JJanuary 7, 2021 1:39 WSPC/INSTRUCTION FILE main
International Journal of Modern Physics D © World Scientific Publishing Company
COSMIC RAY DRIVEN GALACTIC WINDS
SARAH RECCHIA
Universit´e de Paris, CNRS, Astroparticule et Cosmologie, F-75013 Paris, France,[email protected]
Received Day Month YearRevised Day Month YearGalactic winds constitute a primary feedback process in the ecology and evolutionof galaxies. They are ubiquitously observed and exhibit a rich phenomenology, whoseorigin is actively investigated both theoretically and observationally. Cosmic rays havebeen widely recognized as a possible driving agent of galactic winds, especially in Milky-Way like galaxies. The formation of cosmic ray-driven winds is intimately connected withthe microphysics of the cosmic ray transport in galaxies, making it an intrinsically non-linear and multiscale phenomenon. In this complex interplay the cosmic ray distributionaffects the wind launching and, in turns, is shaped by the presence of winds.In this review we summarize the present knowledge of the physics of cosmic rays involvedin the wind formation and of the wind hydrodynamics. We also discuss the theoreticaldifficulties connected with the study of cosmic ray-driven winds and possible futureimprovements and directions.
Keywords : Keyword1; keyword2; keyword3.PACS numbers:
1. Introduction
Galactic winds (GWs) are powerful outflows that originate in the disk of galaxiesand may extend for hundreds of parsecs. The ejected material is mainly composedof very hot dilute gas, often enriched of metals, dust and portions of cooler gas,whose velocity can reach hundreds or even thousands km/s, and, in some cases, thegas may even escape the galaxy and be injected in the intergalactic medium (IGM).The induced mass loss can be comparable to the mass formed in stars.Since their first detection in the M82 starburst galaxy,
1, 2
GWs are ubiquitouslyobserved both locally and at high redshift, especially in galaxies that exhibit ahigh star formation rate (SFR) or an active galactic nucleus (AGN).
Also theMilky Way (MW) might host GWs, as suggested by several observations.
7, 8
GWsrepresent a most important feedback process which deeply affects the evolution andthe structure of galaxies, star formation, and the chemical and physical propertiesof the interstellar medium (ISM) and of the IGM.The formation of GWs is not yet fully understood and it is actively investigatedin astrophysics. GWs may be powered by the energy and momentum injected by a r X i v : . [ a s t r o - ph . H E ] J a n anuary 7, 2021 1:39 WSPC/INSTRUCTION FILE main Authors’ Names stellar winds and supernova explosions or by the energy released during accretiononto supermassive black holes in AGN. They may also be produced by the radi-ation pressure induced by the scattering/absorption of starlight radiation on dustgrains and gas.
11, 12
Thermal and radiation pressure gradients are effective in driv-ing winds in very active galaxies, while in more quiescent galaxies, such as the MW,they are probably not large enough, except maybe for the Galactic Center region.On the other hand, cosmic rays (CRs) escaping from galaxies can effectively pushon the ISM, and eventually launch winds. In fact, CRs are known to be producedin astrophysical sources, likely supernova remnants (SNRs), located in the galacticdisk and to undergo diffusive propagation in the ISM as a result of the scatteringoff plasma waves. Such scattering provides a strong coupling between CRs andthe background plasma, which can allow CRs to produce GWs. In the MW CRsare observed to be energetically in equipartion with the thermal plasma and withthe magnetic field, which suggests their importance in the dynamics of the ISM.The relativistic CR gas exhibits a smaller adiabatic index compared to the thermalplasma, so that the CR pressure can act at relatively large distances from thedisk, making possible the wind launching also in situations where the thermal andradiation pressure would fail. This aspect has been pointed out for the first time inRef. 13, which presented the first study of the CR impact on the launching of GWsin a MW-like galaxy. The dynamical role played by CRs in the wind hydrodynamicshas been subsequently confirmed by many authors and widely investigated in thestationary regime, in time dependent approaches and in simulations. The possibility of CRs to launch winds is intrinsically connected with theirtransport in the ISM, which in turn is affected by the presence of GWs. In fact,CRs escaping from the source region, namely the disk, establish a gradient in theirdistribution function whose consequences are manifold. First of all, such gradientcan lead to the excitation of plasma instabilities, most notably of the streaminginstability of Alfv´en waves,
33, 34 which in turn affects the scattering of CRs, namelytheir diffusive motion, and their coupling with the background gas. Moreover, thecoupling with the ISM and the CR gradient between the disk and the outer halomay induce the formation of GWs, which in turn affect the advective transport ofCRs. Finally, the damping of the CR-induced plasma turbulence and the ionizationlosses of sub-GeV CRs can heat the gas, further affecting the wind launching andthe wind structure. Thus, the dynamics of CR-driven winds is an intrinsically nonlinear process and a multiscale phenomenon, in which the microphysics of the CRtransport, on scales of the Larmor radius of relativistic particles in the galacticmagnetic field, determines the CR distribution function and the wind structure,which in turn affect the CR propagation itself.The multiscale and nonlinear nature of CR-driven winds makes them particu-larly difficult to study, especially when CRs are not treated macroscopically, as arelativistic gas, but the details of the CR microphysics is taken into account. anuary 7, 2021 1:39 WSPC/INSTRUCTION FILE main Instructions for Typing Manuscripts (Paper’s Title) In this review we summarize the relevant properties and the physics involvedin the formation CR-driven winds. The review is organized as follows: 1) in Sec. 2we review the observations, formation mechanisms and the impact of GWs in dif-ferent types of galaxies; 2) in Sec. 3 we briefly introduce the main aspects of CRsastrophysics; 3) in Sec. 4 we illustrate the relevant ingredients of the CR transportin a CR-driven wind; 4) in Sec. 5 we review models of GW hydrodynamics in thepresence of CRs; 5) in Sec. 6 we show in some detail the properties of CR-drivenwinds in the stationary regime, in particular in a MW-like galaxy; 6) in Sec. 7 wedrive some conclusions and discuss future perspectives.
2. Galactic winds
Different gas phases coexist in GWs: dilute hot and warm phases (very hot ∼ K, hot ∼ − K and warm ∼ K), neutral atomic matter ( ∼ K),cold molecular material, dust ( ∼
100 K) and non-thermal particles. Both hot andwarm/cool gas can exhibit very high velocities (hundreds or thousands km/s), andproduce mass loss rates which can be as large as several solar masses per yr. Howthese phases are accelerated to such high velocities is being actively investigated inastrophysics and several aspects remain unclear. Moreover, the multiphase natureof GWs and the fact that a large part of the ouflowing gas is hot and very dilute,make their observation challenging.In this section we briefly summarize the mechanisms responsible for the windformation in different types of galaxies and environments, and the observationaltechniques employed to detect and study GWs. We conclude by stressing the im-portance and implications of GWs in astrophysics and cosmology. For further detailswe refer the reader to the excellent reviews found in Refs. 3,4, 5 and 6.
Production mechanisms
Hot winds are most likely accelerated in starburst regions or in AGN. The me-chanical energy injected in such regions and partially converted to thermal energythrough shocks, creates a bubble of very hot gas whose pressure can be much largerthan that of the surrounding ISM. The dynamical evolution of such bubbles has beenwidely studied, starting form the seminal work of Ref. 9, which showed that, underthe assumption that radiative losses, gravity and other effects can be neglected, theflow solution is self-similar. The bubble may then be able to cross the disk heightand eventually break through the galactic halo. In general, in such energy-drivenflows, the energy input must be large enough, or radiative losses small enough, thatthe wind is not hampered before it can reach the edge of the galactic disk.Winds may also be driven by the mechanical momentum imparted by starburtsand AGN
10, 11 or by radiation. Radiation can heat the ISM via photoionization. Forinstance, UV photons can create regions of ionized hydrogen with temperatures of ∼ K. In addition, radiation pressure on dust grains and gas
11, 12 can effectivelypush the ISM and contribute to the generation of GWs. Radiation pressure may beanuary 7, 2021 1:39 WSPC/INSTRUCTION FILE main Authors’ Names important in different environments, from massive stars to luminous AGNs, fromstarburst to rapidly star-forming galaxies.
A very important and still unclear aspect is how the warm and cold wind phasesinteract with the hot phase and are themselves accelerated to velocities that arecomparable to that of the hot gas, and that can be as high as hundreds km/s. Apossible scenario is that portions of the warm and cold ISM, or even clouds, areadvected in the hot wind and accelerated by its strong ram pressure.
