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NEWTON GRAVITATIONAL CONSTANT CALCULATED BY MEAN OF OTHER PHYSICS CONSTANTS.

MUTUAL RELATIONS BETWEEN THE 4 FORCES OF NATURE AND ITS VARIATION WITH TIME

SECOND PART Ramón Garza Wilmot Sept. 10 of 2015 INTRODUCTION AND PURPOUSE: What I expose here, does not intent in any way to be a theory of the 4 forces of nature. It is only an analysis of the relationships between the size of the coupling parameters of the four fundamental forces and its change with time. The study is not based on any physical theoretical analysis. It is rather a study of the numerical relationships between these parameters, the coupling parameters. It is highly speculative, but with hints of truth on the basis of the results. In this analysis we can get from the ratio of the mass of the proton to that of the electron and that of the neutron to proton, the knowledge of the magnitude of the coupling parameters of the forces, or vice versa. As an example, we can calculate the value of the gravitational constant from the constant of the weak force. We will also see how to calculate the gravitational coupling constant by knowing the mass of the proton, electron and neutron. It is seen how the fundamental forces could be interrelated with the fourth power of the previous less "intense". This propriety of variation with the fourth power, gives a glimpse of the possibility that there are forces of a higher and lower order than the strong force and gravitational force respectively. For example; gravitation, the weakest of the known forces could lead to the existence of another force even weaker with an intensity of down to 1e-256 weaker than the strong force. And even still more, these relationships are determined by the temperature of the cosmic microwave background radiation. The equations obtained with the values of "J" and "D" are so precise, that believe it as a coincidence I considered highly unlikely. The results do not match the traditional cosmology. For example: the standard cosmology said that during or at the time of the Big Bang, the 4 fundamental forces were one and that these were separated later. I deduced that this step was only between some of them. And that the unification of the others has not yet happened, and that it will be until the temperature is closer to the absolute zero when this unification will happen. And I am speaking of a time so remote in the future that the universe will be 10e32 times older than what it is today. Some of the magnitudes which I used are not the conventional or traditional ones, but even so, it is always possible to transform them to those conventional; for example: "m" is not a particle, but the square root of the product of the mass of the proton by the mass of the electron, or α is not the fine-structure constant but 2 π times the inverse of this constant and as well as I will be explaining. However, the end result of the following equations is manifested in an Excel spreadsheet that will be able to locate on the web page of Scribd. http://es.scribd.com/doc/278742010/Universal Write the web address directly in the browser you are using if the link does not operate. In the Excel executable file named Universal within this page, (which in fact includes everything or almost everything that I have written) you can download it and do the calculations for yourselves. With this worksheet, and with only the time as a variable, it may be calculate the following things in the past, present, and in the future. This sheet shows a single column. It is in the green in it part where it says “epoca” where it should be written the time to calculate all the others.

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The Mass, the radius, the density, the number of nucleons, the thermal energy, etc., of the universe. Microwave Cosmic Background Temperature. Charges of each one of the forces Values of the coupling parameters of the forces. Mass of the proton, electron and neutron. Wavelength of the mason. The data from the Planck Epoch The ratio photon/proton The initial conditions of the universe, when it was light for the first time. I am speaking of a time as small as 1.686888763695E-99 seconds, demonstrating that there was never being a moment of infinite density of matter or energy. But very high indeed . Of the order of 10e 204 grams/cm^3 mass density. Finally, remember that this chapter is supported and by necessity with the previous in: http://es.scribd.com/doc/233735619/Exact-Value-of-Hubble-Constant 1.- DESCRIPTION OF THE FORCES a) The strong force: is the most intense of the four, is responsible among other things to keep together the nucleus of the atom in spite of the electrostatic repulsion generated by the electromagnetic force caused by the rejection that between them suffer the protons. It explains the large amount of energy generated by the processes of nuclear fission. Its coupling constant is represented with the symbol " σ " (b) The electromagnetic force: Is the next lower in intensity to the strong force. All the phenomena electrical, magnetic and optical are its manifestation. This is the first of the forces that were unified, because until not long ago it was believed that magnetism and electricity were separate forces. Thanks to the work of J. C. Maxwell proved that both were separate manifestations of a single force the electromagnetic force. Their coupling constant is represented with the symbol "α" (sometimes represented as @) (c) The weak force: it is not a force in the sense of forces of attraction or repulsion between particles. Its role is to transform the identity of subatomic particles during the process of radioactive decay, for example: the transmutation of a neutron into a proton an electron and a neutrino. Its coupling constant will be identified with the letter 'W'. (d) Finally, the weakest and the most famous of the forces, the gravitational force. Is always attractive and is the force that holds the planets to the sun, the stars to galaxies and galaxies to the entire universe. The symbol for the coupling constant is "B" .or β 2.- GENERAL ALGEBRAIC DEFINITION OF THE 4 FORCES. The mathematical definition of these forces or better, the mathematical description of the coupling parameters is expressed in the following way:

