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Flash
154232.  Tue Mar 06, 2007 10:04 am Reply with quote

Why are orbits elliptical?

Quote:
nothing precludes a perfectly circular orbit other than the fact that it would be very unusual for any orbit to be PERFECTLY circular. The situation where the Sun, for instance, would be exactly in the center of a perfectly circular planetary orbit couldn't happen in our Solar System since there are other planets that would gravitationally affect the orbit and cause it to be immediately non-circular by pulling the object to one side or the other. Any other stable orbit type than a perfectly circular one turns out to be an elliptical orbit.

"An ellipse is defined as a figure drawn around two points called the foci in such a way that the distance from one focus to any point on the ellipse and then back to the other focus equals a constant. This makes it very easy to draw ellipses with two thumbtacks and a loop of string (both ends tied together). Press the thumbtacks into a board, loop the string about them, and place a pencil in the loop. If you keep the string taught as you move the pencil, it will trace out an ellipse." - from Michael A. Seeds, "Horizons - Exploring the Universe," Wadsworth Publishing Company. A great freshman non-science major astronomy textbook.

You can think about it in this way... Because of geometry and because of Newton's law of universal gravitation ... , there are only four different paths that an object passing close to the Sun might take. The first would be a hyperbolic orbit - for example, the object comes in from space towards the Sun with so much energy that it swings around the Sun, but still with enough energy to head back out into space on a different course... A path that would look kind of like a stretched out letter "U". Secondly, the object could come in from space without enough energy to escape from the Sun's gravity and would thus be captured in an elliptical orbit with the eccentricity of the orbit dependent upon the object's initial energy. Thirdly, the object might have PRECISELY the amount of energy to be captured in a perfectly circular orbit. And finally fourth, the object might not have enough energy to establish any sort of orbit and it would spiral into the surface of the Sun and burn up.

These are the only theoretical choices: hyperbolic, elliptical, circular, and spiral (impact).

http://www.astro-tom.com/technical_data/why_elliptical.htm

 
dr.bob
154248.  Tue Mar 06, 2007 10:37 am Reply with quote

Much as I hate to disagree with a noted luminary like Astro Tom, some of the things in his article don't quite ring true.

Quote:
Secondly, the object could come in from space without enough energy to escape from the Sun's gravity and would thus be captured in an elliptical orbit with the eccentricity of the orbit dependent upon the object's initial energy.


This is not possible without some further interaction. If an object comes "in from space" then, as it falls into the Sun's gravity well, its potential energy will be converted into kinetic energy and it will get faster and faster. Since energy can be neither created nor destroyed, that kinetic energy will provide it with exactly the right amount of energy to escape from the Sun's gravity well as it gets converted back into potential energy on the way back out. That is unless the object experiences some other force during its fly-by which causes it to lose some of that energy and become captured in an elliptical orbit around the Sun.

Such an interaction is entirely possible and may come from a gravitational interaction with another Solar System body (a bit like those "slingshot" manoeuvres you often hear about to speed up space probes, but in reverse). However, without such an interaction, I don't think it's physicaly possible for an object to "come in from space without enough energy to escape from the Sun's gravity."

Quote:
And finally fourth, the object might not have enough energy to establish any sort of orbit and it would spiral into the surface of the Sun and burn up.


I'm not sure about his use of the word "spiral" here. The first idea that comes to mind is something spinning several times 'round the Sun getting nearer and nearer each time until it eventually hits it. This is physically impossible, for the reasons described above, unless there is some mechanism for removing the excess energy of the object.

Alternatively a "spiral" orbit may just describe less than one complete turn as the object just dives into the Sun. I'm not entirely sure since, despite Astro Tom's assertion (further down the article):

Quote:
there are a lot of objects that fall into the Sun on a spiral path. Comets die this way all of the time.


A quick google of "comet spiral orbit" returns nothing remotely referring to this kind of phenomenon.

