Phase Shift of the Keplerian Orbits inside Dark Matter Halos

Phase Shift of the Keplerian Orbits inside Dark Matter Halos

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.2024.11110038

Received: 07 October 2024; Accepted: 05 November 2024; Published: 06 December 2024

ABSTRACT

A new theory proposed the existence of a new kind of unseen staff named “Zaman”, responsible for the flow of time by its spin. Zaman solves the Dark Matter enigma. Moreover, for an isolated Zaman spherical halo U with differential rotation, if the relation between the radius of each shell R and the length T of its day is Keplerian; that is T²/R³ is constant, then, we proved the known linear Hubble relation is satisfied. In this work, we shall see how some distances can be elongated. We shall prove the Keplerian trajectories are discontinuous. More precisely, after each period, any test particle inside U will receive an instantaneous kick-off to be shifted from its initial orbital phase.

Keywords: Hubble constant, Kepler, Time, phase-shift, Velocity, Zaman Dark Matter

PACS : 95.35.+d, 98.80.Es, 06.30.Ft, 06.30.Gv, 42.87.Bg

INTRODUCTION

Recently, a new model was suggested to solve the enigma of Dark Matter (DM) [Kallel-Jallouli, 2018, 2021a,b,c,d]. Kallel proved the existence of an unusual unseen staff, called Zaman, responsible for the flow of time by its spin. The proposed model offers a solution for the dark matter puzzle. Kallel firstly studied the simple case of a solid body rotating spherical Zaman [Kallel-Jallouli, 2023, 2024c]. She explained the relation between \( U \)-spin and \( U \)-time and aging [Kallel-Jallouli, 2024d].
In reality, solid body rotation is a very special case. In most cases, we also have differential rotation, where, for each shell of radius \( r \) inside \( U \), the length \( T(r) \) of its \( U \)-day depends on \( r \) [Kallel-Jallouli, 2024a].

In the special case when \( \frac{T^2(r)}{r^3} \) is constant, Kallel deduced the Hubble law, without any need for any supplementary hypothesis or any special non-Euclidean geometry, as proposed by Lerner et al. [2014] for their observations to be matched with the laws of nature.
In this work, we shall prove the discontinuity of the Keplerian orbits. More precisely, after each cycle, any test particle receives a kick-off to be shifted from its previous phase position.

Time and Phase RelationFor a Zaman sphere \( U \) with solid body rotation with respect to a non-rotating state \( U_I \), let us select the isotime-disc enclosed in the plane \( (O_{xz}) \) as the semidisc of \( U \)-time 0 (see [Kallel-Jallouli, 2021b, fig. 2]). Any point \( P \) in \( U_I \) with coordinates \( (r, \theta, \phi) \) indicates the space position \( (r, \theta, \phi) \) and the \( U \)-time variation inside \( U \). For the first day, we have the relation [Kallel-Jallouli, 2021a,b,c]:\[
t_U = T – \frac{T}{2\pi} \theta = T \left( 1 – \frac{\theta}{2\pi} \right) \quad \text{(from 0 to T)}. \tag{1}
\]For the \( n \)th day, \( U \)-time is given by:\[
t_U = nT – \frac{T}{2\pi} \theta = T \left( n – \frac{\theta}{2\pi} \right) \quad \text{(between (n-1)T and nT)}. \tag{2}
\]More generally, for a Zaman sphere \( U \) with differential rotation, if we suppose time is universal inside \( U \) and the \( U \)-time \( t_U(R) \) does not depend on \( R \), then, for each spherical shell of radius \( R \) inside \( U \), the \( n \)th day, \( U \)-time \( t_U \) is provided by [Kallel-Jallouli, 2024a]:\[
t_U = nT(R) – \frac{T(R)}{2\pi} \theta = T(R) \left( n – \frac{\theta(R)}{2\pi} \right) \quad \text{(between (n-1)T(R) and nT(R))}. \tag{3}
\]\[
n = \text{Int}\left[\frac{t_U}{T(R)} + 1\right]. \tag{4}
\]The relation (3)(4) can also be written as:\[
\frac{\theta(R)}{2\pi} = \text{Int}\left[\frac{t_U}{T(R)}\right] + 1 – \frac{t_U}{T(R)}. \tag{5}
\]

Special case: Keplerian orbits inside an isolated Zaman DM halo U

For the special case of the differential rotation, where, for each shell of radius \( R \geq R_0 \), the length \( T(R) \) of the \( U \)-day satisfies the Keplerian relation between the radius \( R \) and the period \( T \) [Kallel-Jallouli, 2024a]:

\[
\frac{T^2}{R^3} = k, \tag{6}
\]

where \( k = k(U) \) is a fixed constant. In this case, the test particles move in Keplerian orbits. We can call our system \( U \): an “outer Keplerian universe” (OKU).

Hubble’s Law

If the relation (6) is satisfied, it was shown that the radial velocity \( V_r \) of each test particle, placed at a distance \( R \geq R_0 \) from the central origin, is given, almost everywhere, by the classical Hubble’s law [Kallel-Jallouli, 2024a]:

\[
V_r = \frac{dR}{dt_U} = R’ = H(t_U)R. \tag{7}
\]

The relation (7) is valid for \( t_U \in \bigcup_{n>1} \left] (n-1)\sqrt{kR^3}, n\sqrt{kR^3} \right[ \). The Hubble constant is given by:

\[
H(t_U) = \frac{2}{3t_U}, \tag{8}
\]

which is inversely proportional to the age \( t_U \) of the system \( U \).

