Understanding the Three Body Problem in Space

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In summary, the conversation discusses the orbit of a very light body around a much heavier one, and how its orbit is limited by the Hill sphere and Laplace's sphere of influence. The three body problem in orbital mechanics has not yet been fully solved, but approximations can be made with great accuracy. The conversation also touches on the complexities of posing such a problem and the limitations of defining masses and distances.
  • #1
JANm
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Suppose a body in a circular orbit around a very heavy one. The body itself has a very light one moving in an small orbit in the same plane.

1 What is the orbit of the very light one in space?

very= 1000 times, so the heaviest= 1 million times heavier than the "smallest".

small=1/10, so the smallest distance of the very light one 0,9 times the radius of the orbit of the body around the heaviest and the largest distance of the very light one is 1,1 times the radius of the orbit of the body around the heaviest one, approximately.

2 do these numbers 0,9 and 1,1 need corrections?
 
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  • #2
The largest orbit a one body can have around another body when the 2nd body orbits a third body is limited by the Hill sphere.

This is found by:

[tex]R = a \sqrt[3]{\frac{m}{3M}}[/tex]

In this case, we define "a" as the orbit of the middle sized object around the largest object and assign it a value of 1.

m is the mass of the middle sized object and also will equal 1
M is the mass of the largest object and therefore = 1000.

so:

[tex]R = 1 \sqrt[3]{\frac{1}{3(100)}} = 0.069[/tex]

Which is smaller than the 0.1 you are trying to use for the smallest object's orbit around the middle sized one. IOW, the smallest object cannot orbit at that distance from the middle sized one. It would be pulled away into an independent orbit around the largest object.
 
  • #3
The three body problem in orbital mechanics has, to my knowledge, never been completely solved. It is a humbling reminder of how much we have yet to learn about the universe. We can, however, approximate solutions to problems such as this, as well as much more complicated systems with amazing accuracy.
 
  • #4
Janus said:
The largest orbit a one body can have around another body when the 2nd body orbits a third body is limited by the Hill sphere.

This is found by:

[tex]R = a \sqrt[3]{\frac{m}{3M}}[/tex]
More or less. This is a three body problem, after all. An alternative to the Hill sphere is Laplace's sphere of influence:

[tex]R = a\left(\frac m M\right)^{2/5}[/tex]

With M=1000*m, the sphere of influence is 6.3% of the distance between the primary and secondary bodies (compare to 6.9% for the Hill sphere).

Interestingly, neither the Hill sphere nor the sphere of influence is a sphere.
 
  • #5
Chronos said:
The three body problem in orbital mechanics has, to my knowledge, never been completely solved. It is a humbling reminder of how much we have yet to learn about the universe. We can, however, approximate solutions to problems such as this, as well as much more complicated systems with amazing accuracy.
Hello Chronos
Thanks for this poetical way of understanding three body problems.
I don't quite understand the Lagrange like boarders for posing three-body problems... I have taken care that my problem didnot counteract geometrical possibility by defining radii, I have sayd things about masses, must say have defined them relatively absolute, but the impossibilities of the posing of the question gathers around...
greetings Janm
 

1. What is the Three Body Problem in Space?

The Three Body Problem in Space is a mathematical problem that involves predicting the motion of three objects, such as planets or stars, that are influenced by each other's gravitational forces. It is a complex problem that has been studied by scientists for centuries.

2. Why is the Three Body Problem in Space important?

The Three Body Problem in Space is important because it helps us understand the dynamics of celestial bodies in our universe. By studying this problem, we can make predictions about the behavior of objects in space, such as the orbits of planets and moons.

3. What are some challenges in understanding the Three Body Problem in Space?

One of the main challenges in understanding the Three Body Problem in Space is that it does not have a simple analytical solution. This means that it cannot be solved using traditional mathematical methods and requires advanced computational techniques. Additionally, the problem becomes more complex when more than three bodies are involved.

4. How do scientists study the Three Body Problem in Space?

Scientists use computer simulations and mathematical models to study the Three Body Problem in Space. These simulations allow them to predict the behavior of the three bodies over time and observe how they interact with each other. They also use real-world observations and data from space missions to validate their models.

5. What are some real-world applications of understanding the Three Body Problem in Space?

Understanding the Three Body Problem in Space has many real-world applications, such as predicting the trajectory of spacecraft and satellites, planning space missions, and studying the formation and evolution of planetary systems. It also has applications in other fields, such as physics, astronomy, and engineering.

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