# 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, the simulation does not use Newtonian physics or the physical constants G, h, c), it can translate to the Planck units for comparison with real world orbits.
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 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 only the mass state of the electron is mapped during transition, periodically the spiral angles converge to give an integer orbital radius (360°=4r, 360+120°=9r, 360+180°=16r, 360+216°=25r ... 720°=∞r), the transition steps between these radius can then be used to solve the transition frequency. And so although mapping the atomic electron as a gravitational orbit on a 2-D plane, a radial quantization (as a function of pi and so of geometrical origin) naturally emerges. In this context it is thus not necessary to develop a separate `quantum' theory of gravity.

### 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) = 10$^{23}$ 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 10$^{23}$ 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 10$^{23}$ 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 gravitational orbits that we observe are the time averaging sum of the underlying multiple (n-body) gravitational point-point orbitals rotating at Planck time.

### N-body orbitals

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).
    $\beta ={\frac {1}{r_{orbital}{\sqrt {r_{orbital}}}}}$
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.

#### 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. $r_{s}$ = i; number of center mass points (the orbited object).
2. j$_{max}$ = radius to mass co-efficient.
3. j = number of points, including virtual (for simple 2 body orbits with only 1 orbiting point, j = i + 1 ).
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.
    $r_{orbital}={r_{\alpha }}^{2}\;*\;r_{wavelength}$

#### Orbital formulas (2-D plane)

Outer = orbiting point, inner = orbited center
    $r_{outer}={r_{\alpha }}^{2}\;*\;2({\frac {j_{max}}{i}})^{2}$, orbital radius
    $r_{barycenter}={\frac {r_{outer}}{j}}$, barycenter
    $v_{outer}={\frac {i}{j_{max}r_{\alpha }}}$, orbiting point velocity
    $v_{inner}={\frac {1}{j_{max}r_{\alpha }}}$, orbited point(s) velocity
    $t_{outer}={\frac {2\pi r_{outer}}{v_{outer}}}$, orbiting point period
    $l_{outer}=2\pi (r_{outer}-r_{barycenter})$, distance travelled
1) Simulation (example):
    points = 8 (1 orbiting point and 7 center mass points)
    jmax = 225
    wavelength = 2066.326531
    radius = 566322.241497
Theoretical values (Chat GPT)
    calculated period = 1656793381.3
    calculated orbit length = 3113519.1385389
    calculated orbit velocity = 0.001879244
    calculated barycenter = 70790.2801871
    jmax/j ratio = 28.1250000
Simulation results (data file)
2) Moon orbit.
From the standard gravitational parameters, the earth to moon mass ratio approximates 81:1 and so we can reduce to 1 point orbiting a center of mass comprising i = 81 points, j = i + 1. The simulation itself is dimensionless, here we use the Planck length $l_{p}$ and $c$ to convert to dimensioned SI units.
    ${\frac {3.986004418\;x10^{14}}{4.9048695\;x10^{12}}}=81.2663$
    $r_{earth-moon}$ = 384400km
    $M_{earth}$ = 0.597378 10$^{25}$kg
Solving $j_{max}$
    $r_{outer}={r_{\alpha }}^{2}\;*\;2({\frac {j_{max}}{i}})^{2}={\frac {2r_{earth-moon}m_{P}}{M_{earth}l_{p}}}$
    $j_{max}=1440443$
Gives
    $t_{outer}=4\pi {({\frac {j_{max}{r_{\alpha }}}{i}})}^{3}({\frac {l_{p}}{c}})=0.8643\;10^{-26}$s
    $t_{outer}{\frac {M_{earth}}{m_{P}}}=2371844$s (27.452 days)
    $v_{Moon}=(c){\frac {i}{j_{max}{r_{\alpha }}}}=1018.3m/s$
    $v_{Earth}=(c){\frac {1}{j_{max}r_{\alpha }}}=12.57m/s$
    $r_{barycenter}={\frac {r_{earth-moon}}{j}}=4688km$

