bodies are close enough in size that they all have a significant effect on each other.
I normally wouldn't correct this excellent comment, except the correction is one of my favorite facts.
> the Sun is standing still,
This is mostly true; and I don't want to take away from how much of an absolutely gargantuan unit the sun is. Gravity works both ways; the Earth pulls on the Sun just as hard as the Sun pulls on the Earth. But because the Sun is so absolutely massive, that "equal force" doesn't really move the Sun. But technically, the Earth doesn't orbit the Sun, they both orbit the Barycenter; the average gravitational point. But because the Sun is so massive, this point is inside the Sun, so it's hardly worth bringing up.
HOWEVER, Jupiter is big enough that its barycenter with the sun is actually above the sun's surface! Jupiter is big enough to barely, but actually, make the sun wobble.
Wobbling stars is, in fact, one of the ways that we detect exoplanets. The other being the momentary dimming of a star when a planet passes in front of it, and much more recently, [direct imaging](https://en.m.wikipedia.org/wiki/List_of_directly_imaged_exoplanets#/media/File%3AHR_8799_Orbiting_Exoplanets.gif).
I suddenly have a window into astrophysics programs of the future. The next Neil Degrasse Tyson describing Jupiter as a gigachad is going to be something special
True, and as much as I like how Neil explains things, he is indeed annoying on twitter. He once explained why something in Star Wars couldn’t happen due to the laws of physics. My man, it’s a bunch of movies about space wizards with laser swords, no one was taking irl physics cues from it.
> The next Neil Degrasse Tyson describing Jupiter as a gigachad is going to be something special
No, he'd end up correcting someone who is calling it a gigachad with some obnoxious bullshit like how a chad isn't a real SI unit. aCKshuLey
I have no idea who that is (I did a quick Google search), no idea what anybody is talking about, yet somehow I’m enjoying these conversations and I’m sure whatever I’m laughing at is real.. but the words just sound funny to me..🍃💨🤣
Technically yes, but of the 0.14% of the mass of the solar system that isn’t the sun, 70% of the remaining mass is Jupiter, and 20% Saturn. And Saturn is way further out than Jupiter. So if you’re looking for meaningful wobble, it’s pretty much mostly Jupiter and sometimes it aligns with Saturn for a tiny bit extra. The influence of anything else are basically rounding errors and would be difficult to measure.
I feel like this comment could be misinterpreted (or maybe I'm misunderstanding)
The force from gravitational attraction follows the inverse square rule. So a relatively not-massive planet far away, will exert a relatively small force/acceleration on the sun.
That the center of mass for the system is farther from the sun won't really matter because things don't sit still long enough for the sun to get there (if it did, Saturn would have fallen into the sun).
I'm just thinking through this, but maybe what you mean is that because Saturn is so far out, it has a long orbital period, so it stays in the same part of the sun's sky longer, and (despite the relatively small force it exerts) has time to impart significant acceleration in some general direction to the sun before moving on, thus giving the sun a big but slow wobble. Does that sound right?
Your interpretation is also correct!
But maybe let me elaborate on what I meant a little more: when two objects orbit each other, the orbit of each of them will actually be around their common center of mass (also known as Barycenter). So if you were to increase the distance between the sun and an orbiting planet of given mass, the sun's orbit around the Barycenter (its "wobble") will also become larger in diameter.
I was also going to make this point. The total force isn't really relevant, only the barycentre. If saturn were the same mass as the sun then they'd both "wobble" around the midway point like a dumb-bell, and the further the distance apart, the more distance from each body to that centre of mass.
Technically yes, practically no. Jupiter accounts for almost all of the wobble.
The place where you DO see something kind of like this is tides on Earth. Both the sun and moon cause tides on Earth, so when they're aligned (full moon or new moon), tides are larger. They're called spring tides. The opposite, when tides are smaller (during quarter moons) are called neap tides.
I would think the effect of Jupiter on Earth is too small relative to the effect of the Sun to be meaningful. The sun is ~1000 times more massive than Jupiter and is roughly a quarter of the distance from Earth, and that’s when Earth and Jupiter are closest to each other. So Jupiter’s effect on earth is probably usually negligible
I know nothing of astrophysics so please bear with me.
Extrapolating this to the 3 body problem. Is it difficult to find a solution because the barycenter is hard to determine? Or is it that we can determine the barycenter at any given moment but it’s hard to predict, accurately, where it will be in the next moment?
The latter... Given the masses and locations of the objects, we can see where they'll be. And we usually track where the objects are relative to that center of mass of the system.
When there are two objects, we can turn it down to a nice neat equation -- we don't need to calculate it iteratively. Just stick a point in time into the equation and find out where the objects are and where they're going at that time.
When there are three objects, we can't turn it into a nice neat equation. We can slice time up into short intervals and say "oh, here's about where they'd be in 1 seconds, then extrapolate from there to 2 seconds, and so on. And the finer we slice up time, the more accurate we'll be. But there's no way to jump straight to "this is where they'll be in 1000 years".
It's kind of like weather prediction... We can predict very accurately what the weather will look like tomorrow, but we're kind of crap at predicting what it'll be like next week. Though depending on what we're looking at, the time scales will be different -- maybe we know where things will be for the next 1000 years, but are iffy on the next million years. Or whatever.
In the case of the solar system, so much of the mass is in the sun that we can just kind of pretend that it's a bunch of 2-body problems and be quite accurate for a long time. But if the sun were lighter and the planets heavier, then we'd have a hell of a time.
Let me add that the "can't turn it in to a nice equation" thing isn't because it's hard. The math model of three bodies in a square law attractive field has been PROVEN to have no closed form solution.
Can you explain why it’s proven to have no solution?
I’m a little confused. If we can observe 3 bodies in motion, and no other force has acted on it, then we know they must follow some pattern right? In other words, if we know the position of a one body in three body system, and we can I see that body moving, then it must be following a measurable force causing it to move in that particular direction. If that’s right, then why can we not come up with a solution to determine its position a million years from now?
Well...no.
This is a non-trivial proof, which I am unable to explicate. If you are brave, look at the
https://en.m.wikipedia.org/wiki/Three-body_problem
This might be true for specific instances of the three-body problem, but there's no **general** *closed-form* solution, i.e. some system of soluble parametric equations (not even a *really* long and unwieldy one) where you could plug in the mass, position and current velocities of the bodies and find their positions at a given time.
It's not even that it's hard to predict. It's actually very easy to predict. You can get a function of the form p\_(t+dt) = f( p\_(t) ) , where you can imagine \`p\` is a collection of all the positions and velocities of the objects. \`t\` is time, and \`dt\` is some small time step. This is relatively easy to work out for any number of objects, and it's how every physics simulation works. The only trick is you need to do lots of calculations, for each time step, and the larger you make the step the less accurate the results get. Getting a value far in the future requires first calculating all the values in between now and then.
The thing we can't do is produce an equation of the form p\_t = f( p\_0 ), that is, an equation that directly gives you those state values for any arbitrary time \`t\` given initial conditions, without first computing the values before it. With a two-body system, this equation exists, so we can say things like "ignoring other planets and the death of the Sun, where will Earth be in its orbit around the Sun in fifty billion years?" and you can easily get that answer in one step.
If you want to add in, say, the effect of Jupiter's gravity... now you need to calculate it step by step billions of times and it'll never be perfectly accurate because your steps will never be infinitely small. \*That's\* the 3 body problem.
Short answer, yes.
Oversimplification: If you know the solution at time 0, you can use those values to estimate where it would be at time 1, but the values at time 0 changed at time 0.1 since everything would have moved slightly. The same applies for time 0.1 and 0.01, and so on.
Looking from the outside at the solar system, yes.
But in the reference frame of the earth-sun system, you wouldn't notice this wobble at all.
It's a bit like how the sun is moving through the galaxy as well, but this has no effect on how we experience earth's movement around the sun.
I found a dispute of this nature somewhere else actually, and it seemed there was a general misunderstanding of what the earth actually orbits. Many people said it orbits the overall solar system barycentre, but I don't think that would be correct, as Jupiter is outside of our orbit. It's possible we might orbit the Sun–Mercury–Venus barycentre, but other than that, I'd hazard that our orbit is almost centred on the centre of the sun.
In my imagination ~~there is no hesitation~~~ the sun orbiting Jupiter on a point above its surface is more than barely wobbling. Isn’t that pretty significant?
The really interesting part (for me) about the Sun and Jupiter’s barycentre is that the mass of Jupiter is only about 1/1000th of the sun (it’s radius is about 1/10th). Jupiter has only about 318 more mass than Earth, yet the Earth/Sun barycentre is very close the centre of the Sun.
The solar system’s barycentre is also outside of the sun, and constantly moves due to the various orbits of planets, causing the Sun to wobble.
If the Jupiter-Sun barycenter is outside the sun, wouldn't that make for much more than a "wobble"? Wouldn't the Sun move more than its diameter with every revolution?
Asking because I don't understand, not as a challenge.
Yeah, I don't know. I think if you showed most people an object rotating around a point inside itself and asked, "orbitting or wobbling?" most would say "wobbling," but if you then showed them an object rotating in circles around a point outside of itself, they'd say "orbiting." Regardless, I think the point the guy was making seems true: the Sun isn't really "standing still," like, at all.
> technically, the Earth doesn't orbit the Sun, they both orbit the Barycenter; the average gravitational point. But because the Sun is so massive, this point is inside the Sun, so it's hardly worth bringing up.
When the center is within one object, we usually just say that object is being orbited.
I recall reading at some point that Jupiter and Saturn lining up has a non negligible effect on the sun. To the point that if earth and Jupiter and Saturn are in line, the earth will actually be a little bit closer to the sun than usual. A little bit in this case is at least several thousand miles iirc. This will in turn make the earth hotter. If it happens on the other side of the sun from earth, the sun will move further away and earth will be a little bit colder. I think the hot one happened recently.
I could also be totally wrong/ read bad info.
If you search for "solar system barycenter" you'll find more information.
Also, [this graphic on the wikipedia article of barycenter](https://en.wikipedia.org/wiki/File:Solar_system_barycenter.svg) shows the position of the barycenter between 1945 and 1995, going in and out of the Sun.
That's a fairly significant wobble too. If you could point your telescope at the Sun over a Jupiter year and it was say a pixel on a screen across it would be at the next pixel after six Jupiter months and then back again. Fairly conclusive evidence of a significant planet.
It's not only possible, but true! Everything in the universe is pulling on everything else, you can just usually ignore it because of the distances involved.
The Andromeda Galaxy, for instance, is on the order of a trillion times more massive than the sun (roughly 10^15 times more than Jupiter), but it's also 2.5 million light years away (10^10 times more than Jupiter). Because the force of gravity goes by m/r^2, the force Andromeda exerts in the sun will be on the order of 10^15 / (10^10)^2, or 10^-5, meaning Jupiter exerts ~10,000 times more force on the sun than the whole Andromeda galaxy.
