STEADY-STATE HADRONIC GAMMA-RAY EMISSION FROM 100-MYR-OLD FERMI BUBBLES

The Fermi Bubbles (Dobler et al. 2010; Su et al. 2010) are giant 1–100 GeV γ-ray structures that extend ∼7 kpc north and south of the Galactic nucleus. Structures roughly coincident with the Bubbles are known in X-rays (Bland-Hawthorn & Cohen 2003), total intensity microwaves (Finkbeiner 2004; Dobler & Finkbeiner 2008; Dobler 2012; Ade et al. 2013), polarized intensity microwaves (Jones et al. 2012), and polarized intensity radio continuum (Carretti et al. 2013) emission.

Much theoretical work on the Bubbles has focused on the idea that their γ-ray emission is supplied by the inverse-Compton (IC) up-scattering of ambient light by a cosmic ray (CR) electron population. Given that the Bubbles extend so far into the halo with a relatively fixed hard spectrum, an IC model must invoke up-scattering of the cosmic microwave background to multi-GeV gamma-ray energies, requiring ≳ TeV electron primaries. Such electrons cool quickly, defining a natural timescale ≲ 1 Myr. To explain, then, the large extent and energy content of the Bubbles, some models (e.g., Zubovas & Nayakshin 2012; Guo & Mathews 2012; Yang et al. 2012, 2013; Barkov & Bosch-Ramon 2013) hypothesize recent activity of the super-massive black hole at the Galactic Center (GC) with very fast transport of the electrons. These timing constraints are relaxed if there is distributed acceleration (Cheng et al. 2011; Mertsch & Sarkar 2011; Lacki 2013) throughout the structures. Evidence in support of an active galactic nucleus-like scenario may come from the detection (Su & Finkbeiner 2012) of a jet-like feature in the γ-ray data and the recent claim that the Magellanic stream was bathed in a bright UV flash only a few million years ago (Bland-Hawthorn et al. 2013).

Alternatively, the Bubbles' γ-ray emission may be hadronic in origin with CR protons (and heavier ions)—ultimately energized by nuclear star-formation—accumulated over (much) longer timescales, colliding with ambient gas to supply the γ-rays (Crocker & Aharonian 2011). A number of pieces of evidence are consistent with this scenario. First, the intense star-formation in the inner ∼200 pc diameter region around the GC currently produces a CR power that is elegantly sufficient to supply both the (hadronic) γ-ray luminosity of the Bubbles (Crocker et al. 2011; Crocker 2012) and the 2.3 GHz synchrotron luminosity of the recently discovered, polarized radio lobe counterparts to the Bubbles (Carretti et al. 2013). Second, as we show below (also see Crocker 2012), the mass flux of ∼0.1 M_☉ yr⁻¹ from the nuclear-star-formation-driven outflow is elegantly sufficient to maintain the Bubbles' plasma mass in steady-state against its thermal losses. Last and more speculatively, in the sky distribution of up to 28 neutrino events recently observed by IceCube against a background of 10.6 atmospheric events, there is a hint of an overabundance of neutrinos from the direction of the Inner Galaxy with an inferred flux consistent with a hadronic origin for the γ-rays (Aartsen et al. 2013; Ahlers & Murase 2013; Razzaque 2013).

The apparent cost, however, of any steady state hadronic model is the long timescale implied. The protons and ions fed into the Bubbles collide with ambient gas nuclei over the pp loss time which, on the low volumetric average gas density, is t_pp ≃ 10 Gyr [n_H/(0.005 cm^-3)]^-1. Establishing a steady state requires that the structures have existed for this sort of timeframe, which is difficult to reconcile with other natural timescales relating to the Bubbles/GC. In particular, the 2.3 GHz polarization observations (Carretti et al. 2013) suggest that the outflow feeding electrons into the radio lobes has a vertical speed of ∼1100 km s⁻¹ giving an advective timescale over the ∼8 kpc extent of the radio lobes (marginally larger than the Bubbles) of only 7 × 10⁶ yr. Nuclear star-formation, which supplies a total mechanical power of ∼3 × 10⁴⁰ erg s⁻¹, ∼80% of which goes into heating or moving the outflowing plasma (Crocker 2012), cannot supply all the Bubbles' energy (≳ 5 × 10⁵⁵ erg) within this timescale.

