Gravity via Atomic Orbitals

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Simulating gravitational and atomic orbits via n-body rotating particle-particle orbital pairs at the Planck scale

The following describes a geometrical method for simulating gravitational orbits [1] and atomic orbitals [2] via an n-body network of rotating individual particle-particle orbital pairs . Although the simulation is dimensionless (the only physical constant used is the fine structure constant alpha), it can translate via the Planck units for comparisons with real world orbits. The orbits generated by this dimensionless geometrical approach can be formulated, and despite not using Newtonian physics these formulas demonstrate consistency; for example the derived formulas for radius R, period T and (M + m) will reduce Kepler's formula to G [3]. Likewise the atomic orbital shells naturally quantize according to pi without relying on built-in postulates.

A regular 3-body orbit. The simulation begins with the start (x, y) co-ordinates of each point. No other parameters are required. r0=2*α; x1=1789.5722983; y1=0; x2=cos(pi*2/3)*r0; y2=sin(pi*2/3)*r0; x3=cos(pi*2/3)*r0; y3=sin(pi*2/3)*r0

\[ {\frac {4\pi ^{2}R^{3}}{(M+m)T^{2}}}={\frac {l_{p}c^{2}}{m_{P}}}=G\]

The simulation source code(s) used here are listed below, these give a precise description of this orbital model and so can be used as reference [4].

For simulating gravity, orbiting objects A, B, C... are sub-divided into discrete points, each point can be represented as 1 unit of Planck mass mP (for example, a 1kg satellite would be divided into 1kg/mP = 45940509 points). Each point in object A then forms an orbital pair with every point in objects B, C..., resulting in a universe-wide, n-body network of rotating point-to-point orbital pairs .

Each orbital pair rotates 1 unit of length per unit of time, when these orbital pair rotations are summed and mapped over time, gravitational orbits emerge between the objects A, B, C...

The base simulation requires only the start position (x, y coordinates) of each point, as it maps only rotations of the points within their respective orbital pairs then information regarding the macro objects A, B, C...; momentum, center of mass, barycenter etc ... is not required (each orbital is calculated independently of all other orbitals).

For simulating electron transition within the atom, the same program is used. The electron is assigned as a single point, the nucleus as multiple points clustered together (a 2-body orbit), and an incoming 'photon' (a unit of length) is added to (increases) the electron - proton orbital radius in a series of discrete steps (rather than a single 'jump' between orbital shells). As the electron continues to orbit the nucleus during this transition phase, the electron path traces a hyperbolic spiral. Although we are mapping the electron transition as a gravitational orbit on a 2-D plane, periodically the transition spiral angles converge to give an integer orbital radius (360°=4r, 360+120°=9r, 360+180°=16r, 360+216°=25r ... 720°=∞r), a radial quantization (as a function of pi and so of geometrical origin) naturally emerges. Furthermore, the transition steps between these radius can then be used to solve the transition frequency, replicating the Bohr model. The gravitational orbit returns results analogous to the Bohr model and the 2-photon orbital model approach complements the (electric wave-state) Schrodinger wave equation.

By selecting the start co-ordinates on a 2-D plane for each point accordingly, we can 'design' the required orbits. The 26 points orbit each other resulting in 325 point-point orbitals.

Theory

In the simulation, particles are treated as an electric wave-state to (Planck) mass point-state oscillation, the wave-state as the duration of particle frequency in Planck time units, the point-state duration as 1 unit of Planck time (as a point, this state can be assigned mapping coordinates), the particle itself is a continuous oscillation between these 2 states (i.e.: the particle is not a fixed entity). For example, an electron has a frequency (wave-state duration) = 1023 units of Planck time followed by the mass state (1 unit of Planck time). The background to this oscillation is given in the mathematical electron model.

If the electron has (is) mass (1 unit of Planck mass) for 1 unit of Planck time, and then no mass for 1023 units of Planck time (the wave-state), then in order for a (hypothetical) object composed only of electrons to have (be) 1 unit of Planck mass at every unit of Planck time, the object will require 1023 electrons. This is because orbital rotation occurs at each unit of Planck time and so the simulation requires this object to have a unit of Planck mass at each unit of Planck time (i.e.: on average there will always be 1 electron in the mass point state). We would then measure the mass of this object as 1 Planck mass (the measured mass of an object reflects the average number of units of Planck mass per unit of Planck time). For the simulation program, this Planck mass object can now be defined as a point (it will have point co-ordinates at each unit of Planck time and so can be mapped). As the simulation is dividing the mass of objects into these Planck mass size points and then rotating these points around each other as point-to-point orbital pairs, then by definition gravity is a mass to mass interaction.

Nevertheless, although this is a mass-point to mass-point rotation, and so referred to here as a point-point orbital, it is still a particle to particle orbital, albeit the particles are both in the mass state. We can also map individual particle to particle orbitals albeit as gravitational orbits, the H atom is a well-researched particle-to-particle orbital pair (an electron orbiting a proton) and so can be used as reference. To map orbital transitions between energy levels, the simulation uses the photon-orbital model, in which the orbital (Bohr) radius is treated as a 'physical wave' akin to the photon albeit of inverse or reverse phase. The photon can be considered as a moving wave, the orbital radius as a standing/rotating wave (trapped between the electron and proton).

Orbital momentum derives from this orbital radius, it is the rotation of the orbital radius that pulls the electron, resulting in the electron orbit around the nucleus. Furthermore, orbital transition (between orbitals) occurs between the orbital radius and the photon, the electron has a passive role. Transition (the electron path) follows a specific hyperbolic spiral for which the angle component periodically converges to give integer radius where r = Bohr radius; at 360° radius =4r, 360+120°=9r, 360+180°=16r, 360+216°=25r ... 720°=∞r. As these spiral angles (360°, 360+120°, 360+180°, 360+216° ...) are linked directly to pi, and as the electron is following a semi-classical gravitational orbit, this particular quantization has a geometrical origin.

Although the simulation is not optimized for atomic orbitals (the nucleus is treated simply as a cluster of points), the transition period t measured between these integer radius can be used to solve the transition frequencies f via the general formula

\[ f/c=t\lambda _{H}/(n_{f}^{2}-n_{i}^{2})\]

.

In summary, both gravitational and atomic orbitals reflect the same particle-to-particle orbital pairing, the distinction being the state of the particles; gravitational orbitals are mass to mass whereas atomic orbitals are predominately wave to wave. There are not 2 separate forces used by the simulation, instead particles are treated as oscillations between the 2 states (electric wave and mass point). The transition spiral provides the geometric structure; the photon-orbital hybrid (2-Photon model) provides the angular momentum content.

N-body orbitals

8-body (8 mass points, 28 orbitals), the resulting orbit is a function of the start positions of each point

The simulation universe is a 4-axis hypersphere expanding in increments [5] with 3-axis (the hypersphere surface) projected onto an (x, y) plane with the z axis as the simulation timeline (the expansion axis). Each point is assigned start (x, y, z = 0) co-ordinates and forms pairs with all other points, resulting in a universe-wide n-body network of point-point orbital pairs. The barycenter for each orbital pairing is its center, the points located at each orbital 'pole'.

