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The consequence of the 5D projection theory [1] is extended beyond the Gell-Mann Standard Model for hadrons to cover astronomical objects and galaxies. The proof of Poincare conjecture by Pe-relman’s differential geometrical techniques led us to the consequence that charged massless spinors reside in a 5D void of a galactic core, represented by either an open 5D core or a closed, time frozen, 3D × 1D space structure, embedded in massive structural stellar objects such as stars and planets. The open galactic core is obtained from Ricci Flow mapping. There exist in phase, in plane rotating massless spinors within these void cores, and are responsible for 1) the outward spiral motion of stars in the galaxy in the open core, and 2) self rotations of the massive stellar objects. It is noted that another set of eigen states pertaining to the massless charged spinor pairs rotating out of phase in 1D (out of the 5D manifold) also exist and will generate a relatively weak magnetic field out of the void core. For stars and planets, it forms the intrinsic dipole field. Due to the existence of a homogeneous 5D manifold from which we believe the universe evolves, the angular momentum arising from the rotation of the in-phase spinor pairs is proposed to be counter-balanced by the rotation of the matter in the surrounding Lorentz domain, so as to conserve net zero angular momentum. Explicit expression for this total angular momentum in terms of a number of convergent series is derived for the totally enclosed void case/core, forming in general the structure of a star or a planet. It is shown that the variables/parameters in the Lorentz space-time domain for these stellar objects involve the object’s mass
*M*, the object’s Radius R, period of rotation
*P*, and the 5D void radius
*R*
_{o}, together with the Fermi energy
*E*
_{f} and temperature
*T* of the massless charged spinors residing in the void. We discovered three laws governing the relationships between
*R*
_{o}/
*R*,
*T*,
*E*
_{f} and the angular momentum Iω of such astronomical object of interest, from which we established two distinct regions, which we define as the First and Second Laws for the evolution of the stellar object. The Fermi energy
*E*
_{f} was found to be that of the electron mass, as it is the lightest massive elementary particle that could be created from pure energy in the core. In fact the mid-temperature of the transition region between the First and Second Law regions for this
*E*
_{f} value is 5.3 × 10
^{9} K, just about that of the Bethe fusion temperature. We then apply our theory to analyse observed data of magnetars, pulsars, pre-main-sequence stars, the NGC 6819 group, a number of low-to-mid mass main sequence stars, the M35 members, the NGC 2516 group, brown dwarfs, white dwarfs, magnetic white dwarfs, and members of the solar system. The
*ρ* =
*(R*
_{o}/R) versus
*T*, and
*ρ* versus
*P* relations for each representative object are analysed, with reference to the general process of stellar evolution. Our analysis leads us to the following age sequence of stellar evolution: pulsars, pre-main-sequence stars, matured stars, brown dwarfs, white dwarfs/magnetic white dwarfs, and finally neutron stars. For every group, we found that there is an increasing average mass density during their evolution.

Two years ago, a 125 GeV p-p resonance was forwarded as the probable proof of the existence of the Higgs boson condensed vacuum [

Through several follow-up articles [_{o}. Hence we shall refer R_{o} as the radius of the 4D space void, with time fixed, such that massless spinor states are perpetual, unless thermal energy exchanges between the Lorentz boundary and the void core, inside the Lorentz space-time shell, is implemented.

As we have a spherically shaped mass stellar object model enclosing a 3D × 1D void filled with charged massless spinors satisfying the Fermi distribution, we can connect the physical quantities of the thermal bath of the Fermions in the void and the physical quantities of the matter shell, leading to the discovery of the 1st and 2nd Laws regions for these spinors states. This 4D Riemannian space-time obtained from the superposition of both SU(2) and SU(3), is hence given by [SU(2) + SU(3)] × L, as shown in [

Note that the projection from the 5D space-time onto a 4D Lorentz space-time using the Ricci Flow Theorem, produces a Lorentz 4D, without further mapping the 3D space volume in a doughnut shape, while the doughnut center void remains in the 5D manifold. However when the doughnut 3D volume is transformed by further mapping into a spherical shape, the original 5D void at the center is enclosed into a 4D space void, with time frozen, such that any massless charged spinor states within it must be perpetual. However the Maxwell vector potentials can exist in both 5 and 4 dimensions [_{z} in the void by in phase circulation of the oppositely charged massless spinors, an opposite angular momentum―L_{z} must be generated in the Lorentz spherical mass shell, in order to preserve total zero angular momentum value. In the astronomical scale for stars, this L_{z} leads to a repulsive potential within such a void, leading to the elimination of the gravitational singularity, similar to the action of the gluon repulsive potential within hadrons [

After an introduction of the basic concept 5D to 4D projection above, we follow in Section 2 to present a description of the boundary condition at the 5D - 4D “inter-phase”, leading to a brief sketch of the creation of the universe in view of SU(2), SU(3), and 4D Lorentz group representation. Section 3 is devoted to the derivation of explicit formula (expressed as a number of convergent series of E_{f}/kT) for the angular momentum generated by the spinor pairs rotating in phase. As each type of the spinors is a Fermion system, and the lightest lepton mass energy created by the Po projection is electron, we take the Fermi energy of the spinor to be E_{f} = 0.5 MeV. The radius of the void R_{o} is expressed as an explicit function of the shell mass M, with period of rotation P, and observed radius R together with the Fermi energy E_{f} and temperature T of the massless spinors inside the void core. From this mathematical result, we discovered three laws consequential to the projection theory: 1) At very high temperature such that the angular momentum Lz of the object is mainly contributed by the massless spinors with energies much greater than E_{f}, the normalized void radius R_{o}/R is a linear function of 1/T, with a negative slope, which must represent the early stage of the stellar objects. We call this region the First Law region of angular momentum. 2) At relative low temperature kT = E_{f}, the ratio ρ = (R_{o}/R) of the object is a linear function of 1/E_{f}, and not a function T; thus the ρ versus T relation is a horizontal line. We refer this region as that of the Second Law. Hence, this region must describe the last stages of the stellar object. 3) The “mid-temperature” T_{c} in the transition region between the two laws is a universal constant, dependent only on E_{f} = 0.5 MeV (which is a universal constant in our theory). We name this as the Third Law. These three stages represented by the three Laws are actually shown in this paper to be satisfied by many known stars classifications. Following, in Section 4, we explain why magnetars/pulsars are new-born stars with detailed numerical illustration of some pulsars examples. Combining with the stellar object’s mass density, we open up an analysis of the angular momentum of star groups according to different ranges of mass density of these stellar objects in Sections 5 and 6, and compare the calculated results with many numerical data examples to support the theory. In particular, we analyze numerically the R_{o}-T, and R_{o}-P relations with reference to the general different stages of evolution of these objects. Neutron stars are proposed to be the very oldest stars in Section 7, accompanied with detailed model numerical examples. From purely the view point of angular momentum, planets are similar to stars (see Section 8), but only with smaller values of ρ = R_{o}/R. A general discussion is presented in Section 9, including a summary of the theory presented focusing on some relevant physics concepts involved, giving a sketch of stellar evolution―from pulsars to neutron stars, and providing simple discussions on the Fermi energy, heat bath, degeneracy of an electron gas, as well as possible Bose-Einstein condensation involved in the final stage of stellar objects. The origin of the stellar magnetic field is only very briefly introduced, as we left that discussion to another paper.

Based on the 5D projection theory and Gell-Mann standard model, we put forth the notion that the final major amount of hadron mass comes from Gluon, not from the quark bare masses. The hadrons can only form after grouping through quantum gauge confinement, which must happen sequentially after the existence of quarks on the boundary of the 5D manifold [

From the view point of group symmetry, we would like to point out that the boundary of the finite 5D homogeneous manifold must be obtained from a dimension projection, just like the boundary of a 3D space volume is obtained from a 2D projected surface. Hence this 5D boundary is represented by [SU(3) + SU(2)] × L; here L is the 4D Lorentz space-time. It is this topological realization that dictates the special property that the boundary is being composed of net charge neutral masses, starting with quarks and leptons right at time 0, way before formation of hadrons, etc. Such a property must be maintained as the Lorentz 4D domain and 5D both expand through the continuous rebalancing of energy between them.

In field theory, energy can only be carried by quantum fields irrespective of the domains they belong. However, only photons, meaning vector potential fields can exist in both 5D and 4D manifolds. Thus it must be the photons that act as the medium of energy transport between matter fields in L, and charged massless spinors in 5D. Such a diffusion process between energy exchange of 5D to L and vice-versa obeys entropy theorem and violates time reversal symmetry. However, as the unidirectional time and space expansion is built-in from the homogeneous 5D metric, entropy would naturally be obeyed, leading therefore to the statistical thermodynamic theory of nature. In fact it is the application of this entropy law that provides ground of validity for the second step of the Perelman mapping in his proof of the Poincare Conjecture.

For a stellar mass object, the 5D domain within the Perelman-Poincare void core is frozen in time at t = τ_{0}, thus with or without energy transfer, the expansion in space-time domain occurs only in the L domain surrounding the void core. The opening of the L domain provides a model for the formation of a galaxy, that contains many masses (which we call stars and planets), created by the Ricci Flow of Perelman’s mapping. The galaxy expands in the form of a doughnut 3D space manifold. As the galactic center is in 5D, the galaxy is imbedded in the homogeneous 5D universe. Many galaxies can be created at the same time, on the finite surface area of the so called Creation instant of the 5D manifold. For an averaged galactic core dimension of 100 light years across, it is easy to estimate, based on the domain represented by the Lorentz 4D boundary to the 5D finite domain, that a million galaxies can be simultaneously created by packing the 5D galactic cores together as the entire 5D universe expanded according to the 5D metric from 0 to 1000 light years. Since the centers of the galaxies are connected in the 5D enclosing domain, light can be transmitted between these doughnut galaxies; hence an event at any one point in one Galaxy can be observed by observers in its own, as well as those in other galaxies. A 1000 years is very short as compared to the estimated age of the Milky Way galaxy. Thus if all galaxies were created simultaneously by the Big Bang, then the universe’s age is close to the galaxies age as conjectured by some scientists.

