Kepler's 1st Law - Why Earth revolves in Ellipse,not circle?

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Kepler's 1st Law - Why Earth revolves in Ellipse,not circle?

Postby yusri106 » Sat Nov 07, 2009 11:43 am UTC

Hello ,

Can somebody help me to understand, why Earth revolves the Sun in ellipse shape, not in circle. :roll: :roll:

I understand that, there is no such thing such as perfect circle, but this doesn;t answer why Earth revolves in ellipse.

At first, I assumed the reason why planets (in this case I focus to Earth only), revolves in ellipse are
[*] Due to gravity attraction from other planets
[*] or due to unequal distribution of mass on the Sun, therefore different force of attraction to the Earth,

But I was proven wrong by my lecturer, and my lecturer asked me to find it out myself. The problem is I still can't find the better explanation,

I know this problem is related to Kepler's 1st law, all planets are revolve in ellipse shape and Sun is the focus... but I still couldn't understand his concept.

Hope anyone of you can help me on this...


Thank You
:mrgreen: :D

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Re: Kepler's 1st Law - Why Earth revolves in Ellipse,not circle?

Postby Diadem » Sat Nov 07, 2009 12:00 pm UTC

Well look at the laws of physics. You'll find that possible orbits are ellipses. That's just what gravity allows.

So why do revolve in ellipses rather than circles? Well, first of a all a circle is an ellipse. Just a special case, an ellipse with an eccentricity of 1. So circular orbit is certainly allowed. Just quite unlikely. Getting a circular orbit would be like throwing a trillion-sided die and rolling 1. Quite unlikely.
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Re: Kepler's 1st Law - Why Earth revolves in Ellipse,not circle?

Postby yusri106 » Sat Nov 07, 2009 1:37 pm UTC

Hello Diadem,

Is that means that the Earth rotates in ellipse (with two focus) because to rotates in perfect circle is very rare.

So it is nothing relates to like gravitational force of sun, or maybe any other forces acts on the Earth?

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Re: Kepler's 1st Law - Why Earth revolves in Ellipse,not circle?

Postby Diadem » Sat Nov 07, 2009 1:55 pm UTC

Actually those forces act to make our orbit more circular. Because we're very close to circular. Imagine an solar system that is forming. There's a large chunk of matter in the middle, that's the proto-sun. All around it at other chunks of matter are forming (the planets). They of course inherent the overal rotational direction form the gas cloud that is forming the new solar system, so they all rotate in more orless the same direction. But you expect large random fluctuations. The orbits shouldn't be anywhere near circular.

But over time they smooth out due to the forcing acting on them, and they become more and more circular.
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Re: Kepler's 1st Law - Why Earth revolves in Ellipse,not circle?

Postby DrSir » Sat Nov 07, 2009 5:25 pm UTC

Also, if the orbit is an ellipse, wouldn't it tend to get more ellipsoid? Every orbit it would get pulled more during its closer trajectories, and less pulled on its farther ones. That would make it even more ellipsoidal right?

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Re: Kepler's 1st Law - Why Earth revolves in Ellipse,not circle?

Postby Goemon » Sat Nov 07, 2009 6:28 pm UTC

Imagine you're floating out in empty space far away from the sun, and there aren't any planets circling the sun.

If you throw a baseball directly at the sun, it just falls straight into the sun, yes?

Now throw another baseball so that it just misses hitting the sun. It starts off moving relatively slowly when it leaves your hand, but as it gets closer to the sun, it falls faster and faster. Its path is initially off to one side of the sun, but as it skims past the sun, moving really fast, it gets whirled around the far side and whips back out in your direction. It passes very close to the back side of the sun, being pulled very hard by the sun's gravity. It slows down as it flies away from the sun's gravity. By the time it reaches you, it's moving the same speed it had when you threw it.

The baseball's path in this case is a long thin ellipse: it speeds up as it gets closer to the sun, and the sun's stronger gravitational pull in that region makes its path curve around sharply. But as it's flying away again (and slowing down), the sun's gravity becomes weaker, so its path is much straighter as it comes back to you.

