^{1}

^{*}

^{2}

Beginning with a 5D homogeneous universe [1], we have provided a plausible explanation of the self-rotation phenomenon of stellar objects previously with illustration of large number of star samples [2], via a 5D-4D projection. The origin of such rotation is the balance of the angular momenta of stars and that of positive and negative charged e-trino pairs, within a
3*D* ⊗ 1*D *void of the stellar object, the existence of which is based on conservation/parity laws in physics if one starts with homogeneous 5D universe. While the in-phase e-trino pairs are proposed to be responsible for the generation of angular momentum, the anti-phase but oppositely charge pairs necessarily produce currents. In the 5D to 4D projection, one space variable in the 5D manifold was compacted to zero in most other 5D theories (including theories of Kaluza-Klein and Einstein [3] [4]). We have demonstrated, using the Fermat’s Last Theorem [5], that for validity of gauge invariance at the 4D-5D boundary, the 4
^{th} space variable in the 5D manifold is mapped into two current rings at both magnetic poles as required by Perelman entropy mapping; these loops are the origin of the dipolar magnetic field. One conclusion we draw is that there is no gravitational singularity, and hence no black holes in the universe, a result strongly supported by the recent discovery of many stars with masses well greater than 100 solar mass [6] [7] [8], without trace of phenomena observed (such as strong gamma and X ray emissions), which are supposed to be associated with black holes. We analyze the properties of such loop currents on the 4D-5D boundary, where Maxwell equations are valid. We derive explicit expressions for the dipolar fields over the whole temperature range. We then compare our prediction with measured surface magnetic fields of many stars. Since there is coupling in distribution between the in-phase and anti-phase pairs of e-trinos, the generated mag-netic field is directly related to the angular momentum, leading to the result that the magnetic field can be expressible in terms of only the mechanical variables (mass
*M*, radius
*R*, rotation period
*P*)of a star, as if Maxwell equations are “hidden”. An explanation for the occurrence of this “un-expected result” is provided in Section (7.6). Therefore we provide satisfactory answers to a number of “mysteries” of magnetism in astrophysics such as the “Magnetic Bode’s Relation/Law” [9] and the experimental finding that B-P graph in the log-log plot is linear. Moreover, we have developed a new method for studying the relations among the data (
*M, R, P*) during stellar evolution. Ten groups of stellar objects, effectively over 2000 samples are used in various parts of the analysis. We also explain the emergence of huge magnetic field in very old stars like White Dwarfs in terms of formation of 2D Semion state on stellar surface and release of magnetic flux as magnetic storms upon changing the 2D state back to 3D structure. Moreover, we provide an explanation, on the ground of the 5D theory, for the detection of extremely weak fields in Venus and Mars and the asymmetric distribution of magnetic field on the Martian surface. We predict the equatorial fields B of the newly discovered Trappist-1 star and the 6 nearest planets. The log
*B −* log
*P* graph for the 6 planets is linear and they satisfy the Magnetic Bode’s relation. Based on the above analysis, we have discovered several new laws of stellar magnetism, which are summarized in Section (7.6).

It is a very important step in physics to unify gravity with electrodynamics. Despite many trials, the past endeavors were unsuccessful. It is not correct to think that by adding another dimension to the Lorentz space-time, one can readily bridge gravity and electrodynamics. A unified theory along this line of thought therefore is not one that can be applied separately to a domain with masses in motion, and to another domain pertaining to dynamics of massless photons. The unified theory has to embrace both gravity and electrodynamics in an “inherent” manner. The general method we use is to analyze the physical properties of the universe via projection/mapping between the 4D and 5D space-time, with an analysis of the boundary conditions between the two domains.

In our model, after the absolute time

this time is specified by

energy generated at this instant in the 5D manifold. The 5D manifold has to be homogeneous the reasons of which have been discussed in [

We assume the universe began with a homogeneous 5D space-time structure described by the above metric equation. From the homogeneous 5D manifold, we can apply projection operations [

The above projection reduction approach was presented in [

On the other hand, via the rigorous Perelman-Ricci Flow mapping [

It is our intention to show the mathematical connections between these two mapping/projection procedures, as well as to investigate the physical outcome from such investigation in this paper, with special focus on the origin of magnetic field in the universe. Note that whatever method we employ to analyze the space-time structure of the universe, we always come up with a boundary separating the 5D and 4D domains. We start with the Fermat’s Last Theorem to analyze the space structure in the void and 4D-5D boundary in Section (2).In particular, we show that that breaking of the homogeneity of the 5D space-time, due to the imposition of the lower dimension 4D boundary, would lead to a time-frozen,

We pay special attention to the meaning and mathematical representation of homogeneity. Since other theories relating to transformation between a 5D domain and 4D domain have been published, we give a very shot review on the Kaluza-Klein (K.K.) [^{th} space coordinate in the 5D domain is compacted to reduce dimension. We need to review this issue also because the 4^{th} space variable in our theory is rotated to the radial direction during dimension reduction, together with the emergence of a current loop, (as a consequence of the space-space transformation) in the

Explicit expression for the dipolar magnetic field of a general stellar object is derived in Section (4). Though the classical Biot-Savart law is employed, the quantum signature of the charge current is incorporated. The Three Laws of Dipolar Magnetic Fields of stellar objects, similar to the three laws of Stellar Angular Momentum discovered in [_{s}R^{3} and Iω relation is also linear in a log-log plot, particular for cool stars; here

As we proceed through the paper, the physical properties of the 5D-4D boundary are of crucial importance; we therefore need to high-light some basics, which are consequence of the projection theory and fundamental physics.

a) From the 5D homogeneous metric, one can obtain a 5D second order energy-momentum differential operator without a term pertaining to mass. Solutions of such metric equation represent 5 vector potential fields (including charge-source terms, thus existence of e-trinos) traveling with speed c, similar to the fields associated with the Maxwell potentials in the 4D Lorentz manifold. In the 5D domain, a term pertaining to magnetic monopole exists. In 4D domain, the electric, magnetic symmetry is broken, so that there is no magnetic-mono- pole term in Maxwell equations.

b) In a 5D domain, a Dirac linearization process does not lead to mass creation via projection/mapping, implying the e-trinos must be massless, but charged. These spinor states are equivalent to magnetic monopole states in 5D. In 4D domain, a Dirac linearization process leads to solutions representing the state of massless, charge-neutral neutrino and the state of massive charged lepton (described by the SU(2) group) in pair form.

c) Based on (a) & (b), the states of e-trinos and 4D Maxwell potentials must form the boundary between the 5D & 4D homogeneous manifolds, implying that both classical and quantum representations are allowed in the boundary. A satisfactory quantum theory should also have such representations.

d) The void space is expressed as

e) According to the P1 projection, the 4th space coordinate of e-trinos in the 5D manifold is conformally mapped into quarks of fractional charges. In order to produce a proton (composed of (u, u, d) quarks) and a neutron (composed of (u, d, d) quarks), as an example, we need 3 up and 3 down quarks. Since the 3 up quarks have a total charge of 2e, they are projected via the P1 process from 2 e-trino loop states of one loop. As the 3 down quarks have a total net charge of -e, they can be produced by P1 from just 1 anti-e-trino state from the other loop. To balance the + e charge resulting from the stated P1 projection, the Po projection on an in-phase pair (on the void surface) gives a lepton, such as an electron (exists in Lorentz space L), due to SU(2) symmetry and the unidirectional nature of time in the 5D metric. Effectively, we say that the consequence of Po is specified by the group product

f) After the entropy mapping, or equivalently the combined Po and P1 mapping, we have a spherically shaped mass stellar object model enclosing a

where p, e ,s represent the momentum, charge, and spin respectively. The in phase circulation of the oppositely charged massless spinors (with spin degeneracy

g) In this paper, we analyze the consequence of the out of phase rotating of the spinor pairs specified by the following four state groups:

In the calculation of current, these four states lead to the spin degeneracy factor

Detail of the 4D Maxwell boundary can be explicitly analyzed through Fermat's theorem, in terms of Abelian angles. The Fermat’s sum of quadratic coordinate components has been proved to be rigorous for any number of coordinate dimensions [

Leonhard Euler in 1770 [

In fact, we put forth the notion that n = 2 is the only condition that

By expanding the Lorentz manifold to the 5 space-time, we learn that the universe has a homogenous 5D space-time structure described by the metric

where

In the homogeneous 5D space-time, all 4 orthogonal space axes are exactly equivalent. Thus each axis has a measure r' as represented in the complex phase angle O(1) group [

Since the solutions of a differential equation is totally governed by the boundary conditions that are imposed, the vector and massless charged spinor solutions to the 5D homogeneous metric operator equation are dictated by the 4D Maxwell space-time boundary which forms the enclosure to (or embracing) the 5D manifold. At the absolute time t, the space volume of the homogeneous ND manifold must have a boundary enclosure of

Since the determination of the Ricci Flow direction for the

Our remaining problem is the general spherical shape of the massive stellar objects observed within each galaxy satisfying the Poincare Conjecture. To illustrate this topological mapping process, Perelman introduced an entropy mapping [

We note again that in the 5D manifold, there are 5 vector potential components (instead of 4 in the Maxwell domain), and 2 massless spinors with charges e and -e. [Section 2.1]. In order that these solutions satisfy the boundary conditions imposed by the Maxwell domain, we need to transform the 5 component symmetric vector potentials into the 4 Maxwell vector potentials by breaking the symmetry of solution as carefully illustrated by Maxwell in his thesis, with the emergence of a magnetic mono-pole potential, which corresponds to the 5th component vector potential in the 5D manifold. [

We will find an explicit expression for

While the current loop

Combining these two representations, & letting

which determines the geometry of void boundary.

The rotation of

Therefore,

This is the size of the current loop if a pair of electron and proton is generated simultaneously; in other words, hydrogen atom is generated.

Note that charge, linear momentum, angular momentum, and energy must each be conserved during the generation of matter. We have only the lightest lepton and quarks generated initially, before the gluon potential is in action to produce proton and neutron. A set of (u, u, d) of quarks must be generated (to build up eventually a proton) together with the generation of an electron, according to the 5D projection theory. The gauge requirement for the solution on quark spinors is the charge to mass ratio

where

Since mass is generated from nothing, the uncertainty principle requires that

(as a minimum); where

where

Using the data for

However, there is also the factor due to the reduced-mass effect of the proton-neutron pair, and the reduced mass

Hence, effectively, the time to generate a proton-neutron pair is longer than generating two protons by a factor of 2.225. Now many electrons and protons are generated simultaneously, while the 5D void is expanding until the mass shell is generated. Thus, the void radius

if the mass shell is composed of helium four. If the matter crust is composed of hydrogen and Helium 4, the factor b would be

With expressions (2.3.2) & (2.3.7), we can apply the magnitude of the current loop to calculate the magnetic field generated by such a loop. Before we do the application, we have to derive an explicit representation of the magnetic field generated such a current, one near each magnetic pole, when the particular element(s) is considered to be existing at the Lorentz space-time and void boundary. Such derivation will be carried out in Section (4). At the meantime, we need to analyze in the next Section about the origin of the “key” variable

The original Kaluza-Klein theory (KK theory, see e.g. review in [

where the usual summation rule is understood, and

Equating (3.1.1) to (3.1.2) gives

where

is called the vielbein. For a non-uniform gravitational field, the KK theory introduces the notion that “At every point in a reference frame with an arbitrary gravitational field it is possible to choose a locally inertial (freely falling) reference frame.” In other words, like carrying operation using the concept of calculus, the flat Minkowski metric can be transformed into a curved space-time metric. However, we would remark that in Equations (3.1.3a) & (3.1.3b), it was already assumed that a gravitational field is present due to the presence of the Riemannian tensor. This assumption implies that in the domain considered, mass exists. Thus the 4D is a Lorentz manifold, and by an extension to 5D space- time, such a 5D structure would not be a homogeneous 5D. Hence the Ricci- Flow mapping does not reduce this 5D back to 4D, except simply by closing the extra 4^{th} space dimension into a closed loop.

Without going into further details, we would remark that both the KK theory and Einstein generalized field equations have difficulties to explain experimental data and interpret mathematical singularity: (i) In order to exclude the singularities in the equation set, both K.K. and Einstein introduced the method of compactness. Such an assumption leads to the models of black hole, and dark matter, which to us, are not necessary.; (ii) The electron charge e and mass m_{e} are included in a certain constant κ in K.K. theory which bridges the electromagnetic potentials and some metric tensor components including the 5^{th} dimension. The values of both e & m_{e}_{ }deduced based on the KK theory (or Einstein’s field equations) were not consistent with the well-established experimental values then (see comments in [

Concerning the key difference between their theories and projection theory developed in this series, we need to note that in KK theory, as well as Einstein's 5D metric, the proper time τ is not connected to the 4th space dimension variable. Therefore, the KK 5D is not homogeneous, neither is mass a result of space projection.

