The Galactic Positron Annihilation Radiation & The Propagation of Positrons in the Interstellar Medium
aa r X i v : . [ a s t r o - ph ] A p r Draft version October 25, 2018
Preprint typeset using L A TEX style emulateapj v. 11/26/04
THE GALACTIC POSITRON ANNIHILATION RADIATION & THE PROPAGATION OF POSITRONSIN THE INTERSTELLAR MEDIUM
J. C. Higdon
W. M. Keck Science Center, Claremont Colleges, Claremont, CA 91711-5916and California Institute of Technology, Pasadena, CA 91125 andR. E. Lingenfelter and R. E. Rothschild Center for Astrophysics and Space Sciences, University of California San Diego, La Jolla, CA 92093
Draft version October 25, 2018
ABSTRACTThe ratio of the luminosity of diffuse 511 keV positron annihilation radiation, measured by INTE-GRAL in its four years, from a Galactic “positron bulge” ( < ∼ ∼ ∼ MeV positronsin the various phases of the interstellar medium is taken into consideration, since these relativisticpositrons must first slow down to energies ≤
10 eV before they can annihilate. Moreover, withoutpropagation, none of the proposed positron sources, new or old, can explain the two basic properties onthe Galactic annihilation radiation: the fraction of the annihilation that occurs through positroniumformation and the ratio of the broad/narrow components of the 511 keV line.In particular, we show that in the neutral phases of the interstellar medium, which fill most of thedisk ( > < ∼
80% of them escape down into the HII and HIenvelopes of molecular clouds that lie within 1.5 kpc before they slow down and annihilate, while theremaining ∼
20% escape out into the halo and the disk beyond. This propagation accounts for thelow observed annihilation radiation luminosity of the disk compared to the bulge.In addition, we show that the primary annihilation sites of the propagating positrons in both thebulge and the disk are in the warm ionized phases of the interstellar medium. Such annihilation canalso account for those two basic properties of the emission, the fraction ( ∼ ± ∼ ∼ ∼ . < R < . ∼ R < . Subject headings: (ISM:)cosmic rays — elementary particles — gamma rays: theory — ISM: general– nuclear reactions, nucleosynthesis — (stars:) supernovae: general INTRODUCTION
The discovery (Johnson et al. 1972; Leventhal et al.1979) of the diffuse Galactic positron annihilation radi-
Electronic address: [email protected] address: [email protected], [email protected] Sabbatical 2008-2009 ation at 511 keV from the inner Galaxy has led to ex-tensive studies of the possible origin of the annihilatingpositrons. These studies have shown (e.g. Ramaty & Lin-genfelter 1979; Kn¨odlseder et al. 2005) that, of all thevarious potential sources, the positrons from the decayof radioactive nuclei produced by explosive nucleosynthe-sis in supernovae are the largest and, hence, most likely Higdon, Lingenfelter and Rothschildsource.Recent measurements, however, have raised new ques-tions about the source. Analyses of the measurementsby the gamma-ray spectrometer (SPI) on ESA’s INTE-GRAL spacecraft of the distribution of the diffuse Galac-tic positron annihilation radiation have shown that theluminosity of a Galactic positron bulge within 1.5 kpcof the Galactic Center is ∼ ∼ z ∼
511 keV Line Bulge/Disk Luminosity Ratio
The best-fit analysis of the SPI/INTEGRAL firstyear’s data (Kn¨odlseder et al. 2005, Shells+D1) gave a511 keV line luminosity of L B ∼ (0 . ± . × pho-tons s − in a spherical bulge from 0 to 1.5 kpc and anold, thick disk luminosity of L D ∼ (0 . ± . × photons s − , scaled to a Solar distance of R o ∼ L B /L D ∼ . ± .
8. The subsequent best-fit analyses ofthe first two years’ data, which included more coverage ofthe disk (Weidenspointner et al. 2007, for 4 Nested Shellsand Old Disk), gave a bulge flux of ∼ (0 . ± . × − photons cm − s − in spherical shells out to 1.5 kpc, athick disk flux of ∼ (1 . ± . × − photons cm − s − , and a marginally detected spherical halo flux of ∼ (0 . ± . × − photons cm − s − from 1.5 to8 kpc. To infer the corresponding 511 keV line lumi-nosities, we use the flux to luminosity ratios from theirearlier modeling (Kn¨odlseder et al. 2005 Table 3), whichare essentially just 4 πR where R ∼ R o for the bulge,about 0.57 R o for the thick disk, and about 0.93 R o forthe halo. From the best-fit fluxes, we infer a 511 keVpositron bulge luminosity L B ∼ (0 . ± . × pho-tons s − in the spherical shells out to 1.5 kpc, and a thickdisk luminosity of L D ∼ (0 . ± . × photons s − for a L B /L D ∼ . ± . o , or 0.42 kpc) and wide (FWHM of 11 o ,or 1.5 kpc) spheroidal Gaussian distribution and thickdisk from more extensive observations that increased theeffective sampling distance to 0.75 R o for the disk. Theyfound best-fit fluxes of ∼ (0 . ± . × − photonscm − s − from the bulge and ∼ (0 . ± . × − photons cm − s − from the disk. Scaled to R o ∼ L B ∼ (0 . ± . × photons s − in the spher-ical bulge < L D ∼ (0 . ± . × photons s − . This gives a bulgeto disk 511 keV luminosity ratio of L B /L D ∼ . ± .
