Issue 
A&A
Volume 633, January 2020



Article Number  A55  
Number of page(s)  5  
Section  Astrophysical processes  
DOI  https://doi.org/10.1051/00046361/201936797  
Published online  10 January 2020 
The importance of special relativistic effects in modelling ultrafast outflows
^{1}
Department of Physics, University of Rome “Tor Vergata”, Via della Ricerca Scientifica 1, 00133 Rome, Italy
^{2}
INAF – Osservatorio Astronomico di Roma, Via Frascati 33, 00078 Monteporzio, Italy
email: alfredo.luminari@roma2.infn.it
^{3}
Department of Astronomy, University of Maryland, College Park, MD 20742, USA
^{4}
NASA/Goddard Space Flight Center, Code 662, Greenbelt, MD 20771, USA
^{5}
Department of Physics and Astronomy, James Madison University, Harrisonburg, VA 22807, USA
^{6}
INAF – Osservatorio Astronomico di Trieste, via G.B. Tiepolo 11, 34131 Trieste, Italy
Received:
27
September
2019
Accepted:
25
November
2019
Context. Outflows are observed in a variety of astrophysical sources. Remarkably, ultrafast (v ≥ 0.1c), highly ionised outflows in the UV and Xray bands are often seen in active galactic nuclei (AGNs). Depending on their kinetic power and mass outflow rate, Ė_{out} and Ṁ_{out}, respectively, these outflows may play a key role in regulating the AGN–host galaxy coevolution process through cosmic time and metalfeeding the surrounding circum/intergalactic medium. It is therefore crucial to provide accurate estimates of the wind properties, including Ṁ_{out} and Ė_{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 velocitydependent correction, already for moderate velocities (e.g. v_{out} ≳ 0.05c). This correction linearly propagates to the derived quantities Ṁ_{out} and Ė_{out}.
Conclusions. We demonstrate that special relativistic effects must be considered in order to obtain correct estimates of Ṁ_{out} and Ė_{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 Ṁ_{out} and Ė_{out} for a sample of ∼30 ultrafast outflows (UFOs) taken from the literature and find correction factors of 20 − 120% within the observed range of outflowing velocities (v_{out} ≈ 0.1 − 0.3c). This brings the ratio between Ṁ_{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 highsensitivity Xray 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.
Key words: line: profiles / opacity / radiative transfer / relativistic processes / quasars: absorption lines / accretion / accretion disks
© ESO 2020
1. Introduction
Outflows are ubiquitously observed from a variety of astrophysical sources, and their impact on the surrounding environment depends on their energy. In particular, mildly relativistic and ionised outflows from the innermost regions of active galactic nuclei (AGNs) are often seen in UV and Xray absorption spectra (e.g. Chartas et al. 2002; Tombesi et al. 2010; Rodríguez Hidalgo et al. 2011; Bruni et al. 2019) and may carry sufficient energy to regulate both the growth of the central supermassive black hole (SMBH) and the evolution of the surrounding host galaxy (Costa et al. 2014; FaucherGiguère & Quataert 2012; Tombesi et al. 2015; Zubovas & King 2012). This critically depends on the kinetic power of these outflows, which in turn depends on their velocity and on their mass flux (Di Matteo et al. 2005; King & Pounds 2015).
The lineofsight (LOS) velocity is typically inferred via the blueshift of the absorption features imprinted by the outflowing material onto the continuum emission of the central source, compared to the systemic redshift of the host galaxy. The mass outflow rate ̇M_{out}, instead, for a given covering factor and distance of the outflow, is estimated by measuring the optical depth of the absorption features. The observed optical depth is considered a proxy of the outflow column density N_{H} along the LOS, independently of its outflow velocity v_{out}.
In this work we show that this assumption no longer holds for outflows escaping the central continuum source of radiation with velocities corresponding to a fraction of the speed of light c (e.g. v_{out} ≳ 0.05c). For these outflows the observed (i.e. apparent) optical depth of the spectral features produced by the absorbing material significantly underestimates the intrinsic N_{H} and, consequently, the mass transfer rate of the outflows. Therefore, a velocitydependent correction must be adopted in the estimate of N_{H} to account for this effect.
This pure specialrelativistic effect is universal (i.e. it applies to any fastmoving outflow intercepting the LOS), and affects not only our estimate of the kinetic power of the outflow, but also the ability of the radiative source to effectively accelerate the outflow outwards. For AGN outflows, this may have deep implications on the feedback mechanism and the coevolution with respect to the host galaxy (Kormendy & Ho 2013).
The paper is organised as follows. In Sect. 2 we give an overview of the special relativity treatment for a fastmoving gas embedded in a radiation field. In Sect. 3 we show how to incorporate this treatment in modelling outflow spectra. In Sect. 4 we discuss the results and their implications on estimating Ṁ_{out} and Ė_{out}, and we summarise in Sect. 5.
2. Special relativistic transformation in the outflow reference frame
According to special relativity, the luminosity L′ seen by a clump of gas moving at relativistic speed is reduced of a factor Ψ, with respect to a static gas, as
where L is the luminosity seen by an observer at rest and Ψ (i.e. the deboosting factor) is defined as
where , β = v_{out}/c, v_{out} is the gas velocity, and θ is the angle between the incident luminosity L and the direction of motion of the gas. Figure 1 shows Ψ as a function of v_{out} for θ = 180°, corresponding to a radial outward motion of the gas. The deboosting factor is due to the combination of the spacetime dilatation in the gas reference frame K′, and the relativistic Doppler shift of the received radiation (Rybicki & Lightman 1986).
Fig. 1. Deboosting factor Ψ in the gas reference frame K′ as a function of v_{out} assuming θ = 180°. For speeds lower than 0.1c, the radiation intercepted by the outflow and by the (restframe) observer at infinity are virtually the same. For higher speeds, the fraction of intercepted radiation drops dramatically due to special relativistic effects. 

