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This paper, which was released by Keith Dixon-Roche (one of CalQlata's Contributors) on the 24th of October 2017 defines an exact value for Newton's gravitational constant; 'G'.
whilst also providing a theoretical description of Planck's atom,

Note: All the input data in these calculations has been provided by CalQlata's Constants page.
All non-Planck calculations are the sole copyright priority of Keith Dixon-Roche © 2017
Keith Dixon-Roche is also responsible for all the other web pages on this site related to atomic theory
A 'pdf' version of this paper can be found at: Planck - The Paper

## Planck's Atom (a summary)

The levels of accuracy on this page have been set to assist CalQlata in its effort to establish an accurate value for Newton's gravitational constant (G).
To this end, Calqlata has confirmed all constants used in these calculations via original formulas and indisputable data where possible.

## Unification Theory

The following calculations have not only defined an accurate value for Newton's gravitational constant but also discovered that Planck's theories may provide the basis for a Newtonian atom.

## Constants & Formulas

The following Table, which should be read in conjunction with our Rydberg Atom page, contains modified constants used in the calculations for the Planck atom values:

 Sym (units) Formula Planck Atom Values Rydberg Atom Values G (C/mol) 6.67359232004332E-11 6.67359232004332E-11 F (C) = e.NA 96485.3317942158 5.38005167559927E+25 m₁ (kg) = (ħ.c/G)⁰˙⁵ 2.1765500017459E-08 1.67262163783E-27 m₂ (kg) = (ħ.c/G)⁰˙⁵ 2.1765500017459E-08 9.1093897E-31 h (J.s) = (π.m₂.aₒ.e² / ε₀)⁰˙⁵ 5.02324073024593E-15 6.62607174469163E-34 ħᵨ (J.s) = h / 2π 7.99473592559182E-16 1.05457207144921E-34 ε₀ (s²/m²) = 1 / μ₀.c² 8.85418775855161E-12 8.85418775855161E-12 μ₀ = 4π / 1E+07 1.25663706143592E-06 1.25663706143592E-06 NA 6.02214129E+23 6.02214129E+23 c (m/s) 299792459 299792459 e (C) = Q 89.3378520449704 1.60217648753E-19 k (N.m²/C²) = 1 / 4π.ε₀ 8987551847.32667 8.98755184732667E+09 Rᵧ (J) = R∞.h.c.(Z/n)² 8.76103166894037E+49 2.17987197684936E-18 Rᵧ (eV) = e / Rᵧ 9.80662895782438E+47 13.605691968492 R∞ (/m) = m₂.e⁴ / 8.ε₀².h³.c 5.8176897123571E+55 1.09737269561359E+07 PE (J) = -k.e² / aₒ -1.75220633378807E+50 -4.35974395369872E-18 aₒ (m) = λ / (2π)² 4.0938052242E-37 5.2917721067E-11 Q (C) = Q 89.3378520449704 1.60217648753E-19 Q (J) = (G.m₁² / k.φ)⁰˙⁵ 89.3378520449704 1.60217648753E-19 φ 4.40742111792333E-40 4.40742111792333E-40

The following formulas are provided to assist with the calculation method used to identify G

Planck's original formulas:
Planck's time; tᵨ = (ħ.G / c⁵)⁰˙⁵
Planck's length; λᵨ = (ħ.G / c³)⁰˙⁵
Planck's mass; mᵨ = (ħ.c / G)⁰˙⁵

CalQlata's formulas:
Planck's energy; Eᵨ = (ħ.c⁵ / G)⁰˙⁵
Planck's force; Fᵨ = c⁴ / G

Along with the above formulas ...
ħ = h / 2π
v = 2πR/t
λ = h / m.v
... we can establish the following for a Planck atom:
(ħ.G/c³)⁰˙⁵ = h ÷ (ħ.c/G)⁰˙⁵ ÷ 2πR/t
(ħ.G/c³)⁰˙⁵ = h ÷ 2πR x t ÷ (ħ.c/G)⁰˙⁵
(ħ.G/c³)⁰˙⁵ = ħ/R x (ħ.G/c⁵)⁰˙⁵ ÷ (ħ.c/G)⁰˙⁵
ħ.G/c³ = ħ²/R² x ħ.G/c⁵ ÷ ħ.c/G
ħ.G/c³ = R².ħ² x ħ.G/c⁵ x G/ħ.c
G/c³ = G².ħ / R².c⁶
R² = G.ħ / c³
R = (G.ħ/c³)⁰˙⁵ = λ {λ = 2.π.R/n for non-Planck values}
i.e. in Planck's atom, the radius of separation between its nucleus and its orbiting mass is equal to its wavelength, and its shell number is equal to 2.π
moreover, if R = λ in Planck's atom;
G = λ².c³ / ħ
from which; G = 6.67359232004332E-11 using Rydberg Atom Values (see above Table) verifying the above formula, however, regarding its units:
m² x m³/s³ ÷ J.s = m² x m³/s³ ÷ kg.m².s/s² = m⁵/m² x s²/s⁴ ÷ kg = m³ ÷ kg.s²
are missing; kg/kg
i.e. kg/kg x m³ ÷ kg.s² = kg.m/s² x m²/kg² = N.m²/kg²
In order to create the correct units we need to apply the mass ratio m₁/m₂, which in the Planck atom equals 1.0 (both the force-centre and the orbiting mass are the same)

## Properties and Formulas

The following Tables contain the formulas and properties of a Planck electron in the specified shells (n) orbiting a single Planck proton (Z=1) using the same formulas for a Rydberg Atom and the above constants.

