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QUANTUM UNIVERSE By : Ramón Garza Wilmot November 12 of 2016
INDEX FIRST PART: PREFACE------------------------------------------------------------------------------------------------------------------------2 1.- Units of Planck-------------------------------------------------------------------------------------------------------------6 2. – Some basic calculation of proton and electron properties and the fine structure constant. --–8 3.- Gravity parameter and relation between the electromagnetic and gravitational forces.---------12 4.- Planck units and a different calculation method.--------------------------------------------------------------13 5.- Calculation of the Hubble parameter and the age of the universe ---------------------------------------14 6.- The mass of the universe and the number of protons it has.----------------------------------------------15 7. – The temperature of the universal cosmic background radiation.---------------------------------------17 8.- The "r" the classical radius of mason.-----------------------------------------------------------------------------22 9.- The cosmological constant.-------------------------------------------------------------------------------------------23 10.- Quantitative relation between electromagnetic and gravitational forces and their change with time.-------------------------------------------------------------------------------------------------------------------------25 11.- Origin of the thermal energy of the CBR -----------------------------------------------------------------------28 12.- Variation of values with time.---------------------------------------------------------------------------------------31 13.- Large numbers hypothesis.-----------------------------------------------------------------------------------------32 14.- The Quantum Universe ----------------------------------------------------------------------------------------------- 35 15.- Mathematical definition for each of 2 forces and G calculation -------------------------------------40 16.- Variation of the ratios D = mp/me and J = mn/mp with time. ---------------------------------------42
2 PREFACE
FIRST PART: This work is the gathering and reviewing of my two other works which titles are: EXACT VALUE OF THE HUBBLE CONSTANT. THE MOST PRECISE VALUE OF THE HUBBLE CONSTANT DEDUCED FROM OTHER CONSTANTS OF PHYSICS. And NEWTON GRAVITATIONAL CONSTANT CALCULATED BY MEAN OF OTHER PHYSICS CONSTANTS. MUTUAL RELATIONS BETWEEN THE 4 FORCES OF NATURE AND ITS VARIATION WITH TIME. Plus corrections and actualization of the values of the physics constants as well as a the possible explanation of the results on the analysis of it. This paper is an edition and replacement of a previous one I called “Birth of a Universe” and changed by this “Quatum and Big-Bang” since I think I got better results and better analysis with this one, and because this last name, I think is more appealing or related to the real purpose. Here I decided to put both articles together and at the same time, to correct some mistakes and wrong uses of equations, to change some of the values of the physics constants I used and consequently, the results of the analysis and some new findings. As I said before, this analysis attempts to make a research on the physics history of the universe on the big picture, from the very first instants of its birth and up to now. When I speak about the very first instants, I am speaking of the moment when the universe was born at about 10^(-100)seconds (1/ 10^100) sec. after the moment zero. This means I recognize the Big Bang ideas, but my results are different from the standard theories or has a different interpretation. The pretension of saying that these results represents the actual universe and its birth, is mine. So I will leave my options open by saying that these results are very alike the real universe and I leave to the reader to decide if these are true or not, or at least, to decided how much it seems to be like the real universe. The method is very simple since I don´t use high level mathematics. On the contrary, the math I use is so simple that any high school student will understand it. I use the same equations trough all the history of the Universe unless the calculus sheet of Office allows me to do it since this program it is not good enough for more than 13 decimals in such a manner than precision doesn´t go far than this. Two simple things took me to all of these. First one is that I found the way of calculate the Hubble parameter by mean of some know constants of Physics, and the other was that I got the Gravitational coupling constant by mean of also some simple physics constants, which calculated value is the one I used for all the calculations. This is an analysis without a theory to backup. It is a numerical analysis that is so close to reality (at least speaking of the Hubble constant and the Cosmic Background temperature) that the numbers results are very possible true. One of these results was already investigate by individuals of the size of Paul Dirac, Edward Milne, Arthur Eddington, who also without a big theoretical support for their ideas, tried to explain what now is called the Large Numbers Hypothesis. They were the first modern scientists that tried to specified the universe as a whole, making calculations about its mass, the number of protons it has, its dimensions etc. It is worth mention that all of them, but apparently more with Eddington and Dirac, got involved on the large numbers that the properties of the universe presents, even predicting it from empirical relationships.