Examples

Our universe \( U \) is a good example of (OKU). The Hubble law (7) describes the relationship between a galaxy’s radial velocity and its distance from us. Nevertheless, as Kallel explained [Kallel-Jallouli, 2018], our universe \( U \) is not isolated. Therefore, depending on the impact of the container, the measured Hubble constant can vary [Perivolaropoulos, 2014].Using the approximate value of \( 74 \, \text{km s}^{-1} \text{Mpc}^{-1} \) for \( U \), we can deduce from (8):\[
t_{MW} = \frac{2}{3} \frac{1}{74} \approx 8.5 \, \text{billion years}.
\]

Hubble Tension

We saw how the Hubble constant \( H_U \) depends on the OKU \( U \). Consequently, if we chose different subhaloes \( U_S \), there is no reason for their Hubble constant \( H_{U_S} \) to be equal, if their birth dates are different.Now, we shall use the idea proposed by Kallel [Kallel-Jallouli, 2018], where our universe was born inside the crust of a huge tectonic iceball, named “Feluc,” and is growing from sublimation of the surrounding ice.The recession celerity \( a'(t) \) of the surface of last scattering (related to ice sublimation) remains imprinted in the CMB (Doppler effect) [Planck Collaboration, 2016]. Our universe growth rate \( H_{SLS} \), defined by:

\[
H_{SLS}(t) = \frac{a'(t)}{a(t)}, \tag{9}
\]

can be deduced after taking into account the redshift of photons as they climb out of potential wells to reach us. We need then the use of the Hubble constant \( H_U \) deduced from (7) [Clifton et al. 2012]. The latest estimate of the growth rate \( H_{SLS} \) based on CMB observations by the Planck satellite is \( H_{SLS} = 67.4 \pm 0.5 \, \text{km s}^{-1} \, \text{Mpc}^{-1} \) [Planck Collaboration 2020].

We have two completely different constants: (i) The Hubble constant \( H_U \) (Dark energy manifestation), related to the Kepler’s law (6) and the consequent recessing velocities (7). (ii) The growth rate \( H_{SLS} \) (ice sublimation manifestation), related to the SLS recession velocity (9).

We cannot speak about “\( H_0 \) Tension”. To find the Hubble constant \( H_U \), we only need to search
for the rate of recession of faraway galaxies. However, to find the growth rate \( H_{SLS} \), since
CMB traverses through a radius of the universe to reach us, much work is needed. We need first to calculate \( H_U \).

Distance Elongation

How do we see a disc of radius \( r \) placed in the equator, inside \( U \), at a universal time \( t_U = t_0 \), with the center \( O’ \) placed at a distance \( R > r \) from the central halo origin \( O \)? Suppose the elapsed time \( t_U \) inside \( U \) does not change with \( R \) (time is universal). Since at time \( t_0 \) the least distant point \( P_{\min} \) is at a distance \( (R – r) \) from the center \( O \), and the most distant point \( P_{\max} \) is at a distance \( (R + r) \) from the center, then, by (7), the radial velocity of \( P_{\min} \) at time \( t \) is \( H(t)(R – r) \), while that of \( P_{\max} \) is \( H(t)(R + r) \).

After a small time \( t_1 \), \( P_{\min} \) reaches the position \( P’_{\min} \) so that:

\[
P’_{\min} P_{\min} = \int_{t_0}^{t_0 + t_1} H(s)(R – r) \, ds \tag{10}
\]

And \( P_{\max} \) reaches the position \( P’_{\max} \) so that:

\[
P_{\max} P’_{\max} = \int_{t_0}^{t_0 + t_1} H(s)(R + r) \, ds \tag{11}
\]

Then the new diameter \( P’_{\min} P’_{\max} \), can be expressed with respect to \( r = P_{\min} P_{\max} \) via:

\[
P’_{\max} P’_{\min} = P’_{\max} P_{\max} + P_{\max} P_{\min} – P_{\min} P’_{\min}
= \int_{t_0}^{t_0 + t_1} H(s)(R + r) \, ds + 2r – \int_{t_0}^{t_0 + t_1} H(s)(R – r) \, ds
\]

\[
= 2r \left( 1 + \int_{t_0}^{t_0 + t_1} H(s) \, ds \right)
\]

\[
P’_{\min} P’_{\max} = 2r \left( 1 + \frac{2}{3} \ln \frac{t_0 + t_1}{t_0} \right) \tag{12}
\]

After \( t_1 \) time, the diameter \( 2r \) in the line of sight (aligned with the central origin \( O \)) will
be stretched by the amount:

\[
1 + \frac{2}{3} \ln \frac{t_0 + t_1}{t_0}, \tag{13}
\]

that depends on the elapsed time \( t_1 \) and the age \( t_0 \) of \( U \), when the disc was placed (born)
inside \( U \).

Since we know, the rotational velocity is given by [Kallel-Jallouli, 2024a]:

\[
V_{\text{rot}}(R) = \frac{2\pi R}{T} = \frac{2\pi R}{\sqrt{k R^3}} = \frac{2\pi}{\sqrt{k} R}, \tag{14}
\]

then, the diameter 2r, perpendicular to the line of sight (from the central origin O) remains unchanged. So, by relation (13), more the elapsed time t_1 is higher, more our circle will be elongated in the line of sight (from the origin O), and the eccentricity of our seen “deformed ellipse” will be higher. Consequently, if we place a small subhalo U’ inside a big halo U, far from the center O of U, we have to expect the circular Keplerian orbits inside U’ to become elongated (deformed elliptical trajectories). The elongation is bigger for higher periods (see relation (13)) and then for farther objects from the U’ center. The elongation will be at its maximum (maximum eccentricity) when the plane of the orbit is aligned with O, and the period is high. An orbit perpendicular to the line of sight from O remains circular.
Our seen universe U’ is a subuniverse of a bigger huge tectonic Old-iceball U called Feluc, as proposed by Kallel in her new Big Bang theory [Kallel-Jallouli S, 2018, 2024b]. We can then explain the strongly flattened high-redshift galaxies [Ceverino et al., 2015], confirmed by NASA’s James Webb Space Telescope (JWST) [Pandya et al. 2024] (fig.1) and the absence of any flattened or elongated shapes near our galaxy.

Fig. 1 A distant galaxy identified in Webb’s CEERS Survey (NIRCam Image) [https://webbtelescope.org/contents/news-releases/2024/news-2024-104#section-id-2]

Discontinuity of Trajectories and Phase Shift

We are always interested in an isolated halo \( U \). Let’s consider a test particle \( P \) in a Keplerian circular orbit in the equatorial plane around the origin, with a period \( T_1 \). If at the universal time \( t_U = t_0 \), \( P \) is placed (born) at a distance \( R_1 \) from the origin, with an initial phase \( \theta_{1,0} \), then, at each instant \( t < T_1 \), while \( P \) advances in its orbit around the origin, its distance \( R_1(t) \) from the origin will be stretched(13) via:

\[
R_1(t) = R_1 \left( 1 + \frac{2}{3} \ln \frac{t_U + t}{t_U} \right). \tag{15}
\]

After one revolution (\( T_1 \) time), \( P \) will tend to be at a farther distance \( R_2 \) from the origin,
satisfying:

\[
R_2 = R_1 \left( 1 + \frac{2}{3} \ln \frac{t_0 + T_1}{t_0} \right). \tag{16}
\]

Remarks

1. For the special case when we suppose the period \( T_1 \) equals the age \( t_0 \) of \( U \), when \( P \) (was born and) begun its revolution:

\[
T_1 = t_0, \tag{17}
\]

Then, we get:

\[
R_2 = R_1 \left( 1 + \frac{2}{3} \ln 2 \right). \tag{18}
\]

Reciprocally, if we suppose we have the relation (18), then, the period \( T_1 \) is given by (17).

