{'fraction_non_alphanumeric': 0.035432900432900434, 'fraction_numerical': 0.020876623376623375, 'mean_word_length': 4.5168069735506595, 'pattern_counts': {'":': 0, '<': 14, '<?xml version=': 0, '>': 15, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
A Unified Model of the Fermi Bubbles, Microwave Haze, and Polarized Radio Lobes: Reverse Shocks in the Galactic Center's Giant Outflows

The Galactic Center's giant outflows are manifest in three different, non-thermal phenomena: i) the hard-spectrum, \gamma-ray `Fermi Bubbles' emanating from the nucleus and extending to |b| ~ 50 degrees; ii) the hard-spectrum, total-intensity microwave (~ 20-40 GHz) `Haze' extending to |b| ~ 35 degrees in the lower reaches of the Fermi Bubbles; and iii) the steep spectrum, polarized, `S-PASS' radio (~ 2-20 GHz) Lobes that envelop the Bubbles and extend to |b| ~ 60 degrees. We find that the nuclear outflows inflate a genuine bubble in each Galactic hemisphere that has the classical structure, working outwards, of reverse shock, contact discontinuity, and forward shock. Expanding into the finite pressure of the halo and given appreciable cooling and gravitational losses, the contact discontinuity of each bubble is now expanding only very slowly. We find observational signatures in both hemispheres of giant, reverse shocks at heights of ~ 1 kpc above the nucleus; their presence ultimately explains all three of the non-thermal phenomena mentioned above. Synchrotron emission from shock-reaccelerated cosmic-ray electrons explains the spectrum, morphology, and vertical extent of the microwave Haze and the polarized radio Lobes. Collisions between shock-reaccelerated hadrons and denser gas in cooling condensations that form inside the contact discontinuity account for most of the Bubbles' \gamma-ray emissivity.


Overview
The Fermi Bubbles are enigmatic, giant gamma-ray and radio lobes emanating from the nucleus of the Milky Way (Dobler et al. 2010;Su et al. 2010;Su & Finkbeiner 2012;Yang et al. 2014;Fermi-LAT Collaboration 2014). They emerge north and south from the Galactic disk in the vicinity of l = 0 and extend to heights above the Galactic plane approaching 8 kpc. Features coincident with the Bubbles or some component of the Bubbles have been discovered in total intensity radio and microwave emission, polarised radio and microwave emission, and X-ray emission (Finkbeiner 2004;Dobler & Finkbeiner 2008;Dobler 2012a;Planck Collaboration 2012;Carretti et al. 2013;Bland-Hawthorn & Cohen 2003;Kataoka et al. 2013).
The energy source of the Bubbles remains in contention as does the gamma-ray emission process that illuminates them. Broadly speaking, the Bubbles might be the signatures of a fairly recent (within ∼ Myr timescale) explosive outburst (Su et al. 2010;Zubovas et al. 2011;Mertsch & Sarkar 2011;Zubovas & Nayakshin 2012;Guo & Mathews 2012;Yang et al. 2012;Fujita et al. 2013;Wardle & Yusef-Zadeh 2014;Mou et al. 2014) from the central supermassive black hole with their γ-ray emission originating in inverse Compton collisions of a non-thermal electron population accelerated in association with this activity. Alternatively, the Bubbles may result from the integrated effect of more-or-less secular processes associated with the nucleus like tidal disruption events that occur there every 10 4 −10 5 year (Cheng et al. 2011) or the concentrated and sustained star-formation activity (e.g., Lacki 2014) that occurs throughout the nuclear region (inner 200-300 diameter region around the black hole).
Invoking sustained nuclear processes as the origin of the Bubbles complements the possibility that their γ-ray emission originates from hadronic collisions experienced by a population of cosmic ray protons and heavier ions (Crocker & Aharonian 2011;Crocker 2012;Crocker et al. 2014). Such particles lose energy much more slowly than the high-energy electrons requisite for IC γ-ray production, thereby allowing the total current energy content of the Bubbles to be accumulated more gradually from relatively less powerful processes such as star-formation (the Eddington luminosity of the Galaxy's 4 million solar mass black hole is 5 × 10 44 erg/s; in contrast, the Milky Way's nuclear star formation releases ∼ 3 × 10 40 erg/s: see below).
At present, there are strengths and weaknesses to either of these broad scenarios. Inverse Compton scenarios share the attractive feature that the very same electron population that putatively supplies the IC gammarays can, for magnetic field strengths approaching 10 µG, also supply the hard microwave Haze emission via synchrotron radiation (Dobler et al. 2010;Su et al. 2010;Fermi-LAT Collaboration 2014). On the other hand, if described by a single power law, synchrotron emission from this population of electrons would not explain the comparatively steep radio spectrum measured between 2.3 and 23 GHz in polarisation data (Carretti et al. 2013). Furthermore, the fact that the γ-ray spectrum of the Bubbles does not steepen (Fermi-LAT Collaboration 2014) -may, in fact, harden (Yang et al. 2014;Selig et al. 2014) -with increasing Galactic latitude is difficult to explain naturally within IC scenarios given the attenuation of the optical and infrared background relative to the CMB as the distance from the Galaxy increases.
Another apparent weakness of many explosive, AGNrelated scenarios is that they require that we are viewing the Bubbles at a privileged time at most a few million years since the outburst that created them with the Bubbles currently undergoing rapid expansion at speeds > ∼ 2000 km/s. Rapid expansion is required in order either to transport the > ∼ TeV electrons (required to produce > ∼ GeV γ-rays via IC on the CMB) from the nucleus to the Bubbles' peripheries within their < ∼ 10 6 yr loss times (in which case speeds > ∼ 7000 km/s 0.02 × c are actually required) or to have sufficiently energetic shocks at the expanding edges of the Bubbles that cascading turbulence injected there may energise similarly energetic electrons in situ (Mertsch & Sarkar 2011). Yet X-ray observations of the Bubbles have only indicated weak features associated with the Bubble edges (Su et al. 2010;Kataoka et al. 2013;Fang & Jiang 2014), implying rather mild expansion speeds, < ∼ 300 km/s (close to the sound speed).
Moreover, there are other indications that the Bubbles are rather long-lived structures: 1. The fact that both Bubbles lean away from the l = 0 line towards Galactic west and, moreover, there are extended plumes of polarised, 2.3 GHz emission leaving north west from the high-latitude regions of each bubble (Carretti et al. 2013) suggests that the structures are distorted by a pressure gradient within the Galactic halo set up by the motion of the Milky Way through the local medium towards Andromeda. The ∼ 50 km/s relative speed of this motion implies a rather gentle effect that requires long timescales/low Bubble gas velocities to take effect, consistent with the ∼ 100 − 150 km/s flow velocities we determine below for gas in the Bubbles (cf. Yang et al. 2012, where it was shown that an unreasonably large speed of the Galaxy through the local medium was required to produce the Bubbles' distortion in the case that they are expanding rapidly due to AGN activity).
2. The ∼ 100 km/s rotation of the Bubbles' waist in the plane is apparently inscribed into giant radio structures on the near surface of the Bubbles (Carretti et al. 2013), again requiring that the structures not grow too quickly.
3. The steep spectrum of the Bubbles' polarised radio counterparts between 2.3 and 23 GHz suggests that electrons emitting in this frequency range have had time to cool, mix and accumulate; this requires > ∼ 3 × 10 7 year for reasonable parameters.
4. Finally, observations of UV absorption features towards background quasar and AGB stars have established the existence of individual parcels of ionised gas at low Galactic longitude but varying latitude (Keeney et al. 2006;Zech et al. 2008).
Kinematic analysis points to this gas participating in a nuclear fountain flow. Individual gas parcels seem to have been launched from the nucleus at different times in the past ranging from 20-50 Myr to more than 800 Myr. The parcels are all now either rising to or falling from a maximum heights of ∼ 12 kpc, similar to the vertical extent of the Bubbles. Some of this material, it has been claimed (Zech et al. 2008), is significantly super-solar in metallicity. This phenomenology points to the existence of a long-standing, star-formation-driven, nuclear fountain; as we expand upon later in the paper, we believe this material is condensing in the region leading up to (and possibly on the other side of) the contact discontinuity in each bubble.
Turning now to the star-formation-driven hadronic model (Crocker & Aharonian 2011;Crocker 2012;Crocker et al. 2014), this has the appealing feature that the power currently being delivered into cosmic ray ions by nuclear star-formation is elegantly sufficient to explain the Bubbles' (putatively hadronic) gamma-ray luminosity in steady state. On the other hand, in the original instantiation of this scenario, the low density of target gas nuclei in the Bubble plasma implied a very long pp loss time of a few Gyr, this being the timescale required to establish a steady state in the first place. Some of us have recently shown, however, that accumulation of hadronic cosmic rays into gas condensations formed by local thermal instability in the Bubbles can significantly reduce the timescale associated with the hadronic scenario to ∼ few 10 8 year . Another point concerning the hadronic scenario is that although secondary electrons (and positrons) are a natural concomitant of hadronic gamma-ray production, it is difficult to explain the spectrum of the microwave Haze with such secondaries since IC and synchrotron losses would be expected, in a single zone model, to steepen the steady-state secondary distribution such that it is inconsistent with the hard Haze emission (e.g., Fermi-LAT Collaboration 2014; . It seems, then, that even the hadronic scenario requires primary electrons at some level to explain the totality of Bubble phenomenology. Indeed, the power in cosmic ray primary electrons accelerated in concert with nuclear star formation is a good match to the Bubbles' 2.3 GHz synchrotron luminosity (Carretti et al. 2013). The complement to these points is that the nucleus is underluminous in both diffuse γray and radio continuum emission given how much star formation occurs there (Crocker et al. 2011a,b;Crocker 2012) and this seems to be because both cosmic ray electrons and ions are advected out of the region in the largescale outflow before they can radiate. Thus, these particles radiate on the size scale of the Bubbles into which the nuclear outflow delivers them. As we show below, however, the passage from nucleus to Bubbles is not a smooth journey for these particles.

