Quantum gravity (Planck)

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 and atomic orbitals 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 . 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 .

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 m_P (for example, a 1kg satellite would be divided into 1kg/m_P = 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 electron is assigned as a single mass point, the nucleus as multiple points clustered together (a 2-body orbit), and an incoming 'photon' is added to the 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. In this context the Bohr model is a gravitational model, and thus is not superseded by the Schrodinger wave equation, but rather is complementary to this equation (they each measure different aspects of the transition).

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 gravity-mass Bohr model can then be seen as complementary to the electric-wave Schrödinger equation.

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 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 m_P, Planck length l_p, Planck time t_p), 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 = l_p/t_p) 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.

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).

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

$\beta _{atomic}={\frac {1}{{\sqrt {2\alpha }}r_{orbital}{\sqrt {r_{orbital}}}}}$

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 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 = r_s; 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. k_r 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 $n_{g}=j_{max}/j$.

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

5. r_α = a radius constant, here r_α = sqrt(2α) = 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.

$r_{\alpha }={\sqrt {2\alpha }}$

$r_{orbit}={r_{\alpha }}^{2}\;*\;r_{wavelength}$

6. 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)

$r_{ij}={\sqrt {2\alpha }}$ (for each atomic orbital in the simulation)

Orbital formulas (2-D plane)

$j=i+1$

$r_{orbit}=2\alpha 2{\frac {(k_{r}i+1)^{2}}{i^{2}}}$, orbital radius (center mass to point)

$r_{ij}={\sqrt {\frac {i}{j}}}$ (averaged for each orbit)

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

i = 7, j = 8

k_r = 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

n_g = (k_r 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 k_r. 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 T_orbit as our reference (reversing the orbit period equation).

$T_{o}=T_{orbit}{\frac {m_{P}}{M}}{\frac {c}{l_{p}}}=16\pi {\alpha }^{3/2}{\frac {(k_{r}i+1)^{3}}{i^{5/2}j^{1/2}}}$ (dimensionless orbital period)

$k_{r}={\frac {1}{i}}{({\frac {T_{o}i^{5/2}j^{1/2}}{16\pi {\alpha }^{3/2}}})}^{(1/3)}-{\frac {1}{i}}$ = 12581.4468

Dimensionless solutions

$r_{orbit}=2\alpha 2{\frac {(k_{r}i+1)^{2}}{i^{2}}}$ = 86767420100

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

$k_{r}={\frac {c}{2{\sqrt {\alpha }}}}{({\frac {T_{orbit}}{2\pi \mu _{earth}}})}^{1/3}{\frac {(i+1)}{i}}^{1/6}-{\frac {1}{i}}$ = 12580.3462

$t_{orbit}$ = 0.15956776936 x 10¹⁸

$r_{orbit}$ = 86752239934

Kepler's formula = G

Kepler's formula reduces to G

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

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

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

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

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 Schwarschild radius = $i2l_{p}$, where i is the number of Planck mass points in the center mass and l_p 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 (k_r = 12) .

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

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 r_a and low central masses (M 24...52) were simulated. A longer radius and larger central mass values are required for greater precision. For the simulation program code and the extrapolation code used .

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 α_G 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 α_G 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 ).

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

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 α_G 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 = f_e, 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.

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

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

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

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 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 resource: Relativity (Planck)

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 . 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 (t_d) 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.

The atomic orbital is treated as a single particle to particle orbital pair. 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 can calculate only the mass point-states and so we are mapping only the gravitational orbital component of the electron orbit. For example, in 1 orbit cycle at the lowest energy level in the H atom, the electron will oscillate between wave-state to point-state approximately $2\pi 4\alpha ^{2}$ = 471964 times, and so a plot of the electron as a circular obit around the nucleus will be the sum of 471964 'dots' .

This permits to map the electron orbit around the nucleus as a simple 2-body gravitational orbit with the electron as the orbiting point. Although this (gravitational orbit) approach can only map the electron-as-mass point-state (and so offers no direct information regarding the electron as a wave), during electron transition between n-shell orbitals we find the electron follows a hyperbolic spiral which can be used to derive the transition frequencies (the number of discrete steps required during transition), this is significant because periodically the spiral angle components converge giving integer radius values (360°=4r, 360+120°=9r, 360+180°=16r, 360+216°=25r ... 720°=∞r). As these spiral angles are linked directly to pi via this spiral geometry, we may ask if quantization of the atom has a geometrical origin.

In this context the Bohr model measures the gravitational component (mass point-state) of the atomic orbital and thus complements the (electric wave-state) Schrodinger equation.

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

The basic orbital radius has 2 components, dimensionless (the fine structure constant alpha) and dimensioned (electron + proton wavelength);

wavelength = $\lambda _{H}=\lambda _{p}+\lambda _{e}$

radius = $r_{orbital}=2\alpha n^{2}(\lambda _{H})$

As a mass point, the electron orbits the proton at a fixed radius (the Bohr radius) 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 = $l_{orbital}={\frac {1}{r_{orbital}}}$

Duration = 1 step per wavelength and so velocity

velocity = $v_{orbital}={\frac {1}{2\alpha n}}$

Giving period of orbit

period = $t_{orbital}={\frac {2\pi r_{orbital}}{v_{orbital}}}=2\pi 2\alpha 2\alpha n^{3}\lambda _{H}$

As we are not mapping the wavelength component, a base (reference) orbital (n=1)

$t_{n1}=2\pi 4\alpha ^{2}$ = 471964.356...

