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Post by speakertoanimals on Mar 23, 2011 13:25:49 GMT 1
You have confused the parametric equation for an ellipse, with the actual motion of a planet in an elliptical orbit, which gives a DIFFERENT graph of position against time.
Why? Because the parametric equation plots positions against ANGLE, not against TIME, because the physical path needs the additonal information, of stating HOW the parametric angle varuies with time.
SO, for the parametric equation, we have:
x = acos theta, y = b sin theta
where theta is just a PARAMETER, which when varied from 0 to 2 pi sweeps out the whole ellipse. Hence when we plot either coordinate against theta, we get a sinusoid. In terms of geometry, the angle theta is measured at the CENTRE.
For a physical orbit, we further have to specify HOW theta varies with time. Except according to Keplers laws, we now measure angles wrt one of the foci, to give us the equal areas swept out in equal time result. Hence when plotting projections of planetary motion, we would not necessarily expect to get SAMe projections as we did in the geometric case.
That is why some people seem to think there is a problem, we have a confusion between pure geometry and parametric equations (which just shows what we kind of knew, that an ellipse is just a squashed circle, hence can be written in terms of simple sinusoids), and actual physics, where the geometry of the orbit is the simple ellipse, but the dynamics of the orbit (the actual speed at various points on the path) is determined by central forces (ie directed towards one foci), and newtonian mechanicsand the conservation of momentum.
The exact motion in an orbit is a tad more complicated than simple geometry of ellipses, but is still expressed in terms of the usual trigonometric functions. But is too complicated for this thread, where the simple parametric equation for the path seems to be causing enough trouble....................
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Post by speakertoanimals on Mar 23, 2011 13:36:57 GMT 1
Some of us do have to sleep, and do some other things you know!
Of COURSE orbiting planets vary their linear speed, it is patently obvious that they MUST if their distance from the gravitating mass changes! A force is being exerted, and if they move at all other than totally tangential to the force (that is, if in anything other than a circular orbit), then we have a component of the acceleration in the same direction as the motion, hence we must have change in the linear speed, as well as change in the direction of motion. I short, for anything other than circular, velocity changes in magnitude as well as direction.
Also patently objvious from the fact that moving objects on earth have varying speed as they fall. Because air resistance aside (which gives terminal velocity), they are just moving in orbits as well as any planet, just that the orbit in their case tends to intersect the ground, which usually causes an abrupt stop to orbital motion..................
Conservation of angular momentum tells you the same thing, since angular momentum depends on distance, speed, and direction of velocity, then if distance varies, we would expect other stuff to vary as well to keep angular momentum constant. Which is Kepler and equal areas in a nutshell.
SO, acting as if stating that linear speed varies is somehow surprising and extraordinary, is rather silly. Basic mechanics would lead us to expect this result, and it is the very special case of circular orbits that is extraordinary, NOT the varying speed in ANY OTHER orbit, be it elliptical, hyperbolic, or parabolic. Falling objects increase their linear speed, and it holds for planets falling closer to the sun just as it holds for apples falling to earth.
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Post by carnyx on Mar 23, 2011 15:45:21 GMT 1
STA,
Now, could you answer the remaining and most important question as to whether the changing linear speed in an elliptical orbit is symmetrical about both axes? In other words do the profiles of linear speed of the object as it approaches and recedes from either foci, match?
(A more humourous way of putting it is to ask whether the linear speed at the outer points of either latus rectum, are the same ...)
And incidentally, how will a person in the satellite feel the resulting changes in acceleration?
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Post by speakertoanimals on Mar 23, 2011 16:45:41 GMT 1
symmetrical about BOTH axes? Of course not!
When approaching or receeding from one focus, symmetric since motion under time reversal is just it goes the other way round. Hence same speeds approaching the sun as when departing.
Other, empty focus -- not the same when approaching empty focus as whne approaching full one, I think I said that earlier when I pointed out that motion wrt empty focus is DIFFERENT to that near full focus. Hence the empty one can't be viewed as if it were some shadow object about which the planet is rotating. Its not, because for full focus, greatest speed is at point of closest approach, whereas for empty one, smallest speed is at closest approach to empty focus, which is furthest from full one.