 Shell KE = Rᵧ.(Z/n)²= mₑ.R.(2.π/t)²= mₑ.h² / R³ PE = -2.KE= -h.ƒ= -mₑ.v² E = KE+PE= -KE (J) (J) (J) 1 8.76103166894037E+49 -1.75220633378807E+50 -8.76103166894037E+49 2π 2.21919524656261E+48 -4.43839049312522E+48 -2.21919524656261E+48 Kinetic, Potential and Total Energies in an Atom with one Proton and One Electron

 Shell v = 2.KE / mₑ= 2.π.R / t= √[k.Q₁.Q₂ / mₑ.R] R = aₒ.n² / Z t = v.R= n.h / 2.Rᵧ= n³ / 2.Z².c.R∞= n³ . [π.aₒ]¹˙⁵ . [16.ε₀.mₑ]² / e= n.λ / v (m/s) (m) (s) 1 8.97239322207392E+28 4.09380522418125E-37 2.86680891021906E-65 2π 1.42800073265728E+28 1.61616952231127E-35 7.11112562078409E-63 Orbital Velocities, Radii and Periods

 Shell h = R.v p = mₑ.v (m²/s) (kg.m/s) 1 3.67312302459346E-08 1.95288624831699E+21 2π 2.30789126195887E-07 3.10811499715836E+20 Newton’s Motion Constants and Momenta

 Shell λ = 2πR / n= p / h ƒ = v / λ (m) (Hz) 1 2.57221368350306E-36 3.48819900913307E+64 2π 1.61616952231127E-35 8.83571130963482E+62 Electron Wavelengths and Frequencies

 Shell Fg = G.m₁.m₂ / R²= G.m₁.m₂ / R³.(2.π/t)² Fₑ = k.Q₁.Q₂ / R².ε φ = Fg/Fₑ= G.m₁ / R.(2.π.R/t)²= G.m₁ / R.v²= G.m₁.R / h² (N) (N) 1 1.88643835639276E+47 4.28014093938363E+86 4.40742111792333E-40 2π 1.21038391820525E+44 2.74624068320377E+83 4.40742111792333E-40 Gravitational and Electrostatic Electron Holding Forces and their ratio (φ)

 Shell KEn-1/KEn - 1 = [n/(n-1)]² - 1 KE₁/KEn - 1 = n² - 1 1 2π 38.4784176043574 38 Kinetic Energy Jump Factors Between Shell Numbers (n)n=1 to n: KEn = KE₁ / n²n-1 to n: KEn = KEn-1 . [(n-1)/n]²

### Planck's Atomic Particle

The Planck atom is artificial, equivalent to the Newton atom but based upon Planck's time, length and mass, all the particles of which are identical in size and substance to each other.

Newton's and Planck's atomic particles share a commonality in the product of their volumes:
Vᵨ.Vₑ = 3E-91 (exact) for both Newton's and Planck's atomic particles where: Vᵨ is the volume of the force-centre and Vₑ is the volume of the satellite

The radius (Rᴾ) of a Planck particle can therefore be calculated thus;
Rᴾ = ⁶√[9 x 3E-91 ÷ 16π²] = 5.07563837996471E-16 m
and Vᴾ = Vᵨ = Vₑ = ⁴/₃π.Rᴾ³ = 5.47722557505167E-46 m³

As Planck's mass; mᴾ = 2.1765500017459E-08 kg
Planck's density; ρᴾ = mᴾ/Vᴾ = 3.97381844498046E+37 kg/m³

Planck-Newton Commonality
Fᴾ = c⁴/G = 299792459⁴ / 6.67359232004334E-11 = 1.21038391820525E+44 N
Fᴺ = G.mᵨ.mₑ/aₒ² = 3.63115175E-47 N
Fᴺ/Fᴾ = 3E-91 (exact)
Therefore; Fᴺ/Fᴾ ≡ Vᴺᵨ.Vᴺₑ = Vᴾᵨ.Vᴾₑ = 3E-91 (exact)

## G

Planck's formulas (see above) are used to calculate his electron properties first with 'ħ' for the standard atom and again with 'ħᵨ' (see above), the results from which are summarised below:

 Property Values using Planck's constant 'ħᵨ': A Values using Planck's constant 'ħ': B Ratio A/B t (s) 1.48432887846076E-34 5.39096122598358E-44 2753365895.68949 λ (m) 4.44990604438463E-26 1.61616952231127E-35 2753365895.68949 m (kg) 59.9283854507007 2.17655000174590E-08 2753365895.68949 E (J) 5.38609471364750E+18 1.95618559889903E+09 2753365895.68949 F (N) 1.21038391820525E+44 1.21038391820525E+44 1.0

Note:
E = 2.KE.φ; which means that Planck's energy must be gravitational (Newtonian)
t = λ.(m/E)⁰˙⁵; so it too must be gravitational (Newtonian)

The above ratios reveal that G = 6.67359232004332E-11 N.m²/kg² must be correct because:
2753365895.68949 exactly equals (ħᵨ/ħ)⁰˙⁵ and shows that Planck’s constants vary with mass;
a value of exactly 1.0 also shows that Planck's gravitational force and Newton's gravitational force may be calculated using conventional Newtonian theory.

Using CalQlata’s original estimate of 'G' (6.67128190396304E-11) for the above calculations ...
Ratios = 2752650947.59247 & 2753604252.98478 neither of which ≠ (ħᵨ/ħ)⁰˙⁵
(errors = -0.000173116364635351 & 0.000173146339100150)
and
Ratio (F) = 1.00034632265785 (error = 0.00034632265785)

Using Codata’s estimate of ‘G’ (6.674E-11) for the above calculations ...
Ratios = 2753492044.03629 & 2753323847.52487 neither of which ≠ (ħᵨ/ħ)⁰˙⁵
(errors = 0.000030543799439231 & -0.000030542866544137)
and
Ratio (F) = 0.99993891519978
(error = -0.00006108480022)