# Neil deGrasse Tyson Explains The Three-Body Problem

### Summary

TLDRThe video script offers an insightful explanation of the three-body problem, a concept in astrophysics that describes the difficulty in predicting the motion of three celestial bodies interacting through gravity. The script begins with the simpler two-body problem, which Isaac Newton famously solved using his laws of gravity and mechanics. However, when a third body is introduced, such as Jupiter affecting the Earth's orbit around the Sun, the problem becomes chaotic and unpredictable. The script highlights that Newton himself was concerned about the stability of the solar system due to these gravitational tugs. The solution to this problem was developed over a century later through perturbation theory, which allows for the calculation of the effects of small, repeating tugs on a two-body system. The video also distinguishes between the unrestricted three-body problem, which is mathematically unsolvable due to its chaotic nature, and the restricted three-body problem, where one body is much smaller than the other two and thus its influence is negligible. This simplified version is solvable and is applicable to scenarios like the double-star system portrayed in Star Wars. The script concludes by emphasizing the inherent chaos in celestial mechanics and how scientists model this chaos statistically rather than attempting to predict the exact behavior of each body.

### Takeaways

- ๐ The two-body problem, such as the Earth and the Moon orbiting their common center of gravity, is perfectly solvable using Newton's laws of gravity and mechanics.
- ๐ Newton applied his equations to the Earth-Moon-Sun system, which also worked, but he was concerned about the stability of the solar system due to gravitational tugs from other planets like Jupiter.
- ๐ Newton suggested that God might be responsible for maintaining the stability of the solar system when his calculations couldn't account for all the gravitational interactions.
- ๐ Over a century later, Pierre-Simon Laplace developed perturbation theory, a branch of calculus that helped to understand the net effect of small, repeated gravitational tugs in a two-body system being perturbed by a third body.
- ๐ Perturbation theory showed that the small gravitational tugs from a distant third body, like Jupiter, would often cancel each other out over time, contributing to the stability of the solar system.
- ๐ Napoleon Bonaparte, known for his interest in science, read Laplace's work and questioned why he did not mention the role of God in the stability of celestial bodies; Laplace responded that his calculations did not require such a hypothesis.
- โญ The three-body problem, involving three stars of roughly equal mass, leads to mathematically chaotic and unpredictable orbits, making it unsolvable in the traditional sense.
- ๐ In the context of a double star system with a planet, like in Star Wars, the setup is an example of the restricted three-body problem, which is solvable because the mass of the planet is much less than the two stars.
- ๐ซ The unrestricted three-body problem, where all three bodies have comparable masses, is unsolvable due to the inherent chaos in the system, which leads to exponentially divergent outcomes from minor changes in initial conditions.
- ๐ค The concept of chaos in the three-body problem means that while we can model the general behavior over time, we cannot predict the exact trajectory of each body indefinitely.
- ๐ฎ For systems with more than three bodies, such as star clusters, we can use computational models to simulate their behavior, but the inherent chaos means we cannot predict the precise positions of all bodies at any given time.

### Q & A

### What is the two-body problem in the context of celestial mechanics?

-The two-body problem refers to predicting the motion of two celestial bodies that interact only with each other through gravitational force. It is perfectly solvable using Newton's laws of motion and his law of universal gravitation.

### How does the Earth-Moon system relate to the concept of the two-body problem?

-The Earth-Moon system is an example of a two-body problem where both the Earth and the Moon orbit their common center of gravity. Despite the Earth being much more massive, it still moves in response to the gravitational pull of the Moon.

### What is perturbation theory and how is it related to the three-body problem?

-Perturbation theory is a branch of calculus developed to solve the three-body problem by considering the gravitational influence of a third body as a small 'perturbation' to the otherwise solvable two-body problem. It allows for the approximation of the paths of celestial bodies over time.

### Why did Isaac Newton consider the solar system to be stable despite the presence of multiple bodies?

-Newton believed the solar system to be stable because his laws of motion and gravitation, when applied to the Earth-Moon-Sun system, seemed to hold true. He attributed the stability to divine intervention, suggesting that God corrects any instabilities.

### What is the restricted three-body problem and how does it differ from the general three-body problem?

-The restricted three-body problem is a special case where two bodies have approximately equal masses and the third body has a mass much smaller compared to the other two. This problem is solvable because the smaller body's influence on the two larger bodies can be neglected, simplifying the equations.

### How does the concept of chaos theory apply to the three-body problem?

-Chaos theory is applied to the three-body problem because small changes in the initial conditions can lead to vastly different outcomes over time, making long-term predictions impossible. The system is considered chaotic, meaning it is highly sensitive to initial conditions.

