One evidence of the earth orbiting around the sun is the stellar parallax:

"Stellar parallax is the apparent shift of position of any nearby star (or other object) against the background of distant objects. Created by the different orbital positions of Earth, the extremely small observed shift is largest at time intervals of about six months, when Earth arrives at opposite sides of the Sun in its orbit, giving a baseline distance of about two astronomical units between observations. The parallax itself is considered to be half of this maximum, about equivalent to the observational shift that would occur due to the different positions of Earth and the Sun, a baseline of one astronomical unit (AU).

Stellar parallax is so difficult to detect that its existence was the subject of much debate in astronomy for hundreds of years. Friedrich Bessel made the first successful parallax measurement in 1838, for the star 61 Cygni, using a Fraunhofer heliometer at Königsberg Observatory."

             (from Wikipedia)

Measurement of the distances of the stars (Dyson)