Lester Zick <dontbother@nowhere.net> writes:

On 21 Aug 2006 03:25:34 -0400, Craig Markwardt
<craigmnet@REMOVEcow.physics.wisc.edu> wrote:

Lester Zick <dontbother@nowhere.net> writes:

On 19 Aug 2006 14:19:24 -0400, Craig Markwardt
<craigmnet@REMOVEcow.physics.wisc.edu> wrote:

Lester Zick <dontbother@nowhere.net> writes:
... snip ...

It means I think the sky is falling in gc's just the way it does in
every celestial object whose centers of gravity are stationary with
respect to one another.

... snip ...

What exactly do you mean by,
* "centers of gravity", and that they
* "stationary with respect to one another"

Perfectly reasonable questions.

I'm not asking for an "it's obvious" answer, but rather, what exactly
you think the terms you used mean in terms of geometry and perhaps
mathematics.

I take the term "centers of gravity" to mean the centers through which
universal gravitational attraction operates. Each star in a globular
cluster has its own local internal center of gravity and each star in
various cominations with other stars has a non local center of gravity
presumably lying somewhere between or among them through which
gravitational attraction for the combination acts in relation to
centers of gravity for other combinations.And these centers of gravity
exist for every possible combination of stars within the cluster. And
every possible combination of stationary and moving stars within a
cluster has to be reckoned in terms of all centers of gravity among
all the combinations and not just exclusively between each pair of
stars within the cluster or for the cluster as a whole in calculating
net angular momentum for the cluster.

Wouldn't it be more straightforward to calculate the cluster angular
momentum about a given axis based on a sum of the individual stars'
angular momenta? [*] After all, that is the formal definition of
angular momentum, isn't it?

Regardless of how you add them up the sum is still roughly zero
provided you add them all up and not just a few in relation to a
specific axis.

However, since classical total angular momentum is not defined in the
manner which you describe, you would need to substantiate your claim
with extensive theoretical work, which you have not done.

[*] By that I mean, L_total = Sum[ L_i ] = Sum[ r_i x p_i ], where
r_i and p_i are the ith star's distance from the axis and linear
momentum, and "x" indicates a vector cross product.

And secondly, on what evidence do you claim that "centers of gravity
are stationary with respect to one another?"

Well this may not appear obvious however in calculating net angular
momentum for the cluster as a whole we know two basic things: first
the motion of various stars in the cluster individually and second the
net angular momentum for the cluster as a whole of approximately zero.

Thus we can easily construct an equation where the sum of all angular
momenta for all possible combinations of stars in a globular cluster
equals zero. So regardless how we take various combinations of angular
momenta for various centers of gravity within the cluster we have to
recognize that their sum must be zero for the cluster itself to have a
net angular mometum approximately zero. Which means in turn that all
combinations must offset one another in aggregate. Hence even though
some combinational centers may not offset one another directly, all
centers of gravity within the cluster have to be roughly stationary
with respect to all other centers of gravity regardless of a presence
of non zero angular momentum for certain combinations within the
cluster.

How does your conclusion follow?

Because the net angular momentum for the cluster as a whole is roughly
zero. Consequently regardless of individual angular momenta considered
in isolation the aggregate of all angular momenta must be roughly zero
too. Otherwise the cluster itself must have non zero angular momentum.

Your original conclusion still does not follow. Even if the total sum
angular momentum of an ensemble is zero, one cannot conclude that the
individual components of the sum must be zero. Hence one cannot
conclude that the individual components are "stationary."

First, you've been writing about angular momentum, so what step of
logic allows one to translate from the domain of angular momentum to
"roughly stationary" points? For example, the total angular momentum
of a system can be zero when the total energy or linear momentum is not.

I'm not talking about linear momentum here.As far as different domains
of angular momentum are concerned we're just adding them up. They have
to equal zero whether or not isolated domains have energy or linear
momentum. It's quite possible for moving bodies with kinetic energy
and linear momentum to possess offsetting angular momenta with respect
to various centers of gravity.