4, 5
Anotherpossibility is that the warm/cool gas in GW forms due to the thermal instabilitiesinduced in the gas by radiative cooling
4, 5, 40
Observations
Given the multiphase nature of GWs, multiwavelength observations, of both con-tinuous and line emission and of absorption lines, spanning from radio to X-rays,are necessary in order to identify and characterize the various flow phases.
3, 6, 39
The flow velocity is strongly phase dependent, with the hotter phases showing thehighest speeds (up to thousands km/s), down to the ∼
100 km/s found in the coldmolecular and atomic gas phases. The study of the line profile of emission and ab-sorption lines allows to estimate the fluid velocity, while spatially resolved mappingof the emitting/absorbing material is used to determine the wind morphology.
6, 39
The hot and very hot gas phases are mainly tracked through their X-ray emis-sion. In particular, hard X-rays are expected from the very hot and tenuous gasheated by SNe and stellar winds, and from internal shocks in such hot wind.When the wind collides with denser ambient material or with clouds, the shockheated/evaporated material is expected to produce also soft X-rays. The interaction of a hot wind with a denser and cooler background gas, possiblycharacterized by a mixture of phases, can produce radiative shocks that compressand heat the ambient medium , thus producing the warm wind phases. Such warmphases have been studied through the continuous emission in the vacuum ultravioletand through emission lines, in the optical band (e.g H α , NII, OII, OIII, and SII)and in the near-IR band (e.g the 2 . µ m line of H , and the lines Pa α and Br γ ).
3, 5
The presence of shocks is often revealed by the presence of double-peaked emissionwith typical values of emission-line ratios like SII/H α , NII/H α and OII/H α .
3, 39
Molecular gas is mapped through mm-wave (warm) and mid/near IR (cold)spectroscopy. Atomic hydrogen is tracked by the 21 cm line, but also OI and CIIlines in the far IR are used to identify atomic gas. Finally, dust is studied both inthe IR and in the far-UV band. Radio surveys reveal the presence of synchrotron emitting relativistic electronsin magnetized winds. Complementary to emission techniques, absorption-line techniques are also mas-sively employed, especially for the study of face-on systems and of distant galaxies,where the diffuse emission from the wind is hard to detect. The most used lines arein the UV band: Silicon (Si II λ λ λ λ Instructions for Typing Manuscripts (Paper’s Title) Iron (FeII λ λ λλ , λλ , λλ , As for the Milky Way, the existence of GWs has not been firmly established.However, the recent observation in the X-ray band of Oxygen OVII and OVIIIabsorption lines in the spectra of distant quasars,
7, 8, 41 implies the existence aroundthe Galaxy of a very extended ( ∼ ∼ K) dilute gas, with metallicity larger than 0.3, and likely connected with GWs.Moreover, also the so called Fermi Bubbles, giant structures detected in γ − raysthat extends for several kpc above and below the Galactic Center, might be theresult of outflows. Impact of galactic winds
GWs constitute a most relevant ingredient in the ecology of galaxies, and a primaryfeedback mechanism.In fact, GWs heat the ISM and drain hot gas and metals from the disk of thehost galaxy toward galactic halos or even in the intergalactic space. This limitsthe amount of gas available to form stars. Moreover, the injected material affectsthe surrounding space by changing the temperature, the degree of ionization andthe chemical composition, which especially influence the cooling rate,
8, 45 and sothe time required for the ejected gas to fall back to the disk. All this establishesa gas recycling and a regulation of the SFR.
Models of galactic evolution thatdo not include GWs tend to overestimate the amount of baryons and the SFR. GWs are necessary in order to explain the luminosity function of galaxies and themass/metallicity relation (see Ref. 4 and references therein).In the cases of dense outflows, star formation may even take place within the out-flow, with prominent consequences on the spheroidal component of galaxies and onthe chemistry of the circumgalactic and intergalactic medium. The gas expelled by GWs in the IGM may represent a sizable fraction of the bary-onic matter of the Universe
8, 48 and has also been invoked as a possible importantcontribution to the solution of the missing baryons problem in the local Universe. Finally GWs may also transport magnetic turbulence from the disk to the halo andmay contribute to shaping the large scale magnetic field in galaxies, especially theout of disk components.
3. Cosmic rays in a nutshell
CRs are very high energy charged particles that originate in astrophysical sourcesand reach our planet by propagating and interacting with the ISM. Locally, CRsare revealed with balloon, satellite and air-shower detectors. The observed CR fluxis shown in Fig. 1. Remarkably, it extends as a nearly perfect power-law in energy, ∼ E − γ , from GeV energies up to ∼ eV, with a corresponding energy density of ∼ / cm − , a value comparable with the thermal and magnetic pressure observedanuary 7, 2021 1:39 WSPC/INSTRUCTION FILE main Authors’ Names in the Galactic disk. Protons largely dominate the observed flux, followed by Helium( ∼ ∼ ∼
20 TeV have been measured sofar.
51, 52
Notable spectral breaks occur at: ∼
230 GV, more pronounced in the protonand Helium spectra, but present also for heavier nuclei, where the slope of theproton spectrum changes from γ = 2 . ± .
015 to γ = 2 . ± .
53, 54 the knee ,at ≈ × eV, where the spectral slope changes from γ ≈ . γ ≈ . ankle , at ≈ × eV, wherethe spectral slope flattens to γ ≈ . CRs are also tracked through the radiation produced by their interaction withthe ISM, most notably through γ -rays produced by the decay of π mesons generatedby the interaction of CR protons with the background gas, and by bremsstrahlungand inverse Compton due to CR leptons, and by synchrotron radiation producedby the interaction of CR leptons with the background magnetic field.Together with the flux, the chemical composition and the level of anisotropy ofthe observed CRs, represent precious pieces of information. In particular, the rela-tive abundance of secondary spallation nuclei relative to their primary nuclei (e.gthe Boron/Carbon ratio, see Fig. 2, or the abundance of unstable isotopes relativeto stable isotopes (e.g Be / Be), represent the most reliable measurement of theamount of matter traversed by CRs during their travel through the ISM, and so ofthe residence time of CRs in the Galaxy. The estimated residence time decreaseswith energy and exceeds by orders of magnitudes the ballistic propagation time.This strongly supports the idea that CRs propagate diffusively in the magnetizedISM, as additionally confirmed by the small observed anisotropy ( ∼ .
001 at TeVenergies).
58, 59
Nowadays, the most accredited picture of the origin of Galactic CRs, is theso-called supernova remnant paradigm . In this scenario, the bulk of CRs isthought to be accelerated in the Galactic disk at the shocks of supernova remnants(SNRs) through the mechanism of diffusive shock acceleration (DSA). On the en-ergetic ground, this requires that 3-10% of the mechanical energy released by SNeis channeled into CRs.
61, 62
DSA is rigidity dependent and produces CRs with ap-proximately universal power law energy spectra ≈ E − . . Protons are thought to beaccelerated up to energies (cid:38) × eV, while nuclei with charge Z would reach anenergy Z times larger, namely a factor of 26 for Iron nuclei. In this picture the knee would naturally result from the superposition of the energy cut-offs of the differ-ent nuclear species. Radio, X-ray and γ − ray observations confirmed SNR shocks asacceleration sites of CRs, however it is still unclear whether they can accelerateCRs up to PeV energies (see Refs. 58, 59 and references therein).After leaving their sources, CRs are thought undergo diffusive propagation inthe turbulent Galactic magnetic filed. Magnetic inhomogeneities can scatter CRsand produce their spatial diffusion. The scattering occurs on scales of the Larmorradius r L of the particles and is likely of resonant nature, namely CR with a givenanuary 7, 2021 1:39 WSPC/INSTRUCTION FILE main Instructions for Typing Manuscripts (Paper’s Title) (courtesy Tom Gaisser). larmor radius are scattered by plasma waves whose wave number k satisfies therelation k ∼ /r L .
64, 65
The decrease of the Boron/Carbon ratio with energy canbe accounted for by a diffusion coefficient that increases with energy. The spectralhardening at ∼
230 GV may be due to a change in the type of turbulence relevantanuary 7, 2021 1:39 WSPC/INSTRUCTION FILE main Authors’ Names
Fig. 2. Boron over Carbon ratio as a function of the kinetic energy per nucleon. for CR scattering at ∼ −
300 GV,
66, 67 but other possible interpretations havebeen proposed (see Ref. ).The origin of CRs beyond the ankle is probably extragalactic, since the Larmorradius of such high energy particles in the typical Galactic magnetic field is compa-rable to the size of the Galaxy. On the other hand, the energy of transition betweenGalactic and extragalactic CRs is still largely debated.
