X-ray from Outflow-Cloud Interaction and Its Application in Tidal Disruption Events

Bow shock can accelerate charged particles to relativistic ones (cosmic rays) by first-order Fermi acceleration mechanism. In this work, we assume cosmic ray electrons (CRe) follow a power-law distribution:

with a power index $\Gamma=2$. We also choose $\Gamma=2.5$ and 3.0 for comparison.

We use the precise form of the number of photons produced per unit time per unit energy from ICS (Jones, 1968). The spectrum of Compton-scattered photons resulting from the interaction with electrons of energy $\gamma mc^{2}$ is:

where $E_{\gamma}$ is final photon energy, $E_{ph}$ is incident photon energy, $r_{0}$ is Bohr radius, $F(\zeta_{+})$ and $F(\zeta_{-})$ are equations (24)-(27) in Jones (1968).

The total Compton spectrum would be:

where $E$ is energy of photons after ICS, $N_{e}(\gamma)\textrm{d}\gamma$ is differential number of electrons, ${\textrm{d}N_{\gamma,ph}}/{\textrm{d}E}$ is the spectrum resulting from the interaction of electrons of energy $\gamma mc^{2}$ with an isotropic photon field. Figure 2 shows the spectra of ICS with different power indexes $\Gamma$, where we use a $2\times 10^{4}$ K black body spectrum as the seed photon field.

In addition to the momentum transferred from outflow to the cloud by cloud shock, energy also transferred via thermal conduction between outflow and clouds. The velocity of cloud shock is $v_{\textrm{s,c}}\simeq\chi^{-0.5}v_{\textrm{w}}$, where $\chi\equiv\rho_{\textrm{c}}/\rho_{\textrm{w}}\simeq 10^{4}$ is the density ratio of cloud to outflow (McKee & Cowie, 1975). The velocity of cloud shock is smaller than outflow, and the clouds will soon be immersed in the hot gas behind the outflow front. The windward side of cloud will “feel” both the thermal and the dynamical pressure of outflow, and the shock of TDE outflow would be sufficiently strong so that the initial compression at bow shock will approach 4. The temperature behind the shock front will reach $1.38\times 10^{5}~{}v_{w7}^{2}$ K if particles are fully ionized, where $v_{w7}=v_{w}/(10^{7}$ cm s^-1) and $v_{w}$ is the velocity of outflow (Hollenbach & McKee, 1979). Side and rear faces are primarily under thermal pressure, which is weaker and can be ignored.

In this paper, we try to investigate thermal conduction process and the cooling of clouds. We assume that outflow and clouds have reached a steady-state and the system is at a constant pressure equal to the ram pressure of outflow $\rho_{\textrm{w}}v_{\textrm{w}}^{2}$. We seek the solutions to the time-independent equations of mass, momentum and energy conservation (Hollenbach & McKee, 1979)

where $q$ denotes the heat flux, $\nabla\cdot F=n^{2}\Lambda$ is the cooling term. The equations will reduce to normal hydrodynamic equations provided only that the plasma is approximately isotropic and the stress tensor may be written as $\boldsymbol{\Psi_{ij}}=p\delta_{ij}$ and $u=\textrm{Tr}\boldsymbol{\Psi}/2$ is the internal energy (Hollenbach & McKee, 1979). The thermal conductivity in a fully ionized hydrogen plasma is (Spitzer, 1962)

and heat flux is $q=-\kappa\nabla T$. By solving these equations, we derive the temperature profile of the cloud. Total emission, effective temperature and density profile can be obtained from temperature profile.