As analyzed in Section (2), the 1D space within the Poincare matter sphere is a set of two closed loops that includes the _{ }variable. In his gravity solution, he then further assumed that the loop can be reduced to a structure with zero measure, and ignored the 5D space-time domain external to the doughnut domain by simply changing it to a simple Maxwell 4D space-time without the

Assuming a uniform mass distribution in a plane with the 2D void region bounded by

where_{.}. If we follow the analysis of the metric equation pertaining to the 5D manifold and the 5D-4D boundary as above, the 4D Maxwell space-time does not allow outward flow of e, -e massless spinor pairs that carry outward energy of 2 hν. Then at the origin of the center of the universe, or the center of the galaxy, if a black hole should exist according to those theories, mathematical logic does not indicate there is any outward flow of energy and then becomes a black hole, as we transform 4D Lorentz space-time with mass into the covariant Riemannian curvature space-time to obtain the gravity equation, since energy density seeks for uniformity. On the contrary, according to the 5D projection picture, energy seeks for uniformity by flowing outward to fill the universe at all t. Hence the singularity in the gravity field solution due to mass distribution does not appear. The above statement is both mathematical precise as well as philosophical subtle. Recently, stars of huge masses have been detected by telescopes expositions. For examples, R13601 (M = 365

Moreover, the finding of a

Having analyzed the concrete role played by the space variable _{ }in the conformal mapping, we can now proceed to derive the expression of the intrinsic dipolar magnetic fields appearing in stellar objects.

During the deduction of the three laws of angular momentum in [

As these charges build up a stationary/perpetual current state, the wave function of a spinor pair represented by the symbol

where

Now for non-relativistic particle with mass m moving along a circular orbit so that differentiation with respect to space variable is one-dimensional, we use the symbol

Since the spinor e and spinor?e are two distinguishable particles, in carrying the thermal averaging process later on, we need to calculate the number N for either type of spinors. The electric current, however, is doubled, because they are circulating out of phase in the classical sense. Therefore the overall current density in one ring is finally

We have left our sign convention in deciding the magnetic polarity, as the sign convention is very simple. We have fixed the “classical current” to be one dimensional,

Remark again that the subscript 2 signifies that the 1D space structure has an entangled structure of two loops, composed of both e & -e spinors as explained in Section (2). Note that

The electrodynamics of charged massless particles moving with velocity c is not at all explored much within the frame work of Maxwell equations and the frame work of quantum field theory. An insightful investigation of the exact solution of such particles in Maxwell equations has recently been published [

First, we would emphasize that we proposed that the existence of the massless charged spinors are represented by solution(s) of the 5D metric, rather than the 4D Maxwell potentials generated from classical massless charged particle. In fact, if the spinor solution is obtained from the massless 4D Dirac equation, with the introduction of charge, such solution must be coupled to the Maxwell potentials, and the coupled equation(s) cannot satisfy the Lorentz gauge transformation without a string attachment. Yet when this hypothetical string is reduced to zero, singularities appear in the Maxwell solutions.

It might be simple to analyze the gauge invariance property for charged massless spinor in 4D by applying the projection of the metric from 4D onto 3D (as explained in [_{2} in (4.1.4b) to the relativistic case by simply replacing v by c, obtaining an estimation of the current density:

And the current over the ring is

In passing, we would note that even if there were e & -e charges circulating in opposite directions along the 1D classical current ring, the chance of annihilation is non-zero. In general, the interaction cross-section of massive particles is larger than those of massless particles, such as photons. In fact, the strength of interaction of photons with massive particles depends strongly on the masses of the interacting particles. The cross-section area of a charge-neutral neutrino is well known to be extremely small, so that equipment to detect neutrinos is set in gold mine deep down underground. It has been estimated that there is only an upper limit on the mass of neutrino

A charge current generates magnetic field in space. The Biot-Savart Law expresses the magnetic field in terms of the magnitude, spatial length, direction, and the distance from a reference point (such as the center of a ring current) of a current. According to this law, the magnetic induction field generated by a charge current density J at the space point x is

Using elementary vector analysis,

When the distance between the coordinate origin and the point of observation is much greater the radius of the model ring current, i.e.

Here

where

the statistically averaged energy is

From (4.1.6),

In view of the derivation in Appendix A, we obtain

Then from (4.2.2b), the magnetic filed measured at the equator of the matter star

This equation may be called the Law of Intrinsic Dipole Magnetic Field for Stellar Objects.

Before we proceed to obtain numerical values to illustrate the laws we discover related to the origin of the intrinsic dipolar magnetic field of stellar objects, we need to find explicit expression of

Through computer program, using Simpson’s rule of integration, the integral

We now plot the I versus T graph, since

Such that

For even greater temperature such as

to good approximation. The straight green line in

On the other hand, if the temperature is relatively low, such that

In

temperature

We would emphasize that integral

can also be separated into two regions in temperature. Under the First Law of the angular momentum, i.e.

In view of Equations (4.2.8b) & (4.2.10), we obtain the Law of Intrinsic Dipole Magnetic Field for Hot Stars below:

As P&R are dependent implicitly on T, when T is large, P is small, and one has to carry out detailed numerical analysis before one can learn how B(eq) varies with T in realistic samples. Moreover, all the equations for B(eq) hold under the condition of

equally complicated. Here we want to emphasize that

can be separated into two regions in T. At small T and large T, the asymptotic lines are respectively two straight lines (one horizontal, and one with positive slope) intersecting at

right away that

cool region. Inspection of (4.2.11) reveals that therefore B(eq) depends explicitly only on the mechanical variables in front of the quotient of the state two integrals. This is a very crucial aspect of our discovery.

Under the regime of the Second Law of angular momentum, we present in Appendix B, the approximate expression of statistically weighted over

and Equation (4.2.9) already gives the approximate expression for the integral

where

Recalling that

We may call Equation (4.2.12) as the Law of Intrinsic Dipole Magnetic Field for Cool Stellar Objects. This equation will be used throughout the numerical analysis in this paper.

We have deduced the magnetic expression understand the Second Law condition because we are interested in finding the dipole field when the matter shell has been formed and the temperature of the void-matter boundary is not too high

It has been shown that the Maxwell and Chern-Simon gauge theories are coupled/compatible and the charge flux pinning phenomenon can be realized by the Maxwell-Chern-Simons gauge theory in a 2D system [

Now the confinement of a 3D hydrogen to a 2D state is supported by the fact that exact analytical solutions for 2D hydrogen can be derived for the non-relati- vistic and the relativistic H-atom model [

When a star has a large mass (such as the sun) and a small radius (such as the terrestrial radius), a large gravitation gradient exists along the radial direction, confining the motion of electrons from a 3D space to a

From the symmetry point of view, we say that the 3D space homogeneity is broken into_{. }Setting

via simple expansion of

Hence for non- zero h values, the gravitational potential difference

Let us take some concrete examples. It is well-known that magnetic storms are found to be periodic on the solar surface. We anticipate that the solar motion participates in the occurrence of periodicity. Thus if the gravitational potential difference (as a matter pressure) is the cause of Semion states formation, such states are likely to be formed slightly below the photosphere. Cyclic plasma turbulence could bring the Semion states to the surface, and the

We would note also that the star will be positively charged and would attract negatively charged particles with low kinetic energy. Note for small stars such as Magnetic White Dwarfs

Let us “borrow” some result in condensed matter physics to estimate the magnitude of magnetic field generated by a group of Semions. The 2D number density of composite fermion system σ is related to the magnetic field B by (see e.g. [

where n is the “level” of quantum flux and

Suppose for rough estimation, let us take a magnetic white dwarf WD0011134 associated with the set of data

taking the level n = 1 as the upper limit of the magnetic field B estimation. The 2D density of Semion, from Equation (5.2), is therefore

If Semions in the whole surface layer have enough energy to overcome gravitational and Coulomb attractions, the average magnetic field (which is radial in directions) over the whole stellar surface, being hypothetical released is then

It is known that the magnetic fields of Magnetic White Dwarfs are non-dipo- lar and distributed irregularly [

Note that this final stage of the star still possess the _{o} approaches zero, the explicit expression of B (equatorial) shows that it becomes zero, together with

We have derived the equatorial magnetic surface field in terms of the basic/“raw” data set (M, R, P):

We define the set of variables (M, R, P) as the basics/raw data variables. In some stars, all the three variables can be measured, but the number of such stars is small, and the accuracy of measurement is left with some high degree of uncertainty. In some star groups, the radius can only be theoretically deduced; a typical example is the application of the Hamada-Salpeter equation [

So we begin with the pre-main ?sequence or hollow stars of the Orion Nebula. Data of these stars in the Milky Way are taken from [_{o}, the magnetic parameter

Orion stars | R^{3} (m^{3}) | B (eq, theory, Gauss) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

1 | 1171 | 0.26 | 1.98 | 6.3936 | 2.611 × 10^{27} | 1.497 × 1027 | 3.857 × 10^{42} | 0.5733 | 4.798 × 10^{1} | 1.151 × 10^{6} |

2 | 1219 | 0.40 | 1.96 | 1.1319 | 2.530 × 10^{27} | 7.455 × 1027 | 3.285 × 10^{43} | 2.9430 | 7.502 × 10^{1} | 1.967 × 10^{6} |

3 | 1297 | 1.38 | 1.90 | 5.7280 | 2.308 × 10^{27} | 5.347 × 1027 | 2.104 × 10^{43} | 2.3170 | 2.847 × 102 | 1.761 × 10^{6} |

4 | 1325 | 0.20 | 1.30 | 3.8360 | 7.391 × 10^{26} | 9.594 × 1026 | 2.132 × 10^{42} | 1.2980 | 1.286 × 102 | 9.930 × 10^{5} |

5 | 1354 | 0.23 | 1.67 | 0.6912 | 1.567 × 10^{27} | 5.613 × 1027 | 2.245 × 10^{43} | 3.5820 | 6.973 × 101 | 1.789 × 10^{6} |

6 | 1368 | 0.23 | 1.35 | 2.3846 | 8.277 × 10^{26} | 1.612 × 10^{27} | 2.137 × 10^{42} | 1.9470 | 1.320 × 10^{2} | 1.180 × 10^{6} |

7 | 1396 | 0.28 | 2.03 | 1.6589 | 2.814 × 10^{27} | 4.520 × 10^{27} | 1.683 × 10^{43} | 1.6060 | 4.727 × 10^{1} | 1.665 × 10^{6} |

8 | 1428 | 0.17 | 1.31 | 1.0022 | 7.563 × 10^{26} | 2.352 × 10^{27} | 7.042 × 10^{42} | 3.1100 | 1.068 × 10^{2} | 1.339 × 10^{6} |

9 | 1440 | 0.27 | 1.47 | 1.1750 | 1.069 × 10^{27} | 3.512 × 10^{27} | 1.201 × 10^{43} | 3.2860 | 1.200 × 10^{2} | 1.530 × 10^{6} |

10 | 1453 | 0.23 | 1.66 | 1.1750 | 1.539 × 10^{27} | 3.740 × 10^{27} | 1.305 × 10^{43} | 2.4300 | 7.100 × 10^{1} | 1.562 × 10^{6} |

11 | 1465 | 0.27 | 1.85 | 1.1059 | 2.130 × 10^{27} | 5.176 × 10^{27} | 2.021 × 10^{43} | 2.4300 | 6.000 × 10^{1} | 1.742 × 10^{6} |

12 | 1485 | 0.23 | 1.44 | 5.4778 | 1.005 × 10^{27} | 9.513 × 10^{26} | 2.106 × 10^{42} | 0.9470 | 1.088 × 10^{2} | 0.990 × 10^{6} |

13 | 1500 | 0.24 | 1.90 | 7.6205 | 2.308 × 10^{27} | 1.163 × 10^{27} | 2.750 × 10^{42} | 0.5040 | 4.940 × 10^{1} | 1.058 × 10^{6} |

14 | 1501 | 0.29 | 1.74 | 7.5427 | 1.772 × 10^{27} | 1.183 × 10^{27} | 2.816 × 10^{42} | 0.6674 | 7.770 × 10^{1} | 1.065 × 10^{6} |

15 | 1511 | 0.38 | 1.89 | 1.3306 | 2.271 × 10^{27} | 6.026 × 10^{27} | 1.240 × 10^{43} | 2.6530 | 7.948 × 10^{1} | 1.832 × 10^{6} |

16 | 1522 | 0.35 | 2.09 | 6.2986 | 3.071 × 10^{27} | 2.053 × 10^{27} | 5.872 × 10^{42} | 0.6683 | 5.414 × 10^{1} | 1.279 × 10^{6} |

17 | 1545 | 0.25 | 1.79 | 4.5965 | 1.930 × 10^{27} | 1.602 × 10^{27} | 4.216 × 10^{42} | 0.8300 | 6.155 × 10^{1} | 1.178 × 10^{6} |

18 | 1566 | 0.23 | 1.57 | 6.0480 | 1.302 × 10^{27} | 1.006 × 10^{27} | 2.268 × 10^{42} | 0.7724 | 8.393 × 10^{1} | 1.008 × 10^{6} |

