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 Designing System Orbits, Parts 1 through 5, I discussed several systems for designing planet, asteroid belt, etc., orbits based on a number of numerical calculations.

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.

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

Johannes Kepler, whom I mentioned in the blog on Ellipses and Orbits, in 1596 published

*Mysterium Cosmographicum*, in which he associated the orbits of the six planets known in his time with the five Platonic Solids and the sphere.Taking a page from Kepler's book (metaphorically speaking), this entry discusses using "figurate numbers", which "... can be represented by a regular geometrical arrangement of equally spaced points." [1]

Also known as n-gonal numbers, some familiar sets would be (left-to-right) the triangular numbers, the square numbers, the pentagonal numbers, and the hexagonal numbers. Both Wolfram Mathworld and the Online Encyclopedia of Internet Sequences (OEIS) have extensive equations for a large variety of figurate numbers.

Another kind of figurate number is based on three-dimensional solids, such as cubes, tetrahedrons, dodecahedrons, etc.

Another kind of figurate number is based on three-dimensional solids, such as cubes, tetrahedrons, dodecahedrons, etc.

## Example: Triangular Numbers

The equation for triangular numbers is:

... and the first ten triangular numbers are:

Using triangular numbers, calculate as many numbers as the orbits you wish to fill, divide each calculated value by 10, and use the results as the Astronomical Unit value for system orbits:

This method can be further adjusted by also multiplying the calculated values by the Innermost Stable Orbit, so that the orbits are adjusted for stars much smaller or larger than the Sun.

There is a generic formula for calculating the number of points in a polygon of any number of sides:

Where:

The first 10 numbers for n-gons of 3 through 12 are:

*n*= the index of the polygonal number to be calculated*P*= the number of sides in the polygonThe first 10 numbers for n-gons of 3 through 12 are:

The following table lists the first ten values of several other n-gons and polyhedra:

## Intervals and Gaps

Reminder:

Orbital intervals are calculated by Orbit(n) ÷ Orbit(n-1)

Orbital gaps are calculated by Orbit(n) - Orbit(n-1)

Calculating the orbital intervals and gaps for the triangular numbers reveals that the intervals

*decrease*with distance from the star(s), and that the gaps increase predictably by 0.1 AU per orbit. Neither of these situations is in keeping with measured realities (more on this in the next blog). This turns out to be true regardless of the n-gon or n-hedron used.There is a way to address this: modify the means of calculating the n-gonal or n-hedral number by taking the natural log of the original equation and adding the exponential function of the input number, divided by 100.

For triangular numbers this would result in the following equation:

For triangular numbers this would result in the following equation:

And the orbits, intervals, and gaps become:

Note that this method is only effective for values of n ≥ 2.

The equation(s) can further be modified by multiplying by the Innermost Stable Orbit (

*IS*or*OI*) of the system. For a single-star system with an*Is*= 0.276, the orbits become:For a system based on a 6-gon and with an

*IS*= 1.123, the equation becomes:... and the orbital table looks like: