Keplerian Motion Inside an Isolated Dark Matter Halo and the Hubble’s Law

Keplerian Motion Inside an Isolated Dark Matter Halo and the Hubble’s Law

Saoussan Kallel-Jallouli

Faculté des Sciences de Tunis, Department of Mathematics, Université de Tunnis El Manar, LR03ES04, Tunis, 2092, Tunisia

DOI: https://doi.org/10.51244/IJRSI.2024.11090102

Received: 15 September 2024; Accepted: 26 September 2024; Published: 21 October 2024

ABSTRACT

Recently, the existence of a new kind of staff, named “Zaman”, responsible for time variation by its spin, was established. The proposed model offered a suitable solution for the Dark Matter enigma. In this work, we generalize the previously solid body rotating spherical halo U to a differential rotation, where each spherical shell of radius R has a length of the day T(R) related to R with the Kepler’s relation: . We prove that the linear Hubble’s relation, , is just a consequence of our new “time”-model, and does not need any special hypothesis nor any special non Euclidean topology on U.

Keywords: Dark Matter, Hubble constant, Time, Velocity, Zaman

PACS : 95.35.+d, 98.80.Es, 06.30.Ft, 06.30.Gv

INTRODUCTION

Dark matter (DM) is a major component of the universe, about five times the abundance of ordinary visible matter [Ade, 2016] [Komatsu, 2009]

No direct evidence exists to explain what dark matter consists of. But, we are sure that Dark Matter is not the normal “baryonic” seen matter. We measure the effects of its mass energy, interacting only by the action of its gravitational pull on normal matter. The astronomical community took dark matter seriously, after they ensure that galaxies were rotating so fast that the basic laws of physics imply they would have to rip themselves apart if not kept clumpy together by the presence of a kind of unusual unseen material. Dark matter is inescapable and controls the distribution of visible matter in the universe. In the modern theory of cosmological structure formation, dark matter halos are the basic unit of sufficiently overdense regions into which seen matter collapses [Bullock et al. 2001] [Wechsler & Tinker. 2018]. Many of the fundamental concepts of the current preferred scenario of visible clumpy matter (such as stars, galaxies, clusters) formation are contained in a model firstly proposed by White & Rees in 1978. According to his two-stage theory, dark halos form first, and then luminous dense matter is formed inside the gravitational potential wells these pre-existing halos provide. The gravitational potential wells are the manufacture where known dense matter form and evolve. The most dense matter originate at the center of the biggest dark matter halos [White and Rees, 1978]. Let’s denote each halo by U. Let’s call it: “universe”. U can be an atom or smaller. It can be a solar system, a galaxy cluster. It can be our vast universe or bigger. Let’s begin this note by recalling the new “Zaman” model proposed by Saoussan Kallel [Kallel-Jallouli, 2021a-d], with a generalization to the differential rotation case. In section 3-5, we study a special case of differential rotation. We deduce for that special case the “Hubble’s law”, without any need for any supplementary condition. In section 6, we establish the known relation between the redshift z and the distance from the central origin. In Section 7, we explain how attraction or repulsion inside a halo are related to angular velocity gradients. Section 8 is left for discussions and conclusions.

Zaman Dark Matter Haloes

Recently, Saoussan Kallel proposed a new geometrical model to solve the problem of “Dark Matter” [Kallel-Jallouli, 2021 a-d]. She proved the existence of an unseen matter named “Zaman” responsible by its spin for “time” variations. The proposed model offers a solution for the dark matter enigma, explains how theoretical and observed velocities can be different if the observer and the observed particle does not belong to similar universes [Kallel-Jallouli, 2021d]. The simple case studied by the author, proposed a relation between U-spin and U-time inside a solid body rotating spherical Zaman halo (or DM halo). In reality, to be more realistic, we need to have a differential rotation, where the length T(r) of the U-day depends on the radius r, for each shell inside U. We just need to generalize the definition presented by Kallel-Jallouli [2021 a-d]

Definition of U-day inside U for a solid body rotation

Let us choose the spherical coordinate system (r,θ,φ), given by figure 1

Figure 1. A. Spherical Coordinates. B. Isotime-discs [Kallel-Jallouli, 2021 a-c]

U-time will be the same for a solid body rotation over each semidisc limited by the axis of rotation and a meridian (figure 1). U is spinning with respect to its non spinning U_I state.

