# Quantum gravity (Planck)

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

This orbit model uses a geometrical approach instead of the dimensioned physical constants (G, h, c) to emulate gravitational and atomic orbits via an n-body network of rotating particle-to-particle orbital pairs. Each particle in the 'universe' is connected to every other particle by a discrete circular rotating orbital (representing a unit of momentum) with the particles at each orbital pole, thereby forming a universe-wide network of particle-to-particle orbital pairs [1].

Although dimensionless (the model does not use formulas that require dimensioned constants), orbit period and radius can be measured in Planck units, and so this approach is applicable to modelling gravitational orbitals at the Planck scale. In dimensioned terms, per unit of Planck time tp each orbital pair rotates by 1 unit of Planck length lp at velocity c (v = lp/tp) in 4-axis hypersphere co-ordinates, all orbitals are then summed and averaged to give the new particle co-ordinates. The process then repeats, gravitational orbits between macro-bodies in 3-D space emerging as the averaged (over time) sum of these individual rotating orbitals. The particles (of the orbiting bodies) are connected with each other directly via these orbitals, information regarding the macro orbiting bodies or associated barycenter(s) is not required.

In a simple orbit where a small mass (such as an electron or satellite) is orbiting a larger mass (nucleus, planet ...), the radius of orbit follows the formula

${\displaystyle r_{orbit}=r_{constant}\;*\;r_{wavelength}}$

To emulate gravitational and atomic orbit(al)s, the (inverse) fine-structure constant alpha is used. Although the orbit itself occurs in the hypersphere at the speed of light, this orbital constant returns period and velocity for a 2-D plane (of 3-D space).

${\displaystyle r_{constant}=2\alpha }$

Atomic orbitals are a geometrical subset of these gravitational orbitals, and so likewise can be interpreted as physical units of momentum rather than regions of probability where the particle can be found. They are treated as analogous in physical characteristics to the photon, with electron transition between orbitals occurring when the orbital itself absorbs (or ejects) a photon. The electron has no direct role in the transition.

Although simply the sum of rotating circles, the equations that emerge to describe simple orbits resemble the Bohr model.

## Particle-particle orbitals

Particles are assigned point' (representing a discrete unit of mass) co-ordinates within a 4-axis hyper-sphere universe'. Every particle is then connected to every other particle by a circular orbital (for atomic orbitals, it is the orbital radius which is responsible for the rotation).

The hyper-sphere expands in incremental steps (FOR age = 1 TO ... representing a discrete unit of time), this expansion causes the orbitals to rotate in steps (representing a discrete unit of length) correspondingly.

In a simulation, for each value of incrementing variable age (the simulation clock-rate), all (n-body) orbitals rotate by 1 step (driven by the hypersphere expansion) and the co-ordinates calculated. These are then summed and averaged giving new particle co-ordinates. As this occurs for every particle before the next increment to age, the process can be updated in 'real time' on a serial processor. As the orbitals are circular, the barycenter for each orbital is its center, the particles at each orbital 'pole'.

As these steps are discrete, they may be measured in terms of the Planck units, each point as a unit of Planck mass, with the orbitals rotating 1 unit of Planck length lp per 1 unit of Planck time tp (i.e.: 1 increment to age) at velocity c = lp/tp in hyper-sphere coordinates.

Although orbital and so particle motion occurs at c, the hyper-sphere expansion [2] is equidistant and so invisible' to the observer. Instead observers 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.

## N-body orbit simulation

A simulation program [3] comprising n-body rotating orbitals is described. Each particle in the simulation is assigned initial (x, y) 2-D point co-ordinates (representing 3-D space), forming orbital pairs that rotate around each other on a 2-D plane according to an angle β as defined by the orbital pair radius (the atomic orbital β has an additional alpha term).

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

The total distance travelled, 1 unit of length per increment to age (1 time unit) is given in (x, y, z) co-ordinates, where the (z) axis represents the hypersphere expansion axis.

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

For the following simple orbits, 1 point is assigned as the orbiting point, the remaining points forming the 'central' mass. The only distinction 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). The simulation however treats all points equally, the center points also orbiting each other according to their orbital radius (each point is 1 mass unit, analogous to objects whose mass is a multiple of units of Planck mass that are held together by gravitational forces).

After every orbital has rotated 1 length unit anti-clockwise, 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 tsim (as the number of increments to the simulation clock age) and the (x, y) plane orbit length lsim are noted.

Key:

1. i; number of 'physical' center points in the orbit (the center mass).

2. j = i*x + 1; number of virtual center points (to reduce computation time, i*x virtual points are added to increase center mass up to j = jmax.