I'll do a bit more digging around, though.

 
dr.bob
154259.  Tue Mar 06, 2007 11:39 am Reply with quote

Nope, I can't find anything about spiral orbits.

Kepler said that the orbit of a planet/comet about the Sun is an ellipse with the Sun's centre of mass at one focus. I would contend that objects that impact into the Sun are still describing an elliptical orbit around the Sun's centre of mass, it's just that the path of the orbit intersects the surface of the Sun (i.e. it travels too close to the Sun's centre of mass).

(N.B. orbital dynamics are the same for any two body system, so you can substitute the word "Sun" above for anything else you're considering an orbit around)

Given that a circle is just a special case of an ellipse (in the same was a square is a special case of a rectangle), it's more correct to say that orbits are of two types. Hyperbolic (as explained by Astro Tom) and elliptical (including circular).

 
Flash
154347.  Tue Mar 06, 2007 5:48 pm Reply with quote

If the Sun is at one focus, what's at the other?

 
dr.bob
154425.  Wed Mar 07, 2007 4:33 am Reply with quote

Nothing

If there was something at the other focus, you'd have a three-body gravitational problem which is a highly unstable system.

 
Flash
154430.  Wed Mar 07, 2007 4:50 am Reply with quote

So, to be clear, we can state axiomatically that an elliptical orbit must have an object which exerts a gravitational pull at one focus and (specifically) nothing (or at least nothing significant) at the other?

With regard to three-body gravitational systems, there are exoplanets which are known to orbit binary stars, aren't there?

Are elliptical orbits symmetrical in two planes or can they / must they be symmetrical in one plane only (ie eggshaped)?

And, Dr Bob, I'd just like to say how much we appreciate your patient input.

 
Gray
154437.  Wed Mar 07, 2007 5:07 am Reply with quote

It's not even quite right to say that 'bodies orbit other bodies in elliptical orbits', because, as we all know, the two bodies in question - e.g. the Sun and the Earth - are rotating around each other, and around a mutual centre of gravity.

This is so close to the centre of the Sun's centre of gravity (because the Sun's mass is so much more than the Earth's) that it lies within the body of the sun.

But nevertheless, the Sun does wobble as we go around it.

So the foci of this theoretical orbital ellipse are moving too. And this is where my brain gets off, I think!

 
Flash
154446.  Wed Mar 07, 2007 5:27 am Reply with quote

Quote:
The ellipse has an important property that is used in the reflection of light and sound waves. Any light or signal that starts at one focus will be reflected to the other focus. This principle is used in lithotripsy, a medical procedure for treating kidney stones. The patient is placed in a elliptical tank of water, with the kidney stone at one focus. High-energy shock waves generated at the other focus are concentrated on the stone, pulverizing it.

The principle is also used in the construction of "whispering galleries" such as in St. Paul's Cathedral in London. If a person whispers near one focus, he can be heard at the other focus, although he cannot be heard at many places in between.


The latter phenomenon is said to have been discovered by American President John Quincy Adams, who positioned his desk at one focus of the elliptical ceiling in the Statuary Hall of the Capitol building so as to be able to eavesdrop on Reps talking at the other one. However, the wiki says that
Quote:
this is not possible as the Hall's floor was carpeted at that time and the unusual acoustics of the room were not discovered until the carpet was replaced with tile.


If you are lucky enough to have an elliptical billiard table you can do blindfold trick shots, because if you place the cue ball on one focus and the objective ball on the other you'll always hit it, no matter which way you aim.

http://britton.disted.camosun.bc.ca/jbconics.htm


Last edited by Flash on Wed Mar 07, 2007 6:01 am; edited 1 time in total

 
Gray
154451.  Wed Mar 07, 2007 5:36 am Reply with quote

http://www.newtrier.k12.il.us/academics/math/Connections/curves/ellbilsn.htm

 
Flash
154452.  Wed Mar 07, 2007 5:39 am Reply with quote

Yes. Mind you, on a table that size you could probably hit the bugger even if it wasn't elliptical.

 
dr.bob
154468.  Wed Mar 07, 2007 6:10 am Reply with quote

Flash wrote:
So, to be clear, we can state axiomatically that an elliptical orbit must have an object which exerts a gravitational pull at one focus and (specifically) nothing (or at least nothing significant) at the other?