We show immediately below, however, that both this short advection timescale and the long timescale associated with steady-state hadronic emission need to be reconsidered, given the natural mechanism that is the subject of this Letter. Briefly, we discuss the following: cooling of the Bubbles' interior plasma over 1–2 × 10⁸ yr into cool, overdense filaments leads to the adiabatic compression of CRs and magnetic fields. With the simple prescription that the (dominantly non-thermal) pressure of the filaments reaches equilibrium with the Bubble plasma, we can predict their hadronic emission. As we show below, this prediction is a good match to observations

The ∼1100 km s⁻¹ characteristic speed at the base of the outflow is suggested by two independent analyses of the radio data based on (1) a geometrical analysis of the curvature of one of the strongly magnetized linear "ridge" features seen in the data, and (2) the cooling time of the electrons whose synchrotron emission reaches to the top of the radio lobes. This speed is somewhat in excess of the gravitational escape velocity (∼900 km s⁻¹; Muno et al. 2004), and, if interpreted as purely a plasma velocity, is expected to imply an escaping wind.

In fact, however, the fast flow does not escape to infinity: the 23 GHz polarization data show that the magnetic field lines curl over toward horizontal at the top of the Bubbles, suggesting a vertical speed that is significantly slowing. Thus, there is a structure of a definite, finite height, not a freely expanding wind. The CRs stream ahead of the plasma with a speed limited to the Alfvén velocity because of the streaming instability. This velocity also controls the speed with which torsional Alfvén waves respond to the driving from the rotation of the outflow's base. We thus re-interpret the previously determined 1100 km s⁻¹ characteristic speed as the sum of two components: a plasma wind speed that is somewhat less than the escape velocity (but still capable of delivering gas to ∼few kiloparsec heights into the halo) and the Alfvén speed along the magnetic field lines. Thus, material and energy can be accumulated into the Bubbles for significantly longer than 7 × 10⁶ yr.

The second problem is to reconcile the very long hadronic timescale with other timescales. For instance, the thermally unstable Bubble plasma cools over (1–2) × 10⁸ yr (see below), collapsing into overdense condensations. This concentrates the magnetic field attached to the plasma, thereby mustering in the CRs. Analogous to the processes occurring around the brightest central galaxies (BCGs) of clusters (Sharma et al. 2010), this collapse will continue until the non-thermal pressure in the condensations equilibrates with the external plasma pressure. The CRs and magnetic fields evolve adiabatically at the beginning of the collapse, in contrast to the plasma whose collisional losses rise strongly as n². Thus, the non-thermal interstellar medium phases come to dominate the pressure in the condensations despite their softer equations of state. As for the Hα filaments around BCGs (Sharma et al. 2010), anisotropic heat conduction (due to the suppression of electron motions transverse to magnetic fields) means that the condensations assume the topology of extended filaments with dominantly longitudinal magnetic fields. With this picture, we can calculate how the collapse proceeds.

The Milky Way has an X-ray bulge (XRB) with some individual features clearly correlated with Bubble structures (Su et al. 2010). The bolometric thermal luminosity of the XRB lies in the range L_XRB = (3–9) × 10³⁹ erg s⁻¹ (Snowden et al. 1997; Almy et al. 2000). From this luminosity and assuming a static structure, we can calculate the rate at which material is cooling out of the XRB as $\dot{M}_\mathrm{cool} \simeq 2/3 \ L_\mathrm{XRB} \, \mu {\rm m}_\mathrm{p}/(k_{\rm B} T) \sim (0.07\hbox{--}0.2)\, {M_\odot }$ yr⁻¹ (μ ≃ 0.6 is the mean mass of the plasma constituents in terms of m_p). This $\dot{M}_\mathrm{cool}$ is close to the mass efflux (Crocker 2012) along the SF-driven nuclear outflow. Given that the mechanical energy and mass injected by nuclear SF can sustain the non-thermal luminosity of the Bubbles in steady state, we postulate that the Bubbles are enduring structures inflated and sustained by nuclear star formation. In this circumstance, freshly injected plasma from the nuclear outflow balances the mass drop-out rate, $\dot{M}_\mathrm{in} \equiv \dot{M}_\mathrm{cool}$, resulting in a steady state gas density in the Bubbles of

$$\begin{eqnarray} n_\mathrm{H^+}^{\mathrm {FB}} &=& \left(\frac{\dot{M}_\mathrm{in} \ T_\mathrm{FB} \ k_{\rm {B}} }{(\gamma -1) \ \mu \ 1.22 \ m_\mathrm{p} \ V_\mathrm{FB} \ \Lambda [T]} \right)^{\mathrm 1/2} \nonumber \\ &\simeq &0.003 \ {\rm cm}^{\mathrm -3} \left(\frac{\dot{M}_\mathrm{in}}{0.1\, {M_\odot }\, {\rm yr^{-1}}} \right)^{\mathrm 1/2}, \end{eqnarray} \tag{ 1 }$$