Note that the boundary of the entire 5D universe is represented in terms of the product of three groups: SU(2), SU(3), L, and thus must contain quarks and leptons, plus the 5D voids. Hence, as the universe expands, the density of these massive charges on the boundary of the universe must continue to reduce as the 5D expands, leading to a condition that allows us to treat the entire 5D universe encompassing interior fields of both massive objects and massless vector and spinors to be solutions of the 5D and Lorentz’s metrics operators with open boundary condition. As the 5D space and time dimensions increases, due to the uncertainty principle, with the key parameter specified by the Planck’s constant h, the 5D domain becomes very large, and the fields in the 5D domain will become classical with continuous eigen spectrum energies. Hence astronomical objects obey classical laws, except for neutrinos. The observation of neutrino oscillations and its theoretical explanation is a clear illustration of this boundary effect [

Because of the increase in mass distribution throughout the Lorentz space-time of all created matters, the Riemannian curvature also continuously changes, leading to the increase in the gravitational contracting force acting on the massive shells of stars. Whereas stars with masses smaller than the Chandrasekhar limit will shrink to dwarfs of various colors, those with mass > 1.4M_{?} undergoes gravitational collapse eventually to form neutron stars; more details will be followed up in later sections. The initial formation of a matter shell occurs at extremely high temperatures (see Section 4), and the heat loss to the L thermal bath from the spinor void via diffusion takes a long time. At the very initial formation of 5D space-time, the amount of starting energy is almost infinite, while all the cooling processes take a very long time as the domain expands indefinitely according to the homogeneous 5D metric. In fact, it is this ever expansion of the universe according to the metric, that induces the establishment of statistically generated ensemble theory from which thermal dynamics is realized, with the 5D-4D boundary acting like the wall of a heat bath container.

We may also look at the creation process in terms of the space-time and parity nature of the 5D metric, as if each star starts from a completely new 5D. Such a picture is possible, as 5D is finite with no absolute center point, and can be created from absolute NOTHING. Hence multiple 5D can be created at the Big Bang instant, but these domains must be merged into one eventually. The interesting aspect lies in their boundaries that distinguish them! Each boundary is in a 4D Lorentz domain, characterized by their different quarks and leptons mass values! Hence from Perelman’s mapping, these different Lorentz 4D domains (or one unconnected form of 4D Lorentz boundary) are represented by the different galaxies, within each, via Perelman’s mapping, is further separated into stars and planets, having individual 5D void cores. Depending on the core sizes, and in view of the uncertainty principle, different amounts of energy are created within individual 5D cores. Such amounts of energy are represented in terms of the energies of the massless fields, namely the vector potentials and e-trino, anti-e-trino spinors. Through these massless fields, the L_{z} of the quarks is generated in the Lorentz 4D domain(s). In this sense, we may view the above process as the Big Bang creation of the Universe. However, the mass thus generated was not the final amount of mass in the universe. The total mass is actually changing, as the Gauge Constraint converts multiple quarks into hadrons, then into nuclei via Bethe fusion, and to atoms via Coulomb potential, (including 2D Chern-Simons hydrogens), then via Van der Waal potential to molecules, to gases, and crystals. The above-mentioned series of process of formation is a continuous thermodynamic process, via the continuous application of the Law of Entropy, which is built in by the “non-time reversal” nature form of the 5D metric itself. In another word, the act of projection was automatic due to the very nature of the finiteness of the 5D metric, requiring no further action from the creation process. All the amazing complexities of the Universe hence evolve by itself from the beginning based on the homogeneity of the 5D manifold. Each state―change obeys causality, giving raise to even complex life forms, that would also self evolve―with determination to its own future. In another word, the continuous thermal evolution, when applied to life forms, i.e. Darwin evolution, can be viewed also as part of the evolution of creation of matter in the Lorentz space-time.

We will proceed to derive explicit representations of the angular momentum of the 5D structure inside the void over a wide temperature range, and apply the consequence to analyze different types of stellar objects in later sections.

Consider a system of particles in thermal equilibrium in the void. The density of quantum states within elementary momentum dp and elementary real space dr in the 3D spherical void is gs∙g(p)dr∙dp, where gs measures the spin degeneracy and g(p) is the number of states per unit momentum range. As the quantum unit in phase space

(r, p) is h^{3}, the total number of quantum states within the volume of interest is

Since the void is a sphere with radius R_{o},

As the 5D metric (represented by (ct)^{2} = x^{2}, where x is a four-vector) is homogeneous, when the projection action is taken at t = τ_{0}, the 4D space volume (out of the 5D manifold) as represented by x^{2} is fixed, though the shape may take any “close” form. As the 4D space void is enclosed by the Lorentz space-time, which has only 3 space coordinates, the 4D space void must be expressible as 3D × 1D, and all components of x are equal, with the void radius R_{o} being fixed. Note that a similar statement cannot be applied to the energy-momentum metric E^{2} = (cp)^{2}, because action of projection is not taken at fixed E value. We would also remark again that when mass is created due to projection action, a Lorentz boundary is formed, enclosing the 3D × 1D void. Due to 3D spherical symmetry, all eigenstates of spinors within the void must be represented by spherical symmetric functions, namely L' (quantum states pertaining to spinors rotating along the longitudes, not relevant to nonzero angular momentum generation here), and L_{z} (angular momentum due to spinors rotating along the latitudes). It is the net L_{z} that will lead to the mass shell rotation, such that the total angular momentum in the whole universe (including 4D and 5D) remains zero at all time, as explained in the Section 1. The spinors are Fermions, but of opposite charge, and are strictly speaking, of different kinds of Fermions, which follow the relevant statistical distribution(s). The Fermi-Dirac L distribution, which is expressed generally as

where s is the normalization factor, and L_{f} is the “Fermi angular momentum”, satisfying the property that the probability is unity for angular momentum smaller than L_{f}, but is zero for L > L_{f} at temperature T = 0 K. Since L = (2hν/c)R_{o} = 2Eτ_{0} at time τ_{0} (here ν is the frequency), we define the Fermi angular momentum to be L_{f} = 2E_{f}τ_{0}. The normalized factor s is simply kτ_{0} so that Equation (3.2a) becomes

where k is the Boltzmann constant.

The Fermi distribution in (3.2b) now describes a pair of spinors. For pairs of such spinors in the void rotating in-phase so that each pair has zero charge, the angular momentum generated along the spin axis z by spinor pairs rotating along the latitudes of the void, weighted over the Fermi distribution (pair), is

where r' = r∙sinθ, and θ is the polar angle and ψ is the azimuthal angle so that all spinor pairs generating L_{z} within the void are counted. θ is integrated from 0 to π, and ψ is integrated from 0 to 2π (to avoid over-counting because there are orbits along the longitudes). After integrating over r, θ and ψ,

where z is the unit orientation vector for L_{z}. Noting that p stands for the momentum of a pair, giving

It is shown in Appendix A that the above integral can be expressed as a sum of a number of series, so that we have simply

where _{f}/kT

For very small

, (3.6a)

where_{z} is independent of

_{z} is independent of T (3.7a)

For intermediate values of

(3.8)

We name Equation (3.6a) as the First Law, Equation (3.7a) as the Second Law of Angular Momentum, resulting from the 5D projection theory quantum statistics. L_{z} is to be equated to the mass shell angular momentum Iω of the matter object, where I is the shell’s moment of inertia, and ω is its rate of rotation about the unit vector z, as measured or deduced from astronomical studies. Thus the First Law can be expressed as,

where A = D^{−0.25}{7π^{4}/1920}^{−0.25} = 4.52 × 10^{−18} S.I. units. While according to the 5D projection mapping, the void is fixed at t = τ_{0}, thus the void has a radius R_{o} = cτ_{0}. Hence, the Second Law can be expressed as

Dividing Equation (3.6b) by Equation (3.7b), we arrive at

For fixed E_{f}, the “mid/critical temperature” T_{c} of the “transition region” (that between the First and Second Laws) can be found using (3.9). For example, if we take E_{f} = 0.5 Mev, T_{c} = 5.3 × 10^{9} K. We may consider (3.9) as the Third Law, which is universal according to the 5D model. As the First and Second Laws have simple linear relationships, _{f} in the Second Law region and _{f}, T) about the temperature of the transition region for each E_{f} value.

Hence we must determine the E_{f} value for application. From the 5D E, p metric, with the projection into SU(2) × SU(3) × L, the lowest mass value is that of the electron’s rest mass m_{e}. Thus we have the condition E^{2} >_{f} is chosen as 0.5 Mev, indicating that the lightest lepton is generated (see Sections 1 and 4 for more details).

The mapping of the 5D space-time into a 4D Lorentz space-time (represented by general projection P) using the Ricci Flow Theorem, produces a 3D space of a doughnut structure containing matter, but enclosing a void core (in 5D space-time). It has been noted that P can be represented by a combination of the space to time projection operator Po (or time shift operator) and the space to space conformal projection operator P1. From these projections we obtain the “key stable elementary particles” which build up matter in the 4D Lorentz space-time. These particles are electrons, protons, and neutrons. Keeping in mind that the protons and neutrons are built by quarks, which are fractionally charged. Using a 2D circular coordinate transformation as a simplified example, it has been explained in [_{z}, with the spinning axis z perpendicular to the doughnut/sphere plane (of the galaxy). Hence to conserve angular momentum, the matter in the sphere must move in such a way as to generate the same amount of total angular momentum, but rotating in the opposite sense (i.e. ?L_{z}).

In view of SU(2) symmetry and energy consideration, for every lepton creation with a net charge e, a massless and charge-neutral neutrino must also be created to conserve zero spin. It was argued in [_{z} direction) of a new born star. Such huge electromagnetic fields are observed in pulsars and magnetars. Other models have also proposed the idea that enormous amount of electromagnetic energy is radiated from the outer-shell from a typical pulsar (see e.g. [^{10} - 10^{11} Tesla; (see e.g. [^{9} K, happens to be consistent with our Third Law [^{4} solar mass near the centre of a galaxy, and there are numerous young stars near the galactic centre also [

We would like to remark also that at the birth of a star, there is relatively small amount of (massive) matter, and the electrons and quarks must spin very fast in order to counter-balance the angular momentum of the spinor pairs within the 4D space void of the young star. To form a baryon, the right quark members must be combined in a gauge invariant way (with the “equilateral triangular formation”) described in a recent paper [

The angular momentum of a spherical shell with external radius R_{p}, internal/void radius R_{o} is

Here P is the period of rotation. “d” is the averaged mass density. The asymptotic value of angular momentum Iω_{m} for the pulsar model is thus

Taking the Vela pulsar as an example, with R_{p} = 10^{4} m, P = 0.089 s, Iω_{m} = 0.403 × 10^{40} J-s. From simple mass, density consideration,

where d is the averaged density and M_{p} the mass of pulsar. Based on the above discussion on mass generation, we assume that the mass density is simply ~ nuclear mass density = 3 × 10^{17} kg/m^{3}. This constraint, together with the condition that R_{o} > 0 in our model, there is an upper limit for the mass (called M_{c}) for each R_{p} measured/deduced:

Based on Equations (3.6a) and (4.1), we can calculate the temperature T, in terms of P, R_{p}, and M_{p}.

