Binary systems with orbiting particles: There are both wave and particle discriptions of atoms. We will be looking at particles. The red are electrons and the blue are protons. The red and blue sine waves are an edge view of the orbital plane traced out by the electron and proton pair as they move across the page on a helical path like a spring on a string. Both orbit each other on opposite sides of the center of mass of the system. The orbital period of the proton and electron pair is the same. Looking at a point, in the orbital plane as they orbit, would show alternating charges at the frequency that the electron and proton orbit. Blue-red-blue-red at 6.6E15_hertz or cycles per second. The stationary proton and orbiting electron view of the planetary atom conceals their binary behavior which the red and blue sine waves emphasize.

Rydberg: computed the constant which bears his name and the formula for the known lines in the hydrogen spectrum.
1/wavelength = Rydbergs constant *(1/base²-1/n²)
1/wavelength = 10.973E6_1/m *(1/base²-1/n²)
The base in the visible Balmer series is two and n is greater than two. When n=3 the Balmer series returns a wavelength of 656.3E-9_m, the red of hydrogen-alpha. The base in the ultraviolet Lyman series is one and n is greater than one. When n=2 the Lyman series returns a wavelength of 121.5E-9 _m, called hydrogen Lyman-alpha. We know Rydberg's name because Rydberg's formula fits the wavelength of the hydrogen spectral lines very well.

First: Bohr used Rydberg so Bohr also fits the spectral data. Bohr linked Rydberg's formula with the kinetic energy changes, due to the transition or jumping of an electron between various orbits, in electrons orbiting the nucleus of atoms. These kinetic energy changes emit photons visible in the hydrogen spectrum.

Second: Bohr postulated: The angular momentum of an electron in orbit around a proton equals multiples of Plank's constant divided by 2*pi. We know h_p/(4*pi) as the spin of the electron. The angular momentum of the atom is multiples of twice of the spin of the electron. The angular momentum may be conserved or emitted as photons. Is this the mechanism to shed angular momentum in solar system formation?
me*vt*r = n*h_p/(2*pi)
me*vt*r, is the angular momentum where, me is the mass of the electron, vt is its tangent velocity and r is the orbital radius. n is the multiples of Plank's constant, h_p, divided by 2*pi.
vt = n*h_p/(2*pi*me*r), isolated vt

Centrifugal force: When a force exerts a center seeking centripetal force, inertia opposes this deviation from straight line motion with a center fleeing centrifugal force. The centripetal gravitational force equals the inertial centrifugal force along the orbital path. For every action there is an equal but opposite reaction. Slinging a rock on a rope, around in a circle, demonstrates this centrifugal force which can easily be measured with a spring scale used by fishermen. You and the rock are a binary system. Your centrifugal force, at your distance from the center of mass, equals the centrifugal force of the rock, at its distance from the common center of mass, equals the tension in the rope. If the rope is cut or released both the centripetal and centrifugal forces become zero. The rock continues on its inertial path. You continue on your inertial path. Both paths are in opposite directions. They are determined by momentum prior to release and tangent to the circle at the point of release.

Third: Bohr postulated: The centrifugal force equals the Coulomb force. The Coulomb force is the electrostatic attractive force between an electron in orbit around a proton, both with a charge of ce at a separation of r meters .
me*vt²/r = ce²/(4*pi*e0*r²), The centrifugal force equals the Coulomb force.
r_c = ce²/(4*pi*e0*me*c²), r_c is the classical radius of the electron or me*c²*r_c = ce²/(4*pi*e0),
me*vt²/r = me*c²*r_c/r², substituted me*c²*r_c for ce²/(4*pi*e0)
vt² = c²*r_c/r, isolated vt²

Solve for vt:
vt² = c²*r_c/r, from centrifugal force.
vt = n*h_p/(2*pi*me*r), from angular momentum in increments.
vt²/vt = vt = c²*r_c*2*pi*me*r/(n*h_p*r),
vt = 2*pi*me*c²*r_c/(n*h_p), but h_p = me*c*2*pi*r_c/alpha, alpha is the fine structure constant.
vt = 2*pi*me*c²*r_c*alpha/(n*me*c*2*pi*r_c),
vt = c*alpha/n = c/(137*n), the velocity is a small fraction of the speed of light. Electrons in elliptical orbits move faster when near the nucleus. This is a kinetic energy mass increase which causes a precession of the elliptical orbit. This is expressed as more structure in the hydrogen spectrum as was discussed by Sommerfeld.
vt² = c²*alpha²/n²

Solve for r:
vt²/c² = r_c/r , from the centrifugal force equals the Coulomb force.
r = c²*r_c/ vt², but vt² = c²*alpha²/n², so
r = c²*n²*r_c/(c²*alpha²)
r = n²*r_c/alpha² = n²*5.292E-11_m, this radius is 137 times larger than the ring electron which has radius of r_c/alpha = 3.8616E-13_m.

Substitute values of vt and r in electron momentum:
n*h_p/(2*pi) = me*vt*r
n*h_p/(2*pi) = me *c*alpha/n *n²*r_c/alpha²
h_p/(2*pi) = me *c *r_c/alpha, This is twice the spin of the electron.
h_p = 2*pi *me *c *r_c/alpha, this is true.

The energy of the photon equals the kinetic energy:
h_p *frequency = m *vt²/2
h_p *c/wavelength = m *vt²/2
2*pi*me*c*r_c/alpha *c/wavelength = me *c²*alpha²/n² /2, substituted for h_p, m, vt²
1/wavelength = alpha³/(4*pi*r_c*n²), collected terms.
The Rydberg constant equals alpha³/(4*pi*r_c) = 10.973E6_1/m

Physics seems to be drifting away from a physical reality. Is physics becoming faith based? Do you think we could have arrived at Rydberg's constant, by the above circuitous route, without a substantial portion of Bohr's planetary atom being correct?

*Index* *Next, Inertial inductance*