### Why is the three-body problem considered unsolvable?

-The three-body problem is considered unsolvable because there is no general analytical solution that can predict the motion of three bodies over time due to the inherent chaos in their interactions. Solutions can only be approximated numerically.

### What is the significance of the three-body problem in the context of astrophysics and celestial mechanics?

-The three-body problem is significant because it represents a fundamental challenge in understanding the dynamics of celestial bodies. It highlights the limitations of classical mechanics and the need for advanced mathematical tools like perturbation theory and numerical simulations to study complex systems.

### How did Pierre-Simon Laplace contribute to the understanding of the three-body problem?

-Pierre-Simon Laplace developed perturbation theory, which is a method for approximating the solutions to the three-body problem by treating the gravitational influence of one body as a small perturbation on the two-body problem.

### What is the role of numerical simulations in studying the three-body problem?

-Numerical simulations play a crucial role in studying the three-body problem by allowing scientists to model the system's behavior over time. These simulations can provide insights into the system's evolution, even when an analytical solution is not possible.

### How does the three-body problem relate to real-world celestial systems, such as star clusters?

-The three-body problem is a foundational concept for understanding more complex celestial systems, like star clusters, where the motion of thousands of stars is influenced by gravitational interactions. While individual predictions may not be possible, statistical models can describe the overall behavior of such systems.

### What was Napoleon's involvement with the three-body problem and celestial mechanics?

-Napoleon is noted for having read and appreciated the works on celestial mechanics by Laplace. He questioned Laplace about the absence of any mention of God as the 'architect of the system' in his work, to which Laplace responded that such a hypothesis was unnecessary for his mathematical treatment of the subject.

### Outlines

### ๐ Understanding the Three-Body Problem

This paragraph explains the three-body problem, starting with the two-body problem where the Earth and the Moon orbit their common center of gravity. It highlights how Isaac Newton solved the two-body problem with his laws of gravity and mechanics, and then extended his equations to the Earth-Moon-Sun system. The paragraph then discusses Newton's concerns about the stability of the solar system when considering the gravitational influence of other planets like Jupiter. It mentions Newton's reliance on the idea of a divine intervention to maintain stability, which was later addressed with the development of perturbation theory by Pierre-Simon Laplace, a new branch of calculus that could account for the small, repeating gravitational tugs of a third body on a two-body system.

### ๐ Historical Insight on Celestial Mechanics

This section delves into the historical aspect of the three-body problem, mentioning Napoleon's interest in the field of celestial mechanics. It recounts how Napoleon engaged with Laplace over his work, pointing out the absence of a mention of God as the 'architect' of the system. Laplace's response, indicating that such a hypothesis was unnecessary for his mathematical treatment, is highlighted as a significant moment. The paragraph also touches on the concept of a double star system, like the one depicted in Star Wars, and introduces the idea of a three-body problem involving three stars and a planet, which leads to mathematically chaotic orbits and the inherent instability of such a system.

### ๐ The Restricted Three-Body Problem and Chaos Theory

The final paragraph discusses the restricted three-body problem, where two massive bodies orbit each other and a third, much smaller body orbits these two. This scenario is solvable and was applicable to the Star Wars depiction of a double star system with a planet. It explains that while the restricted three-body problem can be solved, the general three-body problem is unsolvable due to its inherent mathematical chaos. Small changes in initial conditions can lead to exponentially divergent outcomes, making long-term prediction impossible. The paragraph concludes by emphasizing the role of chaos theory in modeling such systems, where we can only predict statistical behaviors over time rather than precise trajectories.

### Mindmap

### Keywords

### ๐กThree-body problem

### ๐กCenter of mass

### ๐กPerturbation theory

### ๐กChaos theory

### ๐กRestricted three-body problem

### ๐กOrbit

### ๐กGravitational forces

### ๐กIsaac Newton

### ๐กCelestial mechanics

### ๐กNapoleon

### ๐กStar clusters

### Highlights

The Moon and the Earth orbit their common center of gravity, not the Earth alone.

The two-body problem is perfectly solvable using equations of gravity and mechanics.

Isaac Newton applied his equations to the Earth-Moon system and then to the Earth-Moon-Sun system.

Newton worried about the stability of the solar system due to gravitational tugs from other planets like Jupiter.

Newton suggested that God might occasionally intervene to stabilize the solar system.

Perturbation Theory, developed by Laplace 113 years after Newton, showed that small, repeated tugs in a two-body system can cancel out over time.

The solar system's stability can be understood through advanced calculus, which was not available to Newton.

