Modern Method
The modern method consists of numerical integration in 3-dimensional space. One starts with a high accuracy value for the position (x, y, z) and the velocity (vx, vy, vz) for each of the bodies involved. When also the mass of each body is known, the acceleration (ax, ay, az) can be calculated from Newton's Law of Gravitation. Each body attracts each other body, the total acceleration being the sum of all these attractions. Next one chooses a small time-step Δt and applies Newton's Second Law of Motion. The acceleration multiplied with Δt gives a correction to the velocity. The velocity multiplied with Δt gives a correction to the position. This procedure is repeated for all other bodies.
The result is a new value for position and velocity for all bodies. Then, using these new values one starts over the whole calculation for the next time-step Δt. Repeating this procedure often enough, and one ends up with a description of the positions of all bodies over time.
The advantage of this method is that for a computer it is a very easy job to do, and it yields highly accurate results for all bodies at the same time, doing away with the complex and difficult procedures for determining perturbations. The disadvantage is that one must start with highly accurate figures in the first place, or the results will drift away from the reality in time; that one gets x, y, z positions which are often first to be transformed into more practical ecliptical or equatorial coordinates before they can be used; and that it is an all or nothing approach. If one wants to know the position of one planet on one particular time, then all other planets and all intermediate time-steps are to be calculated too.
Read more about this topic: Numerical Model Of The Solar System
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