Can warm absorbers be driven by ultra-fast outflows?

Observations show that the mass flux of UFOs is comparable to the accretion rate onto the black hole (Gofford et al. 2015). Therefore, in this paper, we assume that the mass flux of UFOs equals to the accretion rate onto the black hole. We set the accretion rate of the black hole by using the bolometic luminosity ($L_{\rm bol}$) of the AGNs. The black hole accretion rate is calculated by $\dot{M}_{\rm BH}=L_{\rm bol}/(\eta c^{2})$, with $\eta=0.1$ being the radiative efficiency. In some models we set $L_{\rm bol}=0.1L_{\rm Edd}$, with $L_{\rm Edd}$ being Eddingtong luminosity. In other models, we set $L_{\rm bol}=0.5L_{\rm Edd}$ (see Table 1). One can also calculate the accretion rate of the black hole based on the accretion rate at the inner radial boundary. In such case, because our inner radial boundary is far away from the black hole, one need to assume the radial profile of the mass inflow rate in the region between our inner radial boundary and the black hole horizon. This will complicate the problem. Therefore, in this paper, for simplicity, we give a fixed value to the black hole accretion rate (or $L_{\rm bol}$) and assume that it does not vary with time.

Tombesi et al. (2011) found that the mean and maximum velocities of UFOs are $0.1c$ and $0.3c$, respectively. In this paper, we set velocities of UFOs either equal to $0.1c$ or $0.3c$.

The third important parameter of UFOs is its opening angle. We set the opening angle of the UFOs according to the observations of UFOs in some radio-loud AGNs. Tombesi et al. (2014) analyzed the properties of UFOs in some radio-loud AGNs. The inclination angel of the jet in these AGNs is known. Therefore, it is easy to know the angle at which UFOs are present. It is reported that UFOs are detected over a wide range of jet inclination angles ($\sim 10^{\circ}-70^{\circ}$; see also Mehdipour & Costantini 2019). In our models, we inject UFOs at the inner boundary in the region $10^{\circ}\leq\theta\leq 70^{\circ}$ (see the definition of $\theta$ below). The density and velocity of UFOs are assumed to be uniformly distributed in this angular region.

The sound speed of UFOs gas should be significantly smaller than its bulk velocity. Therefore, the internal energy flux of UFOs should be significantly smaller than its kinetic energy flux. In this paper, we set the temperature of the UFOs injected at the inner radial boundary equals to the gas temperature at its adjacent active zone. We have done test by setting a fixed temperature to UFOs, we find that the effects of temperature of UFOs on the results are negligibly small.

The mass of the central black hole is set to be $M_{\rm BH}=10^{9}M_{\odot}$, with $M_{\odot}$ being solar mass. We carry out two-dimensional hydrodynamic simulations by using the ZEUS-MP code (Hayes et al. 2006). The spherical coordinates ($r$,$\theta$,$\phi$) are employed. The equations solved are as follows,

$\rho$, $\bf v$, $e$, $p$, $\Phi$, $\dot{E}$ are respectively gas density, velocity, gas internal energy per unit volume, gas pressure, gravitational potential of the central black hole, and the net heating/cooling rate per unit mass. Ideal gas equation of state $p=(\gamma-1)e$ is adopted, with $\gamma=5/3$ .

For an AGN, it is generally believed that a compact hot corona is above and below the central black hole (Reis & Miller 2013; Uttley et al. 2014; Chainakun et al. 2019). The corona emits X-ray photons. Observations of some luminous AGNs have shown that X-ray photons contribute a little to the bolometric luminosity. Therefore, as in previous pioneer work, we assume that X-rays contribute 5 percents of the bolometric luminosity (e.g., Liu et al. 2013). The left 95 percents of the bolometric luminosity are mainly optical/UV photons. We further assume that the X-ray photons are spherically emitted. Therefore, the X-ray flux is

$\tau_{\rm X}=\int_{0}^{r}\rho\kappa_{\rm es}dr$ is the scattering optical depth, with $\kappa_{\rm es}=0.4\ {\rm cm^{2}\ g^{-1}}$ being the Thomson scattering opacity.