Where: Q is the coupling constant (σ, α, w or B) h is Planck's constant q^2 is the square of the corresponding charge c is the speed of light in vacuum In the particular case of the electromagnetic force, this is represented in this form:

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= 1/ 1.3703599946E 02 Where q^2 is the square of the electric fundamental charge: q^2 = 2.307077352358E-19 eu ħ = h/ 2π = 1.054571725336E-27 gm-cm^ 2/sec c = 2.997924580000E+10 cm/sec Calculated with my procedure it is transformed to: α = h c / q^2 The fact that the constant of the electromagnetic force α known as fine-structure constant is calculated as:

Leads us to express the other coupling parameters in the same way i.e.:

(2.4)

Here, is the product of me (the mass of the electron) and mp (the mass of the

proton.) or . It should be noted that there is not, or at least is not recognize,(as far as I know) the existence of a weak charge (qw) or a strong charge (qs), but the fact that calculations can be made based on this concept, leads me to use the concept of "charges” for these forces. Of course, in the case of the gravitational force, the gravitational charge is:

This represents the gravitational charge between a proton and an electron. I must make notice that the system of units I use, is the cgs (centimeter, gram, second). I will also say that these parameters of the forces in reality are not constants at all, because I have already come to the conclusion in the first part of this analysis, that they change with the age of the universe and in consequence with the energy of the cosmic microwave background, or what is the same, with the energy in that they are involved. In addition, the representation that I'll do for it, has been (in the benefit of the easiness of calculation), represented conversely. This means that the more intense the force, the smaller is its constant or coupling parameter. As an example is B, which in magnitude is the largest of the parameters, but is the weakest of the forces. This is due to the fact that the standard representation is inverted, and in addition is divided by 2 π because of the use of ħ instead of h, Planck's constant. These are the values of the constants I am using. In particular G will be change to the new value I calculate forward. Π = 3.141592653590E+00 G = 6.67191E-08 erg-cm/gm^2 K = 1.380648800000E-16 erg/kelv Z = 4.965196625900E+00

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h = 6.626069570000E-27 erg-seg C = 2.997924580000E+10 cm/seg : G the Newton gravitational constant value of more recent measurement. K the Boltzmann Constant Z one of the constants of the laws of radiation h the Planck constant C the speed of light in vacuum 3.- MATHEMATICAL DEFINITION FOR EACH OF THE 4 FORCES FROM THE STANDARD INFORMATION KNOWN. Now we will get the values that are provided by the various measurements that have been made of these forces, which will help as an initial reference. A) The gravitational force: It is simple to calculate it :

B ≈ 1.954068618890E+42 (3.1 ) Remember that B) The weak force: The parameter that defines it, has a value represented by Gf which is the Fermi´s coupling constant , in such a way that the most accurate values measured for it, give us:

= 1.1663787e-5 GeV^(-2)

= 1/1.1663787e-5 GeV^(2)

= 8.573544767E+04 GeV^2

Now there is a need to transform the value in GeV to ergs in this way:

Then

= 2.200803003E-01 ergs^(-2) (3.2) In consequence of this equation above we have: Gf = 1.435850779E-49 ( 3.3 )