The simulation itself is dimensionless, simply rotating circles. To translate to dimensioned gravitational or atomic orbits, we can use the Planck units (Planck mass mP, Planck length lp, Planck time tp), such that the simulation increments in discrete steps (each step assigned as 1 unit of Planck time), during each step (for each unit of Planck time), the orbitals rotate 1 unit of (Planck) length (at velocity c = lp/tp) in hyper-sphere co-ordinates. These rotations are then all summed and averaged to give new point co-ordinates. As this occurs for every point before the next increment to the simulation clock (the next unit of Planck time), the orbits can be updated in 'real time' (simulation time) on a serial processor.

Setting

\[ \alpha _{inv}={\frac {1}{\alpha }}\]

= 137.035999177⁠ (CODATA 2022)

Orbital pair rotation on the (x, y) plane occurs in discrete steps according to an angle β as defined by the orbital pair radius (the atomic orbital β has an additional alpha term).

\[ \beta _{gravity}={\frac {1}{r_{ij}r_{orbital}{\sqrt {r_{orbital}}}}}\]

\[ \beta _{atomic}={\frac {1}{{\sqrt {2\alpha _{inv}}}r_{orbital}{\sqrt {r_{orbital}}}}}\]

As the simulation treats each (point-point) orbital independently (independent of all other orbitals), no information regarding the points (other than their initial start coordinates) is required by the simulation.

Although orbital and so point rotation occurs at c, the hyper-sphere expansion [6] is equidistant and so `invisible' to the observer. Instead observers (being constrained to 3D space) will register these 4-axis orbits (in hyper-sphere co-ordinates) as a circular motion on a 2-D plane (in 3-D space). An apparent time dilation effect emerges as a consequence.

Symmetrical 4 body orbit; (3 center mass points, 1 orbiting point, 6 orbital pairs). Note that all points orbit each other.

2 body orbits (x, y plane)

For simple 2-body orbits, to reduce computation only 1 point is assigned as the orbiting point and the remaining points are assigned as the central mass. For example the ratio of earth mass to moon mass is 81:1 and so we can simulate this orbit accordingly. However we note that the only actual distinction between a 2-body orbit and a complex orbit being that the central mass points are assigned (x, y) co-ordinates relatively close to each other, and the orbiting point is assigned (x, y) co-ordinates distant from the central points (this becomes the orbital radius) ... this is because the simulation treats all points equally, the center points also orbiting each other according to their orbital radius, for the simulation itself there is no difference between simple 2-body and complex n-body orbits.

The Schwarzschild radius formula in Planck units

\[ r_{s}={\frac {2l_{p}M}{m_{P}}}\]

As the simulation itself is dimensionless, we can remove the dimensioned length component

\[ 2l_{p}\]

, and as each point is analogous to 1 unit of Planck mass

\[ m_{P}\]

, then the Schwarzschild radius for the simulation becomes the number of central mass points. We then assign (x, y) co-ordinates (to the central mass points) within a circle radius

\[ r_{s}\]

= number of central points = total points - 1 (the orbiting point).

After every orbital has rotated 1 length unit (anti-clockwise in these examples), the new co-ordinates for each rotation per point are then averaged and summed, the process then repeats. After 1 complete orbit (return to the start position by the orbiting point), the period t (as the number of increments to the simulation clock) and the (x, y) plane orbit length l (distance as measured on the 2-D plane) are noted.

Key:

1. i = rs; the number of center mass points (the orbited object).

2. j = total number of points, as here there is only 1 orbiting point; j = i + 1

3. kr is a mass to radius co-efficient in the form

\[ j_{max}=(k_{r}i+1)\]

. This function defines orbital radius in terms of the central mass Schwarzschild radius (

\[ i\]

) and the orbiting point (1), thus quantizing the radius. When

\[ k_{r}\]

= 1 then

\[ j_{max}=j\]

, and the radius is at a minimum giving an analogue gravitational principal quantum number

\[ n_{g}=j_{max}/j\]

.

1. x, y = start co-ordinates for each point (on a 2-D plane), z = 0.

2. rα = a radius constant, here rα = sqrt(2/α) = 16.55512; where alpha = inverse fine structure constant = 137.035 999 084 (CODATA 2018). This constant adapts the simulation specifically to gravitational and atomic orbitals.

\[ r_{\alpha }={\sqrt {2\alpha _{inv}}}\]

\[ r_{orbit}={r_{\alpha _{inv}}}^{2}\;*\;r_{wavelength}\]

1. Rotation angle β

\[ \beta _{orbital}={\frac {1}{r_{ij}r_{orbital}{\sqrt {r_{orbital}}}}}\]

\[ r_{ij}={\sqrt {\frac {2j}{i}}}\]

(for each gravitational orbital in the simulation)

\[ r_{ij}={\sqrt {2\alpha _{inv}}}\]

(for each atomic orbital in the simulation)

Orbital formulas (2-D plane)

\[ j=i+1\]

\[ r_{orbit}=2\alpha _{inv}2{\frac {(k_{r}i+1)^{2}}{i^{2}}}\]

, orbital radius (center mass to point)

\[ r_{ij}={\sqrt {\frac {i}{j}}}\]

(averaged for each orbit)

\[ t_{orbit}={\frac {2\pi }{\beta _{orbit}}}=16\pi {\alpha _{inv}}^{3/2}{\frac {{(k_{r}i+1)}^{3}}{i^{5/2}j^{1/2}}}\]

, orbiting point period

\[ r_{barycenter}={\frac {r_{orbit}}{j}}\]

\[ l_{orbit}=2\pi (r_{orbit}-r_{barycenter})\]

, distance travelled by orbiting point

\[ v_{orbit}={\sqrt {\frac {i}{r_{orbit}j}}}\]

, orbiting point velocity

Example (dimensionless). The simulation parameters agree closely with the calculated parameters:

points = 8 (1 orbiting point and 7 center mass points) [7]

i = 7, j = 8

kr = 32

\[ \sqrt {2j/i}\]

= 1.511858

Calculated

calculated orbit period = 2504836149.00059

calculated orbit radius = 566322.241497

calculated orbit length = 3113519.13854

calculated orbit barycenter = 70790.280187, 0

ng = (kr i + 1)/j = 28.125

Simulation

simulation orbit period = 2504836141      
simulation orbit length = 3113519.130546298             
simulation orbit barycenter; x = 70790.28092,  y = 0.000732

Earth moon orbit

The earth to moon mass ratio approximates 81:1 and so can be simulated as a 2-body orbit with the moon as a single orbiting point as in the above example. Here we use the orbital parameters to determine the value for the mass to radius coefficient kr. Planck length

\[ l_{p}\]

, Planck mass

\[ m_{P}\]

and

\[ c\]

are used to convert between the dimensionless formulas and dimensioned SI units.

