On fixing the magnitudes of gravitational constant and strong coupling constant 

U. V. S. Seshavatharam ^{1}*, S. Lakshminarayana ^{2} 

^{1} Honorary faculty, ISERVE, Alakapuri, Hyderabad35,Ttelengana, India ^{2} Dept. of Nuclear Physics, Andhra University, Visakhapatnam03,AP, India *Corresponding author Email: seshavatharam.uvs@gmail.com 
In the earlier published papers the authors suggested that, “Magnitude of the unified force can be assumed to be equal to the classical or astrophysical force limit. Strength of any interaction can be defined as the ratio of the operating force magnitude and the magnitude of. If strength of the Schwarzschild interaction is assumed to be unity, then weak interaction strength seems to be ‘squared Avogadro number’ times less than the Schwarzschild interaction. The characteristic atomic force can be represented by ”. Thinking in this way, atomic gravitational constant can be expressed as With current atomic physical constants and with the assumed two new grand unified back ground numbers analytically  value of can be fixed for 10 digits and can be verified. Inverse of the strong coupling constant can be considered as the ‘natural logarithm of square root of ratio of gravitational and electromagnetic force ratio of down quark mass where the operating gravitational constant is squared Avogadro number times the gravitational constant’. Finally an attempt is made to fit and understand the mystery of Up and Down quarks, nuclear stability, and nuclear binding energy. For medium and heavy atomic nuclides, at the stable mass number, nuclear binding energy seems to be equal to the sum of rest energy of up quarks and down quarks.
Keywords: Gravitational constant; Schwarzschild’s interaction; Astrophysical force limit; Avogadro number; Particle rest masses; Strong interaction; nuclear binding energy; and Electron’s (n^{2}) quantum states.
1. Introduction
From final unification point of view, it is very much essential to couple the universal gravitational constant with the elementary physical constants. Then only the essence of unification can be understood. So far scientists proposed several interesting models (P. A. M. Dirac 1937), (Witten, Edward 1981), (David Gross 2005), (Abdus Salam 1981), (Salam A. & Sivaram C 1993), (Recami E 1994), (Dine, Michael 2007), (Roberto Onofrio 2013). In this context, readers may go through the authors published papers (U. V. S. Seshavatharam & S. Lakshminarayana 2014, 2013, 2011). By introducing two new back ground unified numbers , in the published paper the authors expressed their views (U. V. S. Seshavatharam & S. Lakshminarayana 2014) on final unification and proposed three characteristic relations for connecting, fitting and verifying the Newtonian gravitational constant in a unified approach via the Avogadro number In this paper, the topics covered and reviewed are: Schwarzschild interaction strength, meaning of strength of interaction in atomic physics, significance of Avogadro number, fitting of the gravitational constant, muon an tau rest masses, strong coupling constant, fine structure ratio, reduced Planck’s constant, rest masses of Up &
Down quarks, Nucleon rest masses, rms radius of proton, nuclear charge radius, nuclear stability, nuclear binding energy and unified atomic mass unit. Important points can be expressed as follows.
1) In any inverse square law of force, system is sustained only by means of the central attractive force and it is the root cause of revolving body’s angular momentum. If it is confirmed that, revolving body’s angular momentum is discrete, then it is a clear indication of the discrete nature of the central force acting on the revolving body. If one is willing to think in this direction, the historical mystery of Bohr’s discrete atomic structure and discrete angular momentum can be understood.
2) Note that, as per the basic concepts of final unification, there exists a fundamental unified force from which all the observed forces emerged. If so, magnitude of the unified force can be assumed to be equal to the astrophysical force limit. Note that, magnitude of the radial inward force acting on any black hole surface (U. V. S. Seshavatharam & S. Lakshminarayana 2014) is of the order of.