The distance between the cloud and bow shock is determined by ram pressure balance condition $\rho_{\textrm{w}}V^{2}_{\textrm{w}}=\rho_{\textrm{refl}}V^{2}_{\textrm{refl}}$ (Wilkin, 1996), the ’refl’ means reflected and $\rho_{\textrm{refl}}=\dot{M}/(4\pi R_{\textrm{bow}}^{2}v_{\textrm{refl}})$, where $R_{\rm bow}$ is the distance between cloud and bow shock. Assuming that the reflected shock has a sonic-velocity and all materials hitting clouds are reflected, then $R_{\textrm{bow}}\simeq 5\times 10^{13}~{}\textrm{cm}$ is close to the cloud radius. Thus, we set it equals to the cloud radius for convenience. Because of high temperature $\sim 1.38\times 10^{5}v_{s7}^{2}\simeq 10^{10}~{}{\rm K}$ right behind the bow shock, saturated heat flux will reach $0.4[{2kT_{e}}/{(\pi m_{e})}]^{1/2}n_{e}kT_{e}\simeq 3\times 10^{11}~{}{\rm erg~{}cm^{-2}s^{-1}}$ (Cowie & McKee, 1977). The energy flux under our fiducial parameters is $L_{\rm kin}/(4\pi R_{\rm d}^{2})\simeq 1.5\times 10^{10}~{}\rm{ergs~{}cm^{-2}s^{-1}}$, which is lower than the saturated heat flux and will be the maximum heat flux in our model. Thus, the heat flux is not saturated in our model, and the classical thermal conduction formula (i.e., Equation 5) is applicable. The thermal conductivity is $\kappa\simeq 6\times 10^{18}~{}\rm{ergs~{}s^{-1}deg^{-1}cm^{-1}}$ right behind the bow shock for the temperature $\sim 10^{10}$ K, and energy transfer will be efficient. The heat flux at about $10^{6}$ K is $q=-\kappa\nabla T\simeq 10^{9}~{}{\rm{erg~{}cm^{-2}s^{-1}}}$, when temperature goes below $10^{4}$ K, the heat flux will drop rapidly to $q=10^{7}~{}{\rm erg~{}cm^{-2}s^{-1}}$, which is two orders of magnitude lower and suggests that the thermal conduction below $10^{4}$ K is inefficient. As shown in Figure 3, the temperature will decline sharply below $10^{5}$ K. Therefore, we ignore the thermal conduction below $10^{4}$ K in the following. There will be discontinuities in density and temperature at the surface of the cloud. However, across the discontinuity, the pressure is uniform and the velocity is continuous, implying no interpenetration (Tenorio-Tagle, 1996; Weaver et al., 1977; Borkowski et al., 1990). The discontinued area should be thin and cooling emission is negligible. Since we are interested in the spectrum of cooling process, we ignore this area.

Since windward shock is the main shock, we can simplify this 3-dimension question to a 1-dimension one, which is along the propagation direction of outflow. The energy equation in Equation 4 will be

Together with mass and momentum equations, we can derive temperature profile behind bow shock as shown in Figure 3. The cooling emission for a cloud is $\int n^{2}(r)\Lambda(T)S\textrm{d}r$, where $\Lambda(T)$ is the cooling function (Sutherland & Dopita, 1993). Figure 3 shows temperature profile of cloud and emission as a function of temperature, which peaks at $T\simeq 10^{6}$ K. Figure 3 also gives the overall radiation spectrum of the cloud, which is not a single temperature black-body spectrum. The luminosity for a single cloud is about $L_{\textrm{c}}=1\times 10^{37}$ erg s^-1. Equation 6 is a function of outflow velocity and density. Thus, we change the velocity and density of outflow (the left other parameters in Table 1 unchanged) to investigate the influence of parameters on spectral properties. Figure 4 shows the effective spectra under taking different parameters.

If thermal conduction is not considered, the efficiency of cloud shock is about $\eta_{\textrm{E}}\simeq\chi^{-1/2}=0.01$ for $\chi=10^{4}$, and it will drop by orders of magnitude as $\chi$ could increase by several orders of magnitude if the efficient cooling of the post-shock dense cloud gas is considered (appendix K in Mou & Wang 2021). In contrast, the efficiency for thermal conduction is $\eta=L_{\textrm{c}}/L_{\textrm{p,c}}$, where $L_{\textrm{p,c}}=L_{\textrm{kin}}(\pi R_{c}^{2})/(4\pi R_{d}^{2})$ is the power of the outflow impacting the cloud. More detailed results of the efficiency and effective temperature under different parameters are presented in Appendix A. The effective temperature generally is propotional to the outflow velocity and density. The efficiency will increase (up to $\sim 15\%$) as the outflow density increases, but will decrease firstly and then increase as outflow velocity increases for a given density. If the ram pressure $\rho_{\rm w}v_{\rm w}^{2}$ doesn’t change when varying outflow velocity and density, we will get the same effective temperature.

Thus, we conclude that by means of thermal conduction, the outflow impacting on the cloud can convert $\eta\sim(5-10)\%$ of its energy into cloud radiation, which is much higher than that without thermal conduction. Considering a covering factor of $C_{v}$, the total X-ray luminosity from the cloud can reach $L_{\textrm{tot}}=\eta C_{\textrm{v}}L_{\textrm{kin}}\simeq 1\times 10^{43}$ erg s^-1 $(\eta/0.1)(C_{v}/0.1)(L_{\rm kin}/10^{45})$, implying that this mechanism may account for the delayed X-ray radiation of some TDE candidates.