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Using Eq. (1), the radiative intensity (i.e. the luminosity per solid angle) received by the outflowing gas in K′ can be written as a function of the intensity in the rest frame K, as
where dE, dt, and dΩ corresponds to the energy, time, and solid angle intervals in K. Specifically, in Eq. (3), ψdE is the energy transformation term, which represents the Doppler shift of the wavelengths in K′. The second term, , indicates a reduction of the intensity due to the spacetime dilatation in K′.
It should be noted that Eqs. (1) and (3) also describe the emission from gas moving at relativistic velocity, as usually observed in highvelocity systems such as jets in blazars and gammaray burts (Urry & Padovani 1995; Ghisellini et al. 1993). When radiation is emitted along the direction of motion (i.e. θ ≈ 0°), Ψ increases with increasing v_{out}, while Ψ ≤ 1 when it is emitted perpendicularly or backward (θ = 90° and 180°, respectively). The overall result is to concentrate the emitted radiation into a narrow cone along the direction of motion, an effect known as relativistic beaming (Rybicki & Lightman 1986; Event Horizon Telescope Collaboration 2019).
Another way of describing the reduction of the luminosity seen by the outflowing gas is the following. In K′, the luminosity source appears to be moving away with velocity v_{out} and θ = 180° (for a pure radial motion), which results in a deboosting of the received luminosity due to the relativistic beaming, according to Eq. (1).
3. Modelling outflow absorption spectra including special relativistic effects
We include these special relativistic corrections in modelling spectral absorption features from the outflowing gas, according to the following procedure (see Appendix A for a detailed description):

The incident spectrum S_{I}(K) is transformed from K to K′, obtaining S_{I}(K′), according to Eq. (3);

S_{I}(K′) is given as input to the radiative transfer code to calculate the transmitted spectrum in the outflowing gas frame K′, S_{T}(K′);

The relativisticcorrected transmitted spectrum in K (i.e. S_{out}(K)) is given by
where Δ ≡ 1 − ψ^{3}. The term S_{T}(K′) ⋅ ψ^{−1} indicates the spectrum S_{T}(K′) in Dopplershifted frequencies (from K′ to K).
We note that in the lowvelocity limit v_{out} ≪ c, Ψ ≈ 1, Δ ≈ 0, and the resulting spectrum is S_{out}(K) = S_{T}(K′) ⋅ ψ^{−1}, as is usually calculated. For the opposite highvelocity regime v_{out} → c, Ψ ≈ 0, and the outflowing gas does not interact with the ionising radiation. In fact, S_{I}(K′) and S_{T}(K′) have null intensity (see Eq. (3)), Δ ≈ 1, and S_{out}(K)≈S_{I}(K).
We use the radiative transfer code XSTAR, v2.5 (Kallman & Bautista 2001) to calculate S_{out}(K), which is the spectrum as seen by a rest frame observer in K.
Figure 2 shows the Xray spectrum in the range 6 − 16 keV of a powerlaw continuum source with Γ = 2 and an ionising luminosity L_{ion} = 5 × 10^{46} erg s^{−1} in the 1–1000 Ry (1 Ry = 13.6 eV) energy interval, modified by an absorber with v_{out} = 0.0, 0.3, and 0.5c. In all cases, we assume an absorbing column density of N_{H} = 6 × 10^{23}cm^{−2} and an ionisation parameter , which are typical of ultrafast outflows (UFOs) observed in AGNs (Reeves et al. 2009; Tombesi et al. 2011; Gofford et al. 2013). The middle and right panels of Fig. 2 also report the v_{out} = 0 case for an easier comparison. It can be seen that the absorption features related to the relativistically outflowing gas are both blueshifted and are significantly weaker than the v = 0 case. This effect dramatically increases for increasing velocity, as shown in a more quantitative way in Fig. 3, which displays the column density N_{H} necessary to reproduce outflow absorption features with a fixed optical depth, as a function of v_{out}. The required column density corresponds to N_{H} = 10^{23} cm^{−2} for v_{out} = 0, and it increases by an order of magnitude for v_{out} = 0.8c. It is interesting to note that this effect may introduce an observational bias in current Xray data, which are typically restricted to E < 10 keV, making outflows at higher velocity more difficult to detect due to the weakening of their spectral features at E < 10 keV.
Fig. 2. Absorption spectra for increasing outflow velocity: v_{out} = 0.00 (left panel), = 0.30 (centre panel), = 0.50c (right panel). For comparison, in the centre and right panels we show the absorption spectrum for v_{out} = 0.00c (in light grey). See Sect. 3 for details on the spectral parameters used in this simulation. 