where $n_\mathrm{e} \simeq 1.22 n_\mathrm{H^+}$, (γ − 1) = 2/3, Λ[T] is the plasma thermal cooling function (Raymond et al. 1976), and V_FB = 8.4 × 10⁶⁶ cm³. Reinforcing the steady state picture, this $n_\mathrm{H^+}^{\mathrm {FB}}$ estimate is consistent with X-ray measurements by Suzaku (Kataoka et al. 2013) which give T_FB ≃ 3.5 × 10⁶ K for the Bubble plasma (equal to that of the adjacent halo plasma) and from which we infer $n_\mathrm{H^+}^{\mathrm {FB}} \simeq (1\hbox{--}3) \times 10^{\mathrm -3}$ cm^-3.

For this plasma number density and temperature the cooling time is (1–2) × 10⁸ yr. We show below that the timescale for the formation of the Bubbles is ≳ 2 × 10⁸ yr, only a little longer.

Given the adiabatic compression of relativistic CR protons into the plasma condensations, we can make an estimate of the hadronic γ-ray emission from the Bubbles, finding consistency with observations. The filament pressure, supplied by relativistic CR protons (denoted by p) and magnetic fields (B), equilibrates with the external plasma pressure, $p^{\mathrm {fil}} \equiv p_\mathrm{p}^{\mathrm {fil}} + p_B^{\mathrm {fil}} = p^{\mathrm {FB}} = 2.2 \times T_\mathrm{FB} \ k_\mathrm{{B}} \ n_\mathrm{H^+}^{\mathrm {FB}} = 2.0$ eV cm^-3 and we suppose $p_\mathrm{p}^{\mathrm {fil}} \simeq p_B^{\mathrm {fil}}$. The energy density of the adiabatically accumulated, relativistic CR protons in the filaments is $u_p^{\mathrm {fil}} = u_p^{\mathrm {GC}} (n_\mathrm{H}^{\mathrm {fil}}/n_\mathrm{H}^{\mathrm {GC}})^{\mathrm 4/3}$, where GC denotes a parameter of the nuclear star formation region where the CRs and thermal plasma are energized. With $p_p^{\mathrm {fil}} \equiv 1/3 \ u_p^{\mathrm {fil}} = 1/2 \ p^{\mathrm {fil}}$, we determine the filament filling factor as $f = (3.3 \ k_{\rm B} \ T_\mathrm{FB} n_\mathrm{H^+}^{\mathrm {FB}}/u_p^{\mathrm {GC}})^{\mathrm 3/4} \ n_\mathrm{H^+}^{\mathrm {FB}}/n_\mathrm{H^+}^{\mathrm {GC}}$ and the hadronic luminosity from 1 to 100 GeV of the filaments is

$$\begin{eqnarray} L_\gamma ^{\mathrm {pp}} &\simeq & 3/2 \ 1/3 \ f_\mathrm{bolo} \ \sigma _\mathrm{pp} \ \kappa _\mathrm{pp} \ c \ n_\mathrm{H}^{\mathrm {fil}} \ u_\mathrm{p}^{\mathrm {fil}} \ \ V_\mathrm{fil} \nonumber \\ &\simeq & \frac{3 \ f_\mathrm{bolo} \ \sigma _\mathrm{pp} \ \kappa _\mathrm{pp} \ c \ \dot{M}_\mathrm{in} (k_\mathrm{{B}} \ T_\mathrm{FB})^2}{\Lambda [T_\mathrm{FB}] \ m_\mathrm{p}} \left(\frac{M_{{\rm fil}}}{M_{{\rm pls}}} \right) \nonumber\\ &\simeq & 2 \,{\times}\, 10^{\mathrm 37}\, {\rm erg\,s}^{-1}\! \left(\!\frac{\dot{M}_\mathrm{in}}{ {0.1 {\,M_\odot }\,{{\rm yr^{-1}}}}} \!\right)\! \left(\!\frac{T_\mathrm{FB}}{ {3.5 \times 10^6\, {{{\rm K}}}}}\! \right)^2\! \left(\!\frac{M_{{\rm fil}}}{M_{{\rm pls}}}\! \right)\!,\nonumber\\ \end{eqnarray} \tag{ 2 }$$

where κ_pp ≃ 0.5 is the inelasticity of pp collisions, a factor of 3/2 corrects for the presence of heavy ions (Mori 1997), 1/3 comes from the relative multiplicity of π⁰ amongst all daughter pions, and f_bolo ≃ 0.4 is the fraction of the bolometric luminosity emitted in the 1–100 GeV range. (As in the original hadronic model of Crocker & Aharonian 2011, this scenario reproduces the Bubbles' hard spectrum given the energy independence of the relevant transport processes and near energy independence of σ_pp above the kinematic threshold.) Thus, normalizing to the measured plasma temperature and the mass injection rate required to maintain its density in steady state (Equation (1)), the expected luminosity matches that observed (Lunardini & Razzaque 2012) provided that the filaments' integrated mass, M_fil, approximately equals that in the plasma M_pls(≃ 2 × 10⁷ M_☉).