He won't feel a damn thing, because he is in FREEFALL all the way round the orbit! Come one, basic motion under gravity here, orbits aside!
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Post by carnyx on Mar 24, 2011 10:44:37 GMT 1
@sta,
At last! A satellite will describe an elliptical orbit, with a continous rate of change of velocity that will not be constant i.e it wil not be sinusoidal.
Now, if you imagine a planet with a moon in an elliptical orbit, and move this system laterally. the moon will not trace out a sinusoid waveform at all!
Rather it will be a succession of inverted 'U' shapes .. very simlar to the surface-tension ripples on water nut seen from below.
So if by analogy we look at EM waves including light waves, it may be that they are not actually sinusoidal, either. The implication ought to be mathematically and perhaps physically interesting.
It could even point to the possibility that the frequency of EM waves is not constant but changes with distance travelled.
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Post by speakertoanimals on Mar 24, 2011 13:21:30 GMT 1
Uh, that doesn't make any sense! A constant rate of change of velocity (ie constant accleration) can give a sinusoid for motion in a circle.
Except we already know that EM waves (as for other waves), the basic solutions are sinusoids, and if we add a succession of sinusoids, we can get more complicated shapes. Depends whether or not speed varies with frequency (as in a material), as to whether these complicated waveforms propagate without change of shape.
But the basic sinusoid result isn't in doubt.
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Post by carnyx on Mar 24, 2011 14:17:18 GMT 1
@sta
What bit of this didn't you understand?
Simply that the velocity of a satellite in an elliptical orbit is not symmetrical. As you say, it goes round the planet faster than it does around the 'empty' focus at the other end of the orbit.
So, if you trace this out as a linear waveform, you will not get a sinusoid.
Can you see it now?
Here is another visualisation, .. with the planet at the bottom and the empty focus at the top. If you simulate the orbit with a conductor's baton, ... as you traverse left you will not be tracing a sinusoid in the air, but a series of U shapes.
OK?
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Post by speakertoanimals on Mar 24, 2011 15:25:37 GMT 1
I didn't say you would! Because I was distinguishing the geometry of the orbit (basic sinusoids) from the dynamics of the orbit. So in effect, the non-sinusoidal motion doesn't come direvctly from the geometry, but the combination of the geometry and the mechanics (such as conservation of energy and conservation of angular momentum). in a sense then its not the geometry of the ellipse per se that gives non-sinusoidal displacements, but the physics in terms of the inverse square force.
If you have a different central force, you'll get a different geometry, and a different motion.
I think this caused the confusion, I was initially talking about purely the geometry, which is sinusoidal, separate from the real orbital motion under gravity, which is more complicated.
However, looking up the explicit solution to Keplers problem, the solution for the x coordinate as a function of time is given by:
x = a(cos(epsilon)-e)
where e is the eccentricity. The relation between epsilon and time is:
t proportional to (epsilon - e sin(epsilon))
So although the final relation between x and t can't be written in a simple closed form, the basic parts are still made from sinusoids.
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Post by carnyx on Mar 24, 2011 15:34:31 GMT 1
@sat
Having cleared up the idea that elliptical orbits do not trace out sinusoid waveforms, perhaps we can move on more interesting speculations hinted at in the OP.
If by analogy we look at EM waves including light waves, it may be that they are not actually sinusoids, either. The implication ought to be mathematically and perhaps physically interesting.
(I hasten to say that this observation cannot be new, so I am hoping to discover whole areas of understanding that will be new to me at least)
It could even point to the distant possibility that the frequency of EM waves is not constant but drops over huge distances travelled in some way.
But let's start with the Maths. Will Fourier analysis help us in finding out whether these ellipsoidal waveforms can be synthesised from simple sinewaves? Personally I suspect not. However these waveforms are regular and have a 'loopy' property, and so it may be that geometry could help in resolving them into sets of simple harmonic motions in real and imaginary planes. But there again they may not fully resolve ....
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Post by speakertoanimals on Mar 24, 2011 15:46:07 GMT 1
I've already answered the point about em waves. THe basic equations of em (Maxwell), give fundamental solutions as being sinusoids. Whether you move stuff or not. There is nothing in there to get aves other than sinusoids, unless you do obvious stuff like stick a material in, and modify the basic wave equation so you get other weird solutions. Which is no use at all in empty space where there isn't a material.
