Neil deGrasse Tyson Explains The Three-Body Problem

Summary

The video script delves into the complexities of the three-body problem, a concept in astrophysics that explores the gravitational interactions between three celestial bodies. It begins with the simpler two-body problem, which Isaac Newton successfully modeled using his laws of gravity and motion. However, when a third body is introduced, such as Jupiter's gravitational pull on Earth, the system becomes unpredictable and potentially unstable, leading Newton to invoke divine intervention to explain the observed stability of the solar system. The script then introduces the work of Pierre-Simon Laplace, who developed perturbation theory to address the small, repetitive tugs a two-body system experiences due to a third body. This theory showed that these perturbations often cancel out, maintaining the stability of the solar system over long periods. The video also touches on the restricted three-body problem, where one body is much smaller than the other two, allowing for a solvable model, exemplified by the double-star system in Star Wars. The script concludes by emphasizing the inherent chaos in the three-body problem when all bodies are of similar mass, making long-term predictions impossible due to the system's sensitivity to initial conditions. This chaos is a fundamental aspect of the problem, which is why scientists model it statistically rather than attempting precise predictions.

Takeaways

• 🌌 The two-body problem, where two objects orbit their common center of gravity, is perfectly solvable using Newton's laws of gravity and mechanics.
• 🌟 When a third body, like Jupiter, is introduced into the system, it can exert gravitational forces on the two-body system, causing instability and making the problem more complex.
• 📚 Isaac Newton, despite his work on calculus, did not develop a method to solve the three-body problem analytically due to its inherent complexity.
• 😇 Newton suggested that God might be responsible for the stability of the solar system when considering the influence of additional bodies like Jupiter.
• 🔢 Pierre-Simon Laplace developed perturbation theory, a branch of calculus, to address the gravitational effects of a third body on a two-body system over time.
• 🌐 Perturbation theory allows for the calculation of the net effect of small, repeating gravitational tugs from a third body, showing that they can cancel out over time.
• 🎬 The stability of the solar system, as we observe it, can be explained without needing to invoke divine intervention, contrary to Newton's initial musings.
• ⭐ The three-body problem, where all three objects have approximately the same mass, is mathematically chaotic and cannot be solved analytically due to its sensitive dependence on initial conditions.
• 🌠 In the restricted three-body problem, where one object is much smaller than the other two, the problem becomes solvable, as seen in the simplified model of a planet orbiting two stars, as in the movie Star Wars.
• ⚖️ The presence of a small third body, like a planet, in the solar system doesn't significantly alter the two-body problem of the Earth and the Moon orbiting their common center of mass.
• ⛓ The long-term behavior of the solar system, including the effects of Jupiter, is chaotic, but this chaos unfolds over a much longer timescale than Newton originally considered.

Q & A

What is the two-body problem in astrophysics?

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

How does the three-body problem differ from the two-body problem?

The three-body problem involves three celestial bodies, each with non-negligible mass, interacting with each other through gravity. Unlike the two-body problem, the three-body problem is mathematically chaotic and cannot be solved analytically due to its inherent unpredictability.

What is the center of mass in the context of the Earth and the Moon?

The center of mass is the point at which the combined mass of two or more objects can be considered to be concentrated. For the Earth and the Moon, they both orbit around their common center of mass, which is located about a thousand miles beneath Earth's surface.

Why did Isaac Newton initially believe that the solar system might be unstable?

Newton was concerned that the gravitational pull from larger planets like Jupiter could tug on Earth and disrupt its orbit around the Sun. He thought that these 'tugs' could cause the previously stable orbits to decay into chaos.

What is perturbation theory and how does it relate to the three-body problem?

Perturbation theory is a branch of calculus developed to solve problems where a small disturbance affects a system that is otherwise well understood. It allows for the calculation of the effects of the third body's gravitational pull on a two-body system, assuming the third body's influence is small and repetitive.

How did Napoleon Bonaparte contribute to the discussion of the three-body problem?

Napoleon Bonaparte read the works of Laplace on celestial mechanics and questioned why Laplace did not mention God as the architect of the system. Laplace responded that he had no need for such a hypothesis in his calculations.

