Model-independent determination of the magnetic radius of the proton from spectroscopy of ordinary and muonic hydrogen

To date the magnetic radius of the proton has been determined only by means of electron-proton scattering, which is not free of controversies. Any existing atomic determinations are irrelevant because they are strongly model dependent. We consider a so-called Zemach contribution to the hyperfine interval in ordinary and muonic hydrogen and derive a self-consistent model-independent value of the magnetic radius of the proton. More accurately, we constrain not a value of the magnetic radius by itself, but its certain combination with the electric-charge radius of the proton, namely, ${R}_{E}^{2}+{R}_{M}^{2}$. The result from the ordinary hydrogen is found to be ${R}_{E}^{2}+{R}_{M}^{2}=1.35(12)\text{ }\text{ }{\mathrm{fm}}^{2}$, while the derived muonic value is $1.49(18)\text{ }\text{ }{\mathrm{fm}}^{2}$. That allows us to constrain the value of the magnetic radius of proton ${R}_{M}=0.78(8)\text{ }\text{ }\mathrm{fm}$ at the 10% level.