Let us select the isotime-disc enclosed in the plane \( (Oxz) \) as the semi-disc of U-time 0 \( (t_U \equiv 0) \). This is known as the U-prime meridian. Any point \( P \) in UI with coordinates \( (r, \theta, \phi) \) concurrently indicates the space position and U-time variation. U-time for the first day is given by:

\[
t_U = T – \frac{T}{2\pi} \theta = T\left(1 – \frac{\theta}{2\pi}\right)
\]

For the \( n \)-th day, U-time is:

\[
t_U = nT – \frac{T}{2\pi} \theta \quad \text{for } (n-1)T \leq t_U \leq nT
\]

Differential Rotation for Spherical Shells

For a sphere \( U \) with differential rotation, the U-time for each spherical shell of radius \( R \) is given by:

\[
t_{UR} = nT(R) – \frac{T(R)}{2\pi} \theta = T(R) \left( n – \frac{\theta}{2\pi} \right)
\]

The above relation can also be expressed as:

\[
\frac{\theta}{2\pi} = \text{Int}\left(\frac{t_{UR}}{T(R)}\right) + 1 – \frac{t_{UR}}{T(R)}
\]

Special Case of U-Differential Rotation

Suppose there exists a radius \( R_i < R_0 \) where solid-body rotation occurs. The length of the day \( T \) does not depend on radius \( R \) in this region:

\[
T(R) = T_0
\]

For \( R > R_0 \), differential rotation follows the Keplerian relation:

\[
T^2(R) R^3 = k
\]

or equivalently:

\[
T(R) = k R^{3/2}
\]

The rotational velocity for \( R \geq R_0 \) is:

\[
V_{\text{rot}}(R) = \frac{2\pi R}{T(R)} = \frac{2\pi}{k^{1/2} R^{1/2}} \quad \propto \frac{1}{R^{1/2}}
\]

Fig. 2. Rotational velocity curve

Dynamical Time and Acceleration

The length \( T(R) \) of the day, called dynamical time, depends on radius \( R \). By differentiating the Keplerian relation:

\[
R’ = \frac{2}{3} k^{-1/3} t_U^{-1/3}
\]

The radial acceleration is given by:

\[
R” = -\frac{2}{9} k^{-1/3} t_U^{-4/3}
\]

Thus:

\[
q_0 = -\frac{R” R}{R’^2} = \frac{1}{2} > 0
\]

Hubble’s Law

For particles in Keplerian orbits, the radial velocity follows:

\[
V_r = R’ = H R
\]

where \( H = \frac{2}{3} t_U^{-1} \) is the Hubble constant. This matches Hubble’s law, indicating that particles recede from the center with velocity proportional to their distance from it.

Doppler Effect and Redshift

The rotational Doppler shift is given by:

\[
\Delta f = \frac{\Omega J}{2\pi} = \frac{\Omega (\sigma + l)}{2\pi}
\]

where \( \Omega \) is the angular velocity, \( \sigma \) is spin angular momentum, and \( l \) is orbital angular momentum. For the linear Doppler effect, the observed frequency is:

\[
\nu_{\text{obs}} = \left(1 – \frac{V_r \cdot r}{c}\right) \nu_e
\]

The redshift parameter is defined as:

\[
z = \frac{\nu_e – \nu_{\text{obs}}}{\nu_{\text{obs}}}
\]

Combining both effects, the redshift becomes:

\[
z \approx \frac{H}{c} R
\]

Gravitational Effects and Dark Energy

For a shell radius \( R \), if the length of the day follows \( T = f(R) \), the radial velocity is:

\[
V_r = f(R) f'(R) t_U^{-1}
\]

If \( f'(R) < 0 \), particles experience gravitational attraction (dark matter). If \( f'(R) > 0 \), they experience repulsion (dark energy).

Dark Matter Halos and Regions

An isolated dark matter halo can be divided into three regions:

1. Inner core: Hydrostatic sphere with constant density and no radial or circular velocity.
2. Accretion shell: Region with \( V_r < 0 \), where material falls toward the core.
3. Outflow region: Material with \( V_r > 0 \), expelled from the halo.

Nearly all of the material in region (i) remains bound to the halo. In region (ii), the material feels a gravitational attraction and falls to reach region (i). while essentially all material in region (iii) feels a gravitational repulsion and is expelled out”.