Yes, you have it exactly.

Flash wrote:
With regard to three-body gravitational systems, there are exoplanets which are known to orbit binary stars, aren't there?


Yes. It's hard to keep up with the bewildering array of exoplanets they seem to find these days, but I've found reference to at least one:

http://www.eso.org/outreach/press-rel/pr-1998/pr-18-98.html

And even to one in a three star trinary system:

http://physicsweb.org/articles/news/9/7/6/1

However, I don't think these really count as three-body gravitational problems. In both these cases (and, I think, theory predicts in all other cases as well), for a planet to have a stable orbit around a star in a binary system, it needs to be orbiting one of the stars relatively closely, with the other star relatively far away such that it doesn't influence the planet's orbit significantly.

In that respect it would be a bit like the Moon orbiting the Earth not being significantly affected by the presence of the Sun. Yes, the Sun plays a role (giving rise to Lagrange points and such like), but the Moon is in a stable orbit which can be pretty closely approximated by just solving the two body Earth-Moon gravitational system.

Flash wrote:
Are elliptical orbits symmetrical in two planes or can they / must they be symmetrical in one plane only (ie eggshaped)?


Elliptical orbits are ellipses and are, therefore, symmetrical about two axes. Unless I'm misunderstanding what you mean by "symmetrical in two planes".

Flash wrote:
And, Dr Bob, I'd just like to say how much we appreciate your patient input.


No problem. Just happy to help. You can buy me a drink if I ever manage to make it all the way down to Oxford :)

Gray wrote:
It's not even quite right to say that 'bodies orbit other bodies in elliptical orbits', because, as we all know, the two bodies in question - e.g. the Sun and the Earth - are rotating around each other, and around a mutual centre of gravity.


Yes, that's a very good point, and one that Kepler kind of glossed over. Strictly you should say that the barycentric point, around which both the Sun and Earth orbit, is at one focus of the ellipse.

Gray wrote:
So the foci of this theoretical orbital ellipse are moving too. And this is where my brain gets off, I think!


I don't think that's true. The focus of the theoretical ellipse contains the barycentric point, and that's a fixed point for a two body gravitational system.

Of course, the solar system being rather more complicated than a two body system, the Sun does tend to pirouette around quite a lot, but you can still point to a centre of gravity of the solar system which will remain motionless while all the planets revolve around it.

 
Gray
154472.  Wed Mar 07, 2007 6:23 am Reply with quote

But neither the Sun's nor the Earth's orbits will be an ellipse of which the barycentric point is a focus. I think...

Feynman says this, too, which bolloxes things up even more:
Quote:
"The center of mass is sometimes called the center of gravity, for the reason that, in many cases, gravity may be considered uniform. ...In case the object is so large that the nonparallelism of the gravitational forces is significant, then the center where one must apply the balancing force is not simple to describe, and it departs slightly from the center of mass. That is why one must distinguish between the center of mass and the center of gravity."


The last animation on this page:

http://en.wikipedia.org/wiki/Center_of_mass#Barycenter

shows the case for identical mass elliptical orbits, but I wince when I try to visualise it for non-equal masses... Now there are two ellipses, four foci, and no doubt some unwholesom relativistic effects if you think about Mercury's orbit.

 
Flash
154479.  Wed Mar 07, 2007 6:32 am Reply with quote

dr.bob wrote:
Yes, you have it exactly.


Since I'm doing so well, let me push my luck and speculate thus: is the location of the second focus effectively a function of the sum of all the forces which are operating on the orbiting body other than the gravitational pull of the object which is at the primary focus point? In other words the orbit is a circle distorted (given eccentricity) by a combination of things like the initial impetus of the satellite, asteroid impacts, the competing attractions of other satellites in the same system, etc? All these forces can be agglomerated into one net vector (see, I'm inventing jargon on the fly here), and that gives us the second focus?

 
dr.bob
154486.  Wed Mar 07, 2007 6:49 am Reply with quote

Gray wrote:
But neither the Sun's nor the Earth's orbits will be an ellipse of which the barycentric point is a focus. I think...