3 Their idea was that it must be some kind of connection between the classical physics and the relativity with, the quantum physics. Dirac was upset with the fact that there were 2 different physics without any contact among them, pretending with his “Fundamental Theory” to get a connection among the quantum Physics, relativity and gravitation. These lead him to found quantitative relations between the electrical and gravitational forces for the universe as a whole. All based in a dimensionless very big number, without exposing the physics reasons, except those that they were very similar to the ones found for the universe. Eddington also explore these ideas and more recently Paul Davies whit his very interesting book “The Accidental Universe” Dirac formulated a number without units this way:
Eddington, more accurate, formulated
Note: Ne from Eddington and Nd from Dirac. We must make notice that the number Eddington found is approximately the square of the number that Dirac obtained. The Intention behind all this was that Ne represents the number of protons of the universe. Other scientist saw this relation among the electrical and gravitational forces, had a fundamental importance to relate Cosmology and quantum physics and had being searching that link, that so far doesn´t seems to be explained. The results of this analysis take me to find equations that describe the whole universe since the BB and up to now. We shall see that the properties of the universe, the basic particles, the photons and also the gravitons as well as the CBT (Cosmic background temperature) are all quantized and are multiples or submultiples of the initial values of the quantum values at time t1, the quantum of time. We will also see that all of this depends of the number of the quantum times passed since the BB. Viewed as this, the actual epoch is just a 2.443501428799E+116 times the quantum of time which value is 1.7542350311445E-99 seconds. These initial results for the BB came by accepting that thermal energy of the universe is constant thought all the universe history. I took this premise as correct, since I found that the temperature at the Planck epoch (not the Planck time) is proportional to the actual temperature by a factor of the ratios of epochs raised to the ¾ power, which is naturally depending of the volume (3 power) and the density of thermal energy (4 power). In this manner, the idea of taking the CBT as result of annihilation of particles and antiparticles is incorrect since is not capable of explain the origin of the energy of the universe. The actual idea for the CBT is that at the BB there were 1 billion of antiparticles and 1 billion plus 1 particles without explaining from where they came from and why there where 1 particles more than antiparticles. Then they said the particles and antiparticles annihilated each other leaving as result the CBT and the actual number of particles. My results say this is not necessary. Both, the CBR and the origin of the matter are quantum phenomenon coming from the uncertainty Planck´s principle. But not only that, since is capable of predicting the existence of the graviton and its actual mass (and any epoch mass). I also found that the value of the energy of the CBR has its origin in the value of the Planck´s constant (with the same absolute value of both) by mean of taking the real quantum units, being these, the first instant, the smaller radius of the universe and the smallest value of the mass of the universe. By this mean, I also found the energy value of the average photon, the photon which carries the energy of all the photons in
4 the radiation curves. By mean of equalizing the thermal energy density of the universe to the thermal density of this photon in a spherical space which radius is the wave length of this photon. On this paper, I present the reasons and the numbers I see for these relationships (that is, I try to explain) that in fact, links quantum physics and cosmology, at least in quantitative form. The results are numbers presented with high precision (up to 12 decimals) and they just depend of the values of the physics constants I used to obtain them. In this case, the number I got is in fact the number of particles of what I call “mason” and from which is very simple to get the number of protons of the universe, knowing In this case, the mass of the mason as:
Yet, I consider more relevant, not the number of masons of the universe, but the proportion between electrical and gravitational forces in a number that I identified as “S” which differs from Eddington on the Π^2 factor. This is a simple analysis of the relationships existing among different physics constants that allow us to glimpse the properties of the universe as a whole. Starting from this and especially with the relationship with what has being called Planck units, especially with the mass of Planck. These Planck units are derived starting from what I call “parameters” of the gravitational and electric forces from which are derived the Planck units with easiness. Some algebraic basic knowledge and some simple equations of the classic and quantum physics will be enough to reach the objective of this writing. As I said before, I consider myself that the fundamental element which leads me to these results, has being the determination of the Hubble constant starting from units which relates gravity and quantum physics in a very simple manner. In fact, the result is that Hubble constant identify by H is just a frequency of gravity which reciprocal is the age of the universe. The analysis lead me to calculate the mass the universe, its “radius”, the number of nucleons that it contains, the temperature of the cosmic radiation background, when these things happened and what happened, and how do they change with time. Also which are the values of the parameters of the forces through the time, what relationship are among the parameters of these forces and how can we calculate them as function of the other. And everything starting from the values obtained with the Planck´s units that, after this, have a very clear meaning in our time, which at the same time are derived from these and just these physics constants: - The