If massive neutrinos are Dirac particles, the proposed PTOLEMY experiment will hopefully be able to discover cosmic neutrino background via \(\nu^{}_e + {^3}{\rm H} \to {^3}{\rm He} + e^-\) with a capture rate of \(\Gamma^{}_{\rm D} \approx 4~{\rm yr}^{-1}\). Recently, it has been pointed out that right-handed components of Dirac neutrinos could also be copiously produced in the early Universe and become an extra thermal or nonthermal ingredient of cosmic relic neutrinos, enhancing the capture rate to \(\Gamma^{}_{\rm D} \approx 5.1~{\rm yr}^{-1}\) or \(\Gamma^{}_{\rm D} \approx 6.1~{\rm yr}^{-1}\). In this work, we investigate the possibility to distinguish between thermal and nonthermal spectra of cosmic relic neutrinos by measuring the annual modulation of the capture rate. For neutrino masses of \(0.1~{\rm eV}\), we have found the amplitude of annual modulation in the standard case is \({\cal M} \approx 0.05\%\), which will be increased to \(0.1\%\) and \(0.15\%\) in the presence of additional thermal and nonthermal right-handed neutrinos, respectively. The future detection of such a modulation will be helpful in understanding the Majorana or Dirac nature of massive neutrinos.