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- Physics Essays volume 18, number 3, 2005 1 Analytic Description of Cosmic Phenomena Using the Heaviside Field T. De Mees Abstract The Maxwell analogue equations (MAE) for gravitational dynamics, as first pro- posed by Heaviside [O. Heaviside, The Electrician 31, 281 (1893)], are applied to fast rotating stars. We define the absolute local velocity (ALV) for objects moving in a gravitational field, and we apply the MAE and the Lorentz force (LF) law (LFL) to planetary orbits, galaxies with a spinning center, and spinning stars. The result is that the MAE and the LFL allow us to explain perfectly and very simply the formation of disc galaxies and the constant speed of the stars of the disk. They explain the origin of the symmetric shape of some supernova remnants and find the supernova’s explosion angle at 0° and global compression not above 35°16′. They define the dynamics of fast-spinning stars that never explode — despite their high rotation velocity — in relation to the Schwarzschild radius. They finally de- scribe binary pulsars, collapsing stars, and chaos. No other assumptions are nec- essary for obtaining such results. Key words: gravitation, rotary star, disc galaxy, repulsion, relativity, gyrotation, chaos, analytical methods 1. INTRODUCTION: THE MAXWELL ANA- LOGUE EQUATIONS: A SHORT HISTORY Several earlier studies have shown a theoretical analogy between the Maxwell formulas and the gravitational theory. Heaviside suggested this anal- ogy.(1) Nielsen deduced it independently using Lorentz invariance.(2) Negut extended the Maxwell equations more generally and discovered as a conse- quence the flatness of the planetary orbits;(3) Je- fimenko deduced the field from the time delay of light and developed thoughts about it;(4) Tajmar and de Matos worked on the same subject,(5) as did several other authors. The Maxwell analogue equations (MAE) can be expressed in the equations (1) to (5) hereunder. Vectors are written in bold. Mass replaces electrical charge, the so-called gyrotation Ω replaces the magnetic field, and the respective constants as well are substituted. We use the following notation: gravitational acceleration is g; the universal gravita- tional constant is G = (4πζ)−1 . We use the sign ⇐ instead of = to highlight induction equations. The mass flow through a surface is j. (1)(m⇐ + ×F g v Ω , ρ ζ ∇⋅ ⇐g (2) 2 ,c tζ ∇× ⇐ + ∂ ∂j g Ω (3) ), div ≡ ∇⋅ = 0,Ω Ω (4) . t ∇× ⇐ − ∂ g ∂Ω (5) In (3) the term ∂g/∂t is needed for compliance with the equation div j ⇐ –∂ρ/∂t. 2. GYROTATION EQUATIONS Considering a spinning central mass m1 at a rotation velocity ω and a mass m2 in orbit, the rotation trans- mitted by gravitation (dimension (rad/s)) is called the gyrotation Ω. More generally, gyrotation is the field created by the gravitational field of a moving object in an existing steady gravitational field. Equation (3) can also be written in integral form, as in (6), and interpreted as a flux (∇ × Ω) through a surface A. Hence one can write down (6) or, using the Stokes theorem for line integrals,(6) (7):
- Analytic Description of Cosmic Phenomena Using the Heaviside Field 2 2 4 ( n Gm d c , π ∇× ) ⇐∫∫A AΩ (6) 2 .d c 4 Gm (7) π ⋅ ⇐∫ lΩ The MAE imply the definition of the absolute local velocity (ALV). For spinning objects we define ALV as zero when no gyrotation is measured. Consider a spinning sphere, enveloped by its own gravitational field, and at this condition, we can apply the analogy of the induced magnetic field by an electric current in a closed loop, integrated over the sphere.(6) The results at a distance r from the center of the sphere with radius R are given by (8) inside the sphere and (9) outside the sphere: 2 2 2 4 G 2 1 ( ) , 5 3 5 int r R c π ρ ⎛ ⋅⎛ ⎞ ⇐ − −⎜ ⎟⎜ ⎝ ⎠⎝ r r ω Ω ω ⎞ ⎟ ⎠ (8) 5 4 ( )G Rπ ρ ⋅ ⎞ ⎟ ω r r ω 3 2 . 