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We also note that Schurch & Done (2007) and Saez & Chartas (2011) presented AGN outflow models including special relativistic effects to provide an estimate of both N_{H} and ξ. However, the two studies seem not to account properly for the reduction of the optical depth in the calculation of S_{out}(K). In Eq. (4) the relativisticcorrected optical depth of the wind is preserved by transforming the transmitted spectrum back to the source rest frame K, while this aspect has not been considered in these studies.
4. Discussion
Mass and kinetic energy transfer rates of the outflow (Ṁ_{out} and Ė_{out}, respectively), linearly depends on N_{H}. Specifically, Ṁ_{out} can be calculated as (Crenshaw & Kraemer 2012)
where r, μ, m_{p}, and C_{f} are the distance from the source, the mean atomic weight (≈1.4 for solar abundances), the proton mass, and the covering factor of the outflow, respectively, and Ė_{out} is defined as . Correct estimates of Ṁ_{out} and Ė_{out} are of fundamental importance to testing theoretical models of twophase expansion of AGN outflows towards galaxy scales, in which kiloparsecscale galactic outflows are the result of the shock of ultrafast, accretion discscale outflows onto the ISM (FaucherGiguère & Quataert 2012; Zubovas & King 2012; Menci et al. 2019).
We find that neglecting special relativistic effects will result in an underestimate of N_{H} and, in turn, of Ṁ_{out} and Ė_{out}. As an example, we correct for these effects in the reported values of Ṁ_{out}, Ė_{out} for the UFOs observed in AGNs from Gofford et al. (2015) and Fiore et al. (2017) (see Fig. 3). Specifically, for the UFOs in Gofford et al. (2015) we use the average values between the reported Ṁ_{out}, Ė_{out} calculated using r_{min} and those using r_{max}, where r_{min} (r_{max}) is the minimum (maximum) inferred launching radius. Values of Ṁ_{out}, Ė_{out} reported in Fiore et al. (2017) are calculated in the same way. In Fig. 4 we plot the ratio of the relativisticcorrected energy rates to the original values as a function of v_{out}. The value of is a factor of > 2 higher than for the fastest observed outflows (v_{out} ≥ 0.3c).
Fig. 3. Absorbing gas N_{H} required to reach a given value of the optical depth as a function of v_{out}. Spectral parameters are as in Fig. 2. 