Napoleon criticized Laplace for not mentioning God as the architect of the system, to which Laplace responded that his calculations made such a hypothesis unnecessary.

The three-body problem involves three stars of approximately equal mass and is mathematically chaotic, making it unsolvable.

In the three-body problem, small changes in initial conditions can lead to exponentially different outcomes, a characteristic of chaos.

The restricted three-body problem, where one body is much smaller than the other two, is solvable.

The Star Wars depiction of a double star system with a planet is an example of the restricted three-body problem.

The actual three-body problem is unsolvable due to its inherent chaos, unlike the restricted version.

Chaos theory allows us to model the behavior of the three-body problem statistically over time, rather than predicting exact outcomes.

The problem extends beyond three bodies; systems with more objects also exhibit chaotic behavior.

Star clusters with thousands of stars are an example of systems that we can model but cannot predict with precision.

The essence of the three-body problem is the unpredictability and chaos inherent in systems with three or more bodies.

### Transcripts

you're going to get an astrophysicist

explanation of the literal three-body

problem without reference to anything

that's shown up on streaming services

and that means he's not gonna ruin the

show for you I don't know anything about

I don't know anything about the show but

I do know enough to describe the three

body problem to you coming up

[Music]

let's let's start simple okay okay okay

so as we know the moon orbits the earth

right but that's not the right way to

say it okay okay all right the Moon and

the Earth orbit their common center of

gravity oo so Earth is not just sitting

here right and the moon is going around

going around it they feel in their

Common Center you know where it is it's

a thousand miles beneath earth's surface

along line between the center of the

earth and the center of the Moon gotcha

so as the moon moves here that Center

Mass line shifts

okay so that means Earth is kind of

jiggling like this as the moon goes

around gotta that's their Center of mass

all right this is the two- body problem

it is perfectly solved using equations

of gravity right and mechanics makes

sense perfectly solved yeah Isaac Newton

solved it okay my boy that's your man so

that worked then Isaac applied the

equations to the Earth Moon system going

around the Sun okay okay that worked too

so in that system Let's ignore the moon

for the moment it's earth going around

it's another two- body system two system

all right but then he worried he said

every time Earth comes around the

backstretch and Jupiter's out there

right Jupiter about tug on it a little

bit a lot of gravity a little bit tug on

it as we com around back to the other

side what's up Earth all right and then

it comes around again tugs on it again

what's up earth right and of course

everybody's moving in the same direction

around the Sun so the Earth would have

to go a little farther in its orbit to

be aligned again with Jupiter but it's

going to tug on it right okay he looked

at all these little tugs and he says I'm

worried that the solar system will go

unstable right because it keeps tugging

on it it keeps pulling it away and the

previously stable orbit would just Decay

into chaos okay okay he was worried

about this you know what he said but I

know my stuff works and it's been and

it's looks stable to me right so clearly

it is stable even though it looks like

maybe it wouldn't be stable you know

what he says he said every now and then

God fixes things well there you go

that's the answer even Isaac

Newton wow look at that when in doubt

went in doubt just just let God figure

it out right I can't figure it out God

Did It clearly we're all still here and

we haven't been yanked out of orbit by

Jupiter right but Jupiter is pulling on

us so it's a god correction God God

correction okay this this is the first

hint that a third

body is messing with you right okay in

some way that maybe is harder to

understand fast forward

113 years oh right we get to uh

llas he studied this problem right okay

and he developed I don't think he

invented but he

developed a new branch of calculus oo

called perturbation Theory aha okay

unknown to Newton even though Newton

invented calculus right he invented

calculus right all right so he could

have done it he could have said in order

to solve this problem let me invent more

calcul more calcul just need more calcul

I just need more do do it didn't do it

so LL develops perturbation Theory and

it comes down to we have two bodies the

Sun and the Earth in this case and the

third one the tug is small but it's

repeating it's not a big Jupiter's not

sitting right here it's way way out

there it's just a little tug and so you

can run the equations in such a way and

realize that a two body system that is

tugged Often by something small that it

all cancels out in the end gotcha okay

okay so when it's out here the tug is a

little bit that way but now it's over

here and the tug is less right all right

and then sometimes it's tugging you in

this direction when that's the

configuration you add it all up it all

cancels out Newton could not have known

that without this new branch of calculus

okay okay pertubation Theory so that

took care of that third body gotcha

where solar system is basically stable

okay for the foreseeable future in ways

that Newton had not imagined in ways

that Newton required God right okay oh

by the way just a quick aside this is

now we're up to the year 1800 uh you

know who summoned up these books to read

them immediately because the there a

series of books called Celestial

mechanics okay Napoleon ah na am

Napoleon Napoleon who read all the books