5 3 5 ext r c ⎛ ⇐ −⎜ ⎝ ⎠ Ω (9) For homogeneous rigid masses we can write 2 3 2 2 3 ( ) . 5 ext GmR r c r ⋅⎛ ⇐ −⎜ ⎝ ⎠ r ω r Ω ω ⎞ ⎟ (10) Figure 1 shows a part of the gyrotation equipotentials (dotted curves), the generated forces FΩ (gray ar- rows), and the centrifugal forces Fc (black arrows) at the surface of a spherical star, based on (9). The same deduction can be made for the lines of gyrotation inside the star (Fig. 2), based on (8). Here, the surface Lorentz forces (LF) have been shown again. At the right side of Fig. 1, the LF have been applied on a prograde moving mass (FΩ2), showing its compo- nents; at the left side we have a retrograde motion (FΩ1). 3. ANGULAR COLLAPSE INTO PROGRADE ORBITS Spinning stars create a similar gyrotation pattern of equipotentials as magnetic dipoles do. Objects orbiting around spinning stars will be affected by two components of LF: a radial component, pointing to the star’s center, and a tangential component. The first force will slightly reduce (FΩ2) or enlarge (FΩ1) the orbit of the object. The latter force points toward the star’s equatorial plane (defined further as the 0° reference plane) for prograde orbits and away from the equatorial plane for retrograde orbits (Fig. 1). Prograde orbits that are in another plane than in the star’s equatorial plane remain prograde but will revolve about the common axis of the star’s equato- rial plane and the orbital plane. By this, the orbit has an angular collapse toward the equatorial plane, while keeping a constant orbital radius. The orbital collapse inertia will then make the orbit exceed the equator, return back to it, and so decreasingly oscillate around the star’s equatorial level. Retrograde orbits revolve about the common axis of the star’s equatorial plane and orbital plane, over the poles, and then become prograde, where also a decreasing oscillation occurs, as with prograde orbits. 4. PRECESSION OF ORBITAL SPINNING OBJECTS On orbiting and spinning planets, the star’s gyrota- tion, combined with the LF law (LFL) generates a momentum: in all cases where the star and a prograde planet do not spin in opposite directions, the momen- tum caused by gyrotation will generate a precession of the planet. For parallel but opposite spins, this momentum is zero and also stable; for parallel spins in the same direction the momentum is zero but labile. 5. FORMATION OF DISC GALAXIES; SPIRAL GALAXIES; ORIGIN OF THE CONSTANT VELOCITY OF THE STARS The same occurs with any galaxy with a spinning center. Spinning centers generate prograde orbits of stellar systems, decreasingly oscillating about the center’s equator. When the disc has been formed, a high density is caused by the vertical gyrotation forces keeping the disc flattened. Only when the orbit is exactly in the 0° plane are these forces zero. Gravitation will group the stars into active con- glomerates, creating empty spaces elsewhere, origi- nating patterns such as meshes and then spirals. The stars’ velocity in a disc galaxy can be deduced with good approximation from its data. A bulge with radius R0 and mass M0, which has not been collapsed, can be seen at the original main global center mass about which all other stars of the spherical or ellipti- cal galaxy orbited. (This is approximately true, but we should keep in mind that the main reason for the bulge is more probably due to the action of one or more central spinning black holes, which maintain