Outline of model
Here we briefly explain our model for the Bubbles' structure and emission phenomenology. A schematic showing the configuration of the northern bubble according to our model is shown in fig. 1 for reference.
1. Co-entrained hot plasma, magnetic fields, and cosmic rays are energised within the central ∼ 100 pc (in radius) region of the Galactic nucleus. The pressure gradient supplied by these ISM phases (with the hot plasma dynamically dominant) accelerates a nuclear wind that quickly reaches > ∼ 500 km/s. The dense and massive molecular gas torus circumscribing the injection region collimates this outflow and directs it perpendicular to the plane.
2. The Bubbles are inflated by the injection of hot mass by the nuclear outflow over timescales of at least a few ×10 8 year; characteristic values for the mean mechanical power and mass flux into the bases of the Bubbles are 10 40 erg/s and 0.1 M /year.
3. This outflow, however, is not powerful enoughgiven the strong gravitational field and the thermal pressure of the overlying halo gas -to break out of the Galaxy.
4. Instead, the outflow generates the two Fermi Bubbles. The Bubbles inflate against an atmosphere of non-negligible pressure. An important ingredient of our scenario is that the gas density and pressure in the halo, especially in the vicinity of the Bubbles, is emerging to be larger than has often been assumed in the past (e.g., Stocke et al. 2013;Miller & Bregman 2014) 6 5. The Bubbles develop the classical wind-blown bubble structure of (starting from the smallest scales and working outwards): i) a reverse shock, ii) a contact discontinuity (CD), and a forward shock; see fig. 2.
6. The contact discontinuity separates ISM material swept-up and shocked in the downstream of the forward shock from shocked Bubble material in the downstream of the internal reverse shock. (In our model, the γ-ray edge of the Bubbles is identified with the contact discontinuity.) 7. The Bubbles quickly (within ∼ 10 7 year) reach pressure equilibrium with their environment; at this point the forward shock degenerates into a sound wave (rendering it ineffectual at sweepingup mass). The CD decelerates even faster and is 6 A further consideration is that it is increasingly likely that the gas around the Bubbles is itself in a flow driven into the halo by star-formation occurring in the Milky Way's molecular ring (at Galactocentric radius of ∼ 4 kpc) and/or the Bar (Breitschwerdt et al. 1991(Breitschwerdt et al. , 2002Everett et al. 2008Everett et al. , 2010Kretschmer et al. 2013;de Boer & Weber 2014); halo density models that assume hydrostatic equilibrium may not convey the full story in the inner Galaxy. subsonic after only < ∼ 10 6 year; at the present it is expanding at only ∼ 5 km/s. 8. Thermal plasma and cosmic rays initially suffer adiabatic energy losses as they are blown out of the nucleus on the conically-expanding outflow. However, they encounter giant reverse shocks (one at ∼ kpc heights in each Bubble) where the ram pressure of this flow equilibrates with the thermal pressure of the shocked, downstream gas (cf. the model of Lacki (2013) where the ram pressure of the expanding outflow equilibrates with the thermal pressure of the background Galactic bulge atmosphere). At these shocks, the plasma and cosmic-rays are reheated/re-accelerated.
9. In the immediate downstream of the shocks, there is a systematic sub-sonic flow of v down ∼ 1/4v up ∼ 100-150 km/s. There is energy-independent transport of the non-thermal particles into the Bubbles mediated by this flow in combination with cosmic ray streaming at approximately the Aflven speed. High-energy cosmic-ray electrons produce inverse-Compton and synchrotron emission on photon and magnetic fields in the shock-downstream. Given there is a systematic flow (such that, at any particular distance from the shock, all electrons are the same 'age'), the spectrum of these electrons (and their resultant emission) is hard (mirroring the injection distribution) up to some time (and thus distance)-dependent maximal energy above which the spectrum is exponentially suppressed. Given the rather slow transport speeds and short cooling times, microwave-synchrotronemitting electrons reach only ∼ 1-2 kpc past the shocks and > ∼ GeV inverse-Compton-emitting electrons only reach few ×100 pc. (In contrast, cosmic ray protons and 2.3-GHz-synchrotron-emitting electrons survive to reach the full extent of the Bubbles.) Altogether, we show below that the hard spectrum synchrotron radiation from cosmic ray electrons reaccelerated on the shocks and transported downstream provides an excellent match to the luminosity, spectrum, distribution and morphology of the microwave Haze.
10. As the Bubbles age and their volume slowly increases, the radiative losses of the plasma gradually become more significant. In fact, there is a steadystate limit (at constant radius and mass) where the Bubbles reach both pressure equilibrium with the environment and the cooling radiation from their interior plasma completely saturates the injected mechanical energy. As this limit is approached, an increasingly large fraction of the hot plasma mass injected into the Bubbles is lost by drop-out as cooling leads to local thermal instability that causes individual parcels of gas to cool, collapse, lose buoyancy, and fall out of the Bubbles under gravity.
11. Because of the n 2 dependence of cooling, this mass drop-out process will dominantly occur in the somewhat over-dense shell of material that forms inside of the CD where the slow flow of plasma away from the reverse shock is gradually arrested.
12. While the plasma evolves radiatively in the shell as it cools down into localised, over-dense condensations, cosmic rays hadrons and magnetic fields continue to evolve adiabatically. Their compression in the shell leads to adiabatic energy gain; they are compressed until they equilibrate with the thermal pressure of the volume-filling plasma between the shell and the reverse shock.
13. The ultimate extent and mass of the shell is determined by the condition that the cooling radiation from it saturates the enthalpy flux represented by the slow flow of hot plasma into it from the interior volume.
14. Given that both the mass of shell target material and the hadronic cosmic ray energy density in the shell can be determined from these considerations, we can calculate the hadronic emission from the shell; we find that this essentially saturates the observed γ-ray luminosity of the Bubbles for reasonable parameter values. The fact that the dominant hadronic emission occurs mostly in the shell explains why the volumetric emission from the Bubbles peaks towards their edges (as required to reproduce their flat projected surface brightness: Su et al. 2010).
15. As for the protons and heavier ions, the lowerenergy cosmic ray electrons that live long enough to reach the shell are also adiabatically compressed and reenergised there. At the same time, collisions between the compressed hadron population and the shell gas inject secondary electrons (+ positrons) into the shell. Unlike for the region downstream of the reverse-shocks, however, in the shells there is a mixture of electrons with different ages. Thus the integrated, steady-state shell electron spectrum is steepened by the canonical 1 associated with synchrotron/IC losses. This cooled primary + locally-produced secondary electron population synchrotron-radiates on the adiabaticallycompressed shell magnetic field, explaining the steep-spectrum, polarised emission measured between 2.3 and 23 GHz.
16. The shell/CD does not constitute a perfect boundary, however: observationally, in 2.3 GHz polarisation data, there are plumes of radio emission that extend north-west beyond the γ-ray boundaries of the Bubble. We speculate that these plumes are symptomatic of the break-out of the light fluid of co-mixed cosmic rays, plasma, and magnetic fields through the heavy shell mediated by the Rayleigh-Taylor instability.
1.3. Plan The remainder of this paper is structured as follows. We describe background data in §2. In §3 we set out a semi-analytic calculation describing the evolution of a bubble fed by an outflow of fixed mechanical power and mass flux expanding into an atmosphere of finite, non-negligible pressure. This shows how the Bubblesaccounting for the phenomenon of plasma cooling and drop-out -slowly ( > ∼ few 10 8 year) approach a quasisteady state with size and plasma temperature approximately equal to that observed. This finding removes any fine-tuning 'Why now?' problem and justifies our subsequent steady-state treatment of the Bubbles' nonthermal emission. In §4 et seq. we examine in detail the dynamics and configuration of the reverse shock set up in each of the Fermi Bubbles, particularly examining the non-thermal radiation arising as a result of these shocks. In §5 we describe our procedure for delimiting the region of {Ė,Ṁ } (the mechanical energy and mass flux into the Bubbles) parameter space that can describe the present day Bubbles. In §5.1 we set out the results of this procedure, in particular demonstrating that there is only a small region of this parameter space that is compatible with physically and observationally-motivated constraints and that this region is consistent with the Bubbles' being driven by observed nuclear star formation. In §6 we demonstrate explicitly that, in the small region of parameter space consistent with prior constraints, the Bubbles' i) predicted hadronic γ-ray emission; ii) predicted microwave synchrotron emission; and iii) radio continuum synchrotron emission all match that observed.
In other words, we show that our scenario fundamentally explains the non-thermal phenomena described as the 'Fermi Bubbles', 'Microwave Haze', and 'S-PASS Lobes'. Finally, in §8 we set out implications of our model and future work.

PRELIMINARIES
2.1. Geometry Adopting a distance to the Galactic Center of 8 kpc, geometrically the Bubbles can be approximated as a pair of spheres with radii ∼ 3.2 kpc implying a total volume of V FB 8.4 × 10 66 cm 3 with a combined surface area of A FB 2.5 × 10 45 cm 2 . A better characterisation of the Bubbles' real geometry seems to be that they are bicones emanating north and south from the nucleus that collimate into cylinders of ∼ 3 kpc radius and heights of 2-3 kpc above the plane (Bland-Hawthorn & Cohen 2003;Carretti et al. 2013). The Fermi Bubbles reach ∼ 7 kpc maximal height above the Galactic center; the polarised radio lobes extend up to ∼ 8 kpc. Kataoka et al. (2013) have performed scans across the limbs of both the northern and southern bubbles at high latitudes with the SUZAKU X-ray instrument. In neither scan do these authors find evidence for a change of temperature between the exterior and interior plasma; rather the plasma temperature seems constant at ∼ (3.0 − 3.5) × 10 6 K. For the northern scan, starting in the bright north polar spur feature (located east in projection from the northern bubble) and scanning inwards across the Bubble edge, these authors find a drop in emission measure EM at the edge to an interior value of (3 − 4) × 10 −2 cm −6 pc. In the south, the emission measure is approximately constant at (1−2)×10 −2 cm −6 pc across the scan. Adopting a total projected area of A FB = 7.0×10 44 cm 2 , we find an average line-of-sight distance through the Bubbles l LOS ≡ V FB /A FB 3.7 kpc.