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As shown in Fig. 3, we expect even higher ratios for higher velocity outflows. In this respect, the improved sensitivity and resolution of the new generation of Xray telescopes, such as XRISM and Athena, will be particularly promising and it will allow us to partially overcome the observational bias discussed in Sect. 3. Interestingly, evidence for velocities ≥0.4 − 0.5c have indeed already been reported for some highluminosity quasars, such as PDS 456 and APM 08279+5255 (see e.g. Reeves et al. 2018; Chartas et al. 2009).
Figure 5 shows the ratio of the relativisticcorrected mass loss rate to the mass accretion rate Ṁ_{acc}, as a function of λ_{Edd} ≡ L_{bol}/L_{Edd} (i.e. the ratio of the bolometric and Eddington luminosities). We derive the mass accretion rate as , assuming η = 0.1 as in a standard Shakura & Sunyaev (1973) accretion disc. We note that for almost half of the sources , indicating that is comparable to (or higher than) the mass accretion rate of the disc. This may indicate a limit for the outflow lifetime, after which the accretion disc is depleted and it can no longer sustain the outflow (see e.g. Belloni et al. 1997). The plot also shows an apparent lack of sources with at high λ_{Edd}. However, the sample is too small to allow us to draw any conclusions. Future observations of high λ_{Edd} AGNs are needed to shed light on this aspect.
Finally, we compare the outflow momentum rate, defined as Ṗ_{out} = Ṁ_{out}v_{out}, with the momentum rate of the radiation of the AGN (i.e. ). We obtain a median value of 0.64 for the original sample, and 0.96 after the relativistic correction. Interestingly, the latter value is consistent with unity, as expected for outflows accelerated through the continuum radiation pressure (the socalled Eddington winds; see e.g. King & Pounds 2015).
It is worth noting that in Eq. (2) we use the total gas velocity v_{out}. We assume that the velocity v_{los} along the LOS coincides with v_{out}. As a result, the derived relativistic correction must be regarded as a conservative limit. The correction would increase in the presence of an additional velocity component v_{⊥} perpendicular to the LOS, which implies .
As an example, we consider the MHD model presented by Fukumura et al. (2010, 2014), where the outflowing gas is launched from the accretion disc at Keplerian velocity. Close to the launching radius, most of the velocity is in the direction of the disc rotation ϕ, and it is converted to radial velocity at higher distances (i.e. close to the Alfven point) thanks to MHD effects. For a wind launched at r_{0} = 10r_{G}^{1}, the rotational speed has a roughly constant value of v_{ϕ} = 0.3c until r ≈ 100r_{G}, while the radial velocity (i.e. the component parallel to the LOS) has an average value of v_{LOS} ≈ 0.2c. In Fig. 1 we show that when v_{out} = 0, Ψ = 1 and the relativistic effects are absent. On the other hand, when v_{out} → c, Ψ ≈ 0 and the effects are the highest. Using v_{LOS} as a proxy for v_{out} in Eq. (2) yields Ψ = 0.8, while using the total velocity (i.e. the composition between v_{r} and v_{ϕ}) gives Ψ = 0.6, a factor of 0.25 lower.
It is worth noting, as Abramowicz et al. (1991) already pointed out, that the observed optical depth of the gas depends on the velocity of the outflow relative to the source of radiation (see their Eqs. (2.1) and (2.2)). Specifically, they concentrated on an outflowing wind that is optically thick with respect to Thompson scattering, and calculated the integrated luminosity of its photosphere. Moreover, Sumitomo et al. (2007) and Fukue & Sumitomo (2009) considered the impact of special relativistic effects on the emitted radiation from a fast spherical wind in stars and accreting sources, such as quasars and ULXs. These works further underline the importance of relativistic effects for radiationmatter interaction at high speeds, along with the photoelectric and resonant absorption we investigate in this work.
5. Conclusions
In this work we have shown that special relativistic effects are of fundamental importance for a correct modelling of the outflow spectral features, even for mildly relativistic velocities (v_{out} ≳ 0.05c, see Fig. 1). We have also provided a simple procedure that can be implemented in any radiative transfer code to take into account these effects.
We observe a significant reduction of the optical depth of the outflowing gas for fixed N_{H} and increasing v_{out} (Figs. 2 and 3). This indicates that it is necessary to include a velocitydependent correction when estimating N_{H} of the outflow from the optical depth derived by spectral fitting. This correction is already significant (a factor of ≈0.5) for an outflow velocity of v_{out} = 0.1c and reaches values of a factor of ≳10 for v_{out} ≥ 0.8c (see Fig. 3).
The derived mass and kinetic energy transfer rates linearly depend on N_{H} (see Eq. (5)), and hence must be corrected accordingly. For AGN outflows, this correction can significantly increase both Ṁ_{out} and Ė_{out} and, in turn, the impact of the outflow on the surrounding environment and on the feedback mechanism. We plot in Figs. 4 and 5 the relativisticcorrected Ė_{out}, Ṁ_{out} for a sample of ultrafast outflows (UFOs) in AGNs reported in the literature. These pictures further underline the importance of relativistic corrections for a correct assessment of the outflow properties. Furthermore, these corrections are increasingly important in view of the next generation of highsensitivity Xray telescopes, which will increase the accuracy of the detection of mildly relativistic outflows, as discussed in Sect. 4.
Fig. 4. Ratio of the relativisticcorrected energy transfer rates to the original values as a function of v_{out}, for a sample of ultrafast outflows observed in AGNs (see Sect. 4 for details). 