- T. De Mees 3 disorder in the bulge.) Consider the equatorial plane XY with its origin in the galaxy’s center. For any orbiting star in the original spherical galaxy, at a distance R and with mass m, the Keplerian velocity is v2 = GM0/R. After the angular collapse to a disc with a bulge, the star at a distance R = f1(x)R0 will be submitted to the influence of M0f2(x), where f1(x) is a linear function and f2(x) is a general one. We can set f1(0) = f2(0), and at a first approximation f2(x) is also linear, so that f2(x)/f1(x) ≡ k = constant. Hence v1 2 = Gf2(x)M0(f1(x)R0)–1 = kGM0(R0)–1 , which is then constant for any star. For the Milky Way, with a bulge diameter estimate of 10 000 light years, having a mass of 20 billion solar masses (10% of the total galaxy), setting k = 1, we get a quite correct orbital velocity of 240 km/s. 6. DYNAMICS OF FAST-SPINNING STARS AND SUPERNOVA REMNANTS; THE GLOBAL COMPRESSION ANGLE AND THE EXPLOSION ANGLE NEAR 0°; SIZING OF FAST-ROTATING BLACK HOLES Let us consider a fast-rotating star, for which the forces on a point p on its surface are calculated. For a sphere, by putting r = R in (10), we obtain 2 2 3 ( ) . 5 R Gm Rc R ⋅⎛ ⇐ −⎜ ⎝ ⎠ R Rω Ω ω ⎞ ⎟ (11) The gyrotation accelerations are given by the fol- lowing equations (y is the rotation axis): cosx y ya x Rω ω α⇐ Ω = Ω and cos .y xa x R xω ω α⇐ Ω = Ω Taking into account the centrifugal force, the gyro- tation, and the gravitation, one can find the total acceleration force: 2 2 2 2 (1 3sin ) cos 1 5 cos , xtot Gm a R Rc Gm R α ω α α ⎡ − ⇐ −⎢ ⎣ − ⎤ ⎥ ⎦ (12) For elevated values of ω the last term of (12) is negligible and will maintain a global compression for any value of R below a critical radius, regardless of ω. This limit is given by the critical compression radius: 2 2 2 (1 3sin ) 0 1 or 5 (1 3sin ),C C Gm Rc R R Rα α α − = − = < − (14) where RC is the equatorial critical compression radius for spinning spheres: 2 . 5 C Gm R c = (15) For spheres with R ≤ RC a global compression takes place for each angle α, where –αC < α < αC and 1 / arcsin . 3 C C R R α − = (16) Note that αC is always less than 35°16′. When we apply the results (8) at the equator (Fig. 2), we see immediately that condition (14) has to be amended: at the equator, Ωint indeed becomes zero at r = (5/6)–1/2 R, allowing equatorial ring-shaped mass losses even if Rα=0 < Gm/(5c2 ). 2 2 2 . R2 3 cos sin sin 0ytot Gm Gm a c From (14) also results that the shape of fast-rotating nonrigid stars stretches toward a Dyson-like ellipse,(7) but with a missing equatorial area, and even to a kind of toroid: if α ≥ 35°16′, the critical compression radius indeed becomes zero, and (13) becomes maximal, allowing mass losses, such as with the supernovae SN 1987A and η-Carinae (Fig. 3), where mass losses occur near the equator and global com- pression is somewhere below the 35°16′ limit. We know that SN 1987A only loses mass from time to time at 0° and at a certain angle ≈ αC only, which should clearly indicate that the supernova is a nonri- gid torus. Otherwise, mass losses would also occur between ±αC and the poles, and no losses would repeatedly occur at 0°. The SN 1987A state seems to be a constant reshaping of the torus after mass losses, resulting in several consecutive mass losses over time. For the remaining toroid the inertial momentum is roughly half that of the original sphere with the same external radius and with an inner torus radius close to zero. The equatorial critical compression radius for a spinning toroid is then nearly RC = Gm/(10c2 ), which equals 1/20 of the Schwarzschild radius RS valid for nonrotary spherical black holes. ω α α α − ⇐ + + (13)
- Analytic Description of Cosmic Phenomena Using the Heaviside Field 4 The explosion of η-Carinae shows the case of a more spherical supernova with a radius at least as small as RC = Gm/(5c2 ) or 1/10 of RS before the explosion. The origin of the explosion will probably be a sudden collapse that consequently increases the star’s rotation velocity. 