speed of the light in vacuum c - The Planck´s constant h - The Boltzmann’s constant K - The gravitational constant G We will also see that is possible to calculate the G Newton gravitational constant by knowing the Planck´s constant and the masses of the proton, the electron and the neutron. So It is also required the value of the proton, neutron and the electron mass at the present time, although I don't consider them as constants. Nevertheless its simplicity, it has allowed me to obtain some remarkable results of the properties of universe in general , but fundamentally to be able to calculate with a high precision the temperature of the cosmic background radiation and to explain the meaning of the so call units of Planck. Such as the mass, the time, the temperature, etc. of Planck and its relationship with the properties of the proton and the electron and neutron. There is another important constant I got which is constant thorough almost all the epochs of the universe. This is the actual mason radius which obtained from actual values, represents the radius of the universe when the time was the Planck´s epoch (no the Planck´s time as it is known)
5 This dimension “r” has the value of 6.5762360983485E-15 cm. I must add that this analysis will take me more, and more back on time of what has being called Planck time that as we will see, does not represent an epoch back in time, but just the reciprocal of the Planck frequency. This is why the term epoch, is referring to real time, and making a difference with the Planck´s. remembering what I said before that real time is a multiple of the t1 time The whole result, will take us to know the values of the two of two of the fundamental coupling parameters of the forces as function of the constants I mentioned with time, That is, I will be able to know their value by just knowing the elapsed time since de B.B at any epoch, the actual, the very old past, the first relevant instant which I call “ψ1” , and the future, the far away future. I also explain from where the thermal energy of the universe arose (which explanation is quite different of the standard cosmology) and the origin of matter. I will go far before the Planck´s time. When time had a value of 1.7542350311445E-99 sec., which I define as the moment when the first photon emerge, carrying with it all the energy of the cosmic background radiation today and always, which value comes from the equation:
This epoch and this primordial thermal energy I got it because I obtain with high precision the actual volume of the universe and the cosmic background temperature from some physics and mathematical constants. And with the help of masses and charge of proton and electron which I don´t include as constants this analysis has being possible. This simple previous equation shows the origin of the cosmic radiation. It doesn´t came from the annihilation of particles and antiparticles as the standard theories say. It comes from nothing less or more, than from a quantum leap that created the first super energetic photon from which all the rest of photons come to be. We will also see, that when this first photon began to exist, it also exist another kinds of energy that I think are associated with particles of so very small mass that seems to be the origin of the actual mass of the universe. I will try not to speculate freely and I will let the numbers speak for themselves, That is, from the results, I will try to explain the meanings. Note that I will keep the equations always the same at any time except where the calculus capacity of Office Excel allow me and the small correction to the Wien´s law at time t1. What could be happening, as it is seeing from the results, is that the initial photon breaks in “sons” photons and these “sons” photons do that also, but with a difference in energy, lesser than the “father” photon and so on. Even so, this does not impede the creation of real particles do to the same cause, because the results tells me that spontaneous matter creation or the increment on the number of massive particles is happening since those moments. Of course, these equations also solve the cosmic background radiation in any time of the universe and of course on the actual time. I define the Planck epoch not to what has being call “Planck time” but the real time when gravitational parameter (or coupling parameter) had a unitary value. All this data can be deduced from the equations of the section that defines all the parameters as function of time and from which Planck´s data can be obtained just by making B = 1
6 I will begin exposing the today known as Planck time and how it can be obtained without making the analysis that Planck used. For all these, I will begin exposing what I think are really physics constants in the sense of their invariability with time to differentiate them from those which are not constant. That comes from the fact that the real constants are not properties of matter but truly single conversion factors among those properties. Seen on this way, I enumerate again the invariable physics constants I will use: - The speed of the light “c” that relates mass and energy. - The gravitational constant “G” that relates the mass with the force of gravity. - The constant of Planck “h” that relates the energy with time. - The constant of Boltzmann “K” that relates the heat energy with temperature. You must notice that I have NOT included as constants, the mass of the proton “mp”, and the mass of the electron “me” or the neutron mass mn, and the fundamental electrical charge “qe” or just “q” Although I took these values, as characteristic of the current age. (About 13,600 million of years after the Big Bang). Lastly, I manifest that all this is based on the cgs system of units, where the units are: the centimeter, the gram and the second. The unit of electric charge is that of the electron or electrostatic unit of charge. The temperature is shown in degrees Kelvin or absolute. And to end this preface, I will also clear up that here, I have taken as postulate that the universe is tri dimensionally spherical, since I use for its volume 4π/3 R^3.. So at this time, I would do the calculations, assuming universe it is flat.
