Tracking the earth's position relative to the sun as a fixed object.

[xonto]

The earth orbits the sun elliptically such that, counting the sun as a fixed object, the earth ends up in a different place than where it started. For a visual aid, this video depicts the kind of motion I am talking about:

I am curious about the math involved in determining where the earth is located relative to where the earth was the previous year (relative to the sun), and how many years will it take for the earth's orbit it place it back at an arbitrary starting point.

For example, today, April 14, 2011, 00:05edt (as of this writing) the earth is at it's present location relative to the sun. On April 14, 2012, where will the earth be relative to it's position right now? How many years until it will be at this position on April 14 again, or when was it here last?

Thanks for the input.

[johnrt]

The motion of planets in elliptical orbits are described by Kepler's laws.

Kepler's laws of planetary motion - Wikipedia, the free encyclopedia

[acey]

The phenomenon you're interested in is called precession.

Apsidal precession - Wikipedia, the free encyclopedia

Quote: "It takes about 21,000 years for the ellipse to revolve once."

The perihelion precession of Mercury couldn't be adequately explained using Newtonian mechanics - it became a classic test of general relativity. In the case of Earth you would likewise need general relativity for the most precise answer, but the relativistic effect is far weaker because Earth is further from the Sun.