3. They also found a best-fit combined halo and bulge lu-minosity of ∼ (1 . ± . × photons s − for similarscaling, and subtracting the bulge luminosity, suggests aspherical halo luminosity of L H ∼ × photons s − beyond the bulge, for a total Galactic 511 keV luminosity L G ∼ (2 . ± . × photons s − .Best-fit SPI/INTEGRAL spectral analyses by Weiden-spointner et al. (2008b) also show an asymmetry in thepositron annihilation line flux from two opposing innerdisk components at 0 o to 50 o to either side of the Galac-tic bulge. These show an 80 ±
40% excess in the 511keV line flux at the negative longitudes compared to thepositive.Assuming simply that the positron annihilation andproduction rates are in local equilibrium, these obser-vations of the luminosities have been taken to imply asimilar bulge/disk ratio and asymmetric spatial distri-bution for Galactic positron production. This suggestedproduction ratio and distribution has challenged super-nova source models, since it is much larger than that ex-pected from the distribution and mean bulge/disk ratioof Galactic supernovae (e.g. Kn¨odlseder et al. 2005; Wei-denspointner et al. 2008b). As we show, however, thesespatial properties can be fully explained by positronpropagation.
511 keV Line Width & Positronium Fraction
We also consider the other two fundamental spectralproperties of the Galactic positron annihilation: the ra-tio of the broad to narrow 511 keV line emission and thefraction ( f P s ) of the annihilation that occurs via positro-nium (Ps) formation. As shown by Jean et al. (2006)neither of these can be explained by any of the proposedpositron sources, either old or new, without extensivepositron propagation, since all of these sources are ex-pected to produce most of the positrons in the hot ten-uous phases of the interstellar medium, which cannot betheir primary annihilation site. For this reason alone adetailed treatment is required of the production, propa-gation and annihilation in each phase and region of theGalaxy.Studies (e.g. Guessoum, et al. 1991; Guessoum, et al.2005) show that positron annihilation in different phasesof the interstellar medium occurs in differing ratios of di-rect annihilation on both free and bound electrons, to in-direct annihilation via positronium formation, dependingon the ionization fraction and temperature. Direct anni-hilation produces two photons at 511 keV, while positro-nium annihilation produces either two 511 keV photonsor a three photon continuum depending on the spin state.Positronium is formed 25% of the time in the singletstate, parapositronium, which annihilates with a meanlife of ∼ . × − s into two 511 keV photons, while75% of the time it forms in the triplet state, orthopositro-nium, which annihilates with a mean life of ∼ . × − s into a three photon continuum between 0 and 511 keV(e.g. Guessoum et al. 1991).The SPI/INTEGRAL measurements of the ratio of theGalactic 2 γ
511 line flux to that of the 3 γ continuum inthe bulge and disk shows that the bulk of the positronsannihilate via positronium (Ps) with f P s ∼ ± ∼ ±
9% (Weidenspointneret al. 2006), and ∼ ±
3% (Jean et al. 2006). Theweighted mean positronium fraction of ∼ ±
4% alsoalactic Positron Annihilation 3allows us to determine the total Galactic positron an-nihilation rates, A , in the bulge and disk, since A =( e + /γ ) L . From the above we see that ( e + /γ ) =[2( f P s /
4) + 2(1 − f P s )] − , which equals 1.69 ± A B ∼ (0 . ± . × e + s − and A D ∼ (0 . ± . × e + s − using the previ-ously derived L B and L D . In the hot tenuous plasmaof the halo, however, the positronium fraction dependson the refractory grain abundance, ranging from onlyabout 18% with narrow (FWHM ∼ ∼
11 keV) if the grains all disin-tegrated (Jean et al. 2006). Thus we assume a possi-ble range of ( e + /γ ) ∼ . ± .
07 there. From thebest-fit halo flux, we thus infer a halo annihilation rateof A H ∼ (0 . ± . × e + s − , implying a totalGalactic positron production rate of ∼ (2 . ± . × e + s − .The width of the 511 keV line also depends on the tem-perature of the medium and its state of ionization whicheffects the fraction of positronium formed by charge ex-change with H, H , and He by superthermal, ∼
10 eV,positrons as they slow down (Guessoum et al. 1991;Guessoum et al. 2005). The prompt annihilation of thisfast moving positronium produces a characteristic broad(FWHM ∼ < T ∼ K), tenuous ionized medium which would pro-duce a broader ( ∼
10 keV) line from the hot free elec-tions, can instead produce a narrow line if most of thepositrons annihilate on refractory dust grains (Guessoumet al. 1991; Guessoum et al. 2005). The dust, as itdoes with interstellar molecule formation, provides a 2-dimensional regime with much higher interaction ratesfor surface chemistry.The width of the Galactic bulge 511 keV line mea-sured by SPI/INTEGRAL from the first years’ data hasbeen recently fitted (Churazov et al. 2005; Jean et al.2006) by two components with about 67 ±
10% of theemission in a narrow line with a width of 1.3 ± ±
10% in a broad line with a widthof 5.4 ± ∼ Positron Production & Propagation
The large difference between the ratio of the 511 keVannihilation line bulge/disk luminosity and that of theexpected positron production by supernovae has led tosuggestions there is some new, unrecognized source ofpositrons in the Galactic bulge. This seems unlikely,however, since various recent reviews (e.g. Dermer &Murphy 2001; Kn¨odlseder et al. 2005; Guessoum, Jean &Prantzos 2006) of the potential Galactic positron sourcesall conclude that supernovae are still the most plausi-ble source and that other suggested sources, includingcosmic-ray interactions, novae, and various exotic pro- cesses, all seem to be weaker and much less certain.However, as we have shown (e.g. Guessoum, Ramaty& Lingenfelter 1991), the 511 keV annihilation emissiononly illuminates the annihilation sites, not the sources,of the positrons, since these positrons must first slowdown to energies ≤