Reference values

\[ M\]

= 5.9722 x 1024kg (earth)

\[ m\]

= 7.346 x 1022kg (moon)

\[ T_{orbit}\]

= 27.321661*86400 = 2360591.51s

To simplify, we assume a circular orbit which then gives us this radius

\[ R_{orbit}=({\frac {G(M+m)T_{orbit}^{2}}{4\pi ^{2}}})^{(1/3)}\]

= 384714027m

\[ G={\frac {l_{p}c^{2}}{m_{P}}}\]

= 0.66725e-10

The mass ratio

\[ i={\frac {M}{m}}\]

= 81.298666, j = i + 1

We then find a value for

\[ k_{r}\]

using Torbit as our reference (reversing the orbit period equation).

\[ T_{o}=T_{orbit}{\frac {m_{P}}{M}}{\frac {c}{l_{p}}}=16\pi {\alpha _{inv}}^{3/2}{\frac {(k_{r}i+1)^{3}}{i^{5/2}j^{1/2}}}\]

(dimensionless orbital period)

\[ k_{r}={\frac {1}{i}}{({\frac {T_{o}i^{5/2}j^{1/2}}{16\pi {\alpha _{inv}}^{3/2}}})}^{(1/3)}-{\frac {1}{i}}\]

= 12581.4468

Dimensionless solutions

\[ r_{orbit}=2\alpha _{inv}2{\frac {(k_{r}i+1)^{2}}{i^{2}}}\]

= 86767420100

\[ t_{orbit}=16\pi {\alpha _{inv}}^{3/2}{\frac {{(k_{r}i+1)}^{3}}{i^{5/2}j^{1/2}}}\]

= 0.159610040233 x 1018

\[ r_{barycenter}={\frac {r_{orbit}}{j}}\]

= 1054299229.62

\[ l_{orbit}=2\pi (r_{orbit}-r_{barycenter})\]

= 538551421685

\[ v={\sqrt {\frac {i}{r_{orbit}j}}}\]

= 0.33741701 x 10-5

Converting back to dimensioned values

\[ R=r_{orbit}l_{p}{\frac {M}{m_{P}}}=R_{orbit}\]

= 384714027m

\[ T=t_{orbit}{\frac {l_{p}}{c}}{\frac {M}{m_{P}}}=T_{orbit}\]

= 2360591.51s

\[ B={\frac {R}{j}}\]

= 4674608.301m (barycenter)

\[ L=2\pi (R-B)\]

= 2387858091.51m (distance moon travelled around the barycenter)

\[ V=c{\sqrt {\frac {i}{r_{orbit}j}}}\]

= 1011.551m/s (velocity of the moon around the barycenter)

If we expand the velocity term

\[ v_{orbit}=c{\sqrt {\frac {i}{r_{orbit}j}}}\]

\[ v_{orbit}^{3}={\frac {GM}{T_{orbit}}}2\pi {\frac {i^{2}}{j^{2}}}\]

Note: The standard gravitational parameter μ is the product of the gravitational constant G and the mass M of that body. For several objects in the Solar System, the value of μ is known to greater accuracy than either G or M.

\[ \mu _{earth}\]

= 3.986004418(8)e14

\[ \mu _{moon}\]

= 4.9048695(9)e12

\[ i={\frac {\mu _{earth}}{\mu _{moon}}}\]

= 81.2662685

\[ k_{r}={\frac {c}{2{\sqrt {\alpha _{inv}}}}}{({\frac {T_{orbit}}{2\pi \mu _{earth}}})}^{1/3}{\frac {(i+1)}{i}}^{1/6}-{\frac {1}{i}}\]

= 12580.3462

\[ t_{orbit}\]

= 0.15956776936 x 1018

\[ r_{orbit}\]

= 86752239934

Kepler's formula = G

Kepler's formula reduces to G [8]

\[ R=2\alpha _{inv}2({\frac {k_{r}i+1}{i}})^{2}l_{p}{\frac {M}{m_{P}}}\]

\[ T=16\pi {\alpha _{inv}}^{3/2}{\frac {{(k_{r}i+1)}^{3}}{i^{5/2}(i+1)^{1/2}}}{\frac {l_{p}}{c}}{\frac {M}{m_{P}}}\]

\[ M+m=M({\frac {i+1}{i}})\]

\[ {\frac {4\pi ^{2}R^{3}}{(M+m)T^{2}}}={\frac {l_{p}c^{2}}{m_{P}}}=G\]

Maple code

R:=(2/alpha_{inv})*2*((kr*i+1)^2/i^2)*lp*(M/mP):

T:=(16*Pi/alpha_{inv}^(3/2))*((kr*i+1)^3/(i^(5/2)*(i+1)^(1/2)))*(lp/c)*(M/mP):

Mm:=M*(i+1)/i:

simplify(4*Pi^2*R^3/(Mm*T^2));

Output: lp*c^2/mP

8-body circular orbit plus 1-body with opposing orbitals 1:2

Orbital alignment

Orbital trajectory is a measure of alignment of the orbitals. In the above examples, all orbitals rotate in the same direction = aligned. If all orbitals are unaligned the object will appear to 'fall' = straight line orbit.

In this example, for comparison, onto an 8-body orbit (blue circle orbiting the center mass green circle), is imposed a single point (yellow dot) with a ratio of 1 orbital (anti-clockwise around the center mass) to 2 orbitals (clockwise around the center mass) giving an elliptical orbit.

The change in orbit velocity (acceleration towards the center and deceleration from the center) derives automatically from the change in the orbital radius (there is no barycenter).

An orbital drift (as determined where the blue and yellow meet) naturally occurs; the eccentricity (shape) of the orbit a function of center mass and the ratio of alignment of the orbitals. A near straight line orbit will have a greater drift and a greater eccentricity than a near circular orbit. The elliptical orbit has a longer period than the circular orbit (which has a 360 degree orbit, the sidereal period). The additional period is known as the anomalistic period and includes the precession angle (360 + precession angle). Note: in these simulations there are only 2 orbital types; clock-wise and anti-clockwise ... in a real world orbit there will be a mixture.

Precession

Precession is a change in the orientation of the rotational axis of a rotating body. The first of three tests to establish observational evidence for the theory of general relativity, as proposed by Albert Einstein in 1915, concerned the "anomalous" precession of the perihelion of Mercury. This precession is not predicted by Newtonian gravity.

The formula for precession uses the semi-major axis a (the maximum distance between center of mass) and the semi-minor axis b (the minimum distance between center of mass).

\[ e={\sqrt {1-{\frac {b^{2}}{a^{2}}}}}\]

\[ \theta ={\frac {6\pi GM}{a(1-e^{2})c^{2}}}\]

Where e is the eccentricity of the orbit and θ is the precession angle

The frequency of the center mass Schwarzschild radius =

\[ i2l_{p}\]

, where i is the number of Planck mass points in the center mass and lp is Planck length; a and b become

\[ a=r_{a}i2l_{p}\]

\[ b=r_{b}i2l_{p}\]

The formula for

\[ r_{a}\]

\[ r_{a}=2\alpha 2{\frac {(k_{r}i+1)^{2}}{i^{2}}}\]

For example, the precession angle for Mercury

\[ \theta ={\frac {6\pi GM}{a(1-e^{2})c^{2}}}={\frac {6\pi \lambda _{sun}a}{2b^{2}}}\]

= 0.501866 x 10-6 radians

The Schwarzschild radius of the sun

\[ \lambda _{sun}=i2l_{p}\]

= 2953.25m

The eccentricity of Mercury

\[ e={\sqrt {1-{\frac {b^{2}}{a^{2}}}}}\]

= 0.203225 (where a = 57909050km and b = 56671523km)

In this simulation the ratio of anti-clockwise:clockwise orbitals = 108:1 with orbiting mass = 1 mass unit (kr = 12) [9].