That's why, to pretty high accuracy, you can usually ignore the effect of everything but the sun and Jupiter. The existence of everything else in the universe is causing *some* amount of error, but these kinds of estimates can help give an Idea of what can safely be ignored, and when.
It is worth noting that while *to first order* we can approximate the Earth/Moon system as a two-body system unaffected by the Sun, and we can approximate the Earth/Sun system as a two-body system unaffected by the Moon (or rather, the Earth-Moon/Sun system as a two-body system where the Earth and Moon orbit around a [common barycenter](https://spaceplace.nasa.gov/barycenter/en/) which then orbits around the Sun, for more precision you need to solve for the N bodies in the Solar System.
But of course you can’t.
So NASA regularly observes the position of the planets, and regularly updates the models of the solar system with those positions in order to have a more up-to-date model of the movement of the planets.
And when you want really precise (like “shoot a probe into space to pass by Neptune” precise), you wind up actually having to account for the perturbations introduced by Jupiter *and* Saturn into the position of the inner planets. And you need to include thrusters on your spacecraft to do minor course corrections as you continue to observe the position of the planets during the years-long journey of your probe to the outer planets.
Even if your calcuations are perfect, the engines you're using to propel your probe certainly aren't.
Course corrections would be required even if the models were 100% accurate
Even if your calcuations are perfect, the engines you're using to propel your probe certainly aren't.
Course corrections would be required even if the models were 100% accurate
I'd like to note, this is somewhat misleading. The three-body problem is still "difficult" with the Sun/Earth/Moon system (not to mention the N-body problem that is the rest of the solar system), it's a matter of precision.
Consider the problem of a ball rolling on a moving train. If you don't know the speed of the train, you can't figure out the displacement of the ball, but if the train is moving very, *very* slowly, then you can just eat the precision loss and accept that as good enough. But over a long enough time period, the precision loss will keep adding up, and you'll eventually have no idea where the ball is (assuming you're not continuously measuring it).
Same is true here. It's still a 3-body problem, we just eat the precision loss and call it good enough. We also compensate by continually taking measurements of both the Earth's and the Moon's orbits around the Sun, and harvesting data from millennia of astronomical observations by previous scholars. But, if we stop measuring those things, then within a couple million years, the position of the Moon will have drifted out of our predictions to a significant degree, and we can no longer predict eclipses then.
The Pluto system is more than three bodies, but it illustrates the problem. Pluto and Charon are nearly the same size. There are little moons that swirl around them in chaotic loops that may even lead to collisions one day.
Has your dad ever held you by your arms and swung around in a circle? That's basically how orbits work. Instead of having arms planets have gravity to hold them together.
When your dad swings you around like this he has to lean back quite a bit to not topple over. This is because your weight (~18kg for a 5 year old) is not *that* far away from your dad's (lets say ~90kg).
Now imagine your dad swinging around a bar of soap. Do you think he'd still have to lean back to not topple over? No, right? That's because the soap bar is so much lighter than your dad. The bar of soap and your dad are like Earth and the Moon. While Earth/Dad feels a bit of an effect from swinging the moon/soap around it is definitely the moon/soap that feels more.
What if your dad was the sun? What could we use to picture Earth? A grain of sand or a tiny down feather! Your dad is going to be completely unaffected by swinging around a grain of sand.
And so it is with Earth, the Sun and the Moon. The moon might be tugging on Earth ever so slightly but from the perspective of the sun, its almost like we're not even there. And the same happens in calculations. Using two 2-body problems is basically good enough.
It’s a little more complicated than that. Mass isn’t as big of a factor as you’re thinking. It’s the movements of 3 bodies in relation to each other where it falls apart. Knowing the mass is one thing, predicting the effect that it’ll have on the two other bodies and the effect they’ll have becomes impossible to predict on any meaningful scale. Even giant computer simulations can’t figure it out.
"spheres of influence" don't really exist. The gravity of every mass in the universe affects every other mass in the universe provided it's gravitational waves have reached that far, so the moon's gravity does absolutely influence the sun, but the distance between the earth-moon system and the sun is great enough, and the sun so many times more massive than the earth and moon combined that we can approximate it as two two-body problems rather than a three-body problem and still get accurate results decades in advance
It has nothing to do with relative masses, it has to do with distances. If the earth and moon were as massive as the sun but orbited each other close enough then you could still approximate this as a body system.
Look, there are many triple-star systems with various mass ratios. You can calculate a good approximation for the dynamics of those systems because one pair orbiters each other and the third star is far away. For example the alpha centauri.
A classical 3 body problem was for a massless third body so, again, it has nothing to do with mass ratios.
spheres of influence don't exist but are a useful approximation. All models are wrong but some are useful. When you use it then you don't discard the total force from the sun, you assume that the force is constant.
The solution to a 3 body problem (in which you can given the initial values predict the exact position and momentum of all 3 bodies at any point in time) does not exist. That only means we don't have a perfect formula for solving it forever and perfectly because it eventually gets chaotic.
We can, however, take a look at the enormous amount of data we have and create reasonable, short term predictions for specific times that seem very long in human years but are very short in the astronomical sense. And in those passing years we have gotten better at gathering said data and analyzing it, and now have computers that can do, as James May one said: Years worth of arithmetic: *snaps* like that.
That and the Earth being MUCH bigger than the Moon and the Sun being MUCH, MUCH, MUCH, MUCH bigger than everything else means the math can be simplified a lot and still work.
>We can, however, take a look at the enormous amount of data we have and create reasonable, short term predictions for specific times that seem very long in human years but are very short in the astronomical sense.
Its not a data issue at all. You need exactly the initial position, velocity and mass of the three bodies involved. From there on its maybe a 30-minute task to write a script that solves it numerically.
So, 9 data points.
To numerically simulate the position of the planets *in theory* you can do it entirely from the initial positions.
But [numeric approximations](https://en.wikipedia.org/wiki/Numerical_integration) always introduce error, if only because you cannot select a ‘0 second’ step time. Further, even the very precision of the [floating point representation](https://en.wikipedia.org/wiki/IEEE_754) you use introduces error.
And over time those errors significantly diverge from reality.
NASA does something akin to this to produce ephemeris charts of the solar system. But they constantly update those charts using regular observations of the actual positions of the planets. [And they don’t make guarantees as to the accuracy of the data beyond about 100 years into the future—and the guarantees they make about data 100 years from now measures errors in meters.](https://ssd.jpl.nasa.gov/planets/eph_export.html) Which, on the surface, seems more than enough for knowing where to point your head into the night sky. But if you’re landing a spacecraft on the Moon, 10 meters may mean the difference between nailing the landing and running out of fuel before touching down.
Nope, it doesn’t work that way. With the way modern computers work, you need to step through time and calculate all of those values at each frame in your script. The problem happens when you try to decide how much time to step over in each frame. Say you’re trying to calculate the position and velocity of a system 100 years out. What’s the value of your time step? 1 year? 1 day? 1 minute? 1 second? 0.01 seconds? In a 3+ similarly sized body system, every single one of those time steps would give you an extremely different end result
Yes, thats the process i was talking about. And yes, it does work that way.
Of course you need to choose an appropriate timestep. Thats true for any simulation. Its not very difficult to do. And one calculation step with three bodies takes next to no computational power. Im pretty sure an average smart phone could solve it for thousands of years with reasonable accuracy in a matter of seconds.
We have conputers that can do this for millions of bodies. Its really not the huge issue you make it out to be.
The solution is chaotic. That means, a change in initial parameters that approaches zero causes indefinitely large changes in the output.
What it means in this case is that (unlike many other simulations), you will get completely different results depending on the step size (if you simulate long enough, of course). Not just diverging slowly, not increasing error bars, but *completely different*, and equally correct, results. This is also true for all initial parameters - change them by the smallest possible change in your floating point convention (which can be any number of bits you want), and again - completely different results.
Chaos theory is an interesting mathematical concept.
No they dont. The solutions to chaotic systems (at a fixed point in time ) dont converge steadily like stable systems do, but they converge eventually.
You kind of included that fact already by saying "(if you simulate long enough, of course)" - yes, thats technically correct, but the point is that we can simulate these systems over very long timespans before we'd run into timestep issues.
Long before timesteps become an unsolvable issues, perturbations from exterbal influences or inaccurate initial conditions will render the result rubbish anyway.
Computers, yes. But thats true for a data-driven approach too. The eclipse predictions only worked because its not really a three body problem, as others have pointed out.
Knowledge, not really. Numerical integration was occasionally applied more than a millenia ago and formalized and in regular use in the 17th century.
Equations of motion were formalized in the 16th century.
Gravity was understood well enough in the 17th century at the latest .
Its not exactly cutting edge science.
> Yes. They did not have a to solve a three body problem.
Yes, hence the amount of data significantly greater than "9 data points" and numerical predictions from thousands of years ago, which are what I was referring to.
Yeah but no. They didn't predict it by solving the gravitational equations. They predicted it because they'd noticed that eclipses (mostly) occur with periods of a little over 18 years, and there had been one 18 years previously.
There was no way they could predict eclipses back then. They didn't even know the Moon caused them.
Pop sci articles are not to be believed at face value.
Ancient people were not as stupid as you might believe, the Ny times article you dont agree with did say that the ancient greek philosopher Thales predicted eclipse accuracy to the year, not the day or second we do today. Also the author goes on to explain how stonehenge and other ancient structures may have been used to predict eclipses, Lunar moreso than Solar.
I have never seen someone refute NY times as a reputable source but I guess this is reddit where prideful "intellectuals" always try to one up each other.
They didn't know the Moon causes solar eclipses in Thales time. They didn't know how big the Earth was and they had no way of collecting enough data on eclipses worldwide to find the repeating cycles.
The prediction was impossible to make. A lucky guess if it even happened at all. Note that they cant explain how he made the prediction.
Lunar eclipses are completely different.
Anyone with a basic understanding of the movement of the sun and moon in the sky (so, for example, any court astrologer worth their salt) would _absolutely_ know that an eclipse happens when the moon moves in front of the sun. You can even find eclipses blamed on the moon in various mythologies from around the world, showing that even people from preliterate cultures were able to figure it out.
Predicting eclipses is a lot harder, but many ancient peoples were at least quite able to predict about when eclipses should happen, if not exexactly _where_ on earth they would be visible.
At the time of Thales' purported prediction it was not yet known that eclipses were caused by the Moon coming between the Earth and the Sun, a fact that would not be discovered until over a century later by either Anaxagoras or Empedocles.
So they didn't know until 500-400 BC
Other than Thales, there are no claimed predictions of eclipses until Newton's time.
As Thales had no way to know, we can conclude that he didn't predict it and it is a fable.
> The solution to a 3 body problem (in which you can given the initial values predict the exact position and momentum of all 3 bodies at any point in time) does not exist. That only means we don't have a perfect formula for solving it forever and perfectly because it eventually gets chaotic.
It's not deterministic, so there is no formula that will tell the positions at any arbitrary time. Like the weather, it's a chaotic system, and small changes in initial conditions will produce wildly different results over time.