2. The classical limits of force and power
To unify cosmology, quantum mechanics and the four observed fundamental cosmological interactions certainly a ‘unified force’ is required. In this connection can be considered as the classical force or astrophysical force limit. Similarly can be considered as the classical power limit. If it is true that and are fundamental physical constants in physics, then and can also be considered as fundamental compound physical constants. These classical limits are more powerful than the Uncertainty limit. Without considering the current notion of black hole physics, Schwarzschild radius of black hole (Roger Penrose 1996), (Subrahmanyan Chandrasekhar 1983) can be understood with the characteristic astrophysical limiting force of magnitude. Note that by considering the famous Planck mass can be obtained very easily.
2.1. Simple applications of
a) Magnitude of force of attraction or repulsion between any two charged particles never exceeds.
b) Magnitude of gravitational force of attraction between any two massive bodies never exceeds.
c) Magnitude of mechanical force on a revolving/rotating body never exceeds.
d) Magnitude of electromagnetic force on a revolving body never exceeds.
2.2. Simple applications of
a) Mechanical power never exceeds
b) Electromagnetic power never exceeds
c) Thermal radiation power never exceeds
d) Gravitational radiation power never exceeds
3. Understanding the role of in black hole formation and Planck mass generation
3.1. Schwarzschild radius of a black hole
The four basic physical properties of a rotating black hole are its mass, size, angular velocity and temperature. Without going deep into the mathematics of black hole physics in this section an attempt is made to understand the Schwarzschild radius of a black hole. In all directions, if a force of magnitude acts on the massenergy content of the assumed celestial body it approaches a minimum radius of in the following way. Origin of the force may be due to selfweight or internal attraction or external compression or something else.
(1)
If no force (of zero magnitude) acts on the mass content of the assumed massive body, its radius becomes infinity. With reference to the average magnitude of, the presently believed Schwarzschild radius can be obtained as
(2)
This proposal is very simple and seems to be different from the existing concepts and may be a unified form of the Newton’s law of gravity, Special theory of relativity and General theory of relativity.
3.2 To derive the Planck mass
So far no theoretical model proposed a derivation for the Planck mass. To derive the Planck mass the following two conditions can be given a chance.
Assuming that gravitational force of attraction between two Planck particles of mass separated by a minimum distance (r_{min}) be,
(3)
With reference to wave mechanics, let
(4)
Here, represents the wavelength associated with the Planck mass. With these two assumed conditions Planck mass can be obtained as follows.
(5)
3.3. Understanding the strength of any interaction
From the above relations it is reasonable to say that,
1) If it is true that and are fundamental physical constants, then can be considered as a fundamental compound constant related to a characteristic limiting force.
2) Black holes are the ultimate state of matter’s geometric structure.
3) Magnitude of the operating force at the black hole surface is the order of.
4) Gravitational interaction taking place at black holes can be called as ‘Schwarzschild interaction’.
5) Strength of ‘Schwarzschild interaction’ can be assumed to be unity.
6) Strength of any other interaction can be defined as the ratio of operating force magnitude and the classical or astrophysical force magnitude.
7) If one is willing to represent the magnitude of the operating force as a fraction of i.e , where , then
(6)
If is very small, becomes very large. In this way, can be called as the strength of interaction. Clearly speaking, strength of any interaction is times less than the ‘Schwarzschild interaction’ and effective becomes.
4. Basic concepts and relations final unification
The following concepts and relations can be given a chance in final unification program.
1) With reference to the elementary charge and with mass similar to the Planck mass, a new mass unit can be constructed in the following way.
(7)
It can be called as the Stoney mass (G. J. Stoney, 1881). It is well known that play a vital role in fundamental physics. With these 3 constants, spacetime curvature concepts at a charged particle surface can be studied. It was first introduced by the physicist George Johnstone Stoney. He is most famous for introducing the term ‘electron’ as the ‘fundamental unit quantity of electricity’. In unification program, with this mass unit and with a suitable proportionality ratio characteristic mass of any elementary charge can be generated.
2) Avogadro number is an absolute number and it is having no units like ‘per mole’.
3) Atomic interaction strength is times less than the Schwarzschild interaction and hence atomic gravitational constant can be expressed as:
(8)
4) Similar to the classical force limit , in atomic system there exists a characteristic force of magnitude:
(9)
5) Independent of system of units and without considering the Avogadro number, unified atomic mass unit (P.J. Mohr et al 2010), (B. Andreas et al 2011), (B P Leonard 2007),(K.A. Olive et al 2014) can be fitted as follows.
