Three Body Problem: Dark Matter's Impact

[kurious]

Would three galaxies, held together by dark matter, be an example of a three body problem?

 

[selfAdjoint]

No, because they are too loose constructed. The Sun, Earth, and Moon are a good three body problem. Galaxies are a many body problem.

 

[Jenab]

No, because they are too loose constructed. The Sun, Earth, and Moon are a good three body problem. Galaxies are a many body problem.

Galaxies have so many bodies that they can probably be treated as a fluid dynamical problem of the non-ideal gas variety: gravity being analogous to VanderWaals forces.

Jerry Abbott

 

[Nereid]

Back in the bad old days, AFAIK, modelling of galaxies was done analytically ... with some success (e.g. spiral density wave). These days the preferred approach is simulation; the Toomre brothers did some of the first numerical modelling and showed nice tidal tails from collisions. These days the number of point masses ('stars') is in the millions.

 

[mathman]

If the galaxies are far enough apart (i.e. compared to their sizes), then their relative motions could be looked at as a three body problem, as long as there are no other galaxies nearby.

 

[Nereid]

If the galaxies are far enough apart (i.e. compared to their sizes), then their relative motions could be looked at as a three body problem, as long as there are no other galaxies nearby.

... or dark matter concentrations

 

[Chronos]

The classic 3 body problem is the homework equivalent of violating the Geneva convention. Solving it for 3 galaxies is a physics 101 assignment at Hades University.

 

[Jenab]

The classic 3 body problem is the homework equivalent of violating the Geneva convention. Solving it for 3 galaxies is a physics 101 assignment at Hades University.

I wonder how many more schools exist down there? The University of Hell, Brimstone Technical School...

 

[kjvonly]

Hi,

I'ld like to propose a relatively simple solution to this 3 body problem. Suppose we had a 'satellite' orbitting around the earth. It is affected by both the Earth andthe sun.

Now, there are 4 forces which affect the firmament's orbit around
earth.
1. Fge - Earth's gravity
2. Fgs - sun's gravity
3. Fae - acceleration around the earth
4. Fas - acceleration around the sun

Fge + Fas = Fgs + Fae --- equation 1

The standard physics textbook 'Geostationary satellite orbit' simply
equates:
Fge = Fae

So, substituting this into equation 1, we get:

Fge + Fas = Fgs + Fge

and we are left with:

Fas = Fgs

When this is done, the orbit is like 20,000 to 35,000 miles in space
--- depending on the speed (and thus the period). However, what if,
the Satellite's orbit was designed so that the Earth's gravity was way
larger than the satellite's acceleration around the earth? -- that is:

Fge >> Fae

Well, consider equation 1:

Fge + Fas = Fgs + Fae

rewritting it:

Fge - Fae = Fgs - Fas

Since, Fge >> Fae, this reduces to:

Fge = Fgs - Fas --- equation 2

Does this jive so far?

Toby

 

[tony873004]

I know there is no solution to the 3-body problem except numeric integration. But does this just mean nobody has found the solution yet? I know many prominent mathimaticians from years ago tried to solve it. Were they wasting their time? Has it been proven that the 3-body problem has no solution? Or is it possible that I might wake tommorow to find on page 9 of my newspaper "Mathamatician solves 3-body problem".

 

[Nylex]

gravity being analogous to VanderWaals forces.

Jerry Abbott

How is that? I thought van der Waals forces were an electric dipole thing.

 

[BobG]

I know there is no solution to the 3-body problem except numeric integration. But does this just mean nobody has found the solution yet? I know many prominent mathimaticians from years ago tried to solve it. Were they wasting their time? Has it been proven that the 3-body problem has no solution? Or is it possible that I might wake tommorow to find on page 9 of my newspaper "Mathamatician solves 3-body problem".

Yes, people have solved it.

That's how you find your Lagrange points, for one thing. One of the simple ones would be directly between the Sun and Earth. Your satellite's angular velocity around the Sun is dependent upon the strength of the Sun's gravitational attraction at that distance. The closer to the Sun, the greater the angular velocity. If the satellite is directly between the Earth and Sun, the net gravitational attraction would be the the Sun's minus the Earth's. In essence, you've given the satellite a weaker gravitational attraction to the Sun than it would normally have at that distance - weak enough that the satellite's angular velocity matches the Earth's angular velocity around the Sun giving you a satellite that always stays between you and the Sun (like ACE, for example).

Others can get much more difficult to do by hand, since your angles and distances are always changing, but not so much so for a computer which can do repetitive tasks very well.

 

[tony873004]

Yes, people have solved it...
...Others can get much more difficult to do by hand, since your angles and distances are always changing, but not so much so for a computer which can do repetitive tasks very well.

But a computer doing this as a repetitive task is an example on a numerical integration. And numerical integrations are subject to truncation errors (if you make the time step too fast) and round-off errors.