19 | 1627 | 0.26 | 1.93 | 8.7264 | 2.419 × 10^{27} | 1.141 × 10^{27} | 2.685 × 10^{42} | 0.4719 | 5.107 × 10^{1} | 1.052 × 10^{6} |

20 | 1753 | 0.16 | 3.23 | 3.6288 | 1.134 × 10^{28} | 3.315 × 10^{27} | 1.113 × 10^{43} | 0.2924 | 6.582 × 10^{1} | 1.500 × 10^{6} |

21 | 1760 | 0.20 | 1.57 | 5.2963 | 1.302 × 10^{27} | 1.000 × 10^{27} | 2.252 × 10^{42} | 0.7680 | 7.298 × 10^{1} | 1.007 × 10^{6} |

22 | 1805 | 0.30 | 2.03 | 4.5965 | 2.814 × 10^{27} | 2.217 × 10^{27} | 6.507 × 10^{42} | 0.7880 | 5.064 × 10^{1} | 1.312 × 10^{6} |

23 | 1966 | 0.22 | 1.86 | 6.1430 | 2.165 × 10^{27} | 1.240 × 10^{27} | 3.000 × 10^{42} | 0.5730 | 4.828 × 10^{1} | 1.081 × 10^{6} |

24 | 2037 | 0.21 | 1.73 | 1.8490 | 1.742 × 10^{27} | 2.643 × 10^{27} | 8.224 × 10^{42} | 1.5170 | 5.727 × 10^{1} | 1.392 × 10^{6} |

25 | 2168 | 0.21 | 1.63 | 5.2877 | 1.457 × 10^{27} | 1.099 × 10^{27} | 2.550 × 10^{42} | 0.7540 | 6.847 × 10^{1} | 1.039 × 10^{6} |

26 | 2246 | 0.33 | 1.63 | 8.1734 | 1.457 × 10^{27} | 1.113 × 10^{27} | 2.595 × 10^{42} | 0.7636 | 1.076 × 10^{2} | 1.043 × 10^{6} |

27 | 2301 | 0.15 | 1.42 | 0.7344 | 9.633 × 10^{26} | 3.051 × 10^{27} | 9.964 × 10^{42} | 3.1676 | 7.398 × 10^{1} | 1.460 × 10^{6} |

28 | 2425 | 0.13 | 1.44 | 1.4774 | 1.005 × 10^{27} | 1.655 × 10^{27} | 4.414 × 10^{42} | 1.6472 | 6.148 × 10^{1} | 1.191 × 10^{6} |

29 | 2744 | 0.43 | 1.79 | 5.6419 | 1.930 × 10^{27} | 2.063 × 10^{27} | 5.910 × 10^{42} | 1.0689 | 1.059 × 10^{2} | 1.281 × 10^{6} |

30 | 2784 | 0.22 | 1.26 | 3.4214 | 6.730 × 10^{26} | 1.072 × 10^{27} | 2.470 × 10^{42} | 1.5930 | 1.553 × 10^{2} | 1.030 × 10^{6} |

31 | 2913 | 0.21 | 1.65 | 4.5014 | 1.511 × 10^{27} | 1.263 × 10^{27} | 3.073 × 10^{42} | 0.8356 | 6.601 × 10^{1} | 1.088 × 10^{6} |

32 | 2470 | 0.23 | 1.55 | 2.4278 | 1.253 × 10^{27} | 1.956 × 10^{27} | 5.506 × 10^{42} | 1.5613 | 8.722 × 10^{1} | 1.259 × 10^{6} |
---|---|---|---|---|---|---|---|---|---|---|

33 | 3014 | 1.13 | 1.74 | 6.7565 | 1.772 × 10^{27} | 3.562 × 10^{27} | 1.225 × 10^{43} | 2.0100 | 3.029 × 10^{2} | 1.537 × 10^{6} |

34 | 3115 | 0.27 | 1.91 | 5.8147 | 2.344 × 10^{27} | 1.567 × 10^{27} | 4.098 × 10^{42} | 0.6686 | 5.472 × 10^{1} | 1.169 × 10^{6} |

35 | 3142 | 0.23 | 1.62 | 7.4650 | 1.430 × 10^{27} | 9.001 × 10^{26} | 1.956 × 10^{42} | 0.6293 | 7.639 × 10^{1} | 0.972 × 10^{6} |

36 | 3161 | 0.27 | 1.89 | 0.7258 | 2.271 × 10^{27} | 7.345 × 10^{27} | 3.215 × 10^{43} | 3.2344 | 5.647 × 10^{1} | 1.957 × 10^{6} |

37 | 3189 | 0.30 | 1.54 | 6.0480 | 1.229 × 10^{27} | 1.158 × 10^{27} | 2.846 × 10^{42} | 0.9428 | 1.160 × 10^{2} | 1.078 × 10^{6} |

38 | 3217 | 0.23 | 1.35 | 3.2314 | 8.277 × 10^{26} | 1.283 × 10^{27} | 3.138 × 10^{42} | 1.5500 | 1.320 × 10^{2} | 1.094 × 10^{6} |

39 | 3314 | 0.20 | 1.42 | 4.5446 | 9.633 × 10^{26} | 9.652 × 10^{26} | 2.147 × 10^{42} | 1.0020 | 9.864 × 10^{1} | 0.995 × 10^{6} |

40 | 3341 | 1.12 | 1.80 | 1.4256 | 1.962 × 10^{27} | 1.196 × 10^{28} | 6.158 × 10^{43} | 6.0970 | 2.712 × 10^{2} | 2.302 × 10^{6} |

41 | 3406 | 0.54 | 2.03 | 2.4106 | 2.814 × 10^{27} | 5.589 × 10^{27} | 2.233 × 10^{43} | 1.9860 | 9.115 × 10^{1} | 1.786 × 10^{6} |

42 | 3438 | 0.14 | 1.55 | 2.2032 | 1.253 × 10^{27} | 1.450 × 10^{27} | 3.693 × 10^{42} | 1.1572 | 5.309 × 10^{1} | 1.139 × 10^{6} |

43 | 3613 | 0.22 | 1.87 | 0.9763 | 2.200 × 10^{27} | 4.965 × 10^{27} | 1.906 × 10^{43} | 2.2570 | 4.750 × 10^{1} | 1.717 × 10^{6} |

45 | 3668 | 0.66 | 1.20 | 7.0675 | 5.813 × 10^{26} | 1.318 × 10^{27} | 3.253 × 10^{42} | 2.2670 | 5.394 × 10^{2} | 1.104 × 10^{6} |

46 | 3672 | 0.30 | 1.27 | 6.4627 | 6.891 × 10^{26} | 8.497 × 10^{26} | 1.811 × 10^{42} | 1.2330 | 2.068 × 10^{2} | 0.953 × 10^{6} |

47 | 3678 | 0.36 | 1.61 | 5.6419 | 1.404 × 10^{27} | 1.539 × 10^{27} | 4.001 × 10^{42} | 1.0965 | 1.218 × 10^{2} | 1.162 × 10^{6} |

48 | 3756 | 0.60 | 1.76 | 4.2422 | 1.834 × 10^{27} | 3.197 × 10^{27} | 1.060 × 10^{43} | 1.7430 | 1.554 × 10^{2} | 1.483 × 10^{6} |

sity of the stars in

Substituting (6.1.2) into (6.1.1), we arrive at

Or

Further analysis shows that there is no clear mathematical relation of the P-R plot, nor the P-M plot, using the raw data. In other words, we cannot compare the theoretical prediction and experimental relations for this group of stars. Such a result is to be expected, because they are halo stars and the theory assumes that the angular momentum expression to be represented by

We proceed with another group of stars with larger density, since the magnitude of B is sensitive to density values. Parameters include mass M in units of solar mass

No. | NGC 6819 stars | M/M_{?} | R(m) | P(s) | R^{3} (10^{26}m^{3}) | G-R^{3} (G-m^{3}) | Iω (J-s) | B (eq, G) | D (10^{2} kg/m^{3}) | R_{o}(m) |
---|---|---|---|---|---|---|---|---|---|---|

1 | 5,111,207 | 1.405 | 1.0166 × 10^{9} | 4.568 × 10^{5} | 10.507 | 4.33 × 10^{27} | 1.59 × 10^{43} | 4.123 | 6.19 | 1.64 × 10^{6} |

2 | 5,023,899 | 1.37 | 9.9267 × 10^{8} | 4.156 × 10^{5} | 9.7820 | 4.40 × 10^{27} | 1.62 × 10^{43} | 4.500 | 6.52 | 1.65 × 10^{6} |

3 | 5,023,760 | 1.355 | 9.8239 × 10^{8} | 4.130 × 10^{5} | 9.4810 | 4.32 × 10^{27} | 1.58 × 10^{43} | 4.556 | 6.79 | 1.64 × 10^{6} |

4 | 5,024,227 | 1.355 | 9.8239 × 10^{8} | 4.370 × 10^{5} | 9.4810 | 4.14 × 10^{27} | 1.52 × 10^{43} | 4.366 | 6.79 | 1.62 × 10^{6} |

5 | 5,024,122 | 1.3 | 9.4470 × 10^{8} | 5.500 × 10^{5} | 8.4300 | 3.19 × 10^{27} | 1.06 × 10^{43} | 3.781 | 7.33 | 1.48 × 10^{6} |

6 | 5,112,499 | 1.28 | 9.3093 × 10^{8} | 3.840 × 10^{5} | 8.0680 | 4.04 × 10^{27} | 1.45 × 10^{43} | 5.003 | 7.54 | 1.60 × 10^{6} |

7 | 5,113,601 | 1.28 | 9.3093 × 10^{8} | 6.060 × 10^{5} | 8.0680 | 2.87 × 10^{27} | 9.16 × 10^{42} | 3.552 | 7.54 | 1.43 × 10^{6} |

8 | 5,026,583 | 1.228 | 8.9520 × 10^{8} | 4.230 × 10^{5} | 7.1730 | 3.43 × 10^{27} | 1.16 × 10^{43} | 4.776 | 8.13 | 1.52 × 10^{6} |

9 | 4,938,993 | 1.21 | 8.8273 × 10^{8} | 1.030 × 10^{6} | 6.8787 | 1.71 × 10^{27} | 4.60 × 10^{42} | 2.484 | 8.36 | 1.20 × 10^{6} |

10 | 5,111,834 | 1.101 | 8.0740 × 10^{8} | 1.200 × 10^{6} | 5.2635 | 1.08 × 10^{27} | 2.99 × 10^{42} | 2.051 | 9.94 | 1.03 × 10^{6} |

11 | 5,111,908 | 1.09 | 7.9975 × 10^{8} | 1.500 × 10^{6} | 5.1160 | 1.02 × 10^{27} | 2.32 × 10^{42} | 1.998 | 10.1 | 1.01 × 10^{6} |

12 | 5,024,856 | 1.037 | 7.6296 × 10^{8} | 1.570 × 10^{6} | 4.4420 | 8.88 × 10^{26} | 1.92 × 10^{42} | 1.999 | 11.1 | 9.67 × 10^{5} |

13 | 5,024,280 | 1.026 | 7.5531 × 10^{8} | 1.500 × 10^{6} | 4.3100 | 8.99 × 10^{26} | 1.95 × 10^{42} | 2.085 | 11.3 | 9.71 × 10^{5} |

14 | 5,112,507 | 1.026 | 7.5531 × 10^{8} | 1.570 × 10^{6} | 4.3093 | 8.68 × 10^{26} | 1.86 × 10^{42} | 2.013 | 11.3 | 9.60 × 10^{5} |

15 | 5,023,796 | 1.012 | 7.4558 × 10^{8} | 1.580 × 10^{6} | 4.1450 | 8.38 × 10^{26} | 1.78 × 10^{42} | 2.023 | 11.6 | 9.49 × 10^{5} |

16 | 5,024,008 | 1.00 | 7.3723 × 10^{8} | 1.590 × 10^{6} | 4.0070 | 8.13 × 10^{26} | 1.71 × 10^{42} | 2.030 | 11.9 | 9.40 × 10^{5} |

17 | 5,023,724 | 0.99 | 7.3030 × 10^{8} | 1.560 × 10^{6} | 3.8943 | 8.09 × 10^{26} | 1.70 × 10^{42} | 2.078 | 12.1 | 9.38 × 10^{5} |

18 | 5,023,875 | 0.978 | 7.2193 × 10^{8} | 1.580 × 10^{6} | 3.7620 | 7.77 × 10^{26} | 1.61 × 10^{42} | 2.066 | 12.4 | 9.26 × 10^{5} |

19 | 5,112,268 | 0.972 | 7.1775 × 10^{8} | 1.620 × 10^{6} | 3.6970 | 7.56 × 10^{26} | 1.55 × 10^{42} | 2.045 | 12.5 | 9.17 × 10^{5} |

