NH$_{3}$ (1,1) hyperfine intensity anomalies in the Orion A molecular cloud

In LTE, the two inner satellite lines (ISLs) and the two outer satellite lines (OSLs) of the NH$_{3}$ (1,1) transition are each predicted to have equal intensities. However, hyperfine intensity anomalies (HIAs) are observed to be omnipresent in star formation regions, which is still not fully understood. In addressing this issue, we find that the computation method of the HIA by the ratio of the peak intensities may have defects, especially when being used to process the spectra with low velocity dispersions. Therefore we define the integrated HIAs of the ISLs (HIA$_{\rm IS}$) and OSLs (HIA$_{\rm OS}$) by the ratio of their redshifted to blueshifted integrated intensities and develop a procedure to calculate them. Based on this procedure, we present a systematic study of the integrated HIAs in the northern part of the Orion A MC. We find that integrated HIA$_{\rm IS}$ and HIA$_{\rm OS}$ are commonly present in the Orion A MC and no clear distinction is found at different locations of the MC. The medians of the integrated HIA$_{\rm IS}$ and HIA$_{\rm OS}$ are 0.921$\pm$0.003 and 1.422$\pm$0.009, respectively, which is consistent with the HIA core model and inconsistent with the CE model. Selecting those 170 positions where both integrated HIAs deviate by more than 3-$\sigma$ from unity, most (166) are characterized by HIA$_{\rm IS}$ 1, which suggests that the HIA core model plays a more significant role than the CE model. The remaining four positions are consistent with the CE model. We compare the integrated HIAs with the para-NH$_{3}$ column density ($N$(para-NH$_{3}$)), kinetic temperature ($T_{\rm K}$), total velocity dispersion ($\sigma_{\rm v}$), non-thermal velocity dispersion ($\sigma_{\rm NT}$), and the total opacity of the NH$_{3}$ (1,1) line ($\tau_{0}$). Their correlations can not be fully explained by neither the HIA core nor the CE model.