Deduction of Orbital Velocities in Disk Galaxies. "Dark Matter": a myth?
| Scientific Paper | |
|---|---|
| Title | Deduction of Orbital Velocities in Disk Galaxies. "Dark Matter": a myth? |
| Read in full | Link to paper |
| Author(s) | Thierry De Mees |
| Keywords | Maxwell Analogy, gravitation, galaxy, dark matter, gravitomagnetism, rotary star, black hole, gyrotation, Kepler laws, GRT, angular momentum |
| Published | 2007 |
| Journal | General Science Journal |
| No. of pages | 15 |
Read the full paper here
Abstract
In my paper "A coherent dual vector field theory for gravitation" is explained how simply the Gravitation Theory of Newton can be extended by transposing the Maxwell Electromagnetism into Gravitation. There exists indeed a second field, which can be called: co-gravitation-, Gyrotation- (which I prefer), gravito-magnetic field and so on. In this paper, I will call this global theory the Maxwell Analogy for Gravitation (MAG) "Gyro-Gravitation". One of the many consequences of this Gyro-Gravitation Theory that I have written down, is that Dark Matter does not exist. At least far not in the quantities that someones expect, but rather in marginalized quantities. Many researchers suppose that disk galaxies cannot subsist without missing mass that, apparently, is invisible, and which has to be taken into account in the classic Newton-Kepler model to better explain the disk galaxies' shapes. A remarkable point is that Gyro-gravitation Theory is not only very close to GRT, but more important, easy to calculate with, and coherent with Electromagnetism. It is no coincidence that nobody found the same result with GRT, not because GRT would obtain some other result, but because it is almost impossible to calculate with it. A demonstration is again given in this paper, where I deduce the general equations for the orbital velocities of stars in disk galaxies, based on the assumption of a simple mass distribution of the initial spherical galaxy.
Overview
Thierry De Mees, a Belgian engineer, wrote a long series of papers between 2003 and the early 2010s developing what he calls the Maxwell Analogy for Gravitation (MAG), or "Gyro-Gravitation". The core idea is old and respectable in itself: alongside the ordinary Newtonian field g there exists a second, magnetic-like field — De Mees names it "Gyrotation" Ω — generated by moving mass, obeying equations obtained from Maxwell's equations by substituting mass for charge and G = (4πζ)−1 for the electric constant. A test mass then feels a force analogous to the Lorentz force, F ⇐ m(g + v×Ω). De Mees credits Oliver Heaviside (1893) and Oleg Jefimenko as predecessors.
This particular paper applies the scheme to the flat rotation curves of disk galaxies, the observation that made dark matter a standard ingredient of galactic astronomy. De Mees's claim is that the missing mass is a "myth": once one accounts for how gyrotation converts an originally spherical galaxy into a disk, the observed velocities follow from ordinary gravitation with no invisible component. He is explicit that his theory is non-relativistic — "not really relativistic", he says, preferring "Dopplerian" — and that it reinstates a "locally absolute velocity" defined within whatever bounded system is under study, so that the solar system, or the Milky Way, can each be treated as closed for the purposes of a calculation.
The argument
The problem as stated
De Mees reproduces Burton's 1976 Milky Way rotation curve and the Rubin, Ford and Thonnard (1978) curves for several disk galaxies. In every case the stellar velocity rises steeply near the nucleus, dips slightly before about 5 kpc, and then flattens to nearly a single value across the disk. Kepler's third law for circular orbits, v2 = GM/r, requires velocity to fall with radius. De Mees frames the standard response as a syllogism of elimination: Newton's law contains only mass and distance; distance is measured; G is fixed; therefore the mass distribution must be different from what is seen, hence invisible mass. Against this he sets the observation that v2 = ar is merely the geometry of a centripetal acceleration, so that "any change of the acceleration allows a change of the velocity and/or the radius". His programme is to compute the actual acceleration in a real disk rather than assume a central point mass.
Why disks are flat and prograde
The dynamical role of gyrotation in the paper is to explain disk formation rather than the velocities themselves. A spinning central body generates a gyrotation field; the analogue Lorentz force on an orbiting mass has a tangential component that slowly swivels a prograde orbit toward the spin equator, while a retrograde orbit is pushed away until it too becomes prograde and then converges on the equator. Because the gyrotation force is much weaker than gravitation, the orbital radius barely changes during this swivelling. Applied to a galaxy, the summed gyrotation of thousands of fast-spinning stars and black holes in the bulge yields a fuzzy net field perpendicular to the eventual disk; the residual randomness gives, De Mees argues, a Gaussian thickness profile about the plane.