58, 59
Cosmic ray propagation: building blocks
The propagation of CRs in the Galaxy is an intricate, non-linear, phenomenonwhich depends on several processes that are often not understood. The details ofCR propagation are intimately connected with the formation of CR-driven winds.CRs are thought to undergo a diffusive motion during their escape from theGalaxy due to the scattering on magnetic waves, as suggested by the measuredabundances of secondary nuclei and unstable isotopes and by the observed smalllevel of anisotropy. In addition, CRs may be advected out of the Galaxy due to theplasma waves themselves and to large scale outflows, such as GWs. Finally, otherprocesses, such as ionization and Coulomb losses, bremsstrahlung, synchrotronand inverse Compton, turbulent reaccelaration, nuclear fragmentation and radioac-tive decay, proton-proton interactions, may affect the transport, depending on theCR energy and species. All these processes depend on the specific conditions of the ISM, such as thedensity, temperature and degree of ionization of the background medium, the mag-nitude and the level of turbulence of the magnetic field, the background photonfield, and, in the case of GWs, also on the Galactic gravitational potential. Suchconditions can vary from point to point in the Galaxy and CRs can play an activeanuary 7, 2021 1:39 WSPC/INSTRUCTION FILE main
Instructions for Typing Manuscripts (Paper’s Title) role in determining the propagation parameters. The last aspect is particularly relevant in the case of CR-driven GWs. In fact,the CR density gradient that exists between the Galactic disk, where the sources ofthe bulk of CRs are probably located, and the halo, together with the fact that CRsare observed to be roughly in equipartition with the thermal plasma and with thebackground magnetic field, suggest that CRs may play a relevant dynamical role.It is known that spatial gradients in the CR density can lead to the excitationof Alfv´en waves, at the CR Larmor radius scale, through the so called streaminginstability ,
34, 69 which propagates along the magnetic field lines in the direction ofthe decreasing CR density. The scattering of CRs on such waves affects their dif-fusive propagation and contribute to determine the diffusion coefficient, togetherwith other sources of magnetic turbulence. The level of plasma waves is limited dueto different damping processes, such as the turbulent cascading,
70, 71 the non linearLandau damping,
72, 73 the turbulent damping
74, 75 and the ion-neutral damping, depending on the conditions of the background medium. A larger CR gradient leadsto a higher level of turbulence, thus to a stronger CR scattering and to a smallerdiffusion coefficient. Moreover, in the case of self-generated Alfv´en waves, that movein the direction opposite to the CR gradient, CRs experience also convection withsuch waves at roughly the Alfv´en speed. It has been shown
66, 77 that streaming in-stability of Alfv´en waves may dominate the CR propagation up to ∼
200 GeV, whileat higher energies other sources of turbulence are probably at work. The nature ofsuch additional types of turbulence is still debated. In fact, for a long time Alfv´enicturbulence injected on large scale (10 −
100 pc) by astrophysical sources, such asSNe, and cascading to the resonant scales, was thought to be responsible for the CRscattering. However, subsequent studies showed that Alfv´enic turbulence cascadingto small scales becomes anisotropic and very inefficient at scattering CRs.
Thisproblem does not affect self-generated Alfv´en waves, which are already producedat the resonant scale. Recently it has been suggested that compressive MHD wavescould provide a relevant contribution to CR scattering.
75, 81, 82
In most cases relevant for galactic propagation, the level of magnetic turbulenceon scales resonant with the Larmor radius of CRs is (cid:104) δ B (cid:105) / B (cid:28)
60, 65
This implies that CRs dif-fusion is anisotropic, with particles diffusing preferentially along rather than acrossthe magnetic field lines, which translates in a parallel and perpendicular diffusioncoefficient D (cid:107) (cid:29) D ⊥ . In this case, the CR transport perpendicular to the meanbackground field may be provided by the wandering of field lines.
64, 83, 84
The coupling established by CRs and the ISM by scattering and the presence ofa CR gradient between the disk and the halo, imply that CRs exert a force on thebackground plasma directed toward the Galactic halo. If this force is large enough asto win against the gravitational force, a GW may be launched. The hydrodynamicsof such a wind is determined by CRs. On the other hand, the advective transportanuary 7, 2021 1:39 WSPC/INSTRUCTION FILE main Authors’ Names of CRs is influenced by the presence of a wind.Since CR protons are the most abundant species, all these phenomena, thatdepend on the CR density gradient, are mostly determined by protons. Moreover,the wind launching depends on the CR pressure, which is dominated by 1-10 GeVprotons.3.1.1.
Basic model of proton transport
The propagation phenomenology described above is very rich and complicated. How-ever it is possible to understand some basic aspects of CR propagation relevant inthe case CR-driven winds, with the help of a simplified stationary model of protontransport. In such basic model the CR propagation is assumed to be 1D and to oc-cur only perpendicular to the Galactic disk and solely along a constant backgroundmagnetic field. The diffusion coefficient is assumed to be spatially homogeneous andis treated as a fitting parameter. The specific nature of the turbulence responsiblefor the CR scattering is ignored. The injection of CRs occurs in the Galactic disk,assumed to be infinitely thin, while also the advection velocity (assumed to be con-stant) and the size of the propagation region are preassigned. Other possible effects,such as ionization and Coulomb losses, are neglected.In this scenario the stationary CR advection-diffusion equation reads − ∂∂z (cid:20) D ∂f∂z (cid:21) + u ∂f∂z − d ud z p ∂f∂p = Q ( z, p ) , (1)where f ( z, p ) is the CR distribution function, which is function of the distancefrom the Galactic disk z and of the particle momentum p . Under the followingassumptions this equation is easily solved: the diffusion coefficient D ( p ) ∝ p δ isa power-law in momentum, the injection occurs in the disk Q ( z, p ) = Q ( p ) δ ( z )with a power law injection spectrum Q ( p ) ∝ p − γ (as expected for DSA ), theadvection velocity u is constant and directed away from the disk ( d ud z = 2 uδ ( z )),and CRs freely escape at | z | = H, namely f ( ± H, p ) = 0. The free escape boundaryH, namely the size of the propagation region, if fixed, and it must be not infinity inorder to guarantee the existence of a stationary solution.A standard way to solve Eq. 1 is to integrate it around the disk, between z = 0 − and z = 0 + , and between z = 0 + and z , with the free-escape boundary condition.The solution reads f ( z, p ) = f ( p ) 1 − e − ζ ( − | z | H )1 − e − ζ (2) f ( p ) = 32 u (cid:90) ∞ p dp (cid:48) p (cid:48) Q ( p ) exp (cid:34)(cid:90) p (cid:48) p d p (cid:48)(cid:48) p (cid:48)(cid:48) λ ( p (cid:48)(cid:48) ) (cid:35) , (3)where ζ ( p ) = u HD ( p ) and λ ( p ) = 1 − e ζ ( p ) .For particle momenta for which advection dominates, namely u HD ( p ) (cid:29)
1, the CRanuary 7, 2021 1:39 WSPC/INSTRUCTION FILE main
Instructions for Typing Manuscripts (Paper’s Title) spectrum becomes f adv ( p ) = Q ( p )2 u ∼ p − γ . (4)Thus, in this regime the equilibrium spectrum has the same slope of the injectionspectrum.When diffusion dominates, namely u HD ( p ) (cid:28)
1, the CR spectrum becomes f diff ( p ) = Q ( p ) H D ( p ) ∼ p − γ − δ (5)and the equilibrium spectrum is steeper than the injection spectrum due to the en-ergy dependent diffusive propagation. Since, as inferred, for instance, from the theBoron/Carbon ratio, the diffusion coefficient is an increasing function of momentum,advection(diffusion) dominates for low(high) enough particle momenta. Typical val-ues used to fit CR data are: slope of the injection spectrum γ ∼ . − .
4, slope ofthe diffusion coefficient δ ∼ . − . ∼ × cm /s at ∼ H ∼ ∼
20 km/s.
58, 67
It is interesting to notice that in a GW scenario the size of the diffusive regionemerges naturally, without the need of imposing a free-escape boundary. This is dueto the fact that, as we will see in the remaining of this review, in a stationary GWthe wind velocity increases with the distance from the disk and eventually tends toa constant value. This means that, for a particle of momentum p , advection startsto dominates over diffusion at a distance from the disk, s ∗ ( p ), given by s ∗ ( p ) D ( s ∗ , p ) = s ∗ ( p ) U ( s ∗ ) . (6)Since D generally increases with momentum, the size of the diffusion region alsoincreases with momentum. Above s ∗ ( p ∗ ) particles of momentum p ∗ are basicallyadvected out of the Galaxy. All this may have relevant consequences on the CRspectrum, anisotropy and production of secondaries.
