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I would like to write a program that visualizes the solar system. To do this, I would need equations for calculating the coordinates of each solar system object at a given time. Where could I obtain such equations? I hope they would be as accurate as possible. If there are such equations, it would be also nice to see the raw data they are based on and a description how they were computed. I would be grateful for all help.

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It depends a bit on how precise you would want to be. A very good discussion on how to calculate the orbits of solar system objects is given in the book by Jean Meeus, Astronomical Algorithms (1999), which is at an advanced amateur level. At professional level you have the Explanatory Supplement to the Astronomical Almanac by Urban and Siedelmann.

For precise calculations Meeus uses the VSOP theory from Bretagnon (1987) (wikipedia link). You can download all files needed for these calculations from VizieR. These files contain a lot of numerical terms that are needed in the equations. The resulting positions are however very precise.

For the highest accuracy, you can download the predicted positions from the Jet Propulsion Laboratory (JPL) Horizon system.

There are also quite a few software library projects that implement the equations from Meeus. This might be the best option. Then you do not have to implement the equations yourself. For C/C++ you have for instance LibNova.

For highest precision you have professional software libraries such as for instance the NOVAS libraries from the Naval Observatory for Fortran, C, or Python. But to use that correctly you will have to have a good understanding of celestial mechanics.

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