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Paul Davies defines in his book "The Accidental Universe" the equation (1.12 ) (of his book) as: (3.4) In this analysis, what Davies calls gw is the previous constant Gf. I have already mentioned the traditional equations for this type of relations. I used other units and wherever it appears mp or me, instead I use m^2 = mp x me, and instead of ħ, I use h . The idea is that because the coupling parameters are dimensionless, they retain its value either for any of the particles or units. In this case which is used the mass of the proton, I change it by m keeping constant Gf (or gw) and calculate the equation as: (3.5 ) Of course, expressed in the form of (3.4), this previous value will be 1.0243656141E-05 Here ( 3.5 ) would be the inverse of the coupling constant "W" (3.6) Note that I am transforming or better said using, the equations of the forces by replacing the variable mass “mp” by "m" and “ħ” by “h”. The reason to do this is just that in all of these analysis as numerical it is, I want to use the same system, and keep it, because on this way, I found some sense on all of these. C) The electromagnetic force: This is the force known in more detail. Its coupling constant is calculated easily knowing the value of the elementary electric charge in ues. (Electrostatic units) α = h c/ qa^2 (3.7 )

α = 8.610225783333E 02 It should be remembered that this is usually represented as: 2 π/ 8.610225783333E 02 = 7.2973525495E-03 D) The strong force. First of all, I should say that the value of this constant is not well defined, because I have read for it different values depending on another term that is often described as

αs which value is not define with precision.

The given values ranging from 14.4 in http://rickbradford.co.uk/CCC_AppF_StrongCouplinggs.pdf , to 14.6 in Wikiversity and up to 16 (without units). This is due to the fact that its measurement depends of the energy of the process. Somewhat complicated processes, which analysis is well beyond the scope of this writing.

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As a first approximation, I will take αs = 15 as indicated by Paul Davies in his Accidental Universe. However, this constant is defined by : αs = qs^2/ħc (3.8 ) Being qs the representation of the "strong charge" as I had already explained. Depending on the definition of σ = hc/qs^2 is that:

qs^2 = ħc. s = hc/ = 2π ħ c / and:

s = 2π/ (3.9) In consequence :

≈ 2π/15

Then: ≈ 4.18e-1 (3.10) I write the value of σ as approximate (≈) due to the uncertainty of the value of αs. 4.- SYMMETRY OF THE PARAMETERS OF THE FORCES Then, it comes some interesting thing. Note that by comparing the magnitudes of the parameters it is found that:

≈ ^4

W ≈ ^4 B ≈ W^4

= ^4/x1 (4.1) W = ^4/x2 (4.2) B = W^4/x3 (4.3) We also see that:

= yo F^(1/4) (4.4) = y1 F (4.5) W = y2 F^4 (4.6) B = y3 F^16 (4.7) As part of this analysis is the comparison of the forces with the other´s values and knowing ratios, I have found that the ones which conforms to these values is the 4th power of D, the ratio of the mass of the proton to that of the electron, leaving to deduction of the values of the "y" and “x” to solve. That is to say; y0 , y1, y2, y3 and x1 , x2, x3 Where F = D/4 being D = 1.836152671949E+03 = mp/me and F = 4.590381679872E+02 I also included as a factor J = mn/mp Now, it is to find out which are the values of the "x" and the "y" and see if they have any meaning. Finding any significant relationship between the constants (or parameters) of the forces, requires the knowledge of its numeric values with high accuracy, since otherwise you could lose some meaning, if any. Unfortunately, the basic constants of nature are not measured with the accuracy required. As well as there is no other alternative but to use the information that is best known and from it, infer the rest,

