Re: [Phys-l] When is g = GM/r^2?

[John Mallinckrodt <ajm@csupomona.edu>]

Thanks, David, for your critique.

I wrote:

> > The acceleration of the Moon as determined in a nonrotating  
> > reference frame attached to the Earth is approximately G(M_earth 
> > +M_moon)/r_earth-to-moon^2, toward the Earth.  ... The formula is  
> > approximate only because I  have ignored the influence of other  
> > bodies like the Sun, which, in  fact, turn out to have a fairly  
> > significant effect.

To which you responded:

> That may be somewhat of an understatement in that the acceleration  
> of the moon (along with the earth) toward the sun in a frame in  
> which the Sun is at rest is about 2.17 times *greater* than the  
> acceleration of the moon relative to a frame in which the earth is  
> at rest.  In fact the orbit of the moon is concave toward the sun  
> at all points of its orbit, regardless of lunar phase, as seen in a  
> frame in which the Sun is at rest.

All true, however, since (as you note) the Sun also pulls on the  
Earth, then to first order, there is no effect on the result I am  
speaking about.

Instead, it is the tidal variation (as you effectively noted later in  
your message) of the Sun's gravitation over the spatial extent of the  
Moon's orbit that is relevant here.  In fact, that effect serves to  
cancel about 92% of the difference predicted by my formula at the new  
and full moon phases and to augment it by about 46% at the first and  
last quarters.  (And it is no accident that the ratio of the Sun's  
influence on Earth ocean tides is also about 46% that of the Moon.)   
In any event, it's certainly fair to say that I "understated" the  
effect of the Sun.  Still, that effect is "merely" comparable, not  
quite dominant!

I went on to detail the method for determining g in the reference  
frame of any one of an isolated set of massive particles and wrote:

> > Practically speaking, the condition of "isolation" requires that  
> > no particle and no position of interest be close enough to any  
> > other particles massive enough that including them in the  
> > calculations above significantly affects the results.

To which you responded:

> In light of my observation above, how about saying that the  
> condition of 'isolation' requires that the magnitude of the  
> *change* in the external gravitational field (produced by all  
> ignored bodies outside the collection of interest for the system)  
> across the relevant spatial extent of the motions of the bodies in  
> their mutual CM-at-rest
> frame, be an insignificant fraction of the magnitude of the  
> internal gravitational field produced by the constituent particles  
> of that system?

Yes.  That is what I had in mind and probably a better, more specific  
way of saying it.

Finally, you wrote:

> John, what is your opinion on also making a correction for the  
> acceleration of the CM of the system relative to a frame that  
> includes the other ignored bodies in the calculation of the CM  
> which is to be taken as effectively not accelerating?

I'm not sure I understand the point here.  Doesn't the isolation  
condition already make any further corrections unnecessary?  One  
could, I suppose, use the gradient of the external field if it were  
somehow known, to make a next order correction, but I think that is  
beyond the scope of my intended use--developing an alternate "modern  
Newtonian" pedagogy that does away with the first law and the ill- 
defined baggage of Newtonian "inertial frames" and substitutes the  
principle of equivalence.

I'd like students to come away from such a course with a clear but  
primarily conceptual understanding of 1) the frame dependence of g  
and 2) the way in which g varies spatially, i.e., the idea of  
"locality" and the centrality of *tidal* effects rather than ordinary  
*forces* in gravitation.

John Mallinckrodt

Professor of Physics, Cal Poly Pomona
<http://www.csupomona.edu/~ajm>

and

Lead Guitarist, Out-Laws of Physics
<http://outlawsofphysics.com>