I was referring to an analytical solution to the 3-body problem. A solution where the problem was not divided into n time steps, solve for each time step and add together. For example: How far will a car moving at 60 mph travel in 200 hours? D = V * T = 12000 miles. This is an anaytical solution that would work equally well for any value of T or V. There's no need to break this problem into 1-second time intervals, solve for each interval and add them together, although you could do it that way. But doing it that way does introduce error into the problem that isn't there in a purely analytical solution.

In the case of planetary motion, there are analytical methods that provide good approximations. VSOP87 is an example of one. This method will predict the position of a planet as viewed from Earth to an acrsecond of accuracy for hundreds of years. A similar method is used for the Moon. This is how solar eclipses and planetary occulations are predicted so well. But these methods fail miserably if trying to determine the long-term state of the solar system over millions or billions of years. That is because these solutions must have all known pertubations spelled out exactly in mathamatical terms. There may be pertabutions that we don't know about that don't have any significant effect in the short term. And these methods are only effective because the planets' orbits are close to circular, and not very chaotic in the short term.

Take for example this diagram http://www.orbitsimulator.com/gravity/capture.jpg of the Earth's Moon capturing an asteroid into Earth orbit. An analytical solution should be able to produce a position for the asteroid as a function of time. This diagram was created using the time-step method.

 

[enigma]

Yes, people have solved it.

Numerically only. There aren't enough constants to solve it analytically.

The only reason we're able to solve the 2 body problem analytically is because you can make the assumption that 'body 2's mass' << 'body 1's mass'.

 

[tony873004]

Numerically only. There aren't enough constants to solve it analytically.

The only reason we're able to solve the 2 body problem analytically is because you can make the assumption that 'body 2's mass' << 'body 1's mass'.

I might be wrong, but I think the 2-body is also analytically solveable even if both bodies have significant mass. They would both be tracing perfect unperturbed ellipses around their barycenter. A 3rd body introduces pertubations and messes the whole thing up.

But do you know if a 3-body analytical solution is an impossiblilty, or just a solution that nobody's discovered yet?

 

[pervect]

The two body problem can be solved analytically. The three body problem can be solved in special cases, such as the Lagrange points of the restricted three body problem (note the word: restricted). The general three body problem cannot be solved analytically, there are two few constants and too many variables:

http://scienceworld.wolfram.com/physics/BrunsTheorem.html

but I've never personally seen the proof.

 

[Rothiemurchus]

Here's an idea for solving the three body problem:

Instead of three masses have two masses for which the force of attraction
varies in magnitude which is what would happen as the third mass moves around them.So Newton's law Gm1m2/r^2 would not be valid but would be
replaced by a time dependent function e.g if r = t^2 + t + 1
we would get Gm1m2 / (t^2 + t + 1)^2

We could find this function by asking "how do we minimise r ?"
i.e by using Euler-Lagrange equations.

 

[pervect]

There's a little more about Brunn's theorem at

http://www.wolframscience.com/reference/notes/972d

Proposals that try to solve the three body problem without addressing the roadblock of this theorem are just ignoring the math.

Note that in some sense there is a "solution", though not an analytic one, to the planar three body problem, as there is a series that converges due to Sundman. But it's convergence is very very slow

http://scienceworld.wolfram.com/physics/RestrictedThree-BodyProblem.html

talks about the need for 10^8,000,000 terms of this series to achieve the accuracy needed for astronomy.

 

[Rothiemurchus]

If one of the bodies in a three body problem has a large mass - like a galaxy -
and this mass is in reality made of many masses,what determines when two much smaller masses become part of the many body probelm that the galaxy already is in its own right? Where is the borderline?

 

[tony873004]

If one of the bodies in a three body problem has a large mass - like a galaxy -
and this mass is in reality made of many masses,what determines when two much smaller masses become part of the many body probelm that the galaxy already is in its own right? Where is the borderline?

As long as you have more than 2 bodies, and 2 of your bodies have mass, then you have an n-body problem. In certain instances, you can treat an n-body problem like a bunch of 2 body problems and get good results, but they'll just be approximations. So a galaxy can not be considered a body because it is many bodies.

 

[Jenab]

I might be wrong, but I think the 2-body is also analytically solveable even if both bodies have significant mass. They would both be tracing perfect unperturbed ellipses around their barycenter. A 3rd body introduces pertubations and messes the whole thing up.

But do you know if a 3-body analytical solution is an impossiblilty, or just a solution that nobody's discovered yet?

It's analytically impossible to solve any system of equations when there are more unknowns than you can eliminate. That's what shuts the door on closed form (analytic) solutions. Closed form solutions are desirable, when they are possible, because they permit those "long flying leap" predictions without having to crunch tediously all the way there. Open form solutions (stepwise integrations) may remain possible, if you can divide the problem into steps small enough to let simplifying assumptions be valid to a good approximation.

The constant acceleration formulae of Physics 101 kinematics are the usual simplifying assumptions used in time-step integrations in celestial mechanics. Others (for various applications) are Simpson's Rule, Lagrange Interpolating Polynomials, and Runga-Kutta. The idea behind all of them is, "When you don't really know, take the best guess you can." And generally the more work you do in the guessing (e.g., the smaller the time step), the better the guess will be.