36, 37
4. Cosmic ray driven winds: cosmic ray transport
The inclusion of the CR transport in models of GWs is made extremely difficult byits non linear nature and by the fact that the scales relevant for the CR propaga-tion (Larmor radius of the particles) are much smaller than the those relevant forthe wind ( (cid:38) kpc). For this reason, in most wind models and simulations CRs aretreated as a fluid. The first attempt to account for the effects of a stationary CRdriven wind on the CR transport was by Ref. 36, where the authors used a simplifiedmodel of the wind structure (as inferred from hydrodynamic calculations) in orderto deduce some general implications for the observed CR spectrum. More recentlyRef. 37. presented a semi-analytic, self-consistent calculation, of the wind formationtogether with the propagation of CR protons in the wind. The work is restrictedanuary 7, 2021 1:39 WSPC/INSTRUCTION FILE main Authors’ Names to the stationary regime, and includes the self-generation of Alfv´en waves by CRstreaming instability, the damping of such waves and the resulting gas heating, theCR scattering parallel to the field lines on the self-generated waves, and the CRadvection with such waves and with the wind. The CR pressure contributes to thewind formation and is dominated by CR protons, which are the most abundant CRspecies. The authors also performed a parametric study of the wind launching andCR distribution in different galactic environments. Based on such studies, here we briefly summarize the main features of the CR trans-port in a CR-driven wind that have been explored in the literature, and how theyare linked to the wind hydrodynamics, which is described in the next section. Pos-sible future improvements of the present models are illustrated in the conclusions.In the assumption that the CR transport is dominated by the scattering on self-generated Alfv´en waves and in the presence of a GW, the CR diffusion-advectionequation reads ∂f∂t + ( (cid:126)u + (cid:126)v A ) · (cid:126) ∇ f = (cid:126) ∇ · (cid:104) D (cid:126) ∇ f (cid:105) + (cid:104) (cid:126) ∇ · ( (cid:126)u + (cid:126)v A ) (cid:105) p ∂f∂p + Q c (7)where f ( p, (cid:126)r ) is the CR distribution function and D ( p, (cid:126)r ) the CR diffusion coefficient,while (cid:126)u ( (cid:126)r ) and (cid:126)v A ( (cid:126)r ) are the background plasma velocity (the wind velocity in thiscontext) and the Alfv´en velocity. In the case of self-generation, Alfv´en waves movepreferentially along the magnetic field lines and in the direction of the decreasingCR density, so that CRs will be advected at a speed which is roughly (cid:126)u + (cid:126)v A . Inthe case of an extrinsic turbulence, waves would move along the field lines in bothdirections and CRs would advect at (cid:126)u . The CR diffusion coefficient is given by the Alfv´en wave energy density W ( k, (cid:126)r ),determined by the CR streaming instability and by the possible sources of wavedamping. In the context of galactic propagation the level of turbulence is weak atthe scale resonant with CRs and the diffusion coefficient can be expressed usingquasi-linear theory
60, 65 : D ( (cid:126)r, p ) = 13 v ( p ) r L ( (cid:126)r, p ) k res W ( k res , (cid:126)r ) (cid:12)(cid:12)(cid:12)(cid:12) k res =1 /r L , (8)where v and r L are the particle velocity and Larmor radius, and the wave energydensity is calculated at the resonant wavenumber k res = 1 /r L .The Alfv´en wave transport equation, taking into account the velocity of the back-ground medium, wave generation by CR streaming instability and the relevantdamping processes, is given by ∂W∂t + (cid:126) ∇ · (cid:20)(cid:18) (cid:126)u + (cid:126)v A (cid:19) W (cid:21) = 12 (cid:126)u · (cid:126) ∇ W − [Γ CR + Γ damp ] W (9)where Γ CR is the wave growth rate due to CR streaming instability (the δ − functionanuary 7, 2021 1:39 WSPC/INSTRUCTION FILE main Instructions for Typing Manuscripts (Paper’s Title) encloses the resonance condition):Γ CR = 16 π (cid:126)v A kW B · (cid:90) ∞ dpp v ( p ) δ (cid:18) p − qBkc (cid:19) (cid:126) ∇ f, (10)and Γ damp is the wave damping rate.The damping of waves depends on the conditions of the background plasma and onthe wave density itself. In a hot ionized medium the most relevant processes are theturbulent cascade, the non linear Landau damping and the turbulent damping (seeSec. 3.1).The turbulent cascade of waves to smaller scales is a nonlinear process that canbe described as a diffusion in wavenumber space, with a diffusion coefficient givenby
70, 71 D kk = 5 . × − v A k / W / . (11)At small enough scales waves will eventually dissipate and heat the gas. The typicalrate for this process is Γ casc ∼ D kk /k .Non linear Landau damping is due the interaction between the beat of two Alfv´enwaves and the thermal ions, of temperature T , in the background plasma. Thedamping rate is given by
72, 73 Γ NLL = 12 (cid:115) π k Bol Tm p k Wr L . (12)Turbulent damping is due to the interaction of the self-generated Alfv´en waveswith the background turbulence. Such turbulence can be injected by astrophysicalsources at scales, L inj , much larger than that of the Larmor radius of CR particles,and with a turbulent velocity v T . For waves resonant with CRs with a given Larmorradius, the damping rate is given by
74, 75 Γ turb = (cid:18) v T /L inj r L v A (cid:19) / . (13)In a cold/warm partially ionized medium, a most relevant damping process is theion-neutral damping.
33, 76, 86
For the typical conditions found in the disk of the MW,this process is very effective, so that the damping timescale for waves resonant withCRs of momentum (cid:46)
100 GeV is of the order of tens of years. This means that,probably, the bulk of CRs stream very fast in the disk and is weakly coupled tothe ISM. In the halo, instead, the fraction of neutrals is negligible and CRs can bescattered. The galactic halo is probably the actual diffusion region. In this scenario,the transport of CRs in the disk region is unclear.
87, 88
On the other hand, anotherpossibility is that the neutral phases are mostly confined in clumps/clouds, whilemost of the disk volume is filled with warm/hot ionized gas, so that CR diffusioncould work also in the near-disk region. This aspect is very relevant for the CR-driven wind formation, since the coupling between CRs and the ISM provided bythe CR scattering on plasma waves, is necessary to launch winds.
14, 37 anuary 7, 2021 1:39 WSPC/INSTRUCTION FILE main Authors’ Names
Starting from the CR and wave distribution functions one can define the cor-responding hydrodynamic energy density and pressure that determine the windhydrodynamics: (cid:15) c ( (cid:126)r ) = (cid:90) ∞ dp πp T ( p ) f ( p, (cid:126)r ) , (14) P c ( (cid:126)r ) = (cid:90) ∞ dp π p pv ( p ) f ( p, (cid:126)r ) (15) (cid:15) w ( (cid:126)r ) = (cid:90) ∞ k B π W ( k, (cid:126)r ) dk (16) P w ( (cid:126)r ) = (cid:15) w / , (17)where T ( p ) = (cid:112) p c + ( mc ) − mc is the particle kinetic energy.
