## Titius-Bode Refresher

In a previous blog, I introduced the Titius-Bode relation for determining planetary orbits, plus two modifications to make the relation more useful for serious Worldbuilding. As a refresher, the basic equation for Titius-Bode is:

Where:

m = numbers in the range [-∞, 0, 1, 2, 3, ···) to an arbitrarily chosen maximum.

m = numbers in the range [-∞, 0, 1, 2, 3, ···) to an arbitrarily chosen maximum.

Remember: By definition, any number raised to the power of negative infinity (-∞) is zero, and this is necessary to make the relation return a close approximation of Mercury's semi-major axis.

## TB2: A Modified Titius-Bode Relation

The method described below for determining orbits uses a modification of Titius-Bode. Instead of the two constants (0.4 and 0.3) in the original equation, we substitute values related to the Innermost Stable Orbit (

So, our new equation looks like this:

*IS*or*OI*) for the system in question, and we begin using exponents at zero, rather than negative infinity.So, our new equation looks like this:

Note: The above equation is for single-star systems; for binary systems, substitute OI for IS.

Because the Innermost Stable Orbit is a fundamental value in the equation, itself, we don't generate any orbits smaller than it, and because the equation is based on Titius-Bode, no orbital interval is ever < 1.400 AU or > 2.000 AU, and the intervals (generally) increase from the inner system to the outer, gradually approaching the 2.000 AU limit in successively smaller intervals.

So, for an

So, for an

*IS*value of 0.093, taken to 12 orbits, we get the following orbital table:The twelfth orbit is too large, and we can discard it.

For an

For an

*IS*value of 0.276, our orbital table comes out as:Here, Orbits 11 and 12 are too large, and can be discarded.

For a binary system with an Innermost Stable Orbit of 0.540 AU, we get the following orbital table:

For a binary system with an Innermost Stable Orbit of 0.540 AU, we get the following orbital table:

Note: The larger the value of IS or OI, the sooner the orbits exceed the 35-40 AU upper limit.

In this case, Orbits 10, 11, and 12 need to be discarded.

## TB3: Invoking The Powers

A further modification may be made to the equation by randomizing a number in the range [1.400, 2.000], and using this in place of the constant "2" in the above equation, thus:

The smaller the value assigned to

*ω*, the closer together the orbits will fall, but the intervals will still be between 1.400 AU and 2.000 AU, and increase (generally) from the inner to the outer orbits (toward the limit set by*ω*, itself; to wit, no interval created by this equation will ever exceed the value of*ω*).## Conclusion

This is a good, robust way of calculating planetary orbits around fictional stars, especially when the Worldbuilder already has a specific luminosity in mind for the star.

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

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