20 | 4,937,169 | 0.952 | 7.0870 × 10^{8} | 1.700 × 10^{6} | 3.4853 | 6.97 × 10^{26} | 1.39 × 10^{42} | 2.000 | 13.0 | 8.92 × 10^{5} |

21 | 5,025,271 | 0.952 | 7.0871 × 10^{8} | 1.840 × 10^{6} | 3.4853 | 6.55 × 10^{26} | 1.28 × 10^{42} | 1.880 | 13.0 | 8.74 × 10^{5} |

22 | 5,111,939 | 0.952 | 7.0871 × 10^{8} | 1.880 × 10^{6} | 3.4853 | 6.45 × 10^{26} | 1.26 × 10^{42} | 1.850 | 13.0 | 8.70 × 10^{5} |

23 | 5,112,871 | 0.946 | 6.9953 × 10^{8} | 1.840 × 10^{6} | 3.4234 | 6.47 × 10^{26} | 1.26 × 10^{42} | 1.890 | 13.1 | 8.71 × 10^{5} |

24 | 5,023,666 | 0.93 | 6.8834 × 10^{8} | 1.860 × 10^{6} | 3.2620 | 6.18 × 10^{26} | 1.18 × 10^{42} | 1.894 | 13.5 | 8.57 × 10^{5} |

25 | 5,024,182 | 0.916 | 6.7860 × 10^{8} | 1.840 × 10^{6} | 3.1245 | 6.03 × 10^{26} | 1.15 × 10^{42} | 1.930 | 13.9 | 8.50 × 10^{5} |

26 | 5,023,926 | 0.903 | 6.6949 × 10^{8} | 1.800 × 10^{6} | 3.0000 | 5.94 × 10^{26} | 1.13 × 10^{42} | 1.980 | 14.3 | 8.46 × 10^{5} |

27 | 4,937,149 | 0.883 | 6.5544 × 10^{8} | 1.870 × 10^{6} | 2.8160 | 5.49 × 10^{26} | 1.01 × 10^{42} | 1.950 | 14.9 | 8.24 × 10^{5} |

28 | 4,936,891 | 0.862 | 6.4069 × 10^{8} | 1.900 × 10^{6} | 2.6300 | 5.16 × 10^{26} | 9.32 × 10^{41} | 1.960 | 15.6 | 8.07 × 10^{5} |

29 | 4,937,119 | 0.852 | 6.3367 × 10^{8} | 2.010 × 10^{6} | 2.5445 | 4.81 × 10^{26} | 8.51 × 10^{41} | 1.890 | 15.9 | 7.89 × 10^{5} |

30 | 4,937,356 | 0.847 | 6.3019 × 10^{8} | 1.830 × 10^{6} | 2.5024 | 5.10 × 10^{26} | 9.17 × 10^{41} | 2.038 | 16.1 | 8.04 × 10^{5} |

First, we plot the B − P graph in log scale using the measured basic variables (established expression like mass-radius relation for dwarfs).

Since the correlation is very good, we can assume that

is valid, and Equation (6.2.1) becomes

Since the stars are associated with various values of the sets (M, R, P), if Equation (6.2.2) is to be approximately true, these variables must vary in such a way that the function F_{1} is approximately a constant. We plot in _{1 }for these 30 stars using the raw data from _{1} as a constant.

Equation (6.2.2) can then be approximated by:

Or

Employing raw data from

Or

Or

Note that the above equation is the consequence of the theoretically derived Equation (6.2.1), with the use of the P-R relation from measured data. To test the validity of the theory, we now use the raw data sets (M, R) from

Following,

stars have larger void radius R_{o} when the mass density is low, suggesting that gravity contraction causes the void size to decrease, but at a very slow rate because the value of the slope is not far from unity. We would investigate whether such effect is similar to other star groups, and the power index should in principle, indicative of the mass of the stars involved. As suggested by the crowing of data points in

We follow up to study the B - P graph for Low-to-Mid mass main sequence stars. Parameters mass

Following the argument of the previous sub-section, we can take that the function

is approximately constant, and the above equation gives

Since the M vs R plot is a good straight line with positive slope β = 1.0985(

We require

Or

If one plots the P-R graph (not shown here), the line of best fit has a very large positive slope(specified by angle

Star | M/M_{?} | R (10^{8} m) | P(s) | R^{3} (m^{3}) | B-R^{3} (G-m^{3}) | Iω (J-s) | B (eq, theory, G) | D (10^{3} kg/m^{3}) | R_{o} (10^{6} m) |
---|---|---|---|---|---|---|---|---|---|

Sun | 1.0000 | 6.9550 | 2.1600 × 10^{6} | 3.364 × 10^{26} | 5.9211 × 10^{26} | 1.120 × 10^{42} | 1.760 | 1.3880 | 0.8453 |

KIC892376 | 0.4699 | 3.6112 | 1.3237 × 10^{5} | 4.709 × 10^{25} | 1.0214 × 10^{27} | 2.300 × 10^{42} | 21.690 | 4.7400 | 1.0133 |

1026474 | 0.5914 | 4.4878 | 1.3556 × 10^{5} | 9.038 × 10^{25} | 1.6540 × 10^{27} | 4.394 × 10^{42} | 18.300 | 3.1090 | 1.1897 |

1026146 | 0.6472 | 4.8869 | 1.2866 × 10^{6} | 1.167 × 10^{26} | 3.7140 × 10^{26} | 6.008 × 10^{42} | 3.182 | 2.6344 | 0.7234 |

1162635 | 0.4497 | 3.4643 | 1.3546 × 10^{6} | 4.158 × 10^{25} | 1.6210 × 10^{26} | 1.993 × 10^{4}^{1} | 3.900 | 5.1385 | 0.5490 |

1164102 | 0.5606 | 4.2220 | 2.7210 × 10^{6} | 7.529 × 10^{25} | 1.5270 × 10^{26} | 1.837 × 10^{41} | 2.028 | 3.5400 | 0.5379 |

1027110 | 0.6046 | 4.5824 | 1.4697 × 10^{5} | 9.622 × 10^{25} | 1.6310 × 10^{27} | 4.320 × 10^{42} | 16.950 | 2.9850 | 1.1850 |

1160684 | 0.5239 | 4.0021 | 3.6200 × 10^{4} | 6.410 × 10^{25} | 3.4186 × 10^{27} | 1.159 × 10^{43} | 53.330 | 3.8826 | 1.5163 |

1027277 | 0.6735 | 5.0744 | 5.1960 × 10^{6} | 1.307 × 10^{26} | 1.4210 × 10^{26} | 1.669 × 10^{41} | 1.088 | 2.4488 | 0.5253 |

IM VirB | 0.6644 | 4.7363 | 1.1320 × 10^{5} | 1.063 × 10^{26} | 2.2365 × 10^{27} | 6.585 × 10^{42} | 21.050 | 2.9710 | 1.3166 |

GU BooA | 0.6101 | 4.3608 | 4.2336 × 10^{4} | 8.293 × 10^{25} | 3.8760 × 10^{27} | 1.371 × 10^{43} | 46.740 | 3.4950 | 1.5810 |

UV PscB | 0.7644 | 5.8074 | 6.9120 × 10^{4} | 1.959 × 10^{26} | 4.8855 × 10^{27} | 1.865 × 10^{43} | 24.944 | 1.8540 | 1.7080 |

YY GemA | 0.5992 | 4.3079 | 7.5168 × 10^{4} | 7.995 × 10^{25} | 2.4030 × 10^{27} | 7.399 × 10^{42} | 30.054 | 3.5600 | 1.3554 |

group in the following

With the establishment of (6.3.6), we also deduce that

Likewise, we obtain

The result of Equation (6.3.9) is shown in _{o} and hence current loop size depends also the type of elements generated in the matter shell at the time of observation.

Among this group of stars, we cannot expect all the star samples are built of hydrogen only. Thus the correlation of 0.83 in

M34 stars are pre-dwarfs having mass density slightly greater than that discussed in the last sub-section. Basic data are taken from [

Using raw data from

So far, we can say that Equation (6.4.2) is a consequence of the derived Equation (6.1.1) with input of raw data. The P-R plot in log scale (

Star no. | M/M_{?} | R/R_{?} | P (s) | R^{3} (m^{3}) | BR^{3} (G-m^{3}) | Iω (J-s) | B (Gauss) | D (kg/m^{3}) | R_{o} (m) |
---|---|---|---|---|---|---|---|---|---|

M34-1-304 | 0.90 | 0.87 | 5.250 × 10^{5} | 2.2150 × 10^{26} | 1.2830 × 10^{27} | 3.1390 × 10^{42} | 5.792 | 1.930 × 10^{3} | 1.0940 × 10^{6} |

M34-1-459 | 0.58 | 0.54 | 1.2476 × 10^{5} | 5.2970 × 10^{25} | 1.326 × 10^{27} | 4.4820 × 10^{42} | 25.03 | 5.200 × 10^{3} | 1.1060 × 10^{6} |

M34-1-654 | 0.82 | 0.77 | 8.0836 × 10^{5} | 1.5360 × 10^{26} | 7.209 × 10^{26} | 1.4550 × 10^{42} | 4.694 | 2.540 × 10^{3} | 9.0247 × 10^{5} |

M34-1-1015 | 0.53 | 0.49 | 9.5472 × 10^{4} | 3.9580 × 10^{25} | 1.3094 × 10^{27} | 3.2250 × 10^{42} | 33.082 | 6.361 × 10^{3} | 1.1010 × 10^{6} |

M34-1-1017 | 0.49 | 0.45 | 3.1389 × 10^{5} | 3.0657 × 10^{25} | 4.4500 × 10^{26} | 2.4435 × 10^{42} | 14.52 | 7.593 × 10^{3} | 7.6845 × 10^{5} |

M34-1-1054 | 0.53 | 0.49 | *1.0637 × 10^{6} | 3.9580 × 10^{25} | 2.1472 × 10^{26} | 3.9555 × 10^{41} | 5.425 | 6.361 × 10^{3} | 6.0271 × 10^{5} |

M34-1-1178 | 0.74 | 0.69 | 9.3226 × 10^{4} | 1.1050 × 10^{26} | 2.8605 × 10^{27} | 9.1428 × 10^{42} | 25.90 | 3.182 × 10^{3} | 1.4290 × 10^{6} |

M34-1-1540 | 0.95 | 0.92 | 5.9072 × 10^{5} | 2.6200 × 10^{26} | 1.3300 × 10^{27} | 3.2930 × 10^{42} | 5.078 | 1.850 × 10^{3} | 1.1070 × 10^{6} |

M34-1-1719 | 0.45 | 0.41 | 7.5514 × 10^{4} | 2.3190 × 10^{25} | 1.0570 × 10^{27} | 2.4235 × 10^{42} | 45.58 | 9.220 × 10^{3} | 1.0253 × 10^{6} |

M34-1-1906 | 0.54 | 0.50 | 2.0805 × 10^{5} | 4.2053 × 10^{25} | 7.6310 × 10^{26} | 1.5700 × 10^{42} | 18.145 | 6.100 × 10^{3} | 9.1980 × 10^{5} |

M34-1-2324 | 0.36 | 0.34 | 5.3654 × 10^{4} | 1.3220 × 10^{25} | 8.7230 × 10^{26} | 1.8760 × 10^{42} | 65.984 | 1.294 × 10^{4} | 9.6164 × 10^{5} |

M34-1-2370 | 0.28 | 0.28 | 5.3309 × 10^{4} | 7.3850 × 10^{24} | 5.4260 × 10^{26} | 9.9620 × 10^{41} | 73.47 | 1.800 × 10^{4} | 8.2094 × 10^{5} |

M34-2-599 | 0.67 | 0.63 | 8.6400 × 10^{5} | 8.4123 × 10^{25} | 4.3610 × 10^{26} | 7.4460 × 10^{41} | 5.185 | 3.784 × 10^{3} | 5.1850 × 10^{5} |

M34-2-2676 | 0.47 | 0.43 | 3.9053 × 10^{4} | 2.6750 × 10^{25} | 1.9230 × 10^{27} | 5.3840 × 10^{42} | 71.90 | 8.347 × 10^{3} | 1.2510 × 10^{6} |

M34-2-3071 | 0.91 | 0.88 | 6.7262 × 10^{5} | 2.2930 × 10^{26} | 1.0934 × 10^{27} | 3.4640 × 10^{42} | 4.768 | 1.885 × 10^{3} | 1.0368 × 10^{6} |

*this datum was observed under poor condition.

points are relatively scattered, due to large variation of the mass density among these samples. Just as that demonstrated in the last sub-section, if we have enough samples of narrow density range, the agreement between theory and measured result would be much better. Following, we plot the D-R graph in

NGC 2516 is an open star cluster in the southern sky, also called Southern Beehive [

Substituting Equation (6.5.1) into (6.1.1), we have,

This function F_{5} has been found numerically to be _{5 }to be a constant in our rough estimation, so that

which becomes

Or

From (6.5.3) & (6.5.4), we obtain an equation relating M and P resulting from theory (with information from basic data only):