(10)
Where is the unified atomic mass unit and is the average binding energy per nucleon. If obtained magnitude of Thus it can be suggested that, accuracy of depends only on the accurate ‘average binding energy per nucleon’.
4.1. Semi empirical applications of
There exist two new numbers they can be called as the ‘primordial unified back ground numbers’. They can also be called as the ‘back ground analytical numbers’ using by which micromacro physical constants can be interlinked qualitatively and quantitatively.
Application1: Rest masses of electron and proton
Electron rest mass can be expressed in the following way.
(11)
With proton rest mass can be expressed in the following way.
(12)
Thus,
(13)
Application2: Rest masses of muon and tau
(14)
Where can be called as the electron mass index. It can be estimated as:
(15)
With this number, electron, muon and tau rest masses can be fitted with the semi empirical relation.
(16)
Where Obtained rest energies are 0.511 MeV, 105.95 MeV and 1777.4 MeV respectively (K.A. Olive et al 2014). New heavy charged lepton at may be predicted close to 42262 MeV.
Application3: The reduced Planck’s constant and the rms radius of proton
From above relations,
(16)
If so, Reduced Planck’s constant can be expressed in the following way.
(17)
Characteristic nuclear radii like rms radius of proton (P.J. Mohr et al 2010), (Geiger H & Marsden. E 1909), (Michael O. Distler et al 2011), (Roberto Onofrio 2013) nuclear charge radius etc can be expressed in the following way.
(18)
If so, it is possible to show that,
(19)
Application4: To fit and verify the gravitational constant
In astronomy, the only one available characteristic empirical physical constant is the gravitational constant. Its value has been measured in the lab only within a range of 1 cm to a few meters. Until one measures the value of the gravitational constant with microscopic physical constants, the debate of final unification cannot be stopped up. In this context, G. Rosi et al say (G. Rosi et al 2014): “There is no definitive relationship between and the other fundamental constants, and there is no theoretical prediction for its value, against which to test experimental results. Improving the precision with which we know has not only a pure metrological interest, but is also important because of the key role that has in theories of gravitation, cosmology, particle physics and astrophysics and in geophysical models”. In general, ‘Unification’ means:
a) Understanding the origin of the rest mass of atomic elementary particles.
b) Finding and understanding the critical compositeness of the elementary physical constants.
c) Minimizing the number of elementary physical constants.
d) Merging different branches of physics with possible and suitable physical concepts.
Considering the proposed concepts and relations accurate values of Gravitational constant (L.L. Williams 2009), (George T Gillies 1997), (J Stuhler et al 2003), (Terry Quinn 2013), (J. B. Fixler et al 2007), (Brandenburg, J.E 1992), (Jun Luo and ZhongKun Hu 2000), (St. Schlamminger et al 2002) and Avogadro number can be estimated from elementary atomic physical constants. For the time being (i.e until a perfect model is developed), if one is willing to consider the revolving electron’s angular momentum as a compound physical constant and depends on the protonelectron rest masses, characteristic nuclear charge radius and the proposed discrete force it paves a path for coupling and interconnecting the micromacro elementary physical constants in a consistent manner. Thus it is possible to couple Avogadro number and Gravitational constant in the following way.
(20)
(21)
(22)
(23)
Thus relations (20, 21 and 22) can be considered as the 3 characteristic semi empirical unified relations. This assumed value of may not be absolute but can be given some consideration in unification program for further analysis This entire procedure depends on the two proposed new numbers and needs further research. So far there is no verifying procedure for the measured or estimated magnitude of with this kind of procedure, like other physical constants, value of can be fixed for 10 digits.
Application5: Strong coupling constant, Up and Down quarks and nuclear binding energy
Inverse of the strong coupling constant can be fitted as follows (A.V. Manohar and C.T. Sachrajdahttp 2014):
(24)
Now Down quark mass can be expressed as follows (Halzen, F.; Martin, A. D 1984).