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always and when the results are at least , within the range of accuracy observed experimentally. If not, any result will be doubtful. That is to say, if for example on the achievement on calculating the gravitational constant G, this result must be within the range observed by the best experiment that has been measured of it. However, from some of the basic measurements I have to begin to. And for this, I shall cite as effectively constant those constants which are conversion factors, not properties of matter. For example: C the speed of light is a factor for conversion between mass and energy. h Planck's constant is the conversion factor between energy and frequency or time. The constant K is the Boltzmann factor between temperature and energy 2Π is the relationship between the perimeter of a circle and its radius In addition I will use, because it is necessary, the values of the masses of the proton, the electron and the neutron and the value of the fundamental electrical charge as they are on the present time or epoch. Still my intention is to calculate their values on different age or time of the universe. Again, these are the current values (the current era) that I'll use for these calculations: c = 2.997924580000E+10 cm/sec h = 6.626069570000E-27 erg-sec K = 1.380648800000E-16 erg/kelvin mp = 1.672621777000E-24 grams me = 9.109382910000E-28 grams mn = 1.674927290000E-24 grams From these are calculated : m^2 = 3.903402647729E-26 grams J = Mn/mp = 1.001378382747E 00 D = mp/me = 1.836152671949E 03 F = D/4 = 4.590381679872E 02 Auxiliary I use G the gravitational constant measured recently, but its value will not be definitive because it is one of which I intend to calculate. This has been accepted at least provisionally, taking G = 6.67191E-8 according to the article which is attached to the journal "Nature" that it seems to me that it could be the wisest (from my poor point of view) of all the different measurements made. Note the symmetry of the above equations, which can be deduced from doing the necessary algebraic transformations: y0 = x1^(1/4) x2^(1/16). y2^(1/16) = x1^( 1/4) x2^( 1/16) . y2^(1/16) y1 = 1. x2^(1/4) . y2^(1/4) = x1^( 4/4) x2^( 4/16) . y2^(4/16) y2 = 1. 1. y2^(1) = x1^(16/4) x2^(16/16) . y2^(16/16)/1 y3 = 1. 1. y2^(4) = x1^(64/4) x2^(64/16) . y2^(64/16)/2 So we can see that if "y0" is raised to the 4th power and change the first factor (the first in red) by 1 and you gets "y1 ". Then y1 is raised to the 4th power and changed again the first term (the second in red) by 1 again, and you gets "y2 ". Then "y2" is raised to the 4th power and change the factors that will be in 1, although there is no longer any it would be "y3" except by its division by 2 It should be remembered that the units used are the one from cgs system where it will be use the erg for energy, that is the same as using: gm cm^2 / sec^2

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5.- DETERMINATION OF THE SPECIFIC VALUE FOR EACH FORCE IN THE CURRENT ERA. a) Here I'm going to start with B: Let's begin to see the value of G the Newton constant of gravitation. Here we have variation in its measured value in a way that at first sight, it is not possible to determine with adequate accuracy which one is the good one. The most recent information that I have with regard to its value tells us that this can walk between 6.6709200000 and 6.6729000000E-08 avalue taken from the journal “Nature” in its http://www.nature.com/nature/journal/v510/n7506/full/nature13433.html web page that says: About 300 experiments have tried to determine the value of the Newtonian gravitational constant, G, so far, but large discrepancies in the results have made it impossible to know its value precisely. The weakness of the gravitational interaction and the impossibility of shielding the effects of gravity make it very difficult to measure G while keeping systematic effects under control. Most previous experiments performed were based on the torsion balance pendulum or torsion scheme as in the experiment by Cavendish2 in 1798, and in all cases macroscopic masses were used. Here we report the precise determination of G using laser-cooled atoms and quantum interferometry. We obtain the value G = 6.67191 (99)     × 10- 11m3     kg- 1   s-2 with a relative uncertainty of 150 parts per million (the combined standard uncertainty is given in together within). Our value differs combined by 1.5 standard deviations from the current recommended value of the Committee on Data for Science and Technology. A conceptually different experiment such as ours helps to identify the systematic errors that have supplier elusive in previous experiments, thus improving the confidence in the value of G. There is no definitive relationship between G and the other fundamental constants, and there is no theoretical prediction for its value, against which to experimental test results. Improving the accuracy with which we know G has not only a pure metrological interest, but is also important because of the key role that G you've in theories of gravitation, cosmology, particle physics and astrophysics and in geophysical models.

