Page 1 of 14

European Journal of Applied Sciences – Vol. 10, No. 2

Publication Date: April 25, 2022

DOI:10.14738/aivp.102.11906. Bagdoo, R. (2022). Galaxies Rotation Curves Traced Out by the Theory of Relation. European Journal of Applied Sciences, 10(2). 1-

14.

Services for Science and Education – United Kingdom

Galaxies Rotation Curves Traced Out by the Theory of Relation

Russell Bagdoo

ABSTRACT

Astronomers know since the 1930s that the universe contains more than what

meets the eye. Galaxies within clusters of galaxies and stars within galaxies are

moving around faster than the gravity of the visible matter would imply, indicating

that a huge amount of invisible matter is pulling on them. The theory of Relation

asserts that there are two structures going in opposite directions, the expansion

and the condensation, and also, by means of the principle of Compensation (CP),

that the loss of negative energy of the expanding electromagnetic (EM) wavelength

of flat spacetime is transformed into positive matter and gravific spacetime. The

universal expansion decelerates, unlike the current single thought, and is

counterbalanced by a continual growing global mass; a perpetual creation of

ordinary and dark matter within an incessant big bang process. The global

deceleration takes the form of a local acceleration of every galaxy towards the

internal center, a change of direction from the periphery to the center. It was noted

that the order of magnitude − around 10!"# � �$ ⁄ − of the abnormal rotation curves

observed on the remote galaxies is similar to that of the universal deceleration

�� and not far from the anomalous gravitational variations measured for several

years on Pioneer probes using Doppler information.

Keywords: galaxy rotation curves, dark matter, theory of Relation, two structures,

Compensation principle.

INTRODUCTION

The Swiss astronomer Fritz Zwicky had argued since the 1930's for the existence of unseen

matter, based on the unusual motion of galaxies [1]. Galactic mass appears concentrated

towards the center and diminishes towards the periphery. And yet the stars at the periphery

move as if they are embedded in much greater mass; so much, in fact, that this unseen mass

must extend far beyond the periphery. If this inference is correct, then galaxies are truly not

what they seem; the visible part − the stars we see − must be swamped by cloaks of unseen

matter. Such a notion was so far outside conventional theory that it wasn't taken seriously until

Vera Rubin became interested in it [2]. In 1974, Jeremiah Ostriker, James Peebles, and Amos

Yahi at Princeton University predicted that something like dark matter must exist [3]. Their

calculations on the gravitational stability of spiral galaxies implied that such structures would

fragment as they rotated, because of vibrations triggered by their uneven composition. If,

however, the visible disk was embedded in a much larger and unseen mass, then such

vibrations would be damped, and the spiral would remain stable. The very existence of our

Milky Way implies the reality of unseen matter, they surmised. In 1978, V. Rubin and K. Ford

collected data on ten galaxies emerging with this pattern: Stars at the periphery of spiral

galaxies move too fast − if one assumes the luminous part of these galaxies represents their

entire mass [4].

Page 2 of 14

2

European Journal of Applied Sciences (EJAS) Vol. 10, Issue 2, April-2022

Services for Science and Education – United Kingdom

In 1983, Vera Rubin noted the anomaly is that the rotational velocity � of many galaxies is

constant out to the edges of those galaxies [5]. The rotational velocity � of stars appeared the

same at all distances � from the galactic centers, except for the galactic nuclei themselves. This

means that whatever is �(�1, �2, �3, �4, �5) out to the edges of many galaxies, the rotational

velocity and also the acceleration � of stars appeared the same at all distances from the galactic

centers. The rotational velocity versus distance from the galactic centers gave flat rotation

curves that precipitated the missing mass scenario. The high quality of Rubin and Ford's data,

and the fact that for years there had been scattered suggestions from observation to the effect

that spiral galaxies are surrounded by dark halos which are up to thirty times the diameter of

each visible galaxy, were important factors in the rapid acceptance of the reality of dark matter.

The “dark matter problem” arises because the mass visible (the total of stars, gas and dust

detected at all wavelengths) in large scale dynamical systems is up to one hundred times lesser

than the mass inferred from their dynamics [6, 7]. This “dynamical discrepancy” remains

unresolved today [8]. In principle there are two classes of possible solutions: either there is a

huge quantity of unseen matter of unknown type in large scale astrophysical systems, or

Newton’s law of gravity and Einstein's General Theory of Relativity need to be revised or

replaced.

M. Milgrom suggested that the theories of gravity were incomplete, that Newton's law of

gravitation failed at galactic scales, and advanced a Modified Newtonian Dynamics (MOND) [9]

approach where gravity is modified according to a scale of acceleration, without the

intervention of dark matter. The value of the critical acceleration is: �! = 1.2 × 10"#$�. �"%.