2. If the universe \( U \) contains a massive spinning particle \( P \) in Keplerian orbits, then, since the particle is receding from the halo center, \( P \) gets farther from the halo center. The day becomes longer (time delay) [Kallel-Jallouli, 2024a], leading to an apparent decrease in the rotational speed of the particle, as measured by fixed clocks. If we do not care about the change in the physical “Zaman time”, there will appear to be no conservation of angular momentum!

For \( 0 < t < T_1 \), the phase \( \theta \) satisfies using (5):

\[
\frac{\theta}{2\pi} = \text{Int}\left[\frac{t + t_0}{T_1} + 1\right] – \frac{t + t_0}{T_1}. \tag{19}
\]

At \( t = 0 \), the beginning phase \( \theta_{1,0} \) satisfies, using (5):

\[
\frac{\theta_{1,0}}{2\pi} = \text{Int}\left[\frac{t_0}{T_1} + 1\right] – \frac{t_0}{T_1}. \tag{20}
\]

When \( P \) tends to complete its first revolution, the quantity \( \frac{\theta}{2\pi} \) tends to:

\[
\frac{\theta_{1,f}}{2\pi} = \text{Int}\left[\frac{T_1 + t_0}{T_1} + 1\right] – \frac{T_1 + t_0}{T_1} = \text{Int}\left[\frac{t_0}{T_1} + 1\right] – \frac{t_0}{T_1} = \frac{\theta_{1,0}}{2\pi}. \tag{21}
\]

And the phase tends to \( \theta_{1,0} \). The particle tends to complete one full revolution, with an open orbit.

Unfortunately, the second revolution will have a new period \( T_2 > T_1 \), since the starting new radius \( R_{2,0} \) satisfies (16):

\[
R_{2,0} = R_1 \left( 1 + \frac{2}{3} \ln \frac{t_0 + T_1}{t_0} \right) > R_1. \tag{22}
\]

And using the relation (6), we have:

\[
\frac{T_2^2}{R_{2,0}^3} = \frac{T_1^2}{R_1^3}. \tag{23}
\]

By replacing \( R_{2,0} \) using (22) inside (23), we get:

\[
T_2 = T_1 \left( 1 + \frac{2}{3} \ln \frac{t_0 + T_1}{t_0} \right)^{3/2} > T_1. \tag{24}
\]

\( P \) will restart its second revolution at the new universal time \( T_1 + t_0 \). Unfortunately, the new starting phase \( \theta_{2,0} \) must satisfy the law (5) of its relation to the period:

\[
\frac{\theta_{2,0}}{2\pi} = \text{Int}\left[\frac{T_1 + t_0}{T_2}\right] + 1 – \frac{T_1 + t_0}{T_2}
\neq \frac{\theta_{1,f}}{2\pi} = \text{Int}\left[\frac{t_0}{T_1} + 1\right] – \frac{t_0}{T_1}. \tag{25}
\]

This means that at the universal time \( T_1 + t_0 \), \( P \) will receive a kick-off to change its phase position instantaneously; instead of starting its novel trajectory from the final phase \( \theta_{1,f} = \theta_{1,0} \), \( P \) will jump to restart generally at a different phase \( \theta_{2,0} \neq \theta_{1,0} \), with a starting radius \( R_{2,0} \) given by (22).

Moreover, since the function \( f \) defined for \( x > 0 \), by \( f(x) = \text{Int}[x] – x \), is strictly increasing for \( x \in ]0, 1[ \), and strictly decreasing for \( x > 1 \), then, for \( T_2 < T_1 + t_0 \), we get \( \theta_{2,0} > \theta_{1,f} \), and \( P \) will shift to an advanced phase, instantaneously.

For the second revolution, the starting new universal time is \( t_U = t_0 + T_1 \). For \( 0 < t < T_2 \), the radius \( R_{2,0} \) will be stretched. Using (15)(22), we can write:

\[
R_2(t) = R_{2,0} \left( 1 + \frac{2}{3} \ln \frac{t_0 + T_1 + t}{t_0 + T_1} \right) = R_1 \left( 1 + \frac{2}{3} \ln \frac{t_0 + T_1}{t_0} \right) \left( 1 + \frac{2}{3} \ln \frac{t_0 + T_1 + t}{t_0 + T_1} \right). \tag{26}
\]

\( P \) will continue its third revolution, at the new universal time \( t_U = t_0 + T_1 + T_2 \), with a new orbital period \( T_3 > T_2 \), and a new starting phase \( \theta_{3,0} \), shifted from the previous starting phase \( \theta_{2,0} \).

For \( 0 < t < T_3 \), the distance \( R_3(t) \) is given by (using (15)):

\[
R_3(t) = R_{3,0} \left( 1 + \frac{2}{3} \ln \frac{t_0 + T_1 + T_2 + t}{t_0 + T_1 + T_2} \right). \tag{27}
\]

with

\[
R_{3,0} = R_{2,0} \left( 1 + \frac{2}{3} \ln \frac{t_0 + T_1 + T_2}{t_0 + T_1} \right). \tag{28}
\]

At its \( n \)-th revolution (\( n > 2 \)), the departure of \( P \) will start at the universal time \( t_U = t_0 + \sum_{i=1}^{n-1} T_i \), from the starting phase \( \theta_{n,0} \), shifted from the previous one \( \theta_{n-1,0} \). Moreover, the new period \( T_n \) is greater than the previous one \( T_{n-1} \). The new period \( T_n \) satisfies the recurrent relation:

\[
T_n^2 = T_{n-1}^2 \left( 1 + \frac{2}{3} \ln \frac{t_0 + T_1 + \cdots + T_{n-1}}{t_0 + T_1 + \cdots + T_{n-2}} \right)^3. \tag{29}
\]

