Newton's Atom

{Keith Dixon-Roche © 28/10/2017}

A Newtonian (and Coulomb) model of the atom.

Newton's atom; a very simple system
Fig 1. A Beryllium Atom

Sources: Planck's Atom; Newton's Atom; Laws of Motion; Physical Constants
Related Books: Philosophiæ Naturalis Principia Mathematica Rev. IV; The Atom; The Mathematical Laws of Natural Science
Related Calculators: Atomic Elements; Orbital Motion; Atoms; Physics
CalQlata comment: This is the only study we (at CalQlata) know on this subject that manifestly works.


The purpose of this study is to answer the following question:
1) Can Isaac Newton's laws of orbital motion be used to explain the atom?


The answer to the above question is yes, albeit input from Coulomb, Rydberg and Planck was required to complete it.


In these calculations, the electron and proton in an hydrogen atom have been treated as a satellite orbiting a force-centre.
The purpose of these calculations is to see if Newton's laws of motion can be applied to atoms.

The following Table compares the results from calculations using Newton's laws of motion in combination with Coulomb's law between the earth and a hydrogen atom under gravitational or electrostatic attraction.

Earth Gravitational Force Electrostatic Force Units
G:k 6.67359232E-11 6.67359232E-11 8.98755185E+09 m³/kg/s²
m₁:Q₁ 1.9885E+30 1.67262164E-27 1.60217649E-19 kg:C
m₂:Q₂ 5.96451977E+24 9.1093897E-31 1.60217649E-19 kg:C
T 3.1558118E+07 7239.40504 6.37390647E-11 s
a 1.495945981E+11 5.2917721E-11 5.2917721E-11 m
b 1.495737135E+11 5.2917721E-11 5.2917721E-11 m
e 0.0167091466581 0⁽¹⁾ 0⁽¹⁾
p 1.495528319E+11 5.29177211E-11 5.29177211E-11 m
ƒ 1.47095E+11 5.29177211E-11 5.29177211E-11 m
x' 2.499598079E+09 0 0 m
L 9.398649712E+11 3.32491847E-10 3.32491847E-10 m
K 2.974914364E-19 3.53673364E+38 2.74162451E+10 s²/m³
A 7.029445371E+22 8.79735542E-21 8.79735542E-21
1.47095E+11 5.291772107E-11 5.291772107E-11 m
3.658178805E+22 3.631151755E-47⁽²⁾ 8.238722050E-08⁽²⁾ N
Fc 3.658178805E+22 3.631151755E-47 8.238722050E-08 N
g -0.006133232761 -3.986163590E-17 -5.142206314E+11 m/s²
30286.008788579 4.592806255E-14 5.216453195E+00 m/s
h 4.454920463E+15 2.430408403E-24 2.760428151E-10 m²/s
PE -5.380998113E+33 -1.921522757E-57 -4.359743954E-18 N.m
KE 2.735455E+33 9.607707959E-58 2.179871977E-18 N.m
E -2.645543113E+33 -9.607519611E-58 -2.179871977E-18 N.m
Ř⁽³⁾ 1.520941962E+11 5.291772107E-11 5.291772107E-11 m
Ř⁽³⁾ 1.520941962E+11 5.291772107E-11 5.291772497E-11 m
3.421649078E+22 3.631151755E-47 8.238722050E-08 N
Fc 3.420693766E+22 3.631151755E-47 8.238722050E-08 N
g -5.736671536E-03 -3.986163590E-17 -5.142206314E+11 m/s²
2.929053557E+04 4.592806255E-14 5.216453195E+00 m/s
h 4.454920463E+15 2.430408403E-24 2.760428151E-10 m²/s
PE -5.204129660E+33 -1.921522757E-57 -4.359743954E-18 N.m
KE 2.558586547E+33 9.607707959E-58 2.179871977E-18 N.m
E -2.645543113E+33 -9.607519611E-58 -2.179871977E-18 N.m
Newton's Laws of Motion Applied to the Hydrogen Atom

The above results reveal the following:
1) An eccentricity of '0' (circular) for the orbital path is necessary for compliance with 2) below
2) F̌ (gravitational) = F̌ (electrostatic) x φ
3) Ř = Ř for all three subjects (i.e. the earth and the hydrogen atom under either force), which means that all three orbits comply with Newton's and Coulomb's laws.

That the above calculation technique is exactly the same in all three scenarios proves that Newton's laws of motion do indeed apply to atoms as long as Coulomb's electrostatic charge is included.

You may have noticed that the ratio between the gravitational and electrostatic coupling forces (F̌) in the above atom is 4.4074E-40, which is exactly equal to the coupling ratio; φ

Comparison Between Planetary and Atomic Orbits

In addition to the above proof of Newton's laws when applied to the atom, comparison calculations were carried out between planetary and electron orbits, all of which use identical theories and therefore strengthen the claim that Newton's laws apply to atomic structures.