1.- Units of Planck And from Wikipedia: The Units of Planck or natural units, is a system of units first-time proposed in 1899 by Max Planck. The system measures several of the fundamental magnitudes of the universe as: time, length, mass, electric charge and temperature, by making use of five universal physics constants of the chart to take the value of 1 when equations and calculations are expressed in this system. From: http://estudiarfisica.wordpress.com/2013/08/25/los-systems-of-unit-geometric-natural-andof-planck / The use of this system of units brings several advantages. The first and more obvious is that it simplifies the structure a lot of physics equations because it eliminates the constants of proportionality and makes t the results of the equations does not depend of the value of the constants. On the other hand, it can be compared more easily the magnitudes of a great deal of different units. For example, two protons are rejected because the electromagnetic repulsion is a great deal more strong that the gravitational attraction among them. This can be proven when seeing that the protons have a charge of a natural unit of charge, but their mass is much smaller that the natural unit of mass. It also allows, avoid enough problems of rounding, mainly in calculation. However, they have the inconvenience of that when using them, it is more difficult the notice of dimensional errors. They are popular in the area of investigation of general relativity and the quantum gravity. The Planck´s units usually are named in a humorous form by the scientists as “units of God”, because they eliminate any anthropocentric system of units. Expression of physics laws in Planck´s units from WIKIPEDIA Universal Gravitation Newton law
7
Becomes:
Using Planck´s units . The energy of a particle or photon with radial frequency on its wave function.
Becomes
The famous mass-energy Einstein equation.
Becomes
(As an example, a body has a mass of 5,000 mass Planck´s units has an intrinsic energy of 5,000 energy Planck´s units on its full form.
Becomes
Planck´s units system: The previous system is based on assuming some certain constants equal to the unit (1) by agreement to relate other magnitudes through it. However, one usually finishes wondering why these 5 if we really speak of important constants at fundamental level or if they are the result of other more basic ones. Then, so arise the intent to obtain a unit of longitude starting from the well-known longitude of Planck:
To get the coefficients α, B we just create a vector which have as component the power to which it has to be the exponents of meters, seconds and kilos each one and we set the system of equations. This imply that:
Basic Planck´s Units
8 Giving the value of 1 to the five fundamental constants, the units of time, longitude, mass, it charges and temperature are defined this way: Tabla 2: Unidades de Planck básicas Equivalencia aproximada Nombre Dimensión Expresión en el Sistema Internacional Longitud Planck
de
Longitud (L)
1.616 252(81) 10−35 m [1 ]
×
Masa de Planck Masa (M)
2.176 44(11) × 10−8 kg [2 ]
Tiempo Planck
5.391 24(27) × 10−44 s [3 ]
de
Tiempo (T)
Carga de Planck Carga eléctrica (Q)
1.875 545 870(47) 10−18 C
×
Temperatura de Temperatura (ML2TPlanck 2/k)
1.416 785(71) × 1032 K [4 ]
Notices: Up here, reference to the web pages mentioned ends. 2. – Some basic calculation of proton and electron properties and the fine structure constant. G → Newton constant = 6.67191(99) E-8 cm^3/(gm. sec^2) (latest high accuracy from Nature 510, 518–521 (26 June 2014) doi:10.1038/nature13433) http://www.nature.com/nature/journal/v510/n7506/full/nature13433.html Yet, I will use my own calculation of the G constant (as I will show how I got it) with a value of: G = 6.671850021543E-08 cm^3/(gm. sec^2) As this value of mine and the value accepted by the Nature magazine are not quite different, the results will be differ just a little if I use my own. Except by the Newton constant, I will take these constants as true values from National Institute of Standards and Technology in its last review. h → Planck constant = 6.6260700400E-27 gm-cm^2/sec m → mason = (mp X me)^(1/2) = 3.9034029281817E-26 gm me → electron mass = 9.1093835600000E-28 gm mp → proton mass = 1.6726218980000E-24 gm mn → neutron mass = 1.6749274710000E-24 gm K → Boltzman constant = 1.3806485200E-16 erg/Kelvin Z → factor of Wien´s law = 4.9651142317443 C → Light speed = 2.9979245800E+10 cm/sec q^2 → squared electron charge = 2.3070775131E-19 eu^2 or erg-cm
9 Next I proceed to analyze how to obtain the units of Planck, without appealing to reduce to 1 the 5 basic units that he used. For these, I will use the cgs system of units (centimeter, gram, second) and the electrostatic unit of fundamental charge. The notation A^n will mean that “A” (the magnitude A) is rise to the “n” power. A. Einstein deduced starting from the photoelectric effect that the light exists in discontinuous form, in packages of energy of the so called “quantum”, on which the energy of this “quantum” is defined as function of the frequency of the light and the constant of Planck. This energy can be expressed in two like forms: , h is the constant of Planck and “w” is the angular frequency of the light
being in radians per second. E can be also expressed as;
in this case f is the frequency of the light in cycles per second or Hertz Of course, w = 2.π f
and h is expressed in ergs-seg.