(4.2a) or (4.2b)

where k is the Boltzmann constant. Equation (4.2b) may be called the Lemma of the First Law for spherical shell stellar objects with matter enclosing a 5D void core. The numerical values of the constant D, arising from quantum states in counting the Fermi-Dirac distribution of the spinors, is 6.73726 × 10^{69} S.I. units whereas in (4.2b) I = 0.35514, resulting from the integration over angular momentum under the condition of kT ? E_{f} (see Section (III) and the two Appendices). According to (4.2b), with P, R_{p} fixed, T is a function of M_{p} only. Whereas the rotation period can be measured rather accurately due to the light-house effect, the R_{p} values for pulsars have been commonly assumed to be 1.0 × 10^{4} m. The relevant parameters of some examples of pulsars are listed in _{p} (in units of solar mass) as a parameter and plot T- (M_{p}/M_{?}) graph in _{c} (maximum possible mass) is the same for all pulsars with different rotation periods, but only dependent on R_{p}. With R_{p} = 10^{4} m, all lines in _{c}/M_{?} = 0.631477, which is entered into

Pulsar name | B field (T) | P(s) | Iω_{m} (J-s) | M_{c}/M_{?}(R_{p} = 10^{4} m) | M_{c}/M_{?}(R_{p} = 5000 m) |
---|---|---|---|---|---|

PSR B1937+21 | N | 0.0016 | 1.9739 × 10^{41} | 0.631477 | 0.078935 |

PSR B0833−45(vela) | 6.8 × 10^{8} | 0.089 | 3.5486 × 10^{39} | 0.631477 | 0.078935 |

RX J0822−4300 | 6.5 × 10^{6} | 0.112 | 2.8199 × 10^{39} | 0.631477 | 0.078935 |

SR J1740+1000 | 3.7 × 10^{9} | 0.154 | 2.0508 × 10^{39} | 0.631477 | 0.078935 |

1E 1207.4−5209 | 2.0 × 10^{7} | 0.424 | 7.4488 × 10^{38} | 0.631477 | 0.078935 |

SR B2334+61 | 2.0 × 10^{9} | 0.495 | 6.3804 × 10^{38} | 0.631477 | 0.078935 |

RX J1605.3+3249 | 1.5 × 10^{10} | 3.39 | 9.3165 × 10^{37} | 0.631477 | 0.078935 |

RX J0806.4−4123 | 5.1 × 10^{9} | 11.37 | 2.7777 × 10^{37} | 0.631477 | 0.078935 |

with matter, with the void volume tending to zero as a limit, so that T is approaching infinity. For any mass smaller than the critical mass M_{c}, R_{o} is finite and non-zero, with T also finite.

Any point of a T-(M_{p}/M_{?}) graph for a fixed P tells that to acquire the situation where a shell mass of a certain value (take for example, M_{p}/M_{?} = 0.4 in ^{11} K, so that the in phase spinors rotating would have a total angular momentum of 2.25624 × 10^{37} J-s (calculated using Equation (3.6a)) to balance the Iω of the matter shell according to the First Law. At that situation, the void radius is 7.1567 × 10^{3} m (according to (4.1c)) whereas the radius of the star observed is roughly 10^{4} m. The asymptotic angular momentum calculated according to (4.1b) is entered in

When a pulsar is newly born and evolves, the evolution path cannot be taken to follow a line in _{c}/M_{?} is smaller than ~0.6. Suppose the three pulsars just considered have a common radius of 5000 m instead, and we have the T- (M_{p}/M_{?}) graph in _{?} according to Equation (4.1d). In order to facilitate a qualitative description on the consequence of the 5D theory in some stage of pulsar evolution, let us consider point A in _{p}/M_{?}) graph for M_{p}/M_{?} < 0.01, at point A. Hence this state is represented by the set of numbers (M_{p}/M_{?} = 0.01, P = 0.0016 s, R_{p} = 5000 m, R_{o} = 4.77925 × 10^{3} m, T = 4.0735 × 10^{11} K in the void, according to (4.2b)). The pulsar gains mass after a finite time interval according to this model; also it is observed in general that a pulsar spins down continuously (except for the glitch phenomenon). To obtain the next discrete step in evolution, we need to use another line pertaining to a longer P, bigger R_{p}, and a bigger M_{p}/M_{?} value. Now go back to _{p}/M_{?} = 0.1. According to _{p}/M_{?} = 0.1, P = 0.089 s, R_{p} = 10^{4} m, R_{o} = 9.441529 × 10^{3} m, T = 1.893 × 10^{11} K at the void from (4.2b)). The transition from set one to set two of the above numbers is in line with the model of evolution discussed above. Such a hypothetical evolution step is only a schematic representation. Though the observed P and the rate of change of P of pulsars are well documented, yet accurate experimental results of R_{p} and M_{p} still await, before we can test the theory in details. We wish to point out here that many pulsars could have masses < 1.4 M_{?}, whereas some pulsars having larger masses, should have R_{p} > 10^{4} m. In _{c}/M_{?} versus R_{p} line in log scale. The circle indicates the maximum mass a pulsar can have, irrespective to its P value, if R_{p} is 10^{4} m. The triangle represents that condition that if M_{p} = 1.4 M_{?}, R_{p} should be at least as large as 1.304 × 10^{4} m. We would remark also that the notion of a pulsar’s mass being less than 1.4 M_{?} is not new; in fact, based on X-ray observations of polar cap

characteristics, Pavlov et al. [

Stassun et al. [^{3}) in the range 1) 3.4 - 15.0; 20 stars in the range, 2) 15.1 - 32.0; 47 stars in the range, 3) 32.1 - 90.0; 15 stars in the range, 4) 90.1 - 270.0, and 4 stars in the range, 5) 270.1 - 540. We plot in _{o} is the void radius, the angular momentum Iω is calculated assuming all the mass matter fill up the whole star volume as an approximation, since ρ = R_{o}/R is very small, being ~10^{−3}. We also assume that the star has already cooled down to the region specified by the Second Law: ρ = R_{o}/R = D^{(−0.25)}∙{1/4}^{−}^{0.25}}∙(Iω)^{0.25}/(E_{f}∙R) as given in (3.7b).

It is well established that accurate measurement of stellar spin rates gives useful information to determine their

Star | M/M_{?} | R/R_{?} | P (days) | {Iω}^{0.25}/R | R_{o}/R (10^{−3}) | Density(kg/m^{3}) |
---|---|---|---|---|---|---|