The infalling zone (ii) can miss for certain halos [Diemer & Kravtsov, 2014] [Cuesta et al. 2008]

DISCUSSION AND CONCLUSIONS

In our present study, we used the Zaman Dark Matter solution presented in previous studies [Kallel-Jallouli, 2021a-d, 2024a,b]. We have just generalized the previous definition of time inside a halo, from the case of solid body rotation, to the differential rotation case. Moreover, if we suppose each spherical shell of radius R inside the halo U has a length T(R) of the day related to R by the Keplerian relation (3.2), then, using our definition of time related to phase (2.3), adopted from [Kallel-Jallouli, 2021a-d, 2023, 2024a,b], we proved that the Hubble’s law (5.4)(5.5) is valid for every test particle inside the third zone of U. The halo U can be infinitely small. It can be infinitely big. The diameter of U does not matter. Any particle placed inside U, at a distance R from the center so that the Keplerian relation (3.2) is satisfied, experiences a repulsive force from the center. Moreover, relation (3.2) implies that the length of the day gets longer, farther from the center. So, time gets more stretched (time delay, aging less), farther from the center. If we accept the fact that particles try to move from “more aging” to “less aging”, we also find escaping particles.

If we suppose the length T(R) of the day gets longer, as R gets smaller (T(R) is a decreasing function of R), then, time gets more stretched (time delay, aging less), nearer to the center. If we accept that particles try to move from “more aging” to “less aging”, we also find attracted particles by the halo center as we explicitly found by expression (7.3).

We can then conclude, when U-shells spin faster farther from the origin, or equivalently, the length T(R) of the U-day decreases with R, then any particle placed at a distance R from the origin will feel an attraction to the central point, known as gravity.

In the contrary, when U-shells spin slower farther from the origin, or equivalently, the length T(R) of the U-day increases with respect to R, then, any particle placed at a distance R from the origin will feel a repulsive force from the central point, known us Dark energy effect.

Dark energy and Dark matter are manifestations of the U-spin gradients of Zaman matter. A test particle inside U moves from a fast spinning shell (corresponding to a short day) to a slower-spinning one (corresponding to a longer day, aging less).

The rate of rotation of a red giant seems to remain constant within the core and gradually decreases from the edge of the helium core through the hydrogen-burning shell as the radius increases [Schou et al., 1998][Di Mauro, 2016]. If the Zaman rate of rotation inside the hydrogen-burning shell decreases as the radius increases, similar to matter, then, the outer layers of the star will be expelled and move away from the core, to form a planetary Nebula.

If inside the core, there is an outer shell where Zaman rate of rotation increases as the radius increases, then, the core will begin to collapse. We can then get two remnants: its envelop and its core.

Using the three main regions composing an isolated halo [Gunn & Gott 1972], we can deduce that the rate of rotation  of the shell of radius R depends on R:

1. Ω(R)=0, for R≤r_hs. the length T(R) of the U-day is ∞, for R≤r_hs
2. Ω(R) is increasing for r_hs≤R≤r_ta. T(R) decreases for r_hs≤R≤r_1, reaches its minimum T_min (Ω(R) reaches its maximum Ω_max) at
3. R=r_1 to remain constant till r_ta Ω(R) is decreasing for R≥r_ta . Moreover, T(R) satisfies the Keplerian relation (3.2).

We shall explain in a future work how to choose a mean length T_m of the U-day, T_m=T_min as length of the U-day, and the correspondingly hour equal to 15/360 T_m/T_c , for the physical laws inside U to be universal.

We shall explain in a future work how to choose a mean length   of the U-day,  as length of the U-day, and the correspondingly hour equal to  , for the physical laws inside U to be universal.

Our studied halo U is an isolated one. This case is far from being an ideal case in reality. In fact, about all known haloes are not isolated, they are subhaloes of bigger ones. Even our universe, as explained in the new Big Bang theory [Kallel-Jallouli, 2018, 2024c], can be seen as a subhalo of a bigger one named “Feluc”. Then, U will be deformed by the “gravitational” effect created by the bigger halo inside which the subhalo lives [Kallel-Jallouli, 2024d]. Since radial velocity depends on the position inside U, farther subhaloes from the halo center will be more deformed.

Hubble demonstrated that there is a linear relationship between the recession velocities and distances of galaxies [Hubble, 1929] [Hubble & Humason 1931]. This law has been used by some scientists, erroneously, as an evidence for an expanding Universe. Yet, the classical non-expanding Euclidian (which differs from the traditionally used Einstein–de Sitter static model often used in literature) fits most data, by only espousing a linear Hubble relation (5.4) at all redshifts z [Ellis, 1978][ Harnett, 2011][Crawford, 2011][ Lerner 2006, 2009][Lerner et al., 2014], which is a consequence of the Keplerian relation (3.2), between the radius of a shell and its rate of rotation.

Funding

• No funding was received to assist with the preparation of this manuscript.

Non-financial interests: None.

Nothing to declare:

• The author have no competing interests to declare that are relevant to the content of this article.

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