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Rescuing the intracluster medium of NGC 5813

We use recent X-ray observations of the intracluster medium (ICM) of the galaxy group NGC 5813 to confront theoretical studies of ICM thermal evolution with the newly derived ICM properties. We argue that the ICM of the cooling flow galaxy group NGC 5813 is more likely to be heated by mixing of post-shock jets' gas residing in hot bubbles with the ICM, than by shocks or turbulent-heating. Shocks thermalize only a small fraction of their energy in the inner regions of the cooling flow; in order to adequately heat the inner part of the ICM, they would overheat the outer regions by a large factor, leading to its ejection from the group. Heating by mixing, that was found to be much more efficient than turbulent-heating and shocks-heating, hence, rescues the outer ICM of NGC 5813 from its predestined fate according to cooling flow feedback scenarios that are based on heating by shocks.


INTRODUCTION
A negative feedback mechanism driven by active galactic nucleus (AGN) jets that inflate X-ray bubbles (X-ray deficient cavities) determines the thermal evolution of the intra-cluster medium (ICM) in the inner regions of cooling flow (CF) clusters and groups of galaxies (e.g., Pfrommer 2013; Perucho et al. 2014). In the feedback cycle the ICM feeds the AGN, and the AGN launches jets that heat the ICM.
The processes by which jets heat the ICM, on the other hand, is still in dispute. Based on observations of the galaxy group NGC 5813 Randall et al. (2011) and Randall et al. (2015) argue that shocks heat the ICM; hereafter shocks-heating process. This was put into question by Soker et al. (2013), and is further studied in the present paper. Based on deep X-ray observation of the Perseus and Virgo cooling flow clusters, Zhuravleva et al. (2014) argue that the main heating process is via dissipation of ICM turbulence; hereafter turbulent-heating process. The third heating process is the mixing of cooling ICM with hot shocked jets' gas that reside in the hot bubbles; hereafter mixing-heating process. Gilkis & Soker (2012) and Hillel & Soker (2014) conducted numerical simulations and showed that the mixing-heating process is much more efficient than the shocks-heating process. In a recent paper Hillel & Soker (2015) showed that the mixing-heating is also more efficient than the turbulent-heating when turbulence is driven by AGN activity. The turbulence, as observed in some clusters (e.g., Zhuravleva et al. 2015), is a consequence of the same vortices that cause mixing.
To reach the conclusion that mixing-heating is the dominant process, it is mandatory to inflate bubbles by jets in a self-consistent manner. For example, the vortices that are induced inside and outside the bubbles must be taken into account as they play major roles in the formation of bubbles, their evolution, and their interaction with the ICM, (e.g. Omma et al. 2004;Heinz & Churazov 2005;Roediger et al. 2007;Sternberg & Soker 2008b;Gilkis & Soker 2012;Walg et al. 2013). To inflate bubbles that resemble observed bubbles, our group has employed either slow (sub-relativistic) massive wide (SMW) jets (bipolar outflows), e.g., (Sternberg et al. 2007), or precessing jets, e.g., (Sternberg & Soker 2008a); see also Falceta-Goncalves et al. (2010). SMW jets are supported by observations (e.g., Moe et al. 2009;Dunn et al. 2010;Arav et al. 2013). Both types of inflated bubbles will be described in the present study.
In a recent paper Randall et al. (2015) repeat their claim for shocks-heating in NGC 5813. We here examine some of their arguments along our group findings, and conclude that although shocks play some role, mixing-heating is likely to play a much more significant role in the heating process.

IMPLICATIONS FOR THE OUTER ICM
The inefficiency of the shocks-heating process was discussed in Soker et al. (2013). To excite a shock with a Mach number of Ms at radius r, the energy supplied by the AGN directly to the ICM c 0000 RAS should be where E th (r) is the thermal energy of the ICM inside radius r. According to Randall et al. (2015), to heat the cooling flow region of NGC 5813 inside 30 kpc, about 140 shocks with a Mach number of Ma ≈ 1.2 are required during one cooling time. Removing the dissipated energy in the inner region (up to 30 kpc), the energy that flows to the outer ICM regions within one cooling time at r = 30 kpc is Eout ≈ 100Ein, where we define Ein ≡ E th (r < 30 kpc).