The starting phase \( \theta_{n,0} \) satisfies:

\[
\frac{\theta_{n,0}}{2\pi} = \text{Int}\left[\frac{t_0 + T_1 + \cdots + T_{n-1}}{T_n}\right] + 1
– \frac{t_0 + T_1 + \cdots + T_{n-1}}{T_n}. \tag{30}
\]

For \( 0 < t < T_n \), \( P \) will be at a distance:

\[
R_n(t) = R_{n,0} \left( 1 + \frac{2}{3} \ln \frac{t_0 + T_1 + \cdots + T_{n-1} + t}{t_0 + T_1 + \cdots + T_{n-1}} \right). \tag{31}
\]

The starting distance \( R_{n,0} \) at the universal time \( t_0 + \sum_{i=1}^{n-1} T_i \) is given by the recurrent relation:

\[
R_{n,0} = R_{n-1,0} \left( 1 + \frac{2}{3} \ln \frac{t_0 + \sum_{i=1}^{n-1} T_i}{t_0 + \sum_{i=1}^{n-2} T_i} \right). \tag{32}
\]

Evidently, this spontaneous phase shift, after each revolution, must be accompanied by spontaneous energy exchange. The spontaneous energy release is known in the case of electrons inside an atom. In the case of astronomical haloes, matter ejection was detected to propagate out of the orbital plane of a compact object. A kick can accelerate the rotation of a star by transferring angular momentum to its crust.

The \( n \)-th Scale Factor \( a_n \)

If at the universal time \( t_U = t_0 \), a test particle \( P \) is placed (or born) at a distance \( R_1 \) (with period \( T_1 \) satisfying relation (6)) from the origin, with phase position \( \theta_{1,0} \), then, after \( t’ \) time, the particle will be at a distance \( R(t’) \) from the origin, that can be written in the form:

\[
R(t’) = R_1 a_n(t_0, T_1, t’). \tag{33}
\]

With, for the first period (\( n = 1 \)), \( t’ = t < T_1 \) (see relation (15)):

\[
a_1(t_0, T_1, t) = \left( 1 + \frac{2}{3} \ln \frac{t_0 + t}{t_0} \right) = a_1(t_0, t). \tag{34}
\]

We call the factor \( a_1(t_0, t) \) the “first scale factor.” It depends on the universal time \( t_0 \), and the time \( t \). For any particle \( P \) placed (or born) at a distance \( R_1 \), the “first scale factor” satisfies:

\[
1 \leq a_1(t_0, t) \leq a_1(t_0, T_1). \tag{35}
\]

If we differentiate the first scale factor with respect to time, we obtain the known relation:

\[
\frac{d a_1(t_0, t)}{dt} = \frac{2}{3} \frac{1}{t_0 + t} = H(t_U), \quad \text{for } t < T_1. \tag{36}
\]

The second derivative:

\[
\frac{d^2 a_1(t_0, t)}{dt^2} = -\frac{2}{3} \frac{1}{(t_0 + t)^2} = -\frac{3}{2} H(t_U)^2, \quad \text{for } t < T_1. \tag{37}
\]

The deceleration parameter \( q_0 \) given by:

\[
q_0 = -\frac{a_1”}{a_1 (a_1′)^2} > 0, \tag{38}
\]

is clearly positive.

For the second period (\( n = 2 \)), as we can deduce from (26), if \( T_1 < t’ = T_1 + t < T_1 + T_2 \), then:

\[
a_2(t_0, T_1, t’) = a_1(t_0, T_1) \left( 1 + \frac{2}{3} \ln \frac{t_0 + T_1 + t}{t_0 + T_1} \right)
= a_1(t_0, T_1) a_1(t_0 + T_1, t). \tag{39}
\]

The second scale factor \( a_2(t_0, T_1, t’) \) depends on the universal time \( t_0 \), when the particle \( P \) was at a distance \( R_1 \) from the central origin. Moreover, it also depends on \( R_1 \) (position of \( P \)) (or equivalently on \( T_1 \)), and then on the redshift \( z_1 \) (\( z_1 = H/c.R_1 \), see [Kallel-Jallouli, 2024a]). For any particle \( P \) placed (or born) at a distance \( R_1 \), the “second scale factor” satisfies:

\[
a_1(t_0, T_1) \leq a_2(t_0, T_1, T_1 + t) \leq a_1(t_0, T_1) a_1(t_0 + T_1, T_2). \tag{40}
\]

Differentiating (39) with respect to time, we get for \( T_1 < t’ = T_1 + t < T_1 + T_2 \):

\[
\frac{da_2(t_0, T_1, t’)}{dt} = \frac{2}{3} a_1(t_0, T_1) \frac{1}{t_0 + T_0 + t} = a_1(t_0, T_1) H(t_U). \tag{41}
\]

For the \( n \)-th period (\( n > 2 \)), if \( \sum_{i=1}^{n-1} T_i \leq t’ = t + \sum_{i=1}^{n-1} T_i < \sum_{i=1}^n T_i \), then:

\[
a_n(t_0, T_1, t’) = a_1(t_0 + T_1 + \cdots + T_{n-1}, t) a_1(t_0, T_1) \prod_{i=1}^{n-2} a_1(t_0 + T_1 + \cdots + T_i, T_{i+1}). \tag{42}
\]

This \( n \)-th scale factor \( a_n \) depends on the universal time \( t_0 \), when the particle \( P \) was at a distance \( R_1 \) from the central origin. Moreover, it also depends on the beginning place of each orbit (\( R_j \)) (\( j \leq n-1 \)) (or equivalently on \( T_j \)), then on the redshifts (\( z_j \)) (\( j \leq n-1 \)) (\( z_j = H(t_0 + \sum_{i=1}^{j-1} T_i) / c.R_j \), see [Kallel-Jallouli, 2024a]).