3. jmax; maximum number of mass points per orbital radius.

4. x, y; start co-ordinates for each point (2-D plane).

5. rα; a radius constant, here rα = sqrt(2α) = 16.55512; where alpha = inverse fine structure constant = 137.035 999 084 (CODATA 2018).

${\displaystyle r_{orbital}=r_{constant}\;*\;r_{wavelength}={r_{\alpha }}^{2}\;*\;2({\frac {j_{max}}{i}})^{2}}$
${\displaystyle l_{sim}={\frac {t_{sim}(j-1)}{j_{max}r_{\alpha }}}}$ length
${\displaystyle r_{sim}=({\frac {j}{j-1}}){\frac {l_{sim}}{2\pi }}={\frac {t_{sim}}{2\pi n_{g}r_{\alpha }}}}$ radius
${\displaystyle v_{sim}=({\frac {j}{j-1}}){\frac {l_{sim}}{t_{sim}}}={\frac {1}{n_{g}r_{\alpha }}}}$ velocity

Example:

i = 81, j = jmax = 32*81+1 = 2593 (3321 orbitals)

tsim = 58430803.84

lsim = 3528109.12

This gives the following equations for the 2-D plane

${\displaystyle n_{g}={\frac {j_{max}}{j}}}$ ratio of mass to maximum mass per orbital radius
${\displaystyle r_{outer}=2({\frac {j_{max}}{i}})^{2}{r_{\alpha }}^{2}}$, orbital radius
${\displaystyle r_{barycenter}={\frac {r_{outer}}{j}}}$, barycenter
${\displaystyle v_{outer}={\frac {j}{r_{\alpha }j_{max}}}}$, orbiting point velocity
${\displaystyle v_{inner}={\frac {1}{r_{\alpha }j_{max}}}}$, orbited point(s) velocity
${\displaystyle t_{outer}={\frac {2\pi r_{outer}}{v_{outer}}}={\frac {4\pi {j_{max}}^{3}}{i^{2}j}}{r_{\alpha }}^{3}}$, orbiting point period
${\displaystyle l_{outer}=2\pi (r_{outer}-r_{barycenter})}$, distance travelled

To model a 1kg satellite to earth orbit will require earth mass/Planck mass = 0.2744 x1033 points and 1kg/Planck mass = 45940510 points. We can reduce calculation by using only relative mass and then use the dimensionless ng to assign the start parameters. For example, from the standard gravitational parameters, the earth to moon mass ratio approximates 81:1.

${\displaystyle j={\frac {3.986004418\;x10^{14}}{4.9048695\;x10^{12}}}=81.2663}$

There is 1 orbiting point (distant point) and 81 central points (points in close vicinity)

${\displaystyle i=j-1}$

To calculate ng

${\displaystyle r_{earth-moon}}$ = 384400km
${\displaystyle \lambda _{Earth}}$ = 0.00887m (Schwarzschild radius)
${\displaystyle n_{g}={\sqrt {\frac {2r_{earth-moon}}{\lambda _{Earth}}}}{\frac {1}{r_{\alpha }}}}$ = 17783.25

This gives

${\displaystyle j_{max}=n_{g}j}$ = 1445178.5

Converting from dimensionless numbers to SI Planck units using lp and c;

${\displaystyle t_{outer}=4\pi ({\frac {{j_{max}}^{3}}{i^{2}j}}){r_{\alpha }}^{3}({\frac {l_{p}}{c}})=0.1772\;10^{-25}}$s
${\displaystyle r_{outer}=2({\frac {j_{max}}{i}})^{2}{r_{\alpha }}^{2}(l_{p})=0.2872\;10^{-23}}$m
${\displaystyle v_{Moon}=(c){\frac {j}{j_{max}{r_{\alpha }}}}=1018.3m/s}$
${\displaystyle v_{Earth}=(c){\frac {1}{j_{max}r_{\alpha }}}=12.53m/s}$
${\displaystyle barycenter={\frac {r_{earth-moon}}{j}}=4730km}$

We can use the actual radius and period to translate between values.

${\displaystyle t_{earth-moon}}$ = 27.322 days
${\displaystyle \lambda _{0}={\frac {2j^{2}}{i^{2}}}}$
${\displaystyle {\frac {r_{earth-moon}}{r_{outer}}}\lambda _{0}m_{p}=0.59738\;10^{25}}$kg
${\displaystyle {\frac {t_{outer}0.59738\;10^{25}kg}{\lambda _{0}m_{P}}}=2371851=27.452}$ days

The above assumes a circular orbit, to form an elliptical orbital we can use unaligned orbitals (PE vs KE).