Other Parameters
With this distance scale and assuming for the moment a uniform distribution for the Bubble plasma, a characteristic ionised hydrogen number density in the Bubbles can be estimated as n FB H + ∼ EM/l LOS (1 − 3) × 10 −3 cm −3 . In reality, we show below that the plasma emission from the Bubble is likely to be dominated by a shell of density n shell H + ∼ (2∆r shell/lLOS ) −1/2 n FB H + , and thickness r shell < 3.7/2 kpc, inside the contact discontinuity. The fitted X-ray temperature is thus likely to be characteristic of this shell; the volume-filling, shocked plasma between the shell and the reverse shock will be hotter and more rarefied.
Whatever the exact configuration of plasma within the Bubbles, consistent with our modelling, they are not strongly-over pressured with respect to the surrounding halo (consistent with the absence of obvious X-ray limb brightening in their higher reaches: Kataoka et al. 2013). Working with the characteristic, volumetricaverage number density for the moment, the pressure in the Bubbles k B P FB = k B T F B n tot 2.0 eV cm −3 T F B /(3.5 × 10 6 K) n F B H /(0.003 cm −3 ) with the total plasma number density n F B tot 2.2 n F B H . Independent confirmation that halo pressures approach this scale comes from an analysis of Chandra and XM M -Newton X-ray observations along the 3C 273 sightline (located northwest by ∼ 30 • in projection from the north bubble), from which Fang & Jiang (2014) infer a plasma density of (1.2−1.8)×10 −3 cm −3 and a temperature (1.5−1.8)×10 6 K. The recent modelling of Miller & Bregman (2014) of OVIII emission line data together with independent evidence for halo metallicities of ∼ 0.3 Z , suggests ambient, undisturbed number densities in the halo at 7 kpc of n e +n H 6×10 −3 cm −3 at a characteristic temperature approaching 2×10 6 K. Using the volumetric density estimate, the total plasma mass of the Bubbles is estimated as (1) 2.3. Discussion of Total Energetics and Other Parameters In the limit that radiative losses can be neglected and the atmospheric density treated as constant, the total enthalpy of a slowly inflated bubble is whereĖ is the average mechanical power fed into the bubble over time t. Employing the atmospheric pressure of k B P ext = 2.0 eV cm −3 motivated above, this suggests H FB 6.7 × 10 55 erg for the Bubbles (actually a lower limit as it neglects the z-dependence of the halo density and pressure). Accounting only for core-collapse supernovae, the minimum mechanical power injected by nuclear star-formation can be estimated aṡ where a single core-collapse supernova (SN) requires the formation of ∼ 90 M in stars (for a Kroupa (2001) IMF and assuming the least massive star to explode as a CCSN has 8 M ), we assume 10 51 erg mechanical energy release per supernova, and we have normalised to a nuclear star-formation rate of SFR GC = 0.08 M year (Crocker 2012). Note that there are other significant star-formation-related sources of mechanical energy including stellar winds and proto-stellar outflows that could, in a full accounting, mean a doubling of this estimate. However, we will continue to use this as the fiducial number (cf. Lacki 2014). With this power, the minimum timescale to inflate the Bubbles (assuming an unrealistically high 100% of mechanical energy injected in the nucleus is transferred to the Bubbles and no radiative or other losses) is t = H FB /Ė GC SF 8 × 10 7 year. Given the current surface area of the Bubbles, the expansion speed whereḢ FB saturates the star-formation mechanical power is ∼ 14 km/s; this is an upper limit on the Bubbles' current expansion velocity.
The total plasma mass in the Bubbles of ∼ 3.0 × 10 7 M , if fed by the nuclear outflow at a mass inflow rate,Ṁ , similar to the SFR, requires a minimum timescale to establish of 2.8 × 10 8 year ×Ṁ /(0.08 M /yr) to supply. The cooling time of the plasma in the Bubbles is t cool 5 × 10 8 year ×T FB /(3.5×10 6 K) ×n FB H /(0.003 cm −3 ) 7 using volumetric values. Thus even the minimal inflation time is beginning to approach the radiative cooling time indicating that an adiabatic treatment is inadequate. Note the confluence of timescales around the formation/maintenance of the Bubbles at order ∼ few ×10 8 year (cf. . We show immediately below ( § 3) that the Bubbles can (and do) approach a steady state such that their radii do not grow and the freshly-heated plasma mass being fed into them is matched by the drop-out of cooling plasma that condenses under local thermal instability. In such a steady state model, the rate of mass loss from the nucleus,Ṁ GC ≡ n GC µm p V GC /t wind = n GC µm p 2v wind πr 2 0 is identical to the rate of mass injection into the Bubbles which is, in turn, identical to the rate at which mass is cooling out of the structures, M cool = n F B µm p V F B /t cool . This consideration means that (the Bubbles have actually reached this steady state) we get an estimate for the radius of the nuclear region from which the outflow feeding the Bubbles emerges of This is close to the radius of the Central Molecular Zone (CMZ) molecular torus, 80-100 pc (Molinari et al. 2011). The CMZ is the region of enhanced molecular gas density and star-formation activity surrounding the central supermassive black hole.

EXPANSION OF A RADIATIVE BUBBLE
We model the expansion of a radiative, spherical bubble into a uniform medium of constant given pressure and number density. Generalizing the usual treatment, the bubble is not assumed to be strongly overpressured with respect to the ambient medium and it may be subject to significant cooling. Limitations of the model are that it does not incorporate a gravitational potential (so there is no pressure stratification), neglects the small but finite fraction of the total bubble volume in the region upstream of the internal reverse shock, and neglects the dynamics connected with the formation of over-dense shells, either from swept-up halo gas or due to cooling inside the contact discontinuity of shocked plasma from the nucleus (this latter process is likely important in the Bubbles: see below). Fig. 2 lays out the main features of such a bubble: there is a reverse shock interior to the bubble, a region of shock-heated bubble material in the downstream of the reverse shock leading up to the CD, and then a shell of shocked, ambient matter between the forward shock and contact discontinuity. The equations governing the dynamics of the bubble can be derived following consideration of energy, mass, and momentum conservation. We assume that the kinetic energy inside the bubble is much less than the internal energy density and that a γ equation of state is relevant (p b = (γ − 1) b ). The bubble is fed by mechanical energy injected at a rateĖ and mass at a rateṀ .
It is important to note that the growth of the total mass of the bubble M b does not necessarily satisfẏ M b =Ṁ because of the phenomenon of mass dropout. Energy associated with the flow of hot gas into the bubble can be dissipated by cooling. This sets up local thermal instability of the bubble plasma leading it to collapse into over-dense condensations that, having lost buoyancy (in a real situation that includes a gravitational potential), fall out of the bubble (see Crocker et al. 2014, and references therein). The bubble can thus reach a steady state where its pressure equilibrates with the surrounding atmosphere, the mechanical energy flux into its base is balanced by thermal radiation, and the mass flux by mass dropout. We show below that, for parameters apposite to the Fermi Bubbles, the system is likely approaching this limit.
The complete set of equations describing the bubble is: where we have distinguished between the polytropic index (γ 1 ) for the interior of the bubble and γ 2 for the exterior region, subscripts 'a' refer to the ambient medium, 'b' to the region interior to the contact discontinuity, R b is the radius of the contact discontinuity, R s that of the forward shock, m is the atomic mass unit, and µ is the mean mass of the particles in the plasma (µ = 0.6039 for solar metallicity).

Comparison with Subsequent Modelling
Note that in §5 et seq. we perform independent modelling of the dynamics of the Bubbles' reverse shocks that does incorporate the effects of gravity. On the basis of this subsequent modelling, we identify fiducial values for the rates of mechanical powerĖ and mass injectionṀ at the nucleus that give a good match to Fermi Bubble phenomenology: viz. {Ė,Ṁ } ∼ {10 40 erg/s, 0.08 M /year}. On the basis of this modelling, the reverse shocks are located at ∼ kpc heights in the halo and we find that the flow has typically surrendered ∼ 40% of its mechanical energy to gravity before reaching the shock. For the modeling of the bubble expansion presented in this sectionwhich does not account for the effect of gravity -the relevant comparison is thus not to a bubble subject to the same mechanical energy injection rate in the nucleus but rather to one with the same mechanical energy injection rate at the shock, i.e., ∼ 6 × 10 39 erg/s. Put another way, for purposes of the current bubble modelling, in the absence of other loss processes, once our simulated bubble has reached steady state we have complete thermalisation of the injected mechanical energy. If we demand that the downstream plasma in such a bubble be heated to a temperature 3.5 × 10 6 K (to match X-ray observations), onceṀ = 0.08 M /year is set, we require a mechanical energy injection rate ofĖ = 5/2 T F B k BṀ /(µ m) so thatĖ → 6 × 10 39 erg/s (consistent with a 40% loss to gravity of a total 10 40 erg/s injected at the nucleus).
Figures pertaining to the expansion of a bubble with such a mass and energy injection rate are shown below. As justified above, other relevant parameters are an ambient pressure of 2.0 eV cm −3 (= 3.2 × 10 −12 dyn cm −2 ), and an ambient hydrogen number density 3×10 −3 cm −3 . Note that we assume the plasma has solar metallicity (but verify that assuming twice solar metallicity does not change our results significantly).
As claimed, these figures demonstrate that, for relevant parameters, the contact discontinuity (CD) of the bubble asymptotes to a fixed final radius of ∼ 5 kpc after ∼ 1 Gyr; thereafter the total mass and energy content of the bubble are fixed with the mechanical energy flux into the bubble exactly matched by cooling radiation. The drop out of cooling condensations that form under local thermal instability keeps the total mass of the structure constant. Note, however, that even after only ∼ 10 8 year the CD is growing very slowly and the bubble reaches pressure equilibrium with the environment on timescales as short as ∼ 10 7 year.

