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.

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.

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

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.

"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

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