Re: MOND's similarity to Le Sage's Model
- From: "George Dishman" <george@xxxxxxxxxxxxxxxxx>
- Date: Thu, 28 Sep 2006 10:31:36 +0100
"Aetherist" <TheAetherist@xxxxxxxx> wrote in message
news:v36mh21nd5h20jmp95v8jmm9l4gah6i3ik@xxxxxxxxxx
On 27 Sep 2006 01:29:39 -0700, "George Dishman" <george@xxxxxxxxxxxxxxxxx>
wrote:
On 26 Sep 2006 00:31:53 -0700, "George Dishman"
<geo...@xxxxxxxxxxxxxxxxx> wrote:
On Mon, 25 Sep 2006 20:41:28 +0100, "George Dishman"
<geo...@xxxxxxxxxxxxxxxxx> wrote:
"Aetherist" <TheAether...@xxxxxxxx> wrote in message
news:jgkdh25q1lpcma72t9lafsc04lfhrpsqn5@xxxxxxxxxx
<snip - next bit left for reference below>
Where,
k = µ(x)
and µ and x are arbitrarily determined constants.
<much snipped - mostly confusion>
Just as clearly, x is defined as |a|/a_o where a_o
is an 'arbitrary' constant
Right, it is a_0 that is the constant, not mu or x.
If your statement that mu and z were constants was
just a slip of the finger, why not just say so, I
make typos all the time.
It was such a 'typo' that I didn't realise that it
read literally like it does. Certainly never entered
my mind that µ(x) or x is 'constant'.
OK, but I think you can understand how it reads
now and you reinforced that in your last sentence
"Aetherist" <TheAetherist@xxxxxxxx> wrote in message
news:jgkdh25q1lpcma72t9lafsc04lfhrpsqn5@xxxxxxxxxx
....
Both theories will predicts the same result under similar mathematical
constraints. It also provides an explanation and basis for MOND's
arbitrarily assumes constants...
Let's drop that as there is no real argument now.
[you wrote:]
Given this form, by its very definition, µ(x) MUST!
be non-linear for all x's > 0 & < infinity...
[I wrote:]
µ(x) is a function which is intended to make F=ma
non-linear.
Yes, For all values of a, > 0 and < infinity, plot the
result of the expression
M a µ(x)
Where M is constant, x = |a|/a_o and µ(x) has the
defined characteristics of MOND's definition...
Is the result a straight line?
My point was that we both said exactly the same,
there is no disagreement.
[Paul wrote]:
and µ and x are arbitrarily determined constants.
It is _not_ a constant, arbitrary or otherwise.
That's funny, quote...
"The term a_o is a proposed new constant,
Yes, as I pointed out, a_0 is the constant, not
the function mu or its argument x as you said.
Agreed... That actually never crossed my mind...
If you re-read my original post, that should be
glaringly clear.
Not really if you read your definition of mu and x
above which I left at the top, I can only read it
as saying they are constant and you said the same
again at the bottom. Anyway we have cleared that up
as you say you didn't intend it to read that way so
let's leave that aspect aside, I mention it only
because I don't want anyone left with the impression
that I deliberately took it the wrong way or was
misquoting you.
<snip example>
....... For Newtonian gravity we have:
f = GMm/r^2 (1)
and
f = ma (2)
hence
a = GM/r^2 (3)
For a << a_0 in MOND, eqn (2) becomes:
f = m a^2 /a_0 (4)
hence from (1) and (4) we get:
a^2 = a_0 GM/r^2
a = sqrt(a_0 GM) / r (5)
If you want to show a similarity between MOND and Le
Sage, you need to start with your previous derivation
that showed that shadowing of the momentum flux was
equivalent to eqn (3) for |a| >> a_0 but that it
transitions smoothly to eqn (5) when |a| << a_0. The
nature of that transition will then determine the
function mu.
For those who have studied the underlying basis of
Le Sage's model it is clear. Note that the title is
NOT! MOND 'EQUALS' Le Sage's Model. Certainly MOND
is not founded on the same underpinning basis.
No, I quite understand that but you were arguing for
the two theories giving identical predictions weren't
you:
"Aetherist" <TheAetherist@xxxxxxxx> wrote in message
news:jgkdh25q1lpcma72t9lafsc04lfhrpsqn5@xxxxxxxxxx
....
Since in Le Sage's model it is the net (delta) in the flux ¿ produces
the acceleration (and force) on any mass m at r we find a perfect
one to one correlation in both mathematical form and function to MOND.
Both theories will predicts the same result under similar mathematical
constraints.
It certainly read that way, at least in the general
form or shape of the curves.
What I claim is, (as you should, I hope, be aware of
by now) in Le Sage's model the force, and therefore
the corresponding acceleration is realized by the
differential pressure resulting from a cummulative
shielding effect of the attenuating body(ies). If you
have a single body in the weak limit (then you have
a simple problem which give you your equation 3 above.
I understand Le Sage and agree for two bodies. For
multiple bodies the same should apply provided the
attenuation remains in the linear region, i.e. for
a small exponent in the exponential.
However, where you literally have millons of aggrogate
masses distributed across thousands of light-years
one can get a cummulative effect of blocking that can
depart from this weak limit.
OK, I see where you are going now.
In that case, this results
in the classical exponential form of (1 - e^-x)causing an
increasing weaking in the force (and consequentually, the
observed radial acceleration [when compared to the
unweakened expectation]) of a mass m at r in a fashion
exactly equivalent to MOND's expression, flatting the
rotation profile. Thus the 'similarity'...
I think you haven't grasped the implication of MOND,
it results in an apparent _strengthening_ of the force
at low accelerations hence requiring higher orbital
speed for the centrifugal force to balance gravity.
The galactic velocity curves are higher than predicted,
not lower.
AFAIK the proof Le Sage offerred that his theory was
equaivalent to Newtonian gravity for a >> a_0 holds
for small accelerations, i.e. as r >> infinity so it
does not match MOND.
And hopefully, this also now clearer for you to follow.
Yes it is, and hopefully my response is relevant to
your clarified claim.
George
.
- References:
- MOND's similarity to Le Sage's Model
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