THE FERMI BUBBLES'S GIANT, INTERNAL SHOCKS: BACKGROUND THEORY
In this section and subsequent we proceed to model the Bubbles' giant, reverse shocks. The central temperature of the wind in the injection zone T c is given by Note that choosing these normalising values ofĖ andṀ gives us a central temperature consistent with observations: for the nuclear region, X-ray observations (Muno et al. 2004) point to a diffuse hot, ∼ 10 7 K, plasma of number density n 0.1 cm −3 [ 8 ], about three times the temperature of the Bubble plasma. Asymptotically and neglecting gravity (for the moment) the wind reaches a speed v asym = (2Ė/Ṁ ) 1/2 (Chevalier & Clegg 1985). In these equationsṀ is the total mass efflux in the wind andĖ is the mechanical energy injected at the base of the wind.
Across each shock we have dynamical pressure balance: where at height z with v flow [z] the wind speed at that height; it is assumed that the outflow is collimated into two symmetric cones of solid angle Ω, each receiving half the mechanical power and mass flux injected at the center. Thus, here the factorṀ 1/2 ≡ 1 2Ṁ is the mass injection rate into each outflow cone and, ultimately, into each bubble.
In the freely expanding part of the flow upstream of the shock the plasma (and cosmic rays) cools adiabatically from its central injection temperature according to where ρ[z] and r[z] are the density and radius at height z, respectively, z 0 is the launch height, and the second equality assumes an adiabatic index γ = 5/3 for the (dynamically dominant) thermal plasma. The temperature of the shocked, downstream plasma is where M is the upstream Mach number of the flow. As the upstream flow is dominated by ram pressure and the downstream region is dominated by thermal pressure of the shocked gas, we can roughly locate each shock at a height above the plane: where we normalise to our fiducial values for the temperature and number density of the plasma in the Fermi Bubbles. Within our scenario, the downstream region should also be close to pressure equilibrium with the halo gas; this is consistent with the X-ray observations as already stated (Kataoka et al. 2013).
Adopting the test particle limit (i.e., neglecting the back reaction from the cosmic rays), the shock compression ratio is and the downstream Mach number is In the test particle limit, the spectral index of the nonthermal particle population accelerated at the shock is given by (Bell 1978) where R is, again, the shock compression ratio. In general, perpendicular shocks are less efficient at accelerating non-thermal particle populations (e.g., Pinzke et al. 2013) but note that this is not a concern for our scenario because we have a pre-accelerated (but adiabatically-cooled) particle population emerging from the nucleus.

PROCEDURE
We scan over the (assumedly) steady mass and mechanical energy flux,Ṁ andĖ respectively, into the central acceleration region. As shown above these parameters determine the temperature reached within the central acceleration region following thermalisation of the mechanical energy. The high pressure in this region accelerates the plasma into a flow which is collimated into biconical outflows transverse to the Galactic disk by the surrounding dense molecular torus of the Central Molecular Zone. In this expanding flow, the plasma cools adiabatically until it reaches the reverse shock in each bubble where it is shock (re)heated. For every {Ė,Ṁ } in the parameter space, the asymptotic velocity of the flow, v asym is determined. Note that v asym is reached soon after the flow escapes from the central acceleration zone; subsequent to reaching v asym the adiabatically-cooling flow decelerates in response to the gravitational potential, Φ (Breitschwerdt et al. 1991). Note that, given the high (i.e., supersonic) flow speeds we are interested in, we treat the flow evolution in the ballistic limit. For our fitting, we search over two fit parameters for every sampled {Ė,Ṁ } point: i) the height of the shock z shk and ii) the temperature of the downstream region, T down . There are two conditions that must be met: 1. the Rankine-Hugoniot shock-jump condition enforcing dynamical pressure balance over the shock (eq. 11); 2. the requirement that the downstream plasma number density, n down tot ≡ n FB tot , take on a specified value consistent with the range suggested by the various observations outlined above; we investigate values n FB tot = {2, 4, 6} × 10 −3 cm −3 .

Restriction of Parameter Space
We can exclude much of the {Ė,Ṁ } parameter space with physically-or observationally-motivated constraints; cf. fig. 11. In this figure we show that most of the potential parameter space for the case that the shocked, downstream plasma number density is n F B tot = 4 × 10 −3 cm −3 [ 9 ] can be excluded by only two considerations: 1. The cooling shell should not be too thick; given the Bubbles' radius is ∼ 3 kpc, we employ a practical upper limit on the shell thickness of 1 kpc (excludes the green region; the shell thickness is determined by eq. 23 below).
2. The wind needs to be sufficiently fast that it is sufficiently supersonic at the position of the putative shock; we find a practical lower limit of 500 km/s at the (variable) shock position (excludes the blue region).
We find other physically motivated constraints 10 but these all exclude similar or smaller regions of the parameter space. We show below that, within this restricted parameter space, our model well reproduces the γ-ray, microwave, and radio phenomenology of the Bubbles. First, however, we remark on the fact that theṀ /Ė ratio indicated by this restricted parameter space again suggests that the Bubbles are driven by nuclear star-formation. In fig. 11 the black curves delineate the expectation for the starformation-driven efflux vs. mechanical energy input for mass loading rates, as marked, of β = 1, and 2.8 (here mass loading is defined with respect to the mass injection 9 This number density is slightly smaller than the halo number density at 7 kpc according to the OVIII emission line data modelling of Miller & Bregman (2014) 10 These alternative constraints include: i) that the downstream plasma temperature be at least 3 × 10 6 K; ii) that the cosmic ray spectral index be sufficiently hard -γ CR < ∼ 2.3 -to match the observed γ-ray spectrum; iii) that the total cooling luminosity of the downstream plasma + shell not exceed the upper limit on the bolometric luminosity of the X-ray bulge which, conservatively, is 10 40 erg/s (Snowden et al. 1997;Almy et al. 2000).
rate that accompanies a given rate of star formation not with respect to this star formation rate itself). From modelling of the diffuse, hard X-ray emission emanating from the starburst exemplar M82, Strickland & Heckman (2009) find -for centralised mass loading only -mass loading rates in precisely this range 1 < β < 2.8 (with a practical upper limit at β ∼ 10 for a very hot, 30-70 ×10 6 K, wind).
There is, in general, the possibility for an outflow to also be subjected to distributed (in addition to centralised) mass loading. The population of HI clouds entrained into the nuclear outflow detected by McClure-Griffiths et al. (2013) and the high-speed, warm ionised material found by Fox et al. (2014) together constitute good evidence that exactly such distributed mass loading is taking place in the nuclear outflow, presumably as it scrapes along the inside of the CMZ gas torus. In any case we take the alignment of the allowed, roughly triangular white region in fig. 11 as further good evidence that the outflow is driven by star-formation, though we suspect that both centralised and distributed mass loading should be contributing to generate the final effective β. Note that the pressure in the CMZ H 2 is very close to that in the central X-ray-emitting plasma; the molecular gas is thus able to collimate this plasma and cause it to escape down the steepest density gradient, into the Galactic halo 11 .
We remind the reader that fig. 11 applies to the specific case that n F B tot = 4 × 10 −3 cm −3 ; we have explored smaller and larger values than this. For n F B tot → 2 × 10 −3 cm −3 , the parameter space completely closes off due to an unphysically thick cooling shell (cf. eq. 23 below); on the other hand, for higher densities ∼ 6 × 10 −3 cm −3 , the parameter space opens up but it seems increasingly improbable that, for this and larger downstream densities, the halo could be in pressure equilibrium with the shocked plasma as our scenario demands.
6. NON-THERMAL PARTICLES ACCELERATED AT THE SHOCKS AND THEIR EMISSION 6.1. Non-thermal particle spectra and energetics Within the acceptable parameter space, our modeling shows the upstream flow at the shocks has a Mach number in the range 6-8; this implies (cf. eq. 18) that nonthermal particle populations accelerated there will have spectral indices of ∼ 2.1, exactly as required to explain both the Bubbles' γ-ray and hard microwave emission. Adopting a fiducial magnetic field of 7 µG downstream of the shock ( §6.3.2) the characteristic energy of electrons synchrotron radiating at ν = 23 GHz is E c = 18 GeV (B/7 µG) −1/2 (ν/23 GHz) 1/2 . For IC emission, γ-rays at E γ = 1 GeV are up-scattered by primary electrons of energy E e 640 GeV (E γ /1 GeV) 1/2 (E γ,0 /6.3 × 10 −4 eV) −1/2 adopting an energy for the photon background characteristic of the CMB. Finally, proton primaries of hadronic γ-rays in the observed range of 1-100 GeV have energies covering the rough range 10-1000 GeV.
These characteristic energies should be compared with upper limits to the energies that the giant shocks might accelerate non-thermal particles. The Hillas criterion that a particle's acceleration time be at least equal to the maximal diffusive escape time from an acceleration region, of fiducial size r acc ∼ 100 pc, sets a maximum energy limit of E max 10 15 eV r acc /(100 pc) B/(7 µG) v/(500 km/s). For synchrotron loss-limited acceleration of electrons over a timescale t acc ≡ 4D/v 2 where D is the diffusion coefficient near the shock and v is a characteristic velocity, in the most optimistic case of Bohm diffusion D Bohm = 1/3 r g c, we find a maximum upper limit to the accelerated electron energy of E max e 4 × 10 13 eV (B/(7 µG)) −1/2 (v/(500 km/s)). In this case, the shocks can easily accelerate electrons up to the requisite energies to produce the observed microwave emission and even the IC γ-rays by up-scattering the CMB (though whether such high energy electrons can reach the top of the Bubbles is another matter).
In our modelling we assume that the power injected into freshly-accelerated, non-thermal particles,Ė CR at this shock is 20% of the mechanical power available at the shocks. For the magnetic field amplitudes explored below, we find that electrons are required to take up a fraction 5-20% ofĖ CR in order to explain the radio and microwave emission from the Bubbles, with the remainder going into ions, dominantly protons.

Observations
The measured γ-ray spectrum of the Bubbles is rather hard, consistent with F γ ∝ E −2.1 γ between 1-100 GeV (Su et al. 2010) but with a hardening below ∼ 1 GeV. The Bubbles have a rather flat projected surface brightness (Su et al. 2010). A recent paper from the Fermi collaboration itself (Fermi-LAT Collaboration 2014) gives the 100 MeV -500 GeV luminosity of the Bubbles as 4.4 +2.4 −0.9 × 10 37 erg/s. This publication also reports that the Bubbles' γ-ray spectrum cuts off above ∼ 100 GeV.