1171 | 0.26 | 1.98 | 7.4 | 32.2 | 1.98 | 47.98 |

1219 | 0.4 | 1.96 | 1.31 | 55.5 | 3.42 | 75.02 |

1235 | 0.24 | 2.97 | 1.69 | 37.26 | 2.30 | 12.94 (i) |

1279 | 0.16 | 1.83 | 0.75 | 52.55 | 3.24 | 36.87 |

1297 | 1.38 | 1.90 | 6.63 | 51.26 | 3.16 | 284.70 (v) |

1308 | 0.15 | 3.96 | 8.35 | 19.25 | 1.19 | 34.10 |

1325 | 0.20 | 1.30 | 4.44 | 42.26 | 2.60 | 128.55 (iv) |

1354 | 0.23 | 1.67 | 0.80 | 59.27 | 3.65 | 69.73 |

1357 | 0.30 | 2.56 | 1.37 | 44.77 | 2.76 | 25.25 (ii) |

1368 | 0.23 | 1.35 | 2.76 | 48.37 | 2.98 | 132.00 (iv) |

1396 | 0.28 | 2.03 | 1.92 | 45.36 | 2.80 | 47.27 |

1428 | 0.17 | 1.31 | 1.16 | 56.54 | 3.48 | 106.78 (iv) |

1440 | 0.27 | 1.47 | 1.36 | 57.59 | 3.55 | 120.00 (iv) |

1453 | 0.23 | 1.66 | 1.36 | 52.06 | 3.21 | 71.00 |

1465 | 0.27 | 1.85 | 1.28 | 52.11 | 3.21 | 60.00 |

1474 | 0.23 | 2.30 | 6.03 | 30.48 | 1.88 | 26.70 (ii) |

1485 | 0.23 | 1.44 | 6.34 | 38.84 | 2.39 | 108.80 (iv) |

1500 | 0.24 | 1.90 | 8.82 | 30.82 | 1.90 | 49.40 |

1501 | 0.29 | 1.74 | 8.73 | 33.85 | 2.09 | 77.70 |

1511 | 0.38 | 1.89 | 1.54 | 53.62 | 3.30 | 79.48 |

1522 | 0.35 | 2.09 | 7.29 | 33.86 | 2.09 | 54.14 |

1545 | 0.25 | 1.79 | 5.32 | 36.40 | 2.24 | 61.55 |

1566 | 0.23 | 1.57 | 7.00 | 35.54 | 2.19 | 83.93 |

1568 | 0.25 | 2.14 | 5.06 | 33.71 | 2.08 | 36.00 |

1618 | 0.24 | 2.80 | 4.44 | 30.14 | 1.86 | 15.44 (ii) |

1627 | 0.26 | 1.93 | 10.10 | 30.16 | 1.86 | 51.07 |

1631 | 0.25 | 2.08 | 9.46 | 29.24 | 1.80 | 39.23 |

1692 | 0.21 | 2.35 | 1.99 | 38.90 | 2.40 | 22.85 (ii) |

1694 | 0.22 | 1.89 | 3.09 | 39.30 | 2.42 | 46.02 |

1753 | 0.16 | 3.23 | 4.20 | 25.63 | 1.58 | 65.82 |

1760 | 0.20 | 1.57 | 6.13 | 35.48 | 2.19 | 72.98 |

1805 | 0.30 | 2.03 | 5.32 | 35.60 | 2.19 | 50.64 |

1841 | 0.39 | 2.35 | 7.33 | 32.77 | 2.02 | 42.44 |

1922 | 0.13 | 1.87 | 11.30 | 25.05 | 1.54 | 28.07 (ii) |

1966 | 0.22 | 1.86 | 7.11 | 32.17 | 1.98 | 48.28 |

1982 | 0.37 | 2.72 | 7.78 | 29.61 | 1.82 | 25.95 (ii) |

2036 | 0.21 | 3.89 | 2.26 | 29.29 | 1.81 | 5.04 (i) |

2037 | 0.21 | 1.73 | 2.14 | 44.50 | 2.74 | 57.27 |

2124 | 0.28 | 2.23 | 8.35 | 29.97 | 1.85 | 35.65 |
---|---|---|---|---|---|---|

2168 | 0.21 | 1.63 | 6.12 | 35.26 | 2.17 | 68.47 |

2246 | 0.33 | 1.63 | 9.46 | 35.4 | 2.18 | 107.60 (iv) |

2301 | 0.15 | 1.42 | 0.85 | 56.89 | 3.51 | 73.98 |

2425 | 0.13 | 1.44 | 1.71 | 45.77 | 2.82 | 61.48 |

2428 | 1.29 | 4.92 | 6.83 | 31.09 | 1.92 | 15.00 (i) |

2470 | 0.23 | 1.55 | 2.81 | 44.93 | 2.77 | 87.22 |

2479 | 0.36 | 2.38 | 6.12 | 33.39 | 2.06 | 37.70 |

2583 | 0.27 | 2.39 | 1.07 | 47.95 | 2.96 | 27.93(ii) |

2698 | 0.06 | 2.61 | 2.13 | 26.52 | 1.63 | 4.77 (i) |

2739 | 0.49 | 2.59 | 7.63 | 27.55 | 1.70 | 39.83 |

2744 | 0.43 | 1.79 | 6.53 | 39.60 | 2.44 | 105.87 (iv) |

2784 | 0.22 | 1.26 | 3.96 | 45.24 | 2.79 | 155.30 (iv) |

2816 | 0.16 | 2.58 | 6.29 | 26.00 | 1.60 | 13.16 (i) |

2843 | 0.14 | 1.92 | 6.86 | 28.53 | 1.76 | 27.93 (ii) |

2847 | 0.27 | 2.45 | 7.66 | 28.68 | 1.77 | 25.92 (ii) |

2913 | 0.21 | 1.65 | 5.21 | 36.50 | 2.25 | 66.01 |

2918 | 0.28 | 2.55 | 3.32 | 26.48 | 1.63 | 23.85 (ii) |

2919 | 0.35 | 2.51 | 7.70 | 30.48 | 1.88 | 31.26 (ii) |

2973 | 0.15 | 2.75 | 2.49 | 31.25 | 1.93 | 10.19 (i) |

3007 | 0.16 | 2.43 | 1.71 | 37.11 | 2.29 | 15.53 (ii) |

3014 | 1.13 | 1.74 | 7.82 | 48.88 | 3.01 | 302.88 (v) |

3026 | 0.30 | 2.26 | 7.74 | 30.87 | 1.90 | 36.25 |

3028 | 0.17 | 2.34 | 4.76 | 29.72 | 1.83 | 18.74 (ii) |

3032 | 0.20 | 1.97 | 3.21 | 37.23 | 2.29 | 36.94 |

3082 | 0.31 | 2.51 | 2.84 | 37.95 | 2.34 | 27.68 (ii) |

3087 | 0.22 | 2.04 | 6.43 | 31.49 | 1.94 | 36.59 |

3088 | 0.31 | 2.84 | 4.62 | 31.59 | 1.95 | 19.11 (ii) |

3097 | 0.28 | 2.18 | 3.60 | 37.41 | 2.31 | 38.16 |

3115 | 0.27 | 1.91 | 6.73 | 33.86 | 2.09 | 54.72 |

3122 | 0.15 | 2.25 | 1.14 | 42.00 | 2.59 | 18.60 (ii) |

3142 | 0.23 | 1.62 | 8.64 | 33.20 | 2.04 | 76.39 |

3158 | 0.34 | 2.21 | 8.69 | 31.29 | 1.93 | 44.49 |

3161 | 0.27 | 1.89 | 0.84 | 57.29 | 3.53 | 56.47 |

3178 | 0.20 | 1.98 | 5.03 | 33.20 | 2.05 | 36.38 |

3189 | 0.30 | 1.54 | 7.00 | 38.35 | 2.36 | 115.99 (iv) |

3197 | 0.20 | 2.13 | 6.30 | 30.25 | 1.86 | 29.22 (ii) |

3217 | 0.23 | 1.35 | 3.74 | 44.83 | 2.76 | 132.00 (iv) |

3240 | 0.14 | 2.56 | 5.51 | 26.10 | 1.61 | 11.78 (i) |

3314 | 0.20 | 1.42 | 5.26 | 38.76 | 2.39 | 98.64 (iv) |

3341 | 1.12 | 1.80 | 1.65 | 70.78 | 4.36 | 271.19(v) |

3384 | 0.39 | 2.42 | 5.59 | 34.56 | 2.20 | 38.66 |

3406 | 0.54 | 2.03 | 2.79 | 48.69 | 3.00 | 91.15 (iv) |

3438 | 0.14 | 1.55 | 2.55 | 44.04 | 2.71 | 53.09 |

3447 | 0.16 | 3.87 | 1.16 | 32.40 | 2.00 | 3.90 (i) |
---|---|---|---|---|---|---|

3560 | 0.62 | 3.04 | 1.03 | 52.84 | 3.26 | 31.16 |

3613 | 0.22 | 1.87 | 1.13 | 50.81 | 3.13 | 47.5 |

3662 | 0.26 | 2.24 | 8.18 | 29.50 | 1.82 | 32.67 |

3666 | 0.23 | 2.19 | 4.91 | 32.84 | 2.02 | 30.92 (ii) |

3668 | 0.66 | 1.20 | 8.18 | 50.89 | 3.14 | 539.35 (v) |

3672 | 0.30 | 1.27 | 7.48 | 41.54 | 2.56 | 206.80 (iv) |

3678 | 0.36 | 1.61 | 6.53 | 39.94 | 2.46 | 121.81 (iv) |

3710 | 0.30 | 2.24 | 9.27 | 30.81 | 2.00 | 37.69 |

3756 | 0.60 | 1.76 | 4.91 | 46.61 | 2.87 | 155.41 (iv) |

3885 | 0.16 | 1.84 | 2.99 | 37.08 | 2.29 | 36.27 |

3918 | 0.12 | 1.70 | 1.18 | 45.30 | 2.79 | 34.49 |

4021 | 0.18 | 2.10 | 0.81 | 49.55 | 3.05 | 27.45 (ii) |

4047 | 0.21 | 2.27 | 2.07 | 39.18 | 2.41 | 25.35 (ii) |

4090 | 0.16 | 2.69 | 7.59 | 24.30 | 1.50 | 11.60 (i) |

ages. Very recently, using the methodology of gyrochronology [_{o} and other parameters. The mass is read approximately from the data point of the P - (B-V)_{o}/M graph of [_{?})^{0.945}, for main sequence star M < 1.66 M_{?}, as in reference [^{0.25}/R, mass density, and ρ = R_{o}/R according to the Second Law, where the temperature T = T_{c} = 5.3 × 10^{9} K, in

Star no. | (B-V)o | M/M_{?} | R/R_{?} | P (s) | (Iω)^{0.25}/R | ρ [×10^{−3}] | Den (kg/m^{3}) |
---|---|---|---|---|---|---|---|

5,111,207 | 0.41 | 1.405 | 1.46170 | 4.56829 × 10^{5} | 61.73 | 3.8000 | 6.19 × 10^{2} |

5,023,899 | 0.42 | 1.370 | 1.42727 | 4.15584 × 10^{5} | 63.96 | 3.9400 | 6.52 × 10^{2} |

5,023,760 | 0.43 | 1.355 | 1.41250 | 4.12992 × 10^{5} | 64.21 | 3.9600 | 6.79 × 10^{2} |

5,024,227 | 0.43 | 1.355 | 1.41250 | 4.37184 × 10^{5} | 63.31 | 3.9000 | 6.79 × 10^{2} |

5,024,122 | 0.45 | 1.300 | 1.35826 | 5.49500 × 10^{5} | 60.34 | 3.7190 | 7.33 × 10^{2} |

5,112,499 | 0.46 | 1.280 | 1.338503 | 3.83600 × 10^{5} | 66.24 | 4.0823 | 7.54 × 10^{2} |

5,113,601 | 0.36 | 1.280 | 1.338503 | 6.056640 × 10^{5} | 59.10 | 3.6418 | 7.54 × 10^{2} |

5,026,583 | 0.49 | 1.228 | 1.287059 | 4.233600 × 10^{5} | 65.23 | 4.0198 | 8.13 × 10^{2} |

4,938,993 | 0.50 | 1.210 | 1.269223 | 1.025570 × 10^{6} | 52.46 | 3.2330 | 8.36 × 10^{2} |

5,111,834 | 0.57 | 1.101 | 1.160900 | 1.200000 × 10^{6} | 51.51 | 3.1740 | 9.94 × 10^{2} |

5,111,908 | 0.58 | 1.090 | 1.149940 | 1.504220 × 10^{6} | 48.79 | 3.0066 | 1.01 × 10^{3} |

5,024,856 | 0.62 | 1.037 | 1.097025 | 1.571600 × 10^{6} | 48.79 | 3.0070 | 1.11 × 10^{3} |

5,024,280 | 0.63 | 1.026 | 1.086026 | 1.499900 × 10^{6} | 49.34 | 3.0410 | 1.13 × 10^{3} |

5,112,507 | 0.63 | 1.026 | 1.086026 | 1.571600 × 10^{6} | 48.77 | 3.0055 | 1.13 × 10^{3} |

5,023,796 | 0.64 | 1.012 | 1.072016 | 1.581120 × 10^{6} | 48.85 | 3.0101 | 1.16 × 10^{3} |

5,024,008 | 0.65 | 1.000 | 1.060000 | 1.589760 × 10^{6} | 49.05 | 3.0229 | 1.19 × 10^{3} |

5,023,724 | 0.66 | 0.990 | 1.049980 | 1.555200 × 10^{6} | 49.43 | 3.0462 | 1.21 × 10^{3} |

5,023,875 | 0.67 | 0.978 | 1.037949 | 1.583712 × 10^{6} | 49.34 | 3.0406 | 1.24 × 10^{3} |

5,112,268 | 0.68 | 0.972 | 1.031931 | 1.615680 × 10^{6} | 49.16 | 3.0296 | 1.25 × 10^{3} |

4,937,169 | 0.70 | 0.952 | 1.011854 | 1.695168 × 10^{6} | 48.80 | 3.0073 | 1.30 × 10^{3} |

5,025,271 | 0.70 | 0.952 | 1.011854 | 1.839456 × 10^{6} | 47.81 | 2.9465 | 1.30 × 10^{3} |

