## Method 2: The Titius-Bode Relation

Here we make use of the Titius–Bode relation of planetary orbits. Johann Tietz (1729-1796)—whose name was latinized as Johannes Titius—and Johann Bode (1747-1826) in the late 18th and early 19th centuries each noticed that adding the quantity 4 to each of the sequence of numbers {0, 3, 6, 12, 24, 48, 96, 192, 384}, and then dividing each sum by 10, resulted in a very close (and sometimes exact) prediction of the distances in Astronomical Units from the Sun of the then-known Solar System planets.

This method is most effective for designing planetary orbits for Solar analog stars with luminosities ≤2.8 times that of the Sun; higher luminosity values begin to produce starting orbital distances beginning significantly beyond the Innermost Stable orbit; luminosities more than ~3.5 times that of the Sun begin producing starting orbital distances greater than the nucleal habitable zone orbit.

Below is a table showing the values for the Solar System as predicted by Titius-Bode, compared to the actual orbital distances for the planets:

This method is most effective for designing planetary orbits for Solar analog stars with luminosities ≤2.8 times that of the Sun; higher luminosity values begin to produce starting orbital distances beginning significantly beyond the Innermost Stable orbit; luminosities more than ~3.5 times that of the Sun begin producing starting orbital distances greater than the nucleal habitable zone orbit.

Below is a table showing the values for the Solar System as predicted by Titius-Bode, compared to the actual orbital distances for the planets:

Note: negative numbers in the difference column indicate where the Titius–Bode relation “overestimates” the actual Solar System planetary orbit.

When Uranus was discovered in 1781 (at 19.22 AU) and Ceres in 1801 (at 2.8 AU), their observed orbits fit closely with the orbits predicted by Titius-Bode.

Neptune’s 1846 discovery dropped a fly in to the ointment for Titius-Bode. The next predicted orbit beyond that of Uranus lies at 38.8 AU, but Neptune’s observed orbit was found to be ~30.11 AU.

The reason(s) for Neptune’s “break” with the Titius-Bode relation are still not completely explained to everyone’s satisfaction, though it is probably due to migrations of the giant planets in the outer Solar System. When Pluto [1] was discovered in 1930, its average orbit of 39.44 AU was in closer agreement with Titius-Bode's prediction of a planet at 38.80 AU, but that did nothing to explain Neptune. As a result, the Titius–Bode relation was dismissed for nearly 150 years as being a mathematical coincidence, but not having any real value to astronomers and planetary scientists.

However, Steffen Jacobsen [2] and others revived it in the early 1990s to see if it might be useful as a generalized model to aid in the search for exoplanets. As it turns out, Titius-Bode has been reasonably accurate [3] in predicting where planets would be found around other stars. Whether the effectiveness of generalized Titius-Bode relationships is due to resonances, or some other factor is yet to be clearly defined, however.

The reason(s) for Neptune’s “break” with the Titius-Bode relation are still not completely explained to everyone’s satisfaction, though it is probably due to migrations of the giant planets in the outer Solar System. When Pluto [1] was discovered in 1930, its average orbit of 39.44 AU was in closer agreement with Titius-Bode's prediction of a planet at 38.80 AU, but that did nothing to explain Neptune. As a result, the Titius–Bode relation was dismissed for nearly 150 years as being a mathematical coincidence, but not having any real value to astronomers and planetary scientists.

However, Steffen Jacobsen [2] and others revived it in the early 1990s to see if it might be useful as a generalized model to aid in the search for exoplanets. As it turns out, Titius-Bode has been reasonably accurate [3] in predicting where planets would be found around other stars. Whether the effectiveness of generalized Titius-Bode relationships is due to resonances, or some other factor is yet to be clearly defined, however.

## From Relation to Equation

The Titius–Bode relationship in its fundamental form can be expressed in the equation:

… where

*OP*is the semi–major axis of a planet’s orbit, and the exponent*m*is taken from the technically infinite range of [–∞, 0, 1, 2…). The value -∞ is necessary to account for Mercury’s orbit in the Titius-Bode relation: any number raised to negative infinity is defined as zero, so:…leaving

However, using this equation to determine orbits for Worldbuilding means that the intervals and gaps between orbits will be the same for all the star systems you generate. Also, using this method in its basic form means that no orbit will ever be generated that is closer than 0.4 AU to the central mass of the system.

But, huge numbers of exosystems have revealed planets orbiting

For these reasons, I discuss below a couple of possible modifications to the Titius-Bode equation which make it more generally useful, and take into account relevant characteristics of the star when determining planetary orbits.