ASASSN-15oi was first reported by Holoien et al. (2016), which locates at a luminosity distance $D\simeq 214$ Mpc with a redshift $z=0.0479$ (Holoien et al., 2018). The central black hole has a mass $\sim 10^{6}M_{\sun}$ (Gezari et al., 2017). ASASSN-15oi is associated with radio emission (Horesh et al., 2021), which indicates the presence of a jet or outflow. Unlike other optically discovered TDEs, ASASSN-15oi faded rapidly in the optical and UV, and almost completely disappeared about 3 months after discovery (Holoien et al., 2016). Its UV/optical spectra can be fitted by a black body, and its temperature increased from $2\times 10^{4}$ K to $4\times 10^{4}$ K in the first few months since discovery. In the early stage, the UV/optical luminosity declined following a $t^{-5/3}$ power-law, and $t^{-5/12}$ in later time (Holoien et al., 2018). In contrast to the optical/UV luminosity evolution, the X-ray luminosity remained constant at early time and was about two orders of magnitude weaker than the UV/optical emission. But during $200\sim 400$ days, the X-ray emission increased by more than an order of magnitude and had a luminosity comparable to UV/optical luminosity ($L_{\textrm{opt}}/L_{X}\sim 1$ Gezari et al. 2017). The X-ray emission faded again at later time.

We focus on the observation XMM 0722160701 which was taken about 230 days after the TDE discovery. This observation gave a soft X-ray spectrum that can be fitted by a black body of $0.053\pm 0.02$ keV with a flux $3.2^{+0.9}_{-1.0}\times 10^{-13}$ erg cm^-2 s^-1 and a power-law part with a flux $1.5^{+0.8}_{-0.8}\times 10^{-14}$ erg cm^-2 s^-1 (Holoien et al., 2018). To fit the blackbody part of the spectrum, we take outflow velocity $V_{\textrm{w}}=1.4\times 10^{9}$ cm s^-1 and density $\rho_{\textrm{w}}=1.9\times 10^{7}m_{\textrm{H}}$ cm^-3, cloud density $\rho_{\textrm{c}}=3.6\times 10^{11}m_{\textrm{H}}$ cm^-3. Figure 5 shows overall fitting spectra with different power-law indices. We use a Galactic H${\mathrm{1}}$ column density of $5.6\times 10^{20}$ cm^-2 (Holoien et al., 2016), and photo-electric cross section per hydrogen in Morrison & McCammon (1993). The unabsorbed X-ray flux is about $3\times 10^{-13}$ erg cm^-2 s^-1, which corresponds to a luminosity $L_{X}\sim 10^{42}$ erg s^-1 and is close to the result in Holoien et al. (2018).

The AT2019azh was first discovered on 2019-02-22 (MJD 58529.44) by All Sky Automated Survey for SuperNovae (ASSASN) (Barbarino et al., 2019). AT2019azh locates at a luminosity distance $D\simeq 96$ Mpc, with a redshift $z=0.022$ and a Galactic absorption column density of $N_{\textrm{H}}=4.15\times 10^{20}$ cm^-2 (Liu et al., 2019). Its central SMBH has a mass $M_{\textrm{BH}}\lesssim 4\times 10^{6}$ M_☉. The early-time spectra of AT2019azh have the very blue continuum that is a hallmark of tidal disruption events (Hinkle et al., 2021). Radio emission was detected months after the detection of UV/optical outburst (Goodwin et al., 2022), which indicates the existence of outflow or jet. The UV/optical luminosity declines following an exponential law with time, while the X-rays show a late brightening by a factor of $\sim$ 30-100 at around 250 days after discovery (Liu et al., 2019). The temperature of AT2019azh was around $T\simeq 10^{4}$ K at about 250 days after discovery. The evolution of AT2019azh’s luminosity ratio of X-rays over UV/optical is similar to that of ASASSN-15oi. This ratio rose from $\sim 0.01$ at the early stage to $\sim 1$ at around a year after discovery.

We study on this late-time brightening at around 250 days after the TDE discovery with the X-ray flux of $\sim 5.9\times 10^{-12}$ erg cm^-2 s^-1. The spectrum can be fitted by a blackbody part of $0.053^{+0.006}_{-0.004}$ keV plus a power-law part (Liu et al., 2019). To fit the blackbody part of the spectrum, we take outflow velocity $V_{\textrm{w}}=1.2\times 10^{9}$ cm s^-1 and density $\rho_{\textrm{w}}=2.9\times 10^{7}m_{\textrm{H}}$ cm^-3, and cloud density $\rho_{\textrm{c}}=3.3\times 10^{11}m_{\textrm{H}}$ cm^-3. Figure 6 shows the fitting result based on the outflow-cloud interaction model.