10 eV before they can annihilate.Thus, we examine in detail the propagation and annihi-lation of the β + -decay positrons in the bulge, disk, andhalo, and we show here that the expected positron prop-agation and annihilation are very different in the bulgeand disk, and the cloud and inter-cloud environments.When these differences are taken into consideration, wefind that the spatial distribution, the 511 keV line widthsand the positronium fraction can all in fact be clearly un-derstood in the context of a Galactic supernova origin.Although positron propagation has previously beendiscussed, the arguments range from the simple sug-gestion (Prantzos 2006) that most of the disk positronsmight diffuse along dipolar field lines and annihilate inthe bulge, to Monte Carlo transport simulations (Gillardet al. 2007) suggesting that they travel less than a fewhundred parsecs in all but the hot phase. We thereforealso carefully examine both the physics of MeV electronpropagation and the observational evidence of such prop-agation from extensive measurements of Solar flare andJovian electrons in the heliosphere.Here, we examine in detail the mechanisms of the rela-tivistic positron propagation in the various phases of theinterstellar medium.Overall, we assume that the Galactic stellar and in-terstellar distributions are defined by two superimposedsystems: a stellar bulge-disk-and-halo system of starsand an interstellar bulge-disk-and-halo system of gas andplasma, as shown in Figure 1. The stellar bulge anddisk populations essentially determine the distributionof supernovae from the decay of whose radionuclei thepositrons are born. Following Ferri´ere et al. (2007), theinterstellar bulge, disk and halo define the distribution ofgas and plasma where the positrons die by annihilation.As we discuss in more detail later, the stellar bulgeis confined within the interstellar bulge, which is blownout by a bulge wind to about 3.5 kpc in the Galacticplane and above the disk it feeds and merges into thehalo. The stellar disk extends not only throughout theinterstellar disk out to at least about 15 kpc, but alsoall the way into the interstellar bulge. The interstellardisk, however, extends in only to the outer edge of theinterstellar bulge, and starts in the so-called “molecularring” at around 3.5 kpc, which is defined primarily bythe two opposing innermost spiral arms.Threading the combined stellar and interstellar sys-tems, is the Galactic magnetic field, along whose fluxtubes the positrons travel from their birth to their death.We assume that the flux tubes are nearly vertical (e.g.Beck 2001), perhaps dipolar, in the inner part of theinterstellar bulge and are blown out by the bulge wind(e.g. Bregman 1980; Blitz et al. 1993) roughly radiallyin the outer bulge, out to the molecular ring, where theybegin to be tightly wound into the spiral arms of theinterstellar disk (e.g. Beck 2001). There, in giant starformation regions along these spiral arms, hot, massiveOB star associations are formed, whose collective super-novae generate superbubbles, that blow out plasma aswell as flux tubes into the overlying halo (e.g. Parker Higdon, Lingenfelter and Rothschild1979).Within this framework we model the various aspects ofthe production, propagation and annihilation of Galacticpositrons.We show that roughly half of the Galactic positronsare born in the extended interstellar bulge (e.g. Ferri´ere,Gillard & Jean 2008) in the inner ∼ ∼ MeV positrons created in these SNIa su-pernovae are expected to escape from the remnants intothe surrounding tenuous plasma. There they propagatealong magnetic flux tubes by 1-dimensional diffusion, res-onantly scattered in pitch angle by turbulently generatedmagnetohydrodynamic (MHD) waves that cascade downto the thermal electron gyroradius. In these plasmas thediffusion mean free path is long enough that only a negli-gible fraction annihilate there and ∼
80% of the positronsescape down into the dense HII and HI shells of neighbor-ing molecular clouds within 1.5 kpc, where they quicklyslow down and annihilate, while the remaining ∼ ∼ K) plasma is more modest ( ∼ Aldecay, synthesized by these stars, occurs preferentially inthe superbubbles. The larger bubbles blow out into thehalo, sweeping the magnetic flux tubes up with them.Again the diffusion mean free paths of the positrons aresuch that the positrons escape along the flux tubes, ei-ther up into the halo, or down into the warm ionizedenvelopes, surrounding the base of the superbubble gen-erated chimneys, where they slow down and annihilate.In the warm ( ∼ K), but essentially neutral HI gas,which fills the largest fraction of the disk > > Outline
Although the processes of Galactic positron produc-tion, propagation and annihilation are rather straight-forward, all the details still make the determination ofthe expected 511 keV line emission both lengthly andcomplicated.We consider positron production from the decay ofthree separate long-lived radioisotopes, each synthesizedby distinct classes of stars: the Ni and Ti, respec-tively, by SNIa and SNIp from thermonuclear explosionsof ∼ Gyr-old accreting white dwarves in close binarysystems, and the Al by young ( <
40 Myr), massive( > ⊙ ) stars, most likely through Wolf Rayet (WR)winds and core collapse SNII and SNIbc. The super-novae, SNIa and SNIp, from older stellar populationsare known to occur in galactic bulges and disks, whileWR stars, SNII and SNIbc, formed in extremely youngstellar populations, occur in active, or recent, galacticstar-formation sites, such as spiral arms, molecular-cloudcomplexes, and galactic nuclei. Thus in section 2 webriefly re-examine the production of β + -decay positronsby Galactic supernovae and their spatial and temporaldistribution in the light of recent observations and the-ory.In section 3 we then explore in detail the expected en-ergy loss, scattering, and propagation processes of therelativistic positrons from these supernovae in the dif-ferent phases of the interstellar medium. We considercollisionless scattering of these positrons by small-scalefluctuations generated by magnetohydrodynamic turbu-lence in the ionized interstellar phases, based on exten-sive studies and