### Gravitational coupling constant

In the above, particles were assigned a mass as a theoretical unit of Planck mass (a point). 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;

${\displaystyle \alpha _{G}={\frac {Gm_{e}^{2}}{\hbar c}}={\frac {m_{e}^{2}}{m_{P}^{2}}}=1.75...x10^{-45}}$

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). The above value αG then represents the probability that 2 electrons will be in the mass point-state at any unit of Planck time (wave-particle duality at the Planck level represented by an electric-wave to mass-point oscillation [4]).

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 36 throws (frequency) of the dice will result in a win. The inverse of αG is the frequency of occurence of the mass point-state between the 2 electrons. As 1 second is 1042 units of Planck time, this occurs about once a minute. Gravity now has a similar magnitude to the strong force (at this, the Planck level), albeit this interaction occurs seldom (the Planck level), 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 in the universe will simultaneously be in the mass point-state. For example a 1kg satellite orbits the earth, for any given unit of time, satellite (B) will have ${\displaystyle 1kg/m_{P}=45.9x10^{6}}$ particles in the point-state. The earth (A) will have ${\displaystyle 5.9738\;x10^{24}kg/m_{P}=0.274\;x10^{33}}$ particles in the point-state, and so the number of orbital links (the gravitational coupling constant) between the earth and the satellite will sum to the number of orbitals;

${\displaystyle N_{orbitals}={\frac {m_{A}m_{B}}{m_{P}^{2}}}=0.1261\;x10^{41}}$

Examples:

1. 1kg satellite at a synchronous orbit radius

${\displaystyle j=N_{orbitals}=0.1261\;x10^{41}}$
${\displaystyle i=5.9738\;x10^{24}kg/m_{P}=0.27444\;x10^{33}}$ (earth as the center mass)
${\displaystyle 2il_{p}=\lambda _{earth}=0.00887}$ (Schwarzschild radius)
${\displaystyle r_{o}=42164.17km}$ (synchronous orbit)
${\displaystyle n_{g}={\sqrt {\frac {r_{o}}{il_{p}}}}=5889.674}$
${\displaystyle j_{max}=n_{g}j}$
${\displaystyle t_{outer}=4\pi ({\frac {{j_{max}}^{3}}{i^{2}j}}){r_{\alpha }}^{3}({\frac {l_{p}}{c}})=0.1325265\;x10^{-11}s}$
${\displaystyle r_{outer}=2({\frac {j_{max}}{i}})^{2}{r_{\alpha }}^{2}(l_{p})=0.648515\;x10^{-9}m}$
${\displaystyle v_{outer}={\frac {cj}{j_{max}r_{\alpha }}}=3074.66m/s}$
${\displaystyle \lambda _{0}=2{\frac {j^{2}}{i^{2}}}}$
${\displaystyle {\frac {t_{outer}i}{\lambda _{0}}}={\frac {\pi n_{g}^{3}r_{\alpha }^{3}\lambda _{earth}}{c}}=86164.09165s}$
${\displaystyle {\frac {r_{outer}i}{\lambda _{0}}}={\frac {n_{g}^{2}r_{\alpha }^{2}\lambda _{earth}}{2}}=42164.17km}$

2. 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).

${\displaystyle E_{orbital}={\frac {hc}{2\pi r_{6371}}}-{\frac {hc}{2\pi r_{42164}}}=0.412x10^{-32}J}$ (energy per orbital)
${\displaystyle N_{orbitals}={\frac {M_{earth}m_{satellite}}{m_{P}^{2}}}=0.126x10^{41}}$ (number of orbitals)
${\displaystyle E_{total}=E_{orbital}N_{orbitals}=53MJ/kg}$

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

${\displaystyle N_{sun}={\frac {M_{sun}}{m_{P}}}}$
${\displaystyle N_{planet}={\frac {M_{planet}}{m_{P}}}}$
${\displaystyle N_{orbitals}=N_{sun}N_{planet}}$
${\displaystyle n_{g}={\sqrt {\frac {R_{radius}m_{P}}{2\alpha l_{p}M_{sun}}}}}$
${\displaystyle L_{oam}=2\pi {\frac {Mr^{2}}{T}}=N_{orbitals}n_{g}{\frac {h}{2\pi }}{\sqrt {2\alpha }},\;{\frac {kgm^{2}}{s}}}$

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


Orbital angular momentum combined with orbit velocity cancels ng 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.