The photon doesn't have rest mass, nevertheless, as it poses energy it is possible to attribute “mass” to it, such that: so Now then, the frequency and the speed of light are related by
and in consequence:
The same equation is used to define the wavelength of quantum particles as the electron and the proton, being defined as Compton wavelength of the particle. And then: (2.1) Before going ahead, I will write a very simple explanation of the uncertainty principle of Heisemberg necessary to explain the following equations. Without going into details, this principle says that it is impossible to know simultaneously the impulse and the position of a particle both constant, in such a manner than the uncertainty of the impulse “p” multiply by the uncertainty of its position “x” is equal or higher to ħ/2. Δp . Δx ≥ ħ/2. Going in a differential form we will have. δp X δx ≥ ħ/2 but p = mv and x = vt so: δp = (m. δv + v δm)
and
δ > differential
δx = (v.δt + t.δv)
as m and v are constant, then : (m. δv) X (v.δt) ≥ ħ/2 m vδv ≥ (ħ/2)/ δt by integration of both sides and applying the limits for
0 < v < C and
T< t mn which cannot be. So this assumption does not allow F to be constant. The second assumption is that J is constant and F is the variable. As a result of the foregoing, And
(2.2)
(2.3)
Of course, if ψ = 4.2864758050509E+17 sec (the current era) D will be the current. D
Please follow me in detail to understand how I got this: If B = J^4.F^6/2 And if B changes with time, obviously J or F or both changes with time. Let’s see the moment when B = 1 (Planck epoch) J^4. F^16 = 2B = 2 (J. F)^4 = 2^(1/4) = 1.189207115003 As the first formed atom was the hydrogen atom and this atom doesn´t have neutrons, to assume that in that epoch there were no neutrons is probably right. Also because particles were formed by Planck mass μ which has only protons and electrons. But we can´t infer from this that J = 0 because if this were so, B = 0 , no 1. So J should be equal to 1 Because of this, F = 2^(1/16) = 1.044273782427 as F = D/4 then D = 4.177095129710 Of course μ^2 = μp x μe = D. μe^2 Then μe = μ/D^(1/2) = 2.669795778364E-05 grams and μp = D. μe = 1.115199094312E-04 grams In general, J.F^4 = (2B)^(1/4) So I know the product J.F^(1/4) but I need to know J at every moment to find F. ¿How I do that? We know that J = 1.0013784185193 at the actual epoch and J = 1 at the Planck epoch. My guess here is that J changes lineally with time. Seeing it this way, the increase on its values is proportional to the increase in time between Planck and today epoch. So ∆J = 1.0013784185193 – 1.0000 = 0.0013784185193 ∆ ψ = 4.2864758050509E+17 - 2.1935962439551E-25 ≈≈ 4.2864758050509E+17 seconds Then a constant K4 is established : K4 = 0.0013784185193/4.2864758050509E+17 = 3.215738480739E-21 changes/sec. As ∆J = actual J – 1
then:
44 J = K4.ψ + 1
(2.4)
Today values is : Jact = (3.215738480739E-21 X 4.2864758050509E+17) + 1 = 1.0013784185193 Planck epoch value was: Jp = K4.ψp + 1
= (3.215738480739E-21 X 2.1935962439551E-25) +1 ≈ ≈ 1
Going ahead with F, I have. J.F^4 = (2B)^(1/4)
F = { (2B)^(1/4)/ (K4.ψ + 1) }^(1/4)
(2.5)
Let’s try this equation for the actual epoch and for the Planck epoch. 4. Fact =
D = 1.836152673760E+03
4. Fμ = Dμ =
exactly
4.177095129710E+00 exactly as predicted.
