RADIO GALAXY NGC 1265 UNVEILS THE ACCRETION SHOCK ONTO THE PERSEUS GALAXY CLUSTER

Head–tail radio galaxies show spectacular asymmetric radio morphologies and occur in clusters of galaxies. The favored interpretation of these head–tail sources is radio jets ejected by an active core of a galaxy. The jets are bent at some angle toward one direction giving rise to a "head" structure and fan out at larger distances in a characteristic tail that extends over many tens to hundreds of kpc. If the flow impacting these galaxies is supersonic, ram pressure of the intracluster medium (ICM) causes the jets to bend (Begelman et al. 1979); if the flow is trans-sonic with a Mach number ${\mathcal M}\sim 1,$ the thermal pressure gradient of the interstellar medium of these galaxies due to their motion through the ICM causes the bending (Jones & Owen 1979). In this model, the supersonic inflow is decelerated and heated by a bow shock in front of the galaxy which also generates a turbulent wake that re-accelerates the relativistic particle population in the tail and illuminates the large-scale tail structure of these sources. The Perseus cluster is an outstanding example, as it hosts many of these head–tail sources: most prominently NGC 1265 and IC 310 (Ryle & Windram 1968; Miley et al. 1975). High-resolution observations of the head structure of NGC 1265 reveal two radio jets emerging from the galaxy which are at an angle of 90° to the tail where the two jets are first parallel to each other and then merge (O'Dea & Owen 1986). Gisler & Miley (1979) found an extension of this radio tail to the northeast. Surprisingly, at lower surface brightness, Sijbring & de Bruyn (1998) find an additional large-scale structure that arches toward the east around the steep-spectrum tail of NGC 1265 and forms a closed ring with a very steep, but constant spectral index across. Using rotation measure (RM) synthesis, de Bruyn & Brentjens (2005) find a very weak diffuse polarized structure with a Faraday depth of approximately 50 rad m⁻² at the angular position coincident with the steep-spectrum tail as well as the ring structure of NGC 1265. Follow-up mosaic observations of the area around the Perseus cluster show that most if not all of this emission is of Galactic origin (Brentjens 2011).

Sijbring & de Bruyn (1998) discuss a total of four models to explain this puzzling observation. They immediately discard two models—one model of chance superposition of several independent head–tail galaxies due to the lack of strong radio sources in this field (Model 1) as well as another model that hypothesizes re-acceleration of mildly relativistic electrons in the turbulent wake of a galaxy due to contrived projection probabilities and implausible energetics (Model 2). The previously favored pair of models discussed in Sijbring & de Bruyn (1998)—apparently simple—do in fact appear to have considerable amount of fine-tuning and have severe physical problems associated with them as we will point out here. Model 3: in this model, they postulate a helical cluster wind that has to be aligned with the line of sight (LOS) to produce the observed ring structure in projection: such a configuration is highly unlikely, and this model has problems in explaining the constancy of the spectrum and the surface brightness along the radio ring. To balance the synchrotron and inverse Compton cooling of an electron population, this model would need to postulate a fine-tuned acceleration process that, however, must not fan out the well-confined radio emission along the arc. Model 4: in this model, the "radio tail" would outline the ballistic orbit of NGC 1265. To sustain such a helical orbit of NGC 1265 over 360°, this model would require an undetected dark object of mass M ≳ M_NGC1265 ≃ 3 × 10¹² M_☉ orbiting the galaxy.⁴ The change of the direction in the brighter part of the tail remains unexplainable and there is also the problem in explaining the constancy of the spectrum and the surface brightness along the radio ring. Given these difficulties, it is attractive to consider alternative possibilities that may explore the interaction between a radio galaxy with the outskirts of the Perseus ICM. This has been foreshadowed in a remark by de Bruyn & Brentjens (2005) who speculate whether the large-scale polarized structure that arches around the steep-spectrum tail of NGC 1265 is indeed the remains of an earlier phase of feedback. Our work will demonstrate that this picture is not only the simplest consistent explanation for the radio morphology and spectrum, but we also use it to indirectly infer the presence of a cluster shock wave and measure its properties.

Previously, the morphology of a giant radio galaxy has already been used to indirectly detect a large-scale shock at an intersecting filament of galaxies (Enßlin et al. 2001) by using the radio galaxy as a giant cluster weather station (Burns 1998). Gravitationally driven, supersonic flows of intergalactic gas follow these filaments toward clusters of galaxies that represent the knots of the cosmic web (Bond et al. 1996). The flows will inevitably collide and form large-scale shock waves (Quilis et al. 1998; Miniati et al. 2000; Ryu et al. 2003; Pfrommer et al. 2006). Of great interest is the subclass of accretion shocks that are thought to heat the baryons of the warm–hot intergalactic medium (IGM) when they are accreted onto a galaxy cluster. Formation shocks have also been proposed as possible generation sites of intergalactic magnetic fields (Kulsrud et al. 1997; Ryu et al. 1998a, 2008). Prior to this work, only discrete merger shock waves have been detected in the X-rays (e.g., Markevitch et al. 2002) and there was no characterization possible of the detailed flow properties in the post-shock regime as to test whether shear flows—the necessary condition for generating magnetic fields—are present.

The structure of the paper is as follows. In Section 2, we present the basic picture of our model in a nutshell while we derive the detailed three-dimensional (3D) geometry of NGC 1265 within our model in Section 3. In Section 4, we work out the properties of the accretion shock onto the Perseus cluster including those of the post-shock flow and present the implications for the infalling warm–hot IGM. In Section 5, we carefully examine the underlying physics as well as the hydrodynamic stability of our model and discuss our findings in Section 6. Throughout this work, we use a Hubble constant of H₀ = 70 km s^-1 Mpc⁻¹. For the currently favored ΛCDM cosmology with the present-day density of total matter, Ω_m = 0.28, and the cosmological constant, Ω_Λ = 0.72, we obtain an angular diameter distance to Perseus (z = 0.0179) of D_ang = 75 Mpc; at this distance, 1' corresponds to 21.8 kpc. X-ray data estimate a virial radius and mass for Perseus⁵ of R₂₀₀ = 1.9 Mpc and M₂₀₀ = 7.7 × 10¹⁴ M_☉ (Reiprich & Böhringer 2002).

Before we present the idea of our model, we point out the main morphological and spectral properties of the giant radio galaxy NGC 1265 that every model would have to explain. (1) The synchrotron surface brightness, S_ν, and the spectral index, α, between 49 and 92 cm along the tail of NGC 1265 (starting at the galaxy's head) show a characteristic behavior (as shown in Figure 2 of Sijbring & de Bruyn 1998). In the first part of the tail, both quantities decline moderately in a way that is consistent with synchrotron and inverse Compton cooling of a relativistic electron population that got accelerated at the base or the inner regions of the jet. (2) At the point of the tail where the twist changes in projection from left- to right-handed, both quantities experience a sudden drop—while S_ν changes by a factor of 10, α declines from −1.1 to −2.1. (3) Finally, along the remaining curved arc, S_ν and α stay approximately constant on a total arc length of $l_{\rm arc} = 2\pi R \xi \sqrt{1 + k^2}\gtrsim 700 \;{\rm kpc}$, where R ≃ 150 kpc is the radius of the arc, ξ ≃ 3/4 the projected arc length in units of 2π radians, and k = h/(2πR) ⩾ 0, where h is the height of the 3D helix. This property is in particular puzzling, as there is no visible synchrotron cooling or fanning out of the dilute part of the tail visible.

These findings taken together suggest the presence of two separate populations of relativistic electrons giving rise to the bright and the dim part of the tail, respectively, where the latter must have experienced a coherent energetization event⁶ over a length scale of 2R ≃ 300 kpc and on a timescale that is shorter than the cooling time of the radio emitting electrons of τ_sync, ic ≲ 2.9 × 10⁸ yr. The presence of one radio tail that connects the two electron populations in projection points to a causally connected origin of the synchrotron radiating structure. The most natural explanation that combines these observational requirements are the reminders of two distinct epochs of active galactic nucleus outbursts where the most recent one is still visible as a head–tail radio jet and the older one experienced a recent coherent energetization event. In particular, we propose that such an energizing event could be provided by the passage of a detached radio plasma bubble from a previous outburst through a shock wave. This passage transforms the plasma bubble into a torus (vortex ring) and adiabatically compresses and energizes the aged electron population to emit low surface brightness and steep-spectrum radio emission (as we detail below). If the shock crossing is oblique, then the radio torus and the tail end of the recent outburst experience the same degree of shock deflection since both structures can be regarded as passive tracers of the post-shock velocity field and hence provide a natural explanation for the apparent connectivity of both structures. With this idea in mind, we sketch a schematic of the time evolution in our model in Figure 1.