Center mass	angle θ	θ*i	eccentricity	ra, rb
24	0.001175503	0.028212072	0.194749592	79481.8311615, 77959.9920879
28	0.001009240	0.028022688	0.195433743	79403.2724007, 77872.1317383
32	0.000884077	0.028290464	0.197440737	79344.3788203, 77782.4708225
36	0.000786489	0.028313604	0.197813449	79298.5878077, 77731.6227135
40	0.000708274	0.028330960	0.198373657	79261.9645140, 77686.7492830
44	0.000644252	0.028347088	0.199144931	79232.0062928, 77644.9930740
48	0.000590779	0.028357392	0.200476249	79207.0454340, 77599.0285567
52	0.000545493	0.028365636	0.200748008	79185.9277789, 77573.9329729
...
∞	0.000005888		0.205660603

Extrapolation of simulated elliptical data using the rotating orbital gravity simulator

At a low mass ratio the mass influences the eccentricity, this influence reduces as mass increases and so the ratio 108:1 was chosen because extrapolating to infinity (the sun:mercury mass ratio = 6025000:1) gives an eccentricity e = 0.20566 close to that of the Mercury orbit e = 0.20563. Likewise the extrapolated precession angle = 0.000005888 is only slightly greater than the Mercury orbit angle θ = 0.000005019.

Due to computational limitations, only a short radius ra and low central masses (M 24...52) were simulated. A longer radius and larger central mass values are required for greater precision. Extrapolating over this scale is imprecise however the it does appear that the correlation is within 1-2 orders of magnitude.

For the simulation program code [10] and the extrapolation code used [11].

Frame dragging can also impact the results as the central mass is still of itself a gravitational orbit (the center points also orbit each other), and so the center mass rotates at a relatively high velocity when compared with the orbiting point. Lense-Thirring in dimensionless form;

\[ T=T_{d}{\frac {m_{P}c}{Ml_{p}}}\]

\[ R=R_{d}{\frac {Ml_{p}}{m_{P}}}\]

\[ J=J_{d}{\frac {MR^{2}}{T}}\]

\[ \theta _{f}={\frac {2GJT}{c^{2}r^{2}}}=2{\frac {J_{d}}{R_{d}}}\]

Gravitational coupling constant

In the above, the points were assigned a mass as a theoretical unit of Planck mass. Conventionally, the Gravitational coupling constant αG characterizes the gravitational attraction between a given pair of elementary particles in terms of a particle (i.e.: electron) mass to Planck mass ratio;

\[ \alpha _{G}={\frac {Gm_{e}^{2}}{\hbar c}}=({\frac {m_{e}}{m_{P}}})({\frac {m_{e}}{m_{P}}})=1.75...x10^{-45}\]

For the purposes of this simulation, particles are treated as an oscillation between an electric wave-state (duration particle frequency) and a mass point-state (duration 1 unit of Planck time). This inverse αG then represents the probability that any 2 electrons will be in the mass point-state at any unit of Planck time (wave-mass oscillation at the Planck scale [12]).

\[ \alpha _{G}}^{-1}={\frac {m_{P}^{2}}{m_{e}^{2}}}=0.57...x10^{45\]

As mass is not treated as a constant property of the particle, measured particle mass becomes the averaged frequency of discrete point mass at the Planck level. If 2 dice are thrown simultaneously and a win is 2 'sixes', then approximately every (1/6)x(1/6) = (1/36) = 36 throws (frequency) of the dice will result in a win. Likewise, the inverse of αG is the frequency of occurrence of the mass point-state between the 2 electrons. As 1 second requires 1042 units of Planck time (

\[ t_{p}=10^{-42}s\]

), this occurs about once every 3 minutes.

\[ \frac {{\alpha _{G}}^{-1}}{t_{p}}\]

Gravity now has a similar magnitude to the strong force (at this, the Planck level), albeit this interaction occurs seldom (only once every 3 minutes between 2 electrons), and so when averaged over time (the macro level), gravity appears weak.

If particles oscillate between an electric wave state to Planck mass (for 1 unit of Planck-time) point-state, then at any discrete unit of Planck time, a number of particles will simultaneously be in the mass point-state. If an assigned point contains only electrons, and as the frequency of the electron = fe, then the point will require 1023 electrons so that, on average for each unit of Planck time there will be 1 electron in the mass point state, and so the point will have a mass equal to Planck mass (i.e.: experience continuous gravity at every unit of Planck time).

\[ f_{e}={\frac {m_{P}}{m_{e}}}=10^{23}\]

For example a 1kg satellite orbits the earth, for any given unit of Planck time, satellite (B) will have

\[ 1kg/m_{P}=45940509\]

particles in the point-state. The earth (A) will have

\[ 5.9738\;x10^{24}kg/m_{P}=0.274\;x10^{33}\]

particles in the point-state, and so the earth-satellite coupling constant becomes the number of rotating orbital pairs (at unit of Planck time) between earth and the satellite;

\[ N_{orbitals}=({\frac {m_{A}}{m_{P}}})({\frac {m_{B}}{m_{P}}})=0.1261\;x10^{41}\]

Earthe parameters:

\[ i={\frac {M_{earth}}{m_{P}}}=0.27444\;x10^{33}\]

(earth as the center mass)

\[ i2l_{p}=0.00887\]

(earth Schwarzschild radius)

\[ s={\frac {1kg}{m_{P}}}=45940509\]

(1kg orbiting satellite)

\[ N_{orbitals}=i*s=0.1261\;x10^{41}\]

The energy required to lift a 1 kg satellite into geosynchronous orbit is the difference between the energy of each of the 2 orbits (geosynchronous and earth).

\[ E_{orbital}={\frac {hc}{2\pi r_{6371}}}-{\frac {hc}{2\pi r_{42164}}}=0.412x10^{-32}J\]

(energy per orbital)

\[ E_{total}=E_{orbital}N_{orbitals}=53MJ/kg\]

The orbital angular momentum of the planets derived from the angular momentum of the respective orbital pairs.

\[ N_{orbitals}={\frac {M_{sun}}{m_{P}}}{\frac {M_{planet}}{m_{P}}}\]

\[ n_{g}={\sqrt {\frac {R_{radius}m_{P}}{2\alpha l_{p}M_{sun}}}}\]

\[ L_{oam}=2\pi {\frac {Mr^{2}}{T}}=N_{orbitals}n_{g}{\frac {h}{2\pi }}{\sqrt {2\alpha _{inv}}},\;{\frac {kgm^{2}}{s}}\]

The orbital angular momentum of the planets;

mercury = .9153 x1039  
venus    = .1844 x1041  
earth    = .2662 x1041 
mars     = .3530 x1040 
jupiter   = .1929 x1044   
pluto   = .365 x1039

Comparison of Newtonian and Orbital model simulations

Orbital vs. Newton

A 3-body orbit [13][14] is compared with the equivalent orbit using Newtonian dynamics. The start positions are the same

r0=2*α; x1=3490.3069; y1=0; x2=cos(pi*2/3)*r0; y2=sin(pi*2/3)*r0; x3=cos(pi*2/3)*r0; y3=sin(pi*2/3)*r0

The orbiting point was used to determine the optimal G for the Newtonian orbit (G = 0.4956).