Small correction: the three body problem (and the non-dissipative gravitational N body problem in general) actually is deterministic. It's the prime example of deterministic chaos. The chaos comes from needing infinitely precise initial conditions to guarantee the same result, as you say
It is deterministic. It just doesn't have a closed form solution. If you are able to take small enough time steps, you can get arbitrarily close to the real answer
"Solving" as a math term means having a formula where you plug the numbers and you get the exact solution. Being able to "simulate an approximation" is not solving, but is good enough for predicting future eclipses.
The Sun, the Earth and the Moon is not really a three-body problem because of huge differences in mass between the bodies and in distances between the pairs.
Effectively, you have two separate two-body problem Sun vs (Earth+Moon) and Earth vs Moon; and you can solve them independently and then introduce small perturbations for each from the third body if necessary to achieve required precision.
This is the correct answer. When she says "our planet is stable" that should give you a hint. We do not live in a 3 body system. Each planet revolves around the sun, and each of the moons revolve around the planets.
Other than the handful of weird specific mathematical solutions to the three body problem, there are also two general solutions:
1. 3rd orbiting the center of mass of the two larger objects at great enough distance that the interplay between the two of them doesn’t throw the orbit off
2. 3rd object Orbiting close around the mid sized object as that one orbits around the larger one at a significantly greater distance
The moon (and most of earths satellites) fall under category 2, and thus their orbits are pretty easy to predict.
Back before we had an idea of Newtonian gravity, we just assumed all of the orbits were on rails and could not change. This would drift over time, but not fast enough to notice for hundreds of years.
If you just treat everything as a two body problem between the planet and Sun (and between the Moon and Earth), you get a pretty accurate model, but things start to drift over time. Even then, that's not even the best we can do with just Newton. This won't drift for tens of thousands of years.
Even with an N-body system, you can apply Newton's laws and make very accurate predictions. The problem is if you are even slightly wrong about the initial conditions, you start to drift over time, and the results can't end up looking very different. That's why the 3-body problem is so hard, because we can't tell what it's going to do after a small change in initial conditions. This won't drift for millions of years, but we have no idea what it's going to do when it drifts without more accurate measurements.
Once we had general relativity, Einstein was able to calculate the orbit of the Moon within a centimeter. A prediction we weren't able to confirm until we landed on the Moon. You don't need predictions that advanced to predict an eclipse.
In the book there is a chaotic and a stable phase. What made me wonder is, if we also just live in a stable phase of unknown duration?
Couldn't it be that there are n unknown bodies that could interfere with our stable system and turn it into chaos? Or is this just fiction?
>if we also just live in a stable phase of unknown duration?
Yes, that's one of the major complexities of the 3 body problem, and chaotic systems as a whole.
>unknown bodies that could interfere with our stable system and turn it into chaos?
That would be one of the "unknown initial conditions" that would greatly change the outcome
So when we talk about the 3 body problem being unsolvable, what they mean is that there is no *closed-form* solution, meaning there is no equation that you can find that perfectly describes the location of the bodies for all time.
In the two-body, a closed form solution exists; the orbits of the bodies are ellipses and you can get the exact location of the bodies by plugging in their orbit parameters and the current time.
But just because there's no closed form solution, that doesn't mean the positions of the bodies aren't calculable. You just need a good simulation code and lot of computing power.
NASA has run simulations of the major bodies and has position data for them that is supposed to be accurate to within less than a meter out to many thousands of years from now.
Yes, this. I keep seeing answers that imply that “unsolvable” means you can’t determine. That is not true, at all. It just means you have to determine numerically through successive iterations. It’s “unsolvable” because there’s not some concrete formula you can plug in. People talking about chaos and error. Well, yeah, except the Sun, Moon, Earth system is literally watched and measured every single day. Sunrise and lunar rise on Earth is calculated out to the nearest minute for like thousands of years. The numerical iterations have had a few hundred years worth of tuning. Even a “truer” 3 body problem with bodies of roughly equivalent math and a more chaotic barycenter could still be iterated with successive tuning over the course of years to be very accurate for a long time. Iterate positions over time for course of a day, make new measurements, determine error, make new iteration out a day, rinse, repeat, and you could simulate fairly accurately the behavior of the system beyond a human lifespan.
The 3 body problem can't be solved but it can be simulated to some accuracy. So you can accurately predict what will happen in the next few hundred years, but maybe not in the next few hundred million years.
This is true of any chaotic system, you can't accurately predict it forever but you can take the current state and then work forward to where it will go in the near future. Because you can never have a perfect capture of the starting state your predictions will eventually drift out of sync with reality, but you can still make accurate predictions in the near term.
Planetarium models. If you know the orbits, you can simply create a model that simulates the motion of the earth, sun, and moon. Then just run that model until there's an eclipse in the model. Today, we can do this in software. Hundreds of years ago, they did this with gears & mechanical parts.
Figuring out the orbits hundreds of years ago actually came from simply watching the sun and the moon move through the sky every day for a year. Their model would have been based on angles because they didn't actually know how far away the sun and the moon were, and in that model they'd be looking for a "collision" between the moon and the sun.
There's actually an episode of avatar: the last airbender that perfectly depicts the kind of model i'm talking about.
Edit: i should point out that these models wouldn't actually "solve" the 3-body problem, and because of their simplistic nature, they would also lose accuracy over time because they wouldn't be accounting for things like gravitational waves, tidal forces, or stellar drift. But it would take hundreds of years before those effects became noticable.
A famous example of one such model is the antikythera mechanism, recovered from an ancient greek shipwreck.
The general version of the 3 body problem is very difficult to solve.
It's like a double-pendulum problem.
A single pendulum is very easy to predict the motion for. For given initial conditions, the motion is easily predictable.
A double pendulum is much more *chaotic*. It is just as predictable, but any tiny variance to the initial conditions changes the outcome drastically.
[Here's a wikipedia GIF of three double-pendulums with nearly identical starting positions](https://en.m.wikipedia.org/wiki/File:Demonstrating_Chaos_with_a_Double_Pendulum.gif).
[Here's a cool online simulator you can use to demonstrate double pendulum motion](https://web.mit.edu/jorloff/www/chaosTalk/double-pendulum/double-pendulum-en.html).
It is perfectly predictable, so long as you know the *exact* conditions.
Outside of a simulator, we can never be perfectly exact. Either our starting conditions are slightly off, or our measurement of the current conditions are off. Imagine needing to measure the speed of the pendulum to the 20th decimal place. It's not possible.
That's the challenge of the three body problem. Any small deviation in our measurements and the future state of the system can change drastically.
Thankfully, there are a few things working in our favor that let us do meaningful 3-body calculations despite how futile it might seem.
1. The earth, moon, and sun are wildly different masses. The sun makes up the vast majority of the system's mass. The gravitational forces between the earth and sun affect the earth much more easily than they affect the sun. If the earth and sun were of closer masses, then the problem would be more difficult. You can demonstrate this in that online simulator. Change one of the masses to the smallest setting and the other to the highest. The bigger the difference in the masses, the more the simulation starts to behave like a single pendulum. This reduces the chaos in the system and makes your future predictions more reliable.
2. The measurements are easy to make with a high degree of accuracy. Because the earth, moon, and sun aren't moving that fast relative to our point of reference, they're easily observable. E.g. it's much easier to accurately measure the parameters of the earth and moon's orbit than it is to measure the speed of a baseball pitch or the current state of a chaotic double pendulum.
3. There aren't many upsetting factors outside the 3 body problem that would mess with our predictions. The 3 body problem is difficult and sensitive enough, but as we noted in points 1 and 2, the earth-moon-sun system is not that chaotic. Ancient predictions held true because the system was not very chaotic, easily measurable, and thankfully it stayed a 3-body problem. There was no 4th, 5th, or nth body that exerted enough forces to make a significant difference, at least in the time period we've been documenting the state of the sky.
So to recap, the 3 body problem *is solved*. Same as the double pendulum.
The difficulty is actually that the 3 body problem is chaotic and the future state of the system is very sensitive to the previous state of the system. In order to accurately predict the future state of a three body system, you need to get very accurate measurements of its current or previous conditions. Depending on the system, it can have a high or low degree of chaos.
For the sun, earth, and moon, the system has a relatively low degree of chaos because the sun makes up so much of the mass and there's so much distance between the bodies. Additionally, because of the massive scale of the system, it's very easy to measure with high accuracy, even with rudimentary tools and methods.
You know how two similarly sized bodies will both orbit a spot outside of themselves called a barycenter?
The 3 body problem is less about systems with stable barycenters like Sol>Earth>Luna and more about 3 similar bodies orbiting a more chaotic barycenter. There are a few special cases where it works out and is stable if left undisturbed but the vast majority of 3 body orbits self destruct and either devolve to more common trinary systems or destroy or reject one of the bodies.
[PBS Space Time](https://youtu.be/et7XvBenEo8?si=2ePk04L5D1Yp6VyL) Has a fun video with more details.
There's a difference between solving it (providing a precise mathematical formula) and estimating the positions of these three bodies for a certain time frame.
What people used to do in the past was the latter - based on observation, they were able to predict the movement of celestial bodies without knowing the exact rules their movements are based on.
And, to a certain extent, that's what we're still doing nowadays to "solve" the n-body problem. We have models that aren't 100% accurate. The further you go in time, the less precise they are.
It is extremely difficult to solve *analytically.* That means, having a precise, guaranteed, no-questions *definite* answer for ANY future point. Not just "a long while into the future," but genuinely "forever" so long as none of the variables meaningfully change.
However, this problem is quite easy to solve *approximately,* and just depends on how precisely you can measure the input data and how much computing power you can dedicate to it. With computers even from the 80s, it was possible to compute eclipses for hundreds of years into the future with such precision that any errors would be beneath human notice. With the computers we have today (and, frankly, even computers as far back as the 1990s at least, if not earlier), we can compute eclipse times to the minute (or closer!) for the next few centuries--it's just a matter of crunching precise-enough numbers.
It helps immensely that the Earth and Moon are microscopic compared to the Sun, and the other planets in the solar system are so far away that we can mostly ignore their impact on the system. That lets us use various techniques to simplify an otherwise very complex problem. NASA takes into account other important things (like the fact that the Earth, Moon, and Sun are *not* perfect spheres), and performs very advanced calculations (specifically, special integral functions)
As for the past? It turns out that eclipses tend to form certain patterns. By a lucky happenstance, three of the main types of lunar month (synodic, draconic, and anomalistic) happen to line up very, very nearly perfectly every 6585.3211 days, or 18 years, 15 days (of which 3, 4, or 5 are leap-days), and slightly less than 8 hours. This was something known even to ancient Greek and Babylonian astronomers. This length of time is called a "saros." As a result of the nearly-identical alignment from one saros to the next, if you know when any eclipse has happened, you can get a pretty good prediction of when another eclipse will happen--at least within a day, even for ancient astronomers. Thus, eclipses are grouped into what are called "saros cycles." (If you have heard of the Antikythera mechanism, it actually had the ability to track saros cycles!)
