The importance of special relativistic effects in modelling ultra-fast outflows

Context. Outflows are observed in a variety of astrophysical sources. Remarkably, ultra-fast (v >= 0.1c), highly ionised outflows in the UV and X-ray bands are often seen in active galactic nuclei (AGNs). Depending on their kinetic power and mass outflow rate, E-out and M-out, respectively, these outflows may play a key role in regulating the AGN-host galaxy co-evolution process through cosmic time and metal-feeding the surrounding circum-/inter-galactic medium. It is therefore crucial to provide accurate estimates of the wind properties, including M-out and E-out.Aims. Here we concentrate on special relativistic effects concerning the interaction of light with matter moving at relativistic speed relatively to the source of radiation. Our aim is to assess the impact of these effects on the observed properties of the outflows and implement a correction for these effects in the existing spectral modelling routines.Methods. We define a simple procedure to incorporate relativistic effects in radiative transfer codes. Following this procedure, we run a series of simulations to explore the impact of relativistic effects for different outflow velocities and column densities.Results. The observed optical depth of the wind is usually considered a proxy for its column density N-H, independently of the velocity of the outflow. However, our simulations show that the observed optical depth of an outflow with a given N-H decreases rapidly as the velocity of the wind approaches relativistic values. This, in turn, implies that when estimating N-H from the optical depth, it is necessary to include a velocity-dependent correction, already for moderate velocities (e.g. v(out)greater than or similar to 0.05c). This correction linearly propagates to the derived quantities M-out and E-out.Conclusions. We demonstrate that special relativistic effects must be considered in order to obtain correct estimates of M-out and E-out for an outflow moving at a mildly relativistic speed relative to the illuminating source of radiation. As an example, we calculate the relativistically corrected values of M-out and E-out for a sample of similar to 30 ultra-fast outflows (UFOs) taken from the literature and find correction factors of 20-120% within the observed range of outflowing velocities (v(out)approximate to 0.1-0.3c). This brings the ratio between M-out and the disc accretion rate close or even above unity for the vast majority of the sources of the sample, highlighting the importance of the reported relativistic corrections to understand the growth of the most massive black holes. The next generation of high-sensitivity X-ray telescopes such as XRISM and Athena will provide a much more complete census of UFOs, especially in the fastest velocity regime where the relativistic corrections are increasingly important.