I disagree. Certainly as far as classical Newtonian mechanics is concerned, that's exactly what happens.

Gray wrote:
Feynman says this, too, which bolloxes things up even more:


I've not heard that before, so I'll try and find out more about it. It may be a very small, relativistic type effect (such as mercury's precessional weirdness)

Gray wrote:
The last animation on this page:

http://en.wikipedia.org/wiki/Center_of_mass#Barycenter

shows the case for identical mass elliptical orbits, but I wince when I try to visualise it for non-equal masses... Now there are two ellipses, four foci


Nah, it's dead easy. Just look at the animation of the two bodies of unequal mass in circular orbits, then just stretch the orbits into ellipses.

I've tried looking on google, but the only example I could find is in this pdf of lecture notes:

http://www.physics.hmc.edu/faculty/esin/a062/notes/lecture4.pdf

Page 7 of 16 shows the orbits for the Sirius binary system. You can clearly see two elliptical orbits, thus four foci. However, two of the foci are in the same place, on the centre of mass of the system (marked with a wee "x").

Gray wrote:
and no doubt some unwholesome relativistic effects if you think about Mercury's orbit.


Yeah, but that's just little tweaks around the edges. For a first approximation, Newton and Kepler do perfectly well.

 
dr.bob
154499.  Wed Mar 07, 2007 7:04 am Reply with quote

Flash wrote:
Since I'm doing so well, let me push my luck and speculate thus: is the location of the second focus effectively a function of the sum of all the forces which are operating on the orbiting body other than the gravitational pull of the object which is at the primary focus point? In other words the orbit is a circle distorted (given eccentricity) by a combination of things like the initial impetus of the satellite, asteroid impacts, the competing attractions of other satellites in the same system, etc? All these forces can be agglomerated into one net vector (see, I'm inventing jargon on the fly here), and that gives us the second focus?


In a word, no.

I think you've over-reached yourself there :)

Remember that an elliptical orbit is the solution for a two body gravitational problem. So there will be no "competing attractions of other satellites in the same system" or anything like that. Such forces will act to disrupt the simplicity of the elliptical orbit, for example Cruithne's decidedly non-elliptical orbit because it's heavily influenced by the Earth's gravitational field.

I guess you could say that the location of the second focus is a result of the initial velocity of the orbiting body, since that defines what kind of ellipse it will describe. Also, if there were any impacts which affected the orbiting body's velocity, that would change the ellipse of its orbit and, therefore, the location of the second focus.

However, there's no mystical weirdness about orbits being ellipses rather than circles. Basically all orbits are conic sections, as defined by Newton's reworking of Kepler's first law. So they're free to be circles, ellipses, parabolas, or hyperbolas (hyperbolae? parabolae?).

However, these curves are defined as follows:

Circle : eccentricity = 0
Ellipse : 0 < eccentricity < 1
Parabola : eccentricity = 1
Hyperbola : 1 < eccentricity

So, while orbits can be circular or parabolic, it's fantastically unlikely because their eccentricity has to be one particular value. Any slight deviation (e.g. eccentricity = 0.00001) and it stops being a circle or a parabola and instead becomes an ellipse or a hyperbola. Since nature doesn't really go in for scientific precision, circles and parabolas are pretty much never seen.

Hyperbolic orbits are seen, but these only pass the Sun (or whatever else is being orbited) once, so you can't point to them for very long. Anything with a regular orbit that sticks around, therefore, is almost certain to have an elliptical orbit (unless it's part of a more complicated gravitational system, like Cruithne)

I hope that makes sense. Sometimes I find myself waffling away at a complete tangent to the point I was originally trying to make :)

 

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