#### 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 10$^{42}$ 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 10$^{23}$ 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;
    $N_{orbitals}=({\frac {m_{A}}{m_{P}}})({\frac {m_{B}}{m_{P}}})=0.1261\;x10^{41}$
Examples:
    $i={\frac {M_{earth}}{m_{P}}}=0.27444\;x10^{33}$ (earth as the center mass)
    $i2l_{p}=0.00887$ (earth Schwarzschild radius)
    $s={\frac {1kg}{m_{P}}}=45940509$ (1kg orbiting satellite)
    $j=N_{orbitals}=i*s=0.1261\;x10^{41}$
1) 1kg satellite at earth surface orbit
    $r_{o}=6371000km$ (earth surface)
    $j_{max}={\frac {j}{r_{a}}}{\sqrt {\frac {r_{o}}{il_{p}}}}=0.288645\;x10^{44}$
    $n_{g}={\frac {j_{max}}{j}}=2289.41$
    $r=r_{\alpha }^{2}n_{g}^{2}il_{p}=r_{o}$
    $v={\frac {c}{n_{g}r_{\alpha }}}=7909.7924$ m/s
    $t=2\pi {\frac {r_{outer}}{v_{outer}}}=5060.8374$ s
2) 1kg satellite at a synchronous orbit radius
    $r_{o}=42164.17km$
    $j_{max}={\frac {j}{r_{a}}}{\sqrt {\frac {r_{o}}{il_{p}}}}=0.74256\;x10^{44}$
    $n_{g}={\frac {j_{max}}{j}}=5889.674$
    $r=r_{\alpha }^{2}n_{g}^{2}il_{p}=r_{o}$
    $v={\frac {c}{n_{g}r_{\alpha }}}=3074.66$ m/s
    $t=2\pi {\frac {r_{outer}}{v_{outer}}}=86164.09165$ s
3) 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)
    $N_{orbitals}={\frac {M_{earth}m_{satellite}}{m_{P}^{2}}}=0.126x10^{41}$ (number of orbitals)
    $E_{total}=E_{orbital}N_{orbitals}=53MJ/kg$
4) The orbital angular momentum of the planets derived from the angular momentum of the respective orbital pairs.
    $N_{sun}={\frac {M_{sun}}{m_{P}}}$
    $N_{planet}={\frac {M_{planet}}{m_{P}}}$
    $N_{orbitals}=N_{sun}N_{planet}$
    $n_{g}={\sqrt {\frac {R_{radius}m_{P}}{2\alpha l_{p}M_{sun}}}}$
    $L_{oam}=2\pi {\frac {Mr^{2}}{T}}=N_{orbitals}n_{g}{\frac {h}{2\pi }}{\sqrt {2\alpha }},\;{\frac {kgm^{2}}{s}}$
The orbital angular momentum of the planets;
Orbital angular momentum combined with orbit velocity cancels n$_g$ giving an orbit constant. Adding momentum to an orbit will therefore result in a greater distance of separation and a corresponding reduction in orbit velocity accordingly.
    $L_{oam}v_{g}=N_{orbitals}{\frac {hc}{2\pi }},\;{\frac {kgm^{3}}{s^{2}}}$

#### 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 2$^{32}$ 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.

#### Orbital trajectory (circular vs. straight)

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).
The orbital drift (as determined where the blue and yellow meet) is due to these orbiting points rotating around each other.

#### Precession

Can the orbital plane also rotate?
semi-minor axis: $b=\alpha l^{2}\lambda _{A}$
semi-major axis: $a=\alpha n^{2}\lambda _{A}$
radius of curvature :$L={\frac {b^{2}}{a}}={\frac {al^{4}\lambda _{A}}{n^{2}}}$
    ${\frac {3\lambda _{A}}{2L}}={\frac {3n^{2}}{2\alpha l^{4}}}$
arc secs per 100 years (drift):
    $T_{earth}$ = 365.25 days
drift = ${\frac {3n^{2}}{2\alpha l^{4}}}1296000{\frac {100T_{earth}}{T_{planet}}}$

#### Hyper-sphere orbit

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 B$^1$ to B$^{11}$) 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.
For object B
    $t_{d}=t{\sqrt {1-v_{outer}^{2}}}$
For object A
    $t_{d}=t{\sqrt {1-v_{inner}^{2}}}$

### Atomic orbitals

In the atom we find individual particle to particle orbitals, and as such the atomic orbital is principally a wave-state orbital (during the orbit the electron is predominately in the electric wave-state). The wave-state is defined by a wave-function, we can however map (assign co-ordinates to) the mass point-states and so follow the electron orbit, for example, in 1 orbit at the lowest energy level in the H atom, the electron will oscillate between wave-state to point-state approximately 471960 times, and so a plot of the electron as a circular obit around the nucleus will be the sum of 471960 'dots'.
This permits us to treat 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, 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 (360°, 360+120°, 360+180°, 360+216° ...) are linked directly to pi via this spiral geometry, we may ask if quantization of the atom has a geometrical origin..

