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.