Remarkably, this condition is met: given the steady state, we have M_fil/M_pls ≡ 〈t_fall〉/t_cool with 〈t_fall〉 the mean time for the filaments to fall to the plane at their terminal speed (cf. Benjamin & Danly 1997) through the Bubble plasma of density ρ_pls. For a horizontally aligned filament, this is given by

$$\begin{equation} v_\mathrm{term}^{\rm horiz}[r,z] \simeq \left(\frac{\pi }{2} \frac{\rho _{\rm {fil}}}{\rho _{\rm {pls}}} \frac{r_{\rm {fil}} \ g[r,z]}{c_D} \right)^{1/2}, \end{equation} \tag{ 3 }$$

where g[r, z] is the magnitude of the gravitational acceleration at {r, z} and c_D is the drag coefficient (for a vertically aligned filament of length L_fil, replace π/2 r_fil → L_fil). The filament terminal velocity is super-Alfvénic through the Bubble plasma; this implies draping of the Bubbles' magnetic field around the filaments, the formation of magnetic wakes behind the filaments, and a consequent increase in drag with respect to the hydrodynamic expectation that is accounted for by setting c_D ≃ 1.9, which we adopt here (Dursi & Pfrommer 2008). Employing the potential described by Breitschwerdt et al. (1991), we find the mean time for the filaments to fall from their condensation sites at z_launch to the plane at the terminal speed to be t_fall[z] ≃ 8 × 10⁷ (z_launch/4 kpc)^0.7 yr. Accounting for the fact that filaments do not form below ∼3 kpc (see below), we find that the mean filament falling time satisfies 〈t_fall〉 ∼ t_cool to better than a factor 2. This agreement is not accidental but essentially guaranteed by the following considerations: While buoyancy effects mean that a hot, gravitationally confined, and stratified atmosphere is not generally susceptible to the local cooling instability (Balbus & Soker 1989; Binney et al. 2009), it has been empirically and theoretically established (McCourt et al. 2012; Sharma et al. 2012; Li & Bryan 2014) that cooling filaments can form within a medium in global thermal balance wherever the ratio of the cooling to the free fall time, t_ff, satisfies t_cool/t_ff < (3–10), particularly if the medium is subject to external perturbations. This timescale ratio condition is satisfied in the Bubbles for z ≳ 3 kpc. Thus, because t_fall ∼ few × t_ff, we automatically have t_cool/t_fall ≲ few in any environment where filaments form.

At equilibrium the filaments are compressed to f ∼ 10% and $n_H^{\mathrm {fil}} \sim 0.03$ cm^-3. At any given filament $n_\mathrm{H^+}$, we require that the filaments are warm enough to obey the (conservative) Hα intensity upper limit ∼0.3 Rayleigh (Finkbeiner 2004); this condition means that the filament gas must be warmer than ∼5000 K. At this temperature, their cooling rate is 3 × 10³⁹ erg s⁻¹, ∼10% of the mechanical energy injected into the nuclear outflow. The filament temperature may thus be maintained by an internal agent associated with the outflow—shock heating, thermal conduction, dissipation of hydrodynamical turbulence, magnetic field reconnection, and/or CR excitation of MHD waves or direct CR ionisation—or an external one, e.g., photoionization heating from Lyman continuum photons supplied by the young nuclear stars.

The filament filling factor we favor, ∼10%, corresponds to a layer of thickness ∼100 pc if distributed evenly over the (assumed) spherical volume of each bubble. The compressed magnetic field amplitude is expected to be ∼10 μG. These numbers are a near match to the depth (200–300 pc) and amplitude (∼10 μG) for the magnetized sheath suggested to cover the Bubbles by Carretti et al. (2013).

