Numerical model of the Solar System: Difference between revisions

Revision as of 09:58, 26 March 2015

Revision as of 19:36, 13 April 2015

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<math>\vec{a}_j = \sum_{i \neq j}^n G \frac{M_i}{|\vec{r}_i - \vec{r}_j|^3} (\vec{r}_i - \vec{r}_j)</math>

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:

<math>(a_j)_x = \sum_{i \neq j}^n G \frac{M_i}{( (x_i - x_j)^2 + (y_i - y_j)^2 + (z_i - z_j)^2 )^{3/2}} (x_i - x_j)</math>

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==Complications==

If the highest possible accuracy is needed, things become much more complex. In the case of comets, nongravitational forces (radiation pressure and gas drag) must be taken into account. In the case of Mercury (and other planets for long term calculations), relativistic effects cannot be ignored. Then also the total energy is no longer a constant (because the four vector energy with linear momentum is). The finite speed of light also makes it important to allow for light-time effects, both classical and relativistic. Planets can no longer be considered as particles, but their shape and density must also be considered. For example, the flattening of the Earth causes precession, which causes the axial tilt to change, which affects the long-term movements of all planets.

Long term models (a few tens of millions of years) are not possible due to in [[stability of the Solar System]].

==See also==