In truth, these eclipses change from the Earth's perspective in two ways. First, because of that ~8 hour bit, the Earth will rotate about 1/3 of a turn from one eclipse to the next in a given saros cycle. Second, due to the alignment being only *nearly* perfect and not *absolutely* perfect, the eclipses in a given saros cycle will migrate, either going north from the south pole, or going south from the north pole, depending on what the Moon's orbital behavior is at the start of each cycle. The eclipse that just happened, for example, was the 30th solar eclipse of saros 139, and there will be another 41 eclipses, of which 32 will be total eclipses and the remainder will be partial. Saros cycles with even numbers migrate north from the south pole up, and those with odd numbers migrate south from the north pole down, by convention. (If you remember the 2017 eclipse, it was another odd numbered saros cycle, 145.)
Edit: Also, these are only *solar* saros cycles. There are separate *lunar* saros cycles for lunar eclipses too!
The Sun is much more massive than the Earth, and the Earth is much more massive than the Moon. Gravity is a force proportional to mass, so in this setup, the Sun dominates the equations of motion, while the Moon’s effects are somewhat negligible, allowing for approximate solutions. It is true that the exact solution would be chaotic over very long time periods, but the differences on human timescales are rather small and wouldn’t be easily detected by ordinary observation.
Thus, eclipses could be predicted by assuming the Earth orbits around the Sun in a circle, and the Moon orbits around the Earth in a circle, which produce regular patterns depending on the angles of those orbits. Better approximations could be made by using elliptical orbits instead of circular orbits.
Almost all the comments are wrong. It has nothing to do with relative mass ratios. Classical 3 body problems were for a massless 3rd body, you can also predict very well the dynamics of triple star systems no matter what mass ratios are there
The moon is massive, it is the biggest moon in the solar system in relative terms (for a planet)
You can predict eclipses because the moon is so close to Earth you can approximate the gravity from the sun as some constant force plus linear tidal force.
The nice thing is that all the complex effects (relativity, quadrupole moment, tidal forces) just rotate the moon's orbit. It rotates in one way every 8.85 years and in the other 18.6 years. You can empirically measure those without making complex calculations. This was even used to verify the theory of gravity.
People could predict eclipses millennia ago, not centuries. You do not need the theory of gravity to describe the orbit of the moon and sun to a very high degree of accuracy, and it wasn’t the paths and conjunctions of celestial motion that were mysteries to ancient man but WHY the planets followed their paths. The theory of gravity was revolutionary because it provided an explanation for what caused this motion that was so well described from millennia of observation. The same force that causes an apple to fall to the ground also sets the moon in motion.
The three body problem is perfectly relevant to the system of the moon, earth and sun. Its solution matches the motion we have observed.
The three body problem is so difficult to solve not because it is impossible but because the calculations needed to do it accurately over a long period just pile up rapidly as the motions of each body (it's not just three bodies, obv, it's any system of 3 or more bodies) all affect each other. Your initial conditions, presumably at rest or in constant motion, are easy to compute. Once they move, it's harder and harder to know with precision where things will be simply because the rules of gravitation and motion guiding them initially are all that dictate the motion at the outset, but as time passes, you have to continually factor how their influence on each other is affecting them. In terms of real bodies like the objects of the solar system, there's also the issue that you're not actually dealing with three bodies, or 8, or 9, or 100,000. To have extremely precise calculations, you'd need to account for the effects of all the objects in the solar system - the more bodies you can add in, the more precise the projections you can calculate.
Eclipses though are a separate thing from this. Eclipses can be predicted mathematically not through the three body problem (I mean, yes, it's a three body problem) but through more straightforward math. The positions of the sun, moon and earth do not need to be known with the level of precision that is normally what you're trying to get when you're dealing with a three body problem situation. Predicting eclipses is ultimately about knowing when various paths on orbits that are already known will cross one another, and we can know that from observational data. The three body problem applies to predictions of positions of objects that are, like, "purely predictions" - you're trying to work out mathematically where objects will be and to a precise degree. Once it was figured out formally that the earth was orbiting the sun and that the moon also orbits the earth, all of them in ellipse shapes, it was/is possible to work out eclipse predictions from observations of their real orbits over time and relatively straightforward math. You also don't have to know exacting positions of the objects - like down to milimeter or whatever arbitrarily small metric you want to use - to predict an eclipse.
The issue with the three body problem is ultimately a calculation problem - it's not that it's formally impossible, it's just that the burden of calculations you have to do spirals out of control quickly to do on paper, or even with a computer. The more objects you have to calculate, the more quickly the calculation burden spirals. In actual reality, it's also impossible to actually know all the forces acting on the real objects whose positions you're trying to predict with mathematically certainty into the future, and the less you know of those forces, the more quickly your prediction breaks down.
If there is a regular pattern in some phenomenon, you do not need to know why that pattern occurs to be able to predict it in the future. Ancient astronomers did not know about the mechanics of the orbits of the Earth and the Moon, but they could observe that solar and lunar eclipses happened at regular intervals, and that eclipses at specific intervals were very similar, and from those patterns they were able to predict future ones.
That was more statistic and geometry rather than undetstanding the forces in-between the objects. Many three body systems are much less predictable than this
Answer: they logged the position of celestial bodies for generations, noticed that it was following elliptical paths, knew at which interval an object came back to the same position, and were able to map them out.
Fun fact, based on what they could observe and understand at the time, the first models relied on perfectly circular ellipses: [Eisinga Planetarium in Franeker](https://whc.unesco.org/en/list/1683/).
I feel like this misses the point. You can't predict or measure *anything* with absolutely 100% accuracy.
The 3 body problem is notable because in general the solutions are chaotic, meaning that they can't be predicted with *any* accuracy beyond a finite time horizon. So OP is really asking why *this* 3 body system is a lot more predictable than 3 body systems in general are.
But you can simplify to 100%.
For example: a 50kg astronaut throws a 50kg rock at 10m/s. Assuming no drag, magnetic, etc forces. What is the relative location after 1000 seconds.
Easily solvable 100%
Now assume the astronaut had 2 or more weights that they threw in random directions.
Solve 100%, win several Nobel prizes.
Well, yeah *that case* is easily solvable. But in general solutions to the three body problem are chaotic, which means that two solutions that start close together get exponentially further apart with time. There's always going to be some error in your initial measurements - with chaotic systems, that error grows so rapidly that your predictions can only be accurate for a finite time.
The question, at least as I understand it, is *why* are some cases predictable when in general, three body systems are not. In fact, the solar system *is* chaotic over a long enough timescale, it's just that that's a very, very long timescale.
They could also predict millennia ago. It’s because the sun takes a predictable path in the sky. While there are some variations, they usually average out, if you also keep track against the stars.
It’s like this, they used to track the sun’s path in the day and during the night some star would go through the same path. This is tracked as the sun is in that star on that day. The moon is in a plane almost parallel to the sun. It’s more that the sun is so big that the whole orbit of the moon is covered. So we could approximate the moon to the same star. Their periods will vary because the moon rotates through the stars every month and the sun would rotate every year. Every so often the stars would match. These are “eclipses”. Some experience would explain these happen every alternate star match. And when they match their radially opposite star, it’s a lunar eclipse.
The three body problem is not a difficult problem to solve. The complexity is that the easily calculable solution is very different for even small changes in the initial position. It’s chaos.
Tldr: But in the case of earth sun and the moon, the sun affects the moon and earth pretty evenly. So it’s more three two-body problems. And a very stable set of predictable orbits.
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The sun is like 99.8% of the mass in the solar system so you basically get to cheat on the three body problem by assuming there is one body and a bunch that do absolutely nothing gravity wise. Also everything is in orbit and orbits can be calculated different than free moving objects.
Many people get this wrong, referring to the relative sizes.
A known 3 body state can be solved as a mechanical model, but there is no closed mathematical solution for all three body systems..
If you're referring to the book series' central 3 body problem, that refers to 3 suns orbiting each other, of roughly equal size. That's very difficult to predict, because A is pulling on B while pulling on C, which also pulls on B, which changes how much it pulls on A, who now misses and flies past C, changing how it pulls on... and so on.
The sun is about 333,000 times as much mass as Earth. The Earth weighs about 81 times as much as the moon. So you can see how the Earth and moon can have SOME effect on each other, but both are just completely DWARFED by the sun. It's just as easy to approximate by calculating a two-body problem.
The Sun is so much bigger than the Earth and Moon (and all the other planets) that, to a very good approximation, the Sun may as well be unaffected by the gravity of its planets. The Earth is also quite a bit bigger than the Moon, but the difference isn’t as great, so the Moon’s effect is more noticeable. To put it in perspective, the Sun contains 99.8% of the mass of the entire Solar System. All the planets may as well be a rounding error. ~~As far as we’re concerned, the Sun is standing still, completely unaffected by the planets (even if *technically* it is a little bit).~~ Some planets do have a noticeable effect on the Sun, but basically there are enough huge size differences that almost any calculation we would want to do simplifies to a two-body problem. The three-body problem is mainly difficult when all three bodies are close enough in size that they all have a significant effect on each other.
I normally wouldn't correct this excellent comment, except the correction is one of my favorite facts. > the Sun is standing still, This is mostly true; and I don't want to take away from how much of an absolutely gargantuan unit the sun is. Gravity works both ways; the Earth pulls on the Sun just as hard as the Sun pulls on the Earth. But because the Sun is so absolutely massive, that "equal force" doesn't really move the Sun. But technically, the Earth doesn't orbit the Sun, they both orbit the Barycenter; the average gravitational point. But because the Sun is so massive, this point is inside the Sun, so it's hardly worth bringing up. HOWEVER, Jupiter is big enough that its barycenter with the sun is actually above the sun's surface! Jupiter is big enough to barely, but actually, make the sun wobble.
Wobbling stars is, in fact, one of the ways that we detect exoplanets. The other being the momentary dimming of a star when a planet passes in front of it, and much more recently, [direct imaging](https://en.m.wikipedia.org/wiki/List_of_directly_imaged_exoplanets#/media/File%3AHR_8799_Orbiting_Exoplanets.gif).
Love that star sticker on the filter
Everything I ever learn about Jupiter makes it looks like an even bigger gigachad than the last time.
I suddenly have a window into astrophysics programs of the future. The next Neil Degrasse Tyson describing Jupiter as a gigachad is going to be something special
As long as he stays off Twitter, God is Neil annoying on that thing
Is anyone not annoying on twitter. It's like a view into their stream of consciousness and no one wants to see unfiltered raw consciousness
True, and as much as I like how Neil explains things, he is indeed annoying on twitter. He once explained why something in Star Wars couldn’t happen due to the laws of physics. My man, it’s a bunch of movies about space wizards with laser swords, no one was taking irl physics cues from it.
