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Many-body theory is developed to calculate the $\gamma$-spectra for positronannihilation with valence and core electrons in the noble gas atoms. A properinclusion of correlation effects and core annihilation provides for an accuratedescription of the measured spectra [Iwata \textit{et al.}, Phys. Rev. Lett.{\bf 79}, 39 (1997)]. The theory enables us to calculate the enhancementfactors $\gamma_{nl}$, which describe the effect of electron-positroncorrelations for annihilation on individual electron orbitals $nl$. We findthat the enhancement factors scale with the orbital ionization energy $I_{nl}$(in electron-volt), as $\gamma_{nl}=1+\sqrt{A/I_{nl}}+(B/I_{nl})^{\beta}$,where $A\approx 40$~eV, $B\approx 24$~eV and $\beta\approx 2.3$.

$\gamma$-ray spectra and enhancement factors for positron annihilation spectra with core-electrons