Star no. | M/M_{?} | Radius (m) | P(s) | R^{3} (m^{3}) | BR^{3} (G-m^{3}) | Iω(J-s) | B (Gauss) | D (kg/m^{3}) | R_{o}(m) |
---|---|---|---|---|---|---|---|---|---|

N2516-1-1-784 | 0.20 | 0.24 | 5.61 × 10^{4} | 4.6508 × 10^{24} | 3.220 × 10^{26} | 4.9700 × 10^{41} | 69.254 | 2.430 × 10^{4} | 1.230 × 10^{6} |

N2516-1-1-351 | 0.44 | 0.41 | 2.00 × 10^{5} | 2.3187 × 10^{25} | 5.000 × 10^{26} | 8.9345 × 10^{41} | 21.570 | 9.020 × 10^{3} | 8.000 × 10^{5} |

N2516-1-1-958 | 0.49 | 0.45 | 5.44 × 10^{5} | 3.0660 × 10^{25} | 2.950 × 10^{26} | 4.416 × 10^{41} | 9.615 | 7.593 × 10^{3} | 6.700 × 10^{5} |

N2516-1-1-881 | 0.55 | 0.51 | 6.63 × 10^{5} | 4.4630 × 10^{25} | 3.341 × 10^{26} | 5.2200 × 10^{41} | 7.487 | 5.855 × 10^{3} | 6.980 × 10^{5} |

N2516-1-1-1470 | 0.56 | 0.52 | 7.61 × 10^{5} | 4.7300 × 10^{25} | 3.147 × 10^{26} | 4.8166 × 10^{41} | 6.652 | 5.624 × 10^{3} | 6.850 × 10^{5} |

We use the basic data from

In the following

Nine sets of raw data, each representing an average of 150 stars are entered into

As shown in

This constant F is found to be about

which becomes

Star no. | M/M_{?} | R/R_{?} | P (s) | R^{3} (m^{3}) | Iω(J-s) | BR^{3} (G-m^{3}) | B(eq, G) | D (kg/m^{3}) | R_{o} (m) |
---|---|---|---|---|---|---|---|---|---|

BPL102 | 0.25 | 0.26981 | 7.7040 × 10^{4} | 6.608 × 10^{24} | 5.720 × 10^{41} | 3.577 × 10^{26} | 54.13 | 1.800 × 10^{4} | 7.145 × 10^{5} |

BPL106 | 0.08 | 0.09744 | 1.4688 × 10^{4} | 3.112 × 10^{23} | 1.250 × 10^{41} | 1.145 × 10^{26} | 367.8 | 1.220 × 10^{5} | 4.887 × 10^{5} |

BPL115 | 0.10 | 0.12031 | 1.0476 × 10^{4} | 5.859 × 10^{23} | 3.343 × 10^{41} | 2.392 × 10^{26} | 408.3 | 8.100 × 10^{4} | 6.248 × 10^{5} |

BPL125 | 0.15 | 0.17649 | 6.9660 × 10^{4} | 1.850 × 10^{24} | 1.623 × 10^{41} | 1.428 × 10^{26} | 77.22 | 3.850 × 10^{4} | 5.216 × 10^{5} |

BPL129 | 0.13 | 0.15416 | 3.4700 × 10^{4} | 1.233 × 10^{24} | 2.154 × 10^{41} | 1.721 × 10^{26} | 139.62 | 5.000 × 10^{4} | 5.600 × 10^{5} |

BPL138 | 0.25 | 0.28600 | 9.2916 × 10^{4} | 7.870 × 10^{24} | 5.324 × 10^{41} | 3.392 × 10^{26} | 43.1 | 5.970 × 10^{4} | 7.020 × 10^{5} |

BPL150 | 0.18 | 0.20967 | 6.6456 × 10^{4} | 3.100 × 10^{24} | 2.880 × 10^{41} | 2.140 × 10^{26} | 69.00 | 2.760 × 10^{4} | 6.020 × 10^{5} |

BPL164 | 0.13 | 0.15416 | 7.2576 × 10^{4} | 1.233 × 10^{24} | 1.030 × 10^{41} | 9.894 × 10^{25} | 80.27 | 5.000 × 10^{4} | 4.641 × 10^{5} |

BPL190 | 0.15 | 0.17649 | 1.4497 × 10^{5} | 1.850 × 10^{24} | 7.800 × 10^{40} | 8.030 × 10^{25} | 43.42 | 3.850 × 10^{4} | 4.343 × 10^{5} |

TVLM | 0.07 | 0.10300 | 7.0490 × 10^{3} | 3.676 × 10^{23} | 2.550 × 10^{41} | 1.952 × 10^{26} | 530.94 | 9.047 × 10^{4} | 5.840 × 10^{5} |

J0036 | 0.067 | 0.09500 | 1.1088 × 10^{4} | 2.884 × 10^{23} | 1.320 × 10^{41} | 1.191 × 10^{26} | 413.0 | 1.104 × 10^{5} | 4.952 × 10^{5} |

J1835 | 0.083 | 0.10700 | 1.0224 × 10^{4} | 4.120 × 10^{23} | 2.250 × 10^{41} | 1.777 × 10^{26} | 431.13 | 9.568 × 10^{4} | 5.659 × 10^{5} |

Or

Since the graph of log P against log R is linear with a slope of +2.4462 (see

under the constrain

Or

The log M versus log R plot in _{o}/R graph is shown in

Basic data for a number of White Dwarfs are obtained from [

We have found that numerically, as in other star groups the LHS of the following equation is an approximate constant:

leading to

Equation (6.7.2b) indicates that our theory predicts a large negative slope

Star no. | M/M_{?} | R/R_{⊙} | P (s) | R^{3} (m^{3}) | BR^{3} (G.m^{3}) | Iω (J.s) | B (eq, theory G) | D (kg/m^{3}) | R_{o} (m) |
---|---|---|---|---|---|---|---|---|---|

GD140 | 0.52 | 0.0132 | 1.037 × 10^{3} | 7.738 × 10^{20} | 1.697 × 10^{26} | 2.114 × 10^{41} | 2.193 × 10^{5} | 3.92 × 10^{8} | 5.585 × 10^{5} |

Grw+73 8031 | 0.52 | 0.0132 | 1.296 × 10^{3} | 7.738 × 10^{20} | 1.436 × 10^{26} | 1.691 × 10^{41} | 1.856 × 10^{5} | 3.92 × 10^{8} | 5.280 × 10^{5} |

WD1337+70 | 0.52 | 0.0132 | 1.728 × 10^{3} | 7.738 × 10^{20} | 1.157 × 10^{26} | 1.269 × 10^{41} | 1.495 × 10^{5} | 3.92 × 10^{8} | 4.914 × 10^{5} |

LB253 | 0.52 | 0.0132 | 2.592 × 10^{3} | 7.738 × 10^{20} | 8.535 × 10^{25} | 9.211 × 10^{4}^{0} | 1.103 × 10^{5} | 3.92 × 10^{8} | 4.440 × 10^{5} |

W1346 | 0.52 | 0.0132 | 5.184 × 10^{3} | 7.738 × 10^{20} | 5.077 × 10^{25} | 4.230 × 10^{40} | 6.561 × 10^{4} | 3.92 × 10^{8} | 3.734 × 10^{5} |

G1423−B2B | 0.52 | 0.0132 | 6.998 × 10^{3} | 7.738 × 10^{20} | 4.054 × 10^{25} | 3.132 × 10^{40} | 5.239 × 10^{4} | 3.92 × 10^{8} | 3.464 × 10^{5} |

PG2131+066 | 0.62 | 0.0119 | 1.814 × 10^{4} | 5.669 × 10^{20} | 1.937 × 10^{25} | 1.171 × 10^{40} | 3.417 × 10^{4} | 5.95 × 10^{8} | 2.707 × 10^{5} |

L19-2 | 0.60 | 0.0122 | 9.504 × 10^{4} | 6.109 × 10^{20} | 5.663 × 10^{24} | 2.273 × 10^{39} | 9.270 × 10^{3} | 4.65 × 10^{8} | 1.798 × 10^{5} |

NGC1501 | 0.55 | 0.0128 | 1.011 × 10^{5} | 7.0554 × 10^{20} | 5.447 × 10^{24} | 2.156 × 10^{39} | 7.720 × 10^{3} | 3.03 × 10^{8} | 1.775 × 10^{5} |

Salpeter relation for dwarfs for decades [

In other words, expression (6.7.2b) becomes

For stars at younger age, fusion processes bring in heavier elements, but the density is low so that the gravitation contraction cannot overcome centrifugal force, and positive slopes should result in the P-R plots, as observed experimentally. More detailed discussion requires the analysis of the variation of mass density mathematically as a function of R.

Employing data values in

now. Some detect field~Tesla or more, and some measurements lead to the suggestion that while there are two types of White Dwarfs-one with negligible magnetic field and another type with huge magnetic field. We have devoted one short section on the plausible origin of the sporadic fields, explained in terms of Chern-Simon potential in Secction (5) already. Equations 6.7.2(a) & 6.7.2 (c) might be considered to be the two laws consequential to the new Equation (6.7.1) for White Dwarfs of low masses as stated.

Taking 7 Magnetic Whit Dwarf (MWD) samples [

Leading to;

The function C_{1} for the seven members are respectively 319.864, 318.5664, 319.917, 318.75, 319.635, 319.16, 319.40. The average K_{1} is about 319.33 (see

Therefore (6.8.2) becomes

Or

Now the relation between Mass and radius is assumed to follow the model of Hamada and Salpeter. Using data in

MWD | M/M_{?} | R/R_{⊙} | P(s) | R^{3} (10^{20}m^{3}) | BR^{3} (G-m^{3}) | Iω (J-s ) | B (eq, theory, Gauss) | D (10^{9} kg/m^{3}) | R_{o} (10^{5} m) |
---|---|---|---|---|---|---|---|---|---|

WD0533+053 | 0.71 | 0.0110 | 3.600 × 10^{3} | 4.478 | 6.410 × 10^{25} | 5.773 × 10^{40} | 1.431 × 10^{5} | 0.7532 | 4.028 |

WD1031+234 | 0.93 | 0.0088 | 1.224 × 10^{4} | 2.293 | 2.243 × 10^{25} | 1.42 × 10^{40} | 9.783 × 10^{4} | 1.9270 | 2.839 |

WD0548−001 | 0.69 | 0.0113 | 1.482 × 10^{4} | 4.854 | 2.260 × 10^{25} | 1.438 × 10^{40} | 4.656 × 10^{4} | 0.6752 | 2.846 |

WD0009+501 | 0.74 | 0.0107 | 2.160 × 10^{4} | 4.121 | 1.655 × 10^{25} | 9.489 × 10^{39} | 4.015 × 10^{4} | 0.8530 | 2.564 |

WD0011−134 | 0.71 | 0.0110 | 4.680 × 10^{4} | 4.478 | 9.364 × 10^{24} | 4.441 × 10^{39} | 2.091 × 10^{4} | 0.7530 | 2.121 |

WD1533−057 | 0.94 | 0.0086 | 8.640 × 10^{4} | 2.140 | 5.041 × 10^{24} | 1.947 × 10^{39} | 2.356 × 10^{4} | 2.0870 | 1.726 |

WD0912+536 | 0.75 | 0.0105 | 1.149 × 10^{5} | 3.895 | 4.638 × 10^{24} | 1.741 × 10^{39} | 1.191 × 10^{4} | 0.9150 | 1.678 |

WD1953−011 | 0.74 | 0.0107 | 1.246 × 10^{5} | 4.121 | 4.445 × 10^{24} | 1.645 × 10^{39} | 1.079 × 10^{4} | 0.8530 | 1.655 |

WD1829+547 | 0.90 | 0.0090 | 3.154 × 10^{9} | 2.453 | 1.979 × 10^{21} | 5.593 × 10^{34} | 8.0693 × 10 | 1.7433 | 0.126 |

Into Equation (6.8.3b), we arrive at

So far, we can consider that the theoretical Equation (6.8.1) leads to (6.8.5) using only the stated M-R relation, with input of “raw data”. When we plot

We show the relation between the void radius

It is well known that the three groups of Zeeman components,

On the other hand, measuring directly the features of Zeeman splitting in stellar spectral lines offers another method of detection of surface magnetic field strength of a star. However, such method, which in principle, should give more accurate surface magnetic fields, is limited to field strength greater than

We have analyzed the B (equatorial, theory) ?P relation for 8 groups of stars in the last section. Experimentally, it has long been discovered that the mean projected rotational velocity

Blackett found a positive correlation between the magnetic moment

Now if the “magnetic Bode’s Law” is established, two important issues emerge: (i) Since an electric current or permanent magnet must exist to generate a magnetic field, it is of fundamental interest in physics to know how the Maxwell equations are “hidden behind”/within the magnetic Bode’s law; (ii) On the application side, variables in magnetism (magnetic field, flux) can be calculated from variables of mechanics (period of rotation, radius, mass) and vice versa if only one is unknown; (iii) Various models of dynamo theories have been proposed to explain the source of dipolar field and sporadic fields of stars. We note that long ago, using over ten thousand spot samples, it has been shown, using rather stringent statistics (the Maximum-likelihood analysis) that there is no statistical evidence that sunspots in the northern hemisphere and southern hemisphere are correlated [

In 1996, Baliunas, et al. reported the values of measured magnetic fields of 112 low main sequence stars (type F to late K with one M) [_{HK}’ (called magnetic heating parameter) to approximate the magnetic moment µ_{m} and presented the

It is therefore fruitful to compare our theoretical prediction with measured data within one main star type with more samples than in the last section.