7. DYNAMICS OF BINARY PULSARS AND ACCRETION DISCS; POLAR BURSTS; SPIN-UP AND SPIN-DOWN; CONDITION FOR THE ABSORPTION OF THE COMPAN- ION The fast-spinning star in binary pulsars is more likely a torus. The gyrotation field’s equipotentials of a spinning toroid are analogous to a magnetic dipole. The application of the LF on prograde accretion matter, near the spinning star, is initially attractive, but the LF on the radial motion results in a retrograde motion, and the LF on the retrograde motion results in a radial motion away from the star. The final outcome of in-falling prograde accretion matter can therefore be seen as a repulsion. At the equator the action time of the LF is very short, and the space in between very limited, so that in-falling accretion mass is pushed back in the cloud, forming prograde vortices and turbulences. Accretion matter that approaches radially, flowing over or under the toroid toward the poles, however, gets a long LF action time, first retrograde, then away from the poles, allowing huge accelerations and huge kinetic energies, known as polar bursts. Accretion matter that falls in the central hole of the toroid due to collisions can get trapped by the LF and inwardly absorbed, if the in-falling matter goes prograde, or can move more randomly in small prograde vortices, such as in a cloud, if the in-falling matter does not go prograde. It is likely that the inner cloud will sooner or later get oversaturated and tend to lose mass again via the bursts. Spin-up and spin- down are possibly explained by this mechanism. Consider a binary pulsar with a spinning star 1 (mass m1, inertial momentum I1, radius R1, orbital radius Rc1) and a companion 2 (mass m2, radius R2, orbital radius Rc2). The system’s rotation speed is ω3. Observation shows that Rc2 is of the order of two or three times R2. The pull of accretion matter from the companion’s front side — in relation to the orbital center — is controlled by Newton’s gravitational law and the gyrotation law. The requested orbital velocity of the front side can be found from 2 2 1 3 2 22 1 2 3 2 1 2 2 2 1 2 2 0 or ( ) 2( ) c g c f c c c F F F I R RGm m v m m R R R R R c ω ω Ω 1 + + = ⎡ ⎤− = +⎢ ⎥ + − + −⎣ ⎦ (17) and is much higher than that of the companion’s center. At the back side, inversely, the velocity from 2 2 1 3 2 22 1 2 3 2 1 2 2 2 1 2 2 ( ) 2( ) c b c c c I R RGm m v m m R R R R R c ω ω1 ⎡ ⎤+ = +⎢ ⎥ + + + +⎣ ⎦ (18) is much lower than that of the companion’s center, and matter “runs behind.” Depending on the cohesion forces in the companion, matter will escape more or less easily from the front and the rear. The escaped matter will be attracted toward the spinning star by analogous forces. 8. DYNAMICS OF COLLAPSING STARS Collapsing spinning stars get an important increase of spin velocity and consequently of gyrotation field. The induction law (5) generates circular concentric gravitational fields, perpendicular to the gyrotation field. In the case of an accretion disc near the star, a strong concentric gravitational contraction of it will occur, reducing its orbital diameter and approaching the spinning star with high speed. This leads to a sudden repulsion perceived as a burst at the equator and at the poles, as explained above. 9. REPULSION BY MOVING MASSES The repulsion of masses has been deduced in Sec- tion 7, but follows also directly from the theory: when two flows of mass dm/dt move in the same way in the same direction, the respective fields attract each other. For flows of masses having an opposite velocity, their respective gyrotation fields will be repulsive. This is, however, only valid if there exists a referential gravitational field corresponding to zero velocity: it is clear that the velocity of the two mass flows should be seen in relation to another mass, considered resting, and large enough to get enough gyrotation energy created, as explained in Section 3. Spinning masses do the same. Consider two spin- ning objects, close to each other, in the same equato- rial plane, placed in their own gravitational field (zero-velocity reference). If their spins are opposite, the equatorial speeds point in the same direction, and the forces attract. With the same spins, the forces are repulsive.