If (and only if) you throw the baseball at right angles to the sun, and at just the right speed, the ball will swing all the way around the sun at constant speed in a perfect circle and come back to you from the opposite direction. For any other direction or any other speed, the ball will speed up and curve more sharply as it gets near the sun, and slow down and curve more slowly when it's farther away.
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Re: Kepler's 1st Law - Why Earth revolves in Ellipse,not circle?

Postby You, sir, name? » Sat Nov 07, 2009 7:39 pm UTC

Can't you apply a statistical argument? The set of orbits with eccentricity ~ 1 is much smaller than the whole set of possible orbits. Therefore, most orbits are going to be elliptical.

--edit-- Seems Diadem beat me to this argument. Oh well, superfluous redundancy never killed anyone.
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Re: Kepler's 1st Law - Why Earth revolves in Ellipse,not circle?

Postby doogly » Sat Nov 07, 2009 7:52 pm UTC

A statistical argument would have us living on a comet. Unless you want to make an anthropic selection in there too. Depending on your physics teacher, this may either delight or enrage them.
The most useful form for eccentricity in this case is at http://scienceworld.wolfram.com/physics ... icity.html
Circular orbits have just the right amount of angular momentum to be a circle. A little more, and you're a hyperbola. A little less and you're an ellipse.
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Re: Kepler's 1st Law - Why Earth revolves in Ellipse,not circle?

Postby You, sir, name? » Sat Nov 07, 2009 8:31 pm UTC

doogly wrote:A statistical argument would have us living on a comet. Unless you want to make an anthropic selection in there too. Depending on your physics teacher, this may either delight or enrage them.


Not necessarily. Just toss in some appropriate probability density function, and you will get a behavior that matches our observations.
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Re: Kepler's 1st Law - Why Earth revolves in Ellipse,not circle?

Postby Nintendon't » Mon Nov 09, 2009 6:18 am UTC

Diadem wrote:a circle is ... an ellipse with an eccentricity of 1. Quite unlikely.

I think you mean zero.

And @ OP: you must consider when the solar system was being formed, there were many collisions between chunks of rock and different forms of matter. This was mostly how the planets were created. Having such a haphazard creation story would cause some quantity of eccentricity greater than zero, don't you think?

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Re: Kepler's 1st Law - Why Earth revolves in Ellipse,not circle?

Postby spdqbr » Mon Nov 09, 2009 7:11 am UTC

One of my favorite things from college was deriving the possible shapes of orbits using nothing but Newton's laws and vector calculus. If you have experience with those two things, it's not too difficult and I found it very satisfying...
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Re: Kepler's 1st Law - Why Earth revolves in Ellipse,not circle?

Postby rflrob » Mon Nov 09, 2009 7:43 am UTC

doogly wrote:Circular orbits have just the right amount of angular momentum to be a circle. A little more, and you're a hyperbola. A little less and you're an ellipse.


Actually, a little more or less and you're still an ellipse. I think you might be thinking of a parabolic orbit, which is the lowest energy unbound orbit. A little more and you're a hyperbola, a little less and you're a (very eccentric) ellipse.

@OP: What class are you taking that this comes up? A good calculus-based physics class should probably derive that it's an ellipse (and maybe you just haven't gotten there yet). I seem to recall it's possible to do it without calculus, too, but that way requires a fair amount of cleverness, whereas you can just start with [imath]F = ma = G \frac{M m}{r^2}[/imath] and slog though the math to get it if you do have calculus.
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Re: Kepler's 1st Law - Why Earth revolves in Ellipse,not circle?

Postby Mr Jack » Mon Nov 09, 2009 8:05 am UTC

Goemon wrote:Imagine you're floating out in empty space far away from the sun, and there aren't any planets circling the sun.

If you throw a baseball directly at the sun, it just falls straight into the sun, yes?

Now throw another baseball so that it just misses hitting the sun. It starts off moving relatively slowly when it leaves your hand, but as it gets closer to the sun, it falls faster and faster. Its path is initially off to one side of the sun, but as it skims past the sun, moving really fast, it gets whirled around the far side and whips back out in your direction. It passes very close to the back side of the sun, being pulled very hard by the sun's gravity. It slows down as it flies away from the sun's gravity. By the time it reaches you, it's moving the same speed it had when you threw it.

