On researchgate, I found this interesting paper by I.A. Arbab which proposes an explanation of planetary spin rates by leveraging an analogy with the electron spin-orbit coupling in the Hydrogen atom.

Food for thought. Here’s a taster:

I hope people with greater ability than I have will take a look.

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Massive planets do spin faster in the solar system, as the paper says. However Jupiter is a lot more massive than Saturn but only spins slightly faster, so it’s not straightforward.

Data: http://ssd.jpl.nasa.gov/?planet_phys_par

Other papers by Arbab:

http://scholar.google.co.uk/scholar?q=Arbab%2C+A.I+planet&btnG=&hl=en&as_sdt=1%2C5

Saturn, Jupiter and Neptune have no known surface. Internal conditions are crazy. Neptune iirc has a spin axis on its side.

Fun maths.

Uranus has the side-on spin axis.

http://www.space.com/13231-planet-uranus-knocked-sideways-impacts.html

NB they talk of impacts, but how does an impact on a ball of gas work?

Spin is a vector, has direction. So what about Venus?

Given the near 2:1 resonance of Uranus-Neptune and the near 2:5 resonance of Saturn-Jupiter, they are probably best considered as pairs

Venus is in some sort of coupling which forces it to present the same face to Earth at every Ea-Ve conjunction and nearly the same face to Jupiter at every Ju-Ve conjunction. The only way Venus can fulfill both these conditions is to spin slowly backwards.

Also worth noting that Mercury spins around 4 times faster than venus, and orbits around 4 times faster than Earth.

Uranus not Neptune then oldbrew.

Their supposition is based upon

“The researchers began by model[l]ing the single-impact scenario.” Kind of.

“To account for the discrepancy, the researchers tweaked their simulations’ paramaters a bit.” [s/a/e]

How doth one smite a gurt blog [sic] of gas?

Much the same as diving into the sea from 1000 feet, it is after all only water. Mass times speed, slugs, newtons and all that.

Hidden in this is the strange world of scale. What happens varies by octave or decade, the order of things. What might gently meld under benign rates is awful at fraction speed of light.

If these gas giants are just gas it will nevertheless be in a novel state of matter under the intense pressures deep inside. Maybe behaves as a solid.

TB: Venus length of day (LOD) is in a 3:2 ratio with Mercury LOD*.

Also Mercury’s own spin rate is about 1:2 with that of the Sun (depending on how it’s measured).

*This is only possible because Venus spin is retrograde:

184 spins = 199 orbits = 383 LOD (184 + 199)

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As TB notes, the planet pairings are relevant.

Each of the Earth:Mars, Jupiter:Saturn and Uranus:Neptune pairs are close to matching in their spin rates (within about 2.5%-7%).

The gas giants are further away from the Sun so get less tidal drag force from it. Also being much more massive than the rocky planets they should have more resistance to such drag forces, allowing a faster spin rate.

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TB: ‘Venus is in some sort of coupling which forces it to present the same face to Earth at every Ea-Ve conjunction’

Five Venus LOD = 1 Venus-Earth conjunction, accurate within about 5 hours per conjunction.

25 Venus LOD = ~13 Venus orbits = almost 8 sidereal years.

This was the bit I really liked “When applied to stars…without the need of Dark Matter.

Yes. Dark matter was always a fudge factor to make the numbers fit, just as was Einstein’s gravitational constant. Before that they had the ether, but I think the missing mass is simply photons, which are constantly being recycled by matter.

I’d made the analogy of planet spin orbit coupling to atomic spin orbit coupling some years ago, but got stuck on the math (never did like angular momentum problems…). Maybe I’ll take a look… after a run to Starbucks😉

Intuitively, it ought to work at all scales, I just can’t see how…

OK… read it. Once through quickly…

It is all based on an analogy of electromagnetism to some kind of gavitomagnetics (whatever that is) and makes a lot of “leaps” in the math. Maybe valid, but each one needs unrolling and vetting…

They derive a formula with a “constant” plug number in it, and solves for that number. Planets yielding one value, moon earth a very different value… so how does a constant have multiple values?

It may well be brilliance beyond my ken, but a quick look-over has far more loose ends and questions than I like.