#### Theory
##### 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°)

##### 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 n$^2$ (1, 4, 9, 16 ... ) electrons. Accounting for two states of spin this becomes 2n$^2$ 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
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)
    $t_{ref}=2\pi 4\alpha ^{2}$ = 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' akin to the photon albeit of inverse or reverse phase such that $orbital\;radius+photon=zero$ (cancel).
The photon can be considered as a moving wave, the orbital radius as a standing/rotating wave (trapped between the electron and proton), as such it is the orbital radius 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 itself 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 during the transition phase resulting in the electron following a 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. 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 ($\lambda _{i}+\lambda _{orbital}=zero$ which corresponds to 1 complete n = 1 orbit by the electron, the orbital phase), then the remaining ($\lambda _{f}$) photon continues until it too is absorbed (the transition phase).
    $t_{ref}\sim 2\pi 4\alpha ^{2}$
    $\lambda _{i}=1t_{ref}$
    $\lambda _{f}=4t_{ref}$ (n = 2)
After the ($\lambda _{i}$) photon is absorbed, the ($\lambda _{f}$) photon still has $\lambda _{f}=(n_{f}^{2}-n_{i}^{2})t_{ref}=3t_{ref}$ steps remaining until it too is absorbed.
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);
    $r_{incr}=-{\frac {1}{2\pi 2\alpha r_{wavelength}}}$
In summary; the ($\lambda _{i}$) photon, which has the same wavelength as the orbital radius, deletes the orbital radius in step
$r=r_{orbital}$
WHILE ($\lambda _{i}>0$)
    $r=r+r_{incr}$
    //$\lambda _{i}$ photon
Conversely, because of its minus term, the ($\lambda _{f}$) photon will simultaneously extend the orbital radius accordingly;
WHILE ($r<4r_{orbital}$)
    $r=r-r_{incr}$
    //$\lambda _{f}$ photon
The model assumes orbits also follow along a timeline z-axis
    $t_{orbital}=t_{ref}{\sqrt {1-{\frac {1}{(v_{orbital})^{2}}}}}$
The orbital phase has a fixed radius, however at the transition phase this needs to be calculated for each discrete step as the orbital velocity depends on the radius;
    $t_{transition}=t_{ref}{\sqrt {1-{\frac {1}{(v_{transition})^{2}}}}}$

#### 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 only in 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.
    $\beta ={\frac {1}{r_{orbital}{\sqrt {r_{orbital}}}{\sqrt {2\alpha }}}}$

##### Spiral angle

In this example (see simulation 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
    $\varphi =0,\;r_{orbital}=2\alpha$
    $x=r_{orbital},\;y=0$
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 $r_{incr}$)
    $\varphi =\varphi +\beta$
    $n_{f}^{2}=1+{\frac {t}{2\pi 4\alpha ^{2}}}$ ($n_{f}^{2}$ 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);
    $\beta ={\frac {1}{{r_{orbital}}^{2}n_{f}^{3}}}$
Giving integer values at these spiral angles
    $\varphi =(2)\pi ,\;r=4r_{orbital}$ (360°)
    $\varphi =(8/3)\pi ,\;r=9r_{orbital}$ (360+120°)
    $\varphi =(3)\pi ,\;r=16r_{orbital}$ (360+180°)
    $\varphi =(16/5)\pi ,\;r=25r_{orbital}$ (360+216°)
    $\varphi =(10/3)\pi ,\;r=36r_{orbital}$ (360+240°)
    $\varphi =(7/4)\pi ,\;r=49r_{orbital}$
    $\varphi =(7/2)\pi ,\;r=64r_{orbital}$ (360+270°)

##### Rydberg atom

At the n = 1 orbital, 1 complete rotation becomes (the dimensionless terms are measured on a 2-D plane);
    $t_{ref}={\frac {2\pi r_{orbital}}{v_{orbital}}}=2\pi 2\alpha 2\alpha$
    $1t_{ref}$ = 471964.3563...
    $4t_{ref}$ = 1887857.4255...
    $9t_{ref}$ = 4247679.2074...
    $16t_{ref}$ = 7551429.7021...