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The reason for the transformation of a plasma bubble into a toroidal vortex ring can be easiest seen in the rest frame of the shock, where the ram pressure of the pre-shock gas balances the thermal pressure in the post-shock regime. The bubble is filled with hot (relativistic) and more dilute plasma compared to the surrounding ICM. Once the dilute radio plasma of the bubble comes into contact with the shock surface, the ram pressure is reduced at this point of contact due to the smaller density inside the bubble (Enßlin & Brüggen 2002). The shock and the post-shock gas expand into the bubble and propagate with a faster velocity compared to the incident shock in the ICM. Owing to symmetry, the ambient gas penetrates the line through the center of the bubble first and has a smaller velocity for larger impact parameters. This difference in propagation velocities implies a shear flow and eventually causes a vortex flow around the newly formed torus which stabilizes it as it moves now with the post-shock velocity field. To study the timescale on which this transformation happens, we solve the one-dimensional (1D) Riemann problem of a shock passage through a bubble exactly in Appendix B. These calculations are complemented by a suite of two-dimensional (2D) axisymmetric simulations using a code that employs an upwind, total variation diminishing scheme to solve the hydrodynamical equations of motion (Ryu et al. 1998b). To map out parameter space, we varied the Mach number and initial bubble/ICM density contrast (see Figure 2 for one realization). As a result, we find that an initially spheroidal bubble will then evolve into a torus on a timescale τ_form that is determined by the crossing time of the original bubble–ICM contact discontinuity (CD) through the bubble. For typical numbers, τ_form ≃ 1.4 × 10⁸ yr (Equation (B6)). As we will see in Section 5, this is much faster than any transverse shear on the scale of the bubble or beyond can act to distort it.

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The morphology of the radio torus is consistent with MHD simulations of this effect that assume strong shocks (cf. Figures 9 and 10 in Enßlin & Brüggen 2002). The relativistic plasma within the bubble experiences an adiabatic compression by a factor⁷ C that increases the Lorentz factor of the relativistic electrons as Γ ∝ C^1/3 and the rms magnetic field as B ∝ C^2/3. Hence, the radio cutoff of a cooled electron population increases as ν_max ∝ BΓ² ∝ C^4/3 and illuminates a previously unobservable radio plasma (Enßlin & Gopal-Krishna 2001). Synchrotron and inverse Compton aging develops a steep spectrum with spectral index α ∼ 2 and a low surface brightness S_ν ∝ ν^−α.

The galaxy itself remains on a ballistic orbit that is unaffected by the shock passage that likely triggers a new outflow that is now seen as a head–tail radio emission with the jets being bent by the ram pressure wind. The associated compression wave propagating through the interstellar medium could have triggered a bar-like instability in the inner accretion disk that enabled efficient angular momentum transport outward and accretion onto the super-massive black hole (SMBH) which was responsible for launching the jet.

This model explains the observed spectral steepening of the head–tail radio galaxy of Δα ≃ 0.5 due to radiative cooling along the bright part of the tail. In particular, our model naturally accounts for the observed sudden steepening of the spectral index and dropping synchrotron brightness at the transition (in projection) of the radio tail of the current outburst to the shock-illuminated radio torus and match the observed constancy of spectrum and surface brightness across the torus (see Figure 3 and Sijbring & de Bruyn 1998).

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The shock compression acts mostly perpendicular to the filaments during the formation of the torus. Consequently, the field component parallel to the bubble surface is preferentially amplified and dominates eventually the magnetic energy density (Enßlin & Brüggen 2002). (Provided the initial magnetic field component parallel to the shock was not too weak to be amplified above the perpendicular components.) Hence, our model predicts radial synchrotron polarization vectors relative to the center of the radio torus with a polarization degree of ∼5% for our derived geometry of a viewing angle of χ ≃ 23° (see Section 3). This is consistent with upper limits on the polarization of 10%–25% (Sijbring & de Bruyn 1998). Future low-frequency radio observations with, e.g., LOFAR should be able to detect this signal.

In this slightly technical section, we derive the specific geometry that we will use later on to demonstrate the physical plausibility of our proposed model—if the reader is only interested in the physical ideas and results, this section might be easily skipped without loss of the underlying logic of this work.

In general, the location and orientation of the shock surface with respect to the galaxy's infall velocity vector and the projection geometry of this system are degenerate. Inspired by cosmological simulations that quantify structure formation shocks (Ryu et al. 2003; Pfrommer et al. 2008; Hoeft et al. 2008), we take the plausible assumptions that (1) the shock surface is aligned with the gravitational equipotential surface of the Perseus cluster that we determine by taking concentric spheres around NGC 1275 and (2) the second outburst giving rise to the currently observed head–tail structure has started shortly after shock crossing so that we can identify both events.

3.1. Observables and Simple Derived Quantities

3.1.1. The Perseus Accretion Shock

NGC 1265 has a large radial infall velocity of v_r = 2170 km s^-1 that has been obtained after subtracting the mean heliocentric velocity of the Perseus cluster of v_r,Per = 5366 km s^-1. The Faraday RM across the central few kpc of the two jets (O'Dea & Owen 1986) shows a scatter of 20 rad m⁻² around a mean value of around 25 rad m⁻². The scatter is most probably due to the magnetized plasma in the cocoon surrounding the jet or the interstellar medium of NGC 1265 (O'Dea & Owen 1987). The low value of RM fluctuations strongly argues for an infalling geometry with NGC 1265 sitting on the near side of Perseus and little Faraday rotating material in between us and the source. This argument strongly favors the picture that the shock that transformed the radio bubble is the accretion shock rather than a merger shock. The strongest argument in favor of this picture would be the identification of the filament along which NGC 1265 was accreted in redshift space. Since NGC 1265 is only 600 kpc offset from the center of the Perseus cluster (in projection), the finger-of-God effect makes it very challenging to isolate a galaxy filament. There is however indirect evidence for such a cosmic filament from Perseus X-ray data. Dark Hα filaments near the LOS to the core do not show any foreground cluster X-ray emission and have redshifted velocities of around 3000 km s⁻¹ relative to the Perseus cluster, so they appear to be falling in, too (A. Fabian 2010, private communication).

3.1.2. Electron Cooling Timescales

The radio synchrotron radiating electrons of Lorentz factor Γ emit at a frequency

$$\begin{equation} \nu _{\rm syn} = \frac{3 e B \Gamma ^2}{2\pi m_e c}, \end{equation} \tag{ 1 }$$

where e is the elementary charge, m_e is the electron mass, and c is the light speed. Synchrotron and inverse Compton aging of relativistic electrons occurs on a timescale

$$\begin{equation} \tau _{\rm syn,\,ic} = \frac{6\pi m_e c}{\sigma _{\rm T} \big(B_{\rm cmb}^2+B^2\big) \Gamma }, \end{equation} \tag{ 2 }$$

where B_cmb ≃ 3.2 μG(1 + z)² and σ_T is the Thompson cross section. Combining both equations by eliminating the Lorentz factor Γ yields the cooling time of electrons that emit at frequency ν_syn,

$$\begin{eqnarray} &&\tau _{\rm syn,\,ic} = \frac{\sqrt{54\pi m_e c\, e B \nu _{\rm syn}^{-1}}}{\sigma _{\rm T}\,\big(B_{\rm cmb}^2+B^2\big)} \lesssim 2.9 \times 10^8 \;{\rm yr}. \end{eqnarray} \tag{ 3 }$$

The highest frequency ν_syn = 600 MHz at which the torus can be observed gives the shortest cooling time for any given magnetic field value. Interestingly, τ_syn, ic is then bound from above and attains its maximum cooling time at $B=B_{\rm cmb}/\sqrt{3} \simeq 1.8\;\mu {\rm G}\,(1+z)^2$.