observations of relativistic electrons inthe analgous turbulence in the interplanetary medium.Within this context we adopt a self-consistent transportmodel tied to field-aligned turbulence dependent on theproperties of the ambient interstellar phases.In sections 4, 5, and 6 we determine the expectedspatial distribution of positron production, propagation,slowing down and annihilation in the various phases ofthe positron bulge, disk and halo. In each of thesewe review the fundamental properties of the interstellarmedium, which constrain positron transport as well asleave signatures in the positron annihilation line emis-sion features. Here we explore the very large differ-ences between the interstellar properties, and the result-ing positron propagation and annihilation, in the well-studied local interstellar medium compared to those inthe inner ∼ ∼ MeVpositrons in the various phases of the interstellar mediumis taken into consideration. GALACTIC NUCLEOSYNTHETIC POSITRON SOURCES
Here we briefly re-examine the production of positronsby β + -decay radioactive nuclei produced by differenttypes of Galactic supernovae. We also re-examine therates of such supernovae in the Galaxy and the result-alactic Positron Annihilation 5ing spatial and temporal distribution of their productionof positrons in both the Galactic bulge and disk in thelight of recent observations and theory, in order to deter-mine the expected positron bulge/disk ratio of the suchpositron production, Q B /Q D . Positrons from Supernovae
Positrons resulting from the decay chains of the longer-lived radioactive nuclei, Ni, Ti, Al, all produced byexplosive nucleosynthesis in supernovae, have long beenthought (Colgate 1970; Burger, Stephens & Swanenburg1970; Clayton 1973; Ramaty & Lingenfelter 1979; Chan& Lingenfelter 1993; Milne, The & Leising 1999) to bethe major source of Galactic positrons. The total Galac-tic positron production rates from these radionuclei de-pend on their supernova mass yields M , the supernovaoccurrence rates ν and the survival fraction, f , of theirpositrons in the supernova ejecta as it expands into theinterstellar medium. Ni The decay chain of Ni → Co → Fe produces apositron only 19% of the time, since Ni decays solely byelectron capture with a mean life of 8.8 days into Co,which also decays primarily by electron capture, and just19% by β + emission into Fe with a mean life of 111.4days. Thus the production rate of positrons from Codecay is Q = 0 . f ˙ N , or 130 × f νM e + s − ,where f is the time-integrated survival fraction of Codecay positrons in the expanding supernova ejecta, ˙ N isthe production rate of Ni atoms per s, M is the Niyield in M ⊙ per SN, and ν is the Galactic supernova ratein SN per 100 yr. This Galactic rate alone would greatlyexceed the observed rate of positron annihilation, if allof the positrons escaped from the supernova ejecta, butmost do not.The nucleosynthetic yields of Ni, whose decay energypowers the visible light from the exploding ejecta, havebeen extensively calculated in all different types of super-novae. This is particularly true of the cosmologically im-portant Type Ia thermonuclear supernovae in accretingwhite dwarves, which are best described by the deflagra-tion model W7 of Nomoto, Thielemann, & Yoko (1984)that gives a Ni yield of 0.58 M ⊙ . The peculiar sub-class of Type Ia(bg), also called Type Ip, supernovae inaccreting sub-Chandrasakar white dwarves are expectedto produce 0.44 M ⊙ of Ni from the helium detonationmodel (Woosley, Taam & Weaver 1986).Because of the relatively short decay mean life of Co,most of these positrons lose their energy and annihilatein the ejecta before it becomes either thin enough forthem to escape or sufficiently rarefied for them to survive.Indeed, quantitative studies of the survival of positronsin the expanding supernova ejecta (Chan & Lingenfelter1993; Milne, The & Leising 1999), for various supernovamodels, have shown that essentially all of the positronsproduced by its decay in the most frequent, but massive,core collapse SNII supernovae slow down and annihilateunobserved in their much denser ejecta and do not escapeinto the interstellar medium.In particular, Chan & Lingenfelter (1993) have cal-culated f of positrons from Co decay expected forvarious supernova model distributions of the ejecta den-sity, velocity and magnetic fields. They found that in the limit where the magnetic field is thoroughly tangled, thepositron survival fraction for SNIa deflagration model, f , ranged from 0.1% to 2.5% for unmixed versus uni-formly mixed ejecta. Alternatively in the limit that themagnetic field is fully combed-out (e.g. Colgate, Petschek& Kriese 1980) the SNIa f was much greater, rangingfrom 5% to 13% for unmixed and uniformly mixed ejecta.Following this same procedure Milne, The & Leising(1999) have further shown that a mean survival fraction f ∼ . ±
2% for the deflagration model gave the bestfits to SNIa light curves at late times ( > Co are the dominant source of energy.Using the deflagration model (Nomoto, Thielemann, &Yoko 1984) Ni yield of 0.58 M ⊙ , we calculate the ex-pected positron production rate Q = 75 × f ν Ia e + s − . Ti Chan & Lingenfelter (1993) have shown that thepositrons from Ti → Sc → Ca decay, which producesa positron 95% of the time from β + emission of Sc to Ca, is a significant source of the annihilating positrons.The Solar system abundance ratio of Ca to Fe of1.23 × − (e.g. Lodders 2003) requires a similar rela-tive nucleosynthetic yield ratio of Ti to Ni, sincethese radionuclei are the primary sources of Ca and Fe (Woosley & Pinto 1988). The much longer Tidecay mean life of 89 yr would allow essentially all ( > Ti decay can be scaled by the So-lar system abundance ratio of Ca to Fe, assumingthat half of the Galactic Fe is produced by SNIa (e.g.Timmes, Woosley & Weaver 1995) and f ∼
1, so that Q = 0 . f ˙ N , or 1 . × ν Ia e + s − .The most likely sources of these Ti decay positronsare the peculiar SNIp supernovae, typified by SN 1991bg(Filippenko et al. 1992; Turatto et al. 1996), so that weassume the same spatial distribution as SNIa. In particu-lar, the primary source of Galactic Ca appears to be Hedetonations in accreting sub-Chandrasakar mass whitedwarves, whose calculated (Woosley, Taam & Weaver1986; Woosley & Weaver 1994) Ti yields of M ∼ . M are ∼
