astronomyhistoryobservational-astronomy
Trying to observe the night sky for a few weeks, the motion of the Sun and the stars pretty much fits into the Geocentric Theory i.e. All of them move around the Earth.
What then, which particular observation, made us think that it could be the other way around, that all the planets move around the Sun?
In his lecture The Cosmic Distance Ladder (video), mathematician Terence Tao describes the history of how mankind has successfully mapped the solar system and beyond. In particular, he describes why Copernicus put the sun in the center (reason: he discovered that the sun is dozens of times bigger than the earth) and how Kepler found his laws of planetary motion (He needed a reference point that was fixed in place, but all planets move. His brilliant idea was too choose Mars as the reference point, but only looking at it after a full orbit, when its position returns to the original one).
As you are talking about a "Star map" and "current visible positions", I'll assume you are talking about a star map of the ${\sim} 5000$ stars visible to the naked eye.
Most of those stars are within 1000 light years of he Earth. They have typical velocity dispersions with respect to the Earth of ${\sim} 10$ km/s, with the occasional rare star with a velocity of ${\sim} 100$ km/s. This translates into proper motion on the sky of milli-arcseconds to a few arcseconds per year.
Here is a plot of proper motion versus distance taken from the second version of the Hipparcos catalogue by van Leeuwen (2007). I selected 4022 stars with magnitudes brighter than 6, and with uncertainties in their proper motion of less than 1 milli-arcsecond/year and uncertainty in parallax of less than 20% (to ensure reasonably accurate distances and tangential motions).
To estimate the size of the effect you are talking about, we need to multiply the proper motion by the distance (in light years) to work out how far the stars have moved whilst the light has travelled towards us. The plot below shows by how much a star's position would have shifted from where they would be conventionally plotted in a star chart as a function of stellar magnitude (brightest stars on the left).
Now you could plot a star chart based on these numbers, but the typical deviations are less than 100 arcseconds, which is approximately the same angular resolution as the human eye. So in a revised star chart based on these numbers, which shows where the stars are now, there would be no perceptible difference in the appearance of the constellations.
I have highlighted one outlier in the plot. This is Arcturus, a bright red giant, in the constellation of Bootes (easily found in the night sky). It is at a distance of 37 light years, and in 37 years it moves 83 arcseconds in the night sky.
An issue that does come into play, and I'm not sure what the "rules" of the question are, is the precession of the Earth. This has the effect of changing the positions of stars with respect to our coordinate system (though not with each other). The equinox precesses at around 50 arcseconds per year, so if star maps are plotted on a fixed RA and Dec grid then this is a massive effect compared to what I have described above. This could be calculated, but it feels like cheating because it isn't reflective of any true change in the positions of the stars, just their positions with respect to our coordinate system.
Best Answer
Retrograde motion must be a prime candidate.
As seen from Earth against star background, Mars occasionally slows down and goes backwards. Our moon doesn't.
It probably became clear to people constructing orreries that heliocentric models were enormously simpler and more convincing. They also tied in with simple inverse square laws of gravitation and planetary motion.
The discovery by Galileo Galilei of Jupiter's moons also provided firm evidence of the existence of heavenly objects that, perversely, did not orbit the Earth.
Photo: Thomas Bresson (Galileo probably didn't have a Nikon / mobile phone handy)
Luckily, he had available a corner of a napkin, a goose and some soot (or equivalents)