5. Cosmic ray driven winds: hydrodynamics
The hydrodynamics of CR-driven winds is described by the equations for the con-servation of mass, momentum and energy for the gas, by the evolution of the back-ground magnetic field and by the equations for the CR and magnetic wave pressure.The last are derived by appropriate integrations, respectively in momentum and inwave number, of the transport equation of CR and of the waves (see Sec. 4). Theequations read ∂ρ∂t + (cid:126) ∇ · ( ρ(cid:126)u ) = q (18) ∂ρ(cid:126)u∂t + (cid:126) ∇ · (cid:32) ρ(cid:126)u(cid:126)u + P tot I − ρ (cid:126)B (cid:126)B π (cid:33) = ρ(cid:126)g + (cid:126)m (19) ∂e tot ∂t + (cid:126) ∇ · (cid:126)S = ρ(cid:126)u · ( (cid:126)g + (cid:126)m ) + (cid:15) (20) ∂ (cid:126)B∂t = ∇ × (cid:16) (cid:126)u × (cid:126)B (cid:17) (21) (cid:126) ∇ · (cid:126)B = 0 (22) ∂∂t (cid:18) P c γ c − (cid:19) + (cid:126) ∇ (cid:20) γ c γ c − (cid:126)u + (cid:126)v A ) P c − ¯ Dγ c − (cid:126) ∇ P c (cid:21) = ( (cid:126)u + (cid:126)v A ) · (cid:126) ∇ P c + Q c (23) ∂e w ∂t + (cid:126) ∇ (cid:20)(cid:18) (cid:126)u + (cid:126)v A (cid:19) e w (cid:21) = (cid:126)u · (cid:126) ∇ P w − (cid:126)v A · (cid:126) ∇ P c + Q w , (24)anuary 7, 2021 1:39 WSPC/INSTRUCTION FILE main Instructions for Typing Manuscripts (Paper’s Title) where P tot = P g + P c + P w + B π (25) e tot = 12 ρu + P g γ g − P c γ c − e w + B π (26) (cid:126)S = (cid:18) u + γ g γ g − P g ρ (cid:19) ρ(cid:126)u + 1 γ c − (cid:104) γ c ( (cid:126)u + (cid:126)v A ) P c − ¯ D (cid:126) ∇ P c (cid:105) (27)+ (cid:18) (cid:126)u + (cid:126)v A (cid:19) e w + (cid:126)E × (cid:126)B π . Here ρ , (cid:126)u , P g and γ g are the gas density, velocity, pressure and adiabatic index,respectively. P c , γ c and ¯ D are the CR pressure, adiabatic index and momentumaveraged diffusion coefficient. (cid:126)B is the large scale background magnetic field and e w = 2 P w = (cid:104) δ (cid:126)B (cid:105) / π is the Alfv´en wave energy density, whose speed is (cid:126)v A ( (cid:126)r ) = (cid:126)B ( (cid:126)r ) (cid:112) πρ ( (cid:126)r ) . (28)These equations may also include possible additional sources and sinks of mass,momentum and energy for the wind, deriving, for instance, from supernovae explo-sions and stellar winds, radiative cooling and the damping of plasma waves. g ( (cid:126)r )represents the galactic gravitational acceleration.The hydrodynamic CR and wave equations can be derived from the correspond-ing kinetic transport equations by integration in momentum and wavenumber, re-spectively. The expression for the momentum averaged diffusion coefficient is givenby, ¯ D ( (cid:126)r ) = (cid:82) ∞ dp πp T ( p ) D ( p, (cid:126)r ) (cid:126) ∇ f (cid:82) ∞ dp πp T ( p ) (cid:126) ∇ f , (29)while the CR adiabatic index is defined as (cid:15) c = P c / ( γ c −
1) and γ c = 4 / /
3) forrelativistic(non relativistic) particles. It can be also shown that the hydrodynamiccounterpart of the streaming instability term in the wave transport equation, Γ CR W ,is given by (cid:126)v A · (cid:126) ∇ P c , which appears in the RHS of Eq. 24.In many cases relevant for CR propagation, the wave damping, also in the hotionized medium of galactic halos, is fast enough that waves produced by streaminginstability are practically damped locally and one can safely ignore the actual wavetransport. In this case, the level of magnetic turbulence resonant with CR parti-cles is determined by the balance between the growth and damping rate of waves,Γ CR = Γ damp . The resulting wave pressure is usually orders of magnitude smallerthan that of CRs and of the background plasma and its contribution to the windhydrodynamics can be neglected. However such turbulence is still necessary in or-der to scatter CRs and thus to couple them to the ISM. In addition, the dissipatedwaves heat the gas. This effect is taken into account by adding the the term (cid:126)v A · (cid:126) ∇ P c anuary 7, 2021 1:39 WSPC/INSTRUCTION FILE main Authors’ Names to the the equation for the gas energy density, since all the energy injected by CRsinto waves is transferred to the gas.The hydrodynamic of CR-driven winds has been widely investigated in the lit-erature, starting from the relatively computationally simpler and faster stationaryapproaches, to time-dependent calculations up complex, refined simula-tions.
6. Stationary cosmic ray-driven winds in the Milky Way
So far, the only self-consistent solutions of the coupled system of equations of theCR transport and of the CR-driven wind hydrodynamics, have been obtained solelyin the stationary regime and under some simplifying assumptions, for instance onthe large scale structure of the galactic magnetic field, and neglecting radiative cool-ing.
37, 38
Despite these limitations, the study of stationary winds provides preciousinformation on their properties and on their impact on the CR transport. Moreover,being much easier and faster than simulations and of more complex time-dependentapproaches, it allows to quickly explore wide parameter ranges.
16, 38
In this section we report a common model of stationary winds, illustrating themain physical ingredients, the relevant launching parameters in the MW and somepossible implications on the observed CR spectrum.
The geometry of the magnetic field lines
Radio observations show the presence of galactic magnetic fields, usually of theorder of µ G and with a coherence length that can exceed ∼ kpc, both in the MWand in other galaxies. Their origin is commonly attributed to complex dynamoprocesses (see Refs. 93, 50 and references therein). In the disk region, field lines aremainly parallel to the disk surface, while also X-shaped, out of disk components,have been detected. The activity from several classes of astrophysical sources,such SNe explosions of single stars and cluster of stars, and stellar winds, whichalso contribute to the formation of GWs, can push the field lines in the disk andmake them break through the halo. The pressure support provided by these eventsmay last 10 − yr, but the overlap of the effect of multiple such events, randomlytaking place in the galactic disk, may keep the field line opening up to distancesfrom the disk of the order of the disk radius and for time scales of (cid:38) − yr.Moreover, in the presence of GWs, field lines attached to the outflowing material canbe further transported outward. The rotation of the galaxy additionally contributeto shape the large scale magnetic field.
14, 17
In principle, the magnetic field geometry, which is also essential to CR propaga-tion, should be determined self-consistently with the wind structure, which repre-sents a big complication in models of CR-driven winds. However a good approxima-tion, adopted in several stationary approaches,
14, 16, 37 is to pre-assign the magneticflux tube shape in a way that mimics the geometries found in more complex simula-tions, giving its area as a function of the distance from the disk. With such approachanuary 7, 2021 1:39 WSPC/INSTRUCTION FILE main
Instructions for Typing Manuscripts (Paper’s Title) the equations also becomes one-dimensional and only depend on the coordinateperpendicular to the galactic disk. A typically adopted geometry, that intuitivelyencompass the ingredients illustrated above and confirmed by simulations, is themushroom-like shape in which the magnetic flux tube is roughly cylindrical up to agiven distance from the disk (comparable with the disk radius) and then opens upin a roughly cylindrical way. Stationary equations in the given flux tube geometry
In the stationary regime and assuming a given flux-tube geometry, as explainedabove, the equations greatly simplify and become one-dimensional.
14, 37 Be z thecoordinate perpendicular to the disk, and A ( z ) the area of the mushroom-like fluxtube A ( z ) = A (cid:20) (cid:18) zZ b (cid:19) α (cid:21) , (30)where Z b ∼ disk radius and α = 2 for spherical opening.The hydrodynamic equations and the CR transport equation become
14, 37 ρuA = const , (31) AB = const , (32) dudz = u c ∗ A dAdz − d Φ dz u − c ∗ , (33) dP g dz = γ g P g ρ dρdz − ( γ g − v A u dP c dz (34) dP c dz = γ eff P c ρ u + v A u + v A ) dρdz , (35) c ∗ = γ g P g ρ + γ eff P c ρ (cid:104) − ( γ g − v A u (cid:105) u + v A u + v A ) , (36) γ eff γ eff − γ c γ c − − D ( γ c − u + v A ) P c dP c dz (37) u γ g γ g − P g ρ + Φ + γ eff γ eff − P c ρ u + v A u = const (38)and ∂∂z (cid:20) A D ∂f∂z (cid:21) − A ( u + v A ) ∂f∂z + d A ( u + v A ) dz ∂f∂ ln p + A Q = 0 , (39)where Φ( z ) is the galactic gravitational potential. It has been assumed that Alfv´enwaves produced by CR streaming instability are damped locally and heat thegas.
16, 37 Q ( z, p ) represents the CR injection, which, assuming that the sources aremainly SNRs, can be approximated as a power-law in momentum and to be con-centrated in a thin disk Q ( z, p ) ≈ Q ( p ) δ ( z ). Moreover, Q ∝ ξ CR R SN , where ξ CR is the CR acceleration efficiency and R SN is the SN explosion rate.anuary 7, 2021 1:39 WSPC/INSTRUCTION FILE main Authors’ Names
Eq. 33 is the so-called wind equation , which represents the acceleration of thewind fluid. It contains the generalized sound speed c ∗ , which includes the soundspeed, (cid:112) γ g P g /ρ , modified by the CR pressure through a ”CR sound speed” thatdepends on the Alfv´enic Mach number u/v A . It the case in which Alfv´en waves arenot damped very quickly, one would have an equation for waves and a ”wave soundspeed” in the definition of c ∗ . γ eff is an effective adiabatic index that takes into ac-count the CR diffusivity. It has been shown that in most cases γ eff ∼ . − .
