The paper shows that some astrophysical  phenomena such as the initial TOV limit of neutron stars’ mass and the transition density interval from neutron star to a quark star,  can be explained unitary, by the specific structure and the density levels of the fermionic elementary particles  specific to the particle models of a Cold genesis theory of the author: superdense centroid, kerneloid and photonic shell maintained by etherono-quantonic vortex/vortices of magnetic moment(s), which explain and physical phenomena such as: the connection between the photon’s structure and the electronic neutrinos, the scattering centers experimentally evidenced inside the proton at electron-proton scattering at high and very high energies and inside the electron by X-rays, the Compton effect, the nuclear and the strong force, in a fractalic scenario of particles’ forming, from the considered etherono-quantonic energy.

Keywords: photon; neutrino; preon; electron radius; TOV limit; quark star’s density; quantum vortex

In a Cold Genesis pre-quantum theory of particles and fields, (C.G.T.),^1,2based on the Galilean relativity,  the constituent quarks and the resulted elementary particles are explained as clusters of paired quasielectrons, i.e. pairs of negatrons and positrons with degenerate mas, charge and magnetic moment, named ‘gammons’ (g(e^-e⁺)), resulting that preonic bosons and quarks can be formed also ‘at cold’, as Bose-Einstein condensate of ‘gammons’ which form  stable basic preons z⁰ of mass ~34 m_e, forming constituent quarks, which were evidenced by a research team of Science’ Institute for Nuclear Research in  Debrecen (Hungary),³ as neutral super-light particle with a mass of ~17 MeV/c² , named X17.  Its stability was explained in CGT by the conclusion that it is formed as cluster of an even number n = 7x6 = 42 quasielectrons with mass m_e^* ≈ 34/42 = 0.8095 m_e, i.e. reduced to a value corresponding to the charge e^* = ±(²/₃)e by a degeneration of  the magnetic moment’s quantum vortex Γ_m =  Γ_A +Γ_B generated around superdense centroids by etherono-quantonic winds of the quantum vacuum, i.e. given by ‘heavy’ etherons of mass ms ≈ 10^-60kg and ‘quantons’ of mass m_h = h·1/c² = 7.37x10^-51 kg .

The considered “gammons” were experimentally observed in the form of quanta of “un-matter” plasma.⁴ In CGT, similarly to the S.M.’s constituent quark model, it was considered⁵ that the electron’s mass is formed by a ‘kerneloid’ containing the (super) dense centroid m₀ of radius r₀ £ 10^-18 m and by a shell of bosons which are ‘naked’ photons, in concordance with the evidenced possibility to obtain a B-E condensate of photons,⁶ this electronic kerneloid being equivalent to an ‘impenetrable’ quantum volume (similar to that of the nucleon), having a radius r_ie ≈ 10^-2 fm- in accordance to some high-energy scattering experiments reported by Milonni et al.⁷ The last experimentally determined value for the quark’s radius: ~ 4.3x10^-19 m ⁸ corresponds in this case  to the radius of the super-dense electronic centroid,^1,2 being close to the upper limit determined by X- rays scattering on electron.⁹  It was also concluded¹⁰ that the transition from neutron matter to quark matter begin at densities around (1.5 ÷ 4)x10¹⁸ kg/m3.

In 1939, by neglecting the nuclear forces between neutrons, using the Schwarzchild’s equation and an equation of state P(ρ) specific to a highly compressed cold Fermi gas, P = K· ρ^5/3, (polytropic form in the non-relativistic case of a Fermi gas of neutrons), the mass limit for neutron-degenerate matter was estimated at 0.7 solar masses, (M_TOV = 0.7 M_⊙)- value representing the initial TOV limitTolman–Oppenheimer–Volkoff),^11,12 the corresponding maximum mass before collapse being with ten percent higher than this, (M_S⁰ ≈ 0.77 M_⊙)_.¹² So, the stars more massive than the TOV limit collapse into a black hole and if the mass of the collapsing part of the star is below the TOV limit, the end product is a compact star – either awhite dwarf (for masses below the Chandrasekhar limit) or a neutron star or a (hypothetical) quark star.

In 1996, by an equation of state (EoS) based on a MIT bag-like model of quark’s confining, it was deduced that the upper mass for neutron stars which are not collapsed into a black hole is in a range from 1.5 to 3 solar masses.¹³ In 1932, Louis de Broglie^14-16 suggested that the photon might be the combination of a neutrino and an antineutrino. During the 1930s there was great interest in the neutrino theory of light and Pascual Jordan,¹⁷ Ralph Kronig, Max Born, and others worked on the theory.

