Numerical model of the Solar System: Difference between revisions

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{{Short description|Equations to predict the position of planets}}

{{No footnotes|article|date=April 2009}}A '''numerical model of the Solar System''' is a set of mathematical equations, which, when solved, give the approximate positions of the planets as a function of time. Attempts to create such a model established the more general field of [[celestial mechanics]]. The results of this simulation can be compared with past measurements to check for accuracy and then be used to predict future positions. Its main use therefore is in preparation of almanacs.

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==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 (''v_x'', ''v_y'', ''v_z'') for each of the bodies involved. When also the mass of each body is known, the acceleration (''a_x'', ''a_y'', ''a_z'') can be calculated from [[Newton's Lawlaw of Gravitationgravitation]]. Each body attracts each other body, the total acceleration being the sum of all these attractions. Next one chooses a small time-step Δ''t'' and applies [[Newton's Secondsecond Lawlaw of Motionmotion]]. The acceleration multiplied with Δ''t'' gives a correction to the velocity. The velocity multiplied with Δ''t'' 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 Δ''t''. Repeating this procedure often enough, and one ends up with a description of the positions of all bodies over time.

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likewise for Y and Z.

The former equation (gravitation) may look foreboding, but its calculation is no problem. The latter equations (motion laws) seemsseem simpler, but yet itthey cannot be calculated. Computers cannot integrate, they cannot work with infinitesimal values, so instead of dt we use Δt and bringing the resulting variable to the left:

<math>\Delta v_x = a_{x} \Delta t </math>, and: <math>\Delta x = v_{x} \Delta t </math>