In terms of astronomy, the most key figure in this period is William Herschel.
In which we appear to be at the center of the galaxy this is an illusion
Most importantly: Herschel breaks with the science of astronomy as one of determining positions of objects only and instead embarks on a much broader research program called The Construction of the Heavens
Most Important: Herschel made careful observations of double stars and found they obeyed Kepler's laws Why is this important?
The next great event occurs in 1838/9.
Bessel Measures the First Stellar Parallax
Important Note: While the detection
of Stellar parallax directly confirms the copernicun model there is another,
more complicated method, known as stellar aberration which can be used.
Aberration is related to the earth's finite orbital velocity and a change in
the position of a star due to relative changes in velocity vectors for the
earth and the star. Yes this is all
physics mumbo jumbo the historical point here is that
this effect was first detected in 1727 and that is the year in which science proves
the Earth orbits the sun.
Now we have the concept of the light year as a unit of distance (the star he measured had a distance of 10 light years) note, the distance to Uranus, the furthest known object in our solar system at this time is a mere 3 light hours ... the distance to the nearest star is thus 30,000 times farther away than the distance to Uranus.
This creates an immediate physics problem what is the energy source of these distant objects? That is, we have absolutely no credible idea for why the stars shine and so we fail the alien intelligence test (again ...)
Next, comes the triumph of mathematics and the validation of precision The 1846 Discovery of Neptune:
The Problem With Mercury.
Precision measurements in the 1840's and 1850's showed that the orbit of Mercury did not conform to Newtonian Mechanics. As in the case of Uranus, a suspected perturbing body was the likely explanation.
However, in 1859, Levierre (the same guy involved in the discovery of Neptune) showed, using precision calculations, that this could not be the source of the observed anamoly in the orbit of Mercury.
So what to do Dismiss the Anamoly or Dismiss Newton?
Note the orbital period of Mercury is 88 days, so this Anamoly can be observed quite often with high precision showing its repeatability.
What would Newton Say if presented these data?