Within our model, CRs are initially distributed throughout the volume of the Bubbles and are subsequently gathered in to the filaments by the collapse of the thermally unstable plasma. This process requires that the inward convective velocity exceeds the effective outward velocity associated with CRs' lateral escape. Because the filaments' field lines are largely longitudinal, lateral escape is via cross-field diffusion or, more importantly, field line wandering (Jokipii & Parker 1969). In either case, lateral escape is generically much slower than diffusion along the filaments, which is diffusive with the diffusion coefficient D_||, but limited to the Alfvén velocity because of the streaming instability. In the case of field line wandering, the expectation value of the square of the perpendicular distance reached in time t is given by $\langle r^2 \rangle = 4 \ D_M \sqrt{2 \ D_\mathrm{||} \ t}$ (Nava & Gabici 2013). D_M, the diffusion coefficient for the field lines (with dimensions of length), is poorly constrained. If we adopt D_M = 1 pc from Nava & Gabici (2013) and a parallel diffusion coefficient D_|| similar to that of the Galactic plane, we determine that CR protons up to an energy ∼1 TeV are trapped within the filaments over the ∼10⁸ yr cooling time if their inward transversal collapse (down to final radius r_f) proceeds at ${\sim } \sqrt{f} \ r_f/t_\mathrm{cool} \sim 3$ km s⁻¹ r_f/(100 pc) (t_cool/(10⁸ yr)⁻¹. With these parameter choices, higher-energy CR protons start to escape the filaments within the cooling timescale; consistent with this, there is a steepening in the Bubbles' γ-ray spectrum at ∼100 GeV, corresponding to primary protons of energy ∼1 TeV.

We suggest an intimate connection between the Bubbles and the nuclear molecular torus, which is fed by the Galactic bar and akin to nuclear star-forming rings found in other barred spirals. The torus constitutes much of the mass of the central molecular zone (Molinari et al. 2011) and, with a gravitational potential energy of few × 10⁵⁶ erg, it is the logical candidate to anchor the Bubbles' field lines. The natural timescale associated with the formation of the torus is also ≳ 10⁸ yr, over which it has hosted the formation of ∼10⁷ M_☉ of stars and consequently ∼10⁵ core-collapse supernovae that have released ∼10⁵⁶ erg mechanical energy. The torus seems (Zubovas & Nayakshin 2012; Crocker 2012), moreover, to collimate the GC outflow and material ablated off its inner edge can naturally supply the H i clouds recently detected (McClure-Griffiths et al. 2013) at relatively low latitudes, |b| < 5°, entrained into a nuclear outflow. These n_HI ∼ 1 cm^-3 clouds will help nucleate the condensation of plasma (Marinacci et al. 2010), especially toward the edges of the Bubbles. To match their flat γ-ray surface brightness, a volumetric emissivity that strongly peaks toward the edges of the Bubbles is required (Su et al. 2010); filament nucleation occurring preferentially near the edges may achieve this.

Direct evidence for warm, ionized gas with characteristics very similar to those we infer for the filaments has been obtained via UV absorption studies (Keeney et al. 2006; Zech et al. 2008). These works have uncovered warm, ionized gas clouds high above and below the GC with number densities ≲few × 0.1 cm^-3 and temperatures in the range few × 10⁴ K. These clouds have insufficient velocity to escape the Galaxy and are thus participating in a nuclear fountain. Tantalizingly, all the sight lines that reveal fountaining warm plasma are within the solid angle of the Bubbles or the somewhat larger radio lobes. Of particular interest, the sightline to the Messier 5 globular cluster (Zech et al. 2008) passes very close to the edge of the north Bubble and reveals super-solar metallicity (∼1.6 Z_☉) fountain material, consistent with the scenario of filament nucleation occurring preferentially toward the Bubble edges.

While we believe that nuclear star formation ultimately powers the Bubbles, the mechanism of CR hadron accumulation into cooling filaments does not necessarily require that the CRs are accelerated (only) in the nucleus. They may, for instance, be (re)accelerated on significantly larger scales by shocks in the outflow (cf. Lacki 2013). Indeed, our analysis would still hold in the case that an outburst from Sgr A* ≳ 10⁸ yr ago inflated the Bubbles. Regardless, the fountaining-back of relatively cool and low angular momentum filament gas to the plane may occasionally provide a cold accretion flow on to the black hole or fuel star-formation very close to it.

R.M.C. is the recipient of an Australian Research Council Future Fellowship (FT110100108). R.M.C. gratefully acknowledges conversation or correspondence with Felix Aharonian, Joss Bland-Hawthorn, Vladimir Dogiel, Stefano Gabici, Ortwin Gerhard, Randy Jokipii, Brian Lacki, Federico Marinacci, Mark Morris, Martin Pohl, Vladimir Ptuskin, Mateuz Ruszkowski, Prateek Sharma, Meng Su, and Daniel Wang.