The baseball's path in this case is a long thin ellipse: it speeds up as it gets closer to the sun, and the sun's stronger gravitational pull in that region makes its path curve around sharply. But as it's flying away again (and slowing down), the sun's gravity becomes weaker, so its path is much straighter as it comes back to you.

If (and only if) you throw the baseball at right angles to the sun, and at just the right speed, the ball will swing all the way around the sun at constant speed in a perfect circle and come back to you from the opposite direction. For any other direction or any other speed, the ball will speed up and curve more sharply as it gets near the sun, and slow down and curve more slowly when it's farther away.



This is the best answer. It's doesn't prove it, but it is easier to comprehend than a mathematical argument unless you are some kind of savant who sees a baseball coming and works out where it's going to land with calculus. There is more to physics than equations; there are plenty of people I know who take an interest in physics without even being able to do what we would consider near-elementary maths. To prove that it's an ellipse with maths demonstrates that you are good at maths; to argue it in words demonstrates you understand the physics.

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Re: Kepler's 1st Law - Why Earth revolves in Ellipse,not circle?

Postby doogly » Mon Nov 09, 2009 2:14 pm UTC

False! It means you can wave your hands and say something convincing. The problem is that this sounds great but you can easily do it for wrong statements too.
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Re: Kepler's 1st Law - Why Earth revolves in Ellipse,not circle?

Postby ThinkerEmeritus » Mon Nov 09, 2009 2:38 pm UTC

I agree with Mr. Jack on this one. The OP wanted to understand why planets move in ellipses; proof was not the issue. A good qualitative discussion is the best response.

If you are interested in the ideas and wouldn't be able or willing to follow the math, make sure that the person you listen to has adequate credentials so you don't get fooled. In other words, he should have studied the subject far enough to be trustworthy and not show signs of being a crackpot or otherwise having an ax to grind.

If you are interested in the math, understanding qualitatively what is or at least might be happening is still the first step in constructing a proper mathematical proof. Just diving into the formalism is apt to send you on a long, wandering path before you get to the correct result.
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Re: Kepler's 1st Law - Why Earth revolves in Ellipse,not circle?

Postby Charlie! » Mon Nov 09, 2009 7:08 pm UTC

DrSir wrote:Also, if the orbit is an ellipse, wouldn't it tend to get more ellipsoid? Every orbit it would get pulled more during its closer trajectories, and less pulled on its farther ones. That would make it even more ellipsoidal right?

Nope. An intuitive reason is because as it goes closer it also speeds up, so the "pulling" has proportionately less effect. It all balances out (at least for gravity).
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Re: Kepler's 1st Law - Why Earth revolves in Ellipse,not circle?

Postby douglasm » Mon Nov 09, 2009 10:14 pm UTC

With perfectly Newtonian gravity, no objects but the Sun and a single planet, and ignoring other factors like the Sun turning into a red giant, the planet will orbit in the exact same perfect ellipse forever.

Another explanation for the title question:
Suppose you have a planet that actually is orbiting in a perfect circle. One day something happens to slow that planet down a bit. Acceleration from the Sun's gravity is still the same, though. This planet isn't going fast enough to maintain its circular orbit any more, and gravity will pull it closer to the Sun. As it gets closer, it speeds up - conservation of energy dictates the reduction in gravitational energy is balanced by increased kinetic energy. Eventually it will reach the point where its speed is right for a circular orbit again, but it's been heading towards the Sun to get to that point so it's not going in the right direction for a circle. It keeps on getting closer and faster until it actually goes past the Sun and the direction its inertia is going in sends it back outwards. It heads out away from the Sun, this time slowing down, and again passes the circular orbit point because even though its speed is right at one point its direction is not. The Sun's gravity eventually pulls on it enough that it stops heading away and starts coming back in again, and the math just so happens to work out that it does this at exactly the point where it originally started from and the shape of its path is a perfect ellipse.

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Re: Kepler's 1st Law - Why Earth revolves in Ellipse,not circle?

Postby Mr Jack » Tue Nov 10, 2009 3:19 am UTC

douglasm wrote:, and the math just so happens to work out that it does this at exactly the point where it originally started from and the shape of its path is a perfect ellipse.