Hadronic γ-ray Emission
A successful model of the Bubbles' γ-ray phenomenology would explain the luminosity, spectrum, and morphology (and, most generally, spectral morphology). Generic to any hadronic model is a ∼ GeV-scale low energy down-turn feature, whose position in energy is fixed by the mass of the neutral pion. This is broadly consistent with the spectral phenomenology alluded to above (Crocker & Aharonian 2011).
We show immediately below that our model provides a good match to the overall luminosity of the Bubbles' γ-ray emission. In passing we note our model also well explains the 1-100 GeV hard spectrum of the Bubbles given the reasonably high Mach numbers (6-8) we find at the giant shocks. We also show below that our model predicts the volumetric emission from the Bubbles should peak towards their edges; this is consistent with their flat (projected) surface brightness. We do not explicitly deal with the observed high-energy cut-off here; fairly generically it might indicate either a high-energy cut-off in the accelerated proton spectrum emerging from the shocks or a loss of confinement of the cosmic ray hadrons in the Bubbles at ∼TeV energies.
Following but somewhat modifying the argument presented by Crocker et al. (2014) we can calculate the hadronic luminosity of the Bubbles. Our essential argument is that, given the expansion timescale, the evolution of the plasma in the Bubbles cannot be treated as adiabatic but rather the implication of plasma cooling must be considered. This cooling leads to local thermal instability and the formation of over-dense, cool condensations that, having lost buoyancy, fall out of the Bubbles under gravity (in the steady state limit the drop-out of these condensations completely saturates the injection rate of fresh plasma mass). Globally there is a slow flow in the Bubbles of material downstream of the reverse shock and towards the contact discontinuity. As the CD is approached, this flow must be decelerated with a consequent increase in plasma density. Given the n 2 dependence of the cooling, it is in this somewhat over-dense region that thermal instability will preferentially form the condensations. There will thus be a flow of gas at a rate (saturatingṀ in the steady state limit) of shocked plasma from the from the interior volume of the Bubbles behind the reverse shocks, into this cooling shell, with subsequent thermal collapse, and drop-out under gravity. We review the observational evidence for formation of cool gas condensations near the edges of the Bubbles in §7.2.
We make the approximation that in the interior, volume-filling region, the gas evolves adiabatically whereas in the shell it evolves radiatively. However, a crucial point is that cosmic rays and magnetic fields evolve adiabatically in both shell and volume. Adiabatic compression of these non-thermal ISM components thus raises their relative pressure within the shell. In fact, compression of the shell by the the plasma in the volumefilling interior is arrested when the adiabatic energy-gain of magnetic fields and cosmic rays results in their reaching pressure equilibrium with the interior plasma. At the same time, the size of the shell is controlled by the requirement that the cooling radiation taking place within it (strictly, in the steady state limit) balances the enthalpy flux represented by the flow of hot plasma from the hot interior.
With these considerations, the hadronic γ-ray luminosity of the Bubbles -dominated by collisions in the shell between the adiabatically compressed cosmic rays and the compressed, cooling gas -can be calculated as shown below. The Fermi-LAT band hadronic ('pp') luminosity of the shell is Here, in the first line, the first 3/2 pre-factor corrects for the presence of heavy ions amongst beam and target nuclei (Mori 1997), 1/3 comes from the relative multiplicity of π 0 amongst all daughter pions, f bol is the fraction of the bolometric luminosity emitted in the relevant γ-ray band, u shell p is the energy density of relativistic protons in the shell, σ pp 4 × 10 −26 cm 2 is the total hadronic cross-section and κ pp 0.5 is the hadronic inelasticity. In the second line n H /n tot is the H + to total plasma number density ratio in the shell, m is the atomic mass unit, and µ = 0.6039 is the mean mass of the particles in a solar metallicity plasma. We now assume pressure balance (between the non-thermal shell components and the thermal plasma in the interior) as justified above (and also adopt cosmic ray/magnetic field equipartition in the shell as justified observationally by Carretti et al. 2013) so that We now calculate M shell as follows: mass conservation implies that the flux of mass from interior volume, of number density n F B tot , into the shell is equal (in the steady state limit) to the mass flux into the Bubbles,Ṁ . Thuṡ where v ∼ few × km/s is the effective speed with which plasma is carried across the the surface, of area A shell separating the volume from the shell. The enthalpy flux represented by this flow is dissipated in cooling radiation from the shell, thus is the cooling function at the shell temperature and metallicity (the latter assumed to be solar as for the hot plasma) and, geometrically, we assume a thin shell. As for the shell temperature, as we have emphasised, the X-ray spectrum across the edge of the Bubbles, encompassing the region of the shell, has been successfully fit with a single temperature, collisional ionisation equilibrium plasma model of ∼ 3.5×10 6 K and we therefore adopt T shell = 3.5 × 10 6 K (so that the temperature in the interior volume is necessarily > 3.5 × 10 6 K). In reality, the gas distribution in the shell is more complicated as the gas cascades from the high injection temperature, where it starts to cool, down to optical emission line temperatures. Such a multi-temperature distribution is inferred, e.g., from the coincident X-ray and Hα emission observed from filaments around the bright central galaxies in clusters (e.g., Fabian et al. 2011) and is common to cooling flows. In any case, here we ignore the complication of the multi-temperature nature of the shell gas and assume the effective temperature apposite to the calculation of shell cooling losses is precisely that found from fitting to the observed, apparently thermal, X-ray radiation.
Rearranging and combining eqs. 21 and 22 we find Following eq. 20 and given adiabatic evolution of the cosmic rays, we have that where u F B therm is the internal energy density of the plasma in the Bubbles,Ė CR andĖ therm are, respectively, the power fed into freshly accelerated cosmic rays and freshly heated plasma at the reverse shocks, and the last nearequality relies on adiabatic evolution of both cosmic rays and thermal plasma downstream of the shocks in the interior volume (but before the shell). We thus find ∆r shell 5 2 where we set A shell = 2.5 × 10 45 cm 2 . Then and we may finally write the 1-100 GeV hadronic luminosity of the shell as which compares favourably to the ∼ 2 × 10 37 erg/s 1-100 GeV luminosity of the Bubbles (Su & Finkbeiner 2012) if 10% of the mechanical power at the shock is delivered to cosmic rays (we assume 20% in our plotting as already stated); note that in the interior region, we are normalising to a somewhat hotter temperature than the 3.5 × 10 6 K measured in X-rays which we assume is characteristic of the cooling shell.
The hadronic luminosity of the filaments in the downstream is plotted in fig. 12. Note that there is a subdominant component of the Bubbles' hadronic γ-ray emission due to collisions between the cosmic ray protons and the rarefied, volume-filling plasma but, for fiducial numbers, this provides only 5-10% of the integrated filament luminosity.

Modeled IC Emission
As has been noted, the same population of non-thermal electrons synchrotron radiating at microwave frequencies to explain the haze may emit IC γ-rays in the Fermi band with the correct Bubble spectrum. Within our scenario, however, we find that this process is sub-dominant to the hadronic γ-ray emission explaining < ∼ 10% of the total observed 1-100 GeV γ-ray luminosity of the Bubbles. On the other hand, the concentration of the IC emission to a smaller region downstream of the shocks ( < ∼ 300 pc: cf. fig. 13) means that, within this solid angle, IC contributes a fraction approaching 50% of the total 1-100 GeV γ-ray intensity and we can expect γ-ray substructure around the shocks (preliminarily, we find such: see §7.1.1). In fig 12 we display, in addition to the hadronic γ-ray emission, the IC contribution of giantshock-accelerated electrons to the Bubbles' overall γ-ray luminosity for the case that the power fed into freshly accelerated electrons is 10% of the power fed into all CRs (which is, itself, assumed, to be 20% of the mechanical power at the shock).

Observations
The 23-44 GHz Microwave Haze is found (Finkbeiner 2004;Dobler & Finkbeiner 2008;Dobler 2012a;Planck Collaboration 2012) in total intensity data up to Galactic latitudes |b| < ∼ 35 • above which it becomes severely attenuated though there has been a claim (Dobler 2012b) that faint, Haze-like emission is detected up to the latitudinal extremities of the Bubbles. Here we adopt a solid angle of 0.4 sr for the solid angle of the Haze emission. The Haze has a hard spectrum F ν ∝ ν −0.55 (Planck Collaboration 2012), characteristic of synchrotron emission from a freshly-accelerated population of electrons with spectrum dN e /dE e ∝ E −2.1 e . It has also been claimed (Gold et al. 2011) that the Haze exhibits little coincident polarised emission though high levels of noise in the WMAP polarisation maps would render a putative, hard-spectrum polarised counterpart to the Haze very difficult to detect (Dobler 2012a). Moreover, regions of enhanced polarised emission coincident with sub-structure within the Bubbles have actually been found, in particular, a bright region coincident with part of the the eastern edge of the north bubble (Jones et al. 2012) and the counterparts of the strongly magnetised ridges found in polarised intensity at 2.3 GHz (Carretti et al. 2013).
These ridges appear to wrap around the near surface of the roughly biconical structure circumscribing the Bubbles in a fashion consistent with the rotation of the Galaxy. There is diffuse, polarised 2.3 GHz emission enveloping the entire Bubbles, stretching beyond them, in fact, at high latitudes in two plumes that extend towards Galactic north west and south west. In contrast to the Haze detected in total intensity at microwave frequencies, the spectrum of polarised intensity between 2.3 and 23 GHz is steep, F ν ∝ ν −1.1 (Carretti et al. 2013) consistent with an electron population of mixed ages that has been cooled by synchrotron and/or IC emission.

Magnetic fields
To determine the intensity of synchrotron emission from the Bubbles we need to know the magnetic field amplitude in two environments: in the volume-filling plasma phase interior to the Bubbles and in the denser shell formed by local thermal instability. For the latter, as mentioned, we assume that the pressure of the adiabatically-compressed, non-thermal ISM components, i.e., cosmic rays + magnetic fields, reaches equilibrium with the plasma pressure in the volume, k B P F B > ∼ 2.0 eV cm −3 for fiducial parameters. Given (Carretti et al. 2013) that CRs and magnetic fields are in rough equipartition in the Bubbles (and, therefore, in the filaments adopting an adiabatic index of γ B = 4/3(= γ CR ) for the tangled magnetic fields), then U B fil 3/2k B P F B 3 eV cm −3 so that B fil 11 µG. This is in good agreement with the magnetic field amplitude of ∼ 12 µG suggested by an equipartition analysis of the radio data (and supported by an independent broadband analysis: Carretti et al. 2013) to pertain in a 200-300 pc deep 'sheath' enveloping the Bubbles.
With this determination, we may work backwards to arrive at the adiabatically 'decompressed' field pertaining in the plasma, B FB = (n FB /n fil ) 2/3 B fil (Ė CR /Ė cool ) B fil ∼ (5 − 7) µG. Again, this is consistent with a determination for the volume-filling field amplitude made on the basis on an equipartition analysis of 2.3 GHz radio polarisation data (Carretti et al. 2013).

