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Mean orbital elements
Does anyone have a clear definition of 'mean' used in this context? Obviously, it's some kind of average over time but what I would like to know is how to calculate this averaging from, say, long-term position and velocity data. I understand and can calculate osculating (two-body) elements and maybe you can do the same sort of averaging directly with these. Any ideas? I think this is related to the question: What exactly is the ecliptic? Or somewhat more generally, how do you define an ellipse (and its plane) in three-dimensional space for a body moving in the gravitational field of many other bodies, resulting in a path that is neither an ellipse nor a closed orbit? It's puzzled me for a long time! John Irwin. |
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John Irwin wrote:
Does anyone have a clear definition of 'mean' used in this context? Obviously, it's some kind of average over time but what I would like to know is how to calculate this averaging from, say, long-term position and velocity data. I understand the mean orbital elements are the numbers which describe orbits in the way Kepler described - one body at one focus of an ellipse (eg the sun) and the other orbiting round edge of the ellipse (eg the Earth). In other words, it doesn't take into account perturbations by other bodies (or general relativity). With recent data it's OK for short-term predictions. I think this is related to the question: What exactly is the ecliptic? The ecliptic is the plane of the orbit of the Earth around the sun. Or somewhat more generally, how do you define an ellipse (and its plane) in three-dimensional space for a body moving in the gravitational field of many other bodies, resulting in a path that is neither an ellipse nor a closed orbit? AFAIK you don't. It doesn't work. You model orbits numerically if you want accurate predictions. I suppose what you're getting at is does the ecliptic move and how do we decide what it is for a particular point in time. It does move, look up "planetary precession". I don't know how they decide exactly what the plane of the ecliptic is at a particular moment in time, but it does vary predictably in the short term so I guess it's just a question of combining observations and models. It is hard to get fixed points of reference when everything is moving, so when defining coordinates we use an ecliptic (and celestial equator, which changes through the precession of the Earth's axis relative to the ecliptic) from a fixed point in time (epoch), which is why you will see celestial coordinates listed along with their epoch (J1950, J2000 etc.). Tim -- My last .sig was rubbish too. |
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On Thu, 17 Jun 2004 10:54:36 +0000 (UTC), you wrote:
Does anyone have a clear definition of 'mean' used in this context? Obviously, it's some kind of average over time but what I would like to know is how to calculate this averaging from, say, long-term position and velocity data. I understand and can calculate osculating (two-body) elements and maybe you can do the same sort of averaging directly with these. Any ideas? I *think* mean usually implies constant speed. A body moves faster nearest the focus and slowest farthest from it. The mean values assume the speed is constant and is usually corrected for to produce a 'true' value. I think this is related to the question: What exactly is the ecliptic? Or somewhat more generally, how do you define an ellipse (and its plane) in three-dimensional space for a body moving in the gravitational field of many other bodies, resulting in a path that is neither an ellipse nor a closed orbit? The ecliptic is the path the sun follows in the sky -- or probably average path the sun follows. Its sine curve shape is due to the tilt in the Earth's rotation (relative to its orbit). It's puzzled me for a long time! John Irwin. Robert Chafer |
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Tim Auton wrote:
I understand the mean orbital elements are the numbers which describe orbits in the way Kepler described - one body at one focus of an ellipse (eg the sun) and the other orbiting round edge of the ellipse (eg the Earth). In other words, it doesn't take into account perturbations by other bodies (or general relativity). With recent data it's OK for short-term predictions. A two-body Keplerian ellipse is fixed. But we know the mean elements have long-term changes precisely due to perturbations of various sorts. The ecliptic is the plane of the orbit of the Earth around the sun. But which orbit? The osculating orbit, the mean orbit, or something else? I mentioned the ecliptic because I think it represents an example of an orbit described by mean elements (though I may be wrong). The thing that puzzles me is that we talk about the 'plane of the orbit' but the actual orbit doesn't lie in a plane because the body doesn't follow exactly the same 3D-path each time round the Sun. This is where I think the mean orbit comes in (though I may be wrong) because a mean orbit is described by a set of elements representing an ellipse which does lie in a plane, even though that plane may be changing if the elements are time dependent. So what does the mean orbit actually, erm, mean and how do we calculate it? Thinking more about this, my guess is that the mean orbit is related to the osculating orbit (representing the two-body Keplerian ellipse which is related directly with the instantaneous position and velocity of the orbiting body) by some sort of averaging over time. If this is true then it's not clear to me how this averaging is done. Over what interval do we average to calculate the mean orbital elements at a particular instant of time? I suspect much longer than the orbital period (though I may be wrong). It would be useful to know the details of this calculation. Tim -- Thanks for your input. John. |
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"Robert Chafer" wrote in message ... On Thu, 17 Jun 2004 10:54:36 +0000 (UTC), you wrote: Does anyone have a clear definition of 'mean' used in this context? Obviously, it's some kind of average over time but what I would like to know is how to calculate this averaging from, say, long-term position and velocity data. I understand and can calculate osculating (two-body) elements and maybe you can do the same sort of averaging directly with these. Any ideas? I *think* mean usually implies constant speed. A body moves faster nearest the focus and slowest farthest from it. The mean values assume the speed is constant and is usually corrected for to produce a 'true' value. I think this is related to the question: What exactly is the ecliptic? Or somewhat more generally, how do you define an ellipse (and its plane) in three-dimensional space for a body moving in the gravitational field of many other bodies, resulting in a path that is neither an ellipse nor a closed orbit? The ecliptic is the path the sun follows in the sky -- or probably average path the sun follows. Its sine curve shape is due to the tilt in the Earth's rotation (relative to its orbit). No! Sorry, I can see where the confusion may have arisen. You are talking about mean motion (what you are describing--the angle is called the mean anomaly) but this is nothing to do with mean orbital elements, which refer to the orbital elements at a fixed epoch, referred to ecliptic and equinox of the same date, and omitting certain short-term perturbing effects. -- Mike Dworetsky (Remove "pants" spamblock to send e-mail) |
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