Without solving the solution by mean of equations, but by mean of the attached calculus sheet, the epoch when the mass of the proton and the mass of the electron were the same, was 2.5536821269850E-35 sec. 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 2 forces as function of time elapsed since de big Bang which last 1.7542350311445E-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 4.2864758050509E+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. As the table is wide, I write it down in two parts, going from the present time down to the past. All the calculations were made with the equations of the article and the ones shown on the same table. THE END Ramón Garza Wilmot
[email protected] In the following webpage is the calculus Excel sheet to make the necessary operations. Excepting the first column which was used to know the Kn constants, all the other columns are copy of the first one. In the last two ones, thermal density was calculated by hand, since Excel can´t handle exponents higher than 300. The epochs in seconds should be written by hand and then make the calculations. As I explained, to find the epochs, you must define what do you want to know and move the epoch until you get the number you want. As an example: if you want to know about Planck epoch, move the epoch until B = 1, or if you want to know in which epoch the number of masons was 1, move the epoch until N = 1 and so on . https://www.researchgate.net/publication/310022331_incer1
45 ACTUAL pi 4pi/3 G K z h C b= hc/zK K1 = T ψ^(3/4) K2 = B/@^2 K3 = m^2*ψ K4 = (J - 1)/Ψ ψ Nx = Δψ/Ψo mx = h/ΔΨ/C^2 Mx C^5*Ψ^2/G = u m J D β α S q^2 q l f H R V M E= M C^2 N T λ0 fo ξ fξ vol(ξ) Det Et Mt eξ Deξ (R/ξ) (R/ξ)^4 mf=h*ff/ C^2 Nfξ Nf Nf/Nfξ
3.141592653590E+00 4.188790204786E+00
α= 1
β=1
3.141592653590E+00 4.188790204786E+00 6.671850021543E-08 1.380648520000E-16 4.965114231744E+00 6.626070040000E-27 2.997924580000E+10 2.897772914526E-01
3.141592653590E+00 4.188790204786E+00 6.671850021543E-08 1.380648520000E-16 4.965114231744E+00 6.626070040000E-27 2.997924580000E+10 2.897772914526E-01
4.586081522465E+13 2.635812689234E+36 6.531112187254E-34 3.215738480739E-21
4.586081522465E+13 2.635812689234E+36 6.531112187254E-34
4.586081522465E+13 2.635812689234E+36 6.531112187254E-34
3.215738480739E-21
3.215738480739E-21
4.286475805051E+17 2.443501428799E+116 1.719943733902E-65 4.202684971244E+51 6.668958163108E+94 3.903402928182E-26 1.001378418519E+00 1.836152673760E+03 1.954086043347E+42 8.610226363419E+02 2.269494390588E+39 2.307077352358E-19 4.803204505700E-10 5.662287748095E-12 5.294546503767E+21 2.332918802018E-18 1.285053117754E+28 8.888978345001E+84 1.731076299383E+56 1.555813788859E+77 4.434787314101E+81 2.737577911070E+00
5.781908814872E+11 3.295971584321E+110 1.275097452671E-59 4.202684971244E+51 1.213388056980E+83 3.360918279928E-23 1.000000001859E+00 7.891899115164E+02 2.635812689234E+36 1.000000000000E+00 2.635812689234E+36 1.986445824172E-16 1.409413290760E-08 6.576235640159E-15 4.558724388908E+24 1.729532637090E-12 1.733372655542E+22 2.181546290571E+67 2.335001005447E+50 2.098594246001E+71 6.947508532726E+72 6.916531223591E+04
2.193596243955E-25 1.250457438718E+74 3.360918045761E-23 4.202684971244E+51 1.746508192489E+10 5.456513386798E-05 1.000000000000E+00 4.177095129710E+00 1.000000000000E+00 6.159461256083E-19 1.623518613763E+18 3.225031770774E+02 1.795837345300E+01 4.050606863643E-33 7.401173900407E+42 4.558724071286E+24 6.576236098349E-15 1.191299061166E-42 8.858751666932E+13 7.961848937799E+34 1.623518613763E+18 1.430784577262E+32
