Orbits with Gravity Lab Software

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For centuries, humankind has sought to find order in the universe. In the context of Western thought, in any case, beginning with the Egyptians, Persians, ancient Greeks; in the Americas the Mayans and Azteks, Astronomy evolved out of the necessity to discover a reliable predictor of the seasons for the purposes of agriculture. In most cases, Astronomy takes on a spiritual role in culture as well. The system of accounting that ancient peoples used to measure the seasons evolved, after a great amount of suffering and turmoil, into the physics of Gallileo and the mechanics of Newton. And Newton's remarkable system is still used today, so long as the velocities are not close to the speed of light and the mass vs. density ratio of massive objects is not too great. Linked above is a gravitational simulator upon which several models of celestial motion are explored.

Written in the simple computer language of Q-Basic 4.5, the software is compilable on native systems. At the core is code that generates six n-dimensional arrays. The six arrays correspond to variable requirements in 2-d space, they are velocity vector (in polar coordinates), velocity magnitude, mass, radius, x-position, y-position. "n" corresponds to the number of total objects in the system. Once data is gathered, either entered by hand, loaded from a file, or generated randomly, the simulation can begin. There are three major divisions of the simulation, corresponding to object selection, object position change, and object velocity change, where the actual physics takes place. The simulation begins with object 1, with initial velocity vo, and calculates the next change in velocity of object 1 from the acceleration generated by all other objects. From the gravitational acceleration of object 2, for example, a new velocity vector for object 1 can be determined, and refined until object n's effect on object 1 is considered. The sim goes down the line to object n, correcting the current velocity magnitude and vector until all acceleration effects are accounted for for all objects, then the sim erases the current position of all objects, displaces the objects dependant on their current (freshly calculated) velocities, redraws them, and returns to calculating new accelerations. The result is a fairly accurate model of gravitational motion, in which the orbital properties discussed in mechanics can be seen. Inaccuracies result with high velocities or close interactions (no collision detection is made).

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It is advisable to read the readme.txt file to get a good feeling for how to use the software.

For systems where one mass is significantly greater than the others, the motion of objects around the central mass can be simplified to motion around a fixed point. The type of motion can be subdivided into two main categories - orbits that are bounded and orbits that are unbounded. For bounded orbits, two categories exist - circular and elliptical orbits. In both cases the orbiting body returns to something very close to the initial conditions. The circular orbit exists for initial velocity vector pointing tangential to the line connecting the orbited and orbiting body with magnitude that balances 'centrifugal' acceleration (or acceleration of circular motion) with the acceleration caused by the mass of the orbited body. For these conditions to be met the initial velocity magnitude of the orbiting body will be v=(GM/R)1/2, where M is the orbited body, R is the distance between them, and G is the fundamental constant of gravity.
For the orbit to be elliptical, the same initial conditions apply accept for the initial velocity magnitude - this value must be less than or greater than that of a circular orbit, up to the limiting value of the velocity of a parabolic orbit, where the kinetic energy of motion is equal to the gravitational energy of gravity, to wit:
KE=PE ==> mv2/2 = GMm/R ==> v(parabolic)=(2GM/R)1/2.
Velocities greater than that of parabolic orbits are hyperbolic, and never return to their initial conditions. Ostensibly, a parabolic orbit will return to its initial conditions when time = infinity.

Reduced Mass Theorum

For objects with similar masses, the equations of motion are not as trivial, for each mass is exerting influence on the other, invalidating the equations of motion that hold when assuming that the orbited body is fixed. For example, if we have initial conditions such that two masses are equal at a distance R apart, with initial velocity vectors tangential to the center of mass, they will need velocity magnitudes that are lower than those for fixed orbited bodies, to wit:
mu = m1m2/(m1+m2)
v(circular)=(G(mu)/R)1/2.

With more than one central mass, as in the situation where equal masses are co-orbiting the center of mass, there may exist situations where outside objects orbit the center of mass of another system that is orbiting its own center of mass. In such a situation, the characteristic orbits can exist, however, if the orbital paths come too close to the orbits of the central masses, a gravitational difference which is complicated may disrupt the motion of the orbiting masses leading to chaotic results.

As can be seen, the motion of the close to circular orbiting body is disturbed as well by the oscillation of the central bodies, and the elliptical orbit is greatly disturbed and will soon become irregular, and eventually it will be ejected from the system.

It is possible to have a system where 3 equal mass objects orbit each other, however, since it is impossible to have greater than 3 equidistant objects in 2-d space (the limit for 3-d space is 4), there is a theoretical limit of 3 objects orbiting a common center of mass with exactly the same motion (displaced by their angle distance from each other).