And STILL doesn't help with change of frequency, because basic fact that things have to stay in step means frequency stays constant as you move from one material to another.
Plus WHY should what happens to planets in orbits have anything at all to do with rotating em fields, which after all aren't rotating about anything, there is no analogical link between the two at all. So just because you can have some physical situations where periodic motions can be non-sinusoidal, doesn't mean that em waves can be made non-sinusoidal in a vacuum..................
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Post by carnyx on Mar 24, 2011 16:51:07 GMT 1
@sta,
The analogy between EM waves and planetary orbits is to do with the generation of EM at the molecular and atomic levels.
Given that we know that a pair of orbiting masses even at the subatomic level must be more or less elliptical because a purely circular orbit is not possible ( i.e that there will aways be a 'wobble'). Then, waveforms given off at these levels will not be simple sinusoids,
That Maxwell's formulations work as Simple Harmonic Motions is entirely by the way, as they are descriptions and not the actualité. So it seems reasonable to query the implication that EM waveforms are elliptically generated.
(And whilst you talk about space being essentially empty, we have been here before. Apart from all those fields and bits of matter, Space really does have non-zero vales for permittivity and permeability.)
Can you now make useful comment on the implications of elliptical EM waves?
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Post by speakertoanimals on Mar 24, 2011 18:04:26 GMT 1
Except this sort of quasi-classical picture of em generation by moving charges, in elliptical orbits or not -- IS DOOMED. For starters, it doesn't explain why electrons exist in stable orbits at all! And when we go to the full quantum picture, such simple pictures as electons sitting in nice orbits goes out the window as well.
And your proof for this is? Maxwells equations work at the classical level, and the proof of the pudding is in the fact that they give a correct description for classical waves, with no elliptical nonsense. At the quantum level, the quantum version of Maxwell gives particles which are photons, a frequency determined by the energy, and no elliptical nonsense either.
There is really no point trying to draw false analogies based on classical gravitational orbits, because the mere existence of stable atoms in the first place tells us this is wrong. So hypothesizing some weird elliptical effect on photons based on something we already know is wrong is doomed.
There aren't any because the basic starting position is flawed.
As regards vacuum, the fact that permitivity etc exist DOESN'T say that vacuum contains anything, just says that is you do add something, you change the vacuum values. It doesn't say that if vacuum was really empty, waves would somehow travel instantaneously, as I said before.
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Post by carnyx on Mar 24, 2011 18:54:33 GMT 1
@sta,
I think you have lost the plot. Stable orbits are still stable even if those orbits are elliptical ...
And if moving charges (i.e. moving electrons) are not involved in the production the 'self-propagating disturbances' we call EM waves, or photons ... what is?
How do you get your electrons to move? In straight lines, ellipses or circles?
And how do you know that naturally occurring EM waves are sinusoidal?
(PS, and again you miss the point about permeability and permittivity. These properties define the rate of propagation of EM waves through a medium. And. as in 'free space' they have definite positive values, it is they that define the speed of light in space. And so, 'free space' is just a medium like any other)
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Post by speakertoanimals on Mar 24, 2011 21:24:41 GMT 1
NOT is what is orbiting is a charged particle, because in classical em, an accelerating charge radiates, hence all electrons should radiate away all their energy, and the atom collapse.
A fundamental point which you obviously missed. The killer question for you is HOW any orbiting charge CAN be stable.
That is the why behind quantum theory, the realisation that stability of electron orbits is totally and utterly at variance with classical electromagnetism, that things were so far wrong, that the whole of physics needed to be rewriiten. Whereas you seem to think that all it takes is a bit of magic (if its stable, its stable, but I don't actually know what stable means), plus some sort of electron spiragraph to generate prettier patterns.................
Why don't you learn the basics before trying to discuss these ideas?Like what acceleration is, and what accelerating charges DO..................
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Post by carnyx on Mar 24, 2011 22:12:53 GMT 1
@sta
I'll ask once again; how do you get your electrons to move? In straight lines, ellipses or circles?
But when you mentioned a spirograph, .. are you suggsesting that these charges describe a kind of Euler spiral?
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