3.1.3. Limits on the Galaxy's Inclination Angle

We denote the projected length on the plane of the sky that the galaxy has traveled since shock crossing by L_t,gal. Looking at the bending of the jet of NGC 1265 (O'Dea & Owen 1986) and the overall morphology of the tail (Sijbring & de Bruyn 1998), we conclude that the past orbit of NGC 1265 in the plane of the sky is directed mostly southward. We obtain a lower limit on L_t,gal ≳ 220 kpc by measuring the vertical distance of the "head" to the end of the bright radio tail (Figure 1(a) in Sijbring & de Bruyn 1998). Assuming that the time of shock passage of the galaxy and the radio bubble coincide, we can obtain a lower limit on the inclination of the galaxy's with the LOS,

$$\begin{equation} \theta \ge \arctan \left[\frac{{\rm min}(L_{t,{\rm gal}})}{v_r\,{\rm max}(\tau _{\rm syn,\,ic})}\right] \simeq 19^\circ, \end{equation} \tag{ 4 }$$

where v_r = 2170 km s^-1. We will see that detailed geometric considerations of the overall morphology of NGC 1265 show that there is a time lag between the shock passage of the galaxy and the bubble of τ ≃ 6 × 10⁷ yr that one has to add to τ_syn, ic such that the limit on the inclination weakens slightly and becomes θ ⩾ 16°.

3.2. Geometrical Constraints

It turns out that the morphological richness of NGC 1265 in combination with physical arguments then constrain the geometry and shock properties of this system surprisingly well. To this end, we derive a general expression for the gas velocity after shock passage by taking into account the shock obliquity, ϕ, the shock orientation, χ, and the inclination of the galaxy's orbit, θ (see Table 1 for definitions). Using mass and momentum conservation at an oblique shock with a compression factor C_s = ρ₂/ρ₁, the velocities parallel and perpendicular to the shock read as

$$\begin{equation} {{\bm v}}_{2,\perp } = {{\bm v}}_{1,\perp } = {{\bm v}} + v \cos \phi \, {{\bm n}}_s, \end{equation} \tag{ 5 }$$

$$\begin{equation} {{\bm v}}_{2,\parallel } = \frac{{{\bm v}}_{1,\parallel }}{C_s} = -\frac{v}{C_s}\,\cos \phi \, {{\bm n}}_s, \end{equation} \tag{ 6 }$$

$$\begin{equation} {{\bm v}}_{2} = {{\bm v}} + v \cos \phi \,\frac{C_s-1}{C_s}\, {{\bm n}}_s. \end{equation} \tag{ 7 }$$

In order to connect to observables, we derive the velocity components parallel and perpendicular to the LOS,

$$\begin{equation} v_{2,r} = {{\bm e}}_r\,\cdot {{\bm v}}_2 =v_r\,\left(1 - \frac{\cos \chi \cos \phi }{\cos \theta }\,\frac{C_s-1}{C_s}\right), \end{equation} \tag{ 8 }$$

$$\begin{eqnarray} &&v_{2,t} = \sqrt{({{\bm v}}_{2} - v_{2,r} {{\bm e}}_r)^2} = \left[\frac{v_r^2}{\cos ^{2}\theta }\left(1-\cos ^2\phi \,\frac{C_s^2-1}{C_s^2} \right)\right.\nonumber \\ &&- \left. v_r^2\,\left(1-\frac{\cos \chi \cos \phi }{\cos \theta }\,\frac{C_s-1}{C_s}\right)^2\right]^{1/2}.\quad \end{eqnarray} \tag{ 9 }$$

We will determine the shock compression factor C_s in Section 4 based solely on the volume change of the bubble while assuming that the bubble had negligible ellipticity prior to shock crossing.

Table 1. Definitions

er, e2r = 1	LOS vector pointing to observer
ns,  n2s = 1	Shock normal, pointing outward, away from convexly curved surface
v, v2 = v2	3D proper motion of NGC 1265 relative to Perseus, pointing inward
$\chi =\arccos (\phantom{-}{{\bm e}}_r\cdot {{\bm n}}_s)$	"Shock orientation," angle between LOS and shock normal
$\theta =\arccos (-{{\bm e}}_r\,\cdot \frac{{{\bm v}}}{v})$	"Inclination of the galaxy's orbit," angle between negative LOS and galaxy velocity
$\phi =\arccos (-{{\bm n}}_s\cdot \frac{{{\bm v}}}{v})$	"Shock obliquity," angle between negative shock normal and galaxy velocity
Subscripts t, r	Transverse (on the plane of the sky), radial (along LOS)
Subscripts ⊥, ∥	Perpendicular, parallel to shock normal
Subscripts 1, 2	Pre-shock, post-shock regime; refers to quantities of the gas

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We employ the following strategy to determine the geometry. (1) We choose the accretion shock radius R_s ≳ R₂₀₀ = 1.9 Mpc since the hot gas within the cluster is unlikely to support a shock with Mach number ${\mathcal M}\simeq 4$. (2) A priori, the position of the shock passage of NGC 1265 and its advected radio bubble is unknown. We choose the point where the galaxy passed the shock as the origin of the coordinate system $\mathcal {O}$, use the current galaxy's position as a starting value, and interpolate the past orbits of the galaxy and the tail in the plane of the sky. An iteration of the position of $\mathcal {O}$ (that involves a complete cycle through the points listed here) quickly converges on a consistent model. Subject to our assumption of the shock surface, this yields the shock normal, n_s, the shock orientation, χ, and projected orbit that the galaxy has traveled since shock crossing, L_t,gal. (3) Choosing the inclination of the galaxy's orbit, θ, determines the 3D velocity vector v and the time since shock crossing, τ_s = L_t,gal/(v_rtan θ). (4) Momentum conservation at the shock implies for the post-shock gas that its velocity vector lies in the plane that contains n_s and v and is defined by r · (n_s × v/|v|) = 0. The intersection of this plane and the plane of the sky (given by z = 0 for our choice of coordinates) yields the past transverse orbit of the post-shock gas, v_2,t. (5) Solving Equation (9) for the unknown shock obliquity ϕ enables us to derive the current radial position of the galaxy (relative to the cluster center) according to r_gal = R_s − vτ_scos ϕ with $v=v_r\sqrt{1+\tan ^2\theta }$. (6) We compare the galaxy's velocity, v, to the escape velocity, $v_{\rm esc}(r_{\rm gal})=\sqrt{2G M(<r_{\rm gal})/r_{\rm gal}}$. For the large radial velocity of v_r = 2170 km s^-1, it is not trivial to meet the criterion, v ≲ v_esc, stating that the galaxy is gravitationally bound; hence we prefer small values of R_s and θ ≲ 32° (while still evading the electron cooling bound on θ as derived in Equation (4)). The head–tail morphology of the jet also argues for a large ram pressure that it can only experience inside ∼R₂₀₀ and hence for small values of R_s. We finally check whether the model violates any constraints on the shock obliquity ϕ as will be derived in Section 5.2. If this does not yield a consistent model we vary the shock radius R_s, the inclination of the galaxy's orbit θ, and the position of the galaxy's shock crossing, $\mathcal {O}$, and start the next iteration until we arrive at a self-consistent and physically plausible model.

