Note 10/1: #4 below contains an error. The stars do not orbit along the plane of the ecliptic. Thanks to "CB" at Ethan's blog for pointing this out.
Ethan Seigel has posted a number of purported difficulties with the geocentric hypothesis in a very interesting piece here.
I think all of the reported problems with the geocentric model can be resolved rather easily here, Ethan.
1. Consider the Earth at rest 2. The Moon orbits the Earth. 3. The planets and their moons orbit the Sun. 4. The Sun and stars orbit the Earth along the plane known locally as "the ecliptic".
I believe this resolves all reported difficulties for the geocentric model, and it seems to me it does more than this.
You correctly report the astonishing power and scientific fruitfulness of Newton’s theory of gravity.
It was this astonishing power which persuaded the Republic of letters to adopt the heliocentric model in the first place.
But this theory is wrong- catastrophically wrong- at cosmological scales.
Also, only the geocentric model predicts a cosmological significance to what is locally referred to as the ecliptic.
We now know that the so-called “Axis of Evil” aligns, on a cosmological scale, with this ecliptic.
Score one for Tycho Brahe.
If we suppose two things; a firmament or substance of space which rotates and carries bodies along with it, and a spherically propagating force of gravity, it appears to me we have the basis for explaining both the consistency of gravitational laws at local scales, and the inconsistency of gravitational laws at cosmological scales.