The last term in Equation (3) is the cooling/heating term. The X-rays from the central black hole region can cool/heat gas by Compton process. The gas can also be heated by photoionization process and cooled by recombination process. Bremsstrahlung cooling and line cooling are also considered. The heating/cooling equations used in this paper are same as those in Proga et al. (2000). The cooling/heating rates are for gas with cosmic abundances. They are as follows,

$S_{\rm c}$, $G_{\rm X}$, $B_{\rm r}$ and $L_{\rm line}$ are Compton heating/cooling, photoionization heating-recombination cooling, bremsstrahlung cooling, and line cooling, respectively. In these equations, ${\rm n}=\rho/\mu m_{\rm p}$ is gas number density, with $\mu=1$ being mean molecular weight and $m_{\rm p}$ being proton mass, respectively. ${\rm T}$ is gas temperature. ${\rm T_{X}}$ is the Compton temperature of the X-ray photons. As done by Mościbrodzka & Proga (2013), we set ${\rm T_{X}}=1.16\times 10^{8}$ K. $\xi$ is the ionization parameter which is defined as follows,

The net heating/cooling rate in Equation (3) is expressed as $\rho\dot{E}=S{\rm c}+G{\rm x}-B{\rm r}-L_{\rm line}$.

We study the interaction of UFOs with the ISM in the region from $1{\rm parsec}$ to $1000\ {\rm parsec}$. To make the problem as simple as possible, we assume that the ISM has zero angular momentum. Initially, we put ISM with uniform density ($\rho_{0}$) and temperature ($T_{0}$) in our computational domain. After a period of evolution, a quasi-steady state of accretion towards the center forms. Then, we inject UFOs from the inner radial boundary. We simulate the region above the midplane, therefore, in $\theta$ direction, we have the domain $0\leq\theta\leq\pi/2$. Our resolution is $192\times 64$. In $r$ direction, logarithm grids are adopted in order to well resolve the inner region. In $\theta$ direction, grids are uniformly spaced.

At the inner radial boundary, in the region $\sim 10^{\circ}-70^{\circ}$, we inject UFOs. For other $\theta$ angles, we use outflow boundary conditions. We set the boundary conditions at the outer radial boundary as follows. For a given $\theta$ angle, when we find gas at the last active zone at this angle $v_{r}<0$, then, we inject gas into the computational domain at this angle. The injected gas density and temperature are same as those for the initial conditions. If we find $v_{r}>0$ at the last active zone at a given $\theta$ angle, then at this angle, outflow boundary conditions are employed. The settings can guarantee continuously gas supplies at the outer boundary. At $\theta=0$, axis-of-symmetry boundary conditions are adopted. At $\theta=\pi/2$, reflecting boundary conditions are employed.

We find that the angular location of WAs is changing with time. In order to illustrate this result, we plot Figure 1. This figure shows the distributions of WAs. Colors are for logarithm density of WAs. Arrows show the velocity of WAs. In this figure, the left and right panels are for $5.3\times 10^{7}$ year and $8.4\times 10^{7}$ year, respectively. Note that in this figure, we only show the distribution of WAs. In the regions without WAs, we set the color to dark blue. It does not mean that the regions without WAs have significantly low density. The purpose of plotting this figure in this way is try to protrude the WAs. It can be easily seen that at $5.3\times 10^{7}$ year, WAs are mainly distributed in the region $48^{\circ}<\theta<82^{\circ}$. There is also little amount of WAs in the region $\theta<30^{\circ}$. At $8.4\times 10^{7}$ year, WAs are distributed in the region $\theta<45^{\circ}$. Figure 2 shows the angular distribution of the column density of WAs at $5.3\times 10^{7}$ year (solid line) and $8.4\times 10^{7}$ year (dashed line). This figure also indicates the changing of angular locations of WAs with time. For an AGN, our viewing angle is fixed. Our results indicate that for an AGN, WAs can be observed intermittently. For the observations do not show signiture of WAs, it might not mean that WAs are not present. Instead, WAs may move out of our line of sight.

There are two possible reasons for the changing of angular distributions of WAs. The first reason may be that the angular locations of WAs do vary with time. In order to illustrate the second reason, we plot Figure 3. In this figure, we plot the two-dimensional properties of the flow at $5.3\times 10^{7}$ year. Colors show logarithm density including both inflows and outflows. Arrows show unit velocity vector. Comparing this figure to the left panel of Figure 1, we can see that WAs are just parts of the total outflows. For example, there are outflows at $\sim 38^{\circ}$ close to the outer radial boundary. The outflows at this location do not satisfy the conditions of WAs. We call the outflows do not satisfy the conditions of WAs as non-WAs outflow. If the outflows here keep present but with the properties changing with time, the non-WAs outflows may become WAs. Therefore, the second possible reason for the changing of angular distributions of WAs is that due to the changing of properties, the WAs/non-WAs outflow can become non-WAs/WAs outflows.

The time-averaged covering factors of the WAs listed in Table 1 are calculated as follows. We first calculate the covering factor at each snapshot. Then, we do the time-average for the covering factors at different snapshots. The time-averaged covering factor is $47\%$.

