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
So, our new equation looks like this:
Note: The above equation is for single-star systems; for binary systems, substitute OI for IS.
So, for an IS value of 0.093, taken to 12 orbits, we get the following orbital table:
For an IS value of 0.276, our orbital table comes out as:
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.
TB3: Invoking The Powers
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.