But in 1938, Maurice Pryce¹⁸ brought work on the composite photon theory to a halt. He showed that the conditions imposed by Bose–Einstein commutation relations for the composite photon and the connection between its spin and polarization were incompatible. We will argue that the previous mentioned astrophysical phenomena can be explained unitary, by the specific structure and the density levels of the elementary particles specific to the CGT’s particle models, in a fractalic scenario of particles forming from the considered etherono-quantonic energy.

The density levels of leptons in CGT

For electron, it results in CGT,^1,5,19 that there are three radius specific to the  structure of its inertial mass, corresponding to three levels of mean density of confined ‘naked’ photons, (reduced at their inertial mass: m_f, considered as confined in the photon’s kerneloid, of radius r_f  ≤ 10^-2 fm, for m_f  <m_e and having a G_m^f -vortex sustained by a superdense centroid of radius r₀^f ≤ r₀ = 0.43x10^-3 fm):

a) the super-dense centroid’s radius (r₀^e ≈43x10^-3 fm), corresponding to the highest density level (ρ⁰ ≈ 1.4x10²⁰ kg/m³) and to a mass ~ 0.5x10^-4m_e , (a half of an electronic neutrino having the mass limit: 60 eV/c²,);^1,20

b) the electron kerneloid’ radius (r_ie → 10^-2 fm), given by a dense shell of photons, corresponding to the mean density level, (ρ_i^e ≤ ρ_q⁰ = m_qⁿ/u_q⁰ = 7.5 MeV/c²×91x10^-47 =1.46x10¹⁸ kg/m³ ; m_qⁿ –the current mass of the nucleon’s quarks, contained by its volume u_q⁰ of 0K), and:

c) the electron’s classic radius (a_e ≈41 fm), corresponding to the low density level (ρ_a(a) ≈ 5.16x10¹³ kg/m³) and to a quasi-superficial distribution of the electron’s e-charge.

This electron’s structure explains in CGT the next phenomena evidenced by electron-proton scattering:

1. the a)-level explains the fact that -at very high electron energies λ << r_p, (λ = hc/E –the wavelenght; r_p –the proton’s radius), the proton appears to be a sea of gluons and current quarks evidenced as having a radius r₀ = 0.43x10^-3 fm;⁸
2. the b)-level explains the fact that -at high electron energies λ < r_p, the proton appears to be a cluster of three constituent quarks;
3. the c)-level explains the electron’s charge, the Lorentz force^1,21 and the fact that -at low electron energies λ ~ r_p, the scattering is equivalent to that from a extended charged object.

The electron’s Compton radius r_C^e = λ/2p = h/2pm_ec = 386 fm corresponds in CGT to the radius of an evanescent part containing a spinorial mass m_s » m_e of vector photons vortexed with the light’s speed c, which are weakly bound to the electron’s mass m_e (contained by its volume of classic a-radius) and does not contribute to their inertial mass.

A similar structure is considered in CGT also for heavy and light vector photons, whose inertial mass is considered as contained by a kerneloid of radius r_if → r_ie ⁰ = 10^-2 fm¹⁹ and which contains a super-dense centroid of radius r₀^f < 0.43x10^-3 fm, with the difference that  their kerneloid retains only their evanescent part, containing vortexed quantons and very light vector photons whose inertial mass does not contribute to the inertial mass of that vector photon, they being weakly linked to the vector photon’s kerneloid.

Because in CGT the pseudo-scalar photon is formed as pair of two vector photons with antiparallel spins and magnetic moments, the mentioned similitude can explain the Compton effect (the partial transferring of photon’s kinetic energy to an atomic electron) as consequence of elastic interaction  between the kerneloids of the interacting photon and electron and the upper limit of the electron’s center’ radius determined by X- rays scattering on electron, (~10^-18 m),⁹ –as consequence of elastic interaction between the superdense centroids of X-ray photons and the electron’s centroid.