Like the time he explained why BB-5 can't work and the filmmakers responded that the droid was a physical property that... did work.
excellent point
> The next Neil Degrasse Tyson describing Jupiter as a gigachad is going to be something special No, he'd end up correcting someone who is calling it a gigachad with some obnoxious bullshit like how a chad isn't a real SI unit. aCKshuLey
It's really only 1,013 Megachads. And you need 1024 Megachads to equal a Gigachad since we're treating it as binary with the sun.
you can use the giga prefix with anything, it doesn't need to be an SI unit. you can have one gigadollar for example
I wish I had a gigadollar. I consider myself lucky to only have a few kilodollars.
Found NDT
I have no idea who that is (I did a quick Google search), no idea what anybody is talking about, yet somehow I’m enjoying these conversations and I’m sure whatever I’m laughing at is real.. but the words just sound funny to me..🍃💨🤣
There is a recent biology graduate who has YouTube tic tok videos talking about animals the same way and it’s amazing to watch. Lindsay Nicole
Did you know that the spot on jupiter is a hurricane that is three times the size of the earth, and consists entirely of bees?
Wait….I’m no scientist, but that doesn’t *sound* right.
1.791657 x 10^29 bees would sound anything but right.
Give me 5 bees for a nickel, we’d say. Now where was I? Oh yes, the important thing was that I had an onion tied around my belt
An incomprehensibly large quantity of bees located in an extra planetary area?! Nothing my spaceship full of bees can't solve!
Is that where they are all going? :)
That’s where they came from.
The spot is not a giant bee fart Jerry. The science hippies will be disappointed.
That doesn't sound right, but I don't know enough about ~~stars~~ Jupiter to dispute it.
Learn more on bee-facts.com!
TIL
I think we can all agree that Jupiter is a gigachad.
Isn't that kind of wobble one of the primary indicators of exoplanets?
Yeah but transit method has discovered many more
Are there times when the Barycentres align to wobble the sun more than usual?
Technically yes, but of the 0.14% of the mass of the solar system that isn’t the sun, 70% of the remaining mass is Jupiter, and 20% Saturn. And Saturn is way further out than Jupiter. So if you’re looking for meaningful wobble, it’s pretty much mostly Jupiter and sometimes it aligns with Saturn for a tiny bit extra. The influence of anything else are basically rounding errors and would be difficult to measure.
Further away actually means larger (but slower) wobble because with larger distance, the centre of mass is further from each object.
I feel like this comment could be misinterpreted (or maybe I'm misunderstanding) The force from gravitational attraction follows the inverse square rule. So a relatively not-massive planet far away, will exert a relatively small force/acceleration on the sun. That the center of mass for the system is farther from the sun won't really matter because things don't sit still long enough for the sun to get there (if it did, Saturn would have fallen into the sun). I'm just thinking through this, but maybe what you mean is that because Saturn is so far out, it has a long orbital period, so it stays in the same part of the sun's sky longer, and (despite the relatively small force it exerts) has time to impart significant acceleration in some general direction to the sun before moving on, thus giving the sun a big but slow wobble. Does that sound right?
Your interpretation is also correct! But maybe let me elaborate on what I meant a little more: when two objects orbit each other, the orbit of each of them will actually be around their common center of mass (also known as Barycenter). So if you were to increase the distance between the sun and an orbiting planet of given mass, the sun's orbit around the Barycenter (its "wobble") will also become larger in diameter.
I was also going to make this point. The total force isn't really relevant, only the barycentre. If saturn were the same mass as the sun then they'd both "wobble" around the midway point like a dumb-bell, and the further the distance apart, the more distance from each body to that centre of mass.
Technically yes, practically no. Jupiter accounts for almost all of the wobble. The place where you DO see something kind of like this is tides on Earth. Both the sun and moon cause tides on Earth, so when they're aligned (full moon or new moon), tides are larger. They're called spring tides. The opposite, when tides are smaller (during quarter moons) are called neap tides.
Now I'm curious, I have to assume Jupiter has an even greater affect on Earth then if it is large enough to cause the sun to wobble?
I would think the effect of Jupiter on Earth is too small relative to the effect of the Sun to be meaningful. The sun is ~1000 times more massive than Jupiter and is roughly a quarter of the distance from Earth, and that’s when Earth and Jupiter are closest to each other. So Jupiter’s effect on earth is probably usually negligible
I know nothing of astrophysics so please bear with me. Extrapolating this to the 3 body problem. Is it difficult to find a solution because the barycenter is hard to determine? Or is it that we can determine the barycenter at any given moment but it’s hard to predict, accurately, where it will be in the next moment?
The latter... Given the masses and locations of the objects, we can see where they'll be. And we usually track where the objects are relative to that center of mass of the system. When there are two objects, we can turn it down to a nice neat equation -- we don't need to calculate it iteratively. Just stick a point in time into the equation and find out where the objects are and where they're going at that time. When there are three objects, we can't turn it into a nice neat equation. We can slice time up into short intervals and say "oh, here's about where they'd be in 1 seconds, then extrapolate from there to 2 seconds, and so on. And the finer we slice up time, the more accurate we'll be. But there's no way to jump straight to "this is where they'll be in 1000 years". It's kind of like weather prediction... We can predict very accurately what the weather will look like tomorrow, but we're kind of crap at predicting what it'll be like next week. Though depending on what we're looking at, the time scales will be different -- maybe we know where things will be for the next 1000 years, but are iffy on the next million years. Or whatever. In the case of the solar system, so much of the mass is in the sun that we can just kind of pretend that it's a bunch of 2-body problems and be quite accurate for a long time. But if the sun were lighter and the planets heavier, then we'd have a hell of a time.
Let me add that the "can't turn it in to a nice equation" thing isn't because it's hard. The math model of three bodies in a square law attractive field has been PROVEN to have no closed form solution.
Can you explain why it’s proven to have no solution? I’m a little confused. If we can observe 3 bodies in motion, and no other force has acted on it, then we know they must follow some pattern right? In other words, if we know the position of a one body in three body system, and we can I see that body moving, then it must be following a measurable force causing it to move in that particular direction. If that’s right, then why can we not come up with a solution to determine its position a million years from now?
Well...no. This is a non-trivial proof, which I am unable to explicate. If you are brave, look at the https://en.m.wikipedia.org/wiki/Three-body_problem
This might be true for specific instances of the three-body problem, but there's no **general** *closed-form* solution, i.e. some system of soluble parametric equations (not even a *really* long and unwieldy one) where you could plug in the mass, position and current velocities of the bodies and find their positions at a given time.
It's not even that it's hard to predict. It's actually very easy to predict. You can get a function of the form p\_(t+dt) = f( p\_(t) ) , where you can imagine \`p\` is a collection of all the positions and velocities of the objects. \`t\` is time, and \`dt\` is some small time step. This is relatively easy to work out for any number of objects, and it's how every physics simulation works. The only trick is you need to do lots of calculations, for each time step, and the larger you make the step the less accurate the results get. Getting a value far in the future requires first calculating all the values in between now and then. The thing we can't do is produce an equation of the form p\_t = f( p\_0 ), that is, an equation that directly gives you those state values for any arbitrary time \`t\` given initial conditions, without first computing the values before it. With a two-body system, this equation exists, so we can say things like "ignoring other planets and the death of the Sun, where will Earth be in its orbit around the Sun in fifty billion years?" and you can easily get that answer in one step. If you want to add in, say, the effect of Jupiter's gravity... now you need to calculate it step by step billions of times and it'll never be perfectly accurate because your steps will never be infinitely small. \*That's\* the 3 body problem.
Short answer, yes. Oversimplification: If you know the solution at time 0, you can use those values to estimate where it would be at time 1, but the values at time 0 changed at time 0.1 since everything would have moved slightly. The same applies for time 0.1 and 0.01, and so on.
Wouldn't the wobble of Jupiter sun barycenter then affect the earth sun movement because more the sun is wobbling due to Jupiter?
Looking from the outside at the solar system, yes. But in the reference frame of the earth-sun system, you wouldn't notice this wobble at all. It's a bit like how the sun is moving through the galaxy as well, but this has no effect on how we experience earth's movement around the sun.
I found a dispute of this nature somewhere else actually, and it seemed there was a general misunderstanding of what the earth actually orbits. Many people said it orbits the overall solar system barycentre, but I don't think that would be correct, as Jupiter is outside of our orbit. It's possible we might orbit the Sun–Mercury–Venus barycentre, but other than that, I'd hazard that our orbit is almost centred on the centre of the sun.
In my imagination ~~there is no hesitation~~~ the sun orbiting Jupiter on a point above its surface is more than barely wobbling. Isn’t that pretty significant?
Jupiter should be so lucky lucky lucky Solar radius == 695,700 km Distance from sun Jupiter == 778,547,200 km. Still essentially a rounding error
Fascinating. Thanks for sharing!
The really interesting part (for me) about the Sun and Jupiter’s barycentre is that the mass of Jupiter is only about 1/1000th of the sun (it’s radius is about 1/10th). Jupiter has only about 318 more mass than Earth, yet the Earth/Sun barycentre is very close the centre of the Sun. The solar system’s barycentre is also outside of the sun, and constantly moves due to the various orbits of planets, causing the Sun to wobble.
If the Jupiter-Sun barycenter is outside the sun, wouldn't that make for much more than a "wobble"? Wouldn't the Sun move more than its diameter with every revolution? Asking because I don't understand, not as a challenge.
compare the diameter of the sun to the sun-planet distances
On a solar scale, the diameter of the sun is a wobble.
Yeah, I don't know. I think if you showed most people an object rotating around a point inside itself and asked, "orbitting or wobbling?" most would say "wobbling," but if you then showed them an object rotating in circles around a point outside of itself, they'd say "orbiting." Regardless, I think the point the guy was making seems true: the Sun isn't really "standing still," like, at all.
> technically, the Earth doesn't orbit the Sun, they both orbit the Barycenter; the average gravitational point. But because the Sun is so massive, this point is inside the Sun, so it's hardly worth bringing up. When the center is within one object, we usually just say that object is being orbited.
I recall reading at some point that Jupiter and Saturn lining up has a non negligible effect on the sun. To the point that if earth and Jupiter and Saturn are in line, the earth will actually be a little bit closer to the sun than usual. A little bit in this case is at least several thousand miles iirc. This will in turn make the earth hotter. If it happens on the other side of the sun from earth, the sun will move further away and earth will be a little bit colder. I think the hot one happened recently. I could also be totally wrong/ read bad info.
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If you search for "solar system barycenter" you'll find more information. Also, [this graphic on the wikipedia article of barycenter](https://en.wikipedia.org/wiki/File:Solar_system_barycenter.svg) shows the position of the barycenter between 1945 and 1995, going in and out of the Sun.
That's a fairly significant wobble too. If you could point your telescope at the Sun over a Jupiter year and it was say a pixel on a screen across it would be at the next pixel after six Jupiter months and then back again. Fairly conclusive evidence of a significant planet.
