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

In many of the Orbit Design methods outlined in the past blogs linked above, I made reference to "...interval ratios between [1.4, 2.0] AU," "...add planets at intervals of rand[1.4, 2.0] AU". This is because most of the equations (indeed, all of them in Part 5) are based on the Titius-Bode relation.

Of course, the Titius-Bode relation was first identified in evaluating the orbit spacing of our own Solar System, in which the intervals between the planets' orbits are never narrower than 1.400 AU, nor wider than 2.000 AU [1].

Since I mentioned previously that the Titius-Bode relation/equation has been used by some exoplanet searchers to indicate where in distant star systems to look for potential planets, I wondered how well it fit, and, thence, whether it was borne out by the actual orbital date of so-far identified exoplanet systems.

Of course, the Titius-Bode relation was first identified in evaluating the orbit spacing of our own Solar System, in which the intervals between the planets' orbits are never narrower than 1.400 AU, nor wider than 2.000 AU [1].

Since I mentioned previously that the Titius-Bode relation/equation has been used by some exoplanet searchers to indicate where in distant star systems to look for potential planets, I wondered how well it fit, and, thence, whether it was borne out by the actual orbital date of so-far identified exoplanet systems.

## Intervals in the Exoplanet Catalogue

To explore the real-world situation of orbital spacings and intervals, I surveyed 127 exoplanet systems of three or more planets (two-planet systems would have only had a single orbital interval, and hence, nothing else in the system with which to compare it; however, see the caveats listed below).

The 127 systems comprised 454 orbits, which resulted in a total of 329 between-planet orbital intervals.

The 127 systems comprised 454 orbits, which resulted in a total of 329 between-planet orbital intervals.

Analyzing the basic orbital data revealed that:

Analyzing the orbit intervals revealed the following:

- The minimum orbit was 0.006 AU, in the Kepler-42 system.
- The maximum orbit was 11.600 AU, in the 47 Ursae Majoris system.
- The mean orbital distance was 0.294 AU, which measures 0.093 AU less than Mercury's semi-major axis. The standard deviation of the orbits was 0.837 AU, meaning that 68% of all the orbits lay between 0.006 AU and 1.131 AU from their host star(s), and 95.4% lay between 0.006 AU and 1.969 AU. No orbits fell below -3σ, and only 13 fell above +3σ.

Analyzing the orbit intervals revealed the following:

- The minimum interval was 1.015 AU, in the Kepler-132 system.
- The maximum interval was 13.452 AU, in the Kepler-167 system.
- The mean interval was 2.046 AU, with a standard deviation of 1.539 AU, meaning that 68% of all the intervals measured between 0.507 AU and 3.585 AU, and 95.4% measured between 0.507 AU and 5.125 AU. No orbits fell below -3σ, and only 13 fell above +3σ.

Of the 127 systems surveyed, 49 comprised two-or-more orbital intervals, for a total of 204 intervals available for analysis.

Of the 204 measured orbital intervals across these 49 systems, 106 (51.96%) measured

The sixteen cases of orbital interval variation were:

Of the 204 measured orbital intervals across these 49 systems, 106 (51.96%) measured

*greater-than-or-equal-to*the next farther inward orbital interval. Of these, there were only 2 (12.5%) cases in which the orbital intervals*decreased*consistently from the star(s) outward, and 2 (12.5%) cases in which the orbital intervals*increased*consistently from the star(s) outward. In the remaining 12 cases (75%), the orbital intervals varied between*less-than*and*greater-than-or-equal-to*the preceding orbital interval.The sixteen cases of orbital interval variation were:

In the table above, an

Therefore the first interval between Orbit(2) and Orbit(1) is

Because in 14 out of the 16 cases (87.5%) the orbital intervals vary between

Pruning the 13 orbits which exceeded +3σ, the minimum orbital interval represented is 1.015 AU, and the maximum is 5.125 AU.

**F**indicates that the orbital interval was*less-than*the preceding interval; a**T**indicates that the orbital interval was*greater-than-or-equal-to*the preceding interval. For example, in the sequence**TFFT**—representing a five-planet system—there are four orbital intervals represented; the first**T**indicates the first orbital interval in the system (Orbit(2) ÷ Orbit(1)); the following**F**indicates the second orbital interval, (Orbit(3) ÷ Orbit(2)).Therefore the first interval between Orbit(2) and Orbit(1) is

*greater-than-or-equal-to*the interval between Orbit(3) and Orbit(2). The interval between Orbit(4) and Orbit(3) is*less-than*the interval between Orbit(3) and Orbit(2), and the interval between Orbit(5) and Orbit(4) is*greater-than-or-equal-to*the interval between Orbit(4) and Orbit(3).Because in 14 out of the 16 cases (87.5%) the orbital intervals vary between

*less-than*and*greater-than-or-equal-to*the preceding orbit, my conclusion is that**the Titius-Bode interval bounds of [1.400, 2.000] AU, increasing consistently from the star(s) outward,***does not hold*for the surveyed systems.Pruning the 13 orbits which exceeded +3σ, the minimum orbital interval represented is 1.015 AU, and the maximum is 5.125 AU.

## Caveats

There are a number of qualifications to be borne in mind when considering the above analysis.

- Of the hundreds of exoplanet systems thus far identified, only 127 were surveyed, and only those with three-or-more planets.
- There may be (almost certainly are) more planets in the vast majority of the 127 surveyed systems than those currently known. Each additional planet and its orbital semi-major axis will alter the data and analysis for its particular system, and for the aggregate data.
- Spectral types of the stars were not taken into account.
- Innermost stable orbits were not calculated for the stars in the systems analyzed, which would have provided an additional orbital interval between the first planet's orbit and the most stable inner orbit of the system.

1. Remember that orbital

*intervals*are calculated by Orbit(n) ÷ Orbit(n-1).