The ICM mass residing out at r > 30 kpc can be crudely estimated from the new density profile given by Randall et al. (2015). The electron density profile from r = 22 kpc to r = 35 kpc, the largest radius in their profile, can be fitted with (2) Extrapolating this fit to large distances of hundreds of kpc, we can estimate the mass in the outer region to be Mout ≈ 10 11 M⊙ ≈ 2Min. Randall et al. (2015) do mention that the energy carried by the bubble is transferred out by the buoyant bubbles. But the shocks turn into sound wave that dissipate their energy after a long time in the ICM. By the simple estimate made above the temperature of the outer gas should be It seems that for the shocks-heating process to be efficient in the inner region, the outer ICM would escape in a time much shorter than the cooling time, hence destroying the ICM observed structure. Clearly, a much more efficient heating mechanism than shocks is required to rescue the ICM of NGC 5813. Mixing-heating is this process (Hillel & Soker 2014.

LARGE SCALE FLOW
In the shocks-heating scenario presented by Randall et al. (2011) and Randall et al. (2015) tens of jet-launching episodes along the same axis heat the ICM. They do not consider any flow that might develop in the ICM. We here show that a large-scale meridional flow develops as a result of multi-jet activity cycles. For that we present results from 3D numerical simulations we have carried out recently (Hillel & Soker 2015). We here briefly present the main characteristics of the numerical set up; more details can be found in that paper.
We use the PLUTO code (Mignone et al. 2007) for the hydrodynamic simulations in a three-dimensional Cartesian grid with adaptive mesh refinement (AMR). The computational grid is in the octant where the three coordinates x, y and z are positive. At the x = 0, y = 0 and z = 0 planes we apply reflective boundary conditions. The z coordinate is chosen along the initial axis of the jets. The computational grid spans the cube 0 x, y, z 50 kpc.
At the boundary z = 0 we inject into the grid a jet with a half-opening angle of θj = 70 • (Sternberg et al. 2007). The jet is injected during each active episode lasting 10 Myr, and when the jet is turned off for 10 Myr, reflective boundary conditions apply in the whole z = 0 plane. The initial jet velocity is vj = 8200 km s −1 corresponding to a Mach number of about 10 relative to the ICM. The power of the two jets together (we simulate only one jet) isĖ 2j = 2 × 10 45 erg s −1 (half of it in each direction), and the mass deposition rate isṀ2j = 94M⊙ yr −1 .
The simulation begins with an isothermal box of gas at an initial temperature of TICM(0) = 3 × 10 7 K with a density profile of (e.g., Vernaleo & Reynolds 2006) ρICM(r) = ρ0 with a = 100 kpc and ρ0 = 10 −25 g cm −3 . A gravity field is added to maintain an initial hydrostatic equilibrium, and is kept constant in time. We include radiative cooling in the simulations, where the tabulated cooling function is taken from Table 6 in Sutherland & Dopita (1993).
To reveal the large scale flow we follow an artificial flow quantity called 'tracer.' The tracer is frozen-in to the flow, and hence tells us on the spreading with time of gas starting in a certain volume. A tracer's initial value is set to ξ(0) = 1 in a certain volume and ξ(0) = 0 elsewhere. As the traced gas mixes with the ICM or the jet's material, its value drops to 0 < ξ(t) < 1. We choose to trace the gas starting inside a torus around the z axis. The radius of the cross section of the torus is 2.5 kpc, and it is centered at (y, z) = (20, 15) kpc; or more generally ( x 2 + y 2 , z) = (20, 15) kpc.
In Fig. 1 we present the evolution of the tracer at six times in the meridional plane (y, z). Note that the jets' axis is along the z axis, which is the horizontal axis in the figure. The evolution of other quantities in the (x, z) meridional plane is presented by Hillel & Soker (2015). In the panels on the left column the arrows represent flow velocity, with length linear with velocity up to an upper limit of vm = 400 km s −1 . Faster regions are presented with an arrow length corresponding to vm. In the panels on the right column the arrows represent mass flux φ = vρ.
