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Old 2009-02-24, 00:07   #1
Flatlander
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Default Orbits of comets

Quote:
4. There are no stupid questions, only stupid questioners;
Stupid questioner here!

(Ignore friction, air resistance etc. as necessary.)

If I drop a swing from a certain height it rises up to the same height on the other side and repeatedly goes to the same height.

If I put a heavy ball on a trampoline and roll a lighter ball passed it, it will likewise go back and forwards the same distance either side.

What has changed in the maths/physics that allows a comet to come in from the Oort Cloud, skim passed the sun and then back out to the Oort cloud?

In other words, why don't they 'overshoot' the sun to the same distance the other side?

http://www.st-andrews.ac.uk/~bds2/ltsn/ljm/JAVA/COMETORB/COMET.HTM

Keep it simple now.
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Old 2009-02-24, 03:11   #2
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Quote:
Originally Posted by Flatlander View Post
What has changed in the maths/physics that allows a comet to come in from the Oort Cloud, skim passed the sun and then back out to the Oort cloud?
Nothing changed in the maths or physics; one just has to properly analyze each situation -- they're not identical, because of path constraints present in the first two cases that don't apply to the comet.

Quote:
(Ignore friction, air resistance etc. as necessary.)

If I drop a swing from a certain height it rises up to the same height on the other side and repeatedly goes to the same height.
It behaves as a pendulum. Note that gravitational pull is directed toward the center of the earth throughout, not toward some point within the arc of the swing. The support chains/ropes/whatever prevent the swing from orbiting the Earth's center in an ellipse like the comet's.

Quote:
If I put a heavy ball on a trampoline and roll a lighter ball passed it, it will likewise go back and forwards the same distance either side.
(... and we'll assume the heavy ball remains motionless.) Here, the gravitational force again is always directed toward Earth's center, but the trampoline's curved (dimpled by the heavy ball) surface constrains the light ball's motion to be parallel to it (instead of orbiting Earth's center in an ellipse).

It would be useful to diagram the force(s) acting on the light ball as a right triangle (http://www.glenbrook.k12.il.us/GBSSC...ors/u3l1b.html has illustrations worth more than my following text) with:

the hypotenuse pointing from the light ball toward the center of the earth and its magnitude representing the gravitational force,

one of the two shorter sides directed from the light ball, parallel to the trampoline's surface, toward the center of the depression caused by the heavy ball. (The angle between the trampoline's surface and horizontal = 90 degrees minus the angle between this side and the vertical hypotenuse.) This side's magnitude represents the portion of the gravitational force that can be considered an apparent force directed from the light ball toward the heavy ball.

and

the other side, connecting the hypotenuse to the first short side, perpendicular to the trampoline surface (and, thus, also perpendicular to the first short side). This side's magnitude represents the portion of the gravitational force that can be considered normal (perpendicular) to the trampoline's surface.

By Pythagoras, the square of the hypotenuse's magnitude (= force of gravity) will be equal to the sum of the squares of the magnitudes of the two other sides.

In the swing case, the gravitational force could be resolved as the vector sum (hypotenuse of that right triangle) of a force directed along the path of the swing and a force directed perpendicular to that path (force that the support chains/ropes/whatever exert to hold the swing.

Quote:
In other words, why don't they 'overshoot' the sun to the same distance the other side?

http://www.st-andrews.ac.uk/~bds2/ltsn/ljm/JAVA/COMETORB/COMET.HTM
Short, inadequate answer: because gravity is inversely proportional to the square of the distance, not to the first power of distance.

Longer answer:

(1) Note on that page that the comet's orbit is an ellipse; that wasn't true of the path in either previous case. Also, the Sun is at one of the foci of the ellipse. (The path of the ball along the trampoline's surface might be close to an ellipse in an ideal case, but it would be centered on the large ball rather than having the large ball at one of its foci as in the comet's case.)

(2) For the comet, the gravitational force is always directed toward the Sun at one focus of the ellipse. In the previous two cases, gravity always pulled to one side of the periodic path, not to some point within it. Only some constraint (the swing's support chains, the trampoline's surface) prevented the object from following an elliptical path like the comet's (but with Earth's center at a focus of the ellipse).

Isaac Newton first proved by mathematics (he had to invent calculus to do it, but actually used only plane geometry for most of the proof) that the orbit of a small body under influence of gravity from a large body, in space with no resistance, would be in the shape of an ellipse with the large body at one of the foci of the ellipse. (http://en.wikipedia.org/wiki/Philoso...ia_Mathematica) He also showed that if gravity were inversely proportional to the first power of the distance between the two bodies, the ellipse would be centered on the large body. But being inversely proportional to the square of the distance makes the ellipse shape have the large body at one of its foci instead of being at the center. (http://members.tripod.com/~gravitee/)

Perhaps you can think of it as: inverse-square-of-distance gravity bends the path more strongly near the Sun than an inverse-first-power-of-distance force would.