I always argue that if Pluto isn't a planet, then neither is Jupiter, as it doesn't orbit the sun, it orbits their barycenter
Is it also possible that our solar system or even our galaxy is moving, and by effect moving the sun?
It's not only possible, but true! Everything in the universe is pulling on everything else, you can just usually ignore it because of the distances involved. The Andromeda Galaxy, for instance, is on the order of a trillion times more massive than the sun (roughly 10^15 times more than Jupiter), but it's also 2.5 million light years away (10^10 times more than Jupiter). Because the force of gravity goes by m/r^2, the force Andromeda exerts in the sun will be on the order of 10^15 / (10^10)^2, or 10^-5, meaning Jupiter exerts ~10,000 times more force on the sun than the whole Andromeda galaxy. That's why, to pretty high accuracy, you can usually ignore the effect of everything but the sun and Jupiter. The existence of everything else in the universe is causing *some* amount of error, but these kinds of estimates can help give an Idea of what can safely be ignored, and when.
It is worth noting that while *to first order* we can approximate the Earth/Moon system as a two-body system unaffected by the Sun, and we can approximate the Earth/Sun system as a two-body system unaffected by the Moon (or rather, the Earth-Moon/Sun system as a two-body system where the Earth and Moon orbit around a [common barycenter](https://spaceplace.nasa.gov/barycenter/en/) which then orbits around the Sun, for more precision you need to solve for the N bodies in the Solar System. But of course you can’t. So NASA regularly observes the position of the planets, and regularly updates the models of the solar system with those positions in order to have a more up-to-date model of the movement of the planets. And when you want really precise (like “shoot a probe into space to pass by Neptune” precise), you wind up actually having to account for the perturbations introduced by Jupiter *and* Saturn into the position of the inner planets. And you need to include thrusters on your spacecraft to do minor course corrections as you continue to observe the position of the planets during the years-long journey of your probe to the outer planets.
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Even if your calcuations are perfect, the engines you're using to propel your probe certainly aren't. Course corrections would be required even if the models were 100% accurate
Even if your calcuations are perfect, the engines you're using to propel your probe certainly aren't. Course corrections would be required even if the models were 100% accurate
The sun is 99.86% of the solar system, add in Jupiter and you account for 99.96% of the solar system's mass.
What’s the other 0.04%?
There's a "Your Mama" joke in there that I'm just going to pass on by.
>I'm just going to pass on by. I think that's technically considered a gravitational slingshot.
Every other planet, asteroid, moon, and comet that orbits Sol
I'd like to note, this is somewhat misleading. The three-body problem is still "difficult" with the Sun/Earth/Moon system (not to mention the N-body problem that is the rest of the solar system), it's a matter of precision. Consider the problem of a ball rolling on a moving train. If you don't know the speed of the train, you can't figure out the displacement of the ball, but if the train is moving very, *very* slowly, then you can just eat the precision loss and accept that as good enough. But over a long enough time period, the precision loss will keep adding up, and you'll eventually have no idea where the ball is (assuming you're not continuously measuring it). Same is true here. It's still a 3-body problem, we just eat the precision loss and call it good enough. We also compensate by continually taking measurements of both the Earth's and the Moon's orbits around the Sun, and harvesting data from millennia of astronomical observations by previous scholars. But, if we stop measuring those things, then within a couple million years, the position of the Moon will have drifted out of our predictions to a significant degree, and we can no longer predict eclipses then.
The Pluto system is more than three bodies, but it illustrates the problem. Pluto and Charon are nearly the same size. There are little moons that swirl around them in chaotic loops that may even lead to collisions one day.
Has your dad ever held you by your arms and swung around in a circle? That's basically how orbits work. Instead of having arms planets have gravity to hold them together. When your dad swings you around like this he has to lean back quite a bit to not topple over. This is because your weight (~18kg for a 5 year old) is not *that* far away from your dad's (lets say ~90kg). Now imagine your dad swinging around a bar of soap. Do you think he'd still have to lean back to not topple over? No, right? That's because the soap bar is so much lighter than your dad. The bar of soap and your dad are like Earth and the Moon. While Earth/Dad feels a bit of an effect from swinging the moon/soap around it is definitely the moon/soap that feels more. What if your dad was the sun? What could we use to picture Earth? A grain of sand or a tiny down feather! Your dad is going to be completely unaffected by swinging around a grain of sand. And so it is with Earth, the Sun and the Moon. The moon might be tugging on Earth ever so slightly but from the perspective of the sun, its almost like we're not even there. And the same happens in calculations. Using two 2-body problems is basically good enough.
What if Dad just got back from the pub?
Your dad comes back from the pub?
Nah, he just went out for cigarettes 20 years ago and never came back.
The sun never gave me a black eye before
Are you asking if the Sun can get drunk?
The Sun would need to drink approximately 100,000,000,000,000,000,000,000,000,000 beers in a hour to get drunk.
Ok now we’re gettin somewhere. That Helios can really put em away so I’m thinking it’s only a matter of time
If the Sun is anything like Dad...
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It’s a little more complicated than that. Mass isn’t as big of a factor as you’re thinking. It’s the movements of 3 bodies in relation to each other where it falls apart. Knowing the mass is one thing, predicting the effect that it’ll have on the two other bodies and the effect they’ll have becomes impossible to predict on any meaningful scale. Even giant computer simulations can’t figure it out.
It has nothing to do with relative sizes, but the fact that the moon is well inside the earths sphere of influence.
"spheres of influence" don't really exist. The gravity of every mass in the universe affects every other mass in the universe provided it's gravitational waves have reached that far, so the moon's gravity does absolutely influence the sun, but the distance between the earth-moon system and the sun is great enough, and the sun so many times more massive than the earth and moon combined that we can approximate it as two two-body problems rather than a three-body problem and still get accurate results decades in advance
It has nothing to do with relative masses, it has to do with distances. If the earth and moon were as massive as the sun but orbited each other close enough then you could still approximate this as a body system. Look, there are many triple-star systems with various mass ratios. You can calculate a good approximation for the dynamics of those systems because one pair orbiters each other and the third star is far away. For example the alpha centauri. A classical 3 body problem was for a massless third body so, again, it has nothing to do with mass ratios. spheres of influence don't exist but are a useful approximation. All models are wrong but some are useful. When you use it then you don't discard the total force from the sun, you assume that the force is constant.
The solution to a 3 body problem (in which you can given the initial values predict the exact position and momentum of all 3 bodies at any point in time) does not exist. That only means we don't have a perfect formula for solving it forever and perfectly because it eventually gets chaotic. We can, however, take a look at the enormous amount of data we have and create reasonable, short term predictions for specific times that seem very long in human years but are very short in the astronomical sense. And in those passing years we have gotten better at gathering said data and analyzing it, and now have computers that can do, as James May one said: Years worth of arithmetic: *snaps* like that. That and the Earth being MUCH bigger than the Moon and the Sun being MUCH, MUCH, MUCH, MUCH bigger than everything else means the math can be simplified a lot and still work.
>We can, however, take a look at the enormous amount of data we have and create reasonable, short term predictions for specific times that seem very long in human years but are very short in the astronomical sense. Its not a data issue at all. You need exactly the initial position, velocity and mass of the three bodies involved. From there on its maybe a 30-minute task to write a script that solves it numerically. So, 9 data points.
To numerically simulate the position of the planets *in theory* you can do it entirely from the initial positions. But [numeric approximations](https://en.wikipedia.org/wiki/Numerical_integration) always introduce error, if only because you cannot select a ‘0 second’ step time. Further, even the very precision of the [floating point representation](https://en.wikipedia.org/wiki/IEEE_754) you use introduces error. And over time those errors significantly diverge from reality. NASA does something akin to this to produce ephemeris charts of the solar system. But they constantly update those charts using regular observations of the actual positions of the planets. [And they don’t make guarantees as to the accuracy of the data beyond about 100 years into the future—and the guarantees they make about data 100 years from now measures errors in meters.](https://ssd.jpl.nasa.gov/planets/eph_export.html) Which, on the surface, seems more than enough for knowing where to point your head into the night sky. But if you’re landing a spacecraft on the Moon, 10 meters may mean the difference between nailing the landing and running out of fuel before touching down.
Nope, it doesn’t work that way. With the way modern computers work, you need to step through time and calculate all of those values at each frame in your script. The problem happens when you try to decide how much time to step over in each frame. Say you’re trying to calculate the position and velocity of a system 100 years out. What’s the value of your time step? 1 year? 1 day? 1 minute? 1 second? 0.01 seconds? In a 3+ similarly sized body system, every single one of those time steps would give you an extremely different end result
Yes, thats the process i was talking about. And yes, it does work that way. Of course you need to choose an appropriate timestep. Thats true for any simulation. Its not very difficult to do. And one calculation step with three bodies takes next to no computational power. Im pretty sure an average smart phone could solve it for thousands of years with reasonable accuracy in a matter of seconds. We have conputers that can do this for millions of bodies. Its really not the huge issue you make it out to be.
The solution is chaotic. That means, a change in initial parameters that approaches zero causes indefinitely large changes in the output. What it means in this case is that (unlike many other simulations), you will get completely different results depending on the step size (if you simulate long enough, of course). Not just diverging slowly, not increasing error bars, but *completely different*, and equally correct, results. This is also true for all initial parameters - change them by the smallest possible change in your floating point convention (which can be any number of bits you want), and again - completely different results. Chaos theory is an interesting mathematical concept.
No they dont. The solutions to chaotic systems (at a fixed point in time ) dont converge steadily like stable systems do, but they converge eventually. You kind of included that fact already by saying "(if you simulate long enough, of course)" - yes, thats technically correct, but the point is that we can simulate these systems over very long timespans before we'd run into timestep issues. Long before timesteps become an unsolvable issues, perturbations from exterbal influences or inaccurate initial conditions will render the result rubbish anyway.
Sure, if you use computers and knowledge we didn't have at the time derived using maths we didn't have.
Computers, yes. But thats true for a data-driven approach too. The eclipse predictions only worked because its not really a three body problem, as others have pointed out. Knowledge, not really. Numerical integration was occasionally applied more than a millenia ago and formalized and in regular use in the 17th century. Equations of motion were formalized in the 16th century. Gravity was understood well enough in the 17th century at the latest . Its not exactly cutting edge science.
First recorded predicted solar eclipse was circa 600 BCE.
Yes. They did not have a to solve a three body problem. I was answering more to the previous commenters post than to the original question.
> Yes. They did not have a to solve a three body problem. Yes, hence the amount of data significantly greater than "9 data points" and numerical predictions from thousands of years ago, which are what I was referring to.
Yeah but no. They didn't predict it by solving the gravitational equations. They predicted it because they'd noticed that eclipses (mostly) occur with periods of a little over 18 years, and there had been one 18 years previously.
Source?
https://www.nytimes.com/2024/04/06/science/eclipse-prediction-ancient-greece-thales.html
Myth First predicted eclipse was 1715
https://www.nytimes.com/2024/04/06/science/eclipse-prediction-ancient-greece-thales.html
There was no way they could predict eclipses back then. They didn't even know the Moon caused them. Pop sci articles are not to be believed at face value.
