Poynting-Robertson effect on black-hole-driven winds

Layers of ionized plasma, in the form of winds ejected from the accretion disk of Supermassive Black Holes (SMBHs) are frequently observed in Active Galactic Nuclei (AGNs). Winds with a velocity often exceeding $0.1c$ are called Ultra-Fast-Outflows (UFOs) and thanks to their high power they can play a key role in the co-evolution between the SMBH and the host galaxy. In order to construct a proper model of the properties of these winds, it is necessary to consider special relativistic corrections due to their very high velocities. We present a derivation of the Poynting-Robertson effect (P-R effect) and apply it to the description of the dynamics of UFOs. The P-R effect is a special relativistic correction which breaks the isotropy of the radiation emitted by a moving particle funneling the radiation in the direction of motion. As a result of the conservation of the four-momentum, the emitting particles are subjected to a drag force and decelerate. We provide a derivation of the drag force caused by the P-R effect starting from general Lorentz transformations and assuming isotropic emission in the gas reference frame. Then, we derive the equations to easily implement this drag force in future simulations. Finally, we apply them in a toy model in which the gas particles move radially under the influence of the gravitation force, the radiation pressure and the drag due to the P-R effect. P-R effect plays an important role in determining the velocity profile of the wind. For a wind launched from $r_0=10r_s$ (where $r_S$ stands for the Schwarzschild radius), the asymptotic velocity reached by the wind is between $10$% and $24$% smaller than the one it would possess if we neglect the effect. This shows that the P-R effect should be taken into account when studying the dynamics of high-velocity, photoionized outflows in general.