The math doesn't happen to work out. It has to. It's conservation of energy.

Edit:(disclaimer: I haven't thought much about this, perhaps it isn't. But I'd bet a lot of money that it is; so long as I knew I wasn't betting with a physicist)




Edit 2: to the OP. Not sure whether you know this or not, but Kepler's first law didn't say anything about why it's an ellipse, it just said that it is, without explanation.

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Re: Kepler's 1st Law - Why Earth revolves in Ellipse,not circle?

Postby achan1058 » Tue Nov 10, 2009 3:49 am UTC

DrSir wrote:Also, if the orbit is an ellipse, wouldn't it tend to get more ellipsoid? Every orbit it would get pulled more during its closer trajectories, and less pulled on its farther ones. That would make it even more ellipsoidal right?
No, because when you are closer, you are also moving at a quicker speed.

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Re: Kepler's 1st Law - Why Earth revolves in Ellipse,not circle?

Postby phlip » Tue Nov 10, 2009 4:02 am UTC

Mr Jack wrote:The math doesn't happen to work out. It has to. It's conservation of energy.

It's conservation of energy that it would end up at the same distance from the sun, but not that it would be in the same position... it would be entirely fine by conservation of energy that it reach the same distance from the sun, but in a different direction. It Just So Happens to work out that the whole aphelion/perihelion cycle takes it exactly 360 degrees around the sun. This is just a result of Newtonian mechanics... in General Relativity, IIRC, then a single aphelion/perihelion cycle will take it close to, but not exactly, a full orbit of the sun (though I can't remember offhand whether it's slightly more or slightly less).

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Re: Kepler's 1st Law - Why Earth revolves in Ellipse,not circle?

Postby Twistar » Tue Nov 10, 2009 4:20 am UTC

phlip wrote:
Mr Jack wrote:The math doesn't happen to work out. It has to. It's conservation of energy.

It's conservation of energy that it would end up at the same distance from the sun, but not that it would be in the same position... it would be entirely fine by conservation of energy that it reach the same distance from the sun, but in a different direction. It Just So Happens to work out that the whole aphelion/perihelion cycle takes it exactly 360 degrees around the sun. This is just a result of Newtonian mechanics... in General Relativity, IIRC, then a single aphelion/perihelion cycle will take it close to, but not exactly, a full orbit of the sun (though I can't remember offhand whether it's slightly more or slightly less).


This is why Mercury has perturbations in its orbit?

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Re: Kepler's 1st Law - Why Earth revolves in Ellipse,not circle?

Postby phlip » Tue Nov 10, 2009 4:29 am UTC

It could be... it could also be because the universe isn't ideal, and there are a lot of other variables (Mercury would be gravitationally attracted to other objects too, not just the sun... the sun isn't a static object, and there's a fair bit of fluctuation, which could be relevant that close... I imagine the solar wind would be stronger as you get closer too... umm, magnetism?). I don't really know enough about the topic to say what all the important factors in Mercury's orbit would be, and what things are irrelevant.

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Re: Kepler's 1st Law - Why Earth revolves in Ellipse,not circle?

Postby Diadem » Tue Nov 10, 2009 4:30 am UTC

Isn't orbital precession already present in Newtonian Mechanics? I always thought it was, except for a few anomalies with Mercuries orbit that needed GR to explain.
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Re: Kepler's 1st Law - Why Earth revolves in Ellipse,not circle?

Postby doogly » Tue Nov 10, 2009 6:59 am UTC

Ah yes, there are three conserved quantities in Newtonian Keplerian motion: energy, angular momentum, and the Lens-Runge-Laplace Vector. In GR, only the first two are conserved; LRL is no longer conserved, and the perihelion precesses.
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Re: Kepler's 1st Law - Why Earth revolves in Ellipse,not circle?

Postby Waylah » Tue Nov 10, 2009 8:06 am UTC

Diadem wrote:... a few anomalies with Mercuries orbit that needed GR to explain.


That is so cool :)

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Re: Kepler's 1st Law - Why Earth revolves in Ellipse,not circle?

