(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. Integration In the previous section it was assumed that acceleration remains constant over a small timestep Δt so that the calculation reduces to simply the addition of V × Δt to R and so forth. In reality this is not the case, except when one takes Δt so small that the number of steps to be taken would be prohibitively high. Because while at any time the position is changed by the acceleration, the value of the acceleration is determined by the instantaneous position. Evidently a full integration is needed. Several methods are available. First notice the needed equations: This equation describes the acceleration all bodies i running from 1 to N exercise on a particular body j. It is a vector equation, so it is to be split in 3 equations for each of the X, Y, Z components, yielding: with the additional relationships , likewise for Y and