Our final model that fits best these constraints is shown in Figure 3 and has the parameters C_s = 3.4, R_s = R₂₀₀ = 1.9 Mpc, θ = 32°, ϕ = 9°, χ = 23°. We find the time since shock crossing of NGC 1265 to be 1.8 × 10⁸ yr, the Cartesian vector of the shock normal n = (0.2, 0.34, 0.92), and the velocity vector for NGC 1265 of v = −v(0, sin θ, cos θ) (note that we define the Cartesian coordinate system in Figure 3). The galaxy's velocity has a radial and transverse component of v_r = 2170 km s^-1 and v_t = 1360 km s^-1, respectively, which yields a total velocity of the galaxy, v = 2550 km s^-1. This is only slightly larger than the escape velocity, v_esc(r_gal) = 2350 km s^-1, where we used r_gal = 1.45 Mpc, M(<r_gal) ≃ X_turbM₂₀₀ = 9 × 10¹⁴ M_☉, neglect a logarithmic correction factor of the mass and assume a turbulent pressure support of X_turb = 0.2 (Ryu et al. 2008; Lau et al. 2009; Battaglia et al. 2010). Note that if one relaxes the escape velocity constraint and assumes that tidal processes are able to dissipate more energy to bind the galaxy during first passage, we find consistent solutions at larger shock radii that nevertheless lie close-by in the parameter space $(\theta,\phi,\chi,\mathcal {O})$. The choice of the virial radius as the site of the accretion shock might seem to be too small compared to the typical locations inferred from cosmological simulations (Miniati et al. 2000; Kang et al. 2007; Pfrommer et al. 2008). However, the location of the accretion shock (in particular along a filament) is not stationary but dynamically determined. Depending on the ram pressure of the accreting material and the post-shock pressure, its position will re-adjust dynamically to account for the conservation laws. We have shown that the velocity of NGC 1265 is quite large which implies a large ram pressure and hence should yield a shock position that lies closer toward the cluster center compared to the shock position where dilute IGM accretes from voids with a considerable smaller density and ram pressure. We note that there have been attempts to constrain the 3D velocity of NGC 1265 based on the observed sharpness of its X-ray surface brightness map and the optical velocity dispersion of the Perseus cluster (Sun et al. 2005). To arrive at a lower bound on the orbit inclination, θ_X-ray>45°, these authors use an axially symmetric density profile that however does not take into account magnetic draping. This naturally would account for the sharpened interface between the ICM and the interstellar medium and might break the assumed symmetry depending on the morphology of the ambient magnetic field (Dursi & Pfrommer 2008; Pfrommer & Dursi 2010).

The post-shock gas velocity—as traced by the radio emitting tail—is reduced at the shock by transferring a fraction of the kinetic to internal energy. For our choice of parameters, we find $v_2=v\sqrt{1-\cos ^2\phi \,(C_s^2-1)/C_s^2}\simeq 850\;{\rm km\;s^{-1}}$ that divides into the radial and transverse LOS component of the tail as v_2,r ≃ 520 km s^-1 and v_2,t ≃ 670 km s^-1, respectively. The velocity components parallel and perpendicular to the shock normal are v_1,∥ = vcos ϕ ≃ 2530 km s^-1, v_2,∥ = vcos ϕ/C_s ≃ 740 km s^-1, and v_⊥ = v_1,⊥ = v_2,⊥ = vsin ϕ ≃ 400 km s^-1. Once in the post-shock regime, the radio emitting tail and the torus experience the same large-scale velocity field. To explain the observed direction of the bending of the tail toward west, the shock normal needs to have a component pointing westward (for the observed direction of the galaxy's velocity southward).

What about the orientation of the line connecting the bubble center and NGC 1265 (shown as dashed lines in Figure 3)? Assuming that the spin orientation of the SMBH as possibly traced by the radio jets did not change during shock passage, the previous jet blowing the radio bubbles predominantly had an east–west (EW) component in the plane of the sky. If the jet angle had no component along the LOS, the bubble had crossed the shock earlier than NGC 1265 due to the assumed convex shock curvature. Hence, the bubble would need to have been at a larger distance from NGC 1265 than in our model and would have needed a longer time τ ≃ 3 × 10⁸ yr>τ_syn, ic to reach the current position (extrapolating the position back to the shock with the orientation of the deflection vector given by our model). It follows that the shock-enhanced radio emission would have faded away by now and the bubble would have had to spend at least a factor of 2.5 longer in the post-shock region which would not be consistent with the assumed EW alignment of the previous jet axis. Hence, our model postulates that the eastern bubble was on the front side relative to the observer and had a smaller LOS than NGC 1265 (see Figure 3). This caused a "two-stage process": the bubble continued for τ_2→3 ≃ 6 × 10⁷ yr on its orbit in the filament after NGC 1265 passed the shock until it came into contact with the shock surface. This configuration has the advantage of the faster pre-shock velocity and associated larger length scale in the plane of the sky that the bubble traversed and enables us to fulfill the geometric requirement to cross the shock surface while ending up at current position. After shock passage and its transformation into a torus, it experiences the an almost similar deflection as the tail of NGC 1265 due to the oblique shock passage for the remaining time of τ_3→4 ≃ 1.2 × 10⁸ yr until today. This also solves the problem of the non-observation of the "second" bubble that the previous western jet should have blown: in our proposed geometry, this hypothetical bubble was situated westward of NGC 1265 at the far side of the cut plane in Figure 3. It should have passed the shock τ ∼ 2 × 10⁸ yr earlier than the eastern bubble and faded away by now (Equation (3)). Note that the position and morphology of the accretion shock is a function of time: as the radio bubble comes into contact with the shock surface, the shock expands into the light radio plasma toward the observer and attains additional curvature. For simplicity we suppress this effect in our sketch in Figure 3 and place the torus in the lower panel at a distance from the (initial and unmodified) shock surface which should be equal to that of the modified shock surface.

Using the 49 cm image of NGC 1265 (Figure 1(a) in Sijbring & de Bruyn 1998), we measure the major and minor radius of the torus to R ≃ 7' and r_min ≃ 11–14. We only consider orientations that lie in the northern and southern sectors as the elongated appearance toward the EW might be caused by shear flows as we will argue later on in this section. At the distance of Perseus, this corresponds to physical length scales of R ≃ 150 kpc and r_min ≃ (24...30) kpc, respectively. Assuming that the bubble was spherical before shock crossing and that the major radius did not change (Enßlin & Brüggen 2002), we estimate a compression factor through the associated volume change of

$$\begin{equation} C=\frac{V_{\rm bubble}}{V_{\rm torus}} = \frac{\frac{4}{3}\pi R^3}{2 \pi ^2 R r_{\rm min}^2} = \frac{2}{3\pi }\,\left(\frac{R}{r_{\rm min}}\right)^2\simeq 6\hbox{--}10. \end{equation} \tag{ 10 }$$

The radio plasma is adiabatically compressed across the shock passage according to $P_2 / P_1 = C^{\gamma _{\rm rel}}$, where γ_rel = 4/3 for an ultra-relativistic equation of state. However, the adiabatic index can in principle approach γ_rel ≲ 5/3 if the bubble pressure is dominated by a hot but non-relativistic ionic component. Varying γ_rel ≃ 1.33–1.5 and C ≃ 6–10, we obtain P₂/P₁ ≃ 21.5 ± 10.5. Assuming that the radio bubble is in pressure equilibrium with its surroundings before and after shock crossing, this pressure jump corresponds to the jump at the shock. Applying standard Rankine Hugoniot jump conditions (Landau & Lifshitz 1959) of an ideal fluid of adiabatic index γ = 5/3 yields a shock Mach number of ${\mathcal M}\simeq 4.2_{-1.2}^{+0.8}$, a density jump of 3.4^+0.2_−0.4, and a temperature jump of 6.3^+2.5_−2.7. We assumed a flat prior on the uncertainties of C and γ_rel which yields symmetric error bars around the pressure jump. We quote the jumps in the other thermodynamic quantities corresponding to our mean P₂/P₁. These quantities inherit asymmetric error bars due to the nonlinear dependence of these jumps on Mach number. We caution that we performed these estimates from the 600 MHz map (Sijbring & de Bruyn 1998); future higher resolution and more sensitive radio observations are needed to assess the uncertainties associated with finite beam width and to obtain a more reliable morphology of the low surface brightness regions of the torus.

Extrapolating X-ray profiles of Perseus (Churazov et al. 2003) to R₂₀₀ = 1.9 Mpc and using our mean values for the jumps in thermodynamic quantities, we can derive pre-shock values for the gas temperature and density that reflect upper limits on the gas properties in the infalling warm–hot IGM as predicted by Battaglia et al. (2009). We find kT₁ ≲ 0.4 keV, n₁ = 1.93 n_1,e ≲ 5 × 10⁻⁵ cm⁻³, and P₁ ≲ 3.6 × 10⁻¹⁴ erg cm⁻³. For the cluster temperature profile, we combine the central profile from X-ray data with the temperature profiles toward the cluster periphery according to cosmological cluster simulations (e.g., Pfrommer et al. 2007; Pinzke & Pfrommer 2010), yielding

$$\begin{equation} kT(r) = \left[kT_0 + \frac{kT_{\rm max} - kT_0}{1 + (r/r_{c})^{-3}}\right]\, \left[1 + \left(\frac{r}{0.2\,R_{200}}\right)^2\right]^{-0.3}, \end{equation} \tag{ 11 }$$

where kT₀ = 3 keV, kT_max = 7 keV, and r_c = 94 kpc. This represents the correct temperature profile over the entire range of the cluster when comparing to mosaiced observations by Suzaku (S. Allen 2010, private communication).