${\displaystyle L_{oam}v_{g}=N_{orbitals}{\frac {hc}{2\pi }},\;{\frac {kgm^{3}}{s^{2}}}}$

### 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 (i.e.: in the above example, the earth particles do not follow gravitational orbits around each other). When measuring the orbit of a single point around a larger mass, after each complete orbit we can note that the orbit period and radius reduces (as a function of center mass and start radius distance).

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; i, j, r ... are not preset). 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.

### PE vs. KE (opposing orbitals)

Gravitational potential and kinetic energy are measures of alignment of the orbitals. In the above examples, all orbitals rotate in the same direction = kinetic energy. If all orbitals are unaligned the object will appear to 'fall' = potential energy.

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.

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

### Precession

semi-minor axis: ${\displaystyle b=\alpha l^{2}\lambda _{A}}$

semi-major axis: ${\displaystyle a=\alpha n^{2}\lambda _{A}}$

radius of curvature :${\displaystyle L={\frac {b^{2}}{a}}={\frac {al^{4}\lambda _{A}}{n^{2}}}}$

${\displaystyle {\frac {3\lambda _{A}}{2L}}={\frac {3n^{2}}{2\alpha l^{4}}}}$

arc secs per 100 years (drift):

${\displaystyle T_{earth}}$ = 365.25 days

drift = ${\displaystyle {\frac {3n^{2}}{2\alpha l^{4}}}1296000{\frac {100T_{earth}}{T_{planet}}}}$

Mercury (eccentricity = 0.205630)
T = 87.9691 days
a = 57909050 km (n = 378.2734)
b = 56671523 km (l = 374.2096)
drift = 42.98

Venus (eccentricity = 0.006772)
T = 224.701 days
a = 108208000 km (n = 517.085)
b = 108205519 km (l = 517.079)
drift = 8.6247

Earth (eccentricity = 0.0167)
T = 365.25 days
a = 149598000 km (n = 607.989)
b = 149577138 km (l = 607.946)
drift = 3.8388

Mars (eccentricity = 0.0934)
T = 686.980 days
a = 227939366 km (n = 750.485)
b = 226942967 km (l = 748.843)
drift = 1.351


### Hyper-sphere orbit

An expanding hyper-sphere forms the scaffolding of the universe'. The hyper-sphere expands in uniform incremental 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 [5]. 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. In dimensionless terms;

${\displaystyle d=r_{\alpha }n_{g}}$
${\displaystyle t_{0}=2\pi r=2\pi {\frac {t}{2\pi d}}}$
${\displaystyle v_{outer}={\frac {1}{d}}}$

For object B

${\displaystyle t_{d}={\sqrt {t^{2}-{t_{0}}^{2}}}=t{\sqrt {1-v_{outer}^{2}}}}$

For object A

${\displaystyle t_{d}=t{\sqrt {1-v_{inner}^{2}}}}$

### Planck force

${\displaystyle F_{p}={\frac {m_{P}c^{2}}{l_{p}}}}$
${\displaystyle M_{a}={\frac {m_{P}\lambda _{a}}{2l_{p}}},\;m_{b}={\frac {m_{P}\lambda _{b}}{2l_{p}}}}$
${\displaystyle F_{g}={\frac {M_{a}m_{b}G}{R^{2}}}={\frac {\lambda _{a}\lambda _{b}F_{p}}{4R_{g}^{2}}}={\frac {\lambda _{a}\lambda _{b}F_{p}}{4\alpha ^{2}n^{4}(\lambda _{a}+\lambda _{b})^{2}}}}$

a) ${\displaystyle M_{a}=m_{b}}$

${\displaystyle F_{g}={\frac {F_{p}}{{(4\alpha n^{2})}^{2}}}}$

b) ${\displaystyle M_{a}>>m_{b}}$

${\displaystyle F_{g}={\frac {\lambda _{b}F_{p}}{{(2\alpha n^{2})}^{2}\lambda _{a}}}={\frac {m_{b}c^{2}}{2\alpha ^{2}n^{4}\lambda _{a}}}=m_{b}a_{g}}$

### Atomic orbitals

It has been proposed that a region of space (zero net energy) between a free proton and a free electron can divide into 2 waves of opposite phase (thus still sum to zero). 1 wave escapes (moving-wave), = photon. The other wave is trapped between the electron and the proton (standing-wave) = orbital radius (the Bohr radius is a physical wave). The space encompassing the proton-orbital-electron, by virtual of ejecting the photon, is now a region of lower energy (-hv) than the original proton-zero-electron space, and so is more stable [6].

