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Gravitation Help/Tutorial

This page explains the theory behind gravitational modeling, and offers some useful hints for using the Gravitation program.

  Helpful Hints

  The Theory and Strategies Behind "Gravitation"

The basic gravitational equation is:

where F equals the force between two masses M1 and M2, R is the radius (distance) between the two masses, and G, the universal gravitational constant, is equal to 6.670 x 10-8 dyne cm2 gm-2. This formula gives the force between two masses at a particular instant of time. This formula is based on the work of Isaac Newton and is the basic equation in what is called "Newtonian Gravitation," gravitation in which relativistic effects are not taken into account.

To model the behavior of masses across time, things get a bit more complicated. If there are only two masses being modeled, then a relatively simple set of equations can be used. If more than two masses are involved, one must apply the above formula repeatedly, apply the resulting accelerations to the masses, and repeat the process in a computer simulation. This is what "Gravitation" does.

Numerical gravitation simulations are never more than approximations of the behavior of masses in nature. The limitations are:

  1. One must select a finite time interval for each iteration of the program. If the chosen interval is small, this increases accuracy but requires more computer time. If the chosen interval is large, this speeds up the simulation but at the cost of decreased accuracy.
  2. The floating point numbers used in computers have limited resolution. This resolution limit can degrade the accuracy of the simulation over many iterations.

The effect of these limitations can be dramatically seen in the "Three Mutual Orbit" scenario, where the three bodies gradually begin to lose their original orientation, perform an intricate dance, and finally fly apart at great speed. Some of this behavior can be traced to limitations in the numerical simulation and do not represent real gravitational effects. The rest of the scenario is a valid representation of nature — three mutually attracting bodies of equal masses tend to be unstable.

In some of Gravitation's scenarios, two bodies will approach and then fly apart at high speed. This is a limitation of numerical gravity simulation, it is not a property of natural gravitation — if it were, it would violate the principle of energy conservation. It comes about because a particular body may pass very close to another, experience that body's gravitational attraction, then on the next iteration be located far past that same body, so there is no deceleration to match the acceleration, as there would be in nature. This effect can be minimized by choosing small increments of time, but it cannot be eliminated. This incremental effect, and the errors brought on by computer floating point numbers, are limitations in all numerical gravitational simulations, not just the one performed by Gravitation.

The method used in Gravitation is the same as that used in large research facilities to, for example, model the evolution of star clusters. The computation time required to model a set of N mutually interacting bodies is (more or less) proportional to N2, because each body's gravitational acceleration is the sum of all the other bodies' influences. Therefore, for each body in the system, the program must calculate the sum of the accelerations of all the other bodies. This computational difficulty explains why stellar cluster modeling is an important application for supercomputers.

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