Copyright (c) 1999 Jens Jørgen Nielsen

Congratulations! You now have a copy of GRAVITY 1.4, a fun and easy to use educational celestial mechanics program.

GRAVITY is copyrighted material. You are granted permission to use GRAVITY 1.4 for free, and to make as many backup copies for your own purposes as you want. You are encouraged to distribute copies of GRAVITY.

Gravity is the force of attraction that governs the motion of all larger objects in the Universe, e.g. moons, planets, stars and comets. Newton formulated it as the law of general gravitation:

F = k * m * M / (r*r).

In words it states:

"Two bodies attract each other with a force that is proportional to the mass of each body and is inversely proportional to the square of their distance apart."

If more than two bodies are involved, then every body attracts every other body according to the law of general gravitation. Applying the rule of superposition the resulting force on one of them can be obtained by adding the force experienced from each of the other bodies.

GRAVITY is capable of simulating the motion of bodies, which attract each other according to the law of general gravitation in the three dimensional space.

Input to GRAVITY is a description of what the Universe looks like just before simulation begins. The initial "State of the Universe" is completely specified in an easy to read text file supplied by the user when the program is started.

Several sample files are available, but try creating your own private Universe and observe how it evolves. This is what is really exciting about using GRAVITY.

During simulation the motion of each of the bodies in the Universe is shown as a moving dot or as a line on the screen. Several vision modes are available.

Also during simulation it is possible to move around the perimeter of the Universe using the arrow keys, thereby viewing it from different angles.

The program is started from the DOS prompt, e.g.:

C:> GRAVITY 2PLUS1.BDY

The second parameter on the command line, in this case 2PLUS1.BDY, refers to a text file specifying the start conditions of the Universe to be simulated. The exact format of this file is described below.

Simulation proceeds immediately after the start condition file has been read.

During simulation several keys are active:

In vision mode 1 the bodies have the color specified in the .BDY file. This mode is very useful when the motion of specific objects is observed.

Vision mode 2 visualizes the third dimension using the intensity of the dot on a gray scale. The brighter the object the closer to the viewer it is. With a little practice you will be able to see some depth in the picture.

Vision mode 3 presents a red/green stereogram. When viewed through a pair of colored glasses, red for the right eye and green for the left eye, the feeling of bodies moving in space is surprisingly real.

When the simulation is stopped the current "state of the Universe" is saved in a file named EXIT.BDY. GRAVITY uses this file as its input if no other filename is specified.

A number of .BDY files, each describing a "universe", are supplied with GRAVITY. They demonstrate some of the possibilities of GRAVITY.

4STARS.BDY

Four celestial objects with equal mass. Position and velocity is symmetrical.

8STARS.BDY

Same as 4STARS.BDY but with 4 more objects added. This example reminds me of "les lanciers", a dance I practiced at high school (especially when a small error causes total chaos).

MOON.BDY

A system with a light object (moon) circulating a heavy object (planet) which is circulating a very heavy object (sun). This example uses vision mode 2, where gray scale coding is used for depth visualization.

PLANETS.BDY

This system with four heavy objects circulating a very heavy object is a (very) rough model of our solar system (Sun, Mercury, Venus, Earth, and Mars).

MP1992.BDY

This is an accurate model of our solar system, with the major planets. The position corresponds to 1992. If you want to see the inner planets you have to zoom in.

2PLUS1.BDY

One planet is circulating a double star system. Note the very aperiodic but still stable behavior of the planet.

LAGRANGE.BDY

The famous stable LaGrange points are demonstrated with an object (Moon) circulating a heavy object (Earth). A number of massless particles (dust) initially follow the same trajectory as the Moon. After some time the location of the LaGrange points becomes evident.

CLASH.BDY

Two groups, each comprising 100 stars of equal mass, clash. Although operating in vision mode 2 the arrow keys give a great 3D feeling of what is going on.

3D.BDY

This example showing 41 objects with equal mass, initially distributed on a sphere, demonstrates vision mode 3, the three-dimensional red/green stereogram mode. A pair of colored glasses is highly recommended in order to enjoy the 3D effect. Try the arrow keys.

INTRUDER.BDY

This is a model of a galaxy. A number of massless objects (stars) are circulating a very heavy object (center of galaxy). Another very heavy object passes by at a high speed (another galaxy). Red/green glasses are highly recommended.

164STARS.BDY

164 stars of equal mass flying around in a cluster. It is interesting to both zoom in and out and to pan around.

The initial "State of the Universe" is completely specified in a text file supplied by the user when the program is started. Also specified in this file are a number of parameters controlling the simulation and selecting the vision mode.

The file consists of a special first line containing information common to all bodies, followed by a number of lines each describing the state of a specific body in the Universe.

The format of the first line is:

<timeSlice> <graviConst> <viewMode> <obsDist> <softening>

The format of the following lines is:

<mass> <x1> <x2> <x3> <v1> <v2> <v3> <color> <dotLine>

For implementation purposes the number of bodies is limited to max. 8095. You will probably never experience this limitation as a problem, because the simulation proceeds very slowly when the number of bodies is that large.

The parameters have no dimensions, meaning that no particular unit of time is assumed for the timeSlice parameter. The same is true for the other parameters such as mass, distance and velocity.

The timeSlice parameter determines the step size used in the simulation. In order to work properly the step size must be lower than a certain level, which depends on the value of the other parameters.

The graviConst parameter corresponds to the constant of proportionality, k, of Newton's law of general gravitation.

The obsDist specifies the distance of the observer's eyes from the screen. This parameter is used only in Vision Mode 3. As the screen is approx. 600 units wide, a proper value is in the range of 2000 to 3000.

Softening is a parameter that helps the simulator avoid numerical instability. As two bodies get very close to each other the force of attraction between them gets very large with the result that large errors are introduced to the simulation. Limiting the numerical value of the force of attraction in a smooth manner solves this problem. If a value of 1 is chosen, the limiting only becomes effective when two bodies get very close to each other (distance < 1).

Mass is the weight of the body.

The position of the body is specified using x1, x2 and x3. In order to be within the limit of the screen, the values of x1 and x2 should be in the range of -250 to 250. No two bodies must occupy exactly the same position in space. V1, v2 and v3 specify the velocity.

The color parameter is used in vision mode 1. It ranges from 0 to 15. Depending on the dotLine parameter a dot (0) or a line (1) is drawn.

An easy way to get familiar with the file format is to try modifying one of the .BDY files supplied with GRAVITY.