What is the restricted 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 much smaller mass. In this case, the smaller body is influenced by the gravitational field of the two larger bodies, and the problem is solvable.

Why is the three-body problem considered unsolvable?

The three-body problem is unsolvable because the interactions between three bodies of similar mass result in a system that is mathematically chaotic. Small changes in initial conditions can lead to vastly different outcomes, making long-term predictions impossible.

What does it mean for a system to be 'chaotic' in the context of the three-body problem?

A chaotic system is one where the outcome is highly sensitive to initial conditions. For the three-body problem, this means that even a tiny change in the starting positions or velocities of the bodies can lead to a completely different trajectory over time, making it impossible to predict the system's behavior analytically.

How do astrophysicists approach the modeling of a chaotic system like the three-body problem?

Astrophysicists use statistical methods to model chaotic systems. They can describe the general behavior of the system over time, but they cannot track the exact path of each object indefinitely. The focus is on the statistical properties of the system rather than precise trajectories.

What is the significance of the three-body problem in understanding larger celestial systems, such as star clusters?

The three-body problem is fundamental to understanding the dynamics of larger celestial systems because it illustrates the inherent complexity and unpredictability of gravitational interactions. Even though star clusters contain thousands of stars, the same principles of chaos apply, making precise long-term predictions of individual star trajectories impossible.

Outlines

00:00

🌌 Introduction to the Three-Body Problem

The first paragraph introduces the concept of the three-body problem, explaining that it involves the gravitational interactions between three celestial bodies. It clarifies that the two-body problem is well understood through Newton's laws of gravity and mechanics, but the addition of a third body complicates the system. The speaker humorously recounts Newton's initial concerns about the stability of the solar system when considering the gravitational influence of Jupiter. The paragraph also mentions that despite Newton's belief in divine intervention to maintain the system's stability, the solar system remains stable without the need for such an explanation. It concludes with a teaser about the development of perturbation theory by Pierre-Simon Laplace, which addresses the complexities introduced by the third body.

05:02

📚 Historical Context and the Restricted Three-Body Problem

The second paragraph provides historical context by mentioning Napoleon's interest in celestial mechanics and his interaction with Laplace, highlighting the secular nature of scientific inquiry. It then delves into the complexities of a three-body system, such as a star with two orbiting suns, and introduces the concept of mathematical chaos in the system's orbits when a third body of similar mass is introduced. The paragraph explains that the three-body problem is unsolvable analytically due to its chaotic nature, where minor changes in initial conditions lead to vastly different outcomes. It contrasts this with the restricted three-body problem, which is solvable under the assumption that one body has a much smaller mass than the other two, and uses the example of a planet orbiting two stars in 'Star Wars' to illustrate this concept.

10:04

🚀 The Unsolvable Nature of the Three-Body Problem

The third paragraph emphasizes the chaotic and unsolvable nature of the three-body problem under general conditions. It explains that while we can model the system and make statistical predictions about its behavior over time, we cannot precisely predict the future positions of the bodies involved. The paragraph also touches on the concept of the four-body problem and the challenges it presents, indicating that the complexity increases with each additional body. It concludes by stating that while we cannot track an object's path through the system indefinitely, we can understand its behavior in a statistical sense, acknowledging the inherent chaos in such systems.

Keywords

💡Three-body problem

The three-body problem is a fundamental issue in astrophysics that involves predicting the motion of three celestial bodies interacting through gravitational forces. It is a central theme of the video, illustrating the complexity of celestial mechanics. The video discusses how the three-body problem is mathematically chaotic and cannot be solved analytically due to the unpredictability of the orbits when three bodies of similar mass interact.

💡Center of gravity

The center of gravity is the point where the total gravitational force on a system of bodies is considered to act. It is a key concept in the video, explaining how the Earth and the Moon orbit their common center of gravity. The video uses this concept to transition into the more complex three-body problem, emphasizing how the center of mass shifts as the Moon orbits the Earth.