Regarding to the possible mechanism of the centroids’ and of kerneloids’ forming, in CGT are considered chiral fluctuations in a primordial dark energy medium, formed by a brownian component (generating static pressure) and a dynamic component of the primordial quantum vacuum, given by etherono-quantonic winds which can produce vortices of a fluctuating magnetic-like field B = rot A of a cosmic A-field. Without a dense centroid, these etherono-quantonic vortices are un-stable, but in an energetic Proto-Universe with a high density of the primordial dark energy, they could have been enough strong for generate tiny stable chains of confined quantons, which- in the actual Universe, could be formed at the level of the magnetic field‚ lines’ (etherono-quantonic vortex-tubes – in CGT) of the magnetars’ field.

Also in the Primordial Universe, these quantonic chains could be joined into twisted (chiral) bundles, representing the centroid(s) of light photons which obtained their inertial mass (of their kerneloid) by confining of quantonic clusters  in  a stable etherono-quantonic vortex of their magnetic moment m_f whose sense and value are given by the centroid’s -chirality (§_f = ±1) and mass (m₀^f).

It is logical- in this case, to exist a proportionality between the mass of the vector photon’s kerneloid (its inertial mass) and that of its centroid, the electron’s mass resulting as given by a saturation value of confined photons number. Because the paired electronic centroids with opposed chiralities not ensures the maintained of the electron’s magnetic moment m_e, (i.e. of its etherono-quantonic vortex Γ_m^e), a pair of coupled electronic centroids represents in CGT an electronic neutrino (of Majorana type) and this explains the fact that it can easily penetrate large solid bodies.

Similarly, it results that vector photons of opposed chiralities can annihilate each other's vortexial structure at their collision, phenomenon observed at the interference of two laser radiation waves in antiphase. The remained paired centroids of vector photons can be considered pseudo-neutrins of lower mass, which contribute to the total mass of the Universe’s dark matter and a classic equivalent to the so-named ‚axion’ of the Standard Model (of mass from 10⁻⁵ to 1 eV/c²).²² These pseudo-neutrins explain in CGT the alleged connection between photons and neutrins.^14-18

The density levels of quarks and of mesons and baryons, in CGT

Relating to the mesons and baryons, in CGT these particles are formed by constituent quarks of mass M_q formed by layers of a light mesonic quark M_m^± ≈ [z₂(4z⁰) ± m_e^*] and a number of two preonic bosons: z₂ = 4z⁰ and z_π = 7z⁰ , (Figure 1) composed by z⁰-preons of mass M_z = 34 m_e , obtained as prismatic (crystalline)  arrangement of 21 pairs of quasielectrons of opposed charges: e^* = ±(²/₃)e and mass m_e^* ≈ 0.81 m_e, (Figure 2), the kerneloids of these quasielectrons form a quasi-crystalline cluster representing the kerneloid of the z⁰-preon, the number: n_z = M_q/M_z of z^o-preons’ kerneloids giving the mass m_q of the current quark which –by the total vortical field of  the magnetic moments of its quasielectrons, generate a vortical potential V_Γ and a vortical force F_v(r) which attracts and retains ‚naked’ thermalised photons, forming the quark’s bosonic shell, in CGT’s model:¹

  F_v(r) = -∇ V_Γ = -∇N^e×V_Γ^e(r);     (V_Γ^e = -½u_fρ_sc²)        (1)

(u_f –the volume of the photon’s kerneloid, containing its inertial mass; ½(ρ_sc²)_r –the dynamic etherono-quantonic pressure in the Γ^e –vortex of a bound quasielectron at r -distance).

From figure 2 it is deduced that the radius value:  r_ie⁰ ≈ 10^-2 fm⁷ of the quasielectron’s kerneloid, ensures a mean distance:  d_i ≈ (²/₃)×r_z = 2x10^-2 fm between the electronic centroids m₀ on the radial direction at T = 0K, which gives a value: r_iz = 3x10^-2 fm for the radius of the kerneloid of the cold z⁰-preon, the minimal value of the cold z⁰-preon’s length resulting of value: l_z⁰ = 6xd_i ≈ 0.12 fm and a volume of the cold z⁰-preon’s kerneloid:

u_zi⁰ = 0.34x10^-48 m³.

Figure 1 The cold forming of semi-light quarks.

Figure 2 Kerneloid of a half of  z⁰-preon.

Figure 3 Preonic z_p -layer of a quarcic kerneloid.

Figure 4 Baryonic kerneloid formed by current quark.

