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Saturday, February 6, 2016

One Pager: Proton to Electron Mass Ratio and Conclusion of Proton Radius Puzzle


Equation for the proton to electron mass ratio:
$$\mu={m_p\over m_e} = {\alpha^2\over\pi r_pR_H}=1836.15267$$
$m_p=$ mass of proton
$m_e=$ mass of electron
$\alpha=$ fine-structure constant
$r_p=$ radius of proton
$R_H=$ Rydberg constant

comes from combining the Rydberg equation for electron mass:
$$m_e={2R_Hh\over c\alpha^2}$$
$c=$ speed of light
$h=$ Planck's constant

with Nassim Haramein's equation for the proton mass-radius relationship:
$$m_pr_p=4\ell m_{\ell}$$
$\ell=\sqrt{\hbar G\over c^3}=$ Planck length
$m_{\ell}=\sqrt{\hbar c\over G}=$ Planck mass
$\hbar={h\over 2\pi}$ = Reduced Planck's constant
$G=$ Universal Gravitational constant

CODATA value for ${m_p\over m_e}=1836.15267389(17)$

calculated mass ratio using my derived equation:
1836.15267(421379) <- only good to 9 digits due to limited resolution of CODATA for fundamental constants.

(By the way, this is confirmation to the resolution of the proton radius puzzle:
$$r_p=0.841235640294664 fm$$ )
Mark Rohrbaugh - February 6, 2016

The Surfer, OM-IV