2, while γ c ∼ / The wave heating is encompassed by the term − ( γ g − v A /u dP c /dz in Eq. 34.The wind equation 33 presents different classes of solutions depending onwhether its numerator or its denominator, or both, vanish
13, 14 at a given posi-tion z . The location where both vanish is called the sonic point and it correspondsto the position where the flow velocity equals the compound sound speed, u = c ∗ .The solution relevant for GWs is the one for which the flow is subsonic near thegalactic disk, at the position z where the boundary conditions are specified, then itis accelerated outward and smoothly transit, at the sonic point z c , to the supersonicregime. At large enough distance from the disk, all pressures tend to zero and thewind reaches its terminal velocity, u f . The boundary conditions to be given at z are, for instance, the gas density and temperature, the CR pressure and the valueof the background magnetic field, but not the launching velocity u . In fact, thelaunching velocity is determined by imposing the passage through the sonic point.Since all other quantities are fixed at z , this also fixes the mass and energy fluxes.For launching velocities smaller than u the flow never becomes supersonic andthere is a point where the numerator of the wind equation vanishes. These solutionsare called breezes and are characterized by a non vanishing gas pressure at infinity.For launching velocities larger than u one may either have unphysical solutions,where only the denominator vanishes at some point, or purely supersonic flows.The wind equation is analogous to that of the Parker model for the Solarwind
13, 95 and to that of the De Laval nozzle. In fact, if one defines a ( z ) ≡ A dAdz and g ( z ) ≡ d Φ dz in the wind equation, the last becomes formally identical to that ofthe De Laval nozzle1 u dudz = a ( z ) − g ( z ) c ∗ M − u dudz = a ( z ) M − M = u/c ∗ is the Mach number and a ( z ) ≡ A dAdz for a nozzle of area A ( z ),as shown in Fig. 3. The nozzles is composed by a duct that presents a convergingsection followed by a diverging section. The choke point corresponds to a ( z ) = 0.The geometry of the nozzle allows a smooth transition from the subsonic to theanuary 7, 2021 1:39 WSPC/INSTRUCTION FILE main Instructions for Typing Manuscripts (Paper’s Title) supersonic regime. The flow begins as subsonic in the converging portion, with a ( z ) < M <
1, so that du/dz >
0. At the throat a ( z ) = 0 and M = 1 andthe flow becomes transonic. Note that only in this way du/dz can remain smooth.In the diverging portion a ( z ) > M > du/dz remains positive andthe flow continues to be accelerated (at least until a shock appears).In the case of a wind the numerator a ( z ) − g ( z ) c ∗ plays the role of the duct area in aDe Laval nozzle. In order to keep accelerating the fluid away from the disk: • u < c ∗ (subsonic flow)the gravitational term g ( z ) has to dominate. It behaves as a sort of ”con-verging duct” term, • u > c ∗ (supersonic flow)the area term a ( z ) has to dominate. It behaves as a sort of ”divergingduct” term.This analysis illustrates that if the area of the flux tube is constant, a ( z ) = 0,a stationary wind solution is not possible. Moreover, in an expanding flux tube, a ( z ) >
0, as it happens in the case of a wind, the gravity allows for a stationarysolution.anuary 7, 2021 1:39 WSPC/INSTRUCTION FILE main Authors’ Names
The role of cosmic rays and the cosmic ray spectrum
The possibility to launch a (stationary) wind and its properties strongly dependson the conditions, such as the gas density and temperature, the CR density and themagnetic field at the launching point near the galactic disk. Moreover it also dependson the galactic gravitational potential (and in particular on the Dark Matter halo)and on the geometry of the field lines. As discussed above, the spatial dependentCR spectrum affects the wind launching and is in turn affected by the presence of awind. All these conditions depend on the type of galaxy and on the specific locationin a galaxy.A parametric study of the wind launching is beyond the scope of this review, andthe reader is referred to Refs. 16 and 38, which performed a thorough investigationin the case of the MW. Moreover, Refs. 37 studied how the CR spectrum in the diskgets modified when the launching conditions change and how this compares to theobserved CR spectrum at Earth. Here we briefly summarize the main findings ofthese studies, focusing specifically on the case of MW and on winds launched neardisk region at the galactocentric distance of the Sun.6.3.1.
Wind hydrodynamics
In general, stationary winds can exist only for closed intervals of the launching pa-rameters. In the model presented here, the relevant quantities near the disk are theISM density, n , and temperature, T , the CR pressure, P c , and the magnetic field, B .Winds probably originate from the less dense and hotter phase of the ISM,namely from the HIM,
14, 16 which means for the MW, n ≈ − − − cm − and T ≈ K. The CR pressure is P c ∼ × − erg / cm . The galactic magneticfield is observed to be of the order of few µ G. As for the flux tube area, Z b ≈ α ≈ .
14, 38
The gravitational potential of the MW is mainly contributed by the bulge, the diskand the DM halo. The last is especially relevant for GWs, since it extends for hun-dred kpc from the disk and affects the wind structure up to such large distances. Wave damping is also a relevant ingredient in determining the possibility to launchwinds.
14, 16, 37
In fact, in the absence of damping, Alfv´en waves produced by CRstreaming instability can grow undisturbed, such that their pressure becomes com-parable to that of CRs and contribute to push the gas. This would make the windlaunching easier, also in the case of denser and cooler gas. In the presence of wavedamping the parameter space is additionally reduced by the fact that wave heatingreduces the gas pressure gradient along z , and if it is too strong it may even stallthe outflow. It as been shown that, with launching parameters close to those mentionedabove, stationary winds can exist. An example is reported in Fig. 4, 5 and 6, wherethe chosen launching parameters are listed in Table 1. The resulting Alfv´en speedis v A ∼
28 km/s, the wind launching velocity is u ∼ Instructions for Typing Manuscripts (Paper’s Title) ∼
200 GeV. The re-sulting Alfv´en and wind speeds at the launching point are v A ∼
28 km/s and u ∼ u f ∼
404 km/s, while the mass lossrate per unit area is ˙ m ≈ . × − M (cid:12) kpc − yr − .Parameter Symbol Valuegas density n × − gas temperature T × Kmagnetic field B µG magnetic flux tube scale height Z b
15 kpcmagnetic flux tube opening index α P c × − final velocity is u f ∼
404 km/s, while the mass loss rate per unit area is ˙ m ≈ . × − M (cid:12) kpc − yr − .This set of launching parameters are compatible with the observed conditions inthe local Galactic environment and lead to a reasonably good fit of the observedCR proton spectrum below ∼
200 GeV, as illustrated in what follows.Near the disk, the gas pressure dominates over the CR pressure and the flowis sub-Alfv´enic. A sizable fraction of the CR momentum is employed to produceAlfv´en waves, whose fast damping contribute to keep the gas hot up ≈
100 kpcand makes the gas temperature decrease with z much slower than the gas densityabove ∼
10 kpc. On the other hand, CRs have a smaller adiabatic index than thegas ( γ c = 4 / γ eff ≈ . − . γ g = 5 /
3) so that the CR pressure decreases slowerwith z than the gas pressure. This has prominent consequences on the wind launch-ing: the thermal pressure is effective at driving winds near the disk, but at larger z CRs continue pushing the gas, often making an outflow possible also in situationswhere the thermal pressure would not have been sufficient. It is interesting to notice that the mass loss rate per unit area reported above˙ m ≈ . × − M (cid:12) kpc − yr − , if present on the whole Galactic disk, would leadto a mass loss rate of ˙ M ≈ .