The CGT’s explanation of the initial TOV limit of neutron stars’ mass

Because the quasi-crystalline structure of (u, d)- quark’s kerneloid have three layers- in CGT, (m_1;2 ; z_π ; z_π -Figure 1), with (4; 7; 7) z⁰-preons, it results a length of the (u; d)- quark’ kerneloid at T = 0K: l_q⁰ = 3l_z⁰ = 0.36 fm, and double (l_q⁰’ = 6l_z⁰ ≈ 0.72 fm) for the v-quark of CGT.

The minimal radius of the quark’s kerneloid (specific to its ultra-cold state, T = 0K) results of value:  r_q⁰ » 3xr_z = 0.09 fm - which gives a current quark’s volume: u_q⁰ = πr_q²l_q⁰ = 0.91x10^-47m³.

A cold cluster of three u-d-current quarks will have a radius r_i⁰ ≈ 2r_q⁰ = 0.18 fm at T = 0K.  In report to these theoretic values of T = 0K, the value:  r_qⁱ ≈ r_i/2 = 0.2 fm used in the CGT’s model as radius of a spherical current u/d-quark in concordance with older experiments^23,24 represents a radius of dilated volume of current (u/d)-quark: u_qⁿ ≈ (3.35÷3.38)x10^-47 m³, that corresponds to a small vibration liberty l_v^z of the z⁰-preos inside the quark’s kerneloid, which generate a current quark’s dilated volume and its repulsive shell, of thickness δ_q(l_v^z ) ≈ (0.01¸0.03) fm,²⁵  giving  a scalar repulsive charge, q_s, and an interaction radius: r_qⁱ = r_q + δ_q , (r_q = 0.2 fm), specific to an ordinary temperature associated to the nucleon’s vibration: T_n^j ≈ 1 MeV/k_B.

It can be observed that the density of a black hole corresponding to the initial TOV limit: M_S⁰ = 0.7 M_⊙, by the known Equation:

 r_bh = 3c⁶/32πG³M² = 1.85x10¹⁹/M² ,  (M in M_⊙)    (2)     

is:ρ_bh⁰ = 3.775x10¹⁹ kg/m³ and may be explained in CGT as corresponding to a black hole that resulted by the conversion of a cold t-quark star with current top quarks formed as compact clusters of  z⁰ –preons with inflated volume of their kerneloid, to a mean apparent value: u_z^a = m_t/(ρ_bh⁰n_z) = 0.8x10^-3fm³, (instead of 0.34x10^-3fm³ –for the cold kerneloid of z⁰-preon ), that corresponds- in spherical model, to a radius: r_z^s = 0.058fm, and in a prismatic (cold) form- to an inflated kerneloid’s volume: u_zⁱ = π(3r_ie)²(12r_ie) that corresponds to an apparently inflated volume of the quasielectrons’ kerneloid, of  radius:

r_ie = 1.33x10^-2fm,  given by “zeroth” vibrations of amplitude: δr_ie = (r_ie - r_ie⁰) = 0.33x10^-2 fm, vibrations whose existence is considered also by the Quantum Mechanics.

So, the CGT’s model of quarks can explain microphysically also the initially calculated value of the Tolman–Oppenheimer–Volkoff mass limit  of the neutron stars.^11,12

The CGT’s explanation of the transition density interval at a quark star

For a composite current quark Q , its dilated volume u_Q(T_q) results as sum of apparent volumes u_q(T_z) of its lighter quarks q, and it depends on the current quark’s mass m_q and on its intrinsic temperature T_i^q = T_z given by the vibrations of  the kerneloids k_z of their z₀ –preons, conform to the dilation’ law:

u_Q(T_q) = u_Q⁰(1 +α_QΔT_q) ≈ N_qu_q⁰(1 + α_qΔT_z)          (3)

from Eq.(3) resulting that: α_Q = α_q(ΔT_z/ΔT_q) ≈ Δu_q^a/u_q⁰ΔT_q with T_q –the temperature associated to the q-quarks’ vibrations and T_z –the temperature associated to the z⁰-preons’ vibrations.