To use this value of G to calculate B, I got for B = 1.954068618890E 42 (5.1 ) But from equation where B = y3 F^16 we get : y3 = B/F^16 = 0.502758067 (5.2 ) If we divide y3/J^4 gives us almost exactly 0.5 This is the first case in which I will take this approach as true. It follows that B should be exactly:

(5.3 ) Then B would have the value of:

(5.4 ) Note the remarkable agreement between the value of B when we compare equations (5.1 ) with (5.4). Well, this match for the purposes of this article, is not longer a coincidence, it will be equality. This means that:

(5.5 )

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1.954085733281E+42/1.954068618890E+42 = 1.0000087583E 00 i.e. a difference of about 8 part in 1 million due clearly because the accuracy of G measurement is not good enough. With this in mind, the value of G is simply :

(5.6 )

using the B of (5.4 ) And that is within the range of the measured value for G according to the article of "Nature" mentioned. Obviously G also would be calculated as:

(5.7 )

Note: symbol “/” means a division, to divide by. And expressed as function of gw or Gf:

(5.7 bis) b) Now let us look at W. From (4.3) we have that x3 = W^4/B ≈ 1.980894838 ≈ 2 Accepting that x3 = 2 as we saw in (5.5) we would have :

(5.8 ) Then: W^4 = 2B and as according to (4.6 ) and W = y2 F^4 then:

Accordingly:

(5.9 ) W = 4.446245129e 10 (5.10) Whose proportion with W original (initial) is: 1.0024025063 here and with the equation (3.6 ) it is follows that: Gf = 1.432409406E-49 Gf/ (ħc)ħc) ^3 = 4.5329057195 erg^ (-2) Gf/ (ħc)ħc) ^3 = 1.1635831839e-5 Gev^ (-2) Clarifying that Gf is not the gravitational constant G, but the constant of the weak force.

c) Let's see now to α : Here is very simple. Just the general equation.

(5.11) Where “qa” is the fundamental electric charge. Do not forget at this point a factor a calculate in the first part. This is “r” the classical mason radius, where:

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Constant trough time. d) Now let's look at the strong force. From the equations in section 4 we know then that: x3 = 2, y2 = J, y3 = J^4/2 Lets calculate y0, y1, x2, x1 from equations (4.1 ) to (4.7 ) and using the known values of the parameters of the coupling forces being those accurate or approximate. y0= 9.0305475216E-02

y1 = 1.875710203 x2 = 12.36131721 x1 = 3.54561E-05 Now let us make the operation y1^4/y0 1.3707204142E+02 and 2π (y1^4/y0) = 8.6124903665E+02 ???????

This value is almost equal to α

Well, with this method that I am using, I will make effectively that y1^4/y0 = α / 2π

and because α = y1 F and proceeding in reverse we have:

y1^4/y0 = y1.F/ 2π and y0 = 2π y1^3/ F in addition as y0 = σ/F^(1/4) and

y1 ^3 = (α/F)^3

And we have that: = 4.1810994004E-01 (5.12)

(5.13)

And: (5.14)

It is also deducted:

Note: The concept I am going to consider it as valid for any time or epoch of the universe. It will always be the same even if y1 and y0 change with time. 6.- SUMMARY: All this allows us to deduce with high accuracy equations (4.4 ) to (4.7 ) with the following values:

(6.1)

(6.2)

(6.3)

(6.4) Values for the present time.