You claimed that something was "stationary." In classical mechanics
if a body is stationary, it is not moving. A body cannot have
"kinetic energy and linear momentum" and also be stationary.

Second, it is easy to construct a sum of two functions which is zero
but whose individual values are not.

Sure. The problem is that we have to wind up with centers of gravity
stationary with respect to each other or the angular momentum of the
object as a whole will have some net angular momentum.

For example, consider three identical stars arranged along the X axis
at positions (-1 pc), (0 pc), (+1 pc), [*] with the outer two stars
moving apart and the center star held fixed (with respective
velocities: (-1 pc/yr), (0 pc/yr), (+1 pc/yr)). By any definition,
the total linear and angular momentum of this system is zero. And
yet, the stars are moving apart. The "centers of gravity" as you
define them are also moving apart from each other and thus not
"stationary." Thus, your conclusion is not rigorous.

Which centers of gravity? That for the stars individually considered
in isolation or the center of gravity for the system as a whole? The
fact that the stars themselves are moving apart is irrelevant. The
center of gravity for the system itself is stationary. The centers of
gravity as I define them are only moving apart for the individual
stars and not for the system as a whole.

You claimed that one must consider *all* possible combinations of
"centers of gravity." For the triplet of stars mentioned, there are
three possible pairs of stars, three possible 2-1 combinations, and a
sum of all three, for a total of seven combinations. All of those
"centers of gravity" are moving with respect to each other, even
though the total linear and angular momenta are zero. Thus, your
claim leads to a contradiction.

If you are now changing your claim to only discuss the angular
momentum of the center of mass alone, that is a different matter, but
it is not what you originally claimed.

... snip ...

Of course I maintain that given such an analysis with effectively
stationary centers of gravity overall we're faced with a de facto
necessity for collapse along lines of stationary centers of gravity.
It's what I call the Chicken Little Hypothesis. In other words the sky
is falling.

How does that follow either? Even if, for the sake of argument, the
"centers of gravity" were "stationary," you have defined them as
imaginary points.

It's not a question of how I've defined them.They're defined by Newton
as much as anyone as the centers through which universal gravitational
attraction works. And they're no more imaginary than the local centers
of attraction through which gravitational attraction works inside
stars. Just because you can't associate any material element with them
outside a star doesn't mean they don't actually exist.

That is not relevant. "Centers of gravity" are still imaginary
points. Therefore, they are not attracted to each other by gravity,
nor do they have angular momentum or energy. They do not necessarily
"collapse" by themselves. As you defined them, they merely exist as
points defined by the various stars in the cluster. Thus your
conclusion does not follow. The relevant question is regards the
motions of the individual stars. Then any follow-on discussion of
"centers of gravity" can be computed immmediately.

... snip ...

Let's try a simple example. Suppose we have a globular cluster with
roughly zero net angular momentum. And let's suppose through the magic
of TV we can define opposing centers of gravity in respective
hemispheres A and B around which all stars in those hemispheres have
roughly equal and opposite aggregate angular momentum.

The question then is can A and B be moving with respect to each other?
Obviously not. They have to be stationary with respect to one another.

Your example is not well enough defined. If your "A" and "B" are
defined in terms of a fixed set of stars, initially in two
hemispheres; then of course, as the stars move, the centers of mass of
the two sets of stars will move in some complicated pattern. If your
"A" and "B" are always defined in terms of the two fixed hemispheres,
then of course the hemispheres will not move since you have defined
them to be fixed.

Stars will pass between the boundaries of the two
hemispheres, thus blurring the concept of "center of gravity." Thus,
your example is not sufficient.

On Mon, 21 Aug 2006 10:28:39 -0700, Lester Zick
<dontbother@nowhere.net> wrote:

It's quite possible for moving bodies with kinetic energy
and linear momentum to possess offsetting angular momenta with respect
to various centers of gravity.

This is crazy. On that basis, two objects orbiting something in
opposite directions, in slightly different orbits of course to prevent
collisions, would drop like stones because their angular momenta
cancel out? You're mad.