The Gyro-Gravitational Spin Vector Torque Dynamics of Main Belt Asteroids in relationship with their Tilt and their Orbital Inclination

Scientific Paper
Title	The Gyro-Gravitational Spin Vector Torque Dynamics of Main Belt Asteroids in relationship with their Tilt and their Orbital Inclination
Read in full	Link to paper
Author(s)	Thierry De Mees
Keywords	Main Belt Asteroids, gravitation, gyrotation, prograde, retrograde, orbit, asteroid precession, asteroid nutation, asteroid tilt, angular momentum
Published	2007
Journal	General Science Journal
No. of pages	17

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Abstract

Several observational studies of the main belt asteroids showed a direct link between the evolution of the spin vectors and the inclination of their orbit. A study wherein the evolution of 25 main belt asteroids and 125 synthetic objects was computed over 1Myr (E. Skoglöv, A. Erikson, 2002) clearly quantified this link. Verification of these results with the observation of 73 asteroids confirmed the results. Non-gravitational (YORP-/Yarkovsky-) torques are not considered here. Following observational conclusions have been made by E. Skoglöv and A. Erikson:

• the spin oscillations' amplitude increases with increasing orbital inclination of the asteroid.
• the largest spin oscillations' amplitudes are found if the initial spin vector lays in the orbital plane.
• the spin obliquity differences are generally insensitive to the shape, composition and spin rate of the asteroids.
• there is a significant majority of asteroids with a prograde spin vector compared to retrograde ones.
• the spin vectors of prograde asteroids are more chaotic than the spin vectors of retrograde asteroids.
• there are very few asteroids having a spin vector that lays in the vicinity of the orbital plane.
• the heliocentric distance is relevant for the spin vector behaviour.

In this paper it was found that the gyro-gravitation theory, which is the closest Euclid theory to the General Relativity Theory of Einstein, complies very well with these observations. We find that the asteroid's tilt varies continuously and tends to reach more or less a retrograde spin in relation to the Sun's spin. The tilt varies however spiral-wise due to the gyroscopic effect that causes a motion of precession as well.

Overview

Thierry De Mees takes a body of observational and numerical work on main-belt asteroid spin axes — E. Skoglöv and A. Erikson's 1 Myr integrations of 25 real and 125 synthetic asteroids, and Erikson's spin-vector survey of 73 real objects — and asks whether it can be reproduced analytically from the Maxwell analogy for gravitation rather than from chaotic secular perturbation theory. The seven observational regularities listed in his abstract are the target; each is then checked against his own equations in the closing discussion.

His framework is what he calls gyro-gravitation: a flat, Euclidean field theory in which the Newtonian gravitational field g is supplemented by a second field Ω, the gyrotation, that stands to gravity as the magnetic field stands to the electric field. He attributes the idea to Oliver Heaviside's 1893 gravitational-electromagnetic analogy and describes his own version as "the closest Euclid theory to the General Relativity Theory of Einstein" — that is, he claims general relativity's observational successes for celestial mechanics without curved spacetime. The departure from the mainstream account is that the asteroid spin-axis behaviour usually explained by planetary secular resonances and, over longer times, by the radiation-driven YORP and Yarkovsky effects is here attributed to a torque exerted directly by the Sun's rotation. YORP and Yarkovsky are explicitly excluded from the analysis. Retardation of gravitation is also neglected, on the grounds that it does not noticeably affect the results.

The argument

The gyro-gravitation field equations

De Mees writes a Maxwell-like set with mass in place of charge, gyrotation Ω in place of the magnetic field, and a constant ζ = 4πG/c²:

F ⇐ m(g + v×Ω), ∇·g ⇐ ρ/η, c²∇×Ω ⇐ j/η + ∂g/∂t, ∇·j ⇐ −∂ρ/∂t, ∇·Ω = 0, ∇×g ⇐ −∂Ω/∂t.

He uses ⇐ rather than = to mark that the right-hand side causes the left. The field constant follows from c² = 1/(ηζ) with η⁻¹ = 4πG, and the theory admits gyro-gravitation waves propagating at c. For a spinning sphere of moment of inertia I = (2/5)mR² and angular velocity ω, the external gyrotation at position r is

Ω_ext = (GI/r³c²)[ω − 3(r·ω)r/r²],

falling off as the inverse cube of distance and vanishing in the equatorial plane where r·ω = 0.

Geometry and coordinate transformations

The asteroid's orbit is inclined at i to the solar equatorial plane and its position within the orbit is given by the angle α. The scalar product needed in the gyrotation formula follows from r sin δ = r_z = r sin α sin i, so that cos γ = sin α sin i. Writing the components out gives the solar gyrotation at the asteroid, with position components (r_x, r_y, r_z) = r(cos α, cos i sin α, sin α sin i).

The key definition is the axial tilt, taken as the sum of the obliquity and the orbital inclination:

δ = ε + i.

The tilt is measured against the Sun's spin vector rather than against the asteroid's own orbital plane — a distinction De Mees flags explicitly, since Skoglöv and Erikson refer everything to the ecliptic latitude. Two successive rotations carry the solar frame XYZ into the asteroid's local frame X″Y″Z″, rotated by δ.

Torques

Applying F ⇐ m(g + v×Ω) componentwise and dropping the purely Newtonian part, which exerts no torque, only the gyrotation component perpendicular to the asteroid's equator matters; Ω″_z is irrelevant to the torque. The resulting torques are

his equations (3.20) and (3.21), both carrying the common prefactor 3GmR²I₁ω₁ω/r³c² and depending on δ, α and i through terms in sin 2δ, sin²α sin²i and cos δ sin 2α sin i, with T_z = 0. These define the solar gyrotational torque for every orbital inclination and every point on the orbit.