Because in CGT the volume u_qN of a possible composite current quark: q_N = (u`ud) is approximately equal to the volume of a protonic kernel u_p , we can approximate the value of the quark kerneloid’s radius r_k^z(T_q)^· at an intrinsic temperature T_Q^j ≈ T_n^j corresponding to that of a vibrated nucleon with the energy E_n^j ≈ 1MeV by extrapolating the case of the nucleon’s impenetrable volume u_n(r_iⁿ) at nucleon’s temperature: T_n^j ≈ 1MeV/k_B, considered spherical and filled with dilated kerneloids of z⁰-preons of its dilated q-quarks, to the case of a composite current quark (tri-quark) at ordinary temperature T_Q^j ≈ T_n^j, whose kerneloid’s mass is: m_qⁿ > 3m_q^n-1 by photons acquiring and with the apparent volume u_z^a  approximated by a relation similar to that specific to a nuclear volume:¹⁹

           u_kⁿ ≈ u_z^a··N_zⁿ ;  ⇒   u_k^q ≈ u_z^a··N_z^q                        (4)

            ⇒  r_kⁿ ≈ r_z^a··N_z^1/3;      r_k^q ≈ r_z^a··N_z^1/3 ;                         

(N_z –the number of particle’s z⁰-preons, having an apparent volume u_z^a). A similar relation is also used- in some papers for the strangelet’s radius.²⁶

With: r_iⁿ = (0.44¸0.45) fm ²³ N_z ≈ 1836m_e/34m_e = 54, (for proton), it results by Eq. (4) that: 

r_z^a = 0.118 fm ≈ 0.12 fm, (u_z^a = 0.723x10^-47 m³), at  T_n^j ≈ 1MeV/k_B , the kerneloid of a protonic u/d-quark having -by Eq. (4), at ordinary nucleons’ temperature T_n^j , an apparent radius: r_q^a = (r_q^r + δ_q) = 0.118x18^1/3 ≈ 0.31 fm,  (given by its real radius r_q^r ≈ 0.2 fm and its vibration amplitude δ_q).

For the quarks obtained in CGT: u/d(312; 313 MeV); s-sark (504 MeV); v-vark (574 MeV); c-chark (1700 MeV);  b-bark (5000 MeV); t-top(175 GeV), it results by Eq.(4) the next values of their kerneloid’s volume:  u_u/d(0.2fm) ≈ 0.0335 fm³; u_s·(0.486fm) ≈ 0.2 fm³; u_s(0.5fm) ≈ 0.212 fm³; u_v(0.574fm) ≈ 0.239 fm³; u_c(1.7fm) » 0.696 fm³;  u_v(5fm) ≈ 2.064 fm³.

Also, in CGT was obtained a semi-empiric relation for the current quark’s mass m_q function of the ratio between the mass M_S· of the constituent s^·-quark of the Standard Model (486 MeV) and the mass M_q of the constituent q-quark, in the form:¹⁹                                       

$m_q=M_q-{\Delta }_q=M_q-A_q\cdot e^{k_q\cdot \left(1-\frac{M_{S•}^2}{M_q^2}\right)}MeV/c^2;$          (5)

with M_s· = M_s^·(486MeV) – the constituent mass of s^· –quark.

The constants A_q, k_q, were obtained by taking: m_d = 7.5 MeV/c² for the d-quark having in CGT a constituent mass M_d ≈ 313 MeV/c² and m_S· »

91 MeV/c²  -obtained in CGT,¹⁹ resulting: 

 A_q = Δ_s· = 395 MeV/c² ;   k_q = 0.182  and the values:

(m_s^· = 91; m_s’ = 104; m_v’ = 158; m_c’ = 1233) MeV/c²; m_b’ = 4527 MeV/c², and m_t = 174.5 GeV/c², values which are close to that obtained by the S.M.:

(m_s^· = 92;  m_c^· = 1275 ; m_b^· = 4420) MeV/c² and

 m_t = 173 GeV/c², currently accepted in S.M.²⁷

With the previous values of the bare quarks’ volumes, it results for their densities the values:

ρ_k^s· = 0.8x10¹⁸; ρ_k^s = 0.87x10¹⁸; ρ_k^v = 1.17x10¹⁸; ρ_k^c = 3.15x10¹⁸; ρ_k^b = 3.9x10¹⁸ [kg/m³] and: ρ_k^t ≈ 4.2x10¹⁸ kg/m³ , at T_q ≈ T_q^j .

The obtained values for the mean density of current quarks at T_q ≈ T_q^j , (ρ_k^q = (0.8 ¸ 4.2)x10¹⁸ kg/m³), can also be specific to the density of some quark stars having the same ordinary internal temperature T_q^j  and formed inside a neutron star for which the necessary pressure for its forming is given by the gravitation force and the strong force,  and they are in concordance with previous results, based on theoretical models for the density variation inside a neutron star, which concluded that the transition from neutron matter to quark matter begins at densities around (1.5 ÷ 4)x10¹⁸ kg/m³.¹⁰

The previous result argues the naturalness of the CGT’s model of quarks and indicates that the black holes having heavy mass and a mean density lower than 10¹⁹ kg/m³, even if they can have the Hawking temperature at their surface, as preon stars they have inflated quarks and z⁰-preons, at least in their core, where they have an intrinsic temperature

T_z > 0K.