We now present evidence for post-shock shear flows with an argument that is based on the projected orientation of the ellipsoidally shaped radio torus whose main axis is almost aligned with the EW direction. We consider two cases. (1) If the ellipticity was due to projection of a ring-like torus, the shock normal n_s could not have components in the EW direction. Since the transverse velocity of the galaxy is pointing southward, momentum conservation at the (oblique) shock would imply a deflection of the post-shock gas in the plane that contains n_s and v—without components in the EW direction. Hence, an additional shear flow would be needed to explain the westward bending of the galaxy's tail even though the smooth and coherent bending of the tail might be difficult to reconcile with the turbulent nature of vorticity-induced shear flows. (2) If the bending of the tail was due to oblique shock deflection, n_s would have to have a component pointing westward. Projecting an intrinsically ring-like torus—aligned with the shock surface—would yield an apparent ellipsoidal torus with the main axis at some angle with the EW direction on the plane of the sky. This argues for a shear flow that realigns the orientation of the ellipsoid with the observed EW direction. In fact, assuming that the shock surface is aligned with the gravitational equipotential surface of the Perseus cluster, one can show that the implied curvature of the shock surface causes a vorticity in the post-shock regime that shears the post-shock gas westward.⁸ We cannot completely exclude pre-existing ellipticity in the radio bubble but find it very unlikely to be the dominant source of the observed ellipticity as the same directional deflection of the tail structure as well as the radio torus would then be a pure coincidence rather than a prediction of the model.

The observed flattening of the projected radio torus amounts to f = 1 − b/a ≃ 0.3, where a and b are the major and minor axes of the ellipse, respectively. With the assumption that the shock surface is aligned with the gravitational equipotential surface and using the result from Section 3, the flattening due to projection, f = 1 − cos ϕ ≃ 0.066, falls short of the observed one calling again for shear flows to account for the difference. The amount of vorticity ω injected at a curved shock of curvature radius R_curv should be proportional to the perpendicular velocity v_⊥ over the bubble radius r_bubble (Lighthill 1957),

$$\begin{equation} \omega = \frac{(C_s-1)^2}{C_s} \frac{v_\perp }{R_{\rm curv}} = \tilde{f} \frac{v_\perp }{r_{\rm bubble}}, \end{equation} \tag{ 12 }$$

where the constant of proportionality $\tilde{f}$ depends on the flattening f of the torus. Using $\tilde{f}\simeq f\simeq 0.3$, C_s ≃ 3.4, and r_bubble = 150 kpc, we obtain a rough estimate for the curvature radius of the shock of R_curv ≃ 850 kpc which is also adopted in Figure 3. This is similar to the dimensions of a filament connecting to a galaxy cluster and provides an independent cross check of our picture.

The ratio of the energy density in the shock-injected vorticity (or shear flow) to the thermal energy density is given by

$$\begin{equation} \frac{\varepsilon _{\rm shear}}{\varepsilon _{\rm th,2}} = \frac{\mu m_p v_{\perp }^2}{3 kT_2}\simeq 0.14, \end{equation} \tag{ 13 }$$

where kT₂ ≃ 2.4 keV and v_⊥ ≃ 400 km s^-1. Interestingly, this value is very close to the turbulent-to-thermal pressure support of ε_turb/ε_th< ≃ 0.2 at R₂₀₀ found in cosmological simulations of the formation of galaxy cluster (Ryu et al. 2008; Lau et al. 2009; Battaglia et al. 2010). We caution that our estimates strongly depend on our model assumptions for the orientation and position of the shock surface with respect to the LOS, since projection effects of a circular torus could account for a fraction of the observed ellipticity, reduce the shear contribution, and increase the inferred curvature radius. Another complication is the amount of magnetic helicity inside the torus that acts as a stabilizing agent against the shear that tries to distort and potentially disrupts the plasma torus (Ruszkowski et al. 2007; Braithwaite 2010).

5.1. Examining the Physics of the Model in Detail

Using results on the shock properties and geometric parameters from Sections 3 and 4, we demonstrate the physical feasibility of this model.

5.1.1. The Radio Bubble and Torus

The expansion time of the bubble in the filament (prior to shock passage) is of order the external sound crossing time, τ_exp = R/c₁ ≃ 0.4 Gyr, using a sound speed of the pre-shock region c₁ ≃ 285 km s^-1(kT/0.4 keV)^1/2 and a bubble radius of R ≃ 150 kpc. This assumes that the outer radius R is unchanged upon transformation into a torus (as suggested by numerical simulations of Enßlin & Brüggen 2002) and that R equals the minor axis of the ellipse of the projected torus that is unaffected by the shear. We can estimate the energetics of the previous outflow by calculating the PdV work done in inflating the bubble. Using the pre-shock gas pressure of P₁ ≃ 3.6 × 10⁻¹⁴ erg cm⁻³, we obtain E_bubble = 1/(γ_rel − 1)P₁V ≃ 4 × 10⁵⁸ erg using γ_rel = 4/3. This fits well inside the range of observed jet energies of FRI sources and implies an equipartition magnetic field in the bubble of B_1,eq = (8πP₁/2)^1/2 ≃ 0.66 μG that becomes after shock passage B_2,eq = B_1,eq C^2/3 ≃ 3 μG with C ≃ 10; somewhat higher than B_eq ≃ 1 μG obtained by equipartition arguments applied to the radio flux (Sijbring & de Bruyn 1998). The radio synchrotron radiating electrons of Lorentz factor Γ emit at a frequency ν_syn given by Equation (1). Using a range of magnetic field values of B_eq ≃ (1–3) μG, we require electrons with a post-shock Lorentz factor Γ₂ ≃ 4800–8500 to explain the observed radio torus at 600 MHz and derive a minimum value of Γ₁ = Γ₂/C^1/3 ≃ 2200–4000 for the upper cutoff of the electron population in order to be able to account for the observed radio emission after shock passage. Synchrotron and inverse Compton aging of fossil electrons in the bubble occurs on a timescale

$$\begin{equation} \tau _{\rm syn,\,ic, 1} = \frac{6\pi m_e c}{\sigma _{\rm T} \big(B_{\rm cmb}^2+B_{\rm 1,eq}^2\big) \Gamma _1} = (0.6\hbox{--}1) \;{\rm Gyr} > \tau _{\rm exp}. \end{equation} \tag{ 14 }$$

Calculating the maximum cooling time that the bubble could have been hibernating as given by Equation (14), we show that we meet this criterion for Γ₁ so that the model by Enßlin & Gopal-Krishna (2001) applies.

5.1.2. The Current Outburst and Radio Tail of NGC 1265

Assuming that the twin tails together constitute a cylinder of radius r_tail ≃ 15 and projected length L_t,tail ≃ 11', we can estimate the volume for the radio tail of NGC 1265, V_tail ≃ πr²_tailL_t,tail/tan θ ≃ 3.8 × 10⁷⁰ cm³ using θ ≃ 32°. Taking the minimum energy density from Sijbring & de Bruyn (1998) in the tail of NGC 1265 to be around ε_eq ≃ 10⁻¹² erg cm^-3, we estimate the energy content of the radio tail to be about E_jet ≃ 3.8 × 10⁵⁸ erg which nicely coincides with the result that we derived for the previous outburst. The necessary accreted mass to power the outflow amounts to M_acc = E_jet/(η c²) ≃ 2 × 10⁵ M_☉, where we adopted a typical values of η ≃ 0.1 for the efficiency parameter. Using the time since shock crossing in our model of τ ≃ 1.8 × 10⁸ yr, we estimate the jet power of about 6.7 × 10⁴² erg s^-1 which is in range of observed jet luminosities of FRI sources. Using the 1.4 GHz flux for NGC 1265 of 8 Jy, we obtain a radio luminosity of νL_ν ≃ 7.5 × 10⁴⁰ erg s^-1, or a 1% radiative efficiency which is plausible.

