On the importance of special relativistic effects in modelling ultra-fast outflows

Outflows are observed in a variety of astrophysical sources. Remarkably, ultra-fast ($v\geq 0.1c$), outflows in the UV and X-ray bands are often seen in AGNs. Depending on their energy and mass outflow rate, respectively $\dot{E}_{out}, \dot{M}_{out}$, such outflows may play a key role in regulating the AGN-host galaxy co-evolution process through cosmic time. It is therefore crucial to provide accurate estimates of the wind properties. 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 relativistic correction in the existing spectral modelling routines. 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 these effects on the simulated spectra, for different $v$ and column densities of the outflow. The observed optical depth is usually considered a proxy for the wind $N_H$, independently on its velocity. However, our simulations show that the observed optical depth of an outflow with a given column density $N_H$ 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. $v \geq 0.05c$). This correction linearly propagates to the derived $\dot{M}_{out}, \dot{E}_{out}$. As an example of these effects, we calculate the relativistically corrected values of $\dot{M}_{out}$ and $\dot{E}_{out}$ for a sample of $\sim 30$ Ultra-Fast Outflows taken from the literature, and find correction factors of $20-120 \%$ within the observed range of outflowing velocities.