Quote:
Keep it simple now.
Well, I (or someone) can try to further simplify any of the above if you need it. Isaac Newton did the hard work; that's why the Brits have his picture on their money. But ... it did take Isaac Newton to first work it out, and we're no Isaac Newtons.

Last fiddled with by cheesehead on 2009-02-24 at 03:52
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Old 2009-02-24, 03:11   #3
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So comets do. Many get influnced on their trip by Jupiter and the other planets. This can cause them to not be able to escape back out.

Also a big wild-card is the action of the sun's heat on the comet. The off-gassing that occurs can slow the comet enough as well.
The heat starts on the side of the comet that is facing the sun, which happens to be the leading edge at first. Gas coming out of that side acts like retrorockets, slowing the orbital speed.
(An odd thing about orbital mechanics, change the speed near the sun [or earth in the case of satellites] and you change the total distance you will achieve away from the sun; change the speed away from the sun and you change how close you will get).

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Old 2009-02-24, 04:06   #4
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Quote:
Originally Posted by Flatlander View Post
What has changed in the maths/physics that allows a comet to come in from the Oort Cloud, skim passed the sun and then back out to the Oort cloud?

In other words, why don't they 'overshoot' the sun to the same distance the other side?
Simplest answer: it does overshoot the same distance to the other side, that's why it ends up out in the Oort cloud again. You may be confused by the fact that it doesn't go out in the diametrically opposite direction to that in which it came in. Neither does the ball on the trampoline in your second example. If it is aimed slightly to one side of the ball in the center, i.e., it does not come in directly toward the heavy ball in the center, it will actually follow a curved path and go out at an angle inclined to the direction it came in. The pendulum is constrained in the absence of external forces to swing in a plane, which forces the rise on the outgoing side to be the mirror image of the fall on the incoming side.
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Old 2009-02-24, 10:26   #5
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Quote:
Originally Posted by Flatlander View Post
Keep it simple now.
I have been reviving my pedagogical skills
in the homework part of Physics Forums.
Makes a nice change to have an unpure maths
question raised here.

I hope I'm missing something here, but haven't you
computed orbits in an inverse square law before?

Or heard of Kepler?

David

Last fiddled with by davieddy on 2009-02-24 at 10:26
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Old 2009-02-24, 11:51   #6
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Quote:
Originally Posted by cheesehead View Post
Short, inadequate answer: because gravity is inversely proportional to the square of the distance, not to the first power of distance.
Bingo. This is very different from the pendulum where, to a first approximation, the resultant force is directly proportional to the distance from the equilibrium point. The questioner appears to think that the quite different ways that these forces operate should give rise to similar behaviour. They don't.

In the case of the two balls on a trampoline, I suspect that the behaviour would be much closer to that of orbiting planets. If in practice it isn't then that would be due to friction.
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Old 2009-02-24, 22:05   #7
Flatlander
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Thanks for the answers. I've skimmed through and things are clearer already.

Quote:
I hope I'm missing something here, but haven't you computed orbits in an inverse square law before?
Read about, yes. Computed? Maybe years ago.

And when you mentioned Kepler my first thought was an image (from somewhere) of a segmented ellipse; each segment representing the same period of time and having the same area. (But with the sun not in the middle; which raised the question.)

Some boring background information:
I left school (1980) knowing that I had a job waiting for me as a Painter and Decorator so I became very lazy for the last couple of years. This resulted in my not even taking a physics exam. A week before I was due to take my maths exams I realised that I was going to fail, so I force-fed myself and managed to get a C in O-level maths and a B in O-level sadistics, sorry statistics.

To (mis)quote Douglas Adams:
"I wish I'd listen to my mother."
"Why, what did she say?"
"I don't know, I didn't listen."

Since then I have read (and forgotten) lots of popular science and maths books. So my head is filled with things like spaceships approaching black holes, cats in boxes and fuzzy electrons, but not practical procedures for solving things etc.

Chinese proverb:
“Tell me and I'll forget; show me and I may remember; involve me and I'll understand.”

Things also have to be explained in a certain way for me to understand them. (I have Thickitis.) I have to visualize things. When things become too abstract my eyes glaze over and I go and make tea.

For example: "6 Roots of unity". It was only after reading an explanation in the book "50 Mathematical Ideas you really need to know" that something clicked and I finally understood. (But that was six months ago, and now I've partially forgotten. At least I'll get to re-enjoy the discovery when I read it again.)