Period of orbit (

\[ k_{r}\]

= 2.19006)

\[ t_{calc}\]

= 1122034

\[ t_{orbital}\]

= 1121397

\[ t_{newton}\]

= 1125633

32 mass points (496 orbitals) begin with random co-ordinates, after 232 steps they have clumped to form 1 large mass and 2 orbiting masses.

Freely moving points

The simulation calculates each point as if freely moving in space, and so is useful with 'dust' clouds where the freedom of movement is not restricted.

In this animation, 32 mass points begin with random co-ordinates (the only input parameter here are the start (x, y) coordinates of each point). We then fast-forward 232 steps to see that the points have now clumped to form 1 larger mass and 2 orbiting masses. The larger center mass is then zoomed in on to show the component points are still orbiting each other, there are still 32 freely orbiting points, only the proximity between them has changed, they have formed planets.

Illustration of B's cylindrical orbit relative to A's time-line axis

Hyper-sphere orbit

Main page: User:Platos_Cave_(physics)/Simulation_Hypothesis/Relativity

Each point moves 1 unit of (Planck) length per 1 unit of (Planck) time in x, y, z (hyper-sphere) co-ordinates, the simulation 4-axis hyper-sphere universe expanding in uniform (Planck) steps (the simulation clock-rate) as the origin of the speed of light, and so (hyper-sphere) time and velocity are constants. Particles are pulled along by this expansion, the expansion as the origin of motion, and so all objects, including orbiting objects, travel at, and only at, the speed of light in these hyper-sphere co-ordinates [15]. Time becomes time-line.

While B (satellite) has a circular orbit period on a 2-axis plane (the horizontal axis representing 3-D space) around A (planet), it also follows a cylindrical orbit (from B1 to B11) around the A time-line (vertical expansion) axis (td) in hyper-sphere co-ordinates. A is moving with the universe expansion (along the time-line axis) at (v = c), but is stationary in 3-D space (v = 0). B is orbiting A at (v = c), but the time-line axis motion is equivalent (and so `invisible') to both A and B, as a result the orbital period and velocity measures will be defined in terms of 3-D space co-ordinates by observers on A and B.

Atomic orbitals

fig 5. H atom orbital transitions from n1-n2, n2-n3, n3-n1 via 2 photon capture, photons expand/contract the orbital radius. The spiral pattern emerges because the electron is continuously pulled in an anti-clockwise direction by the rotating orbital.

Gravitational orbit

In the context of the gravitational orbital pair model, the atomic orbital is treated as a 2-body orbit (the electron as a point orbiting an n-body central mass 'nucleus'). To simulate electron transition between orbitals, at each step a unit of length is added. As the electron still continues its orbit, instead of a circular path, it traces a spiral path outwards.

Notes:

1. Seen from the wave-point oscillation cycle model, the electron is predominately in the electric wave-state (which is a locally undefined state), the simulation program can only calculate the mass point-states, and so we are mapping only the gravitational orbital component of the electron orbit. For the gravitational orbit, the orbit co-ordinates were updated per unit of Planck time (with orbitals rotating 1 unit of Planck length). In the atomic orbital we have only 1 point (the electron), and it occurs as a point only after every wavelength cycle, and so we are updating our orbit only after every wavelength cycle. For example we find that at the lowest (n = 1) orbit, there are about 471964 wave-point oscillations, and so we will map a polygon with 471964 sides (the wave state cannot be mapped). To compensate, the orbital rotation angle is modified such that 471964 steps are required to map 1 complete orbit ;

\[ \beta _{orbital}={\frac {1}{r_{ij}r_{orbital}{\sqrt {r_{orbital}}}}}\]

\[ r_{ij}={\sqrt {2\alpha }}\]

1. During transition, at each step 1 unit of the photon is absorbed by the orbital radius, this continues until the photon is completely absorbed, the orbital radius extending proportionately. Photon absorption is not instantaneous, but instead occurs over time according to the photon wavelength. The number of steps required for this absorption then gives us the transition frequency.

Notably, at integer (Bohr) radius (4r, 9r, 16r...) the spiral angle is a function of pi (2pi, (8/3)pi, 3pi ...), and as this is a linear extrapolation (the orbital radius grows in steps), n level orbits can be interpreted as specific orbits (which are inherently stable) occurring along the semi-continuous transition orbital path traced by the electron. The electron does not jump between n-levels, but the interim state (the transition phase) between these quantized n-levels is an orbital-photon hybrid state and thus difficult to detect. We may then have the equivalent of a non-integer n, but as it is interchangeable with spiral angle and radius, we need only 1 parameter to calculate the other 2, and as transition frequency can also be determined in terms of angle and radius, we have a simple geometrical model that provides a framework for the transition process yet uses only pi and alpha as inputs.

2-Photon orbital model

The Lyman series energy formula can be decomposed:

\[ \frac {1}{\lambda }}=R_{\infty }\left(1-{\frac {1}{n^{2}}}\right)=R_{\infty }-{\frac {R_{\infty }}{n^{2}}\]

Mathematically (if not physically) we can divide into 2 waves

\[ \text{Photon}}_{n1}=R_{\infty\]

\[ {\text{Photon}}_{nfinal}=(-{\frac {R_{\infty }}{n^{2}}})\]

\[ {\text{Photon}}_{\text{total}}={\text{Photon}}_{n1}+(-{\text{Photon}}_{n{\text{final}}})\]

This (mathematical) approach permits us to divide the transition into two distinct geometric processes taking place between the incoming photon and the orbital radius, with the electron taking a relatively passive role. Rather than 2 actual distinct photons, we may presume two geometric phases of a single photon absorption, nevertheless the 2-photon image is easier to conceptualize. Note these processes are not instantaneous but rather occur over time in discrete steps;

Process 1 (Cancellation): A photon with energy corresponding to the n=1 orbital frequency cancels the existing orbital structure.

\[ {\text{Photon}}_{n1}+{\text{Orbital}}_{n1}==zero\]

Process 2 (Creation): A (-) photon with energy corresponding to the

\[ n_{\text{final}}\]

orbital creates the new orbital structure.

\[ \text{Photon}}_{nfinal}=={\text{Orbital}}_{nfinal\]

During transition, the system exists in a photon-orbital hybrid state. This is not a

quantum superposition but a geometric intermediate configuration where:

The incoming photon’s momentum is being transferred incrementally

The orbital structure is simultaneously being dismantled and reconstructed

The electron mediates the momentum transfer through its position

Each steps transfers a quantum of momentum

The gravitational orbit returns results analogous to the Bohr model and the 2-photon approach complements the (electric wave-state) Schrodinger equation [16].

Theory

Hyperbolic spiral

Hyperbolic spiral

A hyperbolic spiral is a type of spiral with a pitch angle that increases with distance from its center. As this curve widens (radius r increases), it approaches an asymptotic line (the y-axis) with the limit set by a scaling factor a (as r approaches infinity, the y axis approaches a).

