The Hydrogen atom composed of one proton and one electron has the following classical based governing equation in the scalar field for equating time and length effects due to the mass of the proton and velocity of a classically orbiting electron.
\(
\begin{equation}\label{eqTransformationCompleteHydrogen} \tag{1}
A(r,m,v)= 1 + \frac{e^2}{ 8\pi \epsilon_0 m_e r c^2}+\frac{G M_p}{2 r c^2} + \frac{G M_p}{r c^2}
\end{equation}
\)
Where \(M_p\) is the proton mass, \(\epsilon_0\) is vacuum permittivity, \(r\) the radial distance from the center of the proton, \(e\) is the elementary charge, and \(c\) is the speed of light. Equation \ref{eqTransformationCompleteHydrogen} squared as it was in the celestial orbit derivation is:
\(
\begin{equation}\label{eqTransformationAdvanceSquaredHydrogen}\tag{2}
A(r,m,v)^2= 1 + \frac{e^2}{ 4\pi \epsilon_0 m_e r c^2}+\frac{3 G M_p}{r c^2}
\end{equation}
\)
The second order contributions have been omitted as their contributions are very small compared to other contributions. As with the gravitational orbit, when calculating an advance in orbit, the length traversed over the circumference in a circular orbit is:
\(
\begin{equation}\label{eqTransformationAdvance2Hydrogen}\tag{3}
Advance= 2\pi r + 2\pi \left( \frac{e^2}{ 4\pi \epsilon_0 m_e c^2}+\frac{3 G M_p}{c^2} \right)
\end{equation}
\)
The second part of \ref{eqTransformationAdvance2Hydrogen} has a value on the order of \(10^{-54}\) while the first part has an order of \(10^{-45}\) Thus, it will be omitted for this calculation. The remaining contribution predicts an advance of:
\(
\begin{equation}\label{eqTransformationAdvance3Hydrogen}\tag{4}
Adv = 2\pi \left( \frac{e^2}{ 4\pi \epsilon_0 m_e c^2} \right)
\end{equation}
\)
per revolution.
Equation \((4)\) is for a circular orbit. It is assumed an advanced in a circular orbit is not observable for measurement. It is presumed impossible to tell at what point in a circular orbit an orbiting particle’s advance returns after completing one orbit. It is therefore assumed any observed measurement will predict a return to the same location each completed orbit. This inability to observe an advance in a circular orbit gives way to an unobserved energy within the orbital system. The total binding force of the orbit will be greater than the classically predicted binding energy.
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