Incidentally, more recently, Marsden et al. [

Since the magnetic field measured is the average surface “longitudinal” field, we use the polar magnetic field derived from our theory in Section (5), which is simply twice the equatorial field, and plot the dependent variable B (polar, theory, G) in Gauss against the measured

One can use the “raw data” and plot the mass vs radius graph, for sun-like stars with density in the range

In

HIP no. | Age (Gyr) | M/M_{⊙} | R/R_{⊙} | v sin i (m·s^{−1}) | P (s) | B (measure, G) | BR^{3} (G-m^{3}, measure) | Iω (J-s) | B (eq, theory, G) | D (kg/m^{3}) | R_{o} (m) |
---|---|---|---|---|---|---|---|---|---|---|---|

42,333 | 1.88 | 1.038 | 0.98 | 4.00 × 10^{3} | 1.0706 × 10^{6} | 7.7 | 2.438 × 10^{27} | 2.253 × 10^{42} | 3.160 | 1.560 × 10^{3} | 1.007 × 10^{6} |

42,403 | 1.04 | 1.189 | 1.17 | 6.30 × 10^{3} | 8.1157 × 10^{5} | 4.2 | 2.262 × 10^{27} | 4.850 × 10^{42} | 3.300 | 1.000 × 10^{3} | 1.220 × 10^{6} |

43,410 | 1.88 | 1.211 | 1.24 | 9.00 × 10^{3} | 6.021 × 10^{5} | 8.1 | 5.196 × 10^{27} | 7.482 × 10^{42} | 3.840 | 8.970 × 10^{2} | 1.359 × 10^{6} |

43,726 | 1.32 | 1.056 | 1.00 | 1.20 × 10^{3} | 3.6416 × 10^{6} | 3.7 | 1.245 × 10^{27} | 7.015 × 10^{41} | 1.240 | 1.490 × 10^{3} | 7.520 × 10^{5} |

44,897 | 0.00 | 1.133 | 1.06 | 3.70 × 10^{3} | 1.1877 × 10^{6} | 6.6 | 2.645 × 10^{27} | 2.593 × 10^{42} | 2.775 | 1.340 × 10^{3} | 1.043 × 10^{6} |

46,580 | 0.00 | 0.786 | 0.73 | 3.10 × 10^{3} | 1.0291 × 10^{6} | 13.5 | 1.767 × 10^{27} | 9.847 × 10^{41} | 4.110 | 2.850 × 10^{3} | 8.186 × 10^{5} |

49,908 | N | 0.600 | 0.64 | 1.90 × 10^{3} | 1.4715 × 10^{6} | 3.4 | 2.996 × 10^{26} | 1.601 × 10^{41} | 3.128 | 3.240 × 10^{3} | 6.551 × 10^{5} |

56,242 | 3.8 | 1.0069 | 1.00 | 3.30 × 10^{3} | 1.4567 × 10^{6} | 1.7 | 7.612 × 10^{26} | 2.024 × 10^{42} | 2.061 | 1.070 × 10^{3} | 9.800 × 10^{5} |

56,997 | N | 0.850 | 0.90 | 2.40 × 10^{3} | 1.6387 × 10^{6} | 14.6 | 3.580 × 10^{27} | 1.016 × 10^{43} | 2.246 | 1.650 × 10^{3} | 8.251 × 10^{5} |

57,939 | N | 0.661 | 0.66 | 5.00 × 10^{2} | 5.7684 × 10^{6} | 6.6 | 6.384 × 10^{26} | 1.208 × 10^{41} | 1.153 | 3.250 × 10^{3} | 4.844 × 10^{5} |

62,523 | 1.40 | 1.004 | 0.93 | 2.8 × 10^{3} | 1.4515 × 10^{6} | 3.0 | 8.118 × 10^{26} | 1.4474 × 10^{42} | 2.653 | 1.763 × 10^{3} | 9.013 × 10^{5} |

66,147 | 0.00 | 0.805 | 0.94 | 1.4 × 10^{3} | 2.3098 × 10^{5} | 7.3 | 9.952 × 10^{26} | 4.6173 × 10^{42} | 12.570 | 2.805 × 10^{3} | 1.205 × 10^{6} |

66,275 | 1.64 | 1.341 | 1.46 | 1.5 × 10^{4} | 4.2535 × 10^{5} | 3.2 | 3.350 × 10^{27} | 1.6259 × 10^{43} | 4.210 | 2.549 × 10^{3} | 1.650 × 10^{6} |

68,184 | N | 0.8 | 0.78 | 1.3 × 10^{3} | 2.622 × 10^{6} | 3.7 | 5.907 × 10^{26} | 4.491 × 10^{41} | 1.869 | 2.380 × 10^{3} | 6.730 × 10^{5} |

71,181 | 1.96 | 0.78 | 0.74 | 1.8 × 10^{3} | 1.7965 × 10^{6} | 5.1 | 6.953 × 10^{26} | 5.752 × 10^{41} | 2.636 | 2.718 × 10^{3} | 7.156 × 10^{5} |

113,829 | 1.20 | 1.066 | 1.01 | 3.2 × 10^{3} | 1.3793 × 10^{6} | 2.7 | 9.359 × 10^{26} | 1.9074 × 10^{42} | 2.548 | 1.461 × 10^{3} | 9.656 × 10^{5} |

116,613 | 1.16 | 1.075 | 1.01 | 3.0 × 10^{3} | 1.4712 × 10^{6} | 5.2 | 1.802 × 10^{27} | 1.8033 × 10^{42} | 2.443 | 1.473 × 10^{3} | 9.522 × 10^{5} |

544 | 0.0 | 0.977 | 0.88 | 4.1 × 10^{3} | 8.4000 × 10^{5} | 2.70 | 4.564 × 10^{26} | 1.763 × 10^{42} | 4.930 | 2.750 × 10^{3} | 9.470 × 10^{5} |

3203 | 0.0 | 1.011 | 0.95 | 4.3 × 10^{3} | 9.6546 × 10^{5} | 7.2 | 2.077 × 10^{27} | 2.2864 × 10^{42} | 3.510 | 1.665 × 10^{3} | 1.011 × 10^{6} |

3765 | N | 0.756 | 0.76 | 2.0 × 10^{3} | 1.6606 × 10^{6} | 3.6 | 5.317 × 10^{26} | 6.362 × 10^{41} | 2.625 | 2.432 × 10^{3} | 9.339 × 10^{5} |

3979 | 0.76 | 0.939 | 0.88 | 1.8 × 10^{3} | 1.9300 × 10^{6} | 2.2 | 5.044 × 10^{26} | 9.115 × 10^{41} | 2.579 | 2.639 × 10^{3} | 9.632 × 10^{5} |

7244 | 0.0 | 1.037 | 0.95 | 2.2 × 10^{3} | 1.8870 × 10^{6} | 3.5 | 1.010 × 10^{27} | 1.1999 × 10^{43} | 2.163 | 1.708 × 10^{3} | 8.600 × 10^{5} |

7981 | N | 0.816 | 0.82 | 1.7 × 10^{3} | 2.1079 × 10^{6} | 3.3 | 6.122 × 10^{26} | 6.297 × 10^{41} | 2.074 | 2.090 × 10^{3} | 7.320 × 10^{5} |

10,339 | 0.0 | 0.957 | 0.89 | 6.0 × 10^{3} | 6.4821 × 10^{5} | 10.9 | 2.585 × 10^{27} | 2.829 × 10^{42} | 5.005 | 1.917 × 10^{3} | 1.066 × 10^{6} |

12,114 | 0.54 | 0.809 | 0.76 | 2.9 × 10^{3} | 1.1452 × 10^{6} | 1.2 | 1.772 × 10^{26} | 9.871 × 10^{41} | 3.649 | 2.602 × 10^{3} | 8.191 × 10^{5} |

15,457 | 0.0 | 1.034 | 0.95 | 5.2 × 10^{3} | 7.9836 × 10^{5} | 7.7 | 2.221 × 10^{27} | 2.828 × 10^{42} | 4.114 | 1.703 × 10^{3} | 1.066 × 10^{6} |

16,537 | 0.0 | 0.856 | 0.77 | 2.4 × 10^{3} | 1.4020 × 10^{6} | 10.9 | 1.536 × 10^{26} | 8.758 × 10^{41} | 3.207 | 2.650 × 10^{3} | 7.950 × 10^{5} |

71,631 | 0.0 | 1.044 | 0.97 | 1.68 × 10^{4} | 2.5230 × 10^{5} | 45.3 | 1.391 × 10^{28} | 9.419 × 10^{42} | 9.530 | 1.615 × 10^{3} | 1.440 × 10^{6} |

72,848 | 0.0 | 0.926 | 0.84 | 4.5 × 10^{3} | 8.1573 × 10^{5} | 7.6 | 1.515 × 10^{27} | 1.9378 × 10^{42} | 4.482 | 2.200 × 10^{3} | 9.696 × 10^{5} |

79,578 | 1.92 | 1.042 | 1.00 | 1.4 × 10^{3} | 3.1214 × 10^{6} | 3.4 | 1.144 × 10^{27} | 8.0778 × 10^{41} | 1.377 | 1.472 × 10^{3} | 7.789 × 10^{5} |

81,300 | 0.0 | 0.892 | 0.82 | 2.2 × 10^{3} | 1.6290 × 10^{6} | 0.6 | 1.113 × 10^{26} | 8.9076 × 10^{41} | 2.690 | 1.280 × 10^{3} | 7.983 × 10^{5} |

82,588 | 0.72 | 0.927 | 0.85 | 3.8 × 10^{3} | 9.7749 × 10^{5} | 8.5 | 1.7561 × 10^{27} | 1.6576 × 10^{42} | 3.847 | 2.130 × 10^{3} | 9.324 × 10^{5} |

88,945 | 2.04 | 1.039 | 0.9 | 7.9 × 10^{3} | 5.4763 × 10^{5} | 11.2 | 3.656 × 10^{27} | 4.499 × 10^{42} | 5.149 | 1.510 × 10^{3} | 1.197 × 10^{6} |

88,972 | N | 0.791 | 0.79 | 2.1 × 10^{3} | 1.6439 × 10^{6} | 1.5 | 2.488 × 10^{26} | 7.265 × 10^{41} | 2.580 | 2.270 × 10^{3} | 7.586 × 10^{5} |

91,043 | 0.03 | 1.06 | 1.09 | 39.0 × 10^{3} | 1.2214 × 10^{5} | 89.7 | 3.908 × 10^{28} | 2.4947 × 10^{43} | 13.940 | 1.160 × 10^{3} | 1.837 × 10^{6} |

92,984 | 2.72 | 1.058 | 1.06 | 1.23 × 10^{4} | 3.7660 × 10^{5} | 11.3 | 4.528 × 10^{27} | 7.637 × 10^{42} | 6.239 | 1.250 × 10^{3} | 1.366 × 10^{6} |

96,085 | 0.0 | 0.831 | 0.77 | 3.0 × 10^{3} | 1.1216 × 10^{6} | 1.8 | 2.765 × 10^{26} | 1.063 × 10^{42} | 3.708 | 2.570 × 10^{3} | 8.343 × 10^{5} |

98,921 | 1.76 | 1.065 | 1.01 | 4.3 × 10^{3} | 1.0264 × 10^{6} | 9.8 | 3.397 × 10^{27} | 2.561 × 10^{42} | 3.178 | 1.460 × 10^{3} | 1.049 × 10^{6} |

107,350 | 0.0 | 1.103 | 1.04 | 1.06 × 10^{4} | 4.2875 × 10^{5} | 14.8 | 5.601 × 10^{27} | 6.732 × 10^{42} | 6.009 | 1.390 × 10^{3} | 1.324 × 10^{6} |

109,572 | 2.44 | 1.51 | 2.52 | 1.16 × 10^{4} | 9.4934 × 10^{6} | 0.8 | 4.307 × 10^{27} | 2.444 × 10^{41 } | 0.200 | 1.330 × 10^{2} | 1.027 × 10^{6} |