For the particular spiral that the electron transition path maps, periodically the spiral angles converge to give integer radius, the general form for this type of spiral (beginning at the outer limit ranging inwards);

\[ x=a^{2}{\frac {cos(\varphi )}{\varphi ^{2}}},\;y=a^{2}{\frac {sin(\varphi )}{\varphi ^{2}}},\;0<\varphi <4\pi\]

radius =

\[ {\sqrt {(}}x^{2}+y^{2})r\]

\[ \varphi =(2)\pi ,\;4r\]

(360°)

\[ \varphi =(4/3)\pi ,\;9r\]

(240°)

\[ \varphi =(1)\pi ,\;16r\]

(180°)

\[ \varphi =(4/5)\pi ,\;25r\]

(144°)

\[ \varphi =(2/3)\pi ,\;36r\]

(120°)

Electron at different n level orbitals

Principal quantum number n

The H atom has 1 proton and 1 electron orbiting the proton, in the Bohr model (which approximates a gravitational orbit), the electron can be found at select radius (the Bohr radius) from the proton (nucleus), these radius represent the permitted energy levels (orbital regions) at which the electron may orbit the proton. Electron transition (to a higher energy level) occurs when an incoming photon provides the required energy (momentum). Conversely emission of a photon will result in electron transition to a lower energy level.

The principal quantum number n denotes the energy level for each orbital. As n increases, the electron is at a higher energy level and is therefore less tightly bound to the nucleus (as n increases, the electron orbit is further from the nucleus). Each shell can accommodate up to n2 (1, 4, 9, 16 ... ) electrons. Accounting for two states of spin this becomes 2n2 electrons. As these energy levels are fixed according to this integer n, the orbitals may be said to be quantized.

(Bohr) orbital

Setting

\[ \alpha _{inv}={\frac {1}{\alpha }}\]

= 137.035999177⁠ (CODATA 2022)

The Bohr radius has 2 components, the dimensionless (the fine structure constant alpha) and the dimensioned (electron wavelength);

\[ a_{0}=\alpha _{inv}\times \lambda _{e}\]

The orbital radius of this model also includes the proton wavelength and so is slightly greater than 2x the Bohr radius

\[ \lambda _{0}=\lambda _{p}+\lambda _{e}\]

\[ r_{0}=2\alpha _{inv}\times \lambda _{0}\]

To reduce simulation computing time, only the alpha component is described here, the dimensioned components added later.

\[ r_{orbital}=2\alpha _{inv}n^{2}\]

As a mass point, the electron orbits the proton at a fixed radius (

\[ r_{orbital}\]

) in a series of steps (the duration of each step corresponds to the wavelength component). The distance travelled per step (per wave-point oscillation) equates to the distance between mass point states and is the inverse of the radius.

electron (blue dot) moving 1 step anti-clockwise along the alpha orbital circumference

length =

\[ l_{step}={\frac {1}{r_{orbital}}}\]

Duration = 1 step per wavelength and so velocity

velocity =

\[ v_{orbital}=v_{step}={\frac {1}{2\alpha _{inv}n}}\]

Giving period of orbit

period =

\[ t_{orbital}={\frac {2\pi r_{orbital}}{v_{orbital}}}=2\pi 2\alpha _{inv}2\alpha _{inv}n^{3}\]

The (reference) orbital (n = 1)

\[ t_{n1}=2\pi 4\alpha _{inv}^{2}\]

= 471964.356...

The angle of rotation depends on the orbital radius

\[ \beta ={\frac {1}{r_{orbital}{\sqrt {r_{orbital}}}{\sqrt {2\alpha _{inv}}}}}\]

2-Photon orbitals

The electron can jump between n energy levels via the absorption or emission of a photon. In the 2-photon orbital model [17], the orbital (Bohr) radius is treated as a 'physical wave' (has physical properties) akin to the photon albeit an inverse such that

\[ orbital\;radius+photon=zero\]

(they cancel).

As such it is the orbital radius itself that absorbs or emits the photon during transition, in the process the orbital radius is extended or reduced (until the photon is completely absorbed/emitted). The electron has a `passive' role in the transition phase (mediating the exchange of momentun between the orbital and the photon). It is the rotation of the orbital radius that drives the electron motion, resulting in the electron orbit around the nucleus (orbital momentum comes from the orbital radius), and this rotation continues also during the transition phase resulting in the electron following a hyperbolic spiral path.

Spiral (mass point)

The photon is divided into 2 photons as per the Rydberg formula (denoted initial and final).

\[ \lambda _{photon}=(+\lambda _{i})-(+\lambda _{f})\]

The wavelength of the (

\[ \lambda _{i}\]

) photon corresponds to the wavelength of the orbital radius. The (+

\[ \lambda _{i}\]

) will then delete the orbital radius as described above (orbital + photon = zero), however the (-

\[ \lambda _{f}\]

), because of the Rydberg minus term, will be equivalent to the orbital radius, and so conversely will increase the orbital radius. And so for the duration of the (+

\[ \lambda _{i}\]

) photon wavelength, the orbital radius does not change as the 2 photons cancel each other;

\[ r_{orbital}=r_{orbital}+(\lambda _{i}-\lambda _{f})\]

However, the (

\[ \lambda _{f}\]

) has the longer wavelength, and so after the (

\[ \lambda _{i}\]

) photon has been absorbed, and for the remaining duration of this (

\[ \lambda _{f}\]

) photon wavelength, the orbital radius will be extended until the (

\[ \lambda _{f}\]

) is also absorbed (the transition phase). For example, the electron is at the n = 1 orbital. To jump from an initial

\[ n_{i}=1\]

orbital to a final

\[ n_{f}=2\]

orbital, first the (

\[ \lambda _{i}\]

) photon is absorbed by the orbital radius (

\[ \lambda _{i}+\lambda _{orbital}=zero\]

), but simultaneously the (-

\[ \lambda _{f}\]

) photon adds to the orbital radius, and so the electron follows a normal n = 1 orbit (the orbital phase), then the remaining (

\[ \lambda _{f}\]

) photon continues until it too is absorbed (the transition phase).

\[ \lambda _{i}=1\times t_{n1}\]

(initial orbital)

\[ \lambda _{f}=n^{2}\times t_{n1}\]

(final orbital)

After the (

\[ \lambda _{i}\]

) photon is absorbed, the (

\[ \lambda _{f}\]

) photon still has

\[ \lambda _{f}=(n_{f}^{2}-n_{i}^{2})t_{n1}=3t_{n1}\]

steps remaining until it too is absorbed.

orbital transition during orbital rotation

This process does not occur as a single `jump' between energy levels by the electron, but rather absorption/emission of the photon takes place in discrete steps, each step corresponds to a unit of

\[ r_{incr}\]

(both photon and orbital radius may be considered as constructs from multiple units of this geometry);

\[ r_{incr}=-{\frac {1}{2\pi 2\alpha _{inv}}}\]

At each step 1 unit of

\[ r_{incr}\]

is transferred from each photon to the orbital radius. During the orbital phase the

\[ r_{incr}\]

unit from the (

\[ \lambda _{i}\]

) photon is canceled by the (-)

\[ r_{incr}\]

unit from the (

\[ \lambda _{f}\]

) photon (

\[ r_{incr}\]

• (-)

\[ r_{incr}\]

= zero) and so the orbital radius does not change. After the (

\[ \lambda _{i}\]

) photon is fully absorbed, the orbital radius continues to absorb (-)

\[ r_{incr}\]

units from (what remains of) the (

\[ \lambda _{f}\]

) photon, and as the (

\[ \lambda _{f}\]

) photon is equivalent to the orbital radius, the orbital radius extends in these steps as a consequence. The number of steps

\[ N_{steps}\]

required determines (and so can be used to calculate) the transition frequency.

