SECTION 2-2: PLANK’S GRAVITATIONAL EQUATION

The Bohr model of the hydrogen atom is simple because a large distance separates the proton and electron. The gravitational forces are small and the electrical current repulsive forces are also small. The hydrogen atom has a main attractive force due to the coulomb attraction between the positive proton and the negative electron. It also has the repulsive centrifugal force of the electron spinning around the proton at C/137.036.

The forces within the proton are distributed forces. The dot-waves oscillate from the center of the proton to the radius of the proton. The oscillations are in three different planes due to the three-quark structure of the proton. The planes contract and expand and spin perpendicular to the plane. In addition, the entire proton rotates in a circle perpendicular to the three planes.

This causes each plane to have a complex physical motion. The hydrogen atom is not concerned with the six dimensions of space-time. The distance between the t(+), t(-), and t(o) universes are very small as compared with the Bohr radius. Therefore three-dimensional equations suffice quite well.

The proton oscillates from the proton radius to the Plank radius. In addition it most likely has a logarithmic waveshape during this oscillation. At the Plank radius, the six dimensions meet.

Some of the time, the mass of the dot-waves is at the Plank radius. At other times, the mass is spinning at the Proton radius. We could say that half the mass exists at the Plank radius and half the mass exists at the proton radius. We could also say that most of the time, the Proton masses exists at 0.7071 Rp.

The exact equations of the Proton are left to the mathematicians and physicists. From an engineering perspective, we want to know how the Proton works. Therefore we can produce force equations within the proton by equating all the mass of the proton at the proton radius to all the mass of the proton at the Plank radius. In addition we can say that all the four forces within the proton are equal to each other.

Therefore for the simple model, the gravitational attractive force equals the repulsive centrifugal force. Also the coulomb attraction force equals the magnetic repulsion force. In addition for the simplified equations, we can say that the four forces are all equal to each other.

Scientists and mathematicians can look at many variations of the above. Computer programs can produce complex solutions, which will provide somewhat different answers. However from an Engineering viewpoint we merely want a little understanding of the Proton forces which will agree with Plank’s gravitational equation.

We can now write the four equations of the Proton forces. The four equations must use the Plank equations as the guide. They can be written by inspection since we now know the velocity of the proton and the radius of the proton. Therefore:

11.706238KQQ/2Rp^2 = MpVp^2/ Rp (2-19)

8.911678E2 = 8.911684E2 (2-20)

Equation 2-18 specifies that the total distributed repulsive force within the proton is identical with the total centrifugal force. Both are repulsive forces. Notice that the electrical repulsive forces involve the charge Q and the Plank Charge 11.706238Q.

I do not know how Plank derived his equations. The study of his Equations provides interesting answers for the structure of the Proton. We can now relate the electrical repulsive Coulomb force to the magnetic attractive forces within the proton. Thus:

11.706238KQQ/2Rp^2 = (11.706238)^3 Uo Q^2 Vp^2 / 4π 2 Rp^2 (2-21)

8.911678E2 = 8.911686E2 (2-22)

Equation 2-21 specifies that the distributed coulomb repulsive forces within the proton are equal to the attractive forces due to the current flow within the proton. We now have three forces all equal to each other. The last force is the gravitational force. This force operates at the Plank radius and ties all the dot-waves together. Therefore:

11.706238KQQ/2Rp^2 = 2G Mp^2 / Rpl^2 (11.706238)^3 (2-23)

8.911678E2 = 8.911686E2 (2-24)

In Equation 2-23 we find that the gravitational force operating within the proton is quite large since it involves the centers of masses which are separated by the tiny Plank radius, Rpl.

The equations work as well if we only equate the mechanical forces to each other and the electrical forces to each other. Therefore both sets of forces could have different values. However for the present time, only the simple possibility of four equal forces has been chosen to study.

We can now solve for the gravitational constant in terms of the mass of the proton and the radius of the proton. From Equation 2-23:

G = (QRpl/RpMp)^2 (137.036)^2 K /4 (2-25)

As a check using Rp = 1.230926E-15, pl = 1.616252E-35, Q = 1.602176E-19, Mp= 1.672622E-27, and K = 8.987552E9, we get:

G = 6.67274E-11 (2-26)

We can now simplify the Gravitational constant. Since:

137.036 = hC/ 2π K Q^2 (2-27)

G = (Rpl hC / Q Mp Rp 4π)2 / K (2-28)

G = 6.674284E-11 (2-29)

From Equation 2-6 we obtain:

2π Rp Mp = h/Vp = 11.706238 h/2C (2-30)

G = (Rpl C^2 /11.706238 Q)^2 / K (2-31)

Using Equation 2-27 for 137.036K, we get

G = Rpl C^3 2π/h (2-32)

Solving for the Plank radius, we obtain:

pl = (Gh/2π C3)^0.5 (2-33)

We see that by using 2C/11.706238 for the proton velocity, we obtain the Plank length by rearranging all the equations. Plank came out with a set of equations, which appear to define the Proton very well. Once we add Planks equations to the Bohr orbit, we obtain the most important equations of the proton.

From this analysis we find that the radius and velocity we have chosen agrees with the experimental data for the radius of the proton. It also agrees with the work of Plank. Plank did not finish his work but he left us with a set of equations, which provides us with good answers for the proton.

This concludes the initial study of the proton from a Bohr/Plank perspective.