A interesting question is that how much mass can be carried away by WAs. We plot the radial profile of the mass flux of WAs in the left panel (solid line) of Figure 4. The mass flux of WAs increases with the increase of radius. This means that when WAs move outwards, more and more gas becomes WAs. The mass flux of WAs is a function of radius. The mass flux of WAs listed in Table 1 is its maximum value. The maximum mass flux of WAs can be more than two orders of magnitude higher than the black hole accretion rate. As introduced above, in addition to WAs, there are also non-WAs outflows. In the left panel of Figure 4, we also plot the mass flux of all the outflows (including both WAs and non-WAs) using a dashed line. The mass flux of all the outflows is more than one order of magnitude higher than that of the WAs.

The solid line in the middle panel of Figure 4 denotes the kinetic power of WAs. The kinetic power of WAs is lower than $0.5\%$ of the bolometric luminosity. It has been shown that an outflow with kinetic power larger than $0.5\%L_{\rm bol}$ can give sufficient feedback to its host galaxy (e.g., Di Matteo et al. 2005; Hopkins & Elvis 2010). Therefore, the WAs can not play some important role in regulating the properties of its host galaxy. The kinetic power of the UFOs injected at the inner radial boundary is $1/2\dot{m}_{\rm inject}v_{\rm inject}^{2}\sim 0.5L_{\rm bol}$. The kinetic power of all of the outflows ($\dot{E}_{\rm total}$) is showed by using the dashed line in the middle panel of Figure 4. It can been seen that inside $10\ \rm parsec$, $\dot{E}_{\rm total}$ is comparable to that of the injected UFOs. Also, we note that inside $10\ \rm parsec$, the mass flux of total outflow $\dot{M}_{\rm total}$ is significantly larger than that of UFOs. The results indicate that the kinetic energy of the UFOs is very effectively transferred into the ISM. $\dot{E}_{\rm total}$ decreases with increase of radius. This is because that when moving outwards, the gravity of the black hole do negative work to the outflows. Parts of the kinetic energy of outflows are transferred into their gravitational energy. At the radial outer boundary, $\dot{E}_{\rm total}\sim 10\%L_{\rm bol}$. Therefore, the UFO driven outflows can have a significant influence on the evolution of its host galaxy.

The kinetic power of outflow is $\dot{E}=1/2\dot{M}v^{2}$. The momentum flux of outflow is $\dot{p}=\dot{M}v$. Therefore, we have $\dot{p}=2\dot{E}/v$ and $\dot{p}/(L_{\rm bol}/c)=\frac{2\dot{E}}{L_{\rm bol}}\frac{c}{v}$. From the middle panel of Figure 4, we can see that $\dot{E}_{\rm total}\sim L_{\rm bol}$. Also, the velocity of the outflows is significantly smaller than the speed of light. Therefore, the momentum flux of the outflow $\dot{p}_{\rm total}$ can be significantly higher than $L_{\rm bol}/c$ (see the right panel of Figure 4). The UFOs driven outflow can go to larger radii to interact with the gas in its host galaxy. If the host galaxy gas can very quickly cool down after interacting with the outflows and photons from the central AGN, the behaviour of the gas will be controlled by the momentum carried by the radiation and outflow from the AGN. In such case, the outflows should play much more important role in regulating the properties of the galaxy, because the momentum flux of outflows is significantly higher than that of the radiation.

We now investigate the effects of changing the properties of gas. The results in Table 1 show that generally, with the decrease of the gas density, the strength (mass flux, kinetic power, momentum flux) of WAs decreases. The reason is that with the decrease of gas density, the ionization parameter will become larger. The gas is more easily to have ionization parameter larger than the upper limit of the ionization parameter of WAs ($\log\xi=3$). We take models L0.1v0.3D21T6 and L0.1v0.3D24T6 to do comparisons. Figure 5 plots the two-dimensional distribution of $\log\xi$ at $5.3\times 10^{7}$ year for model L0.1v0.3D21T6 (left panel) and L0.1v0.3D24T6 (right panel). It is clear that the gas in model L0.1v0.3D24T6 has higher ionization parameter. In a significant larger region in model L0.1v0.3D24T6, the gas has ionization parameter much higher than the upper limit of the ionization parameter of WAs ($\log\xi=3$). Therefore, models with lower gas density tend to generate weaker WAs.

From Table 1, we see that the initial gas temperature has very little effects on the strength of WAs. The reason is as follows. The mass flux of WAs increases with the increase of radius (see Figure 4). Therefore, most of the WAs are generated at large radii. At large radii ($>100$ parsec), we find that the cooling/heating timescale calculated as $e/\rho\dot{E}$, is significantly smaller than the escape timescale ($r/v_{r}$) of WAs. Therefore, the internal energy of gas can be quickly changed before gas can move a large distance. In other words, because of the quick cooling/heating, gas can quickly lose the memory of its initial internal energy. Thus, the initial gas temperature has little effects on the strength of WAs.