Furthermore at each step the orbital radius itself continues to rotate, the electron, being pulled along by this rotation according to angle β, thereby traces a spiral path as the orbital radius length changes.

\[ \beta ={\frac {1}{r_{orbital}{\sqrt {r_{orbital}}}r_{\alpha }}}\]

The (accumulated) spiral angle

\[ \Phi\]

at each

\[ n^{2}\]

n = 1 to 2:

\[ \Phi =2\pi \;(r=4\times \;r_{orbital})\]

n = 1 to 3:

\[ \Phi ={\frac {8}{3}}\pi \;(r=9\times \;r_{orbital})\]

n = 1 to 4:

\[ \Phi =3\pi \;(r=16\times \;r_{orbital})\]

n = 1 to 5:

\[ \Phi ={\frac {16}{5}}\pi \;(r=25\times \;r_{orbital})\]

n = 1 to 6:

\[ \Phi ={\frac {10}{3}}\pi \;(r=36\times \;r_{orbital})\]

n = 1 to 7:

\[ \Phi ={\frac {7}{4}}\pi \;(r=49\times \;r_{orbital})\]

n = 1 to 8:

\[ \Phi ={\frac {7}{2}}\pi \;(r=64\times \;r_{orbital})\]

In the mass state we can divide the photon into 2 mathematical structures, the transition frequency is defined as the inverse of one oscillation period at the Compton scale, multiplied by the geometric phase factor (including the dimensioned terms). This gives us the solution for the transition frequency and the Bohr radius with a pi-based spiral acting as the transition 'guard-rail'. We are using only alpha and pi, yet the transition frequency can be solved with high precision using the gravity simulator. This gives a geometrical derivation for the Bohr model, which in this context is the gravitational component of the electron orbit.

Photon-Orbital Hybrid (wave-state)

In this model, atomic orbitals are physical rotating structures rather than probability distributions. The orbital possesses:

Angular momentum: Quantized by discrete rotation steps

Energy: Stored in the rotating configuration

The orbital rotates in discrete steps (β radians per step) rather than continuously. During transition, the system exists in a photon-orbital hybrid state. This is not a quantum superposition but a geometric intermediate configuration where:

The incoming photon's momentum is being transferred incrementally

The orbital structure is simultaneously being dismantled and reconstructed

The electron mediates the momentum transfer through its position

Each step transfers a quantum of momentum

We then note.

Standard QM: Anti-realist---the wavefunction is a calculation tool, not a physical entity. Reality emerges only upon measurement.

2-photon model: Realist---orbitals are real rotating structures. The electron follows definite trajectories, even when unobserved.

The geometric model thus establishes a two-layer architecture for encoding quantum states:

Geometric Framework Layer: The hyperbolic spiral structure provides the spatial scaffold

\[ (r,\theta ,\phi )\]

coordinates with built-in radial quantization

\[ r\propto n^{2}\]

Photon Information Layer: The absorbed photon carries angular momentum quantum numbers

\[ (l,m_{l})\]

that modulate this scaffold

This separation offers that the Bohr model is not replaced by the Schrodinger equation but rather complements it (the mass-state provides the geometric structure; the photon-orbital hybrid provides the angular momentum content).

Simulation

Bohr radius during ionization, as the H atom electron reaches each n level, it completes 1 orbit (for illustration) then continues outward (actual velocity will become slower as radius increases according to angle β)

The gravity simulation ([18]) was setup with the electron as the orbiting point and 65 mass points to represent the nucleus (although the proton-electron mass ratio is 1836:1, this would be become a 1837-body set of orbiting points which is too computationally intensive).

The program was modified with the rotational angle β as

\[ \beta ={\frac {1}{r_{orbital}{\sqrt {r_{orbital}}}r_{\alpha }}}\]

To simulate the transition phase, at each step 1 unit of

\[ r_{incr}={\frac {1}{2\pi 2\alpha _{inv}}}\]

was added to the proton-electron orbital radius.

To convert to frequency in Hz, the dimension components were added

\[ f_{H}={\frac {2c}{\lambda _{e}+\lambda _{p}}}\]

= 0.155184298 1022s-1

For n = 1 to n = n2

\[ f_{n}=2\pi f_{H}\;{\frac {n^{2}-1}{N_{steps}}}\]

Results

\(\displaystyle r/r_{orbital}\)	\(\displaystyle r/r_{orbital}\)	N-steps	\(\displaystyle \Phi\) (deg)	frequency Hz
4.000000115	2.000004018	1887860.649	0.000017120	2466034304131826.5
8.999994875	4.000003286	4247681.247	120.000001964	2922708926063928.0
15.999987119	6.000002004	7551428.532	180.000002514	3082545855782738.5

Experimental values for H(1s-ns) transitions.

H(1s-2s) = 2466 061 413 187.035 kHz [19]

H(1s-3s) = 2922 743 278 665.79 kHz [20]

H(1s-4s) = 3082 581 563 822.63 kHz [21]

Although the simulation was a standard gravitational orbit (charge-less) and without a relativistic correction, the results were quite close;

n = 1 to n = 2: margin = 100 * (experimental - simulated)/experimental = 0.001099%

n = 1 to n = 3: margin = 0.001175%

n = 1 to n = 4: margin = 0.001158%

Rydberg atom

Here a Rydberg atom is defined as an (idealized) atom with a nucleus of point size and an infinite mass and so the barycenter = atom center at co-ordinates (0, 0).

We can simulate this as a simple transition beginning at the initial (ni = 1) orbital.

\[ \varphi =0\]

(start angle)

\[ r_{orbital}=r_{0}=2\alpha\]

(nb. Bohr radius =

\[ \alpha \lambda _{e}\]

= α*electron wavelength)

\[ x=r_{orbital},\;y=0\]

(start co-ordinates)

For each step during transition, setting t = step number (FOR t = 1 TO ...), we can calculate the radius r and

\[ n_{f}\]

.

\[ r=r_{orbital}+{\frac {t}{2\pi 2\alpha }}\]

(number of increments t of

\[ r_{incr}\]

)

\[ \beta ={\frac {1}{r_{orbital}{\sqrt {r_{orbital}}}{\sqrt {2\alpha }}}}\]

\[ \varphi =\varphi +\beta\]

\[ n_{f}^{2}=1+{\frac {t}{2\pi 4\alpha ^{2}}}\]

(

\[ n_{f}^{2}\]

as a function of t)

The spiral angle and

\[ n_{f}^{2}\]

are interchangeable

\[ \varphi =4\pi (1-{\frac {1}{{n_{f}}^{2}}})\]

We can then re-write (

\[ n_{f}\]

is only an integer at prescribed spiral angles);

\[ \beta ={\frac {1}{{r_{orbital}}^{2}n_{f}^{3}}}\]

n orbital transition periods are multiples of the n = 1 orbital. If we include a relativistic term;

\[ t_{n1}=2\pi 4\alpha _{inv}^{2}{\sqrt {(1-{\frac {1}{4\alpha _{inv}^{2}}})}}\]

= 471961.21478

Then we can compare a theoretical Rydberg (also without including charge) and find a further improvement. For ionization, the second photon is of such long wavelength that its momentum contribution is negligible, and so the ionization transition frequency should correspond closely to the n = 1 orbital.