16 times the Solar system M /M ratio.And such models also give the best fits to the light curvesof these peculiar supernovae (e.g. Milne, The & Leising1999; Blinnikov & Sorokina 2004; The et al. 2006). The M /M ratios calculated for other types of supernovaeare much less than the Solar system ratio. Thus we as-sume the spatial distribution of the SNIp is the same asthat of the SNIa and we also scale the Ti productionto that of Ni in SNIa. Al The additional contribution of positrons, produced82% of the time from long-lived ( τ ∼ . × yr)decays of Al to Mg, can be determined much more di-rectly from the measured Galactic luminosity of the 1.809MeV line, which accompanies that decay. This luminos-ity implies a steady-state Galactic mass M ∼ . ± ⊙ of Al (Diehl et al. 2006), or a positron production Q ∼ . M / m p τ ∼ (0 . ± . × e + s − . Higdon, Lingenfelter and RothschildThus they can contribute ∼ ±
4% of the total Galacticpositron annihilation rate of ∼ (2 . ± . × e + s − inferred from the best-fit bulge, disk and halo annihila-tion luminosity discussed above.Kn¨odlseder (1999) has shown that the Al 1.809 MeVline luminosity is strongly correlated with the distribu-tion of massive stars in the disk, which confirms that itis produced in Wolf Rayet winds and/or in core collapsesupernovae, SNII and SNIb/c.With these calculated yields of Ni, Ti and Al invarious types of supernovae, we now estimate the totalGalactic positron production rate from the decay of ra-dionuclei synthesized in supernovae in the bulge and disk,assuming no significant production in the halo, Q B + D = Q + Q + Q = (75 f ν Ia + 1 . ν Ia + 0 . × e + s − (1)scaled to the mean Galactic occurrence rate, ν Ia of SNIasupernovae per 100 yrs. Mean Supernova Rates in Our Galaxy
Galactic supernova occurrence rates depend on theHubble class of the galaxies and are commonly defined(e.g. van den Bergh & Tammann 1991; Cappellaro, Evans& Turatto 1999) in units of SNu, equal to 1 SN per 100 yr,times a factor of h o L BLGal , where h o is the Hubble constantin units of 75 km s − Mpc − and L BLGal is the blue lumi-nosity of the galaxy in units of (10 L BL ⊙ ) − . The Hubbleclassification of our Milky Way Galaxy is Sbc (Binney &Merrifield 1998; Kennicutt 2001). From a study of theextra-galactic observations Cappellaro, Evans & Turatto(1999) find that the SNIa rate is 0.21 ± L BLGal of (1 . ± . × L ⊙ (Freeman 1985), and this ex-tragalactic rate for Sbc galaxies, we estimate the mean supernova rate in our Galaxy for SNIa to be 0.40 ± ∼ Gyr) bursts ofstar formation could cause modest variations in the su-pernova bulge/disk ratio. From observations of oxygen-rich, cool giants, Sjouwerman et al. (1998) conclude thatthe most recent burst of star formation in the inner bulgeoccurred roughly 1 Gyr ago and produced ∼ SNIIand SNIb/c supernovae. Such a burst would then be fol-lowed by a much more extended period of enhanced SNIaand SNIp, occurring with a range of delay times between0.1 and 1 Gyr, calculated from evolutionary models byGreggio (2005). Thus, although the SNII and SNIbc su-pernovae all occurred during the burst of star formationa Gyr ago, we would expect the SNIa and SNIp to still beoccurring at the present time. Using the average ratio ofSNIa to SNII for our Galaxy of 0.25 (e.g. van den Bergh &Tammann 1991; Cappellaro, Evans & Turatto 1999), wewould expect 2.5 × SNIa from this star burst spreadover ∼ present supernova ratein our Galaxy for SNIa to be 0.43 ± ν Ia we then estimate from equa-tion (1) the expected mean Galactic positron production from the decay of supernova produced Co, Ti and Al, Q B + D = [(32 ± f + (0 . ± . . ± . × e + s − . (2)Thus, of the Galactic positron production of ∼ (2 . ± . × e + s − , inferred from the observed bulge,disk and halo annihilation radiation as discussed above, Al decay accounts for ∼ . × e + s − , and Tidecay produces ∼ . × e + s − , nearly all from SNIp,leaving ∼ (1 . ± . × e + s − , which can come from Ni decay positrons, if they have a survival fraction inSNIa ejecta of f ∼ ± ∼ ∼ . ±
2% inferred (Milne, The & Leising1999) from SNIa light curves at late times, when thepositrons become a major source of ejecta heating.
Spatial Distribution of Galactic Supernovae
Here we investigate both the stellar bulge/disk ratio( B ∗ /D ∗ ∼ ( ν B /ν D ) Ia + Ip ) of Galactic supernovae andtheir radial and transverse distributions within the stellarbulge and disk. Of primary concern are the SNIa andSNIp, which appear to be the source of ∼
85% of theannihilating positrons.We first determine the relative contributions of SNIaoccurring in the Galactic stellar bulge and disk. It is wellknown that the SNIa occurrence rate is essentially thesame across the Hubble sequence of galaxy types fromellipticals (E) through late-type spirals (Sd), when suchrates are defined in SNu, the number of supernovae peryear per unit of blue luminosity of the parent galaxy (e.g.Cappellaro, Evans & Turatto 1999). Across this broadHubble sequence, the fraction of galactic blue-band lu-minosities contributed by Galactic disks range from zeroin ellipticals to 95% in late-type Scd galaxies (Simien &de Vaucouleurs 1986), yet the SNIa birth rate per unitblue luminosity of parent galaxies remains the same.Since the SNIa occurrence rates per blue luminosityis independent of the relative contributions of galacticbulges and disks, we scale the ratio of SNIa birthratesin the stellar bulge and disk of our Galaxy by the cor-responding ratio of stellar bulge to disk blue luminosi-ties, L BLB /L BLD . In their classic study Simien & de Vau-couleurs (1986) investigated the systematics of bulge-to-disk ratios in the blue-band. The mean value oftheir tabulations of L BLB /L BLD for our Galaxy is 0.25.Thus, we expect that the Galactic stellar luminositybulge/disk ratio implies a similar time-averaged, meanstellar bulge/disk ratio ( ν B /ν