113,829 | 1.2 | 1.066 | 1.01 | 3.2 × 10^{3} | 1.3793 × 10^{6} | 2.7 | 9.359 × 10^{26} | 1.962 × 10^{42} | 2.550 | 1.460 × 10^{3} | 9.657 × 10^{5} |

HIP no. | Age (Gyr) | M/M_{⊙} | R/R_{⊙} | v sin i (m·s^{−1}) | P(s) | B (measure, G) | BR^{3} (G-m^{3}, measure) | Iω (J-s) | B (eq, theory, G) | D (kg/m^{3}) | R_{o} (m) |
---|---|---|---|---|---|---|---|---|---|---|---|

49,081 | 9.52 | 0.960 | 1.06 | 3.4 × 10^{3} | 1.362 × 10^{6} | 0.8 | 3.2055 × 10^{26} | 1.915 × 10^{42} | 2.210 | 1.138 × 10^{3} | 9.667 × 10^{5} |

9350 | 5.44 | 0.979 | 0.97 | 1.9 × 10^{3} | 2.231 × 10^{6} | 0.5 | 1.5352 × 10^{26} | 9.990 × 10^{41} | 1.770 | 1.515 × 10^{3} | 8.215 × 10^{5} |

49,756 | 8.28 | 0.975 | 1.07 | 1.5 × 10^{3} | 3.117 × 10^{6} | 2.1 | 8.6550 × 10^{26} | 8.663 × 10^{41} | 1.186 | 1.972 × 10^{3} | 7.928 × 10^{5} |

50,316 | 8.00 | 1.070 | 1,46 | 2.6 × 10^{3} | 2.454 × 10^{6} | 3.1 | 3.2460 × 10^{27} | 2.248 × 10^{42} | 0.954 | 4.855 × 10^{2} | 1.006 × 10^{6} |

55,459 | 5.64 | 1.009 | 1.04 | 2.1 × 10^{3} | 4.545 × 10^{6} | 1.6 | 6.0550 × 10^{26} | 5.809 × 10^{41} | 0.957 | 1.270 × 10^{3} | 7.174 × 10^{5} |

56,242 | 3.80 | 1.007 | 1.10 | 3.3 × 10^{3} | 1.457 × 10^{6} | 1.7 | 7.6123 × 10^{26} | 2.024 × 10^{42} | 2.061 | 1.125 × 10^{3} | 9.800 × 10^{5} |

67,422 | N | 0.870 | 1.01 | 0.3 × 10^{3} | 1.471 × 10^{7} | 4.8 | 1.6640 × 10^{27} | 1.594 × 10^{41} | 0.371 | 1.192 × 10^{3} | 5.078 × 10^{5} |

111,274 | 11.4 | 0.979 | 1.38 | 2.0 × 10^{3} | 3.015 × 10^{6} | 1.2 | 1.0610 × 10^{27} | 1.496 × 10^{42} | 0.833 | 5.260 × 10^{2} | 9.509 × 10^{5} |

113,896 | 6.88 | 1.036 | 1.24 | 2.8 × 10^{3} | 1.935 × 10^{6} | 2.5 | 1.6040 × 10^{27} | 1.991 × 10^{42} | 1.422 | 7.673 × 10^{2} | 9.760 × 10^{5} |

114,378 | 6.72 | 1.009 | 1.13 | 1.0 × 10^{4} | 4.794 × 10^{5} | 1.11 | 5.3880 × 10^{27} | 6.502 × 10^{42} | 4.564 | 9.875 × 10^{2} | 1.312 × 10^{6} |

114,456 | 9.16 | 0.939 | 0.97 | 2.1 × 10^{3} | 2.019 × 10^{6} | 1.2 | 3.6846 × 10^{26} | 1.059 × 10^{42} | 1.850 | 1.453 × 10^{3} | 8.336 × 10^{5} |

114,622 | 12.5 | 0.794 | 0.77 | 1.8 × 10^{3} | 3.365 × 10^{6} | 1.1 | 1.5350 × 10^{26} | 3.385 × 10^{41} | 1.572 | 2.456 × 10^{3} | 6.268 × 10^{5} |

115,951 | 6.76 | 1.098 | 1.39 | 2.9 × 10^{3} | 6.074 × 10^{6} | 1.1 | 9.9387 × 10^{26} | 8.449 × 10^{41} | 0.531 | 5.773 × 10^{2} | 7.880 × 10^{5} |

116,106 | 6.28 | 1.015 | 1.15 | 3.7 × 10^{3} | 1.358 × 10^{6} | 2.0 | 1.0233 × 10^{27} | 2.391 × 10^{42} | 2.045 | 9.424 × 10^{2} | 1.022 × 10^{6} |

116,421 | 11.5 | 0.947 | 1.27 | 1.6 × 10^{3} | 5.550 × 10^{6} | 1.0 | 6.8913 × 10^{26} | 7.330 × 10^{41} | 0.582 | 6.530 × 10^{2} | 7.423 × 10^{5} |

682 | 612 | 1.045 | 1.12 | 1.5 × 10^{4} | 3.352 × 10^{5} | 4.4 | 2.0797 × 10^{27} | 9.440 × 10^{42} | 0.621 | 1.050 × 10^{3} | 1.441 × 10^{6} |

1499 | 7.12 | 1.026 | 1.11 | 1.6 × 10^{3} | 3.032 × 10^{6} | 6.6 | 3.0370 × 10^{27} | 1.009 × 10^{42} | 1.190 | 1.059 × 10^{3} | 8.235 × 10^{5} |

1813 | 10.9 | 0.965 | 1.18 | 2.8 × 10^{3} | 1.842 × 10^{6} | 2.4 | 1.3266 × 10^{27} | 1.765 × 10^{42} | 1.510 | 8.294 × 10^{2} | 9.472 × 10^{5} |

4127 | 6.64 | 1.108 | 1.60 | 4.1 × 10^{3} | 1.705 × 10^{6} | 3.5 | 4.8230 × 10^{27} | 4.018 × 10^{42} | 1.122 | 1.600 × 10^{3} | 1.164 × 10^{6} |

5985 | 5.12 | 1.101 | 1.25 | 5.0 × 10^{3} | 1.093 × 10^{6} | 1.4 | 9.1991 × 10^{26} | 3.811 × 10^{42} | 2.250 | 7.960 × 10^{2} | 1.148 × 10^{6} |

6405 | 5.88 | 0.953 | 0.98 | 1.6 × 10^{3} | 2.677 × 10^{6} | 0.8 | 2.5331 × 10^{26} | 8.273 × 10^{41} | 1.420 | 1.430 × 10^{3} | 7.837 × 10^{5} |

7276 | 5.04 | 1.242 | 1.80 | 4.2 × 10^{3} | 1.873 × 10^{6} | 0.8 | 1.5696 × 10^{27} | 5.198 × 10^{42} | 0.955 | 3.007 × 10^{2} | 1.241 × 10^{6} |

7513 | 3.12 | 1.310 | 1.64 | 9.6 × 10^{3} | 7.465 × 10^{5} | 2.5 | 3.7100 × 10^{27} | 1.142 × 10^{43} | 2.280 | 4.194 × 10^{2} | 1.511 × 10^{6} |

7585 | 5.08 | 1.022 | 1.04 | 2.6 × 10^{3} | 1.748 × 10^{6} | 2.5 | 9.4610 × 10^{26} | 1.530 × 10^{42} | 1.980 | 1.283 × 10^{3} | 9.139 × 10^{5} |

7734 | 3.76 | 1.010 | 0.98 | 2.4 × 10^{3} | 1.784 × 10^{6} | 6.6 | 2.0898 × 10^{27} | 1.315 × 10^{42} | 2.110 | 1.515 × 10^{3} | 8.800 × 10^{5} |

8159 | 7.84 | 1.112 | 1.73 | 2.5 × 10^{3} | 3.024 × 10^{6} | 0.4 | 6.9680 × 10^{26} | 2.663 × 10^{42} | 0.651 | 3.033 × 10^{2} | 1.050 × 10^{6} |

8362 | 10.1 | 0.836 | 0.85 | 1.3 × 10^{3} | 2.857 × 10^{6} | 0.9 | 1.8594 × 10^{26} | 5.175 × 10^{41} | 1.600 | 1.945 × 10^{3} | 6.970 × 10^{5} |

9829 | 11.3 | 0.877 | 0.97 | 2.2 × 10^{3} | 1.927 × 10^{6} | 2.3 | 7.0622 × 10^{26} | 1.036 × 10^{42} | 1.820 | 1.357 × 10^{3} | 8.291 × 10^{5} |

10,505 | 7.88 | 1.011 | 1.09 | 1.5 × 10^{3} | 3.176 × 10^{6} | 0.8 | 3.4854 × 10^{26} | 9.151 × 10^{41} | 1.170 | 1.102 × 10^{3} | 8.037 × 10^{5} |

14,150 | 7.60 | 0.962 | 0.99 | 0.8 × 10^{3} | 5.408 × 10^{6} | 0.9 | 2.9380 × 10^{26} | 4.218 × 10^{41} | 0.873 | 1.400 × 10^{3} | 6.622 × 10^{5} |

9911 | 7.16 | 1.080 | 1.36 | 2.7 × 10^{3} | 2.201 × 10^{6} | 0.8 | 6.7704 × 10^{26} | 2.196 × 10^{42} | 1.160 | 6.060 × 10^{2} | 1.000 × 10^{6} |

12,048 | 8.68 | 1.052 | 1.39 | 1.9 × 10^{3} | 3.197 × 10^{6} | 1.0 | 9.0352 × 10^{26} | 1.538 × 10^{42} | 0.832 | 5.530 × 10^{2} | 9.151 × 10^{5} |

113,357 | 6.76 | 1.054 | 1.15 | 2.6 × 10^{3} | 1.933 × 10^{6} | 0.6 | 3.0700 × 10^{26} | 1.745 × 10^{42} | 1.614 | 9.780 × 10^{2} | 9.444 × 10^{5} |

11,548 | 6.76 | 1.101 | 1.81 | 4.3 × 10^{3} | 1.839 × 10^{6} | 0.3 | 5.9850 × 10^{26} | 4.744 × 10^{42} | 0.877 | 2.620 × 10^{2} | 1.213 × 10^{6} |

60,353 | 3.28 | 1.163 | 1.23 | 6.0 × 10^{3} | 8.530 × 10^{5} | 0.7 | 1.8168 × 10^{27} | 4.976 × 10^{42} | 2.902 | 8.810 × 10^{2} | 1.228 × 10^{6} |

74,432 | 10.2 | 0.992 | 1.19 | 1.8 × 10^{3} | 2.889 × 10^{6} | 0.7 | 3.9740 × 10^{26} | 1.176 × 10^{42} | 1.082 | 8.280 × 10^{2} | 8.555 × 10^{5} |

76,114 | 8.44 | 0.957 | 1.05 | 1.0 × 10^{3} | 4.589 × 10^{6} | 1.9 | 2.7230 × 10^{26} | 2.553 × 10^{42} | 0.899 | 1.660 × 10^{3} | 7.091 × 10^{5} |

79,672 | 5.84 | 1.005 | 1.04 | 2.6 × 10^{3} | 1.748 × 10^{6} | 2.3 | 8.6924 × 10^{26} | 1.504 × 10^{42} | 1.608 | 1.260 × 10^{3} | 9.705 × 10^{5} |

109,378 | 10.6 | 0.986 | 1.06 | 1.8 × 10^{3} | 2.563 × 10^{6} | 0.9 | 3.2055 × 10^{26} | 1.046 × 10^{42} | 1.404 | 1.169 × 10^{3} | 8.310 × 10^{5} |

the last section for other star groups. Putting the measured data listed in

The only parameter value we have assumed, based on the argument explained twice before, is the Fermi energy of the spinor to be equal to the rest mass of electron. The cool stars considered in this sub-section and the next is similar to the regime of the Second Law of angular momentum we derived in [

Now we proceed to analyze the relation between the magnetic parameter

We will not take too much space to indicate all graphs above for the older sun- like stars, because there is high similarity in some. Based on data listed in

The variation of the theoretical equatorial fields of old sun-like stars with changing P is demonstrated in

Using raw data in [

We have provided names “young” & “old” for sun-like stars. The separation is not arbitrary. If we go through the data in

Now let us recollect that we have derived an explicit expression for the equatorial field, represented by

And the magnetic parameter is

where C_{1}, C_{2} are constants. So our theoretical derivation tells us that if we plot

Let us finally proceed to obtain demonstration of the Law of Intrinsic Dipolar Magnetic Field for Stellar Objects. When the equatorial field

For comparison, the measured surface magnetic fields published in [

We have shown only samples of one star group as an illustration. Other measured data points of the young stars behave in the same way. This law has been applied to analyze all the 8 star groups in Sections (6) & (7). We shall not repeat to show such similar graphs for other star groups. The reader can test this law readily. Such results show that once the “mechanical” data set (M, R, P) is obtained, the magnetic field can be calculated, without the explicit use of the Maxwell equations. This law is in line with the Law specified in Equation (7.2b) and the linear relation of