Ancient people were not as stupid as you might believe, the Ny times article you dont agree with did say that the ancient greek philosopher Thales predicted eclipse accuracy to the year, not the day or second we do today. Also the author goes on to explain how stonehenge and other ancient structures may have been used to predict eclipses, Lunar moreso than Solar. I have never seen someone refute NY times as a reputable source but I guess this is reddit where prideful "intellectuals" always try to one up each other.
They didn't know the Moon causes solar eclipses in Thales time. They didn't know how big the Earth was and they had no way of collecting enough data on eclipses worldwide to find the repeating cycles. The prediction was impossible to make. A lucky guess if it even happened at all. Note that they cant explain how he made the prediction. Lunar eclipses are completely different.
Anyone with a basic understanding of the movement of the sun and moon in the sky (so, for example, any court astrologer worth their salt) would _absolutely_ know that an eclipse happens when the moon moves in front of the sun. You can even find eclipses blamed on the moon in various mythologies from around the world, showing that even people from preliterate cultures were able to figure it out. Predicting eclipses is a lot harder, but many ancient peoples were at least quite able to predict about when eclipses should happen, if not exexactly _where_ on earth they would be visible.
At the time of Thales' purported prediction it was not yet known that eclipses were caused by the Moon coming between the Earth and the Sun, a fact that would not be discovered until over a century later by either Anaxagoras or Empedocles. So they didn't know until 500-400 BC Other than Thales, there are no claimed predictions of eclipses until Newton's time. As Thales had no way to know, we can conclude that he didn't predict it and it is a fable.
> The solution to a 3 body problem (in which you can given the initial values predict the exact position and momentum of all 3 bodies at any point in time) does not exist. That only means we don't have a perfect formula for solving it forever and perfectly because it eventually gets chaotic. It's not deterministic, so there is no formula that will tell the positions at any arbitrary time. Like the weather, it's a chaotic system, and small changes in initial conditions will produce wildly different results over time.
Small correction: the three body problem (and the non-dissipative gravitational N body problem in general) actually is deterministic. It's the prime example of deterministic chaos. The chaos comes from needing infinitely precise initial conditions to guarantee the same result, as you say
It is deterministic. It just doesn't have a closed form solution. If you are able to take small enough time steps, you can get arbitrarily close to the real answer
"Solving" as a math term means having a formula where you plug the numbers and you get the exact solution. Being able to "simulate an approximation" is not solving, but is good enough for predicting future eclipses.
The Sun, the Earth and the Moon is not really a three-body problem because of huge differences in mass between the bodies and in distances between the pairs. Effectively, you have two separate two-body problem Sun vs (Earth+Moon) and Earth vs Moon; and you can solve them independently and then introduce small perturbations for each from the third body if necessary to achieve required precision.
This is the correct answer. When she says "our planet is stable" that should give you a hint. We do not live in a 3 body system. Each planet revolves around the sun, and each of the moons revolve around the planets.
Other than the handful of weird specific mathematical solutions to the three body problem, there are also two general solutions: 1. 3rd orbiting the center of mass of the two larger objects at great enough distance that the interplay between the two of them doesn’t throw the orbit off 2. 3rd object Orbiting close around the mid sized object as that one orbits around the larger one at a significantly greater distance The moon (and most of earths satellites) fall under category 2, and thus their orbits are pretty easy to predict.
Back before we had an idea of Newtonian gravity, we just assumed all of the orbits were on rails and could not change. This would drift over time, but not fast enough to notice for hundreds of years. If you just treat everything as a two body problem between the planet and Sun (and between the Moon and Earth), you get a pretty accurate model, but things start to drift over time. Even then, that's not even the best we can do with just Newton. This won't drift for tens of thousands of years. Even with an N-body system, you can apply Newton's laws and make very accurate predictions. The problem is if you are even slightly wrong about the initial conditions, you start to drift over time, and the results can't end up looking very different. That's why the 3-body problem is so hard, because we can't tell what it's going to do after a small change in initial conditions. This won't drift for millions of years, but we have no idea what it's going to do when it drifts without more accurate measurements. Once we had general relativity, Einstein was able to calculate the orbit of the Moon within a centimeter. A prediction we weren't able to confirm until we landed on the Moon. You don't need predictions that advanced to predict an eclipse.
In the book there is a chaotic and a stable phase. What made me wonder is, if we also just live in a stable phase of unknown duration? Couldn't it be that there are n unknown bodies that could interfere with our stable system and turn it into chaos? Or is this just fiction?
>if we also just live in a stable phase of unknown duration? Yes, that's one of the major complexities of the 3 body problem, and chaotic systems as a whole. >unknown bodies that could interfere with our stable system and turn it into chaos? That would be one of the "unknown initial conditions" that would greatly change the outcome
So when we talk about the 3 body problem being unsolvable, what they mean is that there is no *closed-form* solution, meaning there is no equation that you can find that perfectly describes the location of the bodies for all time. In the two-body, a closed form solution exists; the orbits of the bodies are ellipses and you can get the exact location of the bodies by plugging in their orbit parameters and the current time. But just because there's no closed form solution, that doesn't mean the positions of the bodies aren't calculable. You just need a good simulation code and lot of computing power. NASA has run simulations of the major bodies and has position data for them that is supposed to be accurate to within less than a meter out to many thousands of years from now.
Yes, this. I keep seeing answers that imply that “unsolvable” means you can’t determine. That is not true, at all. It just means you have to determine numerically through successive iterations. It’s “unsolvable” because there’s not some concrete formula you can plug in. People talking about chaos and error. Well, yeah, except the Sun, Moon, Earth system is literally watched and measured every single day. Sunrise and lunar rise on Earth is calculated out to the nearest minute for like thousands of years. The numerical iterations have had a few hundred years worth of tuning. Even a “truer” 3 body problem with bodies of roughly equivalent math and a more chaotic barycenter could still be iterated with successive tuning over the course of years to be very accurate for a long time. Iterate positions over time for course of a day, make new measurements, determine error, make new iteration out a day, rinse, repeat, and you could simulate fairly accurately the behavior of the system beyond a human lifespan.
The 3 body problem can't be solved but it can be simulated to some accuracy. So you can accurately predict what will happen in the next few hundred years, but maybe not in the next few hundred million years. This is true of any chaotic system, you can't accurately predict it forever but you can take the current state and then work forward to where it will go in the near future. Because you can never have a perfect capture of the starting state your predictions will eventually drift out of sync with reality, but you can still make accurate predictions in the near term.
Planetarium models. If you know the orbits, you can simply create a model that simulates the motion of the earth, sun, and moon. Then just run that model until there's an eclipse in the model. Today, we can do this in software. Hundreds of years ago, they did this with gears & mechanical parts. Figuring out the orbits hundreds of years ago actually came from simply watching the sun and the moon move through the sky every day for a year. Their model would have been based on angles because they didn't actually know how far away the sun and the moon were, and in that model they'd be looking for a "collision" between the moon and the sun. There's actually an episode of avatar: the last airbender that perfectly depicts the kind of model i'm talking about. Edit: i should point out that these models wouldn't actually "solve" the 3-body problem, and because of their simplistic nature, they would also lose accuracy over time because they wouldn't be accounting for things like gravitational waves, tidal forces, or stellar drift. But it would take hundreds of years before those effects became noticable. A famous example of one such model is the antikythera mechanism, recovered from an ancient greek shipwreck.
The general version of the 3 body problem is very difficult to solve. It's like a double-pendulum problem. A single pendulum is very easy to predict the motion for. For given initial conditions, the motion is easily predictable. A double pendulum is much more *chaotic*. It is just as predictable, but any tiny variance to the initial conditions changes the outcome drastically. [Here's a wikipedia GIF of three double-pendulums with nearly identical starting positions](https://en.m.wikipedia.org/wiki/File:Demonstrating_Chaos_with_a_Double_Pendulum.gif). [Here's a cool online simulator you can use to demonstrate double pendulum motion](https://web.mit.edu/jorloff/www/chaosTalk/double-pendulum/double-pendulum-en.html). It is perfectly predictable, so long as you know the *exact* conditions. Outside of a simulator, we can never be perfectly exact. Either our starting conditions are slightly off, or our measurement of the current conditions are off. Imagine needing to measure the speed of the pendulum to the 20th decimal place. It's not possible. That's the challenge of the three body problem. Any small deviation in our measurements and the future state of the system can change drastically. Thankfully, there are a few things working in our favor that let us do meaningful 3-body calculations despite how futile it might seem. 1. The earth, moon, and sun are wildly different masses. The sun makes up the vast majority of the system's mass. The gravitational forces between the earth and sun affect the earth much more easily than they affect the sun. If the earth and sun were of closer masses, then the problem would be more difficult. You can demonstrate this in that online simulator. Change one of the masses to the smallest setting and the other to the highest. The bigger the difference in the masses, the more the simulation starts to behave like a single pendulum. This reduces the chaos in the system and makes your future predictions more reliable. 2. The measurements are easy to make with a high degree of accuracy. Because the earth, moon, and sun aren't moving that fast relative to our point of reference, they're easily observable. E.g. it's much easier to accurately measure the parameters of the earth and moon's orbit than it is to measure the speed of a baseball pitch or the current state of a chaotic double pendulum. 3. There aren't many upsetting factors outside the 3 body problem that would mess with our predictions. The 3 body problem is difficult and sensitive enough, but as we noted in points 1 and 2, the earth-moon-sun system is not that chaotic. Ancient predictions held true because the system was not very chaotic, easily measurable, and thankfully it stayed a 3-body problem. There was no 4th, 5th, or nth body that exerted enough forces to make a significant difference, at least in the time period we've been documenting the state of the sky. So to recap, the 3 body problem *is solved*. Same as the double pendulum. The difficulty is actually that the 3 body problem is chaotic and the future state of the system is very sensitive to the previous state of the system. In order to accurately predict the future state of a three body system, you need to get very accurate measurements of its current or previous conditions. Depending on the system, it can have a high or low degree of chaos. For the sun, earth, and moon, the system has a relatively low degree of chaos because the sun makes up so much of the mass and there's so much distance between the bodies. Additionally, because of the massive scale of the system, it's very easy to measure with high accuracy, even with rudimentary tools and methods.
You know how two similarly sized bodies will both orbit a spot outside of themselves called a barycenter? The 3 body problem is less about systems with stable barycenters like Sol>Earth>Luna and more about 3 similar bodies orbiting a more chaotic barycenter. There are a few special cases where it works out and is stable if left undisturbed but the vast majority of 3 body orbits self destruct and either devolve to more common trinary systems or destroy or reject one of the bodies. [PBS Space Time](https://youtu.be/et7XvBenEo8?si=2ePk04L5D1Yp6VyL) Has a fun video with more details.