5,111,939 | 0.70 | 0.952 | 1.011854 | 1.879200 × 10^{6} | 47.56 | 2.9310 | 1.30 × 10^{3} |

5,112,871 | 0.71 | 0.946 | 1.005826 | 1.836000 × 10^{6} | 47.90 | 2.9520 | 1.31 × 10^{3} |

5,023,666 | 0.73 | 0.930 | 0.989743 | 1.861056 × 10^{6} | 47.92 | 2.9533 | 1.35 × 10^{3} |

5,024,182 | 0.75 | 0.916 | 0.975657 | 1.839456 × 10^{6} | 48.23 | 2.9720 | 1.39 × 10^{3} |

5,023,926 | 0.77 | 0.903 | 0.962567 | 1.798848 × 10^{6} | 48.65 | 2.9981 | 1.43 × 10^{3} |

4,937,149 | 0.80 | 0.883 | 0.942407 | 1.873152 × 10^{6} | 48.40 | 2.9826 | 1.49 × 10^{3} |

4,936,891 | 0.85 | 0.862 | 0.921213 | 1.899070 × 10^{6} | 48.49 | 2.9884 | 1.56 × 10^{3} |

4,937,119 | 0.87 | 0.852 | 0.911111 | 2.011392 × 10^{6} | 47.93 | 2.9534 | 1.59 × 10^{3} |

4,937,356 | 0.89 | 0.847 | 0.906057 | 1.834300 × 10^{6} | 49.11 | 3.0263 | 1.61 × 10^{3} |

HD154708 | N | 1.500 | 1.554930 | 4.636700 × 10^{5} | 60.99 | 3.7584 | 0.5634 × 10^{3} |

the spinor pairs is assumed to be E_{f} = 0.5 MeV, as before. We observe that the mass density varies from 619 kg/m^{3} to 1610 kg/m^{3} in this cluster so far found. The log ρ - log P graph is shown in ^{3}, and we group it within the NGC 6819 group [

We list in _{?}, radius R/R_{?}, period of rotation P(s) of 5 stars in M35 according to [_{o} relation of [_{?})^{0.945}, for M < 1.66 M_{?}, as in reference [_{o}/R as governed by the Second Law (small T = T_{c}) as well as the mean mass density for each star are listed in _{o}/R governed by the Second Law (small T = T_{c}) for 14 members of the main sequence as shown. The quantity (Iω)^{0.25}/R, which has the units of kg^{0.25}∙m^{−0.5}∙s^{−0.25} is also listed for convenience of calculating ρ. We observe that the values of ρ fall into a very narrow range, decreasing for increasing P in

Star no. | (B-V)_{o} | M/M_{?} | R/R_{?} | P(s) | (Iω)^{0.25}/R | ρ[×10^{−2}] | Den (kg/m^{3}) |
---|---|---|---|---|---|---|---|

1 | 0.42 | 1.385 | 1.442035 | 7.60324 × 10^{3} | 173.485 | 1.06900 | 6.522 × 10^{2} |

2 | 0.47 | 1.260 | 1.318730 | 8.29440 × 10^{3} | 173.358 | 1.06832 | 7.758 × 10^{2} |

3 | 0.45 | 1.300 | 1.358258 | 9.24480 × 10^{3} | 167.550 | 1.03250 | 7.326 × 10^{2} |

4 | 0.45 | 1.300 | 1.358258 | 9.67680 × 10^{3} | 165.648 | 1.02080 | 7.326 × 10^{2} |

5 | 0.73 | 0.930 | 0.989743 | 1.08864 × 10^{4} | 173.286 | 1.06787 | 1.355 × 10^{3} |

general, as expected. These stars have densities ranging from 563 to 5140 kg/m^{3}. The ρ-P plot is indicated in ^{3} kg/m^{3} to 9.8 × 10^{4} kg/m^{3}; a dotted line of best fit can be drawn between them. Note that this dotted line (representing stars with higher mass densities) is “above” the solid line. We also include the data of a pre-stellar star named Ap J0323 + 4853 in alpha Persei [_{?} in ^{4} kg/m^{3}, greater than those in the NGC 2516 set, and is anticipated to be above the line of best fit associated with this group.

Note also that the data for many stars in the main sequence have not been used, as there are lots of uncertainties about the periods of rotation P, though the masses can be deduced quite accurately from luminosity-mass relationship. Also, there are different paths of evolution for stars with high masses.

Star | M/M_{?} | Teff (×10^{3} K) | R (10^{8} m) | P(s) | (Iω)^{0.25}/R | ρ[10^{−3}] | Den (10^{3} kg/m^{3}) |
---|---|---|---|---|---|---|---|

Sun | 1.0000 | 5.800 | 6.9550 | 2.16000 × 10^{6} | 46.79 | 2.8760 | 1.3880 |

KIC892376 | 0.4699 | 3.810 | 3.6112 | 1.32365 × 10^{5} | 108.0 | 6.6400 | 4.7400 |

1026474 | 0.5914 | 4.120 | 4.48777 | 1.3556 × 10^{5} | 102.0 | 6.2700 | 3.1090 |

1026146 | 0.6472 | 4.260 | 4.8869 | 1.2866 × 10^{6} | 56.97 | 3.5030 | 2.6344 |

1162635 | 0.4497 | 3.760 | 3.4643 | 1.3546 × 10^{6} | 60.988 | 3.7500 | 5.1385 |

1164102 | 0.5606 | 4.050 | 4.2226 | 2.7210 × 10^{6} | 54.13 | 3.3300 | 3.5400 |

1027110 | 0.6046 | 4.160 | 4.58237 | 1.4697 × 10^{5} | 99.214. | 6.1140 | 2.9850 |

1160684 | 0.5239 | 3.950 | 4.002144 | 3.6200 × 10^{4} | 145.797 | 8.9840 | 3.8826 |

1027277 | 0.6735. | 4.330 | 5.07437 | 5.1960 × 10^{6} | 39.833 | 2.4547 | 2.4488. |

HD154708 | 1.5000 | N | 10.81457 | 4.6367 × 10^{5} | 60.99 | 3.7583 | 0.5634 |

IM VirB | 0.6644 | N | 4.7363 | 1.1320 × 10^{5} | 106.95 | 6.5695 | 2.9710 |

GU BooA | 0.6101 | N | 4.3608 | 4.2336 × 10^{4} | 139.524 | 8.5700 | 3.4950 |

UV PscB | 0.7644 | N | 5.8074 | 6.9120 × 10^{4} | 113.16 | 6.9730 | 0.9310 |

YY GemA | 0.5992 | N | 4.3079 | 7.5168 × 10^{4} | 120.8 | 7.4457 | 3.5600 |

Star | M/M_{?} | R/R_{?} | P(s) | (Iω)^{0.25}/R | ρ[10^{−3}] | Den (10^{3} kg/m^{3}) |
---|---|---|---|---|---|---|

N2516-1-1-784 | 0.20 | 0.24 | 5.6070 × 10^{4} | 159.0 | 9.800 | 2.430 × 10^{4} |

N2516-1-1-351 | 0.44 | 0.41 | 2.0028 × 10^{5} | 107.82 | 6.644 | 9.020 × 10^{3} |

N2516-1-1-958 | 0.49 | 0.45 | 5.4354 × 10^{5} | 82.37 | 5.076 | 7.593 × 10^{3} |

N2516-1-1-881 | 0.55 | 0.51 | 6.6330 × 10^{5} | 75.76 | 4.669 | 5.855 × 10^{3} |

N2516-1-1-1470 | 0.56 | 0.52 | 7.6058 × 10^{5} | 72.84 | 4.489 | 5.624 × 10^{3} |

ApJ0323+4853 | 0.09 | 0.1089 | 2.7360 × 10^{4} | 231.4 | 14.26 | 9.800 × 10^{4} |

Rotation periods for some very low mass stars, anticipated to be brown dwarfs, have been measured and deduced in the Pleiades [_{?}, ρ, R_{o}, and mass density (found to be varying from 1.5 × 10^{4} to 1.22 × 10^{5} kg/m^{3}) are calculated and are entered into ^{9} kg/m^{3}), as expected, because brown dwarfs are slightly “younger” than the old white dwarf stars. The representative star BPL 138 with mass 0.25 M_{?} seems to be out of the line. As the values of ρ are still in the range 10^{−2}, we can simply take the star model as one with mass filling matter almost to the centre, with a small void radius. The ρ-P plots for the ten brown dwarfs are indicated in

Star | M/M_{?} | R/R_{?} | P(s) | (Iω)^{0.25}/R | Ρ = R_{o}/R[×10^{−2}] | Den (10^{4} kg/m^{3}) |
---|---|---|---|---|---|---|

BPL 102 | 0.25 | 0.26981 | 7.7040 × 10^{4} | 142.3 | 0.877 | 1.50 |

BPL 106 | 0.08 | 0.09744 | 1.4688 × 10^{4} | 277.5 | 1.710 | 12.20 |

BPL 115 | 0.10 | 0.12031 | 1.0476 × 10^{4} | 287.4 | 1.771 | 8.10 |

BPL 125 | 0.15 | 0.17649 | 6.9660 × 10^{4} | 163.25 | 1.006 | 3.85 |

BPL 129 | 0.13 | 0.15416 | 3.4700 × 10^{4} | 200.93 | 1.238 | 5.00 |

BPL 138 | 0.25 | 0.28600 | 9.2916 × 10^{4} | 135.8 | 0.837 | 5.97 |

BPL 150 | 0.18 | 0.20967 | 6.6456 × 10^{4} | 158.87 | 0.979 | 2.76 |

BPL 164 | 0.13 | 0.15416 | 7.2576 × 10^{4} | 166.6 | 1.023 | 5.00 |

BPL 190 | 0.15 | 0.17649 | 1.4497 × 10^{5} | 135.14 | 0.839 | 3.85 |

BPL 102 | 0.25 | 0.26981 | 7.7040 × 10^{5} | 142.3 | 0.877 | 1.50 |

When the mass of a matured star is large enough, reaching the critical gravity value, it collapses into a white dwarf (WD) [_{o}/R is taken to be that specified by the Second Law, which represents the stable state where ρ stays constant when the temperature in the void is =T_{c} = 5.3 × 10^{9} K. The mass of the “non- magnetic” WD as published in 2003 have masses within a narrow range of ~0.5 to 0.6 M_{?}, and magnetic data is not available for a number of members considered. We list a number of isolated white dwarfs in _{o}. Notice that whereas the P value covers a relative wide range from ~10^{3} to 10^{5} s, the mass density falls within a very narrow range of several times of 10^{8} kg/m^{3} for WD. Remark also that as the star slows down in rotation, the void radius R_{o} shrinks accordingly. The values of R are deduced from the Hamada-Salpeter relation for dwarfs [^{8} kg/m^{3}, is higher than that of brown dwarfs (BD), this line is “above” that of the one marked BD.