Modeled Synchrotron Emission
Through the GC/Bubble system there are a number of sources of non-thermal electrons potentially capable of producing detectable synchrotron (and/or inverse Compton) radiation. The nucleus itself directly accelerates a population of hard-spectrum primary electrons that is then largely advected from the region before it can lose its energy radiatively in situ (Crocker et al. 2011a,b;Crocker 2012;Carretti et al. 2013). These electrons cool adiabatically and radiatively in the expanding outflow until they encounter the giant reverse shock in each bubble where they are reaccelerated. This reaccelerated population is then systematically transported away from the shock into the downstream region under a combination of streaming and advection in the downstream gas flow. As we show below, higher-energy, microwave-emitting primary electrons from the giant shocks do not live long enough to escape from this systematic flow. This implies that there is a one-to-one correspondence between distance from the shock and the 'age' of the electrons (i.e., the time that has elapsed since they left the acceleration zone). Roughly speaking, while the transport time from the reacceleration point is still sufficiently smaller than the synchrotron+IC cooling timescale (at a given energy) the electron population (around that energy) will retain its hard injection distribution (and is exponentially suppressed once the transport exceeds the cooling time: cf. figs. 13 & 14). The hard-spectrum microwave Haze is largely attributable to the synchrotron radiation from this electron population that is radiating on the volumefilling magnetic field.
Cosmic ray transport downstream of the shock is a combination of the ∼ 100−150 km/s flow of the shocked, downstream plasma and cosmic ray streaming along the field lines at the Alfven velocity (Kulsrud & Pearce 1969) of ∼ 240 km/s. The range microwave-emitting electrons can reach in their loss time is then a function of magnetic field amplitude and reaches a maximum for B 10 µG: see fig. 13. Here the cooling time, t c , is the time taken for an electron injected at the accelerator with the maximal possible energy, E max , to be cooled to the energy at which it synchrotron radiates into the frequency range of concern, E c ≡ m e c 2 ν/ν g [B], where ν g [B] is the electron gyro radius in magnetic field B. The energy of an electron injected at E 0 into a constant magnetic field B evolves according to where τ [E] = 1.3 × 10 8 (B/10 µG) −2 (E/ GeV) −1 yrs, ignoring non-synchrotron losses (particularly IC) for illustrative purposes and assuming pitch angle scattering on timescales short with respect to the loss times (e.g., Reynolds 2009). This means that where E max is the maximal cosmic ray energy the accelerator is capable of attaining. With this description of the energy evolutionof the electrons, one obtains that an electron spectrum injected as a power law by the accelerator, dN e [E, 0]/dE ≡ KE −s , evolves according to The evolution of a cosmic ray electron spectrum, injected with s = 2.1, as the electrons are conveyed downstream of a shock and, simultaneously, suffer synchrotron + IC losses is illustrated in fig. 14. Once the transport time exceeds the cooling time t c a cutoff feature is introduced into the electron spectrum. Depending on the sign of s − 2, a pileup feature may or may not be introduced into the spectrum before the onset of this feature (a negative sign giving rise to this feature). Applied to the outflow population, GeV IC emitting electrons only reach a few ×100 pc past the shocks and microwave-emitting electrons reach a few kpc.
In contrast, lower-energy ∼GHz-emitting electrons survive longer, reach the shell before the contact discontinuity and are accumulated inside it (cf. fig. 13). Compression of the cooling plasma in this shell leads to adiabatic amplification of the frozen-in magnetic field and cosmic rays implying a second re-energization of those electrons that actually survive to this point. The spatiallycoincident, enhanced gas and cosmic ray ion populations close to the Bubbles' edges or in the filaments also lead, however, to significant hadronic collisions which, in addition to producing hadronic gamma-ray emission, also inject secondary electrons (and positrons) via the decay of collisionally-produced charged mesons. The edges/filaments are thus populated by both enhanced primary and secondary electron populations in addition to adiabatically-compressed magnetic fields. The electrons do not escape from these regions but lose their energy in situ to synchrotron radiation. Accumulation and trapping of electrons in this region leads to mixing of cosmic ray electron populations of various ages. Cooling + mixing (in projection) of differently-aged electron populations will be enhanced if there are downdrafts towards the edges of the Bubbles or convection within it. Mixing steepens the overall steady-state population into an ∝ E −3 e -type distribution. The steep-spectrum, polarised emission detected at 2.3 GHz (Carretti et al. 2013) is largely attributable to this population that radiates in the enhanced magnetic field region associated with the cooling shell towards the edges of the Bubbles (and possibly also swept up halo magnetic field outside the CD). We can thus evaluate the 2.3 GHz emission from the Bubbles in a thick target, single-zone model; the emission so calculated is displayed in fig. 15 (green curves). For the magnetic field amplitudes investigated, our calculations of synchrotron emission show that a power going into freshly accelerated electrons of 5-20% the total cosmic ray power at the shock is required to reproduce the observed radio and microwave emission, i.e., , E e (0.05−0.2)×Ė CR . HereĖ CR , the power going into all cosmic rays at the shock, is assumed to be a fixed fraction of 20% of the total mechanical power available at the shock. Fig. 15 shows the 2.3 GHz and microwave synchrotron emission of both primary and secondary electrons. Within our favoured parameter space, secondary electrons generated in the filaments explain 20-30% of the Bubbles' 2.3 GHz luminosity. We will explore this interesting point further in a subsequent publication (Crocker, Taylor et al. in preparation).  1 and fig. S6). There are counterparts to this feature evident in γ-ray (where the feature appears as an edge particularly evident in 10-30 GeV Fermi data; cf. de Boer & Weber 2014, fig. 3a also Ruizhi Yang, private communication) and in H-α emission 12 . At 2.3 GHz, this feature is likely the result of cancellation between intrinsically polarised synchrotron emission on two perpendicular, line-of-sight-superposed magnetic field structures. One of these is the front side sheath of the south bubble whose gross magnetic field structure extends in a roughly south west direction as evident from the 2.3 and 23 GHz polarisation data (reflecting the rotation of the bubble base in the same sense as the Galaxy and conservation of angular momentum in the expanding outflow: Carretti et al. 2013). The cancelling field must run south east along the long axis of the depolarisation feature. We believe that a compelling explanation of this feature is that it represents the pile up of magnetic field lines at the giant shock in the southern bubble (i.e., there is a perpendicular magnetic field configuration at the shock). Adopting a rough distance to the near side of the bubble of 5 kpc, the z-height of the depolarisation feature is 1-3 kpc, similar to the shock suggested by our analysis. In the northern bubble there is only a faint feature corresponding to the depolarisation feature described above (see fig. 16). The north, however, is highly confused and suffers from depolarisation at 2.3 GHz significantly off the plane due to ρ Ophiucus and other local, high latitude gas structures. There is, moreover, a feature of the northern bubble that is highly suggestive of a shock. The Galactic Center Spur (GCS: Sofue et al. 1989;Carretti et al. 2013, see fig. 16) is a highly-collimated total and polarised intensity radio continuum feature that extends in a north-west direction from just east of the Galactic Center to Galactic latitudes of b ∼ 25 • . Carretti et al. (2013) have claimed this feature wraps around the forward surface of the northern bubble, an interpretation we adhere to. On the basis of a geometric analysis of the overall curvature of this structure, Carretti et al. (2013) also claimed, however, that the Spur provides evidence that gas is being driven aloft into the Bubbles at a speed of ∼ 1000 km/s up to heights of at least a few kpc. This analysis was supported by an independent argument that, if the 2.3-GHz-emitting electrons were accelerated (exclusively) in the nucleus, they would have to be transported at a similar speed in order that they reach the top of the Bubbles in their loss time. This high velocity over the entire Bubbles is not consistent with our present results which suggest that while gas may be injected into the base of the bubbles at > ∼ 500 km/s, it is slowed to 100-150 km/s upon encountering the giant shocks at heights of ∼ kpc. But the assumptions leading up to the estimated ∼ 1000 km/s presented by Carretti et al. (2013) are no longer met in our revised scenario, in particular, we now expect injection of high energy electrons at large scales in the Bubbles either via direct acceleration at the large scale shocks or as secondaries and, of course, the shock in the north bubble changes our expectation for the geometry of the GCS. In fact, we expect an inflexion point in the GCS at the shock as the vertical speed of the flow is slowed by ∼ 1/4 crossing the shock. Such an inflexion does seem to be present in the GCS and it has the correct geometry: at b ∼ 17 • the angle between the long axis through the GCS changes from ∼ 17 • to ∼ 48 • (see fig. 16) suggesting a change of speed tan[17 • ]/ tan[48 • ] = 0.27, consistent with a strong shock.
Moreover, an independent determination (Vidal et al. 2014) of the spectral index of the polarized emission between 23 and 33 GHz from the GCS reveals a hardening at higher latitudes (b > ∼ 18 • ) from 3.52 ± 0.51 → 2.62 ± 0.44 (with a similar hardening in a region coincident with part of the eastern edge of the north bubble), consistent with a giant shock -at the correct height above the plane -reaccelerating electrons as we have modeled here. Of course, our scenario does not explain why the reverse shock surfaces fail to run parallel to the Galactic plane but two interesting (and not incompatible) explanations for this phenomenology present themselves: 1. As already remarked, the long axes of the Bubbles do not run perpendicular to the plane in either hemisphere but rather appear to be pushed towards Galactic west; the radio polarisation data only accentuate this impression with long plumes of emission extending beyond the edge of the γ-ray bubbles at high latitudes, but only towards Galactic north-west. Whatever the origin for this apparent distortion, the southern and northern shocks seems to run more-or-less perpendicular to the inferred flow.
2. Currently star-formation activity in the Galactic center is concentrated east of l = 0; there is a preponderance of molecular gas on this side of the region and Sagittarius B2, one of the Galaxy's most active star-forming giant molecular clouds, resides there . Moreover, of the three large stellar clusters in the region, one surrounds the super-massive black hole but the other two, Quintuplet and Arches, are located to the east side. Quintuplet, at ∼ 4 Myr old is likely to have already hosted a number of very massive core-collapse supernovae and there is tantalising evidence that such has occurred (Sofue 2003). So it may well be that, even if on the average, injection of mechanical energy is evenly distributed east and west of the GC, it is now predominantly coming from the east, leading to a localised increase in the outflow velocity tending to push the shocks further into the halo on this side. This localization of the mechanical energy injection associated with star formation may be the origin of the ridge features evident from polarization data on the forward surfaces of both north and south bubbles (Carretti et al. 2013).
We will provide a more detailed and quantitative analysis of this putative shock feature in future work. Lastly here, we note that the putative southern and northern bubble shock features are remarkably mirror symmetric (as are the overall γ-ray Bubbles and radio lobes+plumes).

