What planetary orbit model allows a scientist to predict the exact positions of the planets in the night sky for many years?
There are a number of different models for predicting the position of planets. It depends on how accurate the positions need to be.
Kepler's three laws describe how planets move around the Sun in perfect ellipses. Unfortunately, Kepler's laws only work if there is only one planet in the solar system. If there is a third body the orbits of the planets can't be calculated analytically as the equations are too complex.
A number of approximations have been produced to determine the positions of planets. These are complex power series which sum up the effects of the gravity of the other planets.
The calculations required for high accuracy were too complex for early computers to perform in a reasonable time. A modern laptop can easily perform these calculations in fractions of a second.
The VSOP87 data from Paris is a good approximation of the positions of the 8 planets to a fairly high degree of accuracy.
The definitive model is available from NASA'a jet Propulsion Laboratory. These data sets use numerical integration to combine data from observations from a number of sources including space probes. The data requires complex computations of thousands of terms. The DE430 data set allows the calculation of the position of the Sun, Moon and all of the planets, including Pluto, can be performed to fractions of a millimetre.
A new data set DE432 was created specially for the New Horizons mission to Pluto.