36 M (cid:12) yr − , which is comparable with the Galacticstar formation rate ( ∼ M (cid:12) /yr; see e.g Ref. 96). This confirms the importance ofCR-driven winds in MW-like galaxies.For values of the launching parameters much larger or smaller than the onequoted above, winds might not exist or there could be some type of outflow butit would not be stationary. For instance, if the ISM is too dense and/or cool, theavailable energy could not be enough to drive an outflow, while for very hot anddilute gas the outflow may resemble more to an evaporation.In general, an increase of the gas density leads to winds that have a smaller launchingvelocity and a smaller final velocity, and overall, a smaller mass loss rate. An increaseanuary 7, 2021 1:39 WSPC/INSTRUCTION FILE main Authors’ Names u [ c m / s ] z [kpc] u wind c s v A Fig. 4. Example model: wind velocity u wind , Alfv´en velocity v A and compound sound speed c ∗ ,as function of the distance from the disk. The sonic point is where u wind crosses c ∗ . of the gas temperature and of the CR pressure lead to an increase of u , and, withfixed n , also to an increase of the mass loss rate. However, multiplying T and P c by the same factor, the resulting increase in u is larger in the case of an increaseof T . This is due to the fact that momentum and energy deposited before thesonic point can affect both u (and ˙ m ) and u f , while beyond the sonic point theyaffect only u f , since u is determined by the passage through the critical point. Asillustrated above, the larger adiabatic index of the gas compared to CRs implies thatthe gas pressure get ”exhausted” closer to the disk, while the CR pressure continuesto be effective also at larger distances. Thus, the thermal pressure is generally moreefficient at increasing the u (and ˙ m ). Moreover, the effect of the CR pressure alsodepends on whether the flow is sub-Alfv´enic or super-Alfv´enic. In fact, in these twolimits the CR pressure gradient becomes:
16, 38 lim u<
In the model illustrated here radiative cooling is not included. However it may bea key ingredient in shaping or hampering a wind and deserves some attention.For a hot ISM phase ( T ≈ K), the dominant cooling process is the emission ofanuary 7, 2021 1:39 WSPC/INSTRUCTION FILE main Authors’ Names -4 -3 -2 -1 -3 cm -3 ]T [10 K] Fig. 6. Example model: gas density n and temperature T , as function of the distance from thedisk. Due to wave heating, the gas temperature drops quite slowly with z . In fact the wind isroughly isothermal up to ∼
100 kpc from the disk. forbidden lines and soft X-rays, at a rate given by T d Tdt ∼ Λ( T ) n , (45)where the cooling function is Λ(T ∼ ) ∼ × − erg cm / s for Solar metallic-ity. The cooling rate increases with the gas density.Radiative cooling can be included in the stationary wind model, without affect-ing the topology of the possible solutions by modifying the equation of the gaspressure, Eq. 34, as dP g dz = γ g P g ρ dρdz − ( γ g − v A u dP c dz − ( γ g − T ) n u , (46)and the wind equation, Eq. 33, as1 u dudz = c ∗ a ( z ) − g ( z ) + ( γ g − T ) nu u − c ∗ . (47)Comparing the timescale for radiative cooling and wave heating for the case shownin Fig. 6, one gets τ cool ∼ yr and τ heat ∼ yr.
37, 38
In general, in the absenceof additional heating sources close to the disk, where the gas density is larger, ra-diative cooling might be so intense as to hamper the wind launching. A possibleanuary 7, 2021 1:39 WSPC/INSTRUCTION FILE main
Instructions for Typing Manuscripts (Paper’s Title) heating agent are SN explosions which heat and ionize the surrounding mediumand also inject magnetic turbulence that may eventually dissipate and additionallyheat the gas. Also ionization and Coulomb losses of CRs may represent an im-portant contribution to the gas heating. Finally, radiative cooling may limit the”stationarity” of GWs, making the outflowing material to fall back to the disk,
7, 8 and establishing a gas recycling in the galaxy.6.3.3.
Cosmic ray spectrum
When dealing with GWs launched at the galactocentric distance of the Sun, onehas to keep in mind that, if on the one side the wind is determined by the CR pres-sure, the last depends on the CR spectrum, which is affected by the wind and byself-generated waves in a highly non-linear way, and in the near-disk region shouldmatch the observed CR spectrum.When we speak about the ”observed CR spectrum”, it is important to stressthat the fluxes detected at Earth, for instance those reported by the AMS experi-ment are largely affected by the solar modulation at energies below ∼
10 GeV. On the other hand, the
Voyager I spacecraft has traveled far enough from the Sunthat the measured spectrum, which is reported down to ∼ On the other hand, the CR pressure is largelydominated by ∼ GeV CR protons, which represent the most abundant CR species(see Sec. 3). For this reason , the wind launching is mainly affected by CRs whoselocal measurement is affected by the solar activity. Thus it is important to comparethe computed spectrum in the wind both with the
Voyager I data and with AMSdata, provided that solar modulation is taken into account.Sub-GeV CRs may still affect the wind through the heating produced by ion-ization and Coulomb losses. The importance of low-energy CRs is not restrictedto that. In fact, such CRs are widely recognized as the main ionization agent ofthe interior of molecular clouds (see e.g Refs. 102, 103 and references therein), withimportant implications on the interstellar chemistry and on star formation (see e.g104). Moreover, they play a crucial role in the nucleosynthesis of light elements(lithium, beryllium, and boron).
The requirement that the calculated spectrum matches the observed spectrumprovides a stringent constraint on the wind parameters.
37, 38
In fact, in many cases,winds launched with parameters compatible with the local ISM lead to CR spectramuch different from the observed one. In most cases such spectra result harder thanobservations at low energy, below 100-1000 GeV, and much steeper than observa-tions at higher energies.
37, 38
In Refs. 37 and 38 it is shown that, assuming thatCR sources (say SNRs) inject a power-law spectrum in momentum, Q ( p ) ∝ p − γ ,with γ ≈ . − .
58, 67 the overall normalization of the injection spectrum can beanuary 7, 2021 1:39 WSPC/INSTRUCTION FILE main Authors’ Names chosen in two ways, by suitably adjusting the SNe explosion rate (in the case ofSNRs) and/or the CR acceleration efficiency: i) the calculated and observed protonspectra at Earth coincide at a specific particle momentum. Such momentum shouldbe taken between 10-20 GeV since below that energy the spectrum detected atEarth is affected by solar modulation; ii) the total CR pressure computed from theproton spectrum, using Eq. 16, matches the observed one, P c (cid:12) ∼ × − erg / cm .Notice that, since the CR pressure derives from an integral in momentum of the CRdistribution, different CR spectra may lead to the same pressure. Moreover, sincethe CR spectrum is steeper than ∝ p − , the specific spectral shape above ∼
10 GeVbarely affects the total CR pressure.With both prescriptions on the normalization of the CR injection spectrum thereis no guarantee at all that the computed spectrum matches the observed spectrumat all energies. This is mainly due to two factors which can be better understoodkeeping in mind the benchmark propagation model illustrated in Sec. 3 and againassuming a power-law CR injection spectrum as illustrated above. For particle mo-menta for which advection dominates the spectral slope of the calculated spectrumwill be close to the injection slope, while, where diffusion dominates, the spectralslope will be steeper. In the model presented in this section, the CR diffusion coeffi-cient is solely due to self-generated Alfv´en waves, whose growth rate, that dependson the CR density gradient, decreases with energy (the CR spectrum is steeper that ∝ p − ). This lead to a diffusion coefficient which sharply rises with energy, with aslope larger than the ∼ . − . as shown in Fig. 9. As illustrated above, winds are mainly launchedfrom relatively hot and dilute phases of the ISM. This leads to a CR total advectionvelocity, u + v A , close to the disk which can easily exceed ∼
100 km/s. Such valueincreases with z . The resulting CR transport will be dominated by advection up tohundreds GeV, resulting in a spectrum harder than observations at these energies.A smaller advection speed can be obtained, at fixed T , by considering launching pa-rameters still compatible with observations but with larger gas density and smaller B . This can be seen in Fig. 7, where we show the spectrum obtained with thelaunching parameters: n = 0 .
003 cm − and B = 2 µ G, which gives u + v A ∼ n = 0 .