Similarly, we shall analyze some data of some isolated magnetic white dwarfs (MWD) [_{?} < M < 1M_{?}, and the surface magnetic field varies from 0.07 Tesla to even 1000 Tesla = 10^{7} Gauss, whereas the period of rotation varies from ~10^{3} to longer than one hundred years! The last three columns gives the values calculated for ρ, the mean mass density and the radius of the void core. For the MWD members, the calculated data pair (ρ, P) are indicated by triangles in ^{6} - 10^{9} s), the measured/deduced data point from other groups over the years follow a straight line according to our model. The slope is about −0.24 for the MWD. It was noted in the key reference [^{0.25}. The R_{o} value of MWD also decreases with increasing P.

We propose that (WD) and magnetic white dwarfs (MWD) form two sub-groups of stars [

The relative fast rotation rate ~10^{3} s of MWD & WD (as compared to P = 2.16 × 10^{6} s for the Sun) suggests that they have evolved from very fast rotating stars, such as pulsars. Therefore it is tempting to consider the isolated MWDs (as well as WDs) to be members of the later stage of pulsars. We hypothesize that in the future, periods of rotation > 10^{9} s will be found for MWD/WD with advancement of measurement methodology and more space-flight experimentation. We have already analyzed the data for other groups of stars in the previous sections. Here without showing the data points, we just take the lines of best fit using all the stars in Tables 2-6 (see also Figures 4-6) to represent the ρ-P relations of the pre-main-sequence stars of the Orion Nebula (dotted line), the NGC 2819 group, the M35 group, examples of low-to-mid mass main sequence stars, and members of the NGC 2516 group in

To have some feeling about the transition from the First Law to the Second Law, we indicate in

Star (WD) | M/M_{?} | R/R_{?} | Teff (K) | P (s) | ρ = R_{o}/R | Den (10^{9} kg/m^{3}) | R_{o} (×10^{6} m) |
---|---|---|---|---|---|---|---|

GD140 | 0.52 | 1.32 | 2.30 × 10^{4} | 1.037 × 10^{3} | 1.439 × 10^{−1} | 0.3192 | 1.3210 |

Grw+73 8031 | 0.52 | 1.32 | 1.54 × 10^{4} | 1.296 × 10^{3} | 1.361 × 10^{−1} | 0.3192 | 1.2497 |

WD1337+70 | 0.52 | 1.32 | 2.10 × 10^{4} | 1.728 × 10^{3} | 1.267 × 10^{−1} | 0.3192 | 1.1630 |

LB253 | 0.52 | 1.32 | 1.92 × 10^{4} | 2.592 × 10^{3} | 1.145 × 10^{−1} | 0.3192 | 1.0509 |

W1346 | 0.52 | 1.32 | 2.15 × 10^{4} | 5.184 × 10^{3} | 0.963 × 10^{−1} | 0.3192 | 0.8837 |

G1423-B2B | 0.52 | 1.32 | 1.40 × 10^{4} | 6.998 × 10^{3} | 0.893 × 10^{−1} | 0.3192 | 0.8198 |

PG2131+066 | 0.62 | 1.19 | 8.00 × 10^{4} | 1.814 × 10^{4} | 0.775 × 10^{−1} | 0.5195 | 0.6410 |

L19-2 | 0.60 | 1.22 | 1.22 × 10^{4} | 9.504 × 10^{4} | 0.501 × 10^{−1} | 0.4665 | 0.4255 |

NGC 1501 | 0.55 | 1.28 | 8.10 × 10^{4} | 1.011 × 10^{5} | 0.472 × 10^{−1} | 0.3703 | 0.4199 |

Star | M/M_{?} | R/R_{⊙}(×0.01) | B (100T ) | P (s) | ρ = R_{o}/R(×0.1) | Den (10^{9} kg/m^{3}) | R_{o} (×10^{6} m) |
---|---|---|---|---|---|---|---|

WD0533+053 | 0.71 | 1.10 | 20 | 3.600 × 10^{3} | 1.2486 | 0.7532 | 1.13690 |

WD1031+234 | 0.93 | 0.88 | 500 - 1000 | 1.224 × 10^{4} | 1.0998 | 1.9270 | 0.67312 |

WD0548-001 | 0.69 | 1.13 | 10-20 | 1.482 × 10^{4} | 0.8587 | 0.6752 | 0.67485 |

WD0009+501 | 0.74 | 1.07 | 0.2 | 2.160 × 10^{4} | 0.8173 | 0.8530 | 0.60822 |

WD0011-134 | 0.71 | 1.10 | 16.7 | 4.680 × 10^{4} | 0.6576 | 0.7530 | 0.50307 |

WD1533-057 | 0.94 | 0.86 | 31.0 | 8.640 × 10^{4} | 0.6844 | 2.0870 | 0.40934 |

WD0912+536 | 0.75 | 1.05 | 100.0 | 1.149 × 10^{5} | 0.5451 | 0.9150 | 0.39806 |

WD1953-011 | 0.74 | 1.07 | 0.07 | 1.246 × 10^{5} | 0.5274 | 0.8530 | 0.39249 |

WD1829+547 | 0.90 | 0.90 | 1700 - 1800 | ≥100 | 0.0584 | 1.7433 | 0.03657 |

WD1031+234, WD0912+536. The same graph for the sun is indicated by the dash-dot curve. The linear portion of each line represents the region specified by the First Law, at higher temperatures. Physically, as T decreases, there are more spinors with energies f, and the star enters into the transition region. In the ρ-T representation, the straight line curves up to become a horizontal line. The star is then becoming stable, with fusion taking place to burn up what- ever fuels are available, while the heat energy from the void diffuses to the stellar surface and radiates as luminescent heat. The star cools down and become an old star, while the ratio ρ = R _{o}/R tends to an asymptotic constant.

Mathematically, we wish to point out again the “mid-transition point” indicated by the particular temperature T_{c} =5.3 × 10^{9} K is the intersection of the straight line representing asymptotically the First Law (with finite negative slope) and the horizontal line representing asymptotically the Second Law. The T_{c} value for each stable stellar object is the same, and is therefore universal, for a fixed E_{f}, with reason discussed in earlier sections already.

The ρ -T curve for the Sun is also shown in the same

There are only up to 1000 pulsars found so far, but it is estimated that there are around 10^{9} neutron stars in our galaxy [

influence of the Paczynski galactic gravitational potential, it has been shown in [^{9} to 10^{10} years, similar to that of the galaxy) follow a torus-like shape above the galactic plane. Such a picture is consistent to our model that pulsars are new-born stars, but are aging to become old neutron stars, with age about that of our galaxy.

We distinguish pulsars from the very old neutron stars though both have the same nuclear mass density of ~3 × 10^{17} kg/m^{3}, satisfying

The variation of the void radius R_{o} on changing radius R_{n} of the neutron star is presented in ^{17} kg/m^{3}. For a given mass, the radius of the neutron star must be greater than the “critical radius” R_{nc} so that the density would not be greater than the nuclear density. Such a property is indicated by the M_{n}/M_{?}_{ }− R_{nc} plot in

The angular momentum of a neutron star is also given by the spherical shell model as in the case of pulsar:

In the context of our model, this angular momentum is balanced by that of the spinor pairs in the void; as the neutron star is assumed to be in the final stage of development, its angular momentum is governed by the Second Law (whereas the First Law is applied to study pulsars), and from Equations (3.27a) and (4.2), we have

Leading to

We are interested in the final state where P is large. Therefore we assume that on the left hand side of (7.2c), term (i) =

where we recall that E_{f} = 0.5 MeV and

Solving (7.1) and (7.3c), we obtain the ρ-P relation. Let us take a numerical example to demonstrate how we can deduce the period of rotation P from radius of a neutron star. Consider M_{n} = 1.4 M_{?}. From (7.2c) if we arbitrarily take R_{n} = 1.4 × 10^{4} m, we find R_{o} = 0.8077 × 10^{4} m, leading to ρ = 0.577. Substitute the relevant values into Equation (7.3c), we can solve for P:

_{z} = 2.67096 × 10^{32} J-s, we then test whether the approximation is valid by comparing the terms (i) and (ii); we have found that (i) ? (ii).

As another example, if R_{n} = 1.5 × 10^{4}, R_{o} =1.05 × 10^{4} m, ρ = 0.7, leading to 10^{−52} × (0.7/1.2484)^{4} = 1.875117 × 10^{−51} × (1.5 × 10^{4}/P), or 9.885 × 10^{−54} =1.875117 × 10^{−51} × (1.5 × 10^{4}/P), giving P = 2.8454 × 10^{6} s. For this value of P, (i) = 1.35 × 10^{17} S.I. Units; (ii) = 3.1583 × 10^{18} R_{o}/P = 1.1655 × 10^{16} S.I. Units; this approximation just mentioned is still barely valid. In this case, L_{z} = 7.01292 × 10^{32} z J-s.

As a third example, with R_{n} = 1.35 × 10^{4} m, R_{o} = 0.63 × 10^{4} m, ρ = 0.466666; P =1.2964 × 10^{7}s. The angular momentum of the star is L_{z} = 1.068225 × 10^{32} z J-s, which is 8 orders of magnitude lower than that of the Vela pulsar. Apart from the L_{z} value (effectively the rotation rate), pulsars and neutrons could “appear very similar” to a distant observer.

We can now plot the ρ- P relation for neutron stars with mass = 1.4 solar mass back in _{n} = 1.4M_{?} in this example) does not necessarily follow this line in its evolution. Even if M_{n} remains constant, the star evolves according to the Second law. The temperature cools as the star ages. Since T does not appear explicitly in Equation (3.7a), and we assume E_{f} to be constant, a point on the ρ-P line means that at a certain time, if the rotational period P is measured to be a certain value, the void radius is fixed by the value of ρ on the line. Such a value of ρ (or R_{o}) tells that the angular momentum of the spinor pairs rotating at a certain (yet unknown temperature) T < T_{c}, so that the angular momentum of the spinor pairs have the same magnitude to balance the angular momentum of the matter shell. The numerical example just above already demonstrates the methodology of calculating the angular momentum. In the last numerical example, the spinor pairs follow the Fermi-Dirac equation, and they arrange themselves to such a temperature that gives rise to angular momentum ~10^{32} J-s. Since the spinor pairs are in a heat bath insdie the void, their energy is exchanged between that of the matter star. During evolution, energy is lost eventually through radiation from the star’s surface, and the star cools down. More data measured (e.g. R_{n}, M_{n}) plus numerical analysis like that illustrated in Section 4 might lead us to find the T of the hypothetical neutron star at a certain stage later in the future.