D ) Ia ∼ ν B /ν D ) Ia + Ip ∼ . × . ∼ n D ∗ ( R, z ) Ia , and the number density of SNIa in theGalactic bulge, n B ∗ ( R, z ) Ia , expressed in cylindricallysymmetric Galactocentric coordinates, are n D ∗ ( R, z ) Ia = n o D e − R/R d −| z | /z d ,n B ∗ ( R, z ) Ia = n o B η . e − η r t , (3)where η = p R + ( z/ . , R represents the planar dis-tance from the Galactic center in kpc, z represents thedistance normal to the Galactic plane in kpc, and theconstant r t = 1.9 kpc (Dehnen & Binney 1998). Here thenormalization constants, n o D and n o B are found from thetotal Galactic SNIa birthrate and the ( ν B /ν D ) Ia ratio.Dehnen and Binney ascertained that the more than 90%of the stellar disk population could be represented by z d of 0.180 kpc. They concluded that the most importantmodel parameter was the ratio of the disk scale length, R d , to R ⊙ , the distance of the Sun from the Galactic cen-ter. They found that the observational constraints werebest satisfied by any of four models with R ⊙ = 8 kpcand R d /R ⊙ of 0.25, 0.3, 0.35, & 0.4. Here we will usetheir second model with R d of 2.4 kpc. We also include astar burst contribution to the bulge located in the CMZ(Sjouwerman et al. 1998). Spatial Distribution of Galactic PositronProduction
In Figure 2 we show the relative spatial distributions, F ( < r ), as functions of distance, r , from the Galacticcenter for SNIa residing in the stellar bulge and disk em-ploying the relative number densities, equation (3) for n B ∗ ( R, z ) Ia and n D ∗ ( R, z ) Ia , expressed a functions of R & z in a Galactocentric cylindrical coordinate system, F B ∗ ( < r ) = 2 π Z r n B ∗ ( R ( r ) , z ( r )) Ia r dr,F D ∗ ( < r ) = 2 π Z r n D ∗ ( R ( r ) , z ( r )) Ia r dr,F OB ∗ ( < r ) = 2 π Z r n OB ∗ ( R ( r ) , z ( r )) II & Ibc r dr. (4) F OB ∗ ( < r ) is the relative spatial distribution of mas-sive stars, the expected Wolf-Rayet and SNII & SNIbcprogenitor sources of Al. Although the great majorityof massive stars, which synthesize Al, are located inthe outer ( > < Al ratio. G¨usten (1989) estimated a luminosity ofionizing photons from massive stars in the inner bulge of ∼ . ± . × photons s − . Comparing this to theestimated (McKee & Williams 1997) total Galactic value of ∼ . × photons s − , we expect that roughly 10%of the massive stars and their generated Al occur inthe inner bulge, amounting to ∼ . × e + s − .For the remaining disk portion we use the spatial distri-bution of OB associations, n OB ( R, z ) II & Ibc , of (McKee& Williams (1997, eq. 35) with z d = 0.15 kpc to rep-resent that of the Al made by massive stars. Thesespatial distributions have been normalized at large r totheir expected relative contributions to the total Galac-tic positron production rate fractions: F B ∗ = 0.21, F D ∗ = 0.64, and F OB = 0.15, based on the SNIa and SNIpbulge/disk ratio of 0.33 in the 85% of positrons producedby them.As we see from Figure 2, of the positrons producedfrom SNIa and SNIp in the disk, 13% are born within1.5 kpc. Although the best fit disk model used in theSPI/INTEGRAL analyses (Weidenspointner et al. 2007,2008a) of the annihilation radiation also includes 12%occurring within that radius, in order to treat the prop-agation of the positrons before the annihilate, we obvi-ously need to include them in the total production within1.5 kpc. Once we have determined the distribution ofthe subsequent annihilation of all of the positrons, wethen adjust the estimated bulge ( < > < Q B ∼ [ F B ∗ + F D ∗ ( < .
5) + F OB ( < . Q B + D ∼ [0 .
21 + 0 . .
13) + 0 . . Q B + D ∼ . Q B + D , andthe disk production, including the OB contribution, Q D ∼ [ F D ∗ (1 − F D ∗ ( < . . F OB ] Q B + D ∼ [0 . .
87) + 0 . . Q B + D ∼ . Q B + D . Thus theeffective positron production ratio in the bulge and diskwithin and beyond 1.5 kpc is ( Q B /Q D ) ∼ . / . ∼ Q B and disk Q D we now determine the differential prop-agation of the positrons within and between those regionsand the halo before they annihilate. In simplest termswe define the resulting positron bulge/disk annihilationratio as, A B /A D ∼ ( P B : B Q B + P D : B Q D ) / ( P B : D Q B + P D : D Q D ) , where P B : B and P D : B , are respectively thefractions of the positrons, born in the bulge B and disk D beyond 1.5 kpc, which annihilate in the positron bulge B < . P B : D , and P D : D are the correspond-ing fractions of those positrons which annihilate in thedisk D beyond the positron bulge > . NATURE OF GALACTIC POSITRON PROPAGATION
We now look in detail at the nature of these positrons,and their propagation, slowing down and annihilationin the various phases of the interstellar medium in thebulge, disk and halo. In particular, we investigate theplasma processes that determine the rate of positronpropagation, drawing upon the extensive studies and ob-servations of relativistic electron propagation in the in-terplanetary medium.
Relativistic β + -Decay Positrons from Supernovae Higdon, Lingenfelter and RothschildAssuming that the dominant source of Galacticpositrons is the β + decay of Co, Sc and Al fromsupernovae, the detailed studies of their survival in andescape from the ejecta show that the great bulk of thesurviving and escaping positrons are relativistic .Positrons emitted in the Co → Fe decay are dis-tributed in kinetic energy with a maximum of 1.459 MeVand a mean of 0.630 MeV, and those from Sc → Cadecay have a very similar spectrum. The expected (Chan& Lingenfelter 1993; Milne, The & Leising 1999) meanenergy of the escaping positrons from the shorter-lived(111.4 day mean life) decay of Co is close to 0.5 MeV,reduced from their initial energy by ionization losses inthe still dense ejecta, while the mean energy of those fromthe longer lived (89 yr Ti dominated meanlife) decayof Sc is essentially unchanged at about 0.6 MeV in themuch less dense ejecta. The mean energy of positronsfrom the decay of the very long-lived (1.04 × yr) de-cay of Al is likewise about 0.5 MeV.In order for these surviving relativistic positrons to an-nihilate in the interstellar medium they must first be de-celerated to energies, ≤
10 eV.