The atomic orbital in this model[7] is a specific case of the gravitational orbital. It is also treated as a distinct unit of momentum (rather than simply a region of probability), and has the physical properties of the photon (during orbital transition, it is the orbital radius which absorbs/ejects the photon thereby lengthening or shortening, the electron itself has a passive role). The orbital dimensions are a function of the fine structure constant alpha and the electron-nucleus wavelength. The gravitational orbital simulation can be applied to atomic orbitals, the difference being the angle of rotation β which includes an additional ${\displaystyle {\sqrt {2\alpha }}}$ term.

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

The simulation replaces wave-particle duality with an oscillation between an undefined wave-state (duration particle frequency ${\displaystyle f=\lambda /c}$ in units of Planck time) to a point-state which signifies the completion of 1 wave-state cycle. The gravity simulation maps discrete units of Planck mass as points moving 1 unit of Planck length per unit of Planck time, and so presumes that each orbiting object has a mass as multiples of Planck mass. In atomic orbitals, the point-state is treated as the equivalent of 1 Planck mass (i.e.: mass is not considered a constant property of the particle, but instead signifies the collapse of the wave-state) and so can be assigned co-ordinates, the algorithm need not be modified. The wave-state is an undefined state, albeit where the orbit occurs, and so the orbit is mapped as a series of points separated by this wave-state interval. As the orbital radius rotates around its central axis, it pulls the electron with it.

The orbital radius is divided into sub-segments (alpha units) joined together in series. The wavelength of the orbital radius is the sum of these segments.

${\displaystyle \alpha _{unit}={\frac {\lambda _{e}+\lambda _{(p)}}{2\pi 2\alpha }}}$

The photon physically resembles the orbital radius albeit of inverse phase. During electron transition an incoming photon adds to the orbital radius in discrete steps via transfer of these alpha units, in the process the orbital radius is extended (until the photon is completely absorbed). Conversely the orbital radius may eject a photon, the above in reverse.

During transition the orbital continues rotating (while lengthening or contracting), the electron, being pulled along by this rotation, describes a spiral path as the orbital radius changes (the electron has a passive role in the transition phase). To reduce computation time only the point-state is mapped, with the 2-D x-y orbital plane representing 3-D space and the z-axis the universe timeline, the orbit then reduced to the geometry of the alpha component of the orbital radius.

#### Base orbital

The basic orbital resembles the Bohr model. The simulation assigns the electron point-state co-ordinates at the tip of the orbital radius. After each wave-state oscillation cycle, the electron jumps (the actual motion of the electron occurs during the wave-state) 1 alpha step (1/2αn), plotting over time an orbit around a center.

For the Lyman series, the radius of a basic orbital ${\displaystyle r_{orbital}=2\alpha n^{2}}$ (the Bohr radius) where n is the principal quantum number. On the 2-D (x-y) plane

${\displaystyle r_{orbital}=2\alpha n^{2}}$
${\displaystyle v_{orbital}={\frac {1}{2\alpha n}}}$
${\displaystyle t_{orbit}={\frac {2\pi r}{v}}=2\pi 4{\alpha }^{2}n^{3}}$

The (CODATA 2018 inverse) alpha α = 137.035999084, thus the base alpha orbital (where n = 1), will require about tref ~ 471964 steps to make 1 complete rotation.

${\displaystyle t_{ref}=2\pi 4{\alpha }^{2}}$

We can subdivide this orbital (Bohr) radius into these alpha units.

${\displaystyle \alpha _{unit}={\frac {\lambda _{wave-state}}{2\pi 2\alpha }}}$

As the duration of each step is 1 wave-state oscillation (1${\displaystyle f_{wave-state}}$), the actual orbital period sums the number of these oscillations. The Bohr radius then becomes a physical construct from 471964 alpha units added together in series.

${\displaystyle r_{orbital}=t_{ref}\alpha _{unit}n^{2}=2\alpha n^{2}\lambda _{wave-state}}$

In the following examples using the Lyman series, the wave-state is the electron frequency, the orbital radius including the proton contribution. The proton wavelength can then be added to the wave-state and the results cross-referenced.

#### Orbital transition

The incoming (or ejected photon) is also a construct of alpha units, denoted ${\displaystyle r_{incr}}$ to distinguish from ${\displaystyle \alpha _{unit}}$. Note that the minus sign indicates that the ${\displaystyle r_{incr}}$ unit is of opposite phase to the alpha unit. The wave-state component is added later.

${\displaystyle r_{incr}=-{\frac {1}{2\pi 2\alpha }}=-0.000581}$
${\displaystyle \alpha _{unit}+r_{incr}=0}$

During the transition phase, for each transition step an ${\displaystyle r_{incr}}$ unit is exchanged (transferred) between the orbital radius and the photon.