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Fig. 5. Ratio of the relativisticcorrected outflow mass rate to the inflow mass rate Ṁ_{acc}, as a function of λ_{Edd} ≡ L_{bol}/L_{Edd}. The sample is as in Fig. 4. The dotted line corresponds to . 

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The effects discussed in Sect. 2 have further implications on the radiative driving exerted on the outflowing gas, which will be discussed in a separate work (Luminari et al., in prep.). Moreover, we also plan to present a new version of the Xray spectral modelling code WINE (Luminari et al. 2018), which includes a relativisticcorrected radiative transfer treatment according to the procedure of Sect. 3.
Moreover, in some of the most popular codes, such as XSTAR, the emissivity of all the atomic lines listed in the atomic database is saved in a separated file. Emissivities can be used to build relativisticcorrected outflow emission spectra, as we will illustrate in detail in a forthcoming paper.
Acknowledgments
We thank Stefano Ascenzi for useful discussions and Tim Kallman for having provided custom XSTAR packages. AL, EP, FT, LZ acknowledge financial support from the Italian Space Agency (ASI) under the contract ASIINAF n.201714H.0. FT acknowledges support by the Programma per Giovani Ricercatori – anno 2014 “Rita Levi Montalcini”. FF acknowledges support from INAF under the contract PRININAF2016 FORECAST, and ASI/INAF contract I/037/12/0.
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Appendix A: Algorithm for special relativity corrections
In this appendix we provide a detailed description of the procedure for special relativity corrections outlined in Sect. 3. Following Eq. (3), the incident spectrum in the outflowing gas reference frame S_{I}(K′) is obtained from S_{I}(K) by multiplying the frequencies by a factor ψ and the intensity by a factor ψ^{3}.
For a given set of outflow parameters (N_{H}, log(ξ_{0}),n_{0}, v_{out}), we run a radiative transfer simulation by using S_{I}(K′) as incident spectrum. As a result we obtain the transmitted spectrum S_{T}(K′) displaying the absorption features due to the outflowing gas^{2}.
We then calculate the difference spectrum as follows:
Accordingly, S_{T − I}(K′) represents the absorption features produced by the outflowing gas, with the relativisticcorrected optical depth. As the next step we calculate the relativisticcorrected, restframe absorbed spectrum as
where S_{T − I}(K′) ⋅ ψ^{−1} represents the difference spectrum in rest frame (K) frequencies, which is obtained by dividing the frequencies by a factor ψ. Using Eq. (A.1), we can thus rewrite Eq. (A.2) as
where Δ ≡ 1 − ψ^{3} and S_{I}(K)⋅Δ indicates a scaling of the intensity of the spectrum S_{I}(K) of a factor Δ.
In our calculations we assume that the outflow has a net velocity v_{out} and direction θ. From a physical point of view, v_{out} and θ correspond to the average velocity and direction of the outflow, respectively. Therefore, if a turbulent velocity component is present, the above discussion is still valid, provided that v_{turb} ≪ v_{out}. Furthermore, if the outflowing velocity is a function of the spatial coordinates (i.e. ) the above procedure can be implemented by dividing the outflow into small slabs, and assuming v_{out} to be constant in each of them. Finally, the treatment of more complicated scenarios for requiring a firstprinciple approach are beyond the scope of the present paper.
All Figures
Fig. 1. Deboosting factor Ψ in the gas reference frame K′ as a function of v_{out} assuming θ = 180°. For speeds lower than 0.1c, the radiation intercepted by the outflow and by the (restframe) observer at infinity are virtually the same. For higher speeds, the fraction of intercepted radiation drops dramatically due to special relativistic effects. 

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In the text 
Fig. 2. Absorption spectra for increasing outflow velocity: v_{out} = 0.00 (left panel), = 0.30 (centre panel), = 0.50c (right panel). For comparison, in the centre and right panels we show the absorption spectrum for v_{out} = 0.00c (in light grey). See Sect. 3 for details on the spectral parameters used in this simulation. 

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In the text 
Fig. 3. Absorbing gas N_{H} required to reach a given value of the optical depth as a function of v_{out}. Spectral parameters are as in Fig. 2. 

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In the text 
Fig. 4. Ratio of the relativisticcorrected energy transfer rates to the original values as a function of v_{out}, for a sample of ultrafast outflows observed in AGNs (see Sect. 4 for details). 

Open with DEXTER  
In the text 
Fig. 5. Ratio of the relativisticcorrected outflow mass rate to the inflow mass rate Ṁ_{acc}, as a function of λ_{Edd} ≡ L_{bol}/L_{Edd}. The sample is as in Fig. 4. The dotted line corresponds to . 

Open with DEXTER  
In the text 
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