The Systems

Rydberg: A standard atom of protons and orbiting electrons

Planck: A Planck mass orbiting a Planck mass force centre

Two known planetary force-centres each with a single orbiting planetary body

Constants & Formulas

Newton's gravitational coupling force: Fg = G.m₁.m₂ / R²
Coulomb's electrostatic coupling force: Fₑ = k.Q₁.Q₂ / R²
Centripetal force: Fc = m₂.v² / R
Coupling ratio: φ = Fg/Fₑ = 4.40742111792333E-40
Newton's gravitational constant: G
Coulomb's constant: k
Force-centre mass: m₁
Orbiting mass: m₂
Force-centre elecrical charge: Q₁
Orbiting elecrical charge: Q₂
Separation distance: R
Velocity of orbiting mass: v

Calculation Results

The following Table of results show that the same calculation rules apply to Planck's Atom and planetary systems which are both coupled together with gravitational force only (K=1)

Moon-Earth ⁽¹⁾ Sun-Earth ⁽¹⁾
Planck Atom Std. Atom Perigee Apogee Perigee Apogee
v (m/s) 299792459 2187690.351 1084.034166 958.7083173 30279.07556 29287.2
R (m) 1.61617E-35 5.29177E-11 3.59508E+08 4.06504E+08 1.47095E+11 1.5206E+11
m₁ (kg) 2.17655E-08 1.67262E-27 5.96659E+24 5.96659E+24 1.9885E+30 1.9885E+30
m₂ (kg) 2.17655E-08 9.10939E-31 7.34892E+22 7.34892E+22 5.96659E+24 5.96659E+24
φ 1 4.40742E-40 1 1 1 1
Fg (N) 1.21038E+44 3.63115E-47 2.26408E+20 1.77084E+20 3.65945E+22 3.42437E+22
Fc (N) 1.21038E+44 3.63115E-47 2.40215E+20 1.66162E+20 3.71889E+22 3.36563E+22
Fg:Fc 1 11.00412615 1.000734727
Table 1: Forces
⁽¹⁾ some of these properties have been obtained from planetary systems not yet corrected with the final/actual value for Newton's gravitational constant 'G' and therefore are not expected to provide and exact value of 1.0 for Fg:Fc

As can be seen from the above table, applying the coupling ratio; 'φ' to Rydberg's atom allows us to use Newtonian mechanics to define the properties and behaviour of sub-atomic particles.

Using Henri Poincaré's formula EN = m.v² to convert the above properties to energies:

Moon-Earth ⁽¹⁾ Sun-Earth ⁽¹⁾
Planck Atom Std. Atom Perigee Apogee Perigee Apogee
φ 4.4074E-40 1 1 1 1 1
EN (J) 4.4384E+48 4.35974E-18 8.6359E+28 6.7546E+28 5.4703E+33 5.1178E+33
PEₐ (J) ⁽²ꞌ³⁾ -4.4384E+48 -4.3597E-18 -8.1367E+28 -7.196E+28 -5.381E+33 -5.2047E+33
KEₐ (J) ⁽²⁾ 2.2192E+48 2.1799E-18 4.318E+28 3.3768E+28 2.7352E+33 2.5589E+33
Table 2: Energies
⁽¹⁾ some of these properties have been obtained from planetary systems not yet corrected with the final/actual value for Newton's gravitational constant 'G' and therefore are not expected to provide and exact value of 1.0 for Fg:Fc
⁽²⁾ whilst these values have been calculated using Rydberg's formulas, the energies can be seen to exactly replicate those for Newton's orbits for the earth and its moon
⁽³⁾ these properties include spin induced energy and therefore vary slightly with Poincaré's formula alone for planets. Planck's and standard atoms do not include the effects of spin

As can be seen from the above Table the energies calculated using Newton's formulas replicate the results from Rydberg's, noting that the coupling ratio 'φ' is required for the Planck atom.

Quantity Formula Planck (A) Newton (B) Ratio A/B
t (s) = aₒ/c 5.39096122598358E-44 1.76514516887831E-19 3.05411776948031E-25
λ (m) = aₒ 1.61616952231127E-35 5.29177210670000E-11 3.05411776948031E-25
m (kg) = mN 2.17655000174590E-08 7.12660796350449E+16 3.05411776948031E-25
E (J) = m.c² 1.95618559889903E+09 6.40507585675677E+33 3.05411776948031E-25
F (N) = E/λ 1.21038391820525E+44 1.21038391820525E+44 1.0
Table 3: A Comparison of Newton's and Planck's Atomic Values

If Newton's G were incorrect (i.e. not 6.67359232004332E-11 m³/kg/s²), variations would appear between the Ratios in the above table (see also; Planck's atom).


Newton's laws are universal, i.e. they apply to all orbiting systems irrespective of size. However, whilst you can apply Rydberg's rules to planetary systems (by multiplying the force by 'φ'), you must first assume that they both carry a charge. You must also assume that the charge held by the sun is equal to the combined charge held by all of its orbiting bodies. To the author's knowledge, no such charge has been identified in the earth.

As the above theories are universal, i.e. they also apply to the electron, which is an elementary particle, the author considers it highly likely that Newtonian theory may be applied to all elementary particles and therefore also considers it unlikely that there is a need for the elusive Unification Theory.

Further Reading

This paper should be read in conjunction with:
Laws of Motion
Planetary Spin
Rydberg Atom
Planck Atom
The Atom

You will find further reading on this subject in reference publications(55, 60, 61, 62, 63 & 64)