1.058517057289E-01 2.832192981073E+11 1.027822049707E-01 2.916773950175E+11 4.548229808808E-03 4.249290228758E-13 3.777184882505E+72 4.202684971244E+51
4.189633243674E-06 7.155577602233E+15 4.068141744507E-06 7.369272676026E+15 2.820180066834E-16 1.731425502558E+05 3.777184882505E+72 4.202684971244E+51
2.025303431821E-33 1.480234780081E+43 1.966573434256E-33 1.524440698617E+43 3.185804443740E-98 3.170643716287E+114 3.777184882505E+72 4.202684971244E+51
1.932674848471E-15 4.249290228758E-13 1.250268096623E+29 2.443501428799E+116 2.150390778485E-36 1.954381972474E+87 6.858302903800E+88 3.509192675943E+01
4.882931689521E-11 1.731425502558E+05 4.260846264471E+27 3.295971584321E+110 5.432994218051E-32 7.735485816055E+82 2.714531017056E+84 3.509192675943E+01
1.010105084086E+17 3.170643716287E+114 3.344007390619E+18 1.250457438718E+74 1.123893478429E-04 3.739397951770E+55 1.312226790479E+57 3.509192675943E+01
6.671850021543E-08 1.380648520000E-16 4.965114231744E+00 6.626070040000E-27 2.997924580000E+10 2.897772914526E-01
46 D=1 pi 4pi/3 G K z h C b= hc/zK K1 = T ψ^(3/4) K2 = B/@^2 K3 = m^2*ψ K4 = (J - 1)/Ψ ψ Nx = Δψ/Ψo mx = h/ΔΨ/C^2 Mx C^5*Ψ^2/G = u m J D β α S q^2 q l f H R V M E= M C^2 N T λ0 fo ξ fξ vol(ξ) Det Et Mt eξ Deξ (R/ξ) (R/ξ)^4 mf=h*ff/ C^2 Nfξ Nf Nf/Nfξ
N=1
mx = μ = M
3.141592653590E+00 4.188790204786E+00 6.671850021543E-08 1.380648520000E-16 4.965114231744E+00 6.626070040000E-27 2.997924580000E+10 2.897772914526E-01
3.141592653590E+00 4.188790204786E+00 6.671850021543E-08 1.380648520000E-16 4.965114231744E+00 6.626070040000E-27 2.997924580000E+10 2.897772914526E-01
3.141592653590E+00 4.188790204786E+00 6.671850021543E-08 1.380648520000E-16 4.965114231744E+00 6.626070040000E-27 2.997924580000E+10 2.897772914526E-01
4.586081522465E+13 2.635812689234E+36 6.531112187254E-34
4.586081522465E+13 2.635812689234E+36 6.531112187254E-34
4.586081522465E+13 2.635812689234E+36 6.531112187254E-34
3.215738480739E-21
3.215738480739E-21
3.215738480739E-21
2.553682126985E-35 1.455724051594E+64 2.887006618214E-13 4.202684971244E+51 2.366959970703E-10 5.057200281930E+00 1.000000000000E+00 1.000000000000E+00 1.164153218270E-10 6.645808140004E-24 1.751710542564E+13 2.989020721520E+07 5.467193723950E+03 4.370440034795E-38 6.859548594952E+47 3.915914159530E+34 7.655746417995E-25 1.879538511564E-72 1.031294426291E+04 9.268812064317E+24 2.039259465603E+03 4.037073374243E+39
1.588004853690E-37 9.052406464908E+61 4.642616289421E-11 4.202684971244E+51 9.152945797460E-15 6.413094790570E+01 1.000000000000E+00 7.279563047888E-01 7.239275951835E-13 5.240711995821E-25 1.381353517920E+12 3.790412115292E+08 1.946898075219E+04 3.446415700672E-39 8.698673753764E+48 6.297209971848E+36 4.760718784036E-27 4.519664466402E-79 6.413094790570E+01 5.763802154755E+22 9.999999303264E-01 1.823068139972E+41
1.351137013474E-43 7.702143609530E+55 5.456513386798E-05 4.202684971244E+51 6.626070040000E-27 6.952546084776E+04 1.000000000000E+00 3.038930617034E-01 6.159460826931E-19 4.834082707179E-28 1.274173654039E+09 4.109250802064E+11 6.410343830142E+05 3.179006698642E-42 9.430381449905E+51 7.401173900407E+42 4.050606863643E-33 2.783869623167E-97 5.456513386798E-05 4.904069664231E+16 7.848222708759E-10 6.507545704840E+45
7.177904996759E-41 4.176601085349E+50 6.969759226423E-41 4.301331627977E+50 1.418214546754E-120 2.009634205028E+144 3.777184882505E+72 4.202684971244E+51
1.589503349321E-42 1.886076289981E+52 1.543410736052E-42 1.942402310656E+52 1.540044109693E-125 8.357224104984E+150 3.777184882505E+72 4.202684971244E+51
4.452942854279E-47 6.732456889985E+56 4.323815870697E-47 6.933515833357E+56 3.386025178146E-139 1.356810983917E+169 3.777184882505E+72 4.202684971244E+51
