Physics doesn't depend on the observer
Different descriptions don't change Physics.
You can just use the simplest description (coordinates, reference frames, ...) for your needs, and change to another one via coordinate and reference transformation rules that preserve the invariant nature of the physical process w.r.t. the observer.
A "good" physical theory/model/description:
- is a simple theory/model/description, where simple really means you need the smallest amount of calculations or the easiest calculation as possible
- provides good results when compared with experimental observations within the accuracy needed by the events described
Short story of theories about the motion of the solar system
Prehistory. (Pre)historically, humans were mainly interested in the motion of the Sun and the Moon, and the geocentric system looks quite a natural choice for the description of the system Earth-Sun-Moon. The background of the fixed stars fits this model quite naturally as well.
In the ancient Mesopotamia, the first planets were discovered as celestial bodies "moving differently" if observed from the Earth. Anyway, they were good enough in Math to correctly describe the motion of the planets using a geocentric point of view, and they probably were not so much interested in a easier way to describe their motion, since they were not so affected by them, probably excluding for religion reasons.
First heliocentric models. Around 300 B.C., Greeks astronomer Aristarchus of Samos identified that planets describe orbits around the Sun (however you describe it, planets orbit around the Sun, or the Sun orbits around the planets, the description is relative, the Physics is not), presenting its heliocentric model where planets described circular orbits around the Sun. Anyway, this theory suffered from substantial errors and didn't provide accurate enough predictions: inaccurate predictions and accusations of heresy were enough to keep the geocentric theory alive for some centuries. He had formulated a qualitatively good theory, but not a quantitative good theory (simple model, but not accurate results).
Galileo and Kepler. The use of telescope in '400-'500 provided an instrument for more detailed astronomical observations, that allowed:
- Galileo to observe Jupiter satellites, making him think about if that system could somehow resemble Earth-Moon system;
- Kepler to formulate and accurately test its 3 laws about planetary motion. These three laws provided quite a simple physical model capable of doing accurate predictions of the motion of celestial bodies, if compared with the results of the geocentric theories, thus a good theory (simple description + accurate results).
After that a good heliocentric theory was available, discussions that followed about helio- or geocentrism in '500-'600 were more about philosophy, religion and the role of the human beings in the universe and the history.
Newton's classical mechanics as the first theory of gravitation, through action at disance. During plague lockdown of the 1666 in England, an undergrad student named Newton developed differential calculus for formulating the three principles and developing its theory about Mechanics, including gravity as an action at distance described by its universal gravitation law,
$\mathbf{F} = G m M\dfrac{\mathbf{r}}{r^3}$
Limit of Kepler's laws and classical mechanics: precession of the Mercury perihelium. Kepler's laws are good enough for many applications in celestial motion, but they are not good enough to describe the precession of Mercury perihelium, the closed point of the orbit from the Sun.
We're dealing with a very small prediction error made by classical mechanics and Kepler's laws, about $42.9799$ arcsec/century.
This is one of the classical tests for the validation of Einstein's general relativity, that manages to provide this (more) accurate results, promoting it as the most accurate theory of gravitation, and thus for the motion of celestial objects.
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