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constant with time (6.5 ) Now, with the well known values of the parameters of the forces, we can get the square of the related charges. Making clear that these values are for the present time only because they change with time to change the parameters of the forces. q = (hc/k)^(1/2) all in electrostatic units or (gm cm)^(1/2)/sec

(6.6)

(6.7)

(6.8)

(6.9) If the parameters of the forces are expressed as we usually do (i.e. use h/ 2π instead of only h) these would have the following current values:

k = q^2/ħc

And the actual coefficients are:

and in addition we got that:

7.- THE PARAMETERS OF THE FORCES ON THE BASIS OF THE EPOCH ψ

Ψ is the time elapsed since the Big Bang, in seconds. We see that it is possible to express each of the parameters of the forces as a function of the others:

i.e. a) I will begin with B as a function of time: This is extremely simple since R = c ψ and B = R/r as I explained in the first part of this analysis. (The determination of the Hubble constant from the constants of nature). As a result, B is simply:

(7.1 )

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Being R the radius of the universe at the moment ψ that we are dealing with, and "r" the classic radius of the mason independent of the time (as I showed in the first part of this article). Is necessary to insist that these "constant" or parameter in reality is not constant, and this leads us to another interesting problem. Note: Prior to this, I would like to clarify that due to the introduction of a new and more recent value of the constant of gravitation G, the calculated values in my previous article about how to determine the value of the Hubble constant (article dated on 10 April 2014 published in Scribd and revised on 11 August 2015) are no longer exactly the same. It is in this new revision of that article where I define those, clarifying that the equations of the article of previousr date are correct in spite of slight changes in the absolute values obtained. I said that there is an interesting problem because for example if in this second part is concluded that B = J^4.F^16/2 and I also got that B = R/r or what is the same, B is not really constant because R is growing. Necessarily J and/or F are not constant. The problem that is presented to me here, and given that I intend to find the values of the parameters of the forces in the different epochs of the universe is: which of the two, J or F are variables? Parenthesis: One of the interesting consequences of this analysis is that it can be calculated with great accuracy the mass of the neutron from some physical constants and the masses of the proton and electron. Let's see how to:

With these equations we calculate the mn (neutron mass) and mn as function of ψ (the time). For the actual time:

(7.7) Noting that D does not depend on mn since D = mp/me and G neither, for it can be obtained from other sources or mesurments. You can also see this in reverse, i.e. calculate the value of G from the mass of the particles, which in fact is what I did before. Whatever it is, what I want to show here is the relation among G and mn. Also, by using (5.7) and (5.7bis) making G equal on both, we can get that:

(7.7bis) b) To calculate W from the variations of B with time and knowing the simplicity of the relationship between B and W, it is easy to determine how W changes with time. Let us be clear:

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Resume: (7.2)

(7.3)

(7.4)

c) Calculation for α is simple.

In the first part of these articles I found these 2 equations (10.1 ) and (10.3 ) which are respectively:

(10.3) y (10.1) Equations of the first part. Knowing that B = c Ψ /r according to (7.1 ) is elementary resolve that:

(7.5)

That is α changes directly proportional to the time^(1/2).

Of course, in each case, the value of the charge either electrical, gravitational, weak or strong is deducted fundamentally from its equivalent (hc/K)^(1/2)

(d) Determination of the value of σ depending on time ψ is a bit more complicated.

Since its calculation requires the prior determination of J and D. These is what follows.

Let's see: As B is variable with time, so J or F or the two are also variable. Can even be that both change simultaneously. So I'm going to make a few assumptions whose results will tell me what is the correct one. The first assumption will be that F is constant with time. We must take into account that J must always be equal to or greater than 1, i.e. the neutron must have an equal or greater mass than the proton, or what is the same J ≥ 1 This happen because if the proton would have a mass greater than the neutron´s, it would not be possible disintegration of neutron in a proton and an electron plus a neutrino. Expressed mathematically this would be: J = (2B)^(1/4) /F^4 ≥ 1 The critical case is J = 1 then it should always be that: B ≥ F^16/2 But this is not true since B can take values much lower than F^16/2. For example when m = µ ( Planck mass) and B = 1, because this would make F and D fractional or what is the same : mp> mn which cannot be. So this does not allow F to be constant. The second assumption is that J is constant and F is the variable.

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As a result of the foregoing,

And (7.6) (7.7) Of course, if ψ = 4.286475134213E 17 sec (the current era) D will be the current. D Since D is calculated with time, the values of “mp” and “me” at that time are easily calculable.

As we saw in (5.0 ) ) (7.8)

And from (7.5 )

Then : (7.9)

With the observation that J does not appear in this equation, regardless of its value.

