A Model Universe



W Roy Masefield,

Holland-on-Sea,

Essex



The motion of two bodies under the influence of gravity -

the so-called two-body problem of classical mechanics -

lends itself very well to demonstration by computer. The

mathematics of the problem is quite straightforward,

involving nothing more than Newton's Laws of Force and

Motions. The possible demonstrations include such things

as the capture of one astronomical body by another; orbits

of satellites; behaviour of binary stars and so on, given

only the starting data.

  When the program is Run, the data is called for, and to

give you some idea of suitable values to start with, I

suggest you try out those given in figures 1 to 5. [These

figures were screen dumps of the program in execution.

With the data given below, you can recreate them yourself.]

Then, when you get the feel of it, to try your own values.

Angles, by the way, are entered in degrees - the program

converts these to radians as required by the ZM Spectrum -

with zero being the east direction. Negative values for the

angles are legitimate.

  The initial positions of the two bodies are entered as x

and y co-ordinates, these being the normal pixel co-ordi-

nates of the Spectrum, but do not go above x=170 because

the right-hand quarter of the screen is reserved for

displaying the starting data. When all the data is entered,

the program starts to plot the paths. It is not fast, but

fascinating to watch. Plotting continues until one of the

curves runs off the edge of the screen area, but plotting

can be stopped at any point by Break.

  A look, now, at some of the possible demonstrations.

  Figure 1: This shows a small mass orbiting a much larger

one.

[Data: M1=1000, v1=0,      A1=0, x1=90, y1=90,

       M2=1,    v2=14.142, A2=0, x2=90, y2=140, G=10.

And yes, this orbit is instable. It's surprisingly hard

to find a perfectly stable one.]

  Figure 2: In this, a small body approaches a more massive

body at an oblique angle to the direction of the gravi-

tational force. The larger body is initially at rest. The

smaller body is pulled out of its straight-line path and

traces a near-parabola, and the larger body is given some

motion and it begins to trace out a curved path.

[Data: M1=100, v1=0,   A1=0, x1=110, y1=88,

       M2=10,  v2=2.5, A2=0, x2=0,   y2=140, G=10.]

  Figure 3: Binary stars are illustrated here. One star is

larger than the other, and each has been given the required

tangential velocity for a circular orbit round the common

centre of gravity. Try other velocities and see what

happens.

[Data: M1=100, v1=1.49, A1=0,   x1=80, y1=125,

       M2=50,  v2=2.98, A2=180, x2=80, y2=55, G=10.]

  Figure 4: This is an Earth-Moon simulation. The larger

body is given a velocity as if in orbit round the sun. As

we are looking at only a short piece of this orbit, the

lack of curvature does not matter. The smaller mass is

given the velocity calculated to keep it in orbit, plus

the velocity of the larger.

  The resultant paths are fair approximations to what

actually happesn with the Earth-Moon system: the Moon

performs a series of loops and the Earth, because of the

mutual gravitational action, does a series of cycloid-like

loops. Certainly, the Earth's path is not a smooth one.

[Data: M1=100, v1=12.16, A1=0, x1=0, y1=130,

       M2=600, v2=0.33,  A2=0, x2=0, y2=80,  G=10.]

  Figure 5: For fun, we may see what happens when the force

of gravity is repulsive instead of attractive, i.e., when

we have the anti-gravity of science fiction.

  To do this, enter a negative value for the gravitational

constant. The figure shows two equal masses approaching

each other almost, but not quite, in a direct line. See if

you can interpret what is happening if they /are/ in an

exact head-on course.

[Data: M1=100, v1=4, A1=-45, x1=0,   y1=150,

       M2=100, v2=4, A2=135, x2=140, y2=30,  G=-7.]

  Adventurous programmers might like to develop this

program further by including action under electric forces.

In this case, instead of F_g, we shall have F_E, which

depends on the electric charges (q) carries by the bodies:



                      k q_1 q_2

                F_E = ---------

                           2

                          d



  The motions resulting from this force behave exactly as

for gravitation, i.e., they depend on the masses of the

bodies.