${\displaystyle r_{orbital}=r_{orbital}+r_{incr}}$ (per transition step)

If the wavelength of ${\displaystyle \lambda _{photon}}$ = the wavelength of the orbital radius ${\displaystyle \lambda _{orbital}}$, and as these waves are of inverse phase, the orbital radius will be deleted. This is defined as ionization, returning to the original state; proton-zero-electron.

${\displaystyle \lambda _{orbital}+\lambda _{photon}=zero}$

However an incoming photon is actually 2 photons as per the Rydberg formula.

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

The (+${\displaystyle \lambda _{i}}$) will subtract from the orbital radius as described above, however the (-${\displaystyle \lambda _{f}}$), because of the Rydberg minus term, will conversely increase the orbital radius. And so for the duration of the (+${\displaystyle \lambda _{i}}$) photon wavelength, the orbital radius does not change as the 2 photons cancel;

${\displaystyle r_{orbital}=r_{orbital}+(\lambda _{i})-(\lambda _{f})=r_{orbital}+r_{incr}-r_{incr}}$

The (${\displaystyle \lambda _{f}}$) has the longer wavelength, and so after the (${\displaystyle \lambda _{i}}$) photon has been absorbed, and for the remaining duration of the (${\displaystyle \lambda _{f}}$) wavelength, the orbital radius will be extended in ${\displaystyle -r_{incr}}$ steps (${\displaystyle -r_{incr}}$ = +0.000581).

${\displaystyle r_{orbital}=r_{orbital}+0.000581}$ (per transition step)

For an ${\displaystyle n_{i}}$=1 (${\displaystyle \lambda _{i}=1t_{ref}}$) to ${\displaystyle n_{f}}$=2 (${\displaystyle \lambda _{f}=4t_{ref}}$) orbital transition, the ${\displaystyle \lambda _{i}}$ photon absorption by the ${\displaystyle n_{i}}$=1 orbital requires 1${\displaystyle t_{ref}}$ steps, the remaining ${\displaystyle \lambda _{f}}$ still has 3${\displaystyle t_{ref}}$ segments (of ${\displaystyle r_{incr}}$) left, and so transition continues for another 3${\displaystyle t_{ref}}$ steps. A ${\displaystyle n_{i}}$=2 to ${\displaystyle n_{f}}$=3 transition would require ${\displaystyle t=4t_{ref}+(9-4)t_{ref}}$ steps.

In the above explanation, the (${\displaystyle \lambda _{i}}$) and (${\displaystyle \lambda _{f}}$) were assumed to be identical albeit of different wavelengths. When we actually solve the transitions for the Lyman series, we find that the ${\displaystyle r_{incr}}$ unit for the (${\displaystyle \lambda _{i}}$) photon is slightly longer than its counterpart for the (${\displaystyle \lambda _{f}}$) photon (each transition photon is geometrically distinct), and so they do not cancel completely, but rather after complete absorption of the (${\displaystyle \lambda _{i}}$) photon, the orbital radius is now slightly reduced. We then find a linear relationship between orbital radius and transition frequency.

#### Relativistic orbital

If we include the frequency component (${\displaystyle f_{e}}$) in the orbital radius (which is not practical for a simulation where ${\displaystyle f_{e}}$ ~ 1023), we find that the electron travels 1 Planck length per unit of Planck time in hypersphere coordinates. As with gravitational orbitals, the velocity of rotation derives from the radius of the orbital (from ${\displaystyle \beta }$) and so adjusts as the orbital radius changes. The simulation calculates the relativistic velocity per step to determine the final transition velocity component. The following examples include the (relativistic) z-axis (as with the gravitational orbitals).

#### Transition period

In the classical Bohr model, the electron orbits around the barycenter (center of mass) and for this is used the reduced mass ${\displaystyle \mu }$ (${\displaystyle p_{e}}$ = 1836.15267343 proton/electron mass ratio);

${\displaystyle \mu ={\frac {m_{e}+m_{p}}{m_{p}}}={\frac {p_{e}+1}{p_{e}}}}$ = 1.000544617

However, the ${\displaystyle H_{1s-\infty }}$ (ionization) vs. Rydberg constant shows slight divergence

${\displaystyle \mu ={\frac {10973731.568508}{10967877.174307}}}$ = 1.0005338

Period of orbit

${\displaystyle t_{orbital}=n_{i}2\pi 2\alpha r_{orbital}{\sqrt {1-{\frac {1}{v_{orbital}^{2}}}}}}$

Electron transition can occur when a photon strikes, raising the electron to a higher energy level. For a transition from an (n = i) initial orbital to (n = f) final orbital, the simulation calculates the relativistic velocities for each individual step, ${\displaystyle v_{transition}}$ as the sum of this series.