In order to slow downto thermal energies, these ∼ ∼ − (Berger & Seltzer 1964), or nd sd ∼ × H cm − , in unionized interstellar gas,and this range is reduced by a factor of 1 / (1 + x e ) ifthe gas is ionized with an ionization fraction, x e , relyingsolely on collisional ionization losses. In the process oftraveling this slowing down distance, d sd ∼ /n (1 + x e )kpc, the average velocity of these positrons, ¯ βc is ∼ t sd ∼ d sd / ¯ βc ∼ × /n (1 + x e ) yr , (5)where n is the mean H density in cm − .Thus we see that since the slowing down distance of theradioactive decay positrons in the nominal phases of theinterstellar medium is > n ∼
40 cm − , >
50 kpc in warm media with typical n ∼ − , and > n ∼ × − cm − . Since all of these distancesare far greater than the scales of their correspondinginterstellar components, these fast positrons, streamingalong the large-scale interstellar magnetic field, wouldall escape the Galactic disk before they even slow down, unless there is sufficient plasma turbulence to efficientlyisotropize their trajectories. We thus investigate in de-tail under what conditions the interstellar turbulence canisotropize the trajectories of these relativistic positrons.
Nature of the Turbulence which ScattersInterstellar Positrons
In ionized interstellar phases we expect that the in-terstellar MeV positrons are scattered by resonant in-teractions with ambient turbulent magnetic fluctuations.Consequently, the efficiency of these scatterings in turndepends on the fundamental properties of the MHD tur-bulence at very small spatial scales of the electron gy-roradius, r e . However, the basic properties of MHDturbulence is poorly understood. In a recent review ofturbulence and magnetic fields in astrophysical plasmas,Schekochihin & Cowley (2007) state “despite over fifty years of research and many major advances a satisfac-tory theory of MHD turbulence remains elusive. Indeed,even the simplest (most idealized) cases are not fully un-derstood.”An important advancement was the realization that inmany astrophysical sites MHD turbulence is anisotropic.Such anisotropy is a well-observed phenomenon in solar-wind plasmas (e.g. Belcher & Davis 1971; Matthaeus,Golstein, & Roberts 1990; Osman & Horbury 2007) aswell as in increasingly more accurate, numerical simula-tions (Shebalin, Matthaeus, & Montgomery 1983; Maron& Goldreich 2001; Cho, Lazarian, & Vishniac 2002;Mason, Cattaneo, & Boldyrev 2006; Perez & Boldyrev2008). In a pioneering study Goldreich & Sridhar (1995)developed a sophisticated theory of anisotropic incom-pressible MHD turbulence. They presented the conceptof “critical balance” which predicted in inertial regime ofturbulent MHD flows, first, a filamentary shape for tur-bulent fluctuations, via a relationship between the wavenumbers parallel ( k ) and transverse ( ⊥ ) to the directionof the local mean magnetic field, k k ∝ k / ⊥ and, sec-ond, the modeled energy spectra of transverse turbulentcomponents possesses Kolmogorov (1941) scaling, k − / ⊥ ,observed in low-speed, terrestrial flows.At present, a puzzling dichotomy exists between thesemodel predictions and a plethora of solar-wind observa-tions and numerical simulations. Power spectra of smallscale electromagnetic fluctuations in solar-wind plasmashave been found to have the expected − − p , still have difficulty in representing high-frequency collisionless turbulence evolving on frequen-cies, ω ≈ Ω e , which seem to be most efficient at res-onantly scattering ∼ MeV positrons, where Ω e repre-sents electron gyrofrequency. The application of suchmodels of anisotropic turbulence seems to imply thatcollisionless scattering of MeV electrons would be mod-est and MeV electron transport should be dominatedby streaming at ∼ ¯ βc . Yet a great variety of indepen-dent investigations of MeV electron transport in the in-terplanetary medium irrefutably demonstrate that suchparticles propagate diffusively in the collisionless solarwind plasma, although the nature of the plasma fluctu-ations which actually scatter such electrons remains yetalactic Positron Annihilation 9unknown. Thus, we employ a simple phenomenologicalmodel of collisionless MeV electron scarttering by tur-bulent interstellar fluctuations tied heavily to the welldetermined properties of electron scattering in turbulentsolar-wind plasmas.The propagation of MeV electrons has been well stud-ied in the interplanetary medium employing measure-ments of electrons accelerated by solar particle events aswell as by the Jovian magnetosphere. The propagation ofJovian MeV electrons is, for the most part, dominated bydiffusion (Ferreira et al. 2001a,b, 2003). Fits to intensityand anisotropy time profiles of solar-flare electrons alsoillustrate the dominance of diffusive transport at theseelectron energies, and, consequently, lead to the straight-forward determination of scattering mean free paths viafits to intensity and anisotropy time profiles (Kallenrode1993).Bieber, Wanner, & Matthaeus (1996) related suchphenomenologically-derived scattering mean free pathsfrom solar flare particle measurements to the determina-tion of simultaneous power spectra of magnetic fluctu-ations at MHD spatial scales. Since it is impossible todetermine the full three-dimensional wavevector spectraof turbulent fluctuation from single spacecraft measure-ments (e.g. Fredricks & Coroniti 1976), Bieber, Wanner,& Matthaeus employed a simple two-component repre-sentation of the anisotropic interplanetary MHD turbu-lence. They found that their magnetic-field data con-sisted of two anisotropic populations, fluctuations withlarge correlation lengths transverse to the direction ofthe mean magnetic field (slab turbulence) and fluctua-tions with large correlation lengths parallel to the direc-tion of the mean magnetic field (quasi-two-dimensionalturbulence). In such slab models the wave vectors of thefluctuations are aligned parallel to the direction of themean magnetic field, and a simple interpretation is thatin the inertial range slab turbulence consists of Alfv´enwave propagating along the mean magnetic field. Laterinvestigations (Oughton, Dmitruk, & Matthaeus 2004,2006; Oughton & Matthaeus 2005) have demonstratedthat this simple two-component model of anisotropic tur-bulence seems to model the basic properties of solar windMHD turbulence.Bieber, Wanner, & Matthaeus found that interplane-tary turbulence is dominated by quasi-two-dimensionalturbulence, and the mean ratio of the power in slabturbulence to the power in two-dimensional turbulenceis ∼ Recently, Horbury, Forman & Oughton (2005) have investi-gated in more detail the properties of anisotropic solar wind tur-bulence, implementing a wavelet-based method to track the direc-tion of the local mean magnetic field; such an approach reducesgreatly the noise in the magnetic field measurements. Employ-ing the two-component model, they ascertained that their resultsagreed “remarkably well” with the results of Bieber, Wanner, &Matthaeus.