To have more feeling about the decrease in angular momentum of our neutron star model, we indicate in _{n} for mass equal to (a) 1.4 and (b) 2.0 solar mass as marked, with critical radii R_{nc} = 1.304 × 10^{4} m and 1.46855 × 10^{4} m respectively. When both have the same radius of 1.5 × 10^{4} m, (a) rotates with P = 32.93 days, whereas (b) rotates with P = 325.10 days. Consider the situation where their masses stay constant while cooling down, with associated decreases in angular moment and radius. When the radius of (a) becomes 1.3045 × 10^{4} m, its angular momentum becomes 2.86 × 10^{29} J-s, and P becomes 132.29 years. When model (b) contracts to a radius of 1.469 × 10^{4} m, its angular mo-

mentum becomes 2.84 × 10^{29} J-s (about the same as that of model (a)), but the period of rotation becomes 240.99 years. Note that R_{nc} is the minimum/limiting value of the radius.

In our model, every member of the solar system was created by projection. When matter is created to the shell of a star, the distribution of quarks & elementary particles are in general certainly not even. A young star contains a lot of energy and rotates at a very fast spin as explained in Section 4. With the inhomogeneity of mass distribution, it is therefore likely that a stellar mass structure can be split into two smaller stars, with 5D void in each. We propose this to be the reason of observing so many binary pulsar systems in this universe. [^{0.25}/R for members of the solar system in _{f}. We observe that [(Iω)^{0.25}/R] varies from 11.23 kg^{−0.25}∙m^{−0.5}∙s^{−0.5} [for Venus] to 73.7 kg^{−0.25}∙m^{−0.5}∙s^{−0.5} [for Jupiter] in the solar system. Our model leads to the result that ρ varies only within a narrow range in the solar system, even though other parameters vary significantly. In ^{12} K to T = 10^{8} K for Jupiter (marked J), Earth (marked E), Mars (marked Ma), and Venus (marked V), taking E_{f} = 0.5 MeV. The ρ, density values for members of the solar system are listed in the last two columns of

Planets | Mass (kg) | Radius (m) | P(s) | Iω(J-s) | ρ = R_{o}/R | den(10^{3} kg/m^{3}) |
---|---|---|---|---|---|---|

Sun | 1.99 × 10^{30} | 6.955 × 10^{8} | 2.16 × 10^{6} | 1.100 × 10^{42} | 2.876 × 10^{−3} | 1.410 |

Mercury | 3.3 × 10^{23} | 2.44 × 10^{6} | 5.067 × 10^{6} | 9.745 × 10^{29} | 7.917 × 10^{−4} | 5.427 |

Venus | 4.867 × 10^{24} | 6.05 × 10^{6} | 2.0995 × 10^{7} | 2.133 × 10^{31} | 6.904 × 10^{−4} | 5.204 |

Earth | 5.972 × 10^{24} | 6.37 × 10^{6} | 8.64 × 10^{4} | 6.800 × 10^{33} | 2.755 × 10^{−3} | 5.510 |

Mars | 6.417 × 10^{23} | 3.3895 × 10^{6} | 8.864 × 10^{4} | 2.087 × 10^{32} | 2.179 × 10^{−3} | 3.940 |

Jupiter | 1.8983 × 10^{27} | 6.991 × 10^{7} | 3.573 × 10^{4} | 6.780 × 10^{38} | 4.445 × 10^{−3} | 1.330 |

Saturn | 5.684 × 10^{26} | 5.8232 × 10^{7} | 3.836 × 10^{4} | 1.374 × 10^{38} | 3.539 × 10^{−3} | 0.687 |

Uranus | 8.682 × 10^{25} | 2.54 × 10^{7} | 6.12 × 10^{4} | 2.300 × 10^{36} | 2.972 × 10^{−3} | 1.270 |

Neptune | 1.024 × 10^{26} | 2.46 × 10^{7} | 5.80 × 10^{4} | 2.400 × 10^{36} | 3.197 × 10^{−3} | 1.638 |

Pluto | 1.471 × 10^{22} | 1.184 × 10^{6} | 5.52 × 10^{5} | 9.380 × 10^{28} | 9.113 × 10^{−4} | 1.880 |

show the First two laws by two dotted straight lines (taking J as an example) and the Third Law by the intersection point marked T_{c} = 5.3× 10^{9} K. Note that a model with similar consequential result based on matter in Lorentz space-time alone has been proposed, emphasizing on the orbital architecture of the giant planets of the Solar System [^{3} kg/m^{3}, the ρ-P line falls into the “right place” among groups of stars so far studied.

Perelmann’s proof of the Poincare Conjecture suggests that spherical stellar objects are formed via 5D-4D pro-

jection, with 5D voids in the centre of each object. On the other hand, in view of charge conservation, the spinors are only produced in pairs. Those pairs which are rotating in phase along the latitudes of the spherical void (Section 1 and 2) generate a net angular momentum, which is balanced by the angular momentum of the spinning object in Lorentz manifold. Such a notion provides an explanation on the origin of angular momenta of stars plus various objects in our universe. Since the spinor pairs in the void are Fermion pairs, we assume they satisfy the Fermi-Dirac statistics F(E_{f}, T) in the way explained in Section 3. The limits of integration in the expression of L_{z} are essentially 0 and infinity. Separating this integral into one with limits 0-E_{f} and the other one with E_{f}-infinity, and expanding the function F(E_{f}, T) as a power series, the total angular is then expressible as a number of series which can summed in closed form. The separation of the integration limits has important bearing in physical meaning. It is easy to recognise that such an explicit representation has two asymptotic forms specified by the conditions 1) _{f}/kT = 1, and 2) _{f}. The integral with limits 0 - _{o}∙kT = A∙Iω, with A being a constant. There are data from stellar objects allowing us to calculate the angular momentum Iω observed in the Lorentz space-time. Hence, the radius of the 5D void, R_{o}, is inversely proportional to T (in the 5D void) for an object with well defined angular momentum. In carrying out the above analysis, we discover three laws governing the relationships between R_{o}, T, E_{f} and angular momentum Iω of the astronomical object of interest. The features of these three laws are closely related to the limits of integration mentioned above, and the physics of a Fermion pair system in Section (III). Of importance is that the shape of the ρ- T curve of every stellar object is identical because the function I (^{*}_{fc} at T_{c} = 5.3 × 10^{9} K (called the Third Law) for E_{f} =0.5 MeV, the rest mass of the lightest lepton generated. Note that T_{c} is just greater than Bethe range of fusion.

As stellar objects have different stages of fusion reactions inside the star, the materials formed depend on many parameters such as the dynamics of the plasma surrounding the void, the mass density distribution of the star, the temperature T (in the Lorentz space-time) etc. The 5D void of each star is therefore subjected to different mechanical pressures at different stages of evolution. At present, we do not have enough information on the parameters which would allow us to calculate (with sufficient accuracy to deduce useful physical conclusion) the pressure acting on the void in this model. In order to begin somewhere with this new theory, we employ the normalized void radius R_{o}/R as an independent variable to investigate certain characteristics of the angular momentum as T changes. Certainly ρ = R_{o}/R > 1 is absurdity and this is a condition to check the validity of this theory. While matter is created and each stellar object spins to conserve angular momentum, heat exchange takes place between the 5D - 4D boundary. In general, the temperature of the void containing spinors in each object is much higher than that of the matter space, so that heat energy diffuses via temperature gradient to the Lorentz space, eventually reaching the surface of the object, and there is always a thermal radiation component even in the quiescent state of neutron star, as observed in [

We have explained in Section (1, 2) that the combination projection Po and P1 leads to the creation of all the elementary particles detected/perceived in the 4D manifold in which stars are observed to exist. At the beginning, these particles form a shell enclosing the void. As projection/creation goes on, the shell increases its thickness. Since the temperature at this stage of a star is extremely high (>10^{12} K), the individual quarks can exist, together with the gluon potential fields which can exist in the Lorentz space time structure. It takes a long time before the right combination of the quark members to collide and form hadrons, while emitting large amount of energy in a wide range of the electromagnetic spectrum. Chern-Simons gauge confinement requires that the quark-current rotates in a 2D manner, generating huge magnetic field (with axis not necessarily along the L_{z} direction) of a new born star, as observed in pulsars. We would like to remark also that at the birth of a star, there is relatively small amount of (massive) matter, and the electrons and quarks are spinning very fast to counter-balance the angular momentum of the spinor pairs in the 5D void. Up to the present time, a pulsar named PSR 1937+21 with rotation period of even down to 1.6 ms has been detected [_{o} is expanding), leading inevitably to decrease in spinning rate due to angular momentum conservation, and also leading to decrease in temperature because (heat) energy is lost continuously. Note also that the increase of mass of a stellar object can occur by gaining matter from nearby objects, or due to some unknown reason. Whereas astronomical explosion like the supernova explosion can lead to the formation of lumps of matter which might develop into stars due to gravity, projection theory provides an explanation of the phenomenon “mass generation”. Thus, according to this new model, there are pulsars with a very wide range of masses; they spin down and can form different stages of stars while expanding. It is assumed in the model of pulsar described in (IV) that the mass density of the shell is ~ nuclear density based on the assumption that quarks plus hadrons formed at this stage have such magnitude of density. If other smaller magnitudes of density are considered, the general picture is the same, with different constraints on the radii of these objects.