${\displaystyle t_{transition}=(n_{f}-n_{i})2\pi 2\alpha r_{orbital}{\sqrt {1-{\frac {1}{v_{transition}^{2}}}}}}$

If we use only the electron wavelength to represent transition, then we can replace ${\displaystyle \lambda _{e}}$ with the more precise Rydberg constant R.

${\displaystyle \lambda _{e}={\frac {1}{4\pi \alpha ^{2}R}}}$
${\displaystyle H_{n_{i}-n_{f}}=(n_{f}-n_{i})\;{\frac {8\pi c\alpha ^{2}R}{(t_{orbital}+t_{transition})}}}$

As such, the orbital radius will include the proton wavelength contribution. To account for this, we add an extra term ${\displaystyle r_{o}}$ to the base orbital radius.

${\displaystyle r_{orbital}=2\alpha +r_{o}}$ (n = 1)

Using the proton-electron mass ratio

${\displaystyle r_{o}={\frac {2\alpha }{1836.15267343}}}$ = 0.14926427532
${\displaystyle {\frac {r_{orbital}}{2\alpha }}={\frac {2\alpha +r_{o}}{2\alpha }}={\frac {m_{e}+m_{p}}{m_{p}}}=1.000544617}$

We can use the simulation to determine the optimal period (number of steps that correlate to the experimentally observed transition frequency) for the 1st 3 transition frequencies. Using the Lyman series as example.

${\displaystyle t_{o}=2\pi 2\alpha r_{orbital}}$

${\displaystyle t_{1s-2s}=>t_{o}-3.291074864=>2466\;061\;413\;187.042}$Hz
${\displaystyle {\frac {3(-3.291074864)}{4(2\pi 2\alpha )}}+r_{o}=0.1478309179}$

${\displaystyle t_{1s-3s}=>t_{o}-4.312746113=>2922\;743\;278\;665.80}$kHz
${\displaystyle {\frac {8(-4.312746113)}{9(2\pi 2\alpha )}}+r_{o}=0.1470381130}$

${\displaystyle t_{1s-4s}=>t_{o}-4.609838405=>3082\;581\;563\;822}$kHz
${\displaystyle {\frac {15(-4.609838510)}{16(2\pi 2\alpha )}}+r_{o}=0.1467546294}$

We then repeat using radius to estimate its contribution to period, and find a correlation with the above results, the transition period correction (above) appears to be a function of the orbital radius, and so the (${\displaystyle \lambda _{f}}$) photon ${\displaystyle r_{incr}}$ unit wavelength would be specific for each transition.

${\displaystyle t_{o}=2\pi 2\alpha r_{1s-ns}}$
${\displaystyle r_{1s-2s}=2\alpha +0.14783092037,\;{\frac {r_{1s-2s}}{2\alpha }}=1.0005394}$
${\displaystyle r_{1s-3s}=2\alpha +0.14703811464,\;{\frac {r_{1s-3s}}{2\alpha }}=1.0005365}$
${\displaystyle r_{1s-4s}=2\alpha +0.14675463046,\;{\frac {r_{1s-4s}}{2\alpha }}=1.0005355}$

From these 3 values we can calculate our maximum and minimum (due to limitations of precision of the inputs, ${\displaystyle n_{max}}$, representing ionization, is set to ${\displaystyle 2^{16}}$).

${\displaystyle r_{(1)}}$ = 0.1517159 (n = 1), ${\displaystyle {\frac {2\alpha +r_{(1)}}{2\alpha }}}$ = 1.0005536
${\displaystyle r_{(min)}}$ = 0.1463854446 (nf = 216 ... approximates ionisation), ${\displaystyle {\frac {2\alpha +r_{(min)}}{2\alpha }}}$ = 1.0005341
${\displaystyle n_{(min)}}$ = -0.1162025666 (n when ${\displaystyle r_{o}}$ approaches infinity)

Thus for any ${\displaystyle n_{f}}$ we can derive the ${\displaystyle r_{(f)}}$ term and from there obtain the transition frequency using this approximation.

