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.

==Older efforts==

The simulations can be done in either [[Cartesian coordinate system|Cartesian]] or in [[Spherical coordinate system|spherical]] coordinates. The former are easier, but extremely calculation intensive, and only practical on an electronic computer. As such only the latter was used in former times. Strictly speaking, the latter was not much less calculation intensive, but it was possible to start with some simple approximations and then to add [[Perturbation (astronomy)|perturbations]], as much as needed to reach the wanted accuracy.

In essence this mathematical simulation of the [[Solar System]] is a form of the ''[[N-body problem]]''. The symbol '''''N''''' represents the number of bodies, which can grow quite large if one includes the Sun, 8 planets, dozens of moons, and countless planetoids, comets and so forth. However the influence of the Sun on any other body is so large, and the influence of all the other bodies on each other so small, that the problem can be reduced to the analytically solvable 2-body problem. The result for each planet is an orbit, a simple description of its position as function of time. Once this is solved the influences moons and planets have on each other are added as small corrections. These are small compared to a full planetary orbit. Some corrections might be still several degrees large, while measurements can be made to an accuracy of better than 1″.

Although this method is no longer used for simulations, it is still useful to find an approximate [[ephemeris]] as one can take the relatively simple main solution, perhaps add a few of the largest perturbations, and arrive without too much effort at the wanted planetary position. The disadvantage is that perturbation theory is very advanced mathematics.

==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>

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==References==

*{{Cite book|first=Dan L. |last=Boulet |title=Methods of orbit determination for the microcomputer |publisher=Willmann-Bell, Inc |___location=[[Richmond, Virginia]] |year=1991 |pages= |isbn=978-0-943396-34-7 |oclc=23287041}}{{Page needed|date=September 2010}}

{{Solar System models}}

{{DEFAULTSORT:Numerical Model Of Solar System}}

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[[Category:Dynamical systems]]

[[Category:Dynamics of the Solar System]]

[[Category:Solar System models]]