Re: Determining masses within a rotating reference frame

[John Mallinckrodt <ajmallinckro@CSUPomona.Edu>]

On Wed, 1 May 1996, A. R. Marlow wrote:

> I'm asking for the derivation of a
> SINGLE formula, for either M or M+m, derived by applying Newton's laws in
> the specified frame, and expressed in terms of quantities measured 
> relative to that frame.  

It may be that I don't really understand the assignment ARM has proposed 
for JR, but I'm willing to give it a shot.

First let's recognize that, in reality, there is *no* frame in which both 
Earth and the sun are motionless, so we are dealing with a hypothetical 
situation--one that would correspond to a perfectly circular orbit in the 
proscribed inertial frame.  We should also recognize that we'll want to 
clear out the rest of the universe of any other annoying massive bodies 
so that Earth and the sun are not perturbed.  (We must also beg for Ernie 
Mach's forgiveness because we still want to have our "fictitious," 
"inertial," or whatever you want to call 'em forces.)  I don't think 
there are any real problems with any of the above, I just want to be clear.

So here we are in this frame in which both large bodies are motionless. 
Careful experiments, scrupulously performed in this reference frame,
reveal a general tendency for small objects that are released from rest to
begin moving directly away from a special line (call it L) that is
perpendicular to and intersects the line segment from the center of Earth
to the center of the sun.  This pervasive "background" effect is modified
significantly when the release point is close to one of the two large
bodies.  After exhaustive study we find that all of the data is extremely
well fit by a function that involves only three free parameters: 

  initial acceleration vector       Co times the shortest
  of a small object released   =    displacement vector
  from *any* point P                from L to P
                                 +
                                    Cs divided by the distance 
                                    from P to the center of the
                                    sun squared times a unit 
                                    vector directed toward
                                    the center of the sun.
                                 +
                                    Ce divided by the distance 
                                    from P to the center of 
                                    Earth squared times a unit
                                    vector directed toward
                                    the center of Earth.

where Co, Ce, and Cs are the three parameters of the fit to the data.

We naturally interpret the first term as a background field somehow
associated with the line L and the second two terms as being the result of
some kind of influence from the large bodies.  Finally, armed with
Newton's laws including universal gravitation and the value of G (which I
believe we are allowed and which we could certainly discover and
determine, respectively, even if we aren't), we find the mass of Earth =
Ce/G and the mass of the sun = Cs/G. 

Did I do it?

************************

For those who haven't yet lost consciousness, here is a fractured fairy 
tale based on the observations that assumes no knowledge of Newton's laws.

Joe "Hands" Kepler, the person responsible for the fit described above, 
also pointed out some remarkable unexplained regularities in the fitting 
parameters:

1  The line L, that he only discovered by close attention to the 
   data, passes the center of the sun at a distance equal to 
   the distance from the center of Earth to the center of 
   the sun divided by (1 + Cs/Ce).  (Of course, it also passes the 
   center of Earth at a distance equal to the distance from the 
   center of Earth to the center of the sun divided by (1 + Ce/Cs). 

2  The constant Co is equal to (Ce + Cs) divided by the distance 
   from the center of Earth to the center of the sun cubed.

3  Perhaps most remarkable of all, the first two terms of the 
   initial acceleration equation are equal and opposite at the 
   center of Earth and the first and third terms are equal and 
   opposite at the center of the sun!

Could these be coincidences?  Nick "Cola" Copernicus didn't think so.  He
proposed an alternate frame of reference in which Earth and the sun
actually orbit the line L with an angular velocity of Co^(1/2) and pointed
out that the odd trajectories of objects far away from the two large
bodies become straight lines in this reference frame! 

Then along came Sir "Icesack" Newton who thought that if he invented 
calculus and wrote a long and turgid enough book, people would interpret 
his work as "explaining" the three laws of Joe "Hands" Kepler.  The rest 
could have been history.

John
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