An improvement to Bode’s Law. Why Phi?

I’m working away for the next fortnight, with no internet access. So I thought I’d put up something for the bright denizens of the talkshop to chew on while I’m gone. Bode’s Law is a heuristic equation which gives the approximate distance to the first seven major planets plus Ceres. reasonably well, but then goes completely off the rails as you can see in Figure 1 below.

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I’ve always thought the Titius-Bode equation to be a fudge. It doesn’t relate to any physical concepts that have anything to do with orbits or gravity. So I’ve come up with something better.

My version is based on my previous research which shows that the distribution of planets in the solar system is something to do with power laws (like gravity), phi ratios (resonant fibonacci ratios) and lognormal distribution. Accordingly, I’ve derived an empirical fit which employs these non-linear dimensions and resonance concepts, and come up with this.

I’ve added the small graphic of the asteroid belt to show that the fit to my equation of asteroids Vesta and Sylvia isn’t arbitrary. The positions of both attest to the power of orbital resonance as a factor in shaping the solar system. Vesta (2nd largest asteroid after Ceres) lies at the densest part of the belt, and Sylvia (8th largest) is at its outside edge, the start of the final ‘Kirkwood gap’ between the asteroid belt and Jupiter, so to speak.

Chiron is worth a mention too. Stuart ‘Oldbrew’ spotted that the ratio between its perihelion distance and semi-major axis is phi, and its eccentricity is Phi squared. Its perihelion and aphelion lie close to Saturn and Uranus’ orbits respectively.

Pluto is locked into a 3:2 resonance with Neptune a few percent closer to the Sun than Makemake, and so doesn’t appear on the chart. Saturn is in a near 2:5 resonance with Jupiter, and this has pulled both these major gas giants slightly below and above the model predictions for their positions. Mercury and Venus are also somewhat off beam, perhaps due to Mercury’s proximity to that big non-linear beast, the Sun, and Venus’ 13:8 resonant relationship with earth.

The next step is to try to understand why this equation is as successful as it is. We know that phi is an important number in relation to orbits already (see previous why phi? posts). We also know gravity falls off with the square of the distance, and this equation uses powers of two as input.

But why it is that taking a power of 2, raising it to the power of Phi (~0.618), multiplying that by Phi squared (~0.382) and finally multiplying that by a term equivalent to half of phi plus Phi (~1.118), should quite accurately predict (Pearson R^2 0.9995)  the position of all the major planets plus some important asteroids, a minor planet and a centaur is a mystery.

Please let me have your thoughts below.