5.1.3. Estimates from the Jet Bending

Using the high-resolution VLA observations of the head structure of NGC 1265, O'Dea & Owen (1986) give a minimum jet pressure P_jet, min ≃ (1–3) × 10⁻¹¹ erg cm⁻³ which is about 10 times the ICM estimate at the present position of the galaxy, P_ICM(r_gal) ≃ 1.4 × 10⁻¹² erg cm⁻³ with r_gal ≃ 1.45 Mpc (see Sections 3 and 4). The jet radius is r_jet ≲ 1'' (360 pc), whereas the projected bending radius of the jet is around 30'' (11 kpc), so the physical ratio is r_b/r_jet ∼ 80 when accounting for deprojection and finite resolution effects. Using Equation (A2), this leads to a Mach number ratio of ${\mathcal M}_{\rm jet}/{\mathcal M}_{\rm gal} \simeq 2.8$. The temperature estimate at the position of the galaxy is kT(r_gal) ≃ 3.1 keV, so that we obtain ${\mathcal M}_{\rm gal} \simeq 2.8$ and eventually ${\mathcal M}_{\rm jet} \simeq 7.8$. If we take the jet power to be L_jet/2 ∼ ρ_jetv³_jetπr²_jet (where L_jet is the luminosity of the 2 jets and we assume that most of the jet power is kinetic energy), the power can be expressed in terms of the jet pressure and Mach number as

$$\begin{equation} \frac{L_{\rm jet}}{2} \simeq \gamma _{\rm jet} P_{\rm jet} {\mathcal M}_{\rm jet}^2 v_{\rm jet} \pi r_{\rm jet}^2. \end{equation} \tag{ 15 }$$

Since our estimate of L_jet ≃ 6.7 × 10⁴² erg s^-1, we get v_jet ≃ 7000 km s^-1 which is a typical jet velocity. Solving for the number density of the jet material, we get n_jet ≃ L_jet/(2πm_pv³_jetr²_jet) ≃ 10⁻³ cm⁻³ which is a factor of three larger than the surrounding ICM, n(r_gal) ≃ 3 × 10⁻⁴ cm⁻³ and again a plausible value.

5.2. Stability Analysis of the Torus

There are three mechanisms that have the potential to destroy the torus once it has formed; namely, (1) large-scale shear flows injected at the shock front, (2) ICM turbulence, and (3) Kelvin–Helmholtz instabilities at the interface due to the vortex flow around the minor circle of the torus. By considering each of these processes separately, we will show that none of them can seriously impact the torus on an eddy turnover timescale of the stabilizing vortex flow,

$$\begin{equation} \tau _{\rm eddy}\simeq \frac{2\pi r_{\rm min}}{v_{2,\parallel }} \simeq 2\times 10^8\;{\rm yr}, \end{equation} \tag{ 16 }$$

where r_min ≃ 25 kpc denotes the minor circle of the torus, v_2,∥ = vcos ϕ/C_s ≃ 740 km s^-1 is the post-shock gas velocity parallel to the shock normal, v ≃ 2550 km s^-1 is the total velocity of the galaxy, and ϕ ≃ 9° is the shock obliquity defined in Table 1. Here and in the following, we apply the numerical values of our consistent model as derived in Sections 3 and 4. On a timescale larger than τ_eddy, dissipative effects will thermalize the kinetic energy of this stabilizing flow, and eventually external shear flows and turbulence might start to destroy the torus depending on the magnetic helicity and morphology in the torus (Ruszkowski et al. 2007; Braithwaite 2010).

• 1.  
  The amount of vorticity ω_shear injected at a curved shock with curvature radius R_curv is given by (Lighthill 1957)

  $$\begin{equation} \omega _{\rm shear} = \frac{(C_s-1)^2}{C_s} \frac{v_{\perp }}{R_{\rm curv}} = \frac{(C_s-1)^2}{C_s} \frac{v \sin \phi }{R_{\rm curv}}, \end{equation} \tag{ 17 }$$

  where C_s is the shock compression factor and v_⊥ is the (post-)shock gas velocity perpendicular to the shock normal. Note that for vorticity injection into an irrotational flow encountering a shock, one necessarily needs a curved shock surface according to Crocco's theorem (1937). The generation of vorticity relies on a gradient of either the entropy or stagnation enthalpy along the shock front (perpendicular to the shock normal). This can be easiest seen by considering a homogeneous flow encountering the shock surface at some constant angle (obliquity). It experiences the same "shock deflection" and amount of entropy injection along the shock front. This is because mass and momentum conservation across the shock ensures the conservation of v_⊥ and yields v_2,∥ = v_1,∥/C_s. If the bubble was impacting the shock surface at larger angle, the induced stabilizing vortex flow around the minor circle of radius r_min is accordingly smaller since the perpendicular velocity component cannot contribute to the vortex flow,

  $$\begin{equation} \omega _{\rm torus} = \frac{v_{2,\parallel }}{r_{\rm min}} = \frac{v\cos \phi }{C_s r_{\rm min}}. \end{equation} \tag{ 18 }$$

  We can then derive a criterion for the stability of the torus in the presence of shock injected shear flows, ω_shear < ω_torus, that can be cast into a requirement for the shock obliquity,

  $$\begin{equation} \phi < \phi _{\rm crit} = \arctan \left[\frac{R_{\rm curv}}{r_{\rm min} (C_s - 1)^2} \right] \simeq 80\mbox{$^\circ $}. \end{equation} \tag{ 19 }$$

  Here, C_s ≃ 3.4, R_curv ≃ 850 kpc as inferred from the observed ellipticity of the torus, and r_min ≃ 25 kpc. Since our preferred value of the shock obliquity ϕ = 9° ≪ ϕ_crit, the torus can easily be stabilized by the vortex flow around its minor circle against large-scale shear.
• 2.  
  To assess the impact of ICM turbulence on the torus' stability, we compare the turbulent energy density ε_turb to the kinetic energy density of the torus vortex flow ε_vort,

  $$\begin{eqnarray} \left.\frac{\varepsilon _{\rm turb}}{\varepsilon _{\rm vort}}\right|_{r_{\rm min}}&=& \left.\frac{\varepsilon _{\rm turb}}{\varepsilon _{\rm th, 2}}\right|_{r_{\rm min}} \frac{\varepsilon _{\rm th, 2}}{\varepsilon _{\rm vort}} \nonumber \\ &=& \frac{\varepsilon _{\rm turb}}{\varepsilon _{\rm th, 2}} \left(\frac{R_{\rm curv}}{r_{\rm min}}\right)^{-2/3} \frac{3 kT_2}{\mu m_p v_{2,\parallel }^2}\simeq 0.04.\qquad \end{eqnarray} \tag{ 20 }$$

  Here, ε_th,2 and kT₂ = 2.4 keV denote the post-shock energy density and internal energy, μ = 0.588 denotes the mean molecular weight for a medium of primordial element abundance, and m_p denotes the proton rest mass. To quantify the ratio ε_turb/ε_vort in our model, we assume a turbulent pressure support relative to the thermal pressure at R₂₀₀ of around 20% when correcting for the energy contribution due to bulk flows (Ryu et al. 2008; Lau et al. 2009; Battaglia et al. 2010). We furthermore assume that the turbulent energy density is dominated by the injection scale (which should be comparable to the curvature radius at the shock). Using a Kolmogorov scaling of the turbulence, the power per logarithmic interval in scale length is down by another factor of (R_curv/r_min)^2/3 ≃ 10 at the minor radius r_min compared to the injection scale R_curv. Hence, the energy density of the stabilizing vorticity flow around the minor axis of the torus outweighs any turbulent energy density of the ICM on this scale by a factor of about 25! Equation (20) enables us to solve for the maximum shock obliquity allowed so that the torus is stable against turbulent shear flows,

  $$\begin{equation} \phi < \arccos \left[ \frac{\varepsilon _{\rm turb}}{\varepsilon _{\rm th}}\left(\frac{R_{\rm curv}}{r_{\rm min}}\right)^{-2/3} \frac{3 kT_2 C_s^2}{\mu m_p v^2} \right]^{1/2}\simeq 78\mbox{$^\circ $}, \end{equation} \tag{ 21 }$$