(25)
(26)
Ratio of Up and down quark masses can be guessed as follows.
(27)
Thus Up quark mass can be fitted as follows.
(28)
Note that, these proposed Up and Down quark masses are roughly 2.20 times higher than the current estimates and their proposed mass ratio is matching with the current estimates. In a super symmetric approach, neutron and proton mass difference can be expressed as follows.
(29)
Where can be considered as the super symmetric fermionboson mass ratio (U. V. S. Seshavatharam & S. Lakshminarayana 2010, 2011).
With Up and Down quark masses nuclear binding energy (Chowdhury, P.R. et al 2005), (W.D. Myers & W.J. Swiatecki 1994), (G. Audi & A.H. Wapstra 1993) can be fitted as follows.
Step1: To fit the stable mass number of
(30)
Step2: To fit the nuclear binding energy at stable mass number of
(31)
Where,
See table1 for the estimated nuclear binding energy near to the stable mass number. Considering evenodd corrections on the estimated stable mass number and
With further research, data accuracy can be improved. From the data it is very clear to say that:
1) At the stable mass number, nuclear binding energy seems to be equal to the sum of rest energy of up quarks and down quarks.
2) As per the quark theory proton constitutes two Up quarks and one Down quark. Hence it can be guessed that, near to stable mass number, nuclear binding energy seems to depend only on the proton number.
Step3: To fit the nuclear binding energy above and below the stable mass number of
(32)
Table 1: T_{o} fit the nuclear binding energy near to stable mass number of
Proton number 
Estimated stable mass number

Estimated value of 
Binding energy in MeV (near to stable mass number) 
2 
4 
0.6368 
25.3 
3 
6 
0.6813 
40.5 
4 
8 
0.7148 
56.7 
5 
10 
0.7418 
73.5 
6 
12 
0.7647 
91.0 
7 
14 
0.7846 
108.9 
8 
16 
0.8023 
127.3 
9 
18 
0.8182 
146.0 
10 
21 
0.8327 
165.1 
11 
23 
0.8460 
184.5 
12 
25 
0.8584 
204.2 
13 
27 
0.8699 
224.2 
14 
29 
0.8807 
244.5 
15 
31 
0.8909 
265.0 
16 
34 
0.9005 
285.7 
17 
36 
0.9097 
306.6 
18 
38 
0.9184 
327.8 
19 
40 
0.9267 
349.1 
20 
42 
0.9347 
370.6 
21 
45 
0.9423 
392.4 
22 
47 
0.9496 
414.2 
23 
49 
0.9567 
436.3 
24 
52 
0.9635 
458.5 
25 
54 
0.9701 
480.9 
26 
56 
0.9764 
503.4 
27 
58 
0.9826 
526.0 
28 
61 
0.9886 
548.8 
29 
63 
0.9944 
571.8 
30 
66 
1.0000 
594.8 
31 
68 
1.0000 
614.7 
32 
70 
1.0000 
634.5 
33 
73 
1.0000 
654.3 
34 
75 
1.0000 
674.2 
35 
78 
1.0000 
694.0 
36 
80 
1.0000 
713.8 
37 
82 
1.0000 
733.6 
38 
85 
1.0000 
753.5 
39 
87 
1.0000 
773.3 
40 
90 
1.0000 
793.1 
41 
92 
1.0000 
812.9 
42 
95 
1.0000 
832.8 
43 
97 
1.0000 
852.6 
44 
100 
1.0000 
872.4 
45 
102 
1.0000 
892.3 
46 
105 
1.0000 
912.1 
47 
108 
1.0000 
931.9 
48 
110 
1.0000 
951.7 
49 
113 
1.0000 
971.6 
50 
115 
1.0000 
991.4 
51 
118 
1.0000 
1011.2 
52 
121 
1.0000 
1031.1 
53 
123 
1.0000 
1050.9 
54 
126 
1.0000 
1070.7 
55 
129 
1.0000 
1090.5 
56 
131 
1.0000 
1110.4 
57 
134 
1.0000 
1130.2 
58 
137 
1.0000 
1150.0 
59 
139 
1.0000 
1169.9 
60 
142 
1.0000 
1189.7 
61 
145 
1.0000 
1209.5 
62 
148 
1.0000 
1229.3 
63 
150 
1.0000 
1249.2 
64 
153 
1.0000 
1269.0 
65 
156 
1.0000 
1288.8 
66 
159 
1.0000 
1308.7 
67 
162 
1.0000 
1328.5 
68 
164 
1.0000 
1348.3 
69 
167 
1.0000 
1368.1 
70 
170 
1.0000 
1388.0 
71 
173 
1.0000 
1407.8 
72 
176 
1.0000 
1427.6 
73 