For any particle \( P \) placed (or born) at a distance \( R_1 \), the “\( n \)-th scale factor” satisfies:

\[
a_1(t_0, T_1) \prod_{i=1}^{n-2} a_1(t_0 + T_1 + \cdots + T_i, T_{i+1}) \leq a_n(t_0, T_1, \sum_{i=1}^{n-1} T_i + t)
\leq a_1(t_0, T_1) \prod_{i=1}^{n-1} a_1(t_0 + T_1 + \cdots + T_i, T_{i+1}). \tag{43}
\]

As we can deduce from (42), the derivative of the \( n \)-th scale factor \( a_n(t_0, T_1, t’) \) is given by:

\[
a’_n(t_0, T_1, t’) = C_n(t_0, T_1) H(t_0 + T_0 + \cdots + T_{n-1} + t) = C_n(t_0, T_1) H(t_U). \tag{44}
\]

With \( C_n(t_0, T_1) \) as a constant satisfying the recurrence relation:

\[
C_{n+1}(t_0, T_1) = C_n(t_0, T_1) \left( 1 + \frac{2}{3} \ln \frac{t_0 + T_1 + \cdots + T_n}{t_0 + T_1 + \cdots + T_{n-1}} \right), \quad n > 2. \tag{45}
\]

With

\[
C_2(t_0, T_1) = \left( 1 + \frac{2}{3} \ln \frac{t_0 + T_1}{t_0} \right). \tag{46}
\]

We can see from (44), (45), and (46) that the derivative of the \( n \)-th scale factor, \( a’_n(t_0, T_1, t’) \), is equal to the Hubble constant, only for the case \( n = 1 \) (only for one cycle of a particle). Otherwise, we get a constant \( C_n(t_0, T_1) \) multiplied by the Hubble constant.

To get a good approximation to the Hubble constant in relation (44), \( C_n(t_0, T_1) \) must be close to 1, or equivalently, the time \( t’ \) must be negligible compared to \( t_0 \), i.e., \( t’ \ll t_0 \).

In practice, for example, in the case of our solar system \( U_S \), we can determine, experimentally, for each planet \( P \), the starting position \( R_1(P) \), and the final position \( R_2(P) \) after one complete revolution. Then, since \( k \) is known, we can deduce the period \( T_1 \) from the relation (6). Then, using the relation (33), we can calculate the first scale factor \( a_1(t_0, T_1) \) via:

\[
\frac{R_2}{R_1} = a_1(t_0, T_1) = \left( 1 + \frac{2}{3} \ln \frac{t_0 + T_1}{t_0} \right). \tag{47}
\]

We can then, from relation (47), deduce the age \( t_0 \) of the solar system \( U_S \), when \( P \) started its orbit at a distance \( R_1 \) from the sun, via the relation:

\[
t_0 = \frac{T_1}{e^{\frac{3}{2} \left( \frac{R_2}{R_1} – 1 \right)} – 1}. \tag{48}
\]

Since \( R_2(P) \) is known, we can deduce the period \( T_2 \) from the relation (6). We can also deduce the phase shift \( \theta_{2,0} \) after one orbit via (25).

So, the knowledge of the beginning distance \( R_1 \) from the central origin, and the final distance \( R_2 \), after one complete revolution, is sufficient to deduce: \( T_1 \) and \( T_2 \) via the Kepler relation (6), the age \( t_0 \) of the system \( U \) when the test particle was at \( R_1 \), via the relation (48), then the first scale factor \( a_1(T_1, t) \) via the relation (32). We can also deduce the phase shift \( \theta_{2,0} \) via the relation (25).

The age \( t_0 \) of the system \( U \) can also be deduced via the formula (25), if the phase shift \( \theta_{2,0} \) can be experimentally measured. Some corrections must be taken into consideration since our solar system is not lodged inside an isolated halo.

Some strange astronomical phenomena

Matter ejection from a star in orbit inside a microquasar

Microquasars are binary star systems consisting of a normal star and a compact object, typically a black hole or a neutron star. The material from the normal star is accreted onto the compact object, forming an accretion gaseous disk around it, called an accretion disk.

A complex process, within the accretion disk and between the disk and the compact object, can lead to the emission of intense X-rays and the ejection of material in the form of relativistic jets perpendicular to the plane of the disk at all times. In general, the disc is precessing along with the jets around the same axis and at the same angular frequency. The exact mechanism triggering these ejections and the timescales involved were not yet fully understood.

LS I +61◦303

LS I +61 303 constitutes a high-mass X-ray binary system, part of our own galaxy, the Milky Way. It is situated in the constellation Cassiopeia, which is visible from Earth. LS I +61o303 was located at a distance of 2.0 ± 0.2 kpc [Frail & Hjellming, 1991], but now, it is located at a distance of 2.65 ± 0.09 kpc [Weng et al., 2022].

LS I +61 303 is comprising a compact companion of uncertain identity, potentially a neutron star, more likely a pulsar according to Weng et al. (2022), or conceivably a black hole as suggested by Casares et al. (2005). The compact companion orbits a massive, early-type B0 Ve donor star, characterized by rapid rotation, on an eccentric orbit (e = 0.72 ± 0.15, as reported by Casares et al., 2005), while undergoing mass loss via a high-velocity stable equatorial disk, commonly referred to as an accretion disc [Casares et al. 2005]. The orbital period of the binary system, P = 26.4960 ± 0.0028 d [Gregory 2002], is slightly higher than the previous one 26.4917± 0.0025 [Gregory et al. 1999]. An increasing period by about 3 s per year for SS433 microquasar was identified [Cherepashchuk et al., 2021]

During the orbital motion, the compact object in LS I +61◦303, accreting material from the Be donor, undergoes a periodical (P1~26.49 days) ejection of relativistic outflow of material, at a particular orbital phase around periastron, forming two precessing (P2~26.9±0.1 d) conical jets.

The ejected cloud expands linearly along the direction of the rotation axis of the accretion disc, perpendicular to the accretion disk plane [Massi & Torricelli-Ciamponi, 2014] [Hjellming & Johnston 1988][Martí & Paredes 1995], giving rise to the expanding double radio source observed with radio interferometers [Massi, 1993]. Only one outburst per orbit was observed [Gregory, 2002][Abdo et al. 2009]. Synchrotron emission from relativistic particles refilling the jets, produces a maximum when the jet electron density is at its maximum [Massi et al. 2015].