Postby yusri106 » Tue Nov 10, 2009 11:20 am UTC

Wow.. this topic become more interesting... but at the end of the day, I am still blur why it is so... :roll: :roll: hehe but it is ok.

Is this statement correct? :

since Earth is revolve in ellipse, therefore the centripetal force acting towards the elliptical point is keep on changing due to continuously changing in revolving radius around the elliptical point.

is it correct but negligible due to its tiny difference?

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Re: Kepler's 1st Law - Why Earth revolves in Ellipse,not circle?

Postby phlip » Tue Nov 10, 2009 12:39 pm UTC

That would be correct. When the Earth gets closer to the Sun, the gravitational force increases (by the inverse-square law, at least in Newtonian physics).

It's negligible only because the orbit's only slightly elliptical... other bodies with much more eccentric orbits (like comets) will have a very large difference in the gravitational force as they go around their orbit.

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Re: Kepler's 1st Law - Why Earth revolves in Ellipse,not circle?

Postby PM 2Ring » Tue Nov 10, 2009 2:09 pm UTC

Twistar wrote:This is why Mercury has perturbations in its orbit?
Partly. According to Newtonian theory, Mercury's orbit should precess due to the effects of the other planets in the system, and because the Sun isn't a perfect sphere. But GR predicts the precession should be a little more than what Newtonian theory predicts. And of course, GR agrees with observation.

Remember, we're only talking about a very tiny amount of precession here: the anomaly due to GR is only 43 arc-seconds per century. The precession due to GR is greater for a planet orbiting a heavier star, or with a closer minimum distance to the star, or with a more eccentric orbit. Also, the precession will be greater around a star that spins faster, and is therefore more oblate.

From WIkipedia Sources of the precession of perihelion for Mercury. Amounts are given in arcsec/century.

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Amount         Cause
------          -----
5025.6       Coordinate (due to the precession of the equinoxes)
531.4        Gravitational tugs of the other planets
0.0254       Oblateness of the Sun (quadrupole moment)
42.98±0.04   General relativity
5600.0       Total
5599.7       Observed


Note that the top entry in this table is the largest contribution, but it has nothing to do with the behaviour of Mercury's orbit. It is due to the fact that we base our celestial coordiate system on the orbit of the Earth, and the equinox points. These points slowly travel backwards around the orbit (relative to the stars). In other words, it's a very slowly rotating coordinate system.

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Re: Kepler's 1st Law - Why Earth revolves in Ellipse,not circle?

Postby Diadem » Tue Nov 10, 2009 3:16 pm UTC

What is that total a round number? Just a coincidence? Or some deeper meaning?
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Re: Kepler's 1st Law - Why Earth revolves in Ellipse,not circle?

Postby Mokele » Tue Nov 10, 2009 5:01 pm UTC

Is there any path other than an ellipse/circle that doesn't result in the object just rocketing off into space, never to be seen again? (leaving aside weird crap like figure-8s around multiple objects)
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Re: Kepler's 1st Law - Why Earth revolves in Ellipse,not circle?

Postby douglasm » Tue Nov 10, 2009 6:40 pm UTC

Assuming only two objects and no interactions but gravity, no. If an object's speed is below escape velocity, it will follow an ellipse (a circle is just an ellipse with 0 eccentricity). If an object's speed is exactly escape velocity, it will follow a parabola. If it is greater than escape velocity, it will follow a hyperbola.

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Re: Kepler's 1st Law - Why Earth revolves in Ellipse,not circle?

Postby Diadem » Tue Nov 10, 2009 9:52 pm UTC

Are stable non-elipsoid orbits possible around multiple bodies? I know we can't solve the three body problem. But we must have some solutions for special cases. Does anyone know?
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Re: Kepler's 1st Law - Why Earth revolves in Ellipse,not circle?

Postby PM 2Ring » Wed Nov 11, 2009 6:10 am UTC

Diadem wrote:Are stable non-elipsoid orbits possible around multiple bodies? I know we can't solve the three body problem. But we must have some solutions for special cases. Does anyone know?

Yes:
Mokele wrote:weird crap like figure-8s around multiple objects

John Baez wrote:In 1993 Cris Moore discovered solutions of the gravitational n-body problem where the particles' paths lie in a plane and trace out braids in spacetime! I spoke about these in "week181".