##### 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 = ∞)
R = 10973731.568157 (Rydberg constant)
α =137.035999177 (inverse fine structure constant
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.
$r_{wavelength}=\lambda _{H}={\frac {2c}{\lambda _{e}+\lambda _{p}}}$
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...

##### Simulation atom

The following example simulates an electron transition, the electron begins at radius $r=r_{orbital}$ and makes a 360° rotation at orbital radius (the orbital phase) and then moves in incremental steps to higher orbitals (the transition phase) mapping a hyperbolic spiral path (red line) in the process (photon orbital model).
The period $t_{sim}$ and length $l_{sim}$ are measured at integer $n^{2}r$ (n = 1, 2, 3...) radius. For a Rydberg atom, these radius correspond precisely to the electron path at the (hyperbolic) spiral angles; (360°(1r), 360°(4r), 360+120°(9r); 360+180°(16r), 360+216°(25r), 360+240°(36r) ...) (the angles converge to give integer values at these radius), and we find that as the simulation nucleus mass increases, the integer radius values approach these angles (table 2.). The period $t_{sim}$ can then be used to calculate the transition frequencies.
In this example, the nucleus = 249 mass points (start x, y co-ordinates close to 0, 0), the electron = 1 mass point (at radius x = r, y = 0), with t$_{sim}$ = period and l$_{sim}$ = distance travelled by the electron ($l_{orbital}=l_{sim}$ at n=1), the radius coefficient r$_n$ = radius divided by $r_{orbital}$. As this is a gravitational orbit, although the nucleus comprises 249 points clumped close together, these points are independent of each other (they also rotate around each other), and so the `nucleus' size and shape is not static (the simulation is not optimised for a nucleus). Table 1. gives the relative values and the x, y co-ordinates for the electron, nucleus center and barycenter.
    $j_{atom}=250$ (atomic mass)
    $i_{nucleus}=j_{atom}-1=249$ (relative nucleus mass)
    $r_{wavelength}=2({\frac {j_{atom}}{i_{nucleus}}})^{2}$ = 2.0160965
    $r_{orbital}=2\alpha \;*\;r_{wavelength}$ (radius) = 552.5556
    $t_{n}={\frac {t_{sim}}{r_{wavelength}}}$
    $l_{n}={\frac {l_{sim}}{l_{orbital}}}-l_{orbital}$
    $r_{b}=r_{sim}-{\frac {r_{sim}}{j_{atom}}}$
    $r_{n}={\frac {r_{b}}{r_{orbital}}}$
table 1. Electron transition (mass = 250; r$_n$= 1 to 5) :
| r$_n$ | t$_{sim}$ | l$_n$ | angle | x, y (electron) | x, y (nucleus) | x, y (barycenter) |
| 1 | 471957.072 | 0.9999897 | 360° | 550.334, 0.0036 | -2.2102, -0.00002 | -0.00004, -0.00001 |
| 4 | 1887867.293 | 2.000012 | 359.952489° | 2202.8558, 0.0001 | -7.9565, -1.9475 | 0.8868, -1.9397 |
| 9 | 4247689.502 | 4.000014 | 119.92712° | -2473.180, 4296.283 | 13.558, -10.325 | 3.611, 6.901 |
| 16 | 7551439.538 | 6.000014 | 179.91669° | -8815.254, 12.818 | 25.636, 13.303 | -9.728, 13.301 |
| 25 | 11799118.905 | 8.000014 | 215.9122° | -11158.64, -8081.13 | 16.580, 39.083 | -28.118, 6.602 |
Comparison of the spiral angle at r$_n$ = 4 (360°), 9 (360+120°), 16 (360+180°) with different mass (m = 64, 128, 250, 500, Rydberg). For the proton:electron mass ratio; m = 1836.15267...
table 2. Spiral angle at $r_{n}$ = 4, 9, 16 :
| mass | r$_n$ = 4 | r$_n$ = 9 | r$_n$ = 16 |
| m = 64 | 359.80318° | 119.70323° | 179.66239° |
| m = 128 | 359.90394° | 119.85415° | 179.83377° |
| m = 250 | 359.95249° | 119.92711° | 179.91669° |
| m = 500 | 359.97706° | 119.96501° | |
| Rydberg | 360° | 360+120° | 360+180° |