A similitude between the quark of the  

Standard Model and the CGT’s model

Supposing that at a critical temperature T_c→T_d, (T_c –phase transformation temperature; T_d –the quarks deconfining temperature: ~ 2x10¹² K –for nucleons) some paired kerneloids of paired quasi-electrons (‚gammons’ –in CGT,^1,2) are released and transferred from the quasicrystalline cluster of its kerneloid in the volume of its photonic shell, then their behavior will be relative similar to that of the polarised gluons in S.M. (whose mass has an experimentally determined limit: 1÷ 1.3 MeV/c² -approximately equal to that of an (e^-e⁺)- pair),²⁸ with the difference that these‚ gammons’ will interact by electric and magnetic interactions, (having the tendency to form clusters with 8 quasi-electrons at T →0K) but being maintained inside the constituent quark’s volume by the force generated by the total vortical field of the current quark, (Eq. 1). 

After partial deconfining of a current quark, its reconfining at T < T_c could generate a quasi-crystal or amorphous state- similar to the so-named ‚glasma’ in the S.M.,²⁹ with the difference that this state is considered in S.M. as specific to a saturation state in high energy hadronic collisions and not to a low temperature quarcic state.

It is well known that neutron stars, which are extremely hot when they are formed, (~10¹¹ ÷ 10¹² K), cool down thereafter to ~(10⁷ ¸10⁶)K through processes including thermal radiation, neutrino emission and the formation of a solid crust.³⁰ Comparing the cooling of a rotated neutron star with a cooling metal drop, it results that –because the star’s crust is cooled faster than the star’s interior, the formed solid crust (whose ground state corresponds microscopically to a body-centered cubic (bcc) crystal lattice and macroscopically- to an isotropic bcc poly-crystal with elastic properties, given and by ‘nuclear pastas’)³¹ is contracted conform to Eq. (3) by the aid of the strong forces, given –in CGT by a vortical field which generates an attraction force of the form (1), these forces F_v(y) = -∇ V_Γ(y), generating a superficial tension s_q which –by the aid of the gravitation force F_g(R) equalizes the internal pressure P_i:

$\begin{array}{l}\Delta \text{P}\cdot \text{dV(R') }={\sigma }_{\text{q}}\cdot dS(R\text{'})\text{ ; }\Rightarrow \Delta \text{P }={\text{ P}}_{\text{i}}-G\frac{M(R\text{'})}{R^{\text{'}2}}{\rho }_c(R^{\text{'}})\cdot \delta R=\frac{\text{2}{\sigma }_{\text{q}}}{\text{R'}};\\ \text{ (}{\sigma }_{\text{q}}=\frac{F_n(y)}{2l_y};\text{ R'}=\text{R-}\frac{\delta \text{ R}}{\text{2}})\end{array}$         (6)           

(ρ_c, δR – the solid crust’s density and thickness; M, R – the neutron star’s mass and radius).

 It results that for the same star’ mass M, P_i decreases with R but increases with r_c , dR and s_q.

This analogy is concordant with the known fact that if the conversion of neutron-degenerate matter to quark matter is total, the formed quark star can be imagined as a single gigantic hadron bound by gravity, rather than by the strong force that binds ordinary hadrons. The recent discovery³² of a possible quark star having a radius of about 10.4 kilometers, a surface temperature of approximately 2x10⁶ °C and a mass equal to only 0.77M_⊙ , (almost 1.5 times less than the theoretical limit for neutron stars), corresponding to a mean density:  ρ_m = 3.27x10¹⁷ kg/m³, is in concordance with the conclusion that a such star can have a neutronic inner crust (ρ_s ≈ 2.8x10¹⁷ kg/m³) and a nucleus formed by current u/d-quarks.