The angle of rotation depends on the orbital radius

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

Photon orbital model

The electron can jump between n energy levels via the absorption or emission of a photon. In the photon-orbital model, the orbital (Bohr) radius is treated as a 'physical wave' (has physical properties) akin to the photon albeit of inverse or reverse phase 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. It is the rotation of the orbital radius that pulls the electron, 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.

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

$\lambda _{photon}=R.({\frac {1}{n_{i}^{2}}}-{\frac {1}{n_{f}^{2}}})={\frac {R}{n_{i}^{2}}}-{\frac {R}{n_{f}^{2}}}$

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

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 have the same phase as 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).

$t_{n1}\sim 2\pi 4\alpha ^{2}$

$\lambda _{i}=1t_{n1}$

$\lambda _{f}=4t_{n1}$ (n = 2)

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);

$r_{incr}=-{\frac {1}{2\pi 2\alpha r_{wavelength}}}$

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 the orbital radius does not change. After the ($\lambda _{i}$) photon is fully absorbed, the orbital radius continues to absorb (-)$r_{incr}$ units from the ($\lambda _{f}$) photon, and as the ($\lambda _{f}$) photon has the same phase as the orbital radius, the orbital radius extends in these steps as a consequence. The number of steps required correlates to (and 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.

$\beta ={\frac {1}{r_{orbital}{\sqrt {r_{orbital}}}r_{\alpha }}}$

AI analysis

AI was used to condense the derivation (summarize the method and give conclusions). This presumes familiarity with the Bohr model.

Chat GPT chatgpt.com/share/67ce62fc-8bf8-8012-8622-37a7a4fae6d6

Conclusion : the stability of the n orbital shells (with $m=n^{2}$) arises because at these levels the spiral’s radial and angular increments are in perfect resonance. The electron’s semi-classical motion then traces a repeating, closed path. This geometrical resonance, which directly involves π and the circular nature of the motion, naturally leads to the quantized orbital shells without invoking the usual quantum-mechanical postulates.

Deepseek deepseek-spiral.pdf

Conclusion : The levels $m=n^{2}$ are stable because the spiral’s geometry enforces a self-consistent synchronization between radial growth and angular displacement. This resonance prevents dispersion, creating discrete orbital shells without invoking quantum mechanics. The quantization of m into squares is a geometric constraint ensuring constructive interference in the spiral’s path, much like how integer multiples of wavelengths stabilize standing waves. The stability of $m=n^{2}$ levels arises from geometric resonance in the spiral, where radial expansion and angular progression harmonize to form closed or self-reinforcing paths. This ensures discrete, stable orbital shells.

Qwen chat.qwen.ai/s/9fe132a6-91d7-4ec6-8c82-ebf2e1b2b422

Conclusion : The stability of $m=n^{2}$ orbits is a geometric necessity . The spiral’s radial and angular evolution align to create resonant, non-decaying paths at these specific radii. These orbits are "stable" because they satisfy the minimal condition for periodicity and constructive interference, ensuring the electron’s motion remains bounded without invoking quantum theory.

Claude claude.ai/share/355e21e4-d623-4810-962a-fb1892c2ef3f

Conclusion : The stability of orbits at m = n² emerges from the geometric properties of the spiral itself, specifically from resonances in the relationship between angular rotation and radial expansion. This geometric perspective provides a semi-classical explanation for why certain orbital levels are preferred without invoking quantum mechanical principles.

Grok x.com/i/grok/share/2ERWIbPFpB0wlCsaqcGErGVX2

Conclusion : The integer ( n ) introduces a quantization effect, and at these points, the spiral’s geometry—through the number of steps and the resulting angular position—creates a resonant or balanced configuration. This allows the electron, moving semi-classically, to occupy stable orbital shells, not because of quantum wavefunctions, but due to the inherent structure and symmetry of the spiral’s path at these discrete, integer-squared intervals.

Simulation

The simulation treats the atomic orbital as a 2-body gravitational orbit with the electron (single point) orbiting a central mass - the nucleus. The nucleus is a set of individual points (also orbiting each other) and not a static mass (static entity). The difference between gravitational (between macro objects) and atomic (gravitational) orbits is the angle of rotation $\beta$ which has an additional $r_{\alpha }$ term included as the atomic orbital wavelength component is dominated by the particle wave-state (the mass-state is treated as a point), and so velocity along the 2-D (gravitational) plane (we are only mapping the radial component of the orbital) will decrease proportionately.