There are some mysteries in the properties of the magnetic fields of our planets. Before we analyze them, let us list the relevant data/variables in

The reported measured equatorial fields of 7 planets are indicated by the orange triangles. According to rotation periods and their distances from the sun, it appears that five pairs appear-(i) Mercury (

In our model, the conformal projection of

Planet | Mass (kg) | Radius (m) | P(s) | R^{3} (m^{3}) | BR^{3} (G-m^{3}) | Iω (J-s) | B (eq, theory, G) | D (kg/m^{3}) | R_{o} (m) |
---|---|---|---|---|---|---|---|---|---|

Sun | 1.990 × 10^{30} | 6.955 × 10^{8} | 2.160 × 10^{6} | 3.364 × 10^{26} | 5.921 × 10^{26} | 1.100 × 10^{42} | 1.7600 | 1.410 × 10^{3} | 8.453 × 10^{5} |

Mercury | 3.300 × 10^{23} | 2.440 × 10^{6} | 5.067 × 10^{6} | 1.453 × 10^{19} | 5.343 × 10^{17} | 9.745 × 10^{29} | 0.0368 | 5.427 × 10^{3} | 8.164 × 10^{2} |

Venus | 4.867 × 10^{24} | 6.050 × 10^{6} | 2.100 × 10^{7} | 2.214 × 10^{20} | 5.400 × 10^{18} | 2.133 × 10^{31} | 0.0244 | 5.204 × 10^{3} | 1.766 × 10^{3} |

Earth | 5.972 × 10^{24} | 6.370 × 10^{6} | 8.640 × 10^{4} | 2.585 × 10^{20} | 4.136 × 10^{20} | 6.800 × 10^{33} | 1.6000 | 5.510 × 10^{3} | 7.530 × 10^{3} |

Mars | 6.417 × 10^{23} | 3.390 × 10^{6} | 8.864 × 10^{4} | 3.894 × 10^{19} | 2.992 × 10^{19} | 2.087 × 10^{32} | 0.7680 | 3.940 × 10^{3} | 3.124 × 10^{3} |

Jupiter | 1.898 × 10^{27} | 6.991 × 10^{7} | 3.573 × 10^{4} | 3.417 × 10^{23} | 2.224 × 10^{24} | 6.780 × 10^{38} | 6.5100 | 1.330 × 10^{3} | 1.313 × 10^{5} |

Saturn | 5.684 × 10^{26} | 5.823 × 10^{7} | 3.836 × 10^{4} | 1.975 × 10^{23} | 6.481 × 10^{23} | 1.374 × 10^{38} | 3.2820 | 0.687 × 10^{3} | 8.714 × 10^{4} |

Uranus | 8.682 × 10^{25} | 2.540 × 10^{7} | 6.120 × 10^{4} | 1.639 × 10^{22} | 3.214 × 10^{22} | 2.300 × 10^{36} | 1.9610 | 1.270 × 10^{3} | 3.200 × 10^{4} |

Neptune | 1.024 × 10^{26} | 2.460 × 10^{7} | 5.800 × 10^{4} | 1.489 × 10^{22} | 3.610 × 10^{22} | 2.400 × 10^{36} | 2.4250 | 1.638 × 10^{3} | 3.327 × 10^{4} |

Pluto | 1.471 × 10^{22} | 1.184 × 10^{6} | 5.520 × 10^{5} | 1.660 × 10^{18} | 9.225 × 10^{16} | 9.380 × 10^{28} | 0.0556 | 1.880 × 10^{3} | 4.550 × 10^{2} |

energy and thus rest mass of hadrons are the protons (composed of (u, u, d) quarks) and neutrons (composed of (u, d, d) quarks), meaning that the Lorentz mass shell contains only u and d quarks of charges (2/3)e, and -(1.3)e. To create a proton (u, u, d) and a neutron (u, d, d) from P1, it requires 2 e massless spinors states, but just 1-e massless spinor state. When all the possible electron-proton pairs are created, the system can only generate neutrons, or the sets of (u, d, d) quarks. The quark mass for neutron is

We plot the average surface temperature against distance from the sun in log- log scale (

contribute to the total angular momentum of the Milky Way during the initial epoch. The rotation rate could have been deduced from the temperature if one were to observe these objects passed through a wide temperature range. Now these objects are already in the “cool range”, i.e. under the Second Law regime. Due to different processes of evolution and the fact that these stellar objects have very small masses as compared to the sun, the cooling process is very effective (so that the core temperature of the sun is >> than of the planets), and the mass density is in general much larger than that of the sun for the inner two pairs of planets.

In deriving the angular momentum of the e-trinos states in [

Thus, we consider that the generation of planets via the existence of the star sun is represented by the same group symmetry, and the 9 planets can be counted as 5 twins (with Pluto with its binary as a pair, but lying in the asteroid zone of the solar system). The arrangement of these planets shows Lie Group symmetry, due to SU(3) symmetry. There are two parity of choices in each element of the five (pairs) representation. In other words, there is a(p, n) duplet at the mass level, and the stellar object would split into two in view of, effectively, the SU(3) symmetry. Since the observed data suggests there are 5 pairs, similar to correct for spin multiplicity in a Fermi gas, the possible quantum states need to be divided by 5 when we do the averaging process for the loop current in calculating the dipole magnetic field. Thus we divide the dipole field strength by 5, and show the B-P graph in

Let us first come back to the process of generation of mass in an individual planet, after splitting into pairs. Normally the ec current states of different energy E, and opposite charge and velocity signs, are statistically random, and therefore the parity bias in surface mass creation by P1, is not favored, but non-zero. If such abnormal distribution occurs, the equal number of states within each hemisphere can then be broken. Let us propose a simple scenario for P1 projection in Mars. Suppose the 2 e states came from Northern hemisphere, while the -e state, came from 1 loop in the Southern hemisphere. If that had happened, then the remaining e spinors in the north loop was less than the remaining -e spinors in the south. Such a repeated P1 projection, during Mars cooling (note that there are many volcanoes detected), will result in the magnetic field strength at the South Pole much greater than that at the North Pole [

The dipolar planetary magnetic field shields the stellar particles radiation from impacting the planet’s surface without much energy lost; the magnetic field also serves to sustain the planetary atmosphere and to reduce possible material ejection from the planet’s surface. Thus Mars with an uneven dipole field and a thin atmosphere is known to have sent rocks that hit the earth.

In passing, we would remark that projection Po of the in-phase spinor states on the 3D surface of the void generate electrons in the Lorentz manifold, but leaving behind net positive massless spinor states on the void surface. A net current resulting from such states would generate magnetic field. Similarly, projection P1 of the out- of-phase spinor states in the 1D loops of the void generate quarks in the Lorentz manifold, but leaving behind net negative massless spinor states in the current loops of the void surface. Both events would lead to variation of the surface magnetic fields. Whether this model could explain the reversal of dipole polarity is outside the scope of this paper. We would also emphasize that most quarks and electrons were created when Perelman mappings were being implemented. The number of in-phase and anti-phase spinor states left over within the

According to [

where ^{3} is plotted vs angular momentum

We have sketched the steps that during the mapping from 5D to 4D manifold, the 4^{th} spatial variable in the 5D space-time structure would become the (non- zero) current loop with radius x’ at the 5D-4D boundary (one around each magnetic pole). In the unified 5D theories of Kaluza-Klein and Einstein, this space variable is compacted to zero, leading to gravitational singularity and hence black holes. Our deduction shows that such gravitational singularity does not exist, and the conclusion is well sup-

ported by the numerous detections of stars with masses > 100 M_{⊙} (the largest one reported so far has a mass of

Comparing the time intervals after the Big Bang when the lightest lepton (electron) is generated and that when the three quarks (building up a proton) join to form a proton, we obtain the radius of the loop current x’ in terms of the void radius R_{o}:

for all stellar objects with hydrogen as fuel. The modification on x’ when the next heavier element (Helium) becomes the fuel of the star is also derived.

Analyzing the early stage of formation of a galaxy based on the 5D projection theory, we have provided an answer to the detection of unexpected gamma ray bubbles, one above, and one below a galaxy.

Since Maxwell equations are classical and valid in both the 5D and 4D manifolds, we have derived an explicit expression of the number density of the spinor pairs (which satisfies the Fermi-Dirac distribution but also form the current loop with radius x’ in the classical sense) at the 4D-5D boundary, with quantum signature based on (i) the uncertainty principle & (ii) quantum statistics. Viewed in a 4D Lorentz manifold, this current loop must produce a magnetic field in the classical sense. Therefore we simply apply the Biot-Savart law, with quantum nature incorporated in the number density of the e-trinos, to derive the expression of a distant magnetic field as observed on the matter surface of a stellar object. The surface magnetic field is found to be

The variables are only M, R, P as the Fermi energy

Expanding

as convergent series, we discover that the approximate expression for B can be separated into three regions in temperature domain. In the high temperature domain where

We call Equation (7.6.3) the Law of Intrinsic Dipole Magnetic Field for Hot Stars. This law is applicable to fast rotating pulsars at very high temperature.

On the other hand, if

Thus under the low temperature domain, B is independent of T. We call Equation (7.6.4) as the Law of Intrinsic Dipole Magnetic Field for Cool Stellar Objects. Such an interesting result is obvious because the angular momentum of the object is a function of the void radius R_{o} and the amount of current generated (hence the dipole field strength) near the magnetic poles in the 4D-5D boundary is also a function of R_{o}. The current density, which usually appears in Maxwell equations, is illuminated, giving the “un-ex- pected result” as specified in Equation (7.6.4). The transition temperature between the two domains is

Taking a cool star as a solid sphere as an approximation, it is elementary to show from Equation (7.6.4) that

which is simply the “mysterious Magnetic Bode’s Law” found experimentally for cool stars.

We have found that for samples within a range of mass density, the

Moreover, based on the numerical analysis in Sections (6) & (7) for nine star groups, we have found that once the power index of the

We have proposed that electrons of the elements (atoms), starting with hydrogen could form 2D Semion states, with pinning of magnetic flux. Due to pressure wave or, in fact fluctuations during the natural cooling process, Semion state can be formed on/near the stellar surface when the gravitational force is

very large and the void radius

We have provided an explanation of the asymmetry of magnetic field on the Martian surface and compared the measured and theoretically deduced dipolar magnetic fields of the planets. For the different star groups we studied in Section (6), we cannot compare directly the theoretical magnetic field values with all the stars because we it is difficult to find the published complete set of data for most of these stars. Group comparison (such as Brown Dwarfs group) indicates that our predicted results are in line, in fact, within the same order of magnitude, with observational data and in general trends. For the sun-like group where the whole set of data (B, M, R, P) are available, theoretical predicted values agrees quite well with data (within an order of magnitude), considering the uncertainty and limitation in measurements discussed in details in Section (7.1).

We predict the equatorial fields B of the newly discovered Trappist-1 star and the 6 nearest planets. The B − P graph, and the B R^{3} ? Iω graph in the log-log plots for the 6 planets are both linear.

Based on the homogeneous 5D model of the universe, we have developed explicit expressions to explain quite a number of unanswered queries or mysteries in astrophysics, as outlined in Section (7.6). One step further, we hope that similar study can serve someway in building a part of the bridge to embrace Maxwell electro-magnetism and quantum mechanics in the future.

We wish to express our gratitude to the various types of assistance of Mr. Benjamin Fung and Dr. Junling Gao, Mr. Vincent Wan during the preparation of this manuscript. We appreciate Mr. Cressidy Fan for plotting

Fung, P.C.W. and Wong, K.W. (2017) Origin of Magnetic Fields of Stellar Objects in the Universe Based on the 5D Projection Theory. Journal of Modern Physics, 8, 668-746. https://doi.org/10.4236/jmp.2017.84045

To calculate the void radius

where

If we treat s pair of in-phase spinors of opposite charge as a bound state, the energy is taken to be the additive energy in the pair. We have already noted in [

The constant D is found to be:

Under the region satisfied by the Second Law of Angular Momentum, it has been shown in [

Considering a stellar object to be a solid sphere with mass M, radius R, period of rotation P, conservation of angular momentum among the 5D void and the 4D matter shell requires that

Giving

Numerically,

The dimension of

where

With

is bounded for a wide range of

Now

Integration by parts gives,

Summing all the series, we arrive at

Transforming

From (B.1), (B.12),(B.13),

When

When

Submit or recommend next manuscript to SCIRP and we will provide best service for you:

Accepting pre-submission inquiries through Email, Facebook, LinkedIn, Twitter, etc.

A wide selection of journals (inclusive of 9 subjects, more than 200 journals)

Providing 24-hour high-quality service

User-friendly online submission system

Fair and swift peer-review system

Efficient typesetting and proofreading procedure

Display of the result of downloads and visits, as well as the number of cited articles

Maximum dissemination of your research work

Submit your manuscript at: http://papersubmission.scirp.org/

Or contact jmp@scirp.org