There are several prominent properties of the flow that are relevant to the present study and are revealed in Fig. 1. (i) Mixing. The vortices created during the inflation of the bubble mix hot bubble gas with the ICM gas. This can be clearly seen by following the evolution of the tracer. This is the main heating mechanism of the ICM by jets (Gilkis & Soker 2012;Hillel & Soker 2014. (ii) Outward then inward motion. The shocks running through the ICM followed by sound waves behind them can push gas out. The traced gas presented here is pushed from a torus cross section centered at (y, z) = (20, 15) kpc to an elongated shape centered at (y, z) ≃ (29, 15) kpc at t ≃ 100 Myr. However, then it turns around and flows inward. It is heated only when it starts to be mixed with the hot bubble gas at t ≃ 110 Myr (Hillel & Soker 2015).
(iii) Large scale meridional flow. At late times of t 200 Myr, and after about 10 jet-activity episodes (cycles), a large-scale inflow takes place near the equatorial plane. This results from the dragging of ICM gas by the outflowing jets along their axis, here the horizontal z-axis. It seems though that in reality the jets' axis will change direction and no such large meridional flow will be develope. This change in direction can be seen in the three bubblepairs in NGC 5813 (Randall et al. 2015).
The main conclusion of this section is that one cannot assume that jets activity repeats itself along the same direction for many episodes, while the ICM in the equatorial plane (the plane perpendicular to the jets' axis) does not flow inward. The flow inward will mix ICM gas with bubble gas, leading to heating by mixing that is much more efficient than the shocks-heating process.  Hillel & Soker (2015). At t = 0 the traced gas was contained in a torus whose axis is the z-axis and whose cross section is a circle centered at ( x 2 + y 2 , z) = (20, 15) kpc with a radius of r = 2.5 kpc. Color coding is of the tracer value ξ. The jets' axis is along the horizontal z-axis in the panels. In the three panels on the left the arrow lengths are linear with velocity, up to a maximum value of vm = 400 km s −1 . Faster regions have an arrow length corresponding to vm. The arrows in the right panels show the mass flux φ = vρ. A length of 1 kpc on the map corresponds to φ = 2.3 × 10 −23 km s −1 g cm −3 . Both panels are taken from Sternberg & Soker (2009) where all details are given. The post-shock region at r = 21.5 kpc shows jumps in density, temperature, and pressure. The sound wave just behind the shock at r = 18.5 kpc has density and pressure above the post-shock values, while the temperature is lower than the post-shock value. Randall et al. (2015) find the observed temperature rise across the shocks to be less than what is expected based on the Mach numbers they derive from their calculated density jumps across the shocks. They attribute the discrepancy to inability of the deprojected temperature measurements to resolve the temperature jump due to the narrow width of the shock and the rarefied cool gas behind the shock (Randall et al. 2011).

POST-SHOCK FLOW
We here raise another possibility. When a bubble is inflated self-consistently, sound waves are excited and trail the shock wave (Sternberg & Soker 2009). The sound waves can compress gas behind the shock, and raise the density behind the shock much more than they raise the temperature. In Figs. 2 and 3 we present results from Sternberg & Soker (2009), where all technical details are given.
The relevant properties of these simulations are as follows. The simulation are performed in half of the meridional plane using the two-dimensional version of the VH1 code (Blondin et al. 1990;Stevens et al. 1992) in spherical coordinates with a resolution of 256×256. Gravity was included in these simulations. Two opposite jets are launched along the x-axis.
In the wide jets case two opposite jets were injected in the time period from t = −10 Myr until t = 0, each jet with a half opening angle of α = 70 • . Their combined mass outflow rate waṡ M2j = 10M⊙ yr −1 and the initial velocity was radial with vj = igure 3. Excitation of sound waves by precessing-jet-inflated bubbles as simulated by Sternberg & Soker (2009). Upper panel: Density map for the ICM around the precessing-jet-inflated bubbles at t = 0, just when the jets cease. Lower panel: Density, temperature and pressure along the cut at 80 • as marked on the upper panel by the dashed-dotted line. The pressure is given in units of 9 × 10 −9 erg cm −3 , density in units of 2 × 10 −25 g cm −3 , and temperature in units of 2.7 × 10 7 K. The postshock region at r = 18 kpc shows jumps in density, temperature, and pressure. The sound wave behind the shock at r = 15.5 kpc has density and pressure above the post-shock values, while the temperature is lower than the post-shock value. 7750 km s −1 . The total power of the two jets wasĖ2j = 2 × 10 44 erg s −1 . This case is presented in Fig. 2.