H(1s-∞s) = 3288 086 857 127.60 kHz [22] (n = ∞)

This would give us a H(n1) = 471959.2427762

n = 1: margin = 100 * (H(n1) - t_{n1})/H(n1) = 0.001099%

n = 1 to n = 2: margin = 100 * (experimental - 4 t_{n1})/experimental = 0.000267%

n = 1 to n = 3: margin = 100 * (experimental - 9 t_{n1})/experimental = 0.000398%

n = 1 to n = 4: margin = 100 * (experimental - 16 t_{n1})/experimental = 0.000422%

If we use the non-relativistic

\[ t_{n1}\]

, then our results are remarkably close to the results from the gravity simulation suggesting that central mass size does not have the major role in determining the transition frequency.

n = 1 to n = 2: margin = 0.000932%

n = 1 to n = 3: margin = 0.001063%

n = 1 to n = 4: margin = 0.001088%

Barycenter correction

The experimental hydrogen transition frequencies deviate slightly from the ideal Rydberg formula. When normalized, these deviations exhibit a characteristic pattern: a minimum near n = 2, followed by a rising trend toward the ionization limit as the radius increases.

H(n1) = 471959.242776

H(n2) = 471959.956565

H(n3) = 471959.338749

H(n4) = 471959.222066

The gravity simulation results showed a similar trend suggesting in both cases a barycenter 'wobble'.

sim(n1) = 471964.35638

sim(n2) = 471965.14481

sim(n3) = 471964.88601

sim(n4) = 471964.68921

Therefore, given the possibilities of this gravity simulator as a research tool, the next step is to divide the proton into 3 parts resembling the 3 quarks, and then experiment with different configurations. For example, we could form a U quark cluster and a D quark cluster and then test different arrangements of the clusters (and also different mass ratios) to determine which, if any, reduces the margin. This could give potential insights into the protons internal structure. We can also give the electron further mass points to determine if it also becomes constrained and relaxed.

Geometry coded universe

Modelling a Planck scale simulation universe using geometrical forms. For AI analysis, the following links have been compiled into a single file [23].

• User:Platos_Cave_(physics)/Simulation_Hypothesis/Electron_(mathematical): Mathematical electron from Planck units
• User:Platos_Cave_(physics)/Simulation_Hypothesis/Planck_units_(geometrical): Planck units MLTPA as geometrical objects
• User:Platos_Cave_(physics)/Simulation_Hypothesis/Physical_constant_(anomaly): Anomalies within the physical constants
• User:Platos_Cave_(physics)/Simulation_Hypothesis/Relativity: Relativity as a translation between 2 co-ordinate systems
• User:Platos_Cave_(physics)/Simulation_Hypothesis/Planck_unit_scaffolding: CMB and a Planck unit universe scaffolding
• User:Platos_Cave_(physics)/Simulation_Hypothesis/Sqrt_Planck_momentum: Link between charge and mass
• User:Platos_Cave_(physics)/Simulation_Hypothesis/God_(programmer): Introduction to a Planck scale Programmer God Simulation Hypothesis model

External links

• Planck scale modelling using geometrical objects
• Our Mathematical Universe: My Quest for the Ultimate Nature of Reality -Max Tegmark (Book)
• Dirac-Kerr-Newman black-hole electron -Alexander Burinskii (article)
• Pythagoras "all is number" - Stanford University
• Mathematical universe hypothesis
• Philosophy of mathematics
• Philosophy of physics
• Platonism

References

1. ↑ Macleod, Malcolm J.; "3. Gravitational orbits from n-body rotating particle-particle orbital pairs". SSRN. Dec 2021. doi:10.2139/ssrn.3444571.
2. ↑ Macleod, Malcolm J.; "4. Geometrical origins of quantization in H atom electron transitions". SSRN. Dec 2022. doi:10.2139/ssrn.3703266.
3. ↑ https://codingthecosmos.com/orbitals/maple-code-Kepler.html maple code
4. ↑ https://codingthecosmos.com/orbitals/ Orbital model simulation source codes
5. ↑ Macleod, Malcolm; "1. Planck unit scaffolding correlates with the Cosmic Microwave Background". SSRN. 26 March 2014. doi:10.2139/ssrn.3333513.
6. ↑ Macleod, Malcolm; "2. Relativity as the mathematics of perspective in a hyper-sphere universe". SSRN. 26 March 2014. doi:10.2139/ssrn.3334282.
7. ↑ https://codingthecosmos.com/orbitals/ Gravitational-orbital-simulation-2body.c
8. ↑ https://codingthecosmos.com/orbitals/maple-code-Kepler.html maple code
9. ↑ https://codingthecosmos.com/orbitals/ Orbital model source code repository
10. ↑ https://codingthecosmos.com/orbitals/ Gravitational-orbital-elliptical-orbits.cpp
11. ↑ https://codingthecosmos.com/orbitals/ Gravitational-orbital-ellipse-extrapolate.py
12. ↑ Macleod, M.J. "Programming Planck units from a mathematical electron; a Simulation Hypothesis". Eur. Phys. J. Plus 113: 278. 22 March 2018. doi:10.1140/epjp/i2018-12094-x.
13. ↑ https://codingthecosmos.com/orbitals/ Newton-vs-Orbital_Newton.py
14. ↑ https://codingthecosmos.com/orbitals/ Newton-vs-Orbital_Orbital.py
15. ↑ Macleod, Malcolm; "2. Relativity as the mathematics of perspective in a hyper-sphere universe". SSRN. 2014. doi:10.2139/ssrn.3334282.
16. ↑ Macleod, Malcolm J.; "4. Geometrical origins of quantization in H atom electron transitions". SSRN. Dec 2022. doi:10.2139/ssrn.3703266.
17. ↑ Macleod, Malcolm J.; "4. Geometrical origins of quantization in H atom electron transitions". SSRN. Dec 2022. doi:10.2139/ssrn.3703266.
18. ↑ https://codingthecosmos.com/orbitals/ source code
19. ↑ http://www2.mpq.mpg.de/~haensch/pdf/Improved%20Measurement%20of%20the%20Hydrogen%201S-2S%20Transition%20Frequency.pdf
20. ↑ https://pubmed.ncbi.nlm.nih.gov/33243883/
21. ↑ https://codata.org/
22. ↑ https://codata.org/ (109678.77174307cm-1)
23. ↑ https://codingthecosmos.com/Programmer-God_Simulation-Hypothesis_Complete-Model-2025.pdf Compilation of journal articles 1-7