An improvement to Bode’s Law. Why Phi?

Posted: November 16, 2016 by tallbloke in Analysis, Astronomy, Astrophysics, solar system dynamics

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.

.

titus-bode_law

Figure 1 Titius-Bode equation (red) vs planets (blue)

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 a new equation, as plotted in Figure 2.

sma2

Figure 2. An improvement to the Titius-Bode equation

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, though it’s not far off (11%). 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. It’s only an approximation, but the variances might help us work out what forces shape orbits and how they do it.

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) (also equivalent to root 5 over 2), 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.

Comments
  1. tallbloke says:

    A couple of equivalences to consider:

    phi = 1/Phi (~0.618)
    phi^2 = 1 – phi (~0.382)
    (Phi + phi)/2 = (Square Root 5)/2 = Phi – 0.5 = phi + 0.5 (i.e the arithmetic mean of Phi and phi) (1.118)

  2. oldbrew says:

    Re: ‘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’

    The ratio of Neptune’s aphelion to Pluto’s is very close to 1:1.618 depending on whose data is used.
    http://en.wikipedia.org/wiki/Golden_ratio

    Their perihelion ratio is near 1:1, with Pluto’s perihelion slightly nearer to the Sun than that of Neptune.

    The net result is that 2 Pluto orbits = 3 Neptune orbits timewise.

    Maybe ‘the rails come off’ at Neptune because unlike all the planets nearer to the Sun it has no significant neighbour ‘outside’ its own orbit? Not that we know of anyway.

  3. oldbrew says:

    Terminology note – according to Dr Mae-Wan Ho:
    The numerical value of phi is 0.6180339
    The reciprocal of phi, represented by the same letter in capital Φ (Phi) is also often referred to as the golden ratio, and is equal to:

    1/φ = Φ = (√5+1)/2 = 1.6180339887…= 1 + φ

    http://www.i-sis.org.uk/The_Story_of_Phi-Part_1.php

    Goldennnumber.net says: This site is dedicated to sharing the best information on Phi, the number 1.618
    http://www.goldennumber.net/


    Google translate: ‘In practice, one can choose either upper or lower case Phi and explicitly 1 / Phi.’

  4. tallbloke says:

    Other sites have it the other way round. Uppercase Phi = 0.618 and lowercase phi = 1.618

    If you copy lowercase phi from wordpress, and paste it into a text box in microsoft paint, it turns into something that looks like an uppercase Phi but with a smaller circle across the vertical line.

    Endless fun.

    Which is why I took the trouble to include a definition in my graphic.
    I’ve edited everything so we’re all on the same page.

  5. Chaeremon says:

    Re: taking a power of 2

    Ari Lehto also works with period doubling, see “Quantization of Keplerian systems”, PIRT-X conference in London, September 2006,
    https://arxiv.org/ftp/physics/papers/0611/0611100.pdf

  6. oldbrew says:

    The ratio of Jupiter’s perihelion distance from the Sun to Neptune’s is 1:6
    The same ratio for Uranus:Neptune is 1:1.621

  7. oldbrew says:

    Cosmic ‘barcode’ from distant galaxy confirms Nature’s constancy

    Date: November 15, 2016
    Source: Swinburne University of Technology
    Summary:
    Astronomers have precisely measured the strength of a fundamental force of Nature in a galaxy seen eight billion years in the past. Researchers have confirmed that electromagnetism in a distant galaxy has the same strength as here on Earth. [bold added]

    http://www.sciencedaily.com/releases/2016/11/161115175127.htm

  8. are you sure the planets formed with the masses they have now? sure of the accretion disk? because found exo-planets seem to have busted cold-line theory

    otherwise it seems to me if this would be much more likely if the planets were all roughly the same mass once. and only fringers like J. Marvin Herndon say that (that the four inner planets used to be gas giants that lost their gas).

    total fringe (but maybe there is phi in there) is ‘stellar metamorphosis’, that the inner planets were collected by the sun as dying stars reaching gas giant stage, presumably one at a time. this hypothetically would mean the entire mass of the solar system kept gaining each time a gas giant/dying star was collected. The ratio of the capture to the solar system might be phi-like(?). Let’s pretend the Sun was 10 times the mass of each capture: the first capture 10:1; second capture 11:1, third capture 12:1. And so on. The relative masses would effect the slingshot of the capture, and decided its orbital distance (or something)
    (the distribution of star sizes in the galaxy is here
    http://spiff.rit.edu/classes/phys440/lectures/size/diam_hist_closeup.gif )

    love your site

  9. tallbloke says:

    Hank: interesting thoughts. Mass doesn’t appear in Kepler’s laws, which hold good for orbits regardless. But there are several Phi relationships between the masses and diameters of planetary pairs Oldbrew has discovered. Something fundamental is going on in the solar system (and in other star systems) which science hasn’t got a handle on yet. That’s why we keep asking the question – Why Phi?

  10. JB says:

    Tall, have you read Halton’s Seeing Red? I have a hunch the answer is implied in there. Something to do with the present state of quantized matter in post quasar galaxies, as in the energy/mass ratio.

  11. tallbloke says:

    Some more Transneptunian objects.

    Model SMA (AU)      Object   Observed (AU)
    
    47.52056948	Makemake	45.36
    72.93200849	Eris (Sedna P)	68 (76)
    111.9321153	2013TV158	111.78
    171.787377	2010VZ98	153.9
    263.6500062	2000CR105	230.12
    404.6358177	2009MS9__	349.5
    621.0132415	Planet 9__	665
    953.0976479	Sedna Aphelion  972.8
    
  12. oldbrew says:

    TB: the perihelion of 2009MS9 is only 11 AU, i.e. between Saturn and Uranus. This year it’s at 12 AU. Aphelion is a whopping 684 AU.

    http://en.wikipedia.org/wiki/(418993)_2009_MS9

  13. suricat says:

    Posted: November 16, 2016 by tallbloke in Analysis, Astronomy, Astrophysics, solar system dynamics

    “Please let me have your thoughts below.”

    You use ‘1 AU’ (the distance from Sol to Earth) as a ‘measuring stick’ here when ~3.5 BYA Earth is reputed to have encountered a collision with a ‘Mars like’ planet. This can’t be good for the definition of a ‘yard stick’ for the ‘system’.

    Surely, the ‘most massive planet’ within a ‘stellar system’ would be a better candidate for the ‘yard stick’ to base your calculations upon by way of its ‘massive inertia’ that qualifies it as the ‘most immoveable object’ in the system.

    I’d expect ‘1 JU’ (the distance from Sol to Jupiter) to disclose any ‘system disruption’ from any ‘outside influence’.

    From there, we can begin to understand the more ‘natural’ workings of orbital mechanics.

    Best regards, Ray.

  14. oldbrew says:

    A JU has some merit but the AU is the customary unit of astronomical measurement.

    http://en.wikipedia.org/wiki/Astronomical_unit

  15. JB says:

    @tallbloke Verified. The answer to why Phi Chip covers in chapter 8 Quantizations and 9 Cosmology. Time for a refresher.

  16. tallbloke says:

    Ray: You use ‘1 AU’ (the distance from Sol to Earth) as a ‘measuring stick’ here

    No, I don’t. Look at the equations again.

    JB: Thanks for that, I’ll order a copy.

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