Studying the Doppler shifts and the temperature dependence of Doppler shifts in moss regions can help us understand the heating processes in the core of the active regions. In this paper we have used an active region observation recorded by the Extreme-ultraviolet Imaging Spectrometer (EIS) onboard Hinode on 12-Dec-2007 to measure the Doppler shifts in the moss regions. We have distinguished the moss regions from the rest of the active region by defining a low density cut-off as derived by Tripathi et al. (2010). We have carried out a very careful analysis of the EIS wavelength calibration based on the method described in Young et al. (2012). For spectral lines having maximum sensitivity between log T = 5.85 and log T = 6.25 K, we find that the velocity distribution peaks at around 0 km/s with an estimated error of 4-5 km/s. The width of the distribution decreases with temperature. The mean of the distribution shows a blue shift which increases with increasing temperature and the distribution also shows asymmetries towards blue-shift. Comparing these results with observables predicted from different coronal heating models, we find that these results are consistent with both steady and impulsive heating scenarios. However, the fact that there are a significant number of pixels showing velocity amplitudes that exceed the uncertainty of 5 km s$^{-1}$ is suggestive of impulsive heating. Clearly, further observational constraints are needed to distinguish between these two heating scenarios.
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