Observational Evidence for Condensations
Coincident with the Bubbles' Edges In §1 we described UV absorption evidence (Keeney et al. 2006;Zech et al. 2008) for higher altitude warm ionised material in the Galactic bulge that is participating in a nuclear fountain flow. We believe that this material is the end state of gas condensing in the Bubbles' shell. This warm ionised material seems to be concentrated towards the edges of the Bubbles, just as required if the volumetric γ-ray emissivity is to peak here and as expected for our cooling shell model. Keeney et al. (2006) determine that their clouds are likely launched into a conical outflow that is collimated into a cylinder of radius ∼ 1.6 kpc; with this (somewhat uncertain) radius determination they find that their clouds have z max 12 ± 1 kpc. Given the qualitative resemblance between this geometry and that of the Bubbles, we note that, were the radius apposite to the cloud trajectory calculations presented by Keeney et al. (2006) re-scaled to 3 kpc, the clouds would be inside the Bubbles.
The fountain material found by Zech et al. (2008) is claimed to have super-solar metallicity (∼ 1.6Z ) and is, thus, a very good prospect for having been launched from the GC. Moreover, the illuminating background UV source is a PAGB star in the Messier 5 globular cluster at known distance (7.5 kpc) and height (5.3 kpc) above the plane which, in projection, is coincident with the edge of the north bubble. It seems more than plausible that the cloud is located in the edge and proffers a lower limit to the radius of the northern Bubble of ∼ 2.7 kpc (at z = 5.3 kpc), entirely consistent with the ∼ 3 kpc radius we have adopted elsewhere. , which passes through the conically-expanding lower part of the northern bubble at a height of ∼ 2.3 kpc. These observations find red and blue-shifted warm ionised material which kinematic analysis suggests is moving at ∼ 900 km/s, likely accelerated by the ram pressure of the dynamically dominant 'wind fluid', i.e., the volume-filling, hot plasma phase. Given this supersonic velocity, this sightline must pass somewhere below the giant shocks. Indeed, the northern shock position we infer from the radio polarisation observations discussed in §7.1.2 is safely above the sightline. The ∼ 900 km/s speed inferred by Fox et al. (2014) is somewhat higher than the flow speeds indicated by our model (in the relevant parameter space, our fitting prefers a flow velocity of ∼630-720 km/s at similar heights) and the heights somewhat above where we predict the shocks to lie on the basis of modelling the global flow parameters. It is tantalising, however, that this sightline is on the east side of the north bubble where, observationally, the shock seems to have been pushed furthest from the plane. This could be a spatially and temporarily-localized upwards excursion in the outflow velocity connected to the recent concentration of star-formation activity on the east side of the Galactic Centre as already discussed.

Metallicity considerations
A possible tension between our model and X-ray observations is the following: Modelling of data arising from Suzaku scans across the the limbs of both the north and south bubbles (Kataoka et al. 2013) suggest a significantly sub-solar metallicity Z 0.2Z . This is contrary to the simple expectation were this material to be supplied by the nucleus (in which case it would be expected to be at least solar in metallicity). Interestingly, the warm plasma found by Zech et al. (2008) along the Messier 5 sightline, relatively close to the Suzaku northern scan, is claimed to be super-solar in metallicity as already remarked. One possible explanation for this apparent tension is that the Suzaku observations are, in fact, predominantly revealing emission from swept-up halo material on the other side of the contact discontinuity which various pieces of evidence (as summarised by Miller & Bregman 2014) suggest has a metallicity of ∼ 0.3Z . In any case, the northern Suzaku scan does commence significantly outside the edges of the Bubbles, within the North Polar Spur feature; with 6 of the 8 pointings outside the γ-ray edge and with the metallicity measured only with confidence in the aggregate (and not for individual pointings), there may be a systematic bias here.
Another concern may be the following: these X-ray data have been interpreted by Kataoka et al. (2013) within a 3-component model incorporating i) an absorbed power-law isotropic background; ii) a local (unabsorbed) single-temperature, CIE plasma foreground (representing the local bubble); and iii) an absorbed, singletemperature, CIE plasma that represents the signal region (the north polar spur or around the edge of either the north or south bubble). Such a model may be too simple to adequately describe the putative cooling shell. Indeed, Strickland et al. (2002) and Weaver et al. (2000) demonstrate that fitting to X-ray plasma emission from starburst outflow often leads to unphysically low metallicity estimates. Possible circumstances leading to a toolow metallicity estimate are the existence of neglected multi-temperature plasma distributions; having a large absorbing column; and/or non-thermal contribution to the continuum (that boosts up apparent free-free relative to lines thus making plasma appear to have smaller metallicity). Any or all of these may be relevant effects for the Bubble plasma, in particular, we expect the gas in the cooling shells to be multi-temperature as already discussed.

DISCUSSION AND CONCLUSION
In this paper we have set out what we believe is essentially the correct framework for understanding the Fermi Bubbles. Requiring only that the Bubbles contain the postulated shocks and assuming that their interior plasma density falls in a range consistent with X-ray and other constraints, we have scanned over the parameter space for the nuclear energy and mass injection rates, {Ė,Ṁ }. Remarkably we have found that most of this parameter space can be excluded by measurements and/or physically-motivated constraints. Within the region of parameter space that is not excluded our model reproduces 1. the γ-ray luminosity and spectrum of the Bubbles, mostly via hadronic emission 2. the luminosity, spectrum, and extent of the hard spectrum, total intensity microwave Haze 3. the luminosity, spectrum, and extent of the steep spectrum, polarised ∼GHz S-PASS Lobes 4. the extent, temperature, and density of the Bubbles' plasma 5. the mass flux and initial radius of the nuclear outflow This same, residual region of parameter space presents a compelling fit to a star-formation-driven system with parameters that matches the conditions in the nucleus. From our modelling, presented in §3, of a radiative bubble expanding into an atmosphere of pressure similar to that in the Galactic bulge, for values ofĖ andṀ selected by our reverse shock analysis, the asymptotic radius, temperature, and bolometric cooling luminosity of such a bubble are close to the observed values for the Fermi Bubbles. Our model explains both the hard spectral indices of the CR electron population responsible for the Haze synchrotron emission and the CR proton and ion population responsible for the (hadronic) Bubble γ-rays and why, within uncertainties, these are the same at γ 2.1: these populations are both (re)accelerated at the giant shocks which, are modelling shows, are required to have Mach numbers in the range 6-8 within the allowable {Ė,Ṁ } parameter space. Simultaneously, we explain the steepspectrum, polarised radio emission that subsumes the Bubbles. This has not been adequately addressed in other models: this emission is coming from an aged and mixed population of electrons which survive long enough to travel from the internal shocks out to the edge of the Bubbles where they radiate on the strong and rather regular magnetic field compressed into the cooling shell (from the inside) and possibly also the swept up halo magnetic field on the outside of the CD.
For fiducial parameters, our modelling from §3 suggests that the Bubbles are at least ∼ 5 × 10 8 years old, very similar to the cooling time of plasma at inferred Bubble temperature and number density. The Bubbles may be related to whatever Galaxy-scale process led to the formation of the CMZ gas torus which, at ∼ 3 × 10 7 M , has a very similar H 2 mass to the plasma content of the Bubbles. Sjouwerman et al. (1998) has claimed, on the basis of counting OH/IR stars in the inner 50 pc around the GC, evidence for a starburst in the GC > ∼ 1 Gyr ago which may be a credible event to kick-off the inflation of the Bubbles (though we emphasise that their current non-thermal emission and apparent slow growth can be supported by the current or recent nuclear star-formation rate).
In theory, the Bubbles would reach their asymptotic final radius after ∼Gyr, but they may not survive this long. Indeed, they may already have begun to blow-out as evidenced by the polarised radio plumes that extend significantly beyond their upper γ-ray edges (Carretti et al. 2013). This process may be meditated by the Rayleigh Taylor instability with the buoyant fluid of cosmic rays and hot plasma breaking through the overlying shell of relatively denser shell material at the top of the Bubbles. If so, the plumes might be expected to grow at the expansion speed, v exp ∼ 5 km/s, implying a length scale t cool × v exp ∼ 2.6 kpc which is, indeed, similar to their observed extension.
In any case, the plume phenomenology is consistent with a hadronic origin for the Bubbles' γ-ray emission: in this scenario, although (lower energy) electrons, lower density plasma, and magnetic fields can leak out of the Bubbles and synchrotron radiate beyond them, leaking hadrons will not radiate (or only weakly) beyond the cooling shell/contact discontinuity because of a lack of dense target gas there. (Given the ubiquitous CMB, TeV electrons would also radiate IC from the plumes were they present there, but, as emphasised, in our model these will be concentrated into a zone much further down in the Bubbles close to their acceleration site in the giant shocks). Before moving on from this topic, we again flag the possibility that the overall extension of the Bubbles/Plumes in a direction west and away from the plane is essentially due to the motion of the Galaxy through the local group medium towards the group barycenter in the direction of Andromeda.
On the basis of our modelling we predict there should be significant mass in ∼ 10 4 −10 6 K gas -likely exceeding 10 7 M -concentrated towards the Bubbles' edges. The cooler and denser phases of this gas will be raining down under gravity.

Future Work
In subsequent publications we will present or examine in more detail: the detailed spectral morphology of the synchrotron emission in the Bubbles; the observational evidence for the giant shocks we hypothesise; a detailed treatment of the dynamical production of the dense condensations. We will also examine the origin of the γ-ray spectral downturn at ∼ 100 GeV (Fermi-LAT Collaboration 2014) which is not explicitly dealt with in our current model but may signify the energy where the Bubbles themselves cease to trap cosmic ray protons effectively. Also awaiting a detailed treatment is the dynamical importance of the strong and coherent magnetic field that appears to be wrapped over the surface of the Bubble, particularly in stabilising this surface.