${\displaystyle r_{(f)}=r_{(1)}-(r_{(1)}-r_{(min)}){\frac {(n_{f}^{2}-1)}{(n_{f}^{2}-n_{(min)})}}}$

calculated from ${\displaystyle n_{f}}$
${\displaystyle 1_{s}-n_{f}}$ ${\displaystyle r_{f}}$ frequency (nist.gov) [8] frequency (calc)
${\displaystyle n_{f}=2}$ 0.14783092037 82 258.954399282 82 258.954399280
${\displaystyle n_{f}=3}$ 0.14703811464 97 492.221701 97 492.22172446
${\displaystyle n_{f}=4}$ 0.14675463046 102 823.8530211 102 823.8530211
${\displaystyle n_{f}=5}$ 0.14662233855 105 291.63094 105 291.62973392
${\displaystyle n_{f}=6}$ 0.14655018726 106 632.1498416 106 632.14704984
${\displaystyle n_{f}=12}$ 0.14655018726 108 917.1209 108 917.10882356
${\displaystyle n_{f}=2^{10}}$ 0.14638545054 109 678.63032891

If we include the proton wavelength (the frequency component including a contribution from the nucleus), the radius ${\displaystyle r_{o}}$ term will reduce by an equivalent amount.

${\displaystyle H_{n_{i}-n_{f}}=(n_{f}-n_{i})\;{\frac {8\pi c\alpha ^{2}R}{(t_{orbital}+t_{transition})}}{\frac {p_{e}}{p_{e}+1}}}$
${\displaystyle r_{1s-2s}=2\alpha }$ - 0.0014319826

Taking the difference between the 2 ${\displaystyle r_{1s-2s}}$ radius values returns the reduced mass.

0.14783092037 + 0.0014319826 = 0.149262903 = ${\displaystyle r_{o}}$

${\displaystyle {\frac {2\alpha +0.149262903}{2\alpha }}}$ = 1.000544612
${\displaystyle \mu ={\frac {m_{e}+m_{p}}{m_{p}}}={\frac {p_{e}+1}{p_{e}}}}$ = 1.000544617

The Positronium ${\displaystyle 1s-2s}$ transition correlates period with radius (${\displaystyle r_{o}}$ = 0);

${\displaystyle P(f_{1s-2s})=1233607216.4}$MHz

${\displaystyle r_{orbital}=2\alpha +0.0196784655}$

${\displaystyle t_{transition}=3(t_{o}+45.1830185)}$

${\displaystyle {\frac {3(45.1830185)}{4(2\pi 2\alpha )}}+r_{o}=0.019678499}$

#### Helium

The above considered a charge equivalence, 1 electron to 1 proton. If we expose the electron to more charge, then we may anticipate further changes to the orbital radius. To illustrate, in this example the orbital radius is divided into 4 parts each of ${\displaystyle t_{ref}/4}$ = 117986 and each part is equivalent to 13.59844 eV, the ionisation energy of H. This means that if an electron orbits at a radius where ${\displaystyle t_{orbit}}$ = 117986, then it will require 3*13.59844eV to reach a base H orbital (${\displaystyle t_{ref}}$ = 117986 + 3*117986) and then a further 13.59844eV to ionize from there. Total ionization energy = 4*13.59844eV = 54.4eV.

In this He animation, both orbiting electrons initally occupy the same orbital radius (${\displaystyle t_{orbit}}$ = 247310). As the first He electron (#1 red) is being ionized (absorbing momentum), the remaining He electron (#2 blue) simultaneously drops to a lower orbital (${\displaystyle t_{orbit}}$ = 117986), transferring momentum to electron #1 in the process and thus subsidizing the ionization of electron #1. After dissociation of the red electron, another photon strikes and the blue electron transitions from its now n= 1 to an n= 2 orbital.

#### Diatomic H

Diatomic Hydrogen radius = 37pm. The H Bohr radius was set above at 2α * (λe + λp) = 105.89pm. To simulate as a 'gravitational' orbit using only an anti-clockwise rotation with no allowance for charge, we set;

electrons; mass = 1 point, start co-ordinates (-99, 0) and (99, 0)

protons; mass = 1836 points, (0, 37) and (0, -37)

orbit center = (0, 0)

number of point to point orbitals = 6747301

... thereby setting the distance from each electron to each proton = 105.89 respectively and electron to electron at 2*99.46. The H2 ionization energy (15.426eV) is 1.1344x greater than for the H atom (13.59844eV). Likewise combining the 2 electron-electron radius (105.89/99.46 = 1.06727) gives 1.13454.

If we reduce proton-proton separation, the protons act as a single center mass and the electrons follow a circular orbit. By increasing the proton-proton separation, the electron orbits increase proportionately. This separation distance (74pm) gives a symmetrical orbit.