The mass distribution and the flat curve
The critical step is the assumed density law. Writing the spherical progenitor as galaxy "1" and the disk as galaxy "2", conservation of mass shell by shell gives ρ1dV1 = ρ2dV2. De Mees then postulates, "only in order to get simpler results", that dM1(r)/dr = M0/R0 = constant, where M0 and R0 are the bulge mass and radius — that is, equal mass in each concentric shell of equal thickness, so that M(r) grows linearly with r. Substituting M(r) = M0r/R0 into v2 = GM(r)/r gives immediately
- vg = √(GM0/R0)
— a constant, equal to the Keplerian velocity at the bulge boundary. De Mees at once flags the two idealisations involved: only the mass interior to r was counted, and it was treated as concentrated at a point. He calls this "manipulating formulas mathematically without respecting the full physical meaning".
The detailed integration
The remainder of the paper repairs those two idealisations. Inside the bulge, taken as a homogeneous sphere, the velocity is linear in radius, v ∝ R. Outside, the bulge is treated as a point mass at distance (R2 + H2)1/2, and the disk's own contribution is obtained by integrating the acceleration from infinitesimal rings of the disk over radius r and azimuth α. The radial integration yields expressions in Complete Elliptic Integrals of the First Kind, F(x, π/2), and setting H = 0 for stars in the plane gives a closed-form total acceleration and hence v2 = aR. Evaluating numerically with R0 = 1 and Re = 10 in arbitrary units, the paper's table 5.1 reads: at R = 1, 1.2, 2, 3, 4, 5, 6, 7, 8, 9, 10 the velocity is 1, 0.83, 1.54, 1.75, 1.84, 1.92, 2, 2.07, 2.17, 2.34, 2.78. De Mees describes the resulting profile as matching NGC 4594, NGC 2590 and NGC 1620 "quite well", and christens systems obeying this mass law "galaxies of order zero".
Assessment
Two things in the paper are genuinely worth having. The first is the swivelling mechanism. A weak spin-generated field that torques inclined orbits toward the equatorial plane, and drives retrograde orbits to prograde, is a real physical effect with a real counterpart — the gravitomagnetic sector of general relativity produces Lense-Thirring precession and, in accretion physics, the alignment of inner discs with a spinning hole. De Mees's treatment of it is elementary and calculable, which is his stated selling point over general-relativistic gravitomagnetism. The second is the honesty of the exposition: at every simplification he says what he has thrown away, and he explicitly labels equation (4.2) as chosen for convenience.
That candour, however, exposes the paper's central weakness. The flat rotation curve of section 4 is not deduced from gyrotation at all; it follows in one line from the assumed mass law dM/dr = constant, since M(r) ∝ r gives v2 = GM(r)/r = constant identically. But M(r) ∝ r means ρ ∝ 1/r2, which is precisely the singular isothermal halo profile that dark-matter modelling invokes to produce flat curves. The paper therefore assumes the mass distribution that dark matter was postulated to supply, and then reports that dark matter is unnecessary. The disagreement with the standard account is not about dynamics but about whether the mass in question is luminous; and the observed surface brightness of disk galaxies falls off roughly exponentially, not as 1/r2, which is the actual empirical basis of the missing-mass argument. De Mees reproduces Burton's surface-density curve σ(R) early in the paper but never compares it with the ρ(r) his equation (4.2) requires.
Gyrotation is also numerically absent from the result. De Mees himself notes that the gyrotation force is "of a much smaller order than the gravitation force" and that inside the bulge the randomly oriented fields largely cancel, so "the gravitational acceleration is dominant". Since the second field enters at order v2/c2, and disk stars move at a few hundred km s−1, the correction is of order 10−6 — far too small to reshape a rotation curve. Equations (5.13) to (5.22), the ones that actually produce numbers, contain no gyrotation term at all: they are a Newtonian disk integration.
The numbers themselves do not support the claim of constancy. Table 5.1 rises monotonically from v = 1.54 at R = 2 to v = 2.78 at R = 10, an increase of about 80 per cent across the disk, with a further sharp rise in the outermost bin. Observed curves are flat to within a few per cent over comparable ranges. No quantitative fit, no residuals and no error estimate are offered against the three named galaxies; the agreement is asserted from visual comparison with figure 2.2.
Finally, the paper answers only the rotation-curve evidence for dark matter, and that is now the weakest strand. It says nothing about the mass reconstructions from gravitational lensing — in particular the displacement between the lensing mass peaks and the X-ray gas in the Bullet Cluster 1E 0657-56 — nor about the relative heights of the first three acoustic peaks in the cosmic microwave background power spectrum, which constrain the baryon and non-baryon densities separately, nor about the velocity dispersions of galaxy clusters. A theory that removes dark matter must address these; a rotation-curve model, however elegant, does not by itself settle the question the title raises.