We propose here that those stars with M > 1.4 M_{?}, could suffer from gravitational collapse, and eventually become the “real neutron stars” with mass density ~ nuclear density, as explained in Section (VII). Detailed study of stellar evolution is outside the scope here. We leave out the formation of giants and super-giants, but concentrate on analyzing different groups with respect to their variation of mass densities, void radius R_{o}, and period of rotation P as the stars age. In Sections 5-7, we calculate the ρ = R_{o}/R and density values for the pre-main-sequence stars in the Orion (

To demonstrate what we have said about the ρ-density connection, we indicate in

Many stars with low-to-mid mass have densities from several hundred to ~5 × 10^{3} kg/m^{3}. The NGC 2516 and brown dwarfs are denser―up to 10^{5} kg/m^{3}. We have not included larger stars because there are different pathways as they evolve, and the data on radii are scattered. With increasing data to be obtained later, we can fill in the gap between the range of 10^{5} kg/m^{3} to ~10^{8} kg/m^{3}, as well as the “final range” of 10^{10} to 10^{17} kg/m^{3}. The WDs and MWDs have density up to a few times 10^{9} kg/m^{3}. We observe in ^{17} kg/m^{3}. If we just take the “limiting model” with mass = 1.4 M_{?}, and radius R_{n} = 1.4 × 10^{4} m, ρ is 0.577 (Section 7). We represent this particular neutron star by the little circle in _{?}, and we antic-

ipate that many neutron stars have masses larger than 1.4 M_{?}. Such a very old star group is therefore represented by vertical line passing through the little circle in

Planets are certainly less energetic than the sun and are in the relative “final stage” of development. We conjecture that all members of the solar system could well be within the stable Second law region, with T < T_{c} = 5.3 × 10^{9} K. Note that the Bethe fusion temperature is about 2 × 10^{9} K, confirming that the thermal bath surrounding the void must be that described by Bethe fusion.

The ρ values for planets range from ~10^{−3} to 10^{−4}. The ρ-T and ρ-P graphs for eight planets excluding the binary Pluto are shown respectively in

A detailed study of stellar evolution has been attempted by many astronomers (see e.g. [

We will now discuss the Fermi energy of degenerate electrons in white dwarfs, which are in the final stage of stellar evolution for stars whose masses are smaller than the Chandresekhar limit of 1.4 M_{?}, so that they will not become neutron stars. For every star, the nuclear fusion leads to a temperature greater than around 10^{7} K. At such temperature, a plasma is formed with a huge electron gas. In young stars, there is a large amount of hydrogen nucleus as fusion fuel so that the degenerate electron gas can withstand the gravitational collapse. The size of the star remains for a period of time during which the matter core fuel is being used. Fusion stops when the fuel at this stage ends. However, loss of gravitational energy could result in an increase of kinetic energy of the electrons and ions, offsetting partly the cooling process. In white dwarf, the electron gas pressure prevents the gravitational collapse after a certain stage is reached. Taking the white dwarf WD1829+547 as an example and using parameters specified in ^{20} m^{3}. M = 1.791 × 10^{30} kg. As the period of rotation is estimated to be 100 years, we assume all the hydrogen has fused to become helium. The number of nucleons (protons or neutrons) is Nu = M/mass of proton = 1.791 × 10^{30} kg/[1.66 × 10^{−27}] = 1.0789 × 10^{57}. There are two electrons in a helium four atom, thus the total number of electrons in the Fermi gas is N = Nu/2 = 5.3945 × 10^{56}. Treating the electrons (mass m_{e}) as members of the free Fermi gas, the Fermi energy is simply [_{f} = h^{2}/(2m_{e}).[3N/(8πV)]^{(2/3)} = 5.167 × 10^{−14} J = 0.3225 MeV.

The Fermi temperature T_{f} = E_{f}/k = 3.74 × 10^{9} K. The Fermi gas pressure under the condition T = T_{f} is P_{r} = (2/5) × (N/V) × E_{f} =0.4 × {5.3945 × 10^{56}/[6.4865 × 10^{20}]}. 5.167 × 10^{−14} Pa = 1.72 × 10^{22} Pa. As 1 atmospheric pressure ~ 10^{5} Pascal, P_{r} ~10^{17} atm pressure. Using the notation of this paper, ^{−14} J/[1.38 × 10^{−23 }(J/K) × 10^{7} K] = 374.4, for T ~10^{7} K. So the star is in the Second Law region.

If we take parameters for the white dwarf Sirius B reported (M ~2.09 ´ 10^{30} kg, R ~ 5.6 ´ 10^{6} m [_{f} = 5.3 × 10^{−14} J = 0.33 MeV, T_{f} = 3.84 × 10^{9} K, P_{r} = 1.81 × 10^{22} Pa. Since E_{f} of the degenerate electrons is only a weak function of the star’s mass, it is easy to see that E_{f} ~ 0.2 - 0.3 MeV among most of the white dwarfs so far discovered. In fact, E_{f} of the degenerate electrons in the matter core for WD1748+708, WD0533+053, GD165, GD140 are respectively found, using the above calculation method, to be 0.30, 0.275, 0.246, 0.224 MeV. This is an interesting result, as the Fermi energy of the degenerate electrons in the 4D Lorentz space-time is of the same order of magnitude (in fact very close) to the E_{f} value (=0.5 MeV) we deduce for each member of the spinor pairs in the void! Degenerate Fermions resist strongly further compression because the particles cannot move to lower energy levels which are already filled due to the Pauli Exclusion Principle. As a result, it is difficult to extract thermal energy from these Fermions at this stage. Therefore, at the end stage of a star, there is thermal equilibrium between the void and the matter shell. In other words, the spinors in the void of white dwarfs maintain a certain size and at a fixed T at the end stage of the stellar evolution, and the size of a white dwarf is observed to be constant for a long time. In view of the above analysis, we conjecture that a typical white dwarf has a pressure > 10^{17} atmospheric pressure, a void radius of R_{o} ~ 10^{−1} to 10^{−3} of 6.37 × 10^{6} m. While the R_{o} value remains constant, the temperature is cooled down very slowly, as described by the Second Law. From the angle of thermodynamics, we would also emphasize that the Lorentz boundary domain being an ensemble of energetic massive particles which obey thermodynamics, and thus form a thermal bath enclosing the void. The massless spinors within the void, must then also obey a thermal statistical distribution, processing its own T value. Any T gradient between the boundary domain and the void will leads to energy flow. A thermal gradient implies not just heat flow but also a pressure gradient, as thermal systems are P, V, T mutually dependent. Note also that only vector and spinor solutions exist mathematically in the void. Vector potentials must be generated by current of charges. The solutions to the homogeneous Maxwellian equation (in the 5D void) are plane wave solutions; these solutions represent wave states that would propagate and be dissipated, leaving only macroscopic static states (thermal equilibrium microscopic states) in the void.

As the void contains equal number of e and −e massless spinor pairs moving with c, with R_{o} fixed (Section (II), their quantum states can be divided into (i) in phase orbitals and (ii) opposite phase orbitals. The states of (i) lead to nonzero L_{z}, but magnetic field B = 0, whereas the states of (ii) give L_{z} = 0, but non-zero B_{z}_{’}, where the axis z’ is in general not aligned with z. Furthermore, the numbers of the two types of pairs may not be equal. In our model, we have chosen these numbers to be in the ratio 3D to 1D, minimizing vector potential energy within the void. Note that we do not count the L_{z} states, as the void is being represented by a 3D × 1D manifold.

The origin of magnetic field according to the projection theory will be left to another paper.

When the kinetic energies of degenerate electrons in the matter shell are high, the rate of collision among them is low. They can travel at speeds approaching c to long distances. The spinors in the void have some physical similarity with the degenerate electrons in the 4D matter core: they are charged with positive or negative electronic charge, are degenerate, and have similar values of Fermi energy, even across the 5D - 4D boundary.

Finally, consider the final end stage of a star, namely neutron star. Since the mass is larger, but the radius is much smaller than a WD, gravitational collapse causes the star to contract further. At high pressure, the degenerate electrons bind to the protons, forming more neutrons, with result of fast cooling. Note that the bound electron-proton state is Bosonic, resulting in a Bose-Einstein condensation. Both bosons and neutrons become degenerate gas states, generating a huge outward pressure (due to again the Exclusion Principle) to balance the gravitational force. At the end stage of such heavy stars, the number of neutrons can be much greater than that of protons, as in heavy elements―observers would consider them as “neutron stars”. In fact, there is recent evidence that isolated neutron stars show clear thermal emission in quiescence [

Due to its high mass density, the gravitation gradient of the star is a very steep function of r. Thus all the way to the star surface, the electron orbits are quite 2D! Furthermore, being of very high energy, the electron orbit is likely to be satisfying the Chern-Simons relativistic gauge symmetry. Consider the lightest 2D atomic hydrogen as that near the star surface. This state has a total energy (T.E.) = M_{p} + m_{e}/γ − m_{e}; here M_{p}, m_{e} are respectively the rest masses of proton and electron and γ is the relativistic factor.

Remark that the reduced mass m^{*} is equal to m_{e}, and the Coulomb potential is equal to −m^{*} in the semion limit.

Furthermore, the relativistic factor γ is equal to 0.18 [_{p} + m_{e} (0.82/0.18) = 938.3 + 2.3 = 940.6 MeV which is greater than the neutron rest mass. In fact this object will radiate gamma rays of order of a couple of MeVs or more, with occasional electron jets, whereas a pure neutron surface will not radiate. The same processes happen on regular stars, such as the sun [

Based on the 5D projection theory, we provide an explanation as to how mass and angular momentum can be generated in the universe. We derive explicit expressions relating some relevant quantities in the 5D and 4D manifolds, so that stellar evolution can be analyzed with the model presented in this paper. Using data from quite a number of stellar groups, we have found that the experimental observed data fall into the logic of explanation of our theory.

The authors started to collaborate in various Physics research projects over 4 decades ago. They missed communication for about 20 years. Thanks to Professor W.K. Chow (Poly University of Hong Kong) and Professor Kenneth Young (Chinese University of Hong Kong), they met again in the summer of 2014, and collaborate again on this project. We wish to thank Professor W.K. Chow, Dr. C.L. Pang, Mr. H.K. Tsang and Mr. Benjamin Fung for their kind help in the preparation of this manuscript.

Peter C. W.Fung,K. W.Wong, (2015) On the Origin of Mass and Angular Momentum of Stellar Objects. Journal of Modern Physics,06,2303-2341. doi: 10.4236/jmp.2015.615235

(A.2)

First we would remark that the Fermi distribution function

We derive explicit expressions for the integrals

Define

Similarly, carrying out integration by parts,

(A.9)

(A.11)

∙∙∙

Let

(A.13)

We split

(A.14a) (A.14b)

(A.15a) (A.15b)

(A.16a) (A.16b)

∙∙∙

After summing the series involved, we arrive at a relatively neat form for

For

whereas

so that

We can also start with the integral expression directly. Under the First Law, (A.1) gives

=

From definite integral table,

(B.3) and (B.5) are identical.

Now turn to the Second Law. Result of the series expansion gives

Starting from integral under the condition

The denominator of the integrand is approximately equal to 1, except at very small values of