By this supposition, taking for a single layer of nucleonic quarks: σ_q ≈ 9 MeV/fm² at 0K –considered in some papers for strangelets,³³ for n = dR/2r_q layers of nucleonic quarks (r_q –the u/d- quark’s radius), with M(R’) = 0.77M_⊙ ; R^’ = 10.4 Km, (ρ_m = 3.27x10¹⁷ kg/m³), Eq. (6) can be written in the form:

$\begin{array}{l}\Delta \text{P }={\text{ P}}_{\text{i}}-G\frac{M(R\text{'})}{R^{\text{'}2}}{\rho }_c(R^{\text{'}})\cdot n\cdot d_q=\frac{2n{\sigma }_q}{\text{R'}};{\text{ (d}}_q{\text{=2r}}_q\approx \text{0}\text{.4fm) }\\ {\text{ P}}_{\text{i}}=\frac{{\rho }_{\text{m}}}{{\text{m}}_{\text{q}}}{\text{k}}_{\text{B}}{\text{T}}_{\text{i}}\text{ ; (}\frac{{\rho }_{\text{m}}}{{\text{m}}_{\text{q}}}=\frac{1}{{\upsilon }_q^a};{\sigma }_{\text{q}}\approx 9MeV/f{m}^2;\text{ R'}=\text{R-}\frac{\delta \text{ R}}{\text{2}}=R-n\cdot r_q)\end{array}$             (7)

The volume u_q of an internal current u/d-quark at temperature T_i ≈ 2x10⁶ °K can be approximated by Eq. (3) with the values: u_q⁰ = N_zu_z⁰ ≈ pr_q²l_q⁰ = 0.91x10^-47 m³ and a_q^a obtained in CGT,¹⁹ knowing that –at T_q^j = (m_d/M_n)T_n^j  »  9.27x10⁷ K, (m_d = 7.5 MeV/c²; N_z –the number of q-quark’s z⁰-preons;

M = 938 MeV/c²;  T_n^j » 1MeV/k_B = 1.16x10¹⁰ K),  we have: r_q^r (T_q^j) ≈ 0.2 fm (the dilated real radius) and an apparent radius: r_q^a(T_q^j)  = (r_q^r + δ_q) » 0.31 fm, corresponding to an apparent volume:

 u_j^a(r_q^a) = 0.1247 fm³ ≈ (1/3)u_n(r_i =0.447fm),

(Eq. (4)), resulting that:

Δu_q^a = u_q⁰a_q^aDT_q ;  Þ  Δu₁^a/Δu_j^a = T_q1/T_q^j ;          (8)

(Δu_j^a = u_j^a - u_q⁰)                       

which gives: Δu_j^a = 0.1156x10^-45 m³ ;  Δu₁^a = (u₁^a(T_i) -u_q⁰) = Δu_j^a(T_q1/T_q^j); ⇒ u₁^a(T_i) = 0.0116x10^-45 m³; 

P_i = 86.25x10⁴⁵x1.38x10^-23x2x10⁶ = 2.38x10³⁰ N/m²;

r_q^a = 0.14 fm; (d_q^a = 0.28 fm).

From Eq. (7), with r_c(R’) ≈ m_d/u₁^a(T_i) ≈ 1.15x10¹⁸ kg/m³, by d_q » d_q^a, it results that:

ΔP = 2.38x10³⁰ – 3.08x10¹⁴×n = nx2.77x10¹⁴,  resulting that: n = 4x10¹⁵;   δR = 1.12 m.

The paper shows that the specific structure and the density levels of the fermionic elementary particles  specific to the particle models of CGT: superdense centroid, kerneloid and photonic shell maintained by etherono-quantonic vortex/vortices of magnetic moment(s), can explain unitary and naturally not only physical phenomena such as: the connection between the photons’ structure and the electronic neutrinos, the scattering centers experimentally evidenced inside the proton at electron-proton scattering at high and very high energies and inside the electron by X-rays, the Compton effect, the nuclear and strong interaction,²⁵ but also some astrophysical phenomena such as the initial TOV limit of neutron stars’ mass and the transition density interval from neutron star to a quark star, in an unitary way, in a fractalic scenario of particles’ forming from the etherono-quantonic energy considered in CGT. The mentioned explanations argue the CGT¨s scenario of particles’ cold genesis in a cold Universe, having initially a primordial etherono-quantonic ¨dark¨ energy, which generated initially (pseudo)neutrins and light photons from chiral (vortical) quantum fluctuations and thereafter –electrons and heavier particles. This scenario responds partially to an older question: ¸how was the primordial hot Universe created from almost nothing, i.e. from a cold Universe ?¨.³⁴

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