  where v ≃ 2550 km s^-1 and C_s ≃ 3.4. Interestingly, this is a similar value as ϕ_crit obtained in Equation (19) and is another manifestation of our finding that ε_turb ∼ ε_shear (Equation (13)) implying that a large fraction of post-shock turbulence could have been injected at curved shocks.
• 3.  
  We finally show that the Kelvin–Helmholtz instability at the interface of the torus due to the vortex flow should be suppressed by internal magnetic fields. The shock compression acts mostly perpendicular to the forming toroidal ring. Consequently, the field component parallel to the bubble surface and hence to the surface of the torus is preferentially amplified and dominates eventually the magnetic energy density (Enßlin & Brüggen 2002). Within the torus, we assume equipartition magnetic fields of B_eq,2 ≃ (1–3) μG which correspond to radio and pressure equipartition values, respectively (Section 5.1).⁹ Assuming the torus to be in pressure equilibrium with the surroundings and a relativistic electron component in the bubble whose pressure is dominated by momenta ∼m_ec, we can derive a density contrast between the radio plasma and post-shock density of the ICM,

  $$\begin{equation} \delta = \frac{n_{\rm torus}}{n_{\rm 2, ICM}} \simeq \frac{kT_2}{m_e c^2}\simeq 5\times 10^{-3}, \end{equation} \tag{ 22 }$$

  where kT₂ = 2.4 keV. We can now compare the ratio of the Alfvén velocity, v_A, in the surface layer of the torus to the velocity in the vortex flow around the minor circle of the torus, v_2,∥,

  $$\begin{equation} \frac{v_{{\rm A}}}{v_{2,\parallel }} =\frac{B_{\rm eq}}{\sqrt{4 \pi \delta \rho _{\rm 2,ICM}^{} v_{2,\parallel }^2}} \simeq 4\hbox{--}11, \end{equation} \tag{ 23 }$$

  where n_2,ICM = 2 × 10⁻³ cm⁻³. In such a configuration, magnetic tension is strong enough to suppress the Kelvin–Helmholtz instability of the vortex flow (Chandrasekhar 1961; Dursi 2007). To summarize, if the magnetic field is not too far from equipartition values, B ≳ B_2,eq/10, then the line of arguments presented here guarantees stability of the torus for at least an eddy turnover timescale τ_eddy.

In providing a quantitative 3D model for NGC 1265, we provide conclusive observational evidence for an accretion shock onto a galaxy cluster. The accretion shock is characterized by an intermediate-strength shock with a Mach number of ${\mathcal M}\simeq 4.2_{-1.2}^{+0.8}$. In the context of the IGM, it appears that even weak to intermediate-strength shocks of ${\mathcal M}\sim 2.3$ are able to accelerate electrons as the radio relic in A521 suggests (Giacintucci et al. 2008). This supports the view that these intermediate-strength shocks are very important in understanding non-thermal processes in clusters; in particular as they are much more abundant than high-Mach number shocks (e.g., Ryu et al. 2003; Pfrommer et al. 2006). The presented technique offers a novel way to find large-scale formation shock waves that would likely be missed if we only saw giant radio relics. These are thought to be due to shock-accelerated relativistic electrons at merging or accretion shock waves (Enßlin et al. 1998; Bagchi et al. 2006), whereas the discrimination of the two types of shock waves is difficult on theoretical as well as observational grounds (Enßlin 2006).

The rich morphology of the large-scale radio structure that arches around steep-spectrum tail of NGC 1265 enables a qualitative argument for the need of shear to explain the orientation of its ellipsoidally shaped torus. The energy density of the shear flow corresponds to a turbulent-to-thermal energy density of 14%—consistent with estimates in cosmological simulations. However, simulations are needed to quantify the effect of magnetic helicity internal to the bubble on the torus morphology. In any case, the presence of post-shock shear implies the amplification of weaker seed magnetic fields through shearing motions as well as the generation of magnetic fields through the Biermann battery mechanism if the vorticity has been generated at the curved shock. Even more interesting, the shock passage of multiphase gas with a large density contrast necessarily yields a multiply curved shock surface that causes vorticity injection into later accreted gas on the corresponding curvature scales. Hence, shearing motions are superposed on various scales which implies amplification of the magnetic fields on these scales.

More sensitive polarized and total intensity observations of NGC 1265 are needed to confirm our model predictions with radial polarization vectors of the radio torus and a polarization degree of around 5%. In addition, future observations of different systems similar to NGC 1265 would be beneficial to get a statistical measurement of properties of the accretions shocks and its pre-shock conditions. This provides indirect evidence for the existence of the warm–hot IGM. Our work shows the potential of these kind of serendipitous events as ways to explore the outer fringes of clusters and dynamical features of accretion shocks that are complementary to X-ray observations.

We thank T. A. Enßlin, T. Pfrommer, M. Sun, and an anonymous referee for helpful comments on this manuscript and gratefully acknowledge the great atmosphere at the Kavli Institute for Theoretical Physics program on Particle Acceleration in Astrophysical Plasmas, in Santa Barbara (2009 July 26–October 3) where this project was initiated. That program was supported in part by the National Science Foundation under grant no. PHY05-51164. C.P. gratefully acknowledges the financial support of the National Science and Engineering Research Council of Canada. T.W.J. was supported in part by NSF grant AST0908668 and by the University of Minnesota Supercomputing Institute.

When a low-density bubble crosses a shock of speed v_si in the ICM, the shock accelerates into the bubble, pulling post-shock ambient gas with it. The original bubble–ICM CD follows the shock at a speed intermediate to that of the incident and bubble shocks when the incident shock is at least moderately strong. Because the shock intrusion begins sooner and impacts more strongly on the leading edge of a round bubble than its periphery, the ambient gas penetrates the center of the bubble first. An initially spheroidal bubble will then evolve into a torus (vortex ring) on a timescale determined by the crossing time of the CD through the bubble.

The shock-produced dynamics in the bubble can be estimated from the exact solution to the simple 1D Riemann problem of a plane shock impacting a CD. This is found most simply in the initial rest frame of the bubble, which we assume to be in pressure equilibrium with the unshocked ICM and has a density, ρ_b = δρ_i, with δ < 1. At impact a forward shock will penetrate into the low-density bubble with speed, v_sb, while a rarefaction will propagate backward into the post-shock ICM. The bubble–ICM CD will move forward at the same speed as the post-shock flow in the bubble. The full Riemann solution is obtained by matching the pressure behind the forward shock inside the bubble to the pressure at the foot of the rarefaction in the ICM.

We define the ICM and bubble adiabatic indices as γ_i and γ_b, respectively. Assuming the shock is propagating from the left, we have right to left four uniform states, 0, 1, 2, 3, where 0 and 1 are separated by the forward shock, 1 and 2 are separated by the CD, while 2 and 3 are separated by the reverse rarefaction (note that this numbering scheme differs from the main body of the paper). The state "0" represents the initial conditions in the bubble, while "3" represents conditions in the ICM post-shock flow. We set P₁ = P₂ = P_* and v₁ = v₂ = v_* = v_cd. The initial bubble (and ICM) pressure is P₀, while the initial bubble and ICM sound speeds are related by $c_b = c_0 = \sqrt{\gamma _b P_0/\rho _0} = c_i\sqrt{\gamma _b/ (\gamma _i \delta)}$. The Mach numbers of the external, ICM shock and the internal, bubble shock are $\mathcal {M}_i = v_{{\rm si}}/c_i$ and $\mathcal {M}_b = v_{{\rm sb}}/c_b \equiv \mu \mathcal {M}_i$ and we introduced the Mach number ratio μ.¹⁰

From standard shock jump conditions, we have

$$\begin{eqnarray} P_3 &=& \frac{2}{\gamma _i+1}\left(\gamma _i \mathcal {M}_i^2 - \frac{\gamma _i - 1}{2}\right)P_0\nonumber \\ \rho _3 &=& \frac{(\gamma _i+1)\mathcal {M}_i^2}{(\gamma _i-1)\mathcal {M}_i^2 + 2}\rho _i =\frac{1}{\delta }\frac{(\gamma _i+1)\mathcal {M}_i^2}{(\gamma _i-1)\mathcal {M}_i^2+2}\rho _b\nonumber \\ v_3 &=& \frac{2}{\gamma _i+1}\frac{\big(\mathcal {M}_i^2 - 1\big)}{\mathcal {M}_i}c_{i}\\ P_1 &=& P_2 = P_* = \frac{2}{\gamma _b+1}\left(\gamma _b\mu ^2\mathcal {M}_i^2 - \frac{\gamma _b-1}{2}\right)P_0\nonumber \\ \rho _1 &=& \frac{(\gamma _b+1)\mu ^2\mathcal {M}_i^2}{(\gamma _b-1)\mu ^2\mathcal {M}_i^2 + 2}\rho _b\nonumber \\ v_{\rm CD} &=& v_1 = v_2 = v_* = \frac{2}{\gamma _b+1}\frac{\big(\mu ^2\mathcal {M}_i^2 - 1\big)}{\mu \mathcal {M}_i}c_b\nonumber. \end{eqnarray} \tag{ B1 }$$

