Siphon Analytic Simulation

The physics of the main (1997) siphon 2D simulation model uses Newton's law of gravitation to calculate the forces acting on bodies in motion, and from that their acceleration, and their velocity after some small interval of time. To create radial towers, bodies are strung together with elastic cables, with one end connected to the Earth's equator. These cables introduce new forces acting on moving bodies, which may be found using Hooke's law. The combination of gravitational and elastic forces is used to find new accelerations, velocities, and positions after a small time interval. Neither centrifugal nor coriolis forces were calculated. The siphon effect occurs when high radial towers ( > 150,000 km radius ) are laterally constrained within a rigid radial pipe. In the main simulation model, this effect is produced by simply removing lateral forces acting upon the tower, and using only radial forces to calculate body motions.

Several explanations may be offered for the need for this lateral constraint. Any tower in static equilibrium has no lateral forces acting upon it (assuming the effect of Moon, Sun, etc. are discounted). But a rising tower has coriolis forces acting laterally upon it, that move it from a radial path. In an unconstrained tower, the angular velocity w of rising bodies falls as they rise, since no force is acting laterally to counter coriolis forces, and maintain angular velocity. Another way of thinking about it is to think of a laterally unconstrained high tower in static equilibrium as a store of energy which can be used to do work raising mass from the surface. It only has a limited amount of stored energy, and so can only lift a limited amount of mass. Only by constraining the rising bodies to a radial path, and maintaining the angular velocity of bodies, can a continuous stream of mass be raised.

However, once it is accepted that the tower needs to be constrained to being radial, then the problem becomes essentially a 1D problem.

So another way of considering a tower analytically is to regard it as a 1D string of beads, and consider the radial forces acting on it.

1) Gravitational force: - G . M_e . m / r²

2) Centrifugal force: m . r. w²

3) Tension above: K . d

4) Tension below: -K . d

Where   M_e is the mass of the Earth.
            m is the bead mass.
            G is the gravitational constant.
            r is the bead radial distance from Earth.
            w is bead angular velocity.
            d is string extension beyond unstretched length.
            K is string spring constant.

The sum of these 4 forces is the net radial force acting on a bead, and may be used to calculate its radial acceleration, radial velocity, and distance travelled after a small time interval.

The Java applet below (which may be stopped by moving the cursor over it, and started by moving the cursor off it) is a close replica of the 11th 1997 simulation - the Restricted Base Feed Rigid Tower. It shows the string tensions T, and bead velocities V, and accelerations A along the length of the tower, which is initially in static equilibrium.

In this simulation, as in the original 1997 simulation, a new body is fed in at the base when tension in the string above it is sufficient to lift the body from the planet surface. Thus the lowermost string remains slack until it begins to stretch and exert force on a body at the Earth's surface. Bodies are released from the tower when their radial distance from Earth exceeds 250,000 km.

In addition, because the entire string of beads tends to accelerate upwards, a 'speed limit' of 2 km/s is imposed on the lowest body (where it is assumed easiest to apply braking forces). The braking forces are not at present calculated.

The sudden release and acquisition of bodies, and the imposition of a 'speed limit' on the lowest body, generate forces which ripple up and down the rising column.

Several problems that need solutions:

1. How is the tower to be maintained as radial?
2. How is the tower acceleration to be braked?
3. How are bodies best fed in to minimize forces propagating up the tower?
4. How are bodies best released to minimize forces propagating down the tower?