In addition that comes from

Apparently in these above equations, J seems to be a function of F but in reality it is not so, because with the change ψ , F will change, while J is constant. Without better reason than the one shown. And because the fact that J is very close to 1. I decided to left J as a constant.

The solution for σ where there are only constants (that doesn't change over time) is a

rather algebraically complicated. However the solution whose footsteps I am not going to show, but obtained from the above equations is:

The next table is calculated with the previous equations with Excel from Office. It shows the different values of the most important parameters of the universe and of the 4 forces as function of time elapsed since de big Bang which last 1.68688876E-99 seconds until the first photon was born. Since then, different epochs are shown. When α and B were unified. The Planck time when B = 1, the time when α = 1 and lastly the actual time 4.28647513E+17 seconds. Remember you can make by yourself these calculations by downloading the executable file in Excel where you just have to write the time in seconds. http://es.scribd.com/doc/278742010/Universal On this page the conditions when time was 1 second are shown. If it appears the symbol #¡NUM! on the calculus sheet, it means Excel can´t make the calculation any more. I don´t know if I am going to develop these ideas a little more deep. There are questions on my mind about the reasons for this accidents as the one for B = J^4F^16/2 which is the cause for all of these. If I decided to do so, it will be a third part. END

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ACTUAL alfa = 1 PLANCK UNIFICACION Nfo = 1

km/seg/MP 3.24077929E-20 B = 1 a = B

pi 3.14159265E+00 3.14159265E+00 3.14159265E+00 3.14159265E+00 3.14159265E+00

4pi/3 4.18879020E+00 4.18879020E+00 4.18879020E+00 4.18879020E+00 4.18879020E+00

G 6.67185157E-08 6.67185157E-08 6.67185157E-08 6.67185157E-08 6.67185157E-08

K 1.38064880E-16 1.38064880E-16 1.38064880E-16 1.38064880E-16 1.34063482E-16

z 4.96519663E+00 4.96519663E+00 4.96519663E+00 4.96519663E+00 4.96519663E+00

h 6.62606957E-27 6.62606957E-27 6.62606957E-27 6.62606957E-27 6.62606957E-27

c 2.99792458E+10 2.99792458E+10 2.99792458E+10 2.99792458E+10 2.99792458E+10

j= 32pi^2/9 3.50919268E+01 3.50919268E+01 3.50919268E+01 3.50919268E+01 3.50919268E+01

r 6.57623611E-15 6.57623611E-15 6.57623611E-15 6.57623611E-15 6.57623611E-15

yo 9.03292269E-02 4.15233082E-09 1.23637090E-54 3.68133724E-100 2.81553522E-147

y1 1.87571020E+00 5.07023419E-03 5.90035184E-19 6.86637945E-35 2.21805431E-51

y2=J 1.00137838E+00 1.00137838E+00 1.00137838E+00 1.00137838E+00 1.00137838E+00

y3 5.02762471E-01 5.02762471E-01 5.02762471E-01 5.02762471E-01 5.02762471E-01

x1 3.54933859E-05 5.86326831E-32 3.96020067E-198 0.00000000E+00 0.00000000E+00

x2 1.23613172E+01 6.59954306E-10 1.21035683E-73 2.21979560E-137 2.41707703E-203

x3 2.00000000E+00 2.00000000E+00 2.00000000E+00 2.00000000E+00 2.00000000E+00

y1^4/yo 1.37035999E+02 1.59154943E-01 9.80308714E-20 6.03817359E-38 8.59662017E-57