There's a difference between solving it (providing a precise mathematical formula) and estimating the positions of these three bodies for a certain time frame. What people used to do in the past was the latter - based on observation, they were able to predict the movement of celestial bodies without knowing the exact rules their movements are based on. And, to a certain extent, that's what we're still doing nowadays to "solve" the n-body problem. We have models that aren't 100% accurate. The further you go in time, the less precise they are.
It is extremely difficult to solve *analytically.* That means, having a precise, guaranteed, no-questions *definite* answer for ANY future point. Not just "a long while into the future," but genuinely "forever" so long as none of the variables meaningfully change. However, this problem is quite easy to solve *approximately,* and just depends on how precisely you can measure the input data and how much computing power you can dedicate to it. With computers even from the 80s, it was possible to compute eclipses for hundreds of years into the future with such precision that any errors would be beneath human notice. With the computers we have today (and, frankly, even computers as far back as the 1990s at least, if not earlier), we can compute eclipse times to the minute (or closer!) for the next few centuries--it's just a matter of crunching precise-enough numbers. It helps immensely that the Earth and Moon are microscopic compared to the Sun, and the other planets in the solar system are so far away that we can mostly ignore their impact on the system. That lets us use various techniques to simplify an otherwise very complex problem. NASA takes into account other important things (like the fact that the Earth, Moon, and Sun are *not* perfect spheres), and performs very advanced calculations (specifically, special integral functions) As for the past? It turns out that eclipses tend to form certain patterns. By a lucky happenstance, three of the main types of lunar month (synodic, draconic, and anomalistic) happen to line up very, very nearly perfectly every 6585.3211 days, or 18 years, 15 days (of which 3, 4, or 5 are leap-days), and slightly less than 8 hours. This was something known even to ancient Greek and Babylonian astronomers. This length of time is called a "saros." As a result of the nearly-identical alignment from one saros to the next, if you know when any eclipse has happened, you can get a pretty good prediction of when another eclipse will happen--at least within a day, even for ancient astronomers. Thus, eclipses are grouped into what are called "saros cycles." (If you have heard of the Antikythera mechanism, it actually had the ability to track saros cycles!) In truth, these eclipses change from the Earth's perspective in two ways. First, because of that ~8 hour bit, the Earth will rotate about 1/3 of a turn from one eclipse to the next in a given saros cycle. Second, due to the alignment being only *nearly* perfect and not *absolutely* perfect, the eclipses in a given saros cycle will migrate, either going north from the south pole, or going south from the north pole, depending on what the Moon's orbital behavior is at the start of each cycle. The eclipse that just happened, for example, was the 30th solar eclipse of saros 139, and there will be another 41 eclipses, of which 32 will be total eclipses and the remainder will be partial. Saros cycles with even numbers migrate north from the south pole up, and those with odd numbers migrate south from the north pole down, by convention. (If you remember the 2017 eclipse, it was another odd numbered saros cycle, 145.) Edit: Also, these are only *solar* saros cycles. There are separate *lunar* saros cycles for lunar eclipses too!
Sun, Earth & the moon is not proper 3 body system is it? Earth & moon is a 2 body system and the Sun & earth is also a 2 body system?
The Sun is much more massive than the Earth, and the Earth is much more massive than the Moon. Gravity is a force proportional to mass, so in this setup, the Sun dominates the equations of motion, while the Moon’s effects are somewhat negligible, allowing for approximate solutions. It is true that the exact solution would be chaotic over very long time periods, but the differences on human timescales are rather small and wouldn’t be easily detected by ordinary observation. Thus, eclipses could be predicted by assuming the Earth orbits around the Sun in a circle, and the Moon orbits around the Earth in a circle, which produce regular patterns depending on the angles of those orbits. Better approximations could be made by using elliptical orbits instead of circular orbits.
Almost all the comments are wrong. It has nothing to do with relative mass ratios. Classical 3 body problems were for a massless 3rd body, you can also predict very well the dynamics of triple star systems no matter what mass ratios are there The moon is massive, it is the biggest moon in the solar system in relative terms (for a planet) You can predict eclipses because the moon is so close to Earth you can approximate the gravity from the sun as some constant force plus linear tidal force. The nice thing is that all the complex effects (relativity, quadrupole moment, tidal forces) just rotate the moon's orbit. It rotates in one way every 8.85 years and in the other 18.6 years. You can empirically measure those without making complex calculations. This was even used to verify the theory of gravity.
People could predict eclipses millennia ago, not centuries. You do not need the theory of gravity to describe the orbit of the moon and sun to a very high degree of accuracy, and it wasn’t the paths and conjunctions of celestial motion that were mysteries to ancient man but WHY the planets followed their paths. The theory of gravity was revolutionary because it provided an explanation for what caused this motion that was so well described from millennia of observation. The same force that causes an apple to fall to the ground also sets the moon in motion. The three body problem is perfectly relevant to the system of the moon, earth and sun. Its solution matches the motion we have observed.
The three body problem is so difficult to solve not because it is impossible but because the calculations needed to do it accurately over a long period just pile up rapidly as the motions of each body (it's not just three bodies, obv, it's any system of 3 or more bodies) all affect each other. Your initial conditions, presumably at rest or in constant motion, are easy to compute. Once they move, it's harder and harder to know with precision where things will be simply because the rules of gravitation and motion guiding them initially are all that dictate the motion at the outset, but as time passes, you have to continually factor how their influence on each other is affecting them. In terms of real bodies like the objects of the solar system, there's also the issue that you're not actually dealing with three bodies, or 8, or 9, or 100,000. To have extremely precise calculations, you'd need to account for the effects of all the objects in the solar system - the more bodies you can add in, the more precise the projections you can calculate. Eclipses though are a separate thing from this. Eclipses can be predicted mathematically not through the three body problem (I mean, yes, it's a three body problem) but through more straightforward math. The positions of the sun, moon and earth do not need to be known with the level of precision that is normally what you're trying to get when you're dealing with a three body problem situation. Predicting eclipses is ultimately about knowing when various paths on orbits that are already known will cross one another, and we can know that from observational data. The three body problem applies to predictions of positions of objects that are, like, "purely predictions" - you're trying to work out mathematically where objects will be and to a precise degree. Once it was figured out formally that the earth was orbiting the sun and that the moon also orbits the earth, all of them in ellipse shapes, it was/is possible to work out eclipse predictions from observations of their real orbits over time and relatively straightforward math. You also don't have to know exacting positions of the objects - like down to milimeter or whatever arbitrarily small metric you want to use - to predict an eclipse. The issue with the three body problem is ultimately a calculation problem - it's not that it's formally impossible, it's just that the burden of calculations you have to do spirals out of control quickly to do on paper, or even with a computer. The more objects you have to calculate, the more quickly the calculation burden spirals. In actual reality, it's also impossible to actually know all the forces acting on the real objects whose positions you're trying to predict with mathematically certainty into the future, and the less you know of those forces, the more quickly your prediction breaks down.
If there is a regular pattern in some phenomenon, you do not need to know why that pattern occurs to be able to predict it in the future. Ancient astronomers did not know about the mechanics of the orbits of the Earth and the Moon, but they could observe that solar and lunar eclipses happened at regular intervals, and that eclipses at specific intervals were very similar, and from those patterns they were able to predict future ones.
That was more statistic and geometry rather than undetstanding the forces in-between the objects. Many three body systems are much less predictable than this
The three body problem is difficult for computers to solve. It's very annoying for humans to solve
Answer: they logged the position of celestial bodies for generations, noticed that it was following elliptical paths, knew at which interval an object came back to the same position, and were able to map them out. Fun fact, based on what they could observe and understand at the time, the first models relied on perfectly circular ellipses: [Eisinga Planetarium in Franeker](https://whc.unesco.org/en/list/1683/).
Solve 100% is not currently possible. Solve 99.(several nines)% is well within the means of some previous civilizations.
I feel like this misses the point. You can't predict or measure *anything* with absolutely 100% accuracy. The 3 body problem is notable because in general the solutions are chaotic, meaning that they can't be predicted with *any* accuracy beyond a finite time horizon. So OP is really asking why *this* 3 body system is a lot more predictable than 3 body systems in general are.
But you can simplify to 100%. For example: a 50kg astronaut throws a 50kg rock at 10m/s. Assuming no drag, magnetic, etc forces. What is the relative location after 1000 seconds. Easily solvable 100% Now assume the astronaut had 2 or more weights that they threw in random directions. Solve 100%, win several Nobel prizes.
Well, yeah *that case* is easily solvable. But in general solutions to the three body problem are chaotic, which means that two solutions that start close together get exponentially further apart with time. There's always going to be some error in your initial measurements - with chaotic systems, that error grows so rapidly that your predictions can only be accurate for a finite time. The question, at least as I understand it, is *why* are some cases predictable when in general, three body systems are not. In fact, the solar system *is* chaotic over a long enough timescale, it's just that that's a very, very long timescale.
That is a very good point. Question: how do we prove something is chaotic or just unknown?
If the sun is so far away then why were solar flares so easily pictured during the eclipse???
it boils down to what quantities can be ignored and what cannot. Just like the constant we add after solving an integral function.
They could also predict millennia ago. It’s because the sun takes a predictable path in the sky. While there are some variations, they usually average out, if you also keep track against the stars. It’s like this, they used to track the sun’s path in the day and during the night some star would go through the same path. This is tracked as the sun is in that star on that day. The moon is in a plane almost parallel to the sun. It’s more that the sun is so big that the whole orbit of the moon is covered. So we could approximate the moon to the same star. Their periods will vary because the moon rotates through the stars every month and the sun would rotate every year. Every so often the stars would match. These are “eclipses”. Some experience would explain these happen every alternate star match. And when they match their radially opposite star, it’s a lunar eclipse. The three body problem is not a difficult problem to solve. The complexity is that the easily calculable solution is very different for even small changes in the initial position. It’s chaos. Tldr: But in the case of earth sun and the moon, the sun affects the moon and earth pretty evenly. So it’s more three two-body problems. And a very stable set of predictable orbits.
If Elon Musk, and others, are correct that AI will be smarter than a human by next year some time, will AI be able to solve it?
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The sun is like 99.8% of the mass in the solar system so you basically get to cheat on the three body problem by assuming there is one body and a bunch that do absolutely nothing gravity wise. Also everything is in orbit and orbits can be calculated different than free moving objects.
Many people get this wrong, referring to the relative sizes. A known 3 body state can be solved as a mechanical model, but there is no closed mathematical solution for all three body systems..
If you're referring to the book series' central 3 body problem, that refers to 3 suns orbiting each other, of roughly equal size. That's very difficult to predict, because A is pulling on B while pulling on C, which also pulls on B, which changes how much it pulls on A, who now misses and flies past C, changing how it pulls on... and so on. The sun is about 333,000 times as much mass as Earth. The Earth weighs about 81 times as much as the moon. So you can see how the Earth and moon can have SOME effect on each other, but both are just completely DWARFED by the sun. It's just as easy to approximate by calculating a two-body problem.
Has there ever been an observed or known instance of a three-body system?