Simple Representation of a Planetary Orbit (Demonstration Example 2 for RAT 740)

This is an English translation of the original German document (language: de).

COMMUNICATIONS OF THE ANALOG COMPUTER DEPARTMENT

Special Issue Demonstration Examples for Analog Computer RAT 740 1.11.66

Example 2

Simple Representation of a Planetary Orbit

According to Kepler’s first law, planetary orbits are ellipses with the Sun at one focal point (the center point of the Sun’s orbit).

According to Kepler’s second law, a planet sweeps equal areas in equal times — that is, the orbital velocity is greatest when closest to the Sun (perihelion) and slowest when farthest from the Sun (aphelion). However, the angular velocity ω of a planet around the focal point cannot be kept constant (only in the special case of a circle, where the Winkelgeschwindigkeit ω = const can be achieved, is this possible — this is only possible for very small eccentricities).

For a simplified representation, the following approximation is used: a sine-cosine resolver with outputs (for FG 1) sin(ωt) and cos(ωt), where ω is a “slow” sine-cosine resolver (SCS-1), that on the diagonal of the ellipse generates a point (Par. 4, M II) which, when multiplied by the sine-cosine component with magnitude 1 (Par. 4, M II), generates the orbit.

The semi-minor axis b can be held constant for FG 2 and FG 19 amplifiers. (One can also see the effect of the parameter changes on the orbital shape, for example by simultaneously varying the eccentricity.)

Page 2

[Continuation of the mathematical description of the simplified planetary orbit representation]

Using a two-channel oscilloscope display, one can generate the display using two “actual” sine-cosine resolver outputs (SCS-1, 14, von “einer” “Sonne”-SCS), from which the Keis is held constant. By multiplying the sine component with the magnitude 1 (Par. 4, M II) from the circle, the ellipse is generated.

The size of the semi-axis b allows for FG 2 and FG 19 amplifiers. (One can also see the effect of the parameter change on the orbital shape through simultaneous variation of the eccentricity, for instance by varying the Kreisbahnpunkt for Ellipse e.)

Goals for the Demonstration:

Setting initial conditions: h = 0.2
Setting the orbital parameter (eccentricity) by varying e (0.2)
Setting the elliptical orbit parameters by varying b (0.2)
Displaying the position of the point on the circle in a Lissajous figure on the oscilloscope as the focal point of the orbit
The size of the semi-axis b holds for FG 2 and FG 19. (One can also see the influence of the parameter changes on the elliptical shape, simultaneously varying the Kreisbahnpunkt for Ellipse e.)

Page 3

Circuit Diagram (Patching Diagram)

Demonstrationsbeispiel: Planetenbahn (fur RAT 740)

[Full wiring/patching diagram for the RAT 740 analog computer showing the interconnection of amplifiers, multipliers, and function generators to implement the planetary orbit simulation.]

The diagram shows the analog circuit for computing planetary orbits using:

Multiple integrators and summers (labeled with amplifier gain values)
Sine-cosine function generators
Multipliers (labeled M II)
Potentiometers for setting parameters such as eccentricity (e) and semi-axis length
Output connections to oscilloscope (X-Y display) for visualization of the elliptical orbit

[Translation covers all 3 pages of the original document — this is the complete document.]