In the case of the precessing jets the two opposite jets have the same mass outflow rate and velocity as in the wide-jets case, but they were injected at any given time with a half opening angle of α = 5 • . In the axisymmetric grid employed, the jets are actually precessing very rapidly around the symmetry axis (i.e., in 3D torii were actually injected). The angle between the symmetry axis and the jets' axis θ was varied in a random way. The precession period, i.e., the time the jet returns to the same angle θ, is Tprec = 0.1 Myr. The jet's interaction with the ICM is similar to that of a wide jet with a half opening angle of α ≈ 50 • . The jets were active between t = −18 Myr and t = 0. This case is presented in Fig. 3 The profiles along a cut in each case presented above show a density and temperature jump immediately behind the shock, the right edge in each of the two panels of profiles. Immediately behind the post-shock jump (moving to the left in the panels) there is another increase in the density, but the temperature decreases. The over all density jump overestimates the shock Mach number.
We argue that Randall et al. (2015) somewhat overestimate the Mach numbers in the shocks they study in NGC 5813.

SUMMARY
We reexamined the claim made by Randall et al. (2011) and Randall et al. (2015) that the ICM in the cooling flow group of galaxies NGC 5813 is heated by many shock waves excited by repeated jet-launching episodes. It was already argued by Soker et al. (2013) that the shocks-heating mechanism is inefficient, and cannot account for the NGC 5813 ICM properties. We here strengthened this conclusion.
The energy carried by the shocks and that is not deposited in the inner region, r 30 kpc, is carried to the outer ICM regions and must dissipate there as sound waves. In section 2 we found that for the inefficient shocks-heating mechanism to heat the inner r 30 kpc region, this extra energy heats the outer ICM to high enough temperature for it to escape the group of galaxies. If this was the case, the inner region would have expanded and escaped a long time ago. Clearly a more efficient heating mechanism is required.
In section 3 we showed that after about ten jet-launching episodes along a constant axis a large-scale meridional flow develops (Fig.1). An inflow of mass near the equatorial plane takes place. One cannot regard the ICM for such a case as static. Such a flow leads to a very vigorous mixing of the ICM with the hot bubbles' gas. Heating by mixing becomes the dominate heating mechanism even for gas perpendicular to the jets' axis (Gilkis & Soker 2012;Hillel & Soker 2014. We expect that in reality the jets' axis will change over time, and the meridional flow will not be as prominent as found here In section 4 we showed that sound waves propagating behind a shock wave can form a region behind the shock where the density increases above the post-shock value, but the temperature does not. This might explain the finding of Randall et al. (2015) that the temperature jumps across the shocks in NGC 5813 are lower than those expected for the Mach numbers they deduced based on the density jumps. If true, it implies that Randall et al. (2015) overestimated the Mach numbers in the three shocks they have found in NGC 5813.
Our main conclusion is that shocks cannot be the main heating process of the ICM in NGC 5813. Heating by mixing seems to be the main heating process of the ICM by jets (Gilkis & Soker 2012;Hillel & Soker 2014. The mixing is caused by vortices that are formed during the inflation of the bubbles. To reveal the properties of the vortices in numerical simulations it is crucial to inflate bubbles in a self-consistent manner. When that is done, it is found that the mixing operates also perpendicular to the jets' axis. The vortices lead to turbulence in the ICM. Therefore, the finding of moderate turbulence in the inner region of NGC 5813 by Werner et al. (2009) andde Plaa et al. (2012), as well as in other clusters (Zhuravleva et al. 2014(Zhuravleva et al. , 2015, strengthens the argument that the mixing-heating process is the dominate heating process. Some heating by the dissipation of turbulence takes place as well, but it accounts for at most ≈ 20% of the total heating (Hillel & Soker 2015).