Numerical model of the Solar System: Difference between revisions

Revision as of 19:57, 29 July 2006

Revision as of 14:49, 24 August 2006

Line 32:

likewise for Y and Z.

The former equation (gravitation) may look forbidding, but its calculation is no problem. The latter equations (motion laws) seems simpler, but yet it cannot be calculated. Computers cannot integrate, they cannot work with infinitesimal values, so instreadinstead 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>

Line 60:

To calculate the accelerations the gravitational attraction of each body on each other body is to be taken into account. As a consequence the amount of calculation in the simulation goes up with the square of the number of bodies: Doubling the number of bodies increases the work with a factor four. To increase the accuracy of the simulation not only more decimals are to be taken but also smaller timesteps, again quickly increasing the amount of work. Evidently tricks are to be applied to reduce the amount of work. Some of these tricks are given here.

By far the most important trick is the use of a proper integration method, as already outlined above.

The choice of units is important. Rather than to work in [[SI units]], which would make some values extremilyextremely small and some extremilyextremely large, all units are to be scaled such that they are in the neighbourhood of 1. For example for distances in the solar system the [[astronomical unit]] is most straightforward. If this is not done one is almost for sure to see a simulation aborted in the middle of a calculation on a [[floating point]] [[overflow]] ofor [[underflow]], and if not that bad, still accuracy is likely to get lost due to [[truncation]] errors.

If N is large (not so much in solar system simulations, but more in galaxy simulations) it is customary to create dynamic groups of bodies. All bodies in a particular direction and on large distance from the reference body, which is being calculated at that moment, are taken together and their gravitationgravitational attraction is averaged over the whole group.

The total amount of [[energy]] and [[angular momentum]] of a closed system are conserved quantities. By calculating these amounts after every time step the simulation can be programmed to increase the stepsize Δt if they do not change significantly, and to reduce it if they start to do so. Combining the bodies in groups as in the previous and apply larger and thus less timesteps on the farwawayfaraway bodies than on the closer ones, is also possible.

To allow for an excessively rapid change of the acclerationacceleration when a particular body is close to the reference body, it is customary to introduce a small softness parameter ''e'' so that

<math>a = \frac{G M}{r^2 + e}</math>

==Complications==

In case highest accuracy is needed things get much more complex. In the case of comets non gravitational forces (radiation pressure and gas drags) are to be taken into account. In the case of Mercury relativity effects can be not be ignored. Then also the total energy is no longer a constant (because the four vector energy with linear momentum is). The final speed of light also makes it important to allow for light-time effects, both classical and relativity. Planets can no longer be considered as particles but their shape and density must be considered too. For example the flattening of the Earth causes precession, which causes the axial tilt to change, which has its effects on the longterm movements of all planets.

==References==