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)

Spiral angle

In this example (see H orbital transition spiral n=2 to n=7), for an idealized Rydberg atom (a nucleus of point size, infinite mass and disregarding wavelength), the electron transition starts at the initial (n_i = 1) orbital (applying only the Bohr radius alpha component)

α =137.035999177 (inverse fine structure constant )

$\varphi =0,\;r_{orbital}=2\alpha$

$r_{orbital}=2\alpha$

$x=r_{orbital},\;y=0$

For each step during transition, setting t = step number (FOR t = 1 TO ...), we can calculate the radius r and $n_{f}^{2}$ at each step.

$r=r_{orbital}+{\frac {t}{2\pi 2\alpha }}$ (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

$\varphi =4\pi {\frac {(n_{f}^{2}-n_{f})}{n_{f}^{2}}}$ ($\varphi$ at any $n_{f}^{2}$)

We can then re-write ($n_{f}$ is only an integer at prescribed spiral angles);

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

Setting alpha = 137.03599583054 gives integer transition frequency t values (t combines t orbital and t transition) and integer radius r multiples ;

$\varphi =(2)\pi ,\;r=4r_{orbital},\;t=1887858$ (360°)

$\varphi =(8/3)\pi ,\;r=9r_{orbital},\;t=4247680$ (360+120°)

$\varphi =(3)\pi ,\;r=16r_{orbital},\;t=7551430$ (360+180°)

$\varphi =(16/5)\pi ,\;r=25r_{orbital},\;t=11799109$ (360+216°)

$\varphi =(10/3)\pi ,\;r=36r_{orbital},\;t=16990717$ (360+240°)

$\varphi =(7/4)\pi ,\;r=49r_{orbital},\;t=23126254$

$\varphi =(7/2)\pi ,\;r=64r_{orbital},\;t=30205718$ (360+270°)

H atom

Experimental values for H(1s-ns) transitions (n the principal quantum number).

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

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

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

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

The wavelength of the H atom, for simplification the respective particle wavelengths are presumed constant irrespective of the vicinity of the electron to the proton.

$r_{wavelength}=\lambda _{H}={\frac {2c}{\lambda _{e}+\lambda _{p}}}$ = 0.155184298 10²²

Dividing (dimensioned) wavelength ($r_{wavelength}$) by the (dimensioned) transition frequency returns a dimensionless number (the alpha component of the photon).

$h_{(1s-ns)}=\lambda _{H}{\frac {(n^{2}-1)}{H(1s-ns)}}$

$h_{(1s-2s)}$ = 1887839.82626...

$h_{(1s-3s)}$ = 4247634.04874...

$h_{(1s-4s)}$ = 7551347.55306...

$h_{(1s-\infty )}={\frac {\lambda _{H}}{H(1s-ns)}}$ = 471959.242776...

The Rydberg atom alpha component radius is premised to be $r=2\alpha$, however the H atom orbital barycenter is not in the center of the nucleus for the proton is only 1836x the mass of the electron. Therefore we may anticipate a H orbital radius that is slightly longer than $2\alpha$, this can be tested by running a 2-body simulation using the gravity simulation program modified for atomic orbital transitions where the electron is the orbiting point. Note, to modify the simulation for atomic orbitals, change the angle of rotation mass formula $r_{ij}$ to

$r_{ij}={\sqrt {2\alpha }}$

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

$r_{incr}=-{\frac {1}{2\pi 2\alpha }}$

Setting central mass = 128, the table gives the transition frequencies at the n-levels (1, 2, 3, 4) for the spiral angles (360°, 360°, 360+120°, 360+180°) and the spiral (Bohr) radius (1r, 4r, 9r, 16r). The experimental values are included for reference . Note, for ionisation the $\lambda _{f}$ photon is considered of such long wavelength as to be insignificant in terms of photon momentum and so the n = 1 orbital corresponds to $h_{(1s-\infty )}$. Results also depend on the value for alpha, here is used the CODATA 2022 value 137.035 999 177.

$r=2\alpha +2r_{incr}$	n = 1	n = 2	n = 3	n = 4
Spiral angle	471959.33	1887836.33	4247602.46	7551347.30
Spiral radius	471959.33	1887835.84	4247630.03	7551340.41
Experimental	471959.24	1887839.83	4247634.05	7551347.55

$r=2\alpha +3r_{incr}$	n = 1	n = 2	n = 3	n = 4
Spiral angle	471959.82	1887840.27	4247615.75	7551347.79
Spiral radius	471959.82	1887836.83	4247632.49	7551344.35
Experimental	471959.24	1887839.83	4247634.05	7551347.55

Geometrically coded universe

• Simulation_hypothesis_(Planck): A geometrical Planck scale simulation universe
• Electron_(mathematical): Mathematical electron from Planck units
• Planck_units_(geometrical): Planck units as geometrical forms
• Physical_constant_(anomaly): Anomalies in the physical constants
• Quantum_gravity_(Planck): Gravity at the Planck scale
• Fine-structure_constant_(spiral): Quantization via pi
• Relativity_(Planck): 4-axis hypersphere as origin of motion
• Black-hole_(Planck): CMB and Planck units
• Sqrt_Planck_momentum: Link between charge and mass

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