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*From*: Ludwik Kowalski <kowalskil@mail.montclair.edu>*Date*: Thu, 27 Dec 2012 19:34:30 -0500

I am sorry for forgetting that we are not able to attach files.

On Dec 27, 2012, at 7:23 PM, Ludwik Kowalski wrote:

I am puzzled by the result of a trivial calculation, as described in the attached file. Comments will be appreciated.

Ludwik Kowalski

http://csam.montclair.edu/~kowalski/life/intro.html

Let me paste the content of the file, hoping it will be readable.

A crazy gravitational atom

Newton's law, for point-like particles, is Fg=G*M*m/r2, where k=6.7*10-11, in SI units. Consider a trivially simple solar system--one star and one planet--when M>>m, and when the orbit is circular.

The trajectory of a planet remains circular when the centripetal force is equal to the gravitational force:

m*v2/r = G*M*m/r2

This implies that the product r*v2, for any given M, does not depend on the mass of the planet:

r*v2 =G*M . . . . . . . . . . . . . . . . . . . . . . . . . . . (1)

A satellite remains on a circular orbit of radius r as long as its speed remains sqr(G*M/r). Any radius, satisfying the above relation, is allowed by our classical mechanics model. But this is not true for the semi-classical model, introduced by Niels Bohr. Orbital radii allowed by his model must satisfy an additional condition--the angular momentum, A=p*r=m*v*r, must be a multiple integer of Plank's constant, h_bar (1.05*10-34 J*s).

m*v*r = n*h_bar . . . . . . . . . . . . . . . . . . . . . . (2)

where n is an integer, such as 1,2,3, etc. The smallest allowable radius, corresponding to n=1, is given by

r1=h_bar / m*v

Substituting v given by formula (1), and solving for r, one has:

r1= (h_bar) 2 / (G*M*m2) . . . . . . . . . . . . . . . (3)

Imagine a system in which m= 2*10-27 kg (comparable to the mass of a neutron) and M is one million times larger (the mass of the sun is also much larger than the mass of our planet). The r1 for such system would be 0.41*10-10 meters. This is nearly the same as for the hydrogen atom (where r1=0.53 *10-10 meters). I was surprised by the extremely slow speed v, calculated with the formula 1. It turned out to be 5.7*10-11 m/s. The corresponding period of revolution, along the orbit whose length is only 2.6*10-10 meters, is of 3500 years.

I know how I would react if v turned out to be higher than the speed of light. But how should I react to an extremely slow speed?

**References**:**[Phys-L] A crazy planetary system***From:*Ludwik Kowalski <kowalskil@mail.montclair.edu>

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