Past Blog Tie-ins:
• Designing System Orbits, Part 1: Follow The Giant
• Designing System Orbits, Part 2: Titius-Bode
• Designing System Orbits, Part 3: Center Out
• Designing System Orbits, Part 4: Benford's Law
• Designing System Orbits, Part 5: Titius-Bode Revisited
Orbit Design Refresher
In Part 1, I discussed the Follow the Giant method, designing system orbits based on the location of the largest gas giant in the system.
In Part 2, I covered the long-controversial Titius-Bode method, based on a still-unexplained numerical relationship of the orbits of the planets in the Solar System.
Part 3, titled "Center Out", used the Innermost Stable orbit and random values to create system orbits.
Part 4 discussed using Benford's Law—which describes the distribution of the first digits of multi-digit numbers—as the basis of determining orbital distances from the central star(s).
Finally, in Part 5, I revisited Titius-Bode, in which the constants in the Titius-Bode equation were replaced with the value of the Innermost Stable Orbit, as well as adding a modifier based on the natural logarithm of the orbit's ordinal.
Invoking the Spirit of Kepler
Another kind of figurate number is based on three-dimensional solids, such as cubes, tetrahedrons, dodecahedrons, etc.
Example: Triangular Numbers
n = the index of the polygonal number to be calculated
P = the number of sides in the polygon
The first 10 numbers for n-gons of 3 through 12 are:
Intervals and Gaps
Orbital intervals are calculated by Orbit(n) ÷ Orbit(n-1)
Orbital gaps are calculated by Orbit(n) - Orbit(n-1)
For triangular numbers this would result in the following equation:
Note that this method is only effective for values of n ≥ 2.