This document summarizes research applying Maxwell's analogue equations for gravitation to describe various cosmic phenomena involving spinning or rotating celestial bodies. Key points: 1) The equations allow explaining the formation of disc galaxies from the angular collapse of orbits around spinning galactic centers into the equatorial plane, and the constant velocity of stars in disc galaxies. 2) They describe the dynamics of fast-spinning stars, showing they can maintain a global compression and avoid explosion if their rotation is below a critical rate, and explaining properties of some supernova remnants. 3) Applying the equations to binary pulsars and other systems shows they can account for orbital precession and other effects without additional assumptions.

1. 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):

2. 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

3. 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)

4. 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.

5. 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-

6. 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.