💡Perturbation theory

Perturbation theory is a mathematical technique used to approximate solutions to problems that cannot be solved exactly. In the context of the video, it is used to describe how small, repeating gravitational tugs from a third body can be accounted for in the two-body problem, thus maintaining the stability of the solar system over long periods. The video highlights that this theory was developed after Newton's time and was crucial for understanding the stability of the solar system.

💡Chaos theory

Chaos theory deals with the behavior of dynamic systems that are highly sensitive to initial conditions. The video explains that the three-body problem exhibits chaotic behavior, meaning that slight differences in initial conditions can lead to vastly different outcomes over time. This is a significant aspect of the video's narrative, emphasizing the unpredictability of celestial bodies when three or more are involved.

💡Restricted three-body problem

The restricted three-body problem is a simplification of the three-body problem where one body is considered to have negligible mass compared to the other two. This makes the problem solvable, as the smaller body does not significantly affect the motion of the larger ones. The video uses the example of a planet orbiting two stars in 'Star Wars' to illustrate this concept, where the planet's orbit is stable due to the much larger mass of the two stars.

💡Celestial mechanics

Celestial mechanics is the branch of astronomy that deals with the motion of celestial objects and the forces that cause that motion. The video delves into celestial mechanics by discussing the two-body and three-body problems, and how they relate to the broader field of astrophysics. It also mentions a book on celestial mechanics read by Napoleon, highlighting the historical interest in this field.

💡Orbit

An orbit is the path that an object in space takes around another object due to gravity. The video discusses orbits in the context of the Moon orbiting the Earth, the Earth and Moon orbiting the Sun, and the complexities that arise when a third body is introduced. Orbits are central to understanding the dynamics of the celestial bodies discussed in the video.

💡Gravitational forces

Gravitational forces are the attractive forces that act between bodies with mass. The video explains how these forces influence the motion of celestial bodies, causing them to orbit a common center of gravity in a two-body system and leading to the complexities of the three-body problem where the forces can result in unstable and unpredictable orbits.

💡Isaac Newton

Isaac Newton was an English mathematician, physicist, and astronomer, renowned for his laws of motion and universal law of gravitation. The video references Newton's work in solving the two-body problem and his initial concerns about the stability of the solar system when considering the three-body problem. Newton's contributions are foundational to the discussion of celestial mechanics in the video.

💡Jupiter

Jupiter, the largest planet in our solar system, is used in the video as an example of how a third body can influence the orbits of other celestial bodies, such as the Earth. The video discusses how Jupiter's gravitational pull can tug on the Earth and potentially affect its orbit around the Sun, contributing to the complexities of the three-body problem.

💡Star clusters

Star clusters are groups of stars which orbit around a common center of mass. The video briefly touches on star clusters to illustrate the concept of chaos on a larger scale, where the interactions of thousands of stars in a cluster are too complex to predict with precision. This reinforces the idea that celestial mechanics can be modeled statistically but not predicted exactly for every object.

Highlights

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

The two-body problem is perfectly solvable using Newton's equations of gravity.

Newton applied his equations to the Earth-Moon and Earth-Sun systems successfully.

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

Newton attributed the stability of the solar system to occasional 'corrections by God' when his calculations fell short.

Perturbation Theory, developed 113 years after Newton, allows for the calculation of small, repeating tugs in a two-body system.

The solar system's stability can be understood through perturbation theory, which Newton was unaware of.

Napoleon criticized Laplace for not mentioning the 'Architect of the System' (God) in his work on celestial mechanics.

The three-body problem involves three objects of approximately similar mass and is mathematically chaotic.

The orbits in a three-body problem are unpredictable due to the system's inherent instability and sensitivity to initial conditions.

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

In the restricted three-body problem, the smaller object orbits the combined center of mass of the two larger ones.

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

The four-body problem and beyond follow the same principles of unpredictability as the three-body problem.

Star clusters with thousands of stars are modeled using chaos theory, acknowledging the inability to predict precise positions over time.

The essence of the three-body problem is the unpredictability and chaos inherent in the system's dynamics.

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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