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I would like to know the position and orientation of the Earth, relative to the sun, as a function of time.

I.e. select any coordinate system with the Sun at the origin so that the earth lies in the x-y coordinate plane. Provide a function which takes as input the time and produces

  1. The x-y position of the center of the Earth
  2. The unit vector from the center of the Earth to the North Pole.
  3. The orientation of the Earth around this vector

This function is chaotic in general. I would be happy with a decent approximation.

This problem is solved as part of the "what is the position of the sun in the sky" problem. Solutions to this problem tend not to be well separable (at least I have trouble separating my question out of the complete position-of-sun-in-sky solution.)

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What precision? How broad is the time domain? First approximation is cosine, sine, and constant. –  Andrew Apr 10 '12 at 11:35
    
Ha, yes, I suppose that would satisfy the posed question. I'd like the same precision/range present in solutions to position-of-sun-in-sky problems. Mostly I just want this part of that solution to be cleanly separated out. –  MRocklin Apr 10 '12 at 12:01
    
Why the down-vote? Please help by informing me how I can make the question more clear. –  MRocklin Apr 13 '12 at 16:53
    
It wasn't me, but I'd guess it was the same lack of precision as my comment asked about. You could ask, "What is the dominant correction term to the cosine, sine, constant approximations." –  Andrew Apr 13 '12 at 20:51

1 Answer 1

For a precise measurement of the movement of the Earth around the Sun as a function of time you will need to correct for the changes in the orbit of the Earth caused by the gravitational pull of the different objects on the usual two-body problem.

These perturbations are made to the usual Kepler equation (of course, if these perturbations are long range, i.e., months or years). If all of this is unfamiliar to you, please read Chapter 2 of Murray and Dermott's, 1999 book.

On the following days (if I can fix my old hard drive that seemed to die), I can post a numerical solution to Kepler's equation in Python that I did on an undergraduate course on Extrasolar Planets. However, I must tell that it's not that hard and you can understand easily how to do it if you read Chapter 2 of the book I quoted. If you get stuck, please come back to the post and we'll try to help :-).

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