The AT2018fyk was first discovered by ASASSN survey on 2018 September 8, and this transient is likely a TDE (Wevers et al., 2019). AT2018fyk locates at a luminosity distance 264 Mpc, with redshift $z=0.059$ and a Galactic absorption column density $N_{H}=1.15\times 10^{20}$ cm^-2 (Wevers et al., 2021). The UV/optical temperature is about $4\times 10^{4}$ K during the first several months after discovery. The UV/optical light-curve of AT2018fyk appears a smooth decline. At about 20 days after discovery, the X-ray emission increased by a factor of $\sim 10$ in 6 days and remained unchanged for months. The X-ray evolution may be explained by reprocessing process or delayed accretion (Wevers et al., 2019, 2021). AT2018fyk and ASASSN-15oi have similar evolution of optical to X-ray ratio and late-time X-ray brightening and we try to fit the spectrum with our model.

AT2018fyk was observed by XMM-Newton European Photon Imaging Camera (EPIC) several times, the first observation was taken at about 92 days after discovery. The X-ray of this observation has a flux about $6\times 10^{-13}$ erg cm^-2 s^-1, and can be fitted by a black body part with a temperature $kT\simeq 123$ eV and a power-law part (Wevers et al., 2021). To fit the blackbody spectrum in x-rays, we take the outflow velocity $V_{\textrm{w}}=2.0\times 10^{9}$ cm s^-1 and density $\rho_{\textrm{w}}=9\times 10^{7}m_{\textrm{H}}$ cm^-3, and cloud density $\rho_{\textrm{w}}=1.4\times 10^{12}m_{\textrm{H}}$. Figure 7 shows the fitting result. We notice the spectrum has a peak at about 6-7 keV, which may be the iron K$\alpha$ emission line.

OGLE16aaa was first discovered by OGLE Transient Detection System at the center of its host galaxy on January 2, 2016, and was first reported by Wyrzykowski et al. (2017). OGLE16aaa locates at a luminosity distance $D\simeq 819.4$ Mpc, with a redshift $z=0.1655$ and a Galactic absorption column density of $N_{\textrm{H}}=2.71\times 10^{20}$ cm^-2. Its central SMBH has a mass $M_{\textrm{BH}}\simeq 3.0\times 10^{6}$ M_☉ (Kajava et al., 2020). The UV/optical light-curve of OGLE16aaa indicates a roughly power-law decay with a theoretical power index $\gamma=-5/3$. The temperature remained about $\sim 2\times 10^{4}$ K in the first hundreds of days (Shu et al., 2020). There is no X-ray emission detected in early times of OGLE16aaa, and it was first detected at about 141 days after discovery by XMM-Newton. The X-ray emission rose to a peak at about 300 days after discovery and faded.

XMM-Newton observed OGLE16aaa on November 30 2016 (above 316 days after the TDE discovery) for 36.6 ks, finding a X-ray flux of $\sim 3.2\times 10^{-13}$ erg cm^-2 s^-1. The spectrum can be well described by a two-blackbody model ( $\sim$ 51 eV and $\sim$ 90 eV) or a blackbody part ($\sim 60$ eV) plus a power-law part (Shu et al., 2020). This late-time X-ray brightening is similar to above two TDEs. To fit the blackbody part of the spectrum, we take outflow velocity $V_{\textrm{w}}=1.2\times 10^{9}$ cm s^-1 and density $\rho_{\textrm{w}}=7.5\times 10^{7}m_{\textrm{H}}$ cm^-3, and cloud density $\rho_{\textrm{c}}=6\times 10^{11}m_{\textrm{H}}$ cm^-3. Figure 8 shows the fitting result. In our model, the mass outflow rate is about 38 $M_{\sun}\rm yr^{-1}$ which may be hard to achieve in the TDE based on the present simulations (Curd & Narayan, 2019).

Some other models are also proposed to explain this delayed X-ray brightening in TDEs. The late-time X-ray brightening may arise from a delayed accretion of circularized gas onto the SMBH (Kajava et al., 2020), or possibly the X-ray is blocked by radiation-dominated ejecta in the early phase (Shu et al., 2020). These models would require the enhanced accretion rate in TDEs after hundreds of days, which would dominantly contribute to X-rays not UV/optical bands. The wind-cloud interaction model would naturally produce the delayed X-ray emission properties.