Historically, how do we know that Earth moves around Sun? And it does so in an elliptical orbit? I know the basics of solar system like how Earth moves around Sun, and that we have so many planets, elliptical orbit of earth, and how far is sun from earth etc etc. I want to take a step back and understand how mankind has been able to construct this knowledge of solar system piece by piece. Questions:
-- Sun is something we see on daily basis, so it would not be strange if humankind got curious to know about it, and made theories,observations,maths to prove it. How did Copernicus and others came up with the theory that earth moves around sun in elliptical orbit. How did they prove it? Has there been any actual way by which we have actually observed sun being in center and earth moving around in elliptical orbit. or we made such theory and result of this theory have been proved by other observations like day,night,change in weather at a point and at different points on earth.
 A: The direction in which the celestial bodies are located is seen directly in the telescopes (two angular coordinates); the distance from the Earth is the hard part. For planets, we may measure the distance by waiting for a radar echo.
Aristarchus has invented a useful method to get a third piece of information: when the Moon is exactly half-full, Sun-Moon-Earth form a right angle with the Moon angle being 90 degrees.
The distance from the Earth to the Sun has also been measured by the transit-of-Venus method - to measure the time it takes to cross the Sun. Another method is parallax - measuring and comparing the apparent locations of the celestial objects at the same time, as seen from two places on the Earth.
Tycho Brahe
Still, the angular information - 2/3 of the coordinates - is the most accurate one. Tycho Brahe, a royal astronomer in Prague, measured the positions of planets extraordinarily accurately. The fact that he believed the geocentric system couldn't affect his precision. After all, you may describe everything from the Earth's reference frame.
His young assistant Johannes Kepler - who realized that Copernicus' heliocentric model was right - used those accurate data by Brahe to derive his Kepler's laws. Note that even if you don't know how the distance from the Earth is changing as a function of time, Kepler's laws make many very constraining predictions for each moment of time - where the planets and the Sun should be seen. Kepler has simply made the right guess and verified the predictions of this guess.
In Prague, Brahe was afraid that Kepler could try to get rid of him, so that Kepler would become the main royal astronomer himself. At the end, Brahe died because of not-quite-understood problems with the bladder. I don't claim that Kepler poisoned him. ;-)
A: 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).
A: This is a very general question best answered by reading up on material found in wikipedia 
The short answer is that forces like the gravitational force have orbits for circling masses that are elliptical in nature, where the sun sits at one focus of the ellipse. Circles are an ellipse where the two foci overlap.
If you read the wiki article, you will see that a lot of data were gathered over the centuries on the motion of the planets on the celestial sphere, the geocentric system. Copernicus realized that a heliocentric system would fit the data simply and beautifully instead of the epicycle system that had dominated astronomy till then. The orbits were ellipses.
A: I really liked Ethan Siegel's explanation of this over at ScienceBlogs:
http://scienceblogs.com/startswithabang/2010/09/geocentrism_was_galileo_wrong.php