Precession, nutation and tilt change

De Mees then writes the Euler equations for a cylinder-symmetric body with I₀ = I_x = I_y and I₁ = I_z, using the precession angle φ, nutation angle θ and spin angle ψ. Assuming constant spin rate, the precession velocity reduces to

φ̇ = I₁ω₁/(I₀ − 2I₁),

and the nutation angle to θ = arctan[T_xy(2I₀ − I₁)/(I₀²ω₁²)] with T_xy = √(T_x² + T_y²). Since the nutation and the change of tilt position are equal by definition, θ̇ = δ̇, and differentiating with respect to the orbital angle gives the per-orbit tilt change, his equation (C.6), proportional to GmR²ω sin 2i/r³c² and to a function of δ. The tilt therefore changes by a very small amount over each orbital semicircle and swings back over the next, while the whole torque vector T_xy rotates about the Sun with the orbit — hence the spiral character described in the abstract.

Stability and the comparison with observation

Appendix A derives a neutral tilt angle δ_neutral from the condition Ω″_z = 0, i.e. tan δ (3 cos²α sin²i − 1) = 3 sin α cos α sin i. Plotted against α and i, it shows that for orbital inclinations below about π/8 the tilt is highly stable, both prograde and retrograde, but above π/8 it becomes abruptly unstable until about 7π/8. Crucially the stable zone is much wider in α for prograde orbits than for retrograde ones.

From this asymmetry De Mees claims five of the seven observational points. The inverse-cube distance dependence gives the relevance of heliocentric distance directly. Larger inclination gives larger oscillation amplitude via (C.6). The largest amplitudes occur at δ = π/2, which also accounts for the scarcity of spin vectors lying near the orbital plane, since that is where most of the orbit is unstable. Prograde asteroids being more chaotic follows from the strong instability of prograde high-inclination orbits in Fig. A.1, and the prograde majority (he quotes 64% prograde versus 36% retrograde in Erikson's 73-object sample) follows from the full stability of small prograde inclinations. He extends the same reasoning to planets, suggesting Venus's inverted tilt "might have been originated because of an original orbit inclination wherefore i > π/8, causing a tilt instability and even a tilt switch, without any collision with other bodies."

The sixth point he rejects. Skoglöv and Erikson report that obliquity differences are insensitive to shape, composition and spin rate; De Mees finds the opposite theoretically, and argues that their graphical presentation conceals it. Defining ε = arctan(X₀) − i so that X = cos(arctan(X₀) − i), he shows this purely definitional function reproduces the shape of their Figs. 1.2 and 1.3 with no observational input at all: "the supposed (strong) dependence of the orbit inclination to the spin vector tilt is namely only fictive in that graphic." He recommends the data be presented as δ = f(α, i).

Assessment

The methodological criticism of Skoglöv and Erikson's figures is the most solid contribution here and stands independently of the theory. If a plot of X = cos ε against inclination is generated by a transcription in which i already appears in the definition of ε, then the curve's shape is not evidence of a physical dependence, and De Mees is right to say so. It is also a point against his own case, since it removes some of the observational structure he is trying to explain — and he states it anyway. The analysis is otherwise clean in construction: the field equations are stated up front, the coordinate transformations are given explicitly, the Euler equations are solved with stated simplifications, and the stability result is derived from a single transparent condition Ω″_z = 0. The prediction that the tilt-stability boundary is asymmetric between prograde and retrograde orbits is a real, checkable consequence rather than a restatement of the data.

The difficulties begin with what the framework is. The Maxwell analogy for gravitation is not free-standing; in general relativity the gravitomagnetic field emerges in the weak-field, slow-motion limit with a specific coefficient, and De Mees's Ω_ext is that field written with his own normalisation. He asserts rather than derives the claim that gyro-gravitation is "the closest Euclid theory to the General Relativity Theory of Einstein", and the paper contains no comparison of his coefficients with the standard Lense-Thirring result. This matters because the whole argument is one of magnitude. Solar gravitomagnetic torques in the asteroid belt are extraordinarily small — the effect scales as GI_Sunω_Sun/r³c², and at 2-3 AU the c⁻² suppression is severe. De Mees never evaluates a single number. There is no computed tilt change per orbit for Ceres, no timescale to compare with the 1 Myr integrations he is trying to match, and no comparison with the ordinary Newtonian secular torques from Jupiter and Saturn, which he sets aside at the outset as "not studied here". Since those are precisely the perturbations Skoglöv and Erikson's integrations do include, and which are known to be large, the paper leaves unaddressed why a much weaker effect should dominate the outcome.

The qualitative matching is correspondingly weak as evidence. Five of the seven listed regularities are matched by sign or by trend only — larger inclination gives larger amplitude, prograde is more stable than retrograde — and secular perturbation theory reproduces the same trends. Without a magnitude or a distinguishing prediction, agreement in sign does not discriminate between the two accounts. The Venus suggestion is offered as a possibility with no calculation attached and should be read as such.

There are also internal loose ends. The axial tilt orientation is admitted to be underdetermined — "any tilt orientation upon the cone's surface... will comply with the description" — and results for the x and y axes are simply averaged, which De Mees concedes "could play a role for large orbital inclinations", exactly the regime where his stability switch occurs. Orbital eccentricity is dropped. The precession formula φ̇ = I₁ω₁/(I₀ − 2I₁) has no torque in it at all, so it describes free rather than forced precession, and its relation to the driven problem is not made explicit. Several equations in the text are reproduced from figures and appendices whose plots are not legible in the archived file. Taken together, the paper is best read as a demonstration that a gravitomagnetic torque produces the right kind of behaviour, not that it produces the observed behaviour.

See also

• Thierry De Mees
• Gravitomagnetism
• Oliver Heaviside
• Gravity
• Maxwell's Equations
• General Relativity
• General Science Journal