Speculations
For fiducial parameters, the giant shocks are capable of accelerating cosmic ray protons to energies around the knee in the observed cosmic ray spectrum. Given the very hard spectrum, a total power in these particles that approaches 10 39 erg/s, and that the particles are delivered into the halo but not too far above the disk, these shocks are interesting candidates (cf. Parizot 2014) to be the main accelerators of Galactic cosmic rays in this energy range (cf. Cheng et al. 2012;Lacki 2014).
Our model selects a region of the {Ė,Ṁ } parameter space implying that, if star formation driven, the outflow is mass loaded at a level β ∼ 1 − 2.8, consistent with observations of external, star-formation-driven outflows (Strickland & Heckman 2009). Further, it is remarkable that our results indicate that the mass efflux in the outflow isṀ ∼ 0.03 − 0.1 M /yr, rather close to the current rate of nuclear star formation (Longmore et al. 2013;Crocker 2012, and references therein), in turn close to the long-time-averaged nuclear star formation rate 13 . Moreover, on the basis of the semi-analytic model presented in §3 we can say that the Fermi Bubble system is rather delicately poised in terms of allowable values of the ratioĖ/Ṁ : systems with twice the favoured value for this ratio (of ∼ 1.6 × 10 15 erg/g) do not asymptote to a final radius until ∼ 10 10 year and this radius is 10 kpc, significantly larger than the observed Bubbles (and the plasma they contain is too hot). Nor, significantly, does the mass drop out rate from such systems equilibrate with the mass injection rate until a similar time has elapsed. Conversely, systems withĖ/Ṁ ratios half those favored by our analysis asymptote to final radii within ∼ 10 8 that are smaller than those observed (and contain plasma that is too cold).
These observations suggest the operation of a negative feedback process to maintain the required delicate balance. An obvious mechanism to achieve this -completing the cycle -is that material fountaining back to 13 This can be rather directly and robustly determined from the fact that the total stellar mass of the nuclear bulge (the stellar population coincident with the central molecular zone) is ∼ 10 9 M which has been formed over a timescale 10 10 years Figer et al. 2004).
the plane in the vicinity of the nucleus will trigger further star-formation in this region. Star-formation that is too vigorous or that produces a too large value ofĖ/Ṁ in the outflow (which may currently be happening on the east side of the nucleus) will tend to switch itself off because the outflowing plasma takes too long to condense and fall back from the downstream of the Bubbles. The matter cycle will not be 100% efficient, of course, but the system is not closed: some amount of accretion to the nucleus will continue through the plane and, moreover, interaction and mixing between Bubble matter and the surrounding halo gas around the contact discontinuity may catalyse the condensation of halo gas so that, overall, more gas fountains back to the plane than is lofted upwards (Marinacci et al. 2010), some of it low metallicity.

ACKNOWLEDGEMENTS
RMC is the recipient of an Australian Research Council Future Fellowship (FT110100108). RMC warmly acknowledges enlightening conversations and/or correspondence with Luke Drury, Ed Jenkins, Federico Marinacci, Matt Miller, Christoph Pfrommer, Robin Shelton, Heinz Völk, and Alex Wagner and thanks Doug Finkbeiner for first mentioning to him the possible implications of the Milky Way's motion through the local medium for distorting the Bubbles. Ralph Sutherland is particularly thanked for supplying plasma cooling data generated by MAPPINGS IV code prior to public release, Alex Hill for providing information about H-α emission in the southern Bubble as observed by the Wisconsin H-Alpha Mapper (WHAM) southern sky survey (Haffner et al. 2010), and Ruizhi Yang for providing preliminary maps of the Fermi-detected emission in the southern bubble.    The solid blue curve shows the case for cooling with Z = Z , the green dot-dashed curve is for Z = 2Z . Chosen parameters areĖ = 6 × 10 39 erg/s,Ṁ = 0.08 M /yr, P halo = 3.2 × 10 −12 dyn/cm 2 , n halo tot = 6.6 × 10 −3 cm −3 . For these parameters and solar metallicity, the bubble asymptotes to a final radius of ∼ 5 kpc after ∼Gyr; note that, were it attributed to a single sphere, our assumed Fermi Bubble volume would have an equivalent radius of 4.1 kpc (gray dashed line). The single bubble model reaches this radius after ∼ 5 × 10 8 year, which is approximately equal to the cooling time of the plasma. The dotted (yellow) curve shows the analytical result for the expansion of an over-pressured, adiabatic bubble (OPAB) which severely overestimates the bubble radius at late times in the case of pressure equilibrium.   fig. 3. Note that by few ×10 7 year the forward shock has devolved into a sound wave and the growth of the contact discontinuity is significantly slower than the sound speed. The dotted yellow curve shows the analytical result for the expansion of an over-pressured, adiabatic bubble.   fig. 3. This rapidly reaches approximate equilibrium with the atmosphere at > ∼ few 10 7 . The dashed purple curve is for an over-pressured, adiabatic bubble.   fig. 3. This does not grow after ∼ Gyr; the mechanical energy injected at the base of the outflow is balanced by radiation from the bubble. The dashed purple curve is for an over-pressured, adiabatic bubble; the dotted yellow line is the integrated energy injected into the bubble. -Bolometric cooling luminosity of the bubble; this saturates at ∼ 4 × 10 39 erg/s (injection power is 6 × 10 39 erg/s), consistent, within uncertainties, with the inferred bolometric luminosity of the Galactic X-ray bulge (Snowden et al. 1997;Almy et al. 2000). Parameters as for fig. 3 Fig. 11.-Explored {Ė,Ṁ } parameter space and excluded regions for the case that n F B tot = 4 × 10 −3 cm −3 . Regions that are masked out in green predict an unphysically-thick cooling shell (> 1 kpc); regions that are masked out in blue have winds that are too slow (< 500 km/s) at the position of the putative reverse shock. The thick black dashed curves delineate the expectation for the star-formation-driven mass efflux vs. mechanical energy input for mass loading rates, as marked, of β = 1.0 and 2.8 (cf. Strickland & Heckman 2009). Here mass loading is defined with respect to the mass injection rate (due to supernova ejecta and stellar winds) that accompanies a given rate of star formation (not with respect to this star formation rate itself). The thin, diagonal grey lines are for {Ė,Ṁ } values that give upstream speeds just before the shock of 500 and 900 km/s, respectively. -Predicted 1-100 GeV hadronic and IC emission within the allowed parameter space for the case n F B tot = 4 × 10 −3 cm −3 . Regions that are masked out in grey do not obey all the physical or observational constraints as in fig. 11. The pink/red band is the locus of parameter space points for which the predicted hadronic luminosity is close to that observed (central value of 4.4 × 10 37 erg/s over 100 MeV < Eγ < 500 GeV with 1-σ errors shown as the pink regions: Fermi-LAT Collaboration 2014). The green dashed curves report the total 1-100 GeV Inverse Compton (IC) luminosity as a proportion (0.03, 0.1, or 0.3 from left to right) of 4.4 × 10 37 erg/s; note that though IC is subdominant, it is concentrated within a small solid angle downstream of the shock (cf. fig. 13) and, in this region, contributes a larger proportion of the overall intensity (up to ∼ 50%). The IC curves assume that 20% of the mechanical energy at each shock goes into freshly-accelerated cosmic rays, of which 10% goes into primary electrons. -Ranges of cosmic ray electrons downstream of the shock that emit synchrotron radiation into the specified characteristic frequency or with energies 100 GeV and 1 TeV (dominating IC emission) as a function of downstream region magnetic field. The range is determined by the condition that ttransport = tcool with both synchrotron and IC losses accounted for. Here the downstream gas velocity is assumed to be a fixed 112 km/s (∼ 1/4 vup) and the cosmic rays are taken to stream at the Alfven speed in the specified field and for a density of ρ = 1.4 n H mp where we adopt the n H = 3 × 10 −3 cm −3 . Note that Haze electrons reach a maximum ∼ 3 kpc downstream from the shock and 2.3 GHz electrons can reach a maximum ∼ 10 kpc; electrons IC emitting 1-100 GeV γ-rays are concentrated close to the shock. The vertical line shows the ∼ 10 µG field where the range of the radio and microwave-emitting electrons is approximately maximised. The total vertical extent of the Bubbles is ∼ 8 kpc with the shocks forming at heights ∼ kpc; note that 2.3-GHz-emitting electrons can reach the edges of the Bubbles whereas microwave-emitting electrons do not; from this analysis the vertical extent of the microwave should be ∼ zshk+ range[23 GHz] ∼ 3 kpc; this is consistent with observations that show a drop-off in the intensity of the microwave emission at ∼ 35 • corresponding to the distance of 2.7 kpc for the near-surface of the Bubbles at 5 kpc distance.  Fig. 14.-Spectra of cosmic ray electrons at various distances downstream of a shock. The electrons are injected at the shock with an assumed distribution dNe/dEe ∝ E −2.1 e and are conveyed downstream at a total speed given by the downstream flow speed (112 km/s) + streaming at the Alfven speed (240 km/s). At the same time, they suffer synchrotron (with B = 7µG) and inverse Compton losses (U ISRF = 1.5 eV cm −3 ); the position-dependent spectrum evolves according to Eq. 30. From left to right, the vertical dashed lines indicate the primary electron energy corresponding to i) synchrotron emission at 2.3 GHz; ii) synchrotron emission at 23 GHz; iii) 100 GeV IC emission off starlight; and iv) 10 GeV IC emission off CMB. Note that, for these (realistic) parameters, IC emission is concentrated within < kpc of the shock; in contrast 23-GHz emitting electrons extend ∼ 2 kpc, and 2.3-GHz emitting electrons can reach the full extent of the Bubbles. Note there is a one-to-one correspondence between distance from shock and time since acceleration (electron 'age') with the implication that, provided the emission is resolved, nowhere does the spectrum steepen by the canonical -1 due to the synchrotron + IC losses, rather it maintains its hard injection spectral index up to some distance/time-dependent energy above which is it cut-off. Regions that are masked out in grey do not obey all the physical constraints as described in fig. 11. The green and purple curves show, respectively, where the modelled 2.3 GHz and 23 GHz intensity matches that observed for the nominated percentage of all cosmic ray power going into freshly accelerated electrons, viz. Le = {0.05, 0.1, 0.2} × L CR . The curves account for both primary and secondary electron synchrotron. These curves assume that a fixed 20% of the mechanical energy at each shock goes into freshly-accelerated cosmic rays. The purple curve refers to emission over the Haze region with a volume-filling magnetic field of amplitude of 7 µG (Carretti et al. 2013) and an interstellar radiation field calculated at 3 kpc above the plane (UISRF = 1.7 eV cm −3 : Porter et al. 2008); we adopt a reference value for the Haze intensity at 23 GHz of 1200 Jy/sr (Su et al. 2010) and a reference value for the total intensity at 2.3 GHz of 18.5 kJy/sr (Carretti et al. 2013).