Such a condition would imply, that we occupy a unique position[/b] in the universe and, analogous, in a sense to the ancient conception of a central earth, this hypothesis cannot be disproved..... but it is unwelcome and would only be accepted as a last resort in order to save the phenomenon. Therefore we disregard this possibility......the unwelcome position of a favored location must be avoided at all costs.... such a favored position is intolerable..... therefore in order to restore homogeneity and to escape the horror of a unique position, it must be compensated by spatial curvature. There seems to be no other escape. (1937)

You're really showcasing your ignorance.  Size has absolutely nothing to do with this.  It's possible for the 333,000 smaller body to have a LARGER MASS than the bigger one.  It's about mass and density and not about size.

In this case, however, assuming that the assumption about the MASS of the two planets are indeed, correct, if the sun and earth existed alone ... as an isolated system, then the barycenter would certainly just so happen to reside somewhere within the physical boundaries of the sun.  But the sun and earth are not alone in the solar system.  So even NASA admits that the barycenter of the solar system (depending on the alignment of the planets) CAN exist outside the physical boundaries of the sun.  And then if you expand the system to the entire galaxy and then to the entire universe, no one can prove where the barycenter of the universe is ... or isn't.

I have read the Dimond Brothers attempt to render the 1616 and 1633 decrees no worth the paper they were recorded on. 'Their conclusion is that Geocentrism is NOT an infallible teaching of the Church.' you say and obviously believe. So, here we have the Protestant version of theology, thinking they know more than the judgement of Rome.

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OK, however, it the Earth is orbiting the Sun, it's a Heliocentric solar system ...

I thought this thread was about the Dimond brothers and theology. It is now one that will seen by placed with the flat-earth out of sight.

Yes, all popes who believe the Scriptures do not reveal geocentrism are heretics. But the circuмstances of their heresy is such that it amounts to MATERIAL heresy.

In Catholic theology, the term material heresy refers to an opinion objectively contradictory to the teachings of the Church, which as such is heretical, but which is uttered by a person without the subjective knowledge of its being so.

Beginning with Pope Pius VII, popes were told the heliocentrism of 1820 was not the heretical heliocentrism of 1616 and 1633, but a non heretical heliocentrism. Millions of other Catholics were also material heretics for the same reason. Material heresy is really NON-HERESY.

But you Apollo, have been told the truth, so your choice to accept heliocentrism can be classed as formal heresy because you have rejected the truth after being told it. The same reasoning goes for the Dimond brothers and God knows how many others who have rejected the truth after being told geocentrism was never proven wrong and that no pope every rejected the infallibility of the 1616 decree.

Excellent point.  For all we know, the sun could have a much smaller mass than science claims that it does.