Because of family responsibilities and daily health issues, I have no opportunity to take a course in maths/physics at the local college or even on-line etc*. Also, I've come to realise that my main maths interest is in the properties of numbers themselves. (It would be an exaggeration to call it Number Theory.) Looking back, that is why I did better in Statistics than Maths; I enjoyed more of the lessons. And my main interest in Physics is the bizarre Relativity/Q.M. stuff. But I tend to rush through the groundwork to get to the exciting stuff and so retain relatively little.
I think, also, I like 'to know why', rather than 'to know'. ("Why 'inverse-square' instead of 'inverse' or 'inverse-cube'?" interests me more than "Let's calculate the course of this comet.")

In other words, please don't confuse my inquisitiveness with intelligence or even higher education! (Though that's probably very apparent by now! Maybe next time I'll log-out and pretend I'm a ten year old.)
This is summed up in the anagram below my avatar.

* Maybe it's time for me to take another look at the possibility of taking an on-line/correspondence course.

Appreciatively,
Chris
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Old 2009-02-24, 22:21   #8
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Quote:
Originally Posted by Mr. P-1 View Post
This is very different from the pendulum where, to a first approximation, the resultant force is directly proportional to the distance from the equilibrium point. The questioner appears to think that the quite different ways that these forces operate should give rise to similar behaviour. They don't.
Well... okay.

I would emphasize that the gravitational force is actually operating just the same in all three cases (but in the first two we're considering only Earth's gravity; in the third, only the Sun's gravity), and that only the additional forces exerted by the swing's supports in the first case, or the trampoline surface in the second case, when vectorily added to the gravitational force, cause the resultant force (that's the term that eluded me earlier -- thanks :) to make the path nonelliptical.

Quote:
In the case of the two balls on a trampoline, I suspect that the behaviour would be much closer to that of orbiting planets.
There the shape of the path depends on the shape of the trampoline surface -- it could be (more or less approximately) an ellipse centered on the depression's center (due to the resultant force being approximately inverse to distance from center for a certain surface shape). Or ... with a different certain shape of the trampoline surface, the resultant force could, I think, be approximately inversely proportional to the square of distance from center, shifting the elliptical path to one with a focus at the center!

Last fiddled with by cheesehead on 2009-02-24 at 22:38
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Old 2009-02-25, 17:39   #9
ewmayer
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Quote:
Originally Posted by Flatlander View Post
Stupid questioner here!

What has changed in the maths/physics that allows a comet to come in from the Oort Cloud, skim passed the sun and then back out to the Oort cloud?
Suggest you read up on your basic Newtonian orbital mechanics: Comet orbits, like those of planetary ones (neglecting 3rd-body perturbations, that is) follow the same "ellipse with the sun at one focus" law - just in the case of extremely elliptical orbits like those you describe, the other focus is way out in the Oort cloud, or the Kuiper belt, or whatever.

Any comets for which such an orbit is not quasi-stable, i.e. the "flying off the handle" behavior you describe, would have flown off (either into distant space or into some other orbital trajectory) billions of years ago. The ones we see passing close to the sun - with the exception of recently-perturbed-ones like Shoemaker-Levy - by definition have quasi-stable Newtonian elliptical orbits.

Last fiddled with by ewmayer on 2009-02-25 at 17:40
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Old 2009-02-25, 19:28   #10
cheesehead
 
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Ernst,

I think your response to Flatlander is off-target because it fails to take his quoted question's context into account.

1) That question is contrasted with two preceding examples in which paths were not off-centered ellipses, as they are in the comet case, so referral to "basic Newtonian orbital mechanics" is not appropriate. It is the apparently non-Newtonian-orbital deviations (caused by non-gravitational forces transmitted through path restraints) in the first two examples that needs explaining to show why the paths are so different.

2) The quoted question is followed in the original post by a restatement, "In other words, why don't they 'overshoot' the sun to the same distance the other side?" This shows that Flatlander is looking at the factor that in the first two examples the paths are not noticeably off-centered, whereas the comet's path _is_. Both the pendulum and the light ball on the trampoline make an approach to center (swing) or a closest point to center (trampoline), then continue in the same (swing) or approximately the same (trampoline) direction of travel on the other side of center to recede to approximately the same distance from center as they started.

Neither the comet's orbit, nor overshooting to the same distance but on the other side, is "flying off the handle" behavior. Instead, the comet's orbit is sorta single-handled, whereas the swing's and light trampoline ball's paths are, in effect, two-handled, all relative to the apparent center of attraction in each case.

- - -

Flatlander,

Please intervene if I am misinterpreting your original post.

Last fiddled with by cheesehead on 2009-02-25 at 19:34
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Old 2009-02-25, 20:02   #11
Flatlander
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Yes. It was the off-centredness that I was refering to.
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