The star is encompassed by a disc of accreting material, which rotates at a nearly Keplerian velocity [Chernyakova 2023]. The Keplerian velocity gradually decreases with distance from the star, reaching its minimum at the truncation radius of the disc. The rotation period at the disc truncation radius is [Chernyakova 2023]

The plasma clouds are ejected during the replenishment of the inner part of the accretion disk [Greiner et al. 1996], that follows its sudden disappearance beyond the last stable orbit around the compact companion (probably black hole). Only some fraction is propelled into synchrotron-emitting clouds of plasma .The collision of relativistic ejecta with environmental material has been observed in real time [Hjellming et al. 1998], where the leading edge of the jet decelerates while strongly brightening [Mirabel and Rodríguez,1999]. The size of the jet can be much greater than the orbital size. A tail length of about 6.5mas was detected [Dhawan et al. 2006]. Emitting regions are perpendicular to the jet. The γ-ray emitting region is assumed closer to the compact object than the observed radio jets [Kniffen et al. 1997]. There is a time lag between the start time of the ejection near periastron and the onset of the observed radio-outburst. In the case of LSI+61o303 the radio outbursts occurs about 10 days after the periastron passage. More generally, the onset of the outburst should be observed earlier at higher frequencies [Taylor & Gregory 1982]. The higher frequencies tend to peak earlier and fade more quickly than the lower frequencies. A mean time delay between the 8.3 GHz outburst and the 2.25 GHz outburst of 0.4 days was measured. There is an apparent lengthening of the measured period between outbursts for lower frequencies [Ray et al. 1997][Paredes et al. 1997].

The radio emission appears cometary, with a ‘tail’ pointing away from the high-mass star. the cometary head traces an erratic ellipse, of semi-major axis that can be about 4 times the binary axis [Dhawan et al. 2006].

The ejecta in quasars and microquasars move with relativistic speeds [Mirabel & Rodríguez 1999]. But in microquasars (the black hole is only a few solar masses) the particles ejected at relativistic speeds, from the black hole, can travel up to distances of a few light years only, instead of several million light years as in giant radio galaxies (multiple million solar masses black hole) [Mirabel & Rodríguez, 1998].

What mechanism may cause the disappearance of the inner accretion disk and the sudden matter ejection?

Figure 2: Orbit of the compact object travelling through the halo of the Be star. Radio outbursts in LS I +61◦303 occur around apastron passage (in yellow).

However, near periastron gamma-ray emission is observed (but no radio emission). The yellow curve is inspired from [Massi 2004].

Around periastron, a high-energy outburst was detected, but radio emission was absent [Abdo et al. 2009]

Why no radio bursts are observed at periastron passage? Why does the radio bursts occur only around apastron passage (fig. 2)?

The morphology varies dramatically near periastron, with outflow velocities ∼7500 km/s, and dimunishes gradually around the rest of the orbit, to be ∼1000 km/s near apastron [Dhawan et al. 2006]. A rotation of the inner structure and rapid changes in the orientation of the cometary tail at periastron were detected [Dhawan et al. 2006].

The emission peak intensity positions trace the place downstream in the tail. The peak intensity image is further displaced radially outward from the pulsar orbit, for longer wavelengths, and lags by days in time, compared to shorter wavelengths [Dhawan et al. 2006, figure 4].. Comparing 13 cm and 3.6 cm images, we can deduce the existence of synchrotron opacity gradient along the cometary tail, pointing away from the high-mass star, with the highest-energy particles located in the head [Dhawan et al. 2006].

GRS 1915+105

The microquasar GRS 1915+105 is a binary system, located in the galactic plane inside our Galaxy, in the constellation Aquila, at a distance of approximately 11–12 kpc [Mirabel & Rodríguez, 1994] [Fender et al., 1999]. The normal star is a K iii companion with a mass of \( 0.8 \pm 0.5 \, M_\odot \) [Greiner et al., 2001]. The compact object in GRS 1915+105 is a black hole (BH) with a mass \( M_{\text{BH}} = 12.4_{-1.8}^{+2} \, M_\odot \) [Reid et al., 2014], orbiting the center of mass of the system with a period of \( 33.5 \pm 1.5 \, \text{days} \) [Greiner et al., 2001].

The apparent position of the BH core position in the binary system GRS 1915+105 reveals some “jitter” along the jet axis at certain times [Reid et al. 2014], that can be related to the orbital motion. This jitter could be associated with the production of low-level jet emission [Reid et al. 2014], indicating that the processes driving the jets can influence the apparent BH core position. Despite the orbital motion, the predicted positional jitter due to the black hole’s orbit around the binary system’s center of mass is very small, approximately 0.004 milliarcseconds (mas) [Reid et al. 2014].

At regular intervals, the accretion disk, which surrounds the BH in GRS 1915+105, disrupts and throws off its inner portion. The abrupt vanishing of the inner accretion disk, detected in the X-rays, triggers the formation of relativistic expanding clouds of plasma observed at longer wavelengths [Mirabel et al. 1998] [Mirabel & Rodríguez 1999]. This disk of matter re-forms itself after each powerful jet, as the black hole pulls in more matter from its companion star. The black hole, in the constellation Aquila, throws off, periodically, the mass equal to that of a 100 trillion ton asteroid at nearly the speed of light approximately 650 million miles per hour. This process clearly requires a lot of energy, each cycle. The jet material must come from the inner disk [Greiner et al., 1996].

If we examine the source on a longer timescale, larger superluminal jets appear at the same trigger point, just much less often than the smaller ones. After the flare, we can see slower replenishment of the inner region of the accretion disk. From 1994 January-August data, the time separation between ejections for GRS 1915+105 suggests a quasi-periodicity at intervals in the range of 20-30 days [Rodriguez & Mirabel, 1999‏]. The kinetic energy of the plasma clouds suggests an unclear acceleration mechanism with very large power [Rodriguez et al. 1995]. The inner disk region empties and refills on timescales of seconds. The jets of GRS 1915+105 carry an important amount of energy away from the central source into the interstellar medium.

Pulsar glitches

It’s widely accepted in astrophysics that radio pulsars are rapidly rotating highly magnetized neutron stars formed as a result of supernova explosions from their progenitor stars. They initially have rapid rotation speeds, around 50 Hz, shortly after their formation. However, within a few hundred thousand years, they slow down to longer periods due to the braking torque exerted by their high magnetic field strengths (∼1012 G) [Bailes et al. 2011]. As pulsars rotate, they emit electromagnetic radiations in form of cones out of their magnetic poles tilted with respect to their rotation axes. As these beams sweep across space as the star rotates, much like the beam of a lighthouse, they create pulses of radiation that can be observed when they cross the observer’s line of sight. Once their rotation periods reduce to a few seconds, the majority of pulsars cease emitting radio waves [Bailes et al. 2011].

Pulsars rotation rates can be remarkably stable over long periods. They are used to be the most stable ’clocks’ in the universe. Even as these pulses remain unbelievably regular, pulsars gradually lose their rotational kinetic energy, leading to a smooth decrease in their spin rate over time, as rotational energy transforms into an electromagnetic outflow [Helfand et al. 2001].