Jerry Abbott

 

[Haelfix]

Already the two body general solution is a horrendous chore to do, and took the work of many famous physicists to solve completely (for the case of a 1/r potential). Unfortunately the integral form is nasty (elliptical integrals), but they have been done.

Change the potential, and you can no longer get a closed form stable solution.

The general three body formula is impossible to solve analytically as people mentioned.

Fortunately some people have figured out some pretty heroic many body formalisms for getting decent answers. Field theory, fluid mechanics, hole theory, statistical mechanics, etc etc.

Computers are good enough these days however, that the three or four body problem is pretty well pinned down (for Newtonian potentials of course). Which is why they can tell you where Mars is going to be in the year 2490 +/- a few kilometers.

 

[khavel]

You can download my new e-book 'N Bodies - No Problem' at:
http://www.grevytpress.com/model.html
You can also try there n-body simulations (all done by the same program).

 

[Nereid]

You can download my new e-book 'N Bodies - No Problem' at:
http://www.grevytpress.com/model.html
You can also try there n-body simulations (all done by the same program).

Have you submitted your solution for publication in a peer-reviewed journal?

If so, what response(s) did you get from the reviewers?

If not, why not?

 

[khavel]

I am now sending the printout of my e-book to several journals for a possible review.

Do you have any suggestions as to where I shoudl send it?

 

[remcook]

khavel...if i may ask...how does your results differ from any other n-body simulation?
From what I see of your e-book, you just present equations of motion, not an actual solution.

Could you explain some more?

 

[khavel]

It is the actual solution. The program in the last chapter of my e-book calculates the trajectories of all bodies, which can be then displayed. To understand how the program works in detail, you need to know something about Java language.

All simulations at my website calculate the trajectories of all bodies by this program and display them. That is the solution of n-body problem.

 

[remcook]

..but how is that different from just numerically integrating the equations of motion:

\ddot{\vec{r}_j} = \sum_{i=1}^n -\frac{GM_i}{r_{ij}^3} \vec{r}_{ij}

(for i is not equal to j, no bar means magnitude)

and

\vec{r}_{ij} = \vec{r}_j - \vec{r}_i


according to your definition, this is a solutions as well then:

http://www.cita.utoronto.ca/~dubinski/nbody/

do you agree?

 

[Billy T]

Already the two body general solution is a horrendous chore to do, and took the work of many famous physicists to solve completely (for the case of a 1/r potential). Unfortunately the integral form is nasty (elliptical integrals), but they have been done.

Change the potential, and you can no longer get a closed form stable solution...

Unless I misunderstand you, you are wrong. all that is required for the analytic solution of any "central -force" two-body problem is to convert it to a one body problem with a fixed center (At old center of mass), and the one body with "reduced mass" equal to mM/(m+M). The potential can have any form (power) you like. Check out any high level classical mechanic book - perhaps even a google search on {"reduced mass" AND "central force"} will show you this.

 

[khavel]

Hi Remcook:

I do not know how they do it on that University of Toronto web site. Does anybody know?

khavel

http://www.grevytpress.com/model.html

 

[AntonVrba]

http://count.ucsc.edu/~rmont/papers/list.html

Above a link to papers using mathematics to define the orbits and below a link to Java animation of n-body orbits

http://www.soe.ucsc.edu/~charlie/3body

Khavel how does your work and above correlate?

Some very fancy orbits do exist - completely unexpected, like a celestial ballet

 

[khavel]

Very interesting web sites. Nice simulations!

The unrestricted simulations done by my program are interactive. You can edit the values of the initial conditions of the bodies and observe their amended trajectories.

 

[PeteSF]

Hi khavel,
Your simulations work incrementally, right?
As in each step is calculated from the previous step?
And the smaller the time increment, the more accurate the simulation, right??

 

[PeteSF]

Quote from khavel's site (http://www.grevytpress.com/enbody.pdf ):

In an n-body gravitational system, the trajectory of anybody generally depends on the masses, positions, and velocities of all remaining bodies.
To determine the trajectories of all bodies, we set a time interval deltaT for the recalculation. Obviously, the shorter the time interval results in more precise calculation of the trajectories. At the beginning of the time interval we know the masses, positions, and velocities of all bodies. During time interval deltaT we calculate for each body the sum of accelerations imparted by gravitation of all other bodies, from their masses and positions. Then, for each body, we respectively integrate the sums of the accelerations over time interval deltaT, to obtain the increment of its velocity. We add the increment of its velocity to its previous velocity, to obtain its new velocity. Then we integrate its new velocity over time interval deltaT, to obtain the increment of its position. We add the increment of its position to its previous position, to obtain its new position. Since acceleration, velocity, and position are vectors, all calculations are done with their x and y components.
We keep repeating all calculations, over successive time intervals deltaT, using the new positions and velocities of all bodies. This method of calculation is valid for any number of bodies. Needless to say, the complexity of calculation increases with the number of bodies.

'nuff said. Khavel has produced an incremental numerical simulation, not an analytic one. Nothing new here.