Based on these discussions, we model interstellar
MHDturbulence as composite slab/two-dimensional turbu-lence. We assume here that the ratio of the power ofinterstellar slab turbulence along the mean magnetic fieldto the power of 2-dimensional turbulence normal to thedirection of the mean field is 0.15 following Bieber et al.(1996). Similarly, we assume that the spectrum of in-terstellar slab turbulence steepens from a spectral indexof − − k d > /r p , where r p isthe proton gyroradius, since Leamon et al. (1998) havefound that at higher wave numbers the spectral indexof solar-wind variations varied from − − − l o . For kl o ≥ k ≤ k d , the modeled turbulence possesses aspectral index, s . From the discussion above, we assume s = 5/3 and at k d = 1/r p , we assume the spectral shapesteepens to s = 3. Teufel & Schlickeiser (2002, their eq.58), find that for such slab turbulence the electron scat-tering mean free path along the magnetic field is λ k = 92 (cid:0) B o (cid:14) δB k (cid:1) J k min a K ( a, h, s, I, J ) , (6) k min = 2 πl o , k d = 1 r p , J = ¯ βck min Ω e , I = ¯ βck d Ω e , a = ¯ βcV a ,f ( s, h ) = 2 h − − s ,J ≪ I ≪ ≪ aK = a f J s I − s ( F (1 , h − , hh − , − πaf Q h − ) −
13 2 F (1 , h − , h +2 h − , − πaf I h − ) ) J ≪ ≪ I ≪ aK = π h − s − − s i + a f J s I − s F (1 , h − , ph − , − πaf I )where δB k / π is the energy density of the turbulent mag-netic fluctuation in the direction of the mean magneticfield, c is the speed of light, Ω e is the gyro-frequency ofthe positron, the mean magnetic field is B o , V a is theAlfv´en speed, B o / √ πρ , ρ is the ion mass density, r e isthe electron gyroradius; the average positron speed is ¯ βc ,and K is a dimensionless quantity involving a Gauss hy-pergeometric function F (Table 3 of Teufel & Schlick-eiser 2002).Thus, by such one-dimensional (1-D) diffusion thepositrons in their slowing down time t sd would bedistributed along a flux tube a mean length l sd ∼ (2 t sd λ k ¯ βc/ / in either direction from their point oforigin. Ion-Neutral Damping ), concentrated in acloud population (e.g. Tielens 2005). Moreover, a majorfraction of the disk ( > l pH =5 × /n cm at 8000 K, or if hydrogen is fully ionized,and He is the dominant neutral species, the mean freepath for proton-He collisions is l pHe ∼ . × /n cmwhere n is the H number density, including both ions andneutrals (Lithwick & Goldreich 2001).Consequently, they showed that MHD turbulence cancascade to small spatial scales, much less than thesedamping collision mean free paths, only if the HI frac-tion, n HI /n , is less than a critical value, n HI /n < f crit ≈ l pH /l o ) / . (7)Therefore in predominantly neutral interstellar phasesturbulent MHD cascades are halted by ion-neutral colli-sions at ∼ l pH , spatial scales far greater than the gy-roradii of MeV positrons. Hence we expect collision-less scattering of positrons by such large-scale turbulentMHD fluctuations to be very inefficient. Similarly sim-ple elastic collisions with ambient electrons, whose meanpitch angle scattering is 90 o and mean energy loss is 50%,have a mean free path much longer than slowing downdistance due to ionization losses (e.g. Berger & Seltzer1964), so they too offer no significant scattering.In the absence of such scatterings, ∼ MeV positronsmight be expected to generate plasma waves by a reso-nant streaming instability, similar to the creation (Kul-srud 2005) of ion Alfv´en waves by streaming relativis-tic cosmic ray nuclei. But these do not appear to beeffective either. These ∼ MeV positrons generate res-onant whistler waves (Schlickeiser 2002) at wavelengthssignificantly less than the scale of the thermal proton gy-roradius, r p = u p / Ω p , at frequencies approaching thoseof thermal electrons, Ω e , where u p is the thermal pro-ton speed. The phase speed of whistler waves in thedirection of the mean field is the electron Alfv´en speed, V ae = ( m p /m e ) / V a . If the positron streaming veloc-ity, V s > V ae , and if the whistler waves weren’t dampedso they grew to saturation, then the resulting magneticfield fluctuations in these waves would interact with thestreaming positrons via quasi-linear wave-particle inter-actions that change their pitch angles, reducing V s to alevel just less than V ae . However, in predominantly neu-tral interstellar phases such whistler waves are also sub-ject to severe ion-neutral damping (Kennel 2008 privatecommunication). Thus, we expect that positron streaming velocitiesalong the magnetic flux tubes are comparable to theirparticle velocities, ¯ βc , similar to the very high driftspeeds found (Felice & Kulsrud 2001) for cosmic-ray nu-clei in warm HI regions. Nonetheless, in turbulent mediathe flux tubes themselves may essentially random walk,so that the propagation is described by so-called “com-pound diffusion” (Lingenfelter