179 
1.0000 
1447.4 
74 
182 
1.0000 
1467.3 
75 
184 
1.0000 
1487.1 
76 
187 
1.0000 
1506.9 
77 
190 
1.0000 
1526.8 
78 
193 
1.0000 
1546.6 
79 
196 
1.0000 
1566.4 
80 
199 
1.0000 
1586.2 
81 
202 
1.0000 
1606.1 
82 
205 
1.0000 
1625.9 
83 
208 
1.0000 
1645.7 
84 
211 
1.0000 
1665.6 
85 
214 
1.0000 
1685.4 
86 
217 
1.0000 
1705.2 
87 
220 
1.0000 
1725.0 
88 
223 
1.0000 
1744.9 
89 
227 
1.0000 
1764.7 
90 
230 
1.0000 
1784.5 
91 
233 
1.0000 
1804.4 
92 
236 
1.0000 
1824.2 
93 
239 
1.0000 
1844.0 
94 
242 
1.0000 
1863.8 
95 
245 
1.0000 
1883.7 
96 
248 
1.0000 
1903.5 
97 
252 
1.0000 
1923.3 
98 
255 
1.0000 
1943.1 
99 
258 
1.0000 
1963.0 
100 
261 
1.0000 
1982.8 
5. To understand the discrete behavior and the total energy of electron in hydrogen atom
Step1: To understand the discrete behavior
From Bohr’s theory of Hydrogen atom (N.Bohr 1913) maximum number of electrons that can be accommodated in any principal quantum shell are this proposal can be reinterpreted as follows: In Hydrogen atom, in principal quantum shell, electron can exist in different quantum states. It can be understood as follows. Guess that currently believed s shell is the basic unit of all quantum shells and it constitutes a maximum of 2 numbers of electrons. With reference to the current concept of electrons, there can exit number of sshells. If one sshell represents on quantum state, then with reference to number of sshells, one can expect number of different quantum states with different energy levels.
Step2: To understand the potential energy of different states
Let potential energy of electron at any one quantum state be:
(33)
Where is the distance between electron and proton corresponding to quantum state. Potential energy of possible quantum states can be:
(34)
Based on the Virial theorem (Celso L. Ladera et al 2010) in a central force field, quantitatively kinetic energy is half the potential energy. Following this idea, total kinetic energy of electron for quantum state can be:
(35)
Thus, total energy of electron for quantum states can be:
(36)
If so, potential energy of electron at any one quantum state can be:
(37)
Kinetic energy of electron at any one quantum state can be:
(38)
Total energy of electron at any one quantum state can be:
(39)
Step3: To understand the emitted photon energy
With reference to the jumping nature of electron from one quantum state to another quantum sate, emitted photon energy can be:
(40)
Where
6. Conclusion
So far no model succeeded in coupling and understanding the unified concepts of gravity, electromagnetic and strong interactions. Based on the proposed concepts and accurate relations and with further research and analysis, different models of final unification can be developed with different proportionality ratios and finally a unified model can be standardized. The absolute magnitude of can be fixed and uncertainty in its current recommended magnitude can be minimized
Acknowledgements
The first author is indebted to professor K. V. Krishna Murthy, Chairman, Institute of Scientific Research on Vedas (ISERVE), Hyderabad, India and Shri K. V. R. S. Murthy, former scientist IICT (CSIR) Govt. of India, Director, Research and Development, ISERVE, for their valuable guidance and great support in developing this subject.
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