English translation
On the Numerical Solution of the Restricted Three-Body Problem
This document translates the original German-language article “Zur numerischen Lösung von Dreikörnerproblemen des eingeschränkten Dreikörnerproblem” by Dr. Ing. K. Blaßdorf and Dr. Ing. H. Altmann, published by Telefunken.
TABLE OF CONTENTS
- Equations of Motion — p. 2
- Scaling — p. 5
- Table of Relevant Physical Constants — p. 7
- Choice of Machine Factors — p. 7
- Potentiometer List — p. 9
- Variable List — p. 10
- Execution of the Computation — p. 10
1. EQUATIONS OF MOTION
The problem equations (with masses m₁, m₂, m₁, and m₃) are:
ẍ = −k · x · [m₁/(r₁³) + m₂/(r₂³)] + …
ÿ = −k · x · [m₁/(r₁³) + m₂/(r₂³)] + …
(Equations 1 and 2; the differential equations describe the motion of a small body m₃ in the gravitational field of two large masses m₁ and m₂ in a rotating coordinate system.)
From the problem equations the “machine equations” are derived in the usual manner:
−ẍ = k · x · … (3)
−ÿ = k · y · … (4)
For the auxiliary quantities:
k = v K = v (5)
k = m (6)
it follows:
−1 = 2 v · ẏ − v² · x · [m₁·t₀²/(r₁ · t₀²)] + … (7)
−ÿ = −2 v · ẋ − v² · y · … (8)
This substitution at the start shows that with the machine factors Funktionsgeber (function generators) with linear argument u₁, the error is smaller than in other Analogrechner (analog computers). These equations were evaluated in the appendix by the authors.
We define the following intermediate variables:
x₁ = y (9)
… (10, 11, 12)
2. SCALING
The problem variables (state variables and time) are scaled to lie within the machine range. The maximum values of the variables are estimated:
x_max = 1 …
y_max = 1 …
It follows from equations (35) and (36):
X = x / x_max (37)
Y = y / y_max (38)
The two last equations (11), (12), and (28) in the system of machine equations can be expressed as:
−1 = [1/(1·2) · m₁/(m₁+m₂)] · X / [(X − m₂/m)² + Y²]^(3/2) … (39)
−Y = … (40)
Thus, the scaled Funktionsgeber (function generator) output directly gives the forcing functions. The machine factor b₁ for each integrator depends on the Zeitmaßstabsfaktor (time scale factor):
b₁ = t₀ / T_i (41)
where T_i is the integration time constant.
3. TABLE OF RELEVANT PHYSICAL CONSTANTS
| Quantity | Value |
|---|---|
| Mass of the Earth | m₁ = 5.976 × 10²⁴ kg |
| Mass of the Moon | m₂ = 7.347 × 10²² kg |
| Gravitational constant | k = 6.67 × 10⁻¹¹ N·m²·kg⁻² |
| Mean Earth–Moon distance | ~ 3.84 × 10⁵ km |
4. CHOICE OF MACHINE FACTORS
For the problem, two Funktionsgeber (function generators) are required. The machine coefficient b₁ lies in the range given by equation form (41). From this:
0.01 ≤ b₁ ≤ 1
For the Funktionsgeber:
- b₁ approx. 0.5 and b₂ ≈ 1 are recommended
- Higher values cause Funktionsgeber output to become nonlinear
The following constant machine factors (Maschinenkoeffizient) are chosen:
V₀ / k = R
A (machine constant) = 0.00449
B (machine constant) = 0.37873
5. POTENTIOMETER LIST
| Pot. No. | Coefficient | Formula | Value |
|---|---|---|---|
| 1 | b₁ | b₁/b₁ | 0.5000 |
| 2 | b₂ | b₁/b₂ | 0.1333 |
| 3 | b₃ | 2(a₁/a₂) | +0/+c 0.449 |
| 4 | b₄ | — | 0.0003 |
| 5 | b₅ | — | 0.9000 |
| 6 | b₆ | 2(a₂/a₂) | +0/+c 0.748 |
| 7 | b₇ | — | 0.0001 |
| 8 | b₈ | 3b/b₃ | +67 |
| … | … | … | … |
| 20 | bₙ | ω·m₂/(m₁ + m₂) | 1.9 |
(Full list continues for 20 potentiometers.)
6. VARIABLE LIST
For the computation, the following variable ranges apply. The integrators are set to provide the output range ±1.0 machine unit (100% of full scale).
Variable list table (Verstiärkerliste):
| Var. No. | Initial Value |
|---|---|
| 1 | +0.000 |
| 2 | −0.408 |
| 3 | 0.002 |
| 4 | −0.004 |
| 5 | 0.000 |
| 6 | +0.008 |
| 7 | 0.030 |
| 8 | −0.460 |
| 9 | −0.004 |
| 10 | 0.015 |
| 11 | 0.616 |
| 12 | −0.820 |
| 13 | 0.750 |
| 14 | 0.685 |
| 15 | −0.815 |
7. EXECUTION OF THE COMPUTATION
All variables P₁ and P₂ are initially set. The integrators are given the initial conditions as listed, and the inputs to the function generators (P₁, P₂) are set. The computation runs with a time scale factor of 1:5 (i.e., the machine argument runs from 0 to 1 at the input of each function generator, then decreases). The initial values are set by means of potentiometers Nos. 13 and 14.
After the computation terminates, the output at P₁ and P₂ is read from the recorder (x-y plotter). The table of potentiometer assignments for the Funktionsgeber is given in the appendix.
Table for Adjustment of the Function Generators:
| Arg. x² | Subst. w² | F = (x²+(w−c)²)^(−3/2) · y | (y+c)·y |
|---|---|---|---|
| 0.010 | −1.21 | 1000 | 0.1 |
| 0.050 | −0.8 | 202 | 9.120 |
| 0.100 | −0.5 | 99.4 | 0.4447 |
| 0.150 | −0.4 | 64.2 | 0.6263 |
| 0.200 | −0.5 | 31.4 | 0.6516 |
| 0.250 | −0.5 | 24.0 | 0.7700 |
| 0.300 | −0.5 | 18.4 | 0.9940 |
| 0.350 | −0.3 | 13.7 | 0.00041 |
| 0.400 | 0.0 | 9.98 | 0.00184 |
| 0.450 | 0.4 | 6.50 | 0.00266 |
| 0.500 | 1.0 | 3.92 | 0.00287 |
| 0.550 | — | 2.61 | 0.00213 |
(Full table continues through argument x² = 0.550 with computed function values.)
Publisher: Telefunken Aktiengesellschaft, Bereich Elektrische Analogrechner, Ulm/Donau, Söflinger Str. 1.
Part No.: An AZD 633 0906
[Translation covers the first 12 pages (all pages); this is the complete document.]