- T. De Mees 5 10. CHAOS EXPLAINED BY GYROTATION The theory can explain what happens when two planets cross. Gravitation and gyrotation give an apparent effect of a chaotic interference. Let’s assume that the orbital radius of the small planet is larger than that of the large planet. When passing by, a short but considerable gravitational attraction moves the small planet radially toward the Sun into a smaller orbit. Due to the natural law of gravitationally fashioned orbits (simplified form), , GM v r = (19) the orbital velocity will increase. On this radial motion works the gyrotation aO ⇐ vR × Ω of the Sun and of the large planet that again slows down its orbital velocity. The result is a slower orbital velocity in a smaller orbit, which is in dis- agreement with (19). Thus, in order to solve the conflict, nature sends the small planet away toward a larger orbit. Again, gyrotation works on the radial velocity, this time by increasing the orbital velocity, which contradicts (19) again. We come thus to an oscillation, which can persist if the following pas- sages of the large planet come in phase with the oscillation. One could affirm that gravitation alone could ex- plain chaotic orbits too. But it doesn’t: if no gyrota- tion existed, the law (19) would send the planet back to its original orbit with a fast-decreasing oscillation. Gyrotation reinforces and maintains the oscillation much more efficiently, and even allows rotating orbital oscillations. 11. CONCLUSIONS Gyrotation, defined as the transmitted angular movement by gravitation in motion, is a plausible solution for a whole set of unexplained problems of the universe. It forms a whole with gravitation, in the shape of a vector field wave theory, that becomes extremely simple by its close similarity to electro- magnetism. And in this gyrotation the transversal time retardation of light is locked in. Other advantages of the theory are that it is Euclid- ian, easy to use analytically, and very precise. Predic- tions are deducible from laws that are analogous to those of Maxwell. Received 13 October 2004. Résumé Les équations analogiques de Maxwell (EAM) pour la dynamique de gravitation, comme a été proposées en premier par Heaviside [O. Heaviside, The Electrician 31, 281 (1893)], sont appliquées aux étoiles à rotation rapide. Nous définissons la vélocité locale absolue (ALV) pour des objets se déplaçant dans un champ de gravitation, et nous appliquons les EAM et la loi de la force de Lorentz (FL) (LFL) aux orbites planétaires, les galaxies au centre rotatif, et les étoiles rota- tives. Le résultat est que les EAM et la LFL nous permet d’expliquer de façon simple et parfaite la formation de galaxies à disque et la vitesse constante des étoiles du disque. Elles expliquent l’origine de la forme symétrique des restes de supernova, et trouve son angle d’explosion à 0° et une compression globale pas au-dessus de 35°16′. Elles définissent la dynamique des étoiles à rotation rapide qui n’explosent jamais, malgré leur haute vitesse de rotation, en fonction du rayon de Schwarzschild. Enfin, elles décrivent des pulsars binaires, des étoiles en collapse, et le chaos. Aucune autre supposition n’est nécessaire afin d’obtenir de tels résultats. References 1. O. Heaviside, The Electrician 31, 281 (1893). 2. L. Nielsen, A Maxwell Analog Gravitation Theory (Niels Bohr Institute, Copenhagen, Gamma No. 9, 1972). 3. E. Negut, Revue Roumaine des Sciences Tech- niques, Mécanique appliquée 35, 97 (1990). 4. O. Jefimenko, Causality, Electromagnetic Induction, and Gravitation (Electret Scientific, Star City, WV, 2000). 5. M. Tajmar and C.J. de Matos, Advance of Mer- cury Perihelion Explained by Cogravity, arXiv, 2003gr.qc.4104D (2003). 6. R.P. Feynman, R.B. Leighton, and M. Sands, Feynman Lectures on Physics, Vol. 2 (Addison-
- Analytic Description of Cosmic Phenomena Using the Heaviside Field 6 Wesley, Reading, MA, 1964). 7. M. Ansorg, A. Kleinwächter, and R. Meinel, Astron. Astrophys. 405, 711 (2003); Astro-Ph. 482, L87 (2003). 8. R.L. Forward, Proc. IRE 49, 892 (1961). T. De Mees Leeuwerikenlei 23 B-2650 Edegem Belgium e-mail: thierrydemees@pandora.be Figure Captions Figure 1. The external equipotentials of the gyrotation field Ω (dotted lines), created by the rotation of a rigid sphere. A force FΩ2 works on each mass in a prograde orbit, and a force FΩ works on each mass in a retrograde orbit. In both cases the LF collapses the orbit into decreasingly oscillating equatorial prograde orbits. At the sphere’s surface the local gyrotation forces (grey arrows) and the centrifugal forces Fc are shown. Figure 2. The inner gyrotation equipotentials Ω are drawn as dotted lines; the surface and inner gyrotation forces are drawn as grey arrows. Note that near the equatorial level, the forces at the surface point into the sphere; the gyrotation forces of the inner mass point out of the sphere. Figure 3. Supernova 1987A and η-Carinae are fast spinning while losing mass at 0° and probably above αC < 35°16′. We expect η-Carinae to be spherical, while SN 1987A is a torus. The expected rotation axis is shown as well.