2.850092463224E+24 2.009634205028E+144 1.098423370060E+16 1.455724051594E+64 3.171155538965E+03 1.325285032413E+48 4.650680529281E+49 3.509192675943E+01
1.287049375627E+26 8.357224104984E+150 3.084544297141E+15 9.052406464907E+61 1.432035559935E+05 2.934762996692E+46 1.029864881362E+48 3.509192675943E+01
4.594196153527E+30 1.356810983917E+169 9.368129875959E+13 7.702143609530E+55 5.111732607743E+09 8.221644780242E+41 2.885133564703E+43 3.509192675943E+01
47
pi 4pi/3 G K z h C b= hc/zK K1 = T ψ^(3/4) K2 = B/@^2 K3 = m^2*ψ K4 = (J - 1)/Ψ ψ Nx = Δψ/Ψo mx = h/ΔΨ/C^2 Mx C^5*Ψ^2/G = u m J D β α S q^2 q l f H R V M E= M C^2 N T λ0 fo ξ fξ vol(ξ) Det Et Mt eξ Deξ (R/ξ) (R/ξ)^4 mf=h*ff/ C^2 Nfξ Nf Nf/Nfξ
UNIFICACION a= B 3.141592653590E+00 4.188790204786E+00 6.671850021543E-08 1.380648520000E-16 4.965114231744E+00 6.626070040000E-27 2.997924580000E+10 2.897772914526E-01
Nx = 1
Nfo = 1
3.141592653590E+00 4.188790204786E+00 6.671850021543E-08 1.380648520000E-16 4.965114231744E+00 6.626070040000E-27 2.997924580000E+10 2.897772914526E-01
3.141592653590E+00 4.188790204786E+00 6.671850021543E-08 1.380648520000E-16 4.965114231744E+00 6.626070040000E-27 2.997924580000E+10 2.897772914526E-01
4.586081522465E+13 2.635812689234E+36 6.531112187254E-34
4.586081522465E+13 2.635812689234E+36 6.531112187254E-34
4.586081522465E+13 2.635812689234E+36 6.531112187254E-34
3.215738480739E-21
3.215738480739E-21
3.215738480739E-21
8.322275506312E-62 4.744105184630E+37 8.858751666932E+13 4.202684971244E+51 2.513861908223E-63 8.858751666932E+13 1.000000000000E+00 2.210890345652E-02 3.793895767850E-37 3.793896032184E-37 9.999999303264E-01 5.235899474631E+20 2.288208791748E+10 2.494955430190E-51 1.201594442820E+61 1.201594442820E+61 2.494955430191E-51 6.505444628830E-152 3.360918045761E-23 3.020642498937E-02 3.793895503515E-37 2.959785265621E+59
3.508470062289E-99 1.000000000000E+00 4.202684971244E+51 4.202684971244E+51 4.467790389949E-138 4.314541835155E+32 1.000000000000E+00 1.020051112015E-04 1.599414692634E-74 7.789745489412E-56 2.053231000689E-19 2.550077954243E+39 5.049829654794E+19 5.122720191524E-70 5.852212238647E+79 2.850245213002E+98 1.051812863793E-88 4.874207832712E-264 1.416881757449E-60 1.273429817165E-39 3.283967829875E-93 3.181291310431E+87
1.754235031144E-99 0.000000000000E+00
1.116947597487E-138 6.101683578702E+32 1.000000000000E+00 9.768042913460E-05 7.997073463171E-75 5.508181859281E-56 1.451853563930E-19 3.606354828000E+39 6.005293355032E+19 3.622310185548E-70 8.276277917780E+79 5.700490426004E+98 5.259064318965E-89 6.092759790890E-265 7.084408787245E-61 6.367149085825E-40 1.161057960851E-93 5.350272917639E+87
9.790483614419E-61 3.062080177107E+70 9.506577968579E-61 3.153526526484E+70 3.598829341069E-180 5.806190196080E+223 3.777184882505E+72 4.202684971244E+51
9.108794611248E-89 3.291241825014E+98 8.844656666813E-89 3.389531886804E+98 2.898221317332E-264 7.7493307879808E+335 3.777184882505E+72 4.202684971244E+51
5.416121680397E-89 5.535186904775E+98 5.259064318965E-89 5.700490426004E+98 6.092759790890E-265 6.19946463038464E+336 3.777184882505E+72 4.202684971244E+51
2.089548763748E+44 5.806190196080E+223 2.624451656986E+09 4.744105184630E+37 2.324936549111E+23 1.807655771445E+28 6.343412393781E+29 3.509192675943E+01
2.245927568478E+72 7.749330787980E+235 1.189207115003E+00 2.000000000000E+00 2.498931434959E+51 1.681792830507E+00 5.901735083270E+01 3.509192675943E+01
3.777184882505E+72 6.19946463038471E+336 1.000000000000E+00 1.000000000000E+00 4.202684971244E+51 1.000000000000E+00 3.509192675943E+01 3.509192675943E+01