We discuss the effects of strength of UFOs on the properties of WAs. We can change both the velocity and mass flux of UFOs injected at the inner radial boundary. From Table 1, it is clear that generally when the initial gas density is high ($10^{-21}{\rm g\ cm^{-3}}$), with the increase of the velocity or mass flux of UFOs, the WAs will become stronger. This is easy to understand that stronger UFOs can drive stronger WAs. However, we note that the increase of strength of WAs with the increase of strength of UFOs is not significant. When the initial gas density is lower than $10^{-21}{\rm g\ cm^{-3}}$, the effects of changing the strength of UFOs are extremely weak. The reason is as follows. When the initial gas density is much lower, as introduced above, the ionization parameter of gas will be much higher. In this case, the ionization parameter of gas is too higher that only a tiny fraction of the gas can satisfy the ionization parameter criterion of WAs. For such high ionization parameter gas, the increase of strength of UFOs can hardly increase the strength of WAs. The increase of strength of UFOs does help drive stronger outflows. However, most fraction of the stronger outflows still not satisfy the ionization criterion of WAs.

In this paper, in order to achieve a quasi-steady state, we allow gas to freely move out/into the computational domain at the outer radial boundary (see Section 2.3). This kind of settings can keep continuous supplying of gas into the computational domain from the outer radial boundary. We do find that quasi-steady state can be achieved by using such kind of boundary conditions. The time evolution of the mass flux of the WAs measured at the radial outer boundary for our fiducial model L0.1v0.3D21T6 is shown by the black line in Figure 6. It is clear that the mass flux oscillates around some mean values. A quasi-steady state has been achieved in this model. Note that at some snapshots, the mass flux of WAs at the outer boundary is 0. This is because WAs are not smoothly distributed in the computational domain. They are much clumpy (see Figure 1). There are no WAs at the outer boundary for these snapshots. Another kind of usually adopted boundary condition is the outflow boundary condition. Outflow boundary condition allows gas to freely flow out of the computational domain, but gas is not permitted to go into the computational domain from the boundaries. Therefore, the total mass in the computational domain will keep decreasing with time. We carry out a model L0.1v0.3D21T6-Bn to test the effects of such boundary conditions. The only difference between model L0.1v0.3D21T6-Bn and our fiducial model L0.1v0.3D21T6 is the setting of outer radial boundary conditions. In model L0.1v0.3D21T6-Bn, at the outer radial boundary, we use outflow boundary conditions to not allow gas to flow into our computational domain. The mass flux of WAs measured at the outer radial boundary is shown by the rad line in Figure 6. It is clear that the mass flux decreases with time. After $3\times 10^{7}$ year, the WAs disappear. The reason is as follows. The UFOs blow out the overlying ISM continuously, without newly injected gas into the computational domain, the only left gas is the UFOs injected from the inner radial boundary. From the analysis above, we conclude that the properties of the WAs mainly depend on the density of the ISM injected into the computational domain from the outer radial boundary.

In a real galaxy, there may be two cases for gas supplies from the galaxy scale to the central AGN. In the first case, gas in the galaxy scale continuously supplies to the central AGN. This case is mimicked by the free inflow/outflow outer boundary conditions used in the models in Table 1. In the second case, there is no gas supplies from the galaxy scale to the central AGN. This case is mimicked by the boundary conditions used in model L0.1v0.3D21T6-Bn.

In our models, we assume that UFOs are always present. In reality, the UFOs may be intermittently produced by the AGN. For the intermittently generated UFOs, there are two related timescales. The first one is how long the UFOs can last once it is produced. The second one is that after the UFOs disappearing, how long it will take for another UFOs to be generated. There is no information about the two timescales from both observations and theory.

We discuss the possible effects of intermittent UFOs in the case that there is no continuously gas supplies from the galaxy scale to the central AGN (boundary conditions in model L0.1v0.3D21T6-Bn). If the UFOs do not last sufficiently long time to push the gas away from our computational domain, the gas with velocity smaller than the escape velocity may fall back after UFOs’s shutting down. If the shutting down timescale of UFOs is not so long, the fall back gas may not have sufficient time to go into the region inside 1 parsec. If both the two conditions are satisfied, the WAs may be generated periodically. On a time-averaged sense, this will produce more WAs compared to continuously UFO injection (model L0.1v0.3D21T6-Bn). However, it is not clear whether the two conditions can be satisfied simultaneously. Therefore, it is hard to judge whether can intermittent UFOs produce more WAs.

In the case that there is continuously gas supplies from galaxy scale to the central AGN (the models in Table 1), due to the lack of information about the duration timescale of UFOs, it is also quite hard to guess the effects of intermittent UFOs on the WAs.