I’ll revisit again with Starbucks aboard and see if more becomes clear. As it stands now, I like the ideas, but not the proof…

The big idea is gravitomagnetics, but that is just assumed… though an interesting idea… unify gravity and magnetics (then go for grand unified theory…).

OK, he didn’t make gravitomagnetics up, but incorporates it by reference

http://gravity.wikia.com/wiki/Gravitomagnetism

looks like some digging to do for me to catch up on that axis… so OK, no fault in not explaining it to his audience who can be assumed familiar with it…

As above, so below. It’s resonance all the way down.

The best way to model the solar system is using Hamiltonian mechanics, but it’s deep physics and tricky maths. It’s also the basis of the Schrödinger equation in quantum mechanics so the parallel is clear. I’ve been thinking of writing a revue but still don’t know very much about it – where to start, even.

Parameters expressed as integer ratios are ubiquitous in QM, as in solar system in Table 1 at the end of the Gkolias article.

A few references:

Resonance In the Solar System, Steve Bache, 2012, uncw.edu/phy/documents/BachePHY495.pdf

A simple PPT style overview with historical background going back to Anaximander.

Ioannis Gkolias et.al., The theory of secondary resonances in the spin-orbit problem, arxiv.org/pdf/1603.07760v1.pdf

ABSTRACT

We study the resonant dynamics in a simple one degree of freedom, time dependent Hamiltonian model describing spin-orbit interactions. The equations of motion admit periodic solutions associated with resonant motions, the most important being the synchronous one in which most evolved satellites of the Solar system, including the Moon, are observed. Such primary resonances can be surrounded by a chain of smaller islands which one refers to as secondary resonances. …

Alessandra Celletti, Quasi–Periodic Attractors And Spin/Orbit Resonances, November 2007, researchgate.net[…]

Introduction

Mechanical systems, in real life, are typically dissipative, and perfectly conservative systems arise as mathematical abstractions. In this lecture, we shall consider nearly–conservative mechanical systems having in mind applications to celestial mechanics. In particular we are interested in the spin–orbit model for an oblate planet (satellite) whose center of mass revolves around a “fixed” star; the planet is not completely rigid and averaged effects of tides, which bring in dissipation, are taken into account. We shall see that a mathematical theory of such systems is consistent with the strange case of Mercury, which is the only planet or satellite in the Solar system being stack in a 3:2 spin/orbit resonance (i.e., it turns three times around its rotational spin axis, while it makes one revolution around the Sun).

Eric B. Ford, Architectures of planetary systems and implications for their formation, PNAS vol. 111 no. 35

Abstract

… With the advent of long-term, nearly continuous monitoring by Kepler, the method of transit timing variations (TTVs) has blossomed as a new technique for characterizing the gravitational effects of mutual planetary perturbations for hundreds of planets. TTVs can provide precise, but complex, constraints on planetary masses, densities, and orbits, even for planetary systems with faint host stars. …

Luis Acedo, 2014, Quantum Mechanics of the Solar System, researchgate.net

Abstract

According to the correspondence principle, as formulated by Bohr, both in the old and the modern quantum theory, the classical limit should be recovered for large values of the quantum numbers in any quantum system. … We also consider the perturbed Kepler problem with a central perturbation force proportional to the inverse of the cube of the distance to the central body. …

The Moon is an interesting case. Its rotation period is very similar to that of the Sun.

‘Solar rotation is arbitrarily taken to be 27.2753 days for the purpose of Carrington rotations. Each rotation of the Sun under this scheme is given a unique number called the Carrington Rotation Number, starting from November 9, 1853. ‘

http://en.wikipedia.org/wiki/Solar_rotation#Carrington_rotation

‘It is customary to specify positions of celestial bodies with respect to the vernal equinox. Because of Earth’s precession of the equinoxes, this point moves back slowly along the ecliptic. Therefore, it takes the Moon less time to return to an ecliptic longitude of zero than to the same point amidst the fixed stars: 27.321582 days (27 d 7 h 43 min 4.7 s). This slightly shorter period is known as tropical month; cf. the analogous tropical year of the Sun.’

http://en.wikipedia.org/wiki/Lunar_month#Tropical_month

NB the Moon’s orbit is synchronous i.e. 1 orbit of the Sun = 1 rotation of the Moon.

The Sun-Moon time difference is about 1.1 hours using the Carrington period.