In general, pulsar’s slow-down is smooth and continuous. However, the rotation of young pulsars, such as the Crab pulsar (B0531+21), is occasionally disturbed by discrete spin-up incidents known as glitches. Pulsar glitches are typically characterized by sudden rapid rises in rotation rate or spin frequency (∆ν) [Espinoza et al. 2011][Yu et al. 2013]. This discontinuous increase of the spin-frequency of a pulsar, or fractional decline in rotation period (∆P) is followed by a gradual return to the pre-glitch rotation rate over time.

PSR B2035+36

The pulsar PSR B2035+36 (J2037+3621) is an isolated relatively old radio pulsar discovered in 1985 [Dewey et al. 1985], with 0.6187 s period and -4.5024 × 10−15 s s−1 period derivative [Hobbs et al. 2004].

Figure 3. [Kou et al. 2018, fig.1) Variations of ν and ˙ν of PSR B2035+36. (a) Variations of frequency ν relative to the pre-glitch solutions; (b) variations of the first frequency derivatives. The dashed line is the glitch epoch of MJD 52950.

Around MJD 52950, PSR B2035+36 experienced a glitch (fig.3), characterized by a frequency jump of approximately ∆ν ∼ 12.4(5) nHz [Kou et al. 2018]. The observed large glitches were attributed to a sudden angular momentum transfer deposited into the solid crust causing it to spin-up [Zhao et al. 2017]. A sudden apparent change in a neutron star’s moment of inertia has been detected during a glitch event [Zhao et al. 2017].

PSR B0833-45

Glitches can potentially affect the position of a pulsar, although the effect would be very small and difficult to detect directly. The sky coordinates of the Vela pulsar PSR B0833-45 centroid exhibited sinusoidal wanderings [Helfand et al. 2001].

The two polar jets, seen in the Vela pulsar PSR B0833-45, must be supplied with particles from the pulsar [Helfand et al. 2001]. The energy input from a glitch must be deposited at the base of the crust within the star, the response of the star’s magnetosphere could result in the release of energy to diffuse outward [Helfand et al. 2001] [Kou et al. 2018]. The pulsar spin-down rate, after-glitch, implies an energy loss. What causes a glitch to occur?

Spruit & Phinney suggested that the rotation axes and space velocities of pulsars could be connected through the nature of the “kicks” given to neutron stars [Spruit & Phinney, 1998].

PSR B0919+06

The pulsar PSR B0919+06 , also known as PSR J0923+08, estimated to be located at a distance of about 1,200 light-years from Earth, is part of a binary system, its companion likely being a white dwarf star [Gong et al. 2018]. The orbital period of this binary system, which is the time it takes for the pulsar and its companion to complete one orbit around their common center of mass, is measured to be around 10 minutes.

PSR B0919+06 is characterized by its relatively strong radio emissions, featuring a rotational frequency of ν = 2.3 Hz and a deceleration rate of ν˙ = −7.4 × 10−14 Hz s−1 [Manchester et al. 1978].

An unusual spinning up, with an amplitude of approximately 10 milliseconds occurring over roughly 10-minute intervals (specifically 14 minutes), was initially detected in the singular pulsar PSR B0919+06 [Rankin et al. 2006]. The spin-up can be related to the presence of an orbiting celestial body [Wahl et al. 2016] [Gong et al. 2018].

A kick can accelerate the rotation of a star by transferring angular momentum to it. When we refer to a “kick” in the context of stellar dynamics, we typically mean a sudden increase in momentum, often caused by asymmetric supernova explosions or gravitational interactions in binary systems. This asymmetric ejection can cause the remnant core, such as a neutron star or a black hole, to receive a kick, leading to an increase in its rotational velocity about its own axis rather than its orbital motion around another object. If a kick accelerates the rotation of a star, it primarily affects the star’s rotation rate and do not necessarily cause the orbital motion of surrounding objects to accelerate.

Break-up velocity

The break-up angular velocity (the maximum velocity at which it can rotate) of a star, with mass M, radius R, core rotation Ω, can be calculated by equating centrifugal acceleration with gravity:

\[
\Omega^2 r_* = \frac{GM_*}{r_*^2}. \tag{49}
\]

Most stars in fact rotate well below the break-up rotation rate given by:

\[
\Omega_{\text{br}} \sim \sqrt{\frac{GM}{R^3}}. \tag{50}
\]

Which clearly decreases as the star’s radius R increases.

At the rotating star surface, mass loss happens when the surface rotation rate Ω_s reaches the break-up limit Ω_br, above which the star will be forced to centrifugally expel mass. Basically, this mass outflow arises from the fact that matter is rotating at a lower rate than the star.

In binary systems where the main star has accreted in its disk enough mass from a companion donor star, the small amounts of accreted matter can be enough to replenish the angular momentum of the main star, causing it to increase the its surface rotation rate towards the break-up limit [Ryan et al. 2002], triggering centrifugal mass loss, from their equatorial regions [Hartmann & MacGregor, 1982], and increasing the X-ray luminosity [Bhattacharyya & Chakrabarty, 2017]. A part of the accreted matter (the inner disk) is thrown away from the system by two bipolar outflows [Hartmann & MacGregor 1982]. This ejected matter takes away angular momentum from the main star [Tauris 2012]. Ejected matters are loaded with disc material but their power is borrowed from the stellar rotational energy reservoir [Hartmann & MacGregor 1982]. The ejected mass carries away enough angular momentum and slows down the rotation rate of the star to prevent the star from ever rotating near breakup.

When the accretion disk is emptied by the enhanced accretion rate, the source returns to an extended X-ray quiescent state. A new outburst occurs when sufficient mass accumulates in the disk again [Done et al. 2007].

Accretion from a binary companion can be crucial for supplying enough angular momentum to drive centrifugal mass loss [Bhattacharyya & Chakrabarty, 2017].‏ jets from discs can only carry away the exact fraction of disc angular momentum, leading to a spin-down, and allowing matter to be accreted by the star [Bhattacharyya & Chakrabarty, 2017].