First, we can modify the equation to include the luminosity of the host star(s) as a multiplier:

*OP**= 0.4*, which closely approximates Mercury’s measured average orbital distance of 0.387 AU.However, using this equation to determine orbits for Worldbuilding means that the intervals and gaps between orbits will be the same for all the star systems you generate. Also, using this method in its basic form means that no orbit will ever be generated that is closer than 0.4 AU to the central mass of the system.

But, huge numbers of exosystems have revealed planets orbiting

*much*closer to their host stars than Mercury orbits the Sun: Kepler 42c, for instance, has a semi-major axis of ~0.00585 AU [4], less than*one-sixtieth*of Mercury's orbital distance from the Sun.For these reasons, I discuss below a couple of possible modifications to the Titius-Bode equation which make it more generally useful, and take into account relevant characteristics of the star when determining planetary orbits.

**Alternative Equation 1**First, we can modify the equation to include the luminosity of the host star(s) as a multiplier:

Let’s try this method with a fictional star called Nysheryn, which has a luminosity of 2.76 times that of the Sun, so our equation becomes:

… and our sequence, for a run of 12 planets, works out as:

Several things are immediately obvious:

So, our modified equation would be:

- Starting at Planet 7 (ordinal 5), the orbital intervals all hover in the range [1.92, 2.00], which may seem a bit unnatural; however, this is a recognized feature of the Titius–Bode relation, a feature also mirrored in the Solar System.
- Even more unnatural seeming,
*almost**all*of the orbital gaps are exactly twice the magnitude of the preceding one; nevertheless, this is also a recognized feature of the Titius–Bode relation. - The orbital distances become prohibitively large beyond Planet 7 (ordinal 5).

*IS*) calculated for the star(s) in question.**Alternative Equation 2**So, our modified equation would be:

Nysheryn, with a luminosity of 2.76 times that of the Sun falls into the second luminosity regime, so would have a mass of:

… meaning its Innermost Stable Orbit would be:

… and our equation for Nysheryn becomes:

… which results in the following table:

Our first planet now falls much closer to the Innermost Stable Orbit, with an orbital gap of only 0.029 AU. Note that the orbital

This modification of the equation serves the need to fill in the gap between the first orbit and the Innermost Stable Orbit.

Planets 10, 11, and 12 (ordinals 8, 9, and 10) are

*intervals*have not changed, but the orbital*gaps*have reduced significantly, and we don’t reach an orbit comparable to Neptune until Planet 9 (ordinal 7).This modification of the equation serves the need to fill in the gap between the first orbit and the Innermost Stable Orbit.

Planets 10, 11, and 12 (ordinals 8, 9, and 10) are

*much*farther out than is really useful so, we can simply drop them from the list.## Example 2

This method also works well for stars with a lower luminosity than the Sun.

Let’s say we have a star called Anra, with a luminosity of 0.76 solar units. Because Anra's luminosity is <1.0 times that of the Sun, we use its mass to calculate the Innermost Stable Orbit, which then would be:

Let’s say we have a star called Anra, with a luminosity of 0.76 solar units. Because Anra's luminosity is <1.0 times that of the Sun, we use its mass to calculate the Innermost Stable Orbit, which then would be:

… and its Innermost Stable Orbit would be:

This makes its Titius-Bode equation:

... which produces the following planetary orbital table:

The orbits of the first two planets are now smaller than the innermost stable orbit of 0.093 AU, so we have to drop them out of the sequence, and set the first planetary orbit to 0.10 AU. If we needed more planets, we could calculate more beyond ordinal 10, but since we’re already at the 30-35 AU range, we can just be satisfied with a ten planet system with orbits from 0.10 AU to 30.16 AU.

## Conclusion

The Titius-Bode relation is a valid and useful method for calculating orbits for fictional star systems.

Any of three equations may be used, depending on the needs of the Worldbuilder:

Any of three equations may be used, depending on the needs of the Worldbuilder:

This system shares with the previous one the potential problem of no planet falling at precisely 1.0 AU, and thus, the need to create specialized time-keeping methods and processes.

1. No, Pluto's not a planet, but at the time of its discovery, it was deemed to be one, and its measured orbit

2. www.centauri-dreams.org/?p=32757

3. 83% of the time; 124 out of 151 tested Kepler discovered exoplanet systems have planetary orbits closely in agreement with Titius-Bode predictions.

4. exoplanets.org/table

*seemed*to further invalidate Titius-Bode.2. www.centauri-dreams.org/?p=32757

3. 83% of the time; 124 out of 151 tested Kepler discovered exoplanet systems have planetary orbits closely in agreement with Titius-Bode predictions.

4. exoplanets.org/table