In addition, the Riemann invariant connecting states 2 and 3 through the rarefaction along with the relation $\rho \propto P^{1/\gamma _i}$ inside the rarefaction give

$$\begin{equation} v_* = v_3 + \frac{2}{\gamma _i-1} c_3 \left[1 - \left(\frac{P_*}{P_3}\right)^{\frac{\gamma _i-1}{2\gamma _i}}\right], \end{equation} \tag{ B2 }$$

while

$$\begin{equation} c_3^2 = \frac{\gamma _i P_3}{\rho _3} = \frac{2}{\gamma _i+1}\frac{\left[\gamma _i\mathcal {M}_i^2- \frac{\gamma _i-1}{2}\right]\left[(\gamma _i-1)\mathcal {M}_i^2 + 2\right]}{(\gamma _i+1)\mathcal {M}_i^2}c_i^2. \end{equation} \tag{ B3 }$$

These can be combined to give the following equation for $\mu = \mathcal {M}_b/\mathcal {M}_i$:

$$\begin{eqnarray} 1 &=&\frac{\gamma _i+1}{\gamma _b+1}\frac{\mu ^2\mathcal {M}_i^2-1}{\mu \delta ^{1/2}\big(\mathcal {M}_i^2-1\big)}\left(\frac{\gamma _b}{\gamma _i}\right)^{1/2}\nonumber \\ &&-\frac{1}{\gamma _i-1}\frac{\left[2\gamma _i\mathcal {M}_i^2-\gamma _i+1\right]^{1/2}\left[(\gamma _i-1)\mathcal {M}_i^2+2\right]^{1/2}}{\mathcal {M}_i^2-1}\nonumber\\ &&\times\left[1-\left(\frac{(\gamma _i+1)\big(2\gamma _b\mu ^2\mathcal {M}_i^2-\gamma _b+1\big)}{(\gamma _b+1)\big(2\gamma _i\mathcal {M}_i^2-\gamma _i+1\big)}\right)^{\frac{\gamma _i-1}{2\gamma _i}}\right], \end{eqnarray} \tag{ B4 }$$

which can be solved numerically for μ, given $\mathcal {M}_i$, γ_i, γ_b, and δ. For δ = 1, γ_i = γ_b, the obvious solution is μ = 1. For our problem we expect γ_i = 5/3 and γ_b = 4/3. This gives

$$\begin{eqnarray} 1 &=& \frac{8}{7}\left(\frac{4}{5}\right)^{1/2}\frac{\mu ^2\mathcal {M}_i^2-1}{\mu \delta ^{1/2}\big(\mathcal {M}_i^2-1\big)} -\frac{\left[\big(5\mathcal {M}_i^2-1\big)\big(\mathcal {M}_i^2+3\big)\right]^{1/2}}{\mathcal {M}_i^2-1}\nonumber\\ &&\times\left[1-\left(\frac{4}{7}\right)^{1/5}\left(\frac{8\mu ^2\mathcal {M}_i^2-1}{5\mathcal {M}_i^2-1}\right)^{1/5}\right]. \end{eqnarray} \tag{ B5 }$$

Equation (B5) is displayed for δ = 10⁻² and δ = 10⁻³ in the left panel of Figure 4.

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In the strong (external) shock limit, $\mu \mathcal {M}_i \gg 1$, it is easy to see that approximately $\mu \propto \sqrt{\delta }$. Empirically we find in this limit when δ ≪ 1 that $\mu \sim 2 \sqrt{\delta }$, so that v_sb ≈ 2v_si. As $\mathcal {M}_i \rightarrow 1$, the solution μ → 1, so $\mathcal {M}_b \rightarrow 1$ also applies. The more general solutions for small to moderate incident shock strengths are shown for two values of δ as solid curves in Figure 4. The lower bound for v_sb is given by $c_b = v_{{\rm si}}\sqrt{\gamma _b/\gamma _i} [1/(\mathcal {M}_i\sqrt{\delta })]$ and is shown in each case by a dotted curve. We note that the solution shown in Figure 4 is almost identical to the case of γ_i = γ_b, since the internal shock in the bubble is barely supersonic with $\mathcal {M}_b \gtrsim 1$ due to the large sound speed in the bubble. This can be easily seen by taking the ratio of the corresponding solid-to-dotted lines which provides $\mathcal {M}_b$ for each external Mach number $\mathcal {M}_i$.

We can understand the general behavior of an increasing shock speed inside a low-density bubble relative to the incident shock speed for smaller Mach numbers $\mathcal {M}_i$ by the following line of arguments. For small Mach numbers, both waves are just nonlinear sound waves, so each propagates near the local sound speed, which is much larger in the bubble. It turns out for large Mach numbers $\mathcal {M}_i$ that the pressure just behind the penetrating shock is smaller than the external post-shock pressure by a factor that scales with the density contrast δ. This pressure drop comes from the rarefaction going back into the shocked ICM. For weak shocks that rarefaction is weak (provided the original bubble was in pressure equilibrium).

The right panel in Figure 4 shows the behavior of v_CD corresponding to the shock solutions in the left figure panel. In the limit $\mu ^2 \mathcal {M}_i^2\gg 1$, the expression for v_CD in Equation (B1) takes the form $v_{\rm CD}/v_{\rm si} \approx (2/(\gamma _b+1)) \sqrt{\gamma _b/\delta \gamma _i} \mu$. Note that the limit $\mu ^2 \mathcal {M}_i^2\gg 1$ requires the internal bubble (pseudo-) temperature not to be too high and hence implies a lower limit on the bubble density contrast of δ ≳ 10⁻³ assuming the original bubble was in pressure equilibrium. Applying our empirical result for strong shocks with 10⁻³ ≲ δ ≪ 1 (namely, $\mu \sim 2 \sqrt{\delta }$), we would expect $v_{\rm CD}/v_{\rm si}\rightarrow (4/(\gamma _b+1))\sqrt{\gamma _b/\gamma _i} \sim 1.5$, which is within about 20% of the exact solutions in this limit. This value for v_CD/v_si is also a reasonable estimate at moderate shock numbers, say $\mathcal {M}_i\gtrsim 3$. For $\mathcal {M}_i\gtrsim 2,$ the limit v_CD>v_si still applies. As $\mathcal {M}_i$ approaches one, however, the speed of the CD drops below the incident shock speed, since as the bubble shock becomes a sound wave, the CD's motion vanishes. The results of these considerations are quantitatively confirmed by a suite of 2D axisymmetric simulations where we varied the Mach number $\mathcal {M}_i = \lbrace 2,3,5,10\rbrace$ and initial bubble/ICM density contrast (see Figure 2 for one realization).

For our moderate-strength shock example in Perseus, we can use these results to estimate the time for the CD to cross a bubble diameter, 2R_bubble, transforming the bubble into a vortex ring, with a toroidal topology; namely,

$$\begin{equation} \tau _{{\rm form}}\sim \frac{2 R_{{\rm bubble}}}{1.5 v_{{\rm si}}}\sim 1.4\times 10^{8} \;{\rm yr}. \end{equation} \tag{ B6 }$$

Here, we adopted our fiducial values of R_bubble ≃ 150 kpc, δ ≃ 5 × 10⁻³ (Equation (22)), and $v_{\rm si} = \mathcal {M}_i c_i = \mathcal {M}_i \sqrt{\gamma kT_i / (\mu _{{\rm mw}} m_p)} \simeq 4.2\times 330\;{\rm km\;s^{-1}} \simeq 1400\;{\rm km\;s^{-1}}$ where we used the sound speed in the pre-shock gas in the infalling filament of temperature kT_i ≃ 0.4 keV and μ_mw = 0.588. We emphasize that this estimate is close to the value τ_3–4 = 1.2 × 10⁸ yr that we adopted in our model and consistent with the derived the error bars on $\mathcal {M}_i$.