More recently, Moore and Michael Nauenberg have found solutions with cubic symmetry and vanishing angular momentum, and made movies of these:
Image

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Re: Kepler's 1st Law - Why Earth revolves in Ellipse,not circle?

Postby Diadem » Wed Nov 11, 2009 9:58 am UTC

Wow. That is awesome.

Is that a stable equilibrium? Would be awesome of if that could exist in real life.
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Re: Kepler's 1st Law - Why Earth revolves in Ellipse,not circle?

Postby Talith » Wed Nov 11, 2009 1:21 pm UTC

Something tells me that the set up isn't stable, a small pertubation of one particle into the centre will probably attract it more to the other 3 and so cause a collapse.

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Re: Kepler's 1st Law - Why Earth revolves in Ellipse,not circle?

Postby PM 2Ring » Wed Nov 11, 2009 2:42 pm UTC

Issues of stability are discussed in the various papers linked to on John Baez's site. Just follow the "John Baez" link at the head of the quote in my earlier post. I'm not sure about the setup I posted above, but I'm pretty sure it's believed to be stable, as is the 3 body figure 8.
Cris Moore wrote:We also give numerical evidence that a planar 3-body orbit [...] is dynamically stable.

There are quite a few other orbit anims on Cris's gallery page, some planar, like the 21 body figure 8, and some with cubical symmetry like the one above, so please check out the Baez link I gave earlier. :)

Note that the above diagram is showing the orbits in a rotating framework to make their 3D structure more obvious. In reality, the cubical frame would not be rotating (unless you're in a really fast spaceship :) ).

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Re: Kepler's 1st Law - Why Earth revolves in Ellipse,not circle?

Postby thoughtfully » Wed Nov 11, 2009 9:15 pm UTC

The most well-known solved special cases to the three-body problem are the Lagrange Points.
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Re: Kepler's 1st Law - Why Earth revolves in Ellipse,not circle?

Postby Hellzapoppin » Thu Nov 12, 2009 1:39 am UTC

My first post on xkdc forums! I'll try not to make a tit of myself.

To explain why the earth has an elliptical orbit I'd look at how the earth initially formed. Basically I imagine a turmoil of impacts on the proto planet as it clears it's orbital path of debris, be the impacts small and sustained or large and powerful, they will influence the planet to be's orbit in a near vacuum.

This is favouring the statistical explanations mentioned above, the chances of a perfect circle are just very unlikely, but not impossible. Somebody mentioned statistical explanations could give us a comet like trajectory (potentially). This is unlikely as comets are either captured free roaming bodies or results of ancient collisions. Different rules and constraints for planets and comets.

Hope I'm somewhere in the ball park, didn't notice this version being posted anywhere.

Also: Hi xkdc forums! I hope we can be friends.

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Re: Kepler's 1st Law - Why Earth revolves in Ellipse,not circle?

Postby shadydentist » Thu Nov 12, 2009 5:45 pm UTC

Hellzapoppin wrote:This is favouring the statistical explanations mentioned above, the chances of a perfect circle are just very unlikely, but not impossible.


Well... technically it *is* impossible to get a perfect circle. Whenever you take a measurement of a continuous distribution (in this case, eccentricity), you can only ever get a probability for getting something between two points. So if you wanted to get something very close to a circle, you could ask "what is the probability of getting an ellipse with an eccentricity between 0 and .00001", but the probability of getting exactly zero is meaningless.

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Re: Kepler's 1st Law - Why Earth revolves in Ellipse,not circle?

Postby PM 2Ring » Fri Nov 13, 2009 1:07 am UTC

In a stable system, we expect most inner system orbits to be near circular and mostly in the same plane (the primary's equatorial plane). This is largely due to the way angular momentum works as the cloud of gas & dust forming the system collapses.

As the system condenses, bodies in elliptical paths are more likely to collide with other bodies. Once the planetary orbits are mostly stabilized, those in more elliptical orbits will be more likely to have their orbits perturbed by other bodies, and those in more elliptical orbits will also perturb the orbits of other bodies the most. These two effects combine to gradually make almost all inner system orbits close to circular.


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