ψ 4.28647513E+17 5.78190873E+11 2.19359625E-25 8.32227682E-62 1.68688876E-99

mp 1.67262178E-24 9.44004092E-22 1.11500693E-04 1.31698629E+13 6.14112582E+30

me 9.10938291E-28 1.19658046E-24 2.67025510E-05 5.95886574E+14 6.30452793E+34

D 1.83615267E+03 7.88918194E+02 4.17565696E+00 2.21012916E-02 9.74081786E-05

J 1.00137838E+00 1.00137838E+00 1.00137838E+00 1.00137838E+00 1.00137838E+00

F 4.59038168E+02 1.97229548E+02 1.04391424E+00 5.52532290E-03 2.43520446E-05

m 3.90340265E-26 3.36091780E-23 5.45651256E-05 8.85874964E+13 6.22229051E+32

mn 1.67492729E-24 9.45305290E-22 1.11654383E-04 0.00000000E+00 0.00000000E+00

mnu = mn-mp-me 1.39457471E-27 1.04618498E-25 0.00000000E+00 0.00000000E+00 0.00000000E+00

mc2 3.50820334E-05 3.02064228E-02 4.90406892E+16 7.96184711E+34 5.59231582E+53

q 4.80320451E-10 1.40941324E-08 1.79583727E+01 2.28820861E+10 6.06435399E+19

q^2 2.30707735E-19 1.98644568E-16 3.22503151E+02 5.23589865E+20 3.67763893E+39

l 5.66228775E-12 6.57623611E-15 4.05060719E-33 2.49495583E-51 3.55209853E-70

f 5.29454650E+21 4.55872406E+24 7.40117331E+42 1.20159425E+61 8.43986886E+79

σ 4.18109940E-01 1.55609083E-08 1.24972651E-54 1.00367942E-100 1.97785669E-148

α 8.61022575E+02 1.00000000E+00 6.15946131E-19 3.79389636E-37 5.40141575E-56

W 4.44624513E+10 1.51525642E+09 1.18920712E+00 9.33316330E-10 3.52159928E-19

β 1.95408573E+42 2.63581264E+36 1.00000000E+00 3.79389636E-37 7.69006040E-75

S 2.26949419E+39 2.63581264E+36 1.62351860E+18 1.00000000E+00 1.42371199E-19

16

H 2.33291917E-18 1.72953266E-12 4.55872406E+24 1.20159425E+61 5.92807316E+98

hH/C^2 1.71994388E-65 1.27509738E-59 3.36091780E-23 8.85874964E+13 4.37046997E+51

R 1.28505292E+28 1.73337263E+22 6.57623611E-15 2.49495583E-51 5.05716529E-89

V 8.88897417E+84 2.18154619E+67 1.19129907E-42 6.50544772E-152 5.41763889E-265

1/R^2 6.05561110E-57 3.32825145E-45 2.31230546E+28 1.60647614E+101 3.91008054E+176

M 1.73107563E+56 2.33500043E+50 8.85874964E+13 3.36091780E-23 6.81243197E-61

E= M C^2 1.55581319E+77 2.09859373E+71 7.96184711E+34 3.02064228E-02 6.12270852E-40

N 4.43478622E+81 6.94750830E+72 1.62351860E+18 3.79389636E-37 1.09484312E-93

Dm 1.94744140E-29 1.07034196E-17 7.43620965E+55 5.16631283E+128 1.25745405E+204

T 2.73753184E+00 6.91641409E+04 1.43076033E+32 2.95973476E+59 5.50959374E+87

Det 4.24900854E-13 1.73130999E+05 3.17043202E+114 5.80579981E+223 #¡NUM!

Et 3.77693271E+72 3.77693271E+72 3.77693271E+72 3.77693271E+72 3.77693271E+72

Mt 4.20240440E+51 4.20240440E+51 4.20240440E+51 4.20240440E+51 4.20240440E+51

rg 2.89766598E-54 2.49495583E-51 4.05060719E-33 6.57623611E-15 4.61907755E+04

lg 1.28505292E+28 1.73337263E+22 6.57623611E-15 2.49495583E-51 5.05716529E-89

A 3.36091780E-23 3.36091780E-23 3.36091780E-23 3.36091780E-23 3.36091780E-23

K2 2.63581264E+36 2.63581264E+36 2.63581264E+36 2.63581264E+36 2.63581264E+36

K1 4.58600380E+13 4.58600380E+13 4.58600380E+13 4.58600380E+13 4.58600380E+13