he could on physics and engineering and

metalurgy look at that okay it wasn't

just a tyrant right he was like he was a

smart Tyrant smart Tyrant was all right

so he summons up the book doesn't need

doesn't have to be translated because

they're both in French right he reads it

goes to llas and says Monier this is a

beautiful piece of work brilliant but

you make no mention of the architect of

the system he's referring to God and

llas replied sir I had no need for that

hypothesis oo that's a mic drop oh that

is tough

man you that's a dig on Napoleon and on

new Newton yeah and on Newton I have oh

man look at that yeah all right so let's

keep going go ahead so now let's say we

have not just the planet and one of its

moons but let's say we have a star and

another star double star system famously

portrayed in what film uh Star Wars Star

Wars yeah all right of course so those

two suns and the planet is stable and

I'll tell you why in a minute mhm but if

you take a third sun and put it there

about approximately the same size then

what kind of orbits will they have give

me two fists here okay so I'm feeling

this one but now I feel that where's my

gravitational allegiance to go am I

going to come through but then am I

going to go that that way or this way so

I'm coming into the system and do I go

to you in orbit but wait you're still

coming around here now I feel this

and so it turns out the orbits of a

three-body problem are mathematically

chaotic yes I was about to say that did

not seem very stable SS has to give well

this is this is in the series what talk

something I don't I haven't seen the

series I'm just saying something has to

give that's all two of these are going

to collide one is going to get ejected

right okay that is the classical three

body problem three objects of a

approximately similar mass trying to

maintain a stable orbit and it goes

chaotic with just three objects look at

that it is an unsolvable you can let me

say that differently you can calculate

incrementally what's happening and track

it until the system dies right or splits

apart or whatever but you cannot

analytically predict the future of the

three-body system because what chaos

will do for you in your mathematical

model is if you change the initial

conditions by a little bit right a

little bit the solution diverges further

down the line that goes crazy it's not

just a little bit different later on

down line it is exponentially

exponentially different correct with the

with the smallest increment of distance

so I'll say I'll move you in this

direction in this model and then in a

slightly different direction than the

other model it goes chaotic that's what

we mean by chaos right okay it's

mathematically defined Okay so now

there's something called the restricted

three body problem all right okay okay

the restricted three body problem never

heard you have give me your two your two

things back two plan you got that okay

two bodies you got your two bodies now

the third body is little ah now you two

will orbit each other right okay and

then and then this it's not messing with

them right so so there restricted three

body problem we have two masses of

approximately equal and one that's much

less than the other two that is solvable

right it's called the restricted three

body problem gotcha in the Star Wars

case that's the restricted three body

problem right because you have the two

stars and you have the little planet the

little planet deal and it's even better

because the planet is so far away that

it only really saw one merged gravity of

the two stars right okay you're far

enough away that that difference is not

really mattering to you you maintain one

stable orbit around them both around

both stars both Stars okay now if it got

really close then you'll have issues

because then ites again gravitational

Allegiance matters the stars are not

going to care but you will cuz you

you'll get eat you don't know where to

go you don't know where to go I'm in

love with two stars and I don't know

what to do which way do I

turn so anyhow I so so the three body

problem the takeaway here is it's

unsolvable yes not just because we don't

know how to do it yet because it's

mathematically UNS bu into the system

the system is chaotic yeah okay unless

you make certain assumptions about the

system that you would then invoke so

that you can solve it and so one of them

is a small object around bigger ones

another one oh by the way in this

solution with Jupiter out there slightly

tugging right yes it turns out over a

very long time scale this is chaotic

but much longer time skill than Newton

ever imagined okay okay because yes we

are small compared to the Sun but

Jupiter isn't all right and we're trying

to orbit between them right right so

that's that's all it's not deeper than

that it's not yeah right I could have

said the four body problem but this

problem begins at the three body problem

right right because you're going to have

the same thing in four bodies or five

bodies it's going to be the same we have

star clusters with thousands of stars in

them and they're all just orbiting we

have to we can model it but cannot

predict with Precision where everybody's

going to be at any given time okay CU

it's chaotic the're chaotic so it's

basically it's about the chaos it's

about the chaos it's all about the chaos

yeah so what we do is we model the chaos

right right we say this will be

statistically looking like this over

time you're not going to track one

object through the system exactly for

eternity that's not going to work that's

so cool yeah all right that is so cool

there it is all right another explainer

slipped in from torn from the pages of

Science Fiction yes just the just a

simple description of the three body

problem until next time keep looking up

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