006 cm − and B = 1 µ G, which gives u + v A ∼
30 km/s (thelast values have been used in Fig. 4, 5 and 6, and are reported in Table 1). In thecase of larger density and smaller B , the predicted spectrum matches reasonablywell observations, while with the other set of parameters the predicted spectrum istoo hard up to hundreds GeV. This is a rather general feature of CR-driven winds,in which launching parameters compatible with observations and for which a windcan exist, may lead to a CR spectrum qualitatively different from the observed one,even if the corresponding CR pressure is close to observations.For this reasons, the measurement of the CR spectrum at energies below few hun-dreds GeV represents a precious constraint on the wind parameters. This is surelyanuary 7, 2021 1:39 WSPC/INSTRUCTION FILE main Instructions for Typing Manuscripts (Paper’s Title) -1 -1 F l u x * E k . [ G e V . s - s r - m - ] Ek [GeV] AMS-02VOYAGERfit - solar mod.fit - no mod.large u wind external turbulence
Fig. 7. CR proton spectrum as a function of the particle kinetic energy in the near disk region, ascompared to observations. The green curve shows the spectrum corresponding to the wind modelof Fig. 4, with a total advection velocity near the disk of ∼
30 km/s. The red curve shows the samespectrum with the application of solar modulation. The blue curve shows a spectrum obtainedwith a total advection velocity near the disk of ∼
100 km/s, which is too hard to fit data. Themagenta curve shows qualitatively the effect of an additional, Kolmogorov-like, extrinsic turbulencethat scatter CRs. This leads to a much harder spectrum at energies (cid:38) − true for winds launched at the galactocentric distance of the Sun, where we havea direct measurement of the CR spectrum, but it is applicable also to the cases inwhich an indirect indication of the CR spectrum is available, for instance thanksto γ − ray observations. This is the case for CRs interacting with the ISM in theGalactic disk at different galactocentric distances, or for γ − ray observations inother galaxies. At higher CR energies ( E (cid:38) − D ∝ p . ), may be relevant at those energiesanuary 7, 2021 1:39 WSPC/INSTRUCTION FILE main Authors’ Names f ( p ) p . [ c g s ] p [GeV] diskz=10 kpcz=50 kpcz=100 kpc Fig. 8. CR distribution as function of the particle momentum p , for different distance from thedisk. The corresponding wind structure is shown in Fig. 4. and could even revert the spectral trend illustrated here and lead to a hardening. Such hardening is in fact measured in the CR spectrum (see Sec. 3). The possibleeffect of such an additional turbulence is qualitatively shown in Fig. 7.In Fig. 8 we show the CR distribution as a function of particle momentum p fordifferent distances from the disk. Low energy particles, below ∼
100 GeV, experiencea diffusive halo smaller than ∼
10 kpc. Above that distance they are advected out ofthe galaxy in a progressively larger flux tube. This results in a fast decrease of theCR density with z and in a progressively harder spectrum. Higher energy particlesexperience a much larger diffusive halo (see Fig.10). This leads to a smaller decreaseof the CR density with z . For such energies, the opening of the flux tube is partiallycompensated by a decrease of the diffusion coefficient at large z , also shown inFig. 9. This effect is the result of the non-linear nature of the streaming instabilitycombined with the decrease of the background magnetic field. If fact, if on theone side the CR density is small at high energies and slightly decreases with z ,the background magnetic field also decreases, so that (cid:104) δ B (cid:105) / B in Eq. 8 may becomelarger. This may in fact lead to an overall decrease of the diffusion coefficient, despitethe small amount of produced waves. anuary 7, 2021 1:39 WSPC/INSTRUCTION FILE main
Instructions for Typing Manuscripts (Paper’s Title) D [ c m / s ] p [GeV/c]diskz=10 kpcz=50 kpcz=100 kpcp p Fig. 9. CR diffusion coefficient as function of the particle momentum p , for different distancesfrom the disk. The corresponding wind structure is shown in Fig. 4. Notice the sharp momentumdependence of D ( p ), as compared to the ∝ p . − . usually assumed in standard CR propagationscenarios. This is a normal consequence of streaming instability, for which the level of wavesdepends on the CR density.
7. Conclusions and perspective
GWs represents a most important feedback process in the life of galaxies, and theirstudy constitutes an active research area in astrophysics, both from the observa-tional and theoretical point of view. GWs may be powered by several mechanisms,depending on the specific environment in which they get formed.The key role of CRs in launching GWs, especially in MW-like galaxies, has beenconfirmed by many hydrodynamic studies. The coupling between CRs and the ISM,necessary for CRs to effectively push on the background plasma, is mediated by thescattering of CRs off plasma waves at the very small scales (well below 1 pc, com-pared to the multi kpc scale of the wind structure) of the Larmor radius of the CRparticles in the galactic magnetic field. Such scattering is also responsible for thediffusive motion of CRs. The dynamics of CR-driven winds is a multiscale non-linearphenomenon intrinsically connected with the microphysics of the CR transport. Infact, the CR distribution responsible for the wind launching is often affected byCR-induced plasma instabilities, by the (non-linear) damping of plasma waves, andby the presence of the wind itself. Moreover, the details of the CR propagation indifferent galactic environments is also not well understood. For these reasons, CR-anuary 7, 2021 1:39 WSPC/INSTRUCTION FILE main Authors’ Names S * [ k p c ] p [GeV]effective diffusive halo size Fig. 10. Effective diffusive halo size as function of the particle momentum p . The correspondingwind structure is shown in Fig. 4. Particles of momentum p are basically advected out of the galaxyfor z > S ∗ ( p ). The effective diffusive halo size increases with energy, as shown in Sec. 3 driven winds are particularly difficult to study, and most of current models onlyfocuses on the wind hydrodynamics, treating CR as a relativistic gas and neglect-ing the kinematics of the CR propagation. So far, the only self-consistent modelingof the CR propagation together with the wind structure has been obtained solelyfor CR protons, in the stationary regime, and under the assumption that only self-generated waves contribute to diffusion and with a pre-assigned geometry of thelarge-scale magnetic field. Moreover possible relevant effects, such as radiative cool-ing, the effect of extrinsic sources of turbulence, and ionization and Coulomb lossesfor sub GeV CRs, have been neglected.Despite these limitations, all these studies allowed to investigate the wind structurein a variety of galactic environments and to drive some general conclusions on theCR spectrum expected in the presence of GWs. In particular it has been shownthat the measurement of the CR spectrum below few hundreds GeV represents animportant constraint on the wind structure, since winds launched with parameterscompatible with a given galactic environment generally lead to spectra differentfrom observations.Future improvements of the state of the art should focus on several aspects.First of all, it is of primary importance a better understanding of the sources andanuary 7, 2021 1:39 WSPC/INSTRUCTION FILE main Instructions for Typing Manuscripts (Paper’s Title) types of plasma turbulence that may scatter CRs, in addition to self-generatedAlfv´en waves, together with a deeper study of the possible damping processes.For instance, the plasma turbulence injected by sources in the disk and cascadingto smaller scales may represent a relevant contribution to CR scattering, whosetransport in the halo may be enhanced by GWs. This would require to solve alsothe wave transport equation. The relevance of such investigation is obviously notrestricted to CR-driven winds, and is central to the physics of CRs in general.Ionization and Coulomb losses are important in determining the shape of the CRspectrum below GeV and may represent and important source of heating in thenear-disk region, which may mitigate the possible effects of radiative cooling.
14, 37
Another important aspect is the transport of heavier nuclei in the wind, with possi-ble relevant implications for the production of secondaries, and thus on observablessuch as the B/C ratio. This would require the inclusion of spallation and nucleardecay terms in the equations.An other interesting possibility could be the reacceleration of CRs at the wind ter-mination shock, analogous to the CR acceleration at SNR shocks.
Second, of great importance is to treat the full, hydrodynamic and CR transportproblem, in time-dependent calculations. In fact, while in some interesting cases theassumption of stationarity may be justified, it is inadequate in describing interestingoutflows such as ”galactic fountains”, where the ejected material cools down andfalls back to the disk, or outbursts produced in active galaxies, possibly with thecontribution of CRs. Non stationary winds may originate from the Galactic centerregion, due e.g to a periodic activity of the central supermassive black hole, and pos-sibly connected with the
Fermi Bubbles . Moreover, in time-dependent winds, theformation of multiple shocks may additionally contribute to the CR reacceleration.Of great interest is also the role of CRs in shaping the large scale, rotating, galacticmagnetic field, which in turn affects the wind structure and the CR propagation. In general, the development of a fully time-dependent, self consistent, CR-drivenwind model, would allow to investigate such winds and the resulting CR distribu-tion in detail and in a large variety of environments, from MW-like galaxies, tostarburst and AGN galaxies.Third, GWs populate the galactic halos with hot plasma up to hundreds of kpcfrom the Galactic disk,
7, 8, 41 which act as a target for the interaction of CRs, withthe consequent production of γ − rays and neutrinos. This may lead to a relevantcontribution to the isotropic γ − ray background and to the diffuse neutrino fluxdetected by Ice-Cube. Such and extended γ − ray halo has indeed been detectedin the Andromeda galaxy. As a final remark, it is important to stress the possible prominent role playedby CRs in the evolution of galaxies. In this review we showed the effect of CRdriving in MW-like galaxies, where the mass loss induced by CR-driven winds cananuary 7, 2021 1:39 WSPC/INSTRUCTION FILE main Authors’ Names be comparable to the mass formed in stars. Despite the role of CRs in starburstand AGN galaxies is not that clear, it may also be relevant. This, together with theother possible effects of CRs on the galactic environments discussed above, showsthe importance of including CRs in models of galactic evolution.
Acknowledgments
The author acknowledge support from Agence Nationale de la Recherche (grantANR- 17-CE31-0014).
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