Newton's Law of Gravitation is that the force between masses m₁ and m₂ separated by a distance r is given by

F = Gm₁m₂/r²

where G is a constant equal to 6.67259×10^-11 m³/kg.

The mass of any body is essentially equal to the number of nucleons it contains times the mass of a nucleon; m_i=m_nn_i. Thus the force formula could be represented as

F = Gm_n²n₁n₂/r²

Thus the force constant in units of kg*m³/s² is Gm_n = (6.67259×10^-11)(1.6749×10^-27)² =1.871855×10^-64 kg*m³/s². The ratio of this constant to the product of Planck's constant and the speed of light in a vacuum, hc=1.9865×10^-25 kg*m³/s², is 9.42288×10^-40.

The Electrostatic Force

Coulomb's Law of Electrostatics is that the force between two charges q₁ and q₂ separated by a distance r is given by

F = (1/(4πε))q₁q₂/r²

where (1/(4πε)) is a constant equal to 9×10⁹ kg*m³/s².

The charge of any body is essentially equal to the net number of elementary charges it contains times the value of the elementary charge; q_i=q_en_i. Thus the force formula could be represented as

F = (1/(4πε))q_e²n₁n₂/r²

Thus the force constant in units of kg*m³/s² is (1/(4πε))q_e²=(9×10⁹)(1.60218x10^-19)² = 2.3103×10^-28 kg*m³/s². The ratio of this constant to hc=1.9865×10^-25 kg*m³/s² is 1.163×10^-3 or approximately 1/860. If h-bar, h=h/2π, had been used to compute the ratio the result would have come out 1/137, the familiar fine structure constant.

The Nuclear Force

Suppose the force between two nucleons is given by the formula

F = H*e^−λr/r²

The justification for this formula is that the nuclear force is carried by particles subject to decay; i.e., the π mesons. The population of remaining particles is a negative exponential function the time since emission which translates into a negative exponential function of distance. These remaining particles are spread over an area of 4πr². The intensity is thus proportional to e^−λr/r². For more on this model see Nuclear Force.

In that previous work an estimate was made of the magnitude of the constant H* based upon the physical characteristics of the deuteron. It is 1.92570×10^-25 kg*m³/s². These are the same units as those of hc. The magnitude of hc is 1.986259×10^-25. The ratio is 0.96876=1/1.03225. This is close enough to unity to suggest that the actual value of H* is hc. This would make the nuclear force formula

F = (hc)e^−λr/r²

The near-equality of the nuclear force constant with the value of hc may be merely an accident but it is certainly an intriguing one. It is notable that the mass of the π0 meson is about 3.4 percent less than the masses of the π± mesons.

[Update: The above estimate of H, 1.92570×10^-25 kg*m³/s², was based upon the mass deficit of the deuteron being 2.22457 MeV. It appears that the mass deficit is considerably greater than this value. An alternate estimate of H based upon the separation distance of the nucleons in a deuteron being 3.2 fermi is 3.392372×10^-26 kg*m³/s². This makes H equal to 0.17079hc=(1/5.855)hc.]

Natural Units

The estimate of λ is 6.57×10¹⁴ m^-1=1/(1.522 fermi). This makes ρ₀=1/λ=1.522 fermi, a natural unit of length. This length divided by the speed of light would be a natural unit of time,

τ₀=ρ₀/c=1.522×10^-15/3×10⁸
=5.07×10^-24 seconds.

Planck's constant is a natural unit of action, energy×time, so h/τ₀ would be a natural unit of energy. Its value is

E₀ = 6.626×10⁻³⁴/5.07×10⁻²⁴
= 1.306×10⁻¹⁰ joules (kg*m²/s²)

Through the Einstein relation, E=mc², there would be a natural unit of mass equal to

m₀ = E₀/c² = 1.306×10⁻¹⁰/9×10¹⁶
= 1.45×10⁻²⁷ kg.

This is notably close to the masses of the proton and the neutron; i.e., 1.6726×10⁻²⁷ and 1.6749×10⁻²⁷ kg, respectively. There is only about a 15 percent difference betweeen the computed m₀ and the masses of the proton and neutron.

The explanation for the computed natural unit of mass being close to the mass of the proton and neutron is prosaic. The scale length ρ₀ is computed from Yukawa's relation which involves h-bar, h, whereas in the above computation of the natural unit of energy Planck's constant h was used. This introduced a factor of 2π. If the same version of Planck's constant had been used in both places the mass of the pi meson would have been the result of the computation of a natural unit of mass. Instead what resulted was 2π times the mass of the pi meson. By coincidence (?) the mass of the proton is approximately 2π times the mass of the pi meson.

(To be continued.)