The theory has produced a good agreement with data since its introduction in 1983, but the

lack of a physical mechanism has been a major drawback to its general acceptance, so an

alternative approach was pursued. It claimed that rotation curves could be accounted for if

galaxy mass � increased linearly with distance � from center. The mass distribution indicated

by luminous matter gave a vastly different arrangement of matter. It was concluded that there

must be a hidden mass existing on larger scales of galaxies in order to match the rotation curves

with a suitable matter distribution. However, the search for such “dark” or “missing” matter

since the mid 1980’s has been largely unsuccessful. In this paper we are considering in part 2

the relation between the rotations curves of galaxies, the expansion of the universe and the

Pioneer effect. There is also an approach to gravitation using the theory of Relation and the

principle of Compensation. In part 3 there is an additional term ∆� in the Newtonian equation

for gravitational acceleration. In part 4 we speculate on the nature of dark matter, dark energy

and “dark spaces”.

RELATION BETWEEN THE ROTATIONS CURVES OF GALAXIES, THE EXPANSION OF THE

UNIVERSE AND THE PIONEER EFFECT

The galaxy rotation problem is that the observed velocity of stars and gas beyond a distance of

one to three kiloparsecs from the center of spiral galaxies is found to be constant and

independent of the distance to the galactic center, contrary to the predictions of Newtonian

dynamics. Galaxies are gravitationally bound and their outer members revolve considerably

faster than Newton’s laws say they should. To make the calculations agree with what is

observed, astronomers have been forced to assume that immense amounts of invisible matter

surround all galaxies. An approach of the gravitation by using the theory of Relation [10] and

the principle of Compensation shows that there is an additional term in the Newtonian equation

Page 3 of 14

3

Bagdoo, R. (2022). Galaxies Rotation Curves Traced Out by the Theory of Relation. European Journal of Applied Sciences, 10(2). 1-14.

URL: http://dx.doi.org/10.14738/aivp.102.11906

for the gravitational acceleration. In a previous article [11], we have describe the concepts and

the principle of the theory of Relation in order to give a solution to the observed deviation from

expectations of the trajectories of various unmanned spacecraft visiting the outer solar system,

notably Pioneer 10.

Recall that the theory of Relation gives an interaction between the electromagnetic (EM)

spacetime of expansion and the gravitational matter of condensation. We initially assumed that

our universe is made of two complementary and interpenetrated structures, one for

condensation with a gravific spacetime and EM matter (Einstein), the other for expansion with

a flat EM spacetime and ordinary matter (Lorentz-Maxwell). The theory asserts that the energy

density of the vacuum is negative and is associated with a positive pressure which drives a

decelerated expansion. This universal deceleration is accompanied, in virtue of the principle of

Compensation, with a continual growing global mass: it is a continual creation of matter inside

an after-big bang process. Both structures run in opposite directions, so the outwards

deceleration of the global expansion could be interpreted as a local inwards acceleration of each

galaxy towards its center; a change of direction of outer stars and gas towards the bright

luminous disc.

The observed anomalous rotation curves of galaxies are about the same order of magnitude

than the universal deceleration �� ≈ 10"#$ � �% ⁄ . Hubble discovered that spectra of distant

galaxies are red-shifted proportional to the independently estimated distance to the galaxies. A

plausible model of this effect is that the universe is expanding uniformly. Freedman et al. [12]

estimated the expansion rate (Hubble constant) � as 72 ± 8 km per second per megaparsec, or

2,33 ± 0.26 × 10"#&���"#(�� = ~7 × 10"#$ � �% ⁄ ). In virtue of the principle of Compensation,

the deceleration of the expansion would cause the anomalous acceleration towards the center

of the galaxy (change of direction of the curves swept by distant stars) [13, 14]. Notice that it is

also the order of magnitude measured by the gravimeters during the solar eclipses (Allais

effect) [15, 16] and that of the spacecrafts anomaly [17].

The Pioneer anomaly is seen in radio Doppler and ranging data, yielding information on the

velocity and distance of the spacecraft. The Pioneer 10 spacecraft, at a distance from the Sun of

about 67 astronomical units (AU) or 1.0 × 10#'� , experiences a measured acceleration

towards the Sun of 1.32 × 10"( � �% ⁄ , which is 8.74 ± 1.33 × 10"#$ � �% ⁄ less than the

Newtonian model [18, 19]. So when all known forces acting on the spacecraft are taken into

consideration, a very small but unexplained force remains, causing a constant sunward

acceleration of (8.74 ± 1.33) × 10"#$ � �% ⁄ [20, 21]. The magnitude of Pioneer effect is

numerically quite close to the product of the speed of light and the Hubble constant.

Let us be more specific about the galaxies:

Firstly, galaxies, globally, follow the structure of the decelerated expansion. Their recessional

velocity is decreasing. They are losing ground compared to the current theory of an accelerating

universe. The redshift seems to be higher not because of Hubble effect (speed is slower) or

Doppler shift, but due to tired-light effect [22].

Secondly, what is important to say is that the mass of the galaxies increases with the

cosmological time and the expansion. The mass of the universe is growing with the