DISCUSSIONS AND CONCLUSIONS

In the previous work [Kallel-Jallouli, 2024a], the author studied the case of an isolated Zaman Dark Matter halo \( U \) with differential rotation, where each shell of radius \( R \) rotates with a period \( T(R) \), related to \( R \) via the Kepler relation:

\[
\frac{T^2(R)}{R^3} = k, \quad \text{for } R > r_0.
\]

Let us call this halo \( U \): an “outer Keplerian universe” (OKU). Kallel proved that the Hubble’s law (7)(8) is valid inside any OKU, without any need for any supplementary hypothesis or any non-Euclidean geometry on \( U \) [Kallel-Jallouli, 2024a]. She has just used her proposed Zaman Dark Matter (DM) model
[Kallel-Jallouli, 2021a,b,c,d]. Moreover, let us consider the more general case when the period \( T \) of each shell of radius \( R \) is related to \( R \) via [Kallel-Jallouli, 2024a].

T=f(R), (51)

with f is a one-to-one positive function. Then, if f is a decreasing function, any test particle inside U, satisfying relation (51) feels a gravitational attraction to the Halo center (Dark Matter effect). If f is an increasing function, any test particle inside U, satisfying relation (48) feels a gravitational expulsion from the Halo center (Dark energy effect). Therefore, Zaman spin gradient results in the appearance of Dark Matter and Dark Energy.

In this present study, we proved the Keplerian trajectories inside an OKU do not fully overlap. Moreover, after each period, any test particle \( P \) inside \( U \) receives a kick-off and instantaneously shifts to a phase position higher than the previous one, to continue its next cycle. The phase shift is a well-known phenomenon. For Mercury, the advance of the perihelion is measured to be about 9.55 arc minutes per century. This measured phase shift can be deduced by only knowing the distance \( R_1 \) from Mercury to the sun and the age \( t_0 \) of the solar system \( U_S \) at the time of measurement.

We can then deduce the period \( T_1 \) via the Kepler formula (6), then the distance \( R_{2,0} \) after one period via (22).
The new period \( T_2 \) can be calculated via (23). We can deduce the phase shift \( \theta_{2,0} \) via the relation (25).

We can also use the knowledge of the distance between the moon and our Earth to deduce the age of our Earth-moon system \( U_E \). It is not difficult to measure the distances \( R_1 \) and \( R_2 \). It is not difficult to measure the periods \( T_1 \) and \( T_2 \). Equation (48) allows us to deduce the age \( t_E \) of the system \( U_E \) when the moon was at the distance \( R_1 \) from Earth. We can also deduce the phase shift \( \theta_{2,0} \) of the moon after each orbit, via the relation (25). Evidently, since \( U_E \) is not an isolated halo (it is a subhalo of the solar system halo \( U_S \)), some corrections must be added to the theoretical solutions.

The scale factor \( a_n(t_0, T_1, t’) \) depends on the universal time \( t_0 \), when the particle \( P \) was placed (or born) at a distance \( R_1 \) (related to \( T_1 \) via the Kepler relation (6)) from the center of \( U \). When we differentiate the scale factor with respect to time, we find the Hubble constant only for an elapsed time \( t’ < T_1 \). Otherwise, we get the Hubble constant multiplied by a constant \( C_n(t_0, T_1) \), which can be close to 1 only if the time \( t’ \) is negligible with respect to the age \( t_0 \) of \( U \).

Since after each complete cycle, a particle \( P \) in its Keplerian orbit is instantaneously shifted, this will give a tremendous energy to the kicked particle. It will spin up and we can explain most of the strange astronomical phenomena given in paragraph 8. The spin-up of the star crust will cause it to rotate much faster than the surrounding magnetosphere and inner accretion disk. The crust length of the day will be smaller than the surrounding inner accretion disk length of the day. The created high gravity (by Zaman spin gradient) will be directed outward as Kallel explained [Kallel-Jallouli, 2024a], and the nearest inner part will be ejected outward.

Now, we shall explain the absence of radio wavelengths when the associated star is nearer to the center, in its trajectory inside a binary system.

Fig. 4: The jet emission region \( e \) that produces Gamma rays at periastron, at a distance \( d \) from the accretor star center \( S \).

Suppose the gamma photons are ejected at periastron, from a source \( e \) at a distance \( d \) to the accretor star \( S \). Since the distance between the two stars is at the minimum \( a \) (fig. 2), the distance between \( e \) and the central star will be \( a_1 = a + d \). When the accretor star is at apastron, at a distance \( b > a \) from the central star (fig. 2), the distance between \( e \) and the central star will be \( b_1 = b + d > a_1 \) (The jet is always directed outward). Suppose the detector is aligned with the apastron-periastron position, in the periastron direction. When passing from the distance \( a_1 \), where the radial velocity \( v_{1r} \) is subjected to the relation [Kallel-Jallouli, 2024a]:

\[
v_{1r} = H_1 a_1,
\]

and the circular velocity is given by [Kallel-Jallouli, 2024a]:

\[
v_{1c} = \frac{2\pi}{a_1^{1/2}},
\]

to the distance \( b_1 \) where the radial velocity \( v_{2r} \) satisfies the relation:

\[
v_{2r} = H_1 b_1,
\]

and the circular velocity is given by:

\[
v_{2c} = \frac{2\pi}{b_1^{1/2}},
\]

neglecting the effect of circular velocity, the observed frequency from periastron passage is given by [Kallel-Jallouli, 2024a]:

\[
\nu_{p,\text{obs}} = \left( 1 – \frac{v_{1r}}{c} \right) \nu_e = \left( 1 + \frac{v_{1r}}{c} \right) \nu_e. \tag{52}
\]

And the observed frequency from apastron passage is given by:

\[
\nu_{a,\text{obs}} = \left( 1 – \frac{v_{2r}}{c} \right) \nu_e. \tag{53}
\]

The difference between the two frequencies when passing from periastron to apastron is given by:

\[
\Delta \nu = \nu_{p,\text{obs}} – \nu_{a,\text{obs}} = \frac{(v_{1r} + v_{2r})}{c} \nu_e = \frac{H_1}{c} (a_1 + b_1) \nu_e. \tag{54}
\]

This relation means that, when the star passes from periastron to apastron, the light emitted from point \( e \) will be redshifted. Reciprocally, when the star passes from apastron to periastron position, the received light will be blueshifted. This explains why in some binary systems, we detect gamma rays at periastron passage but detect longer waves (radio waves) at apastron passage, from the same emitter source.

Moreover, from relation (54), if we know the emitted and observed light frequencies, and the distances \( a_1 \) and \( b_1 \), then, we can calculate \( H_1 \), and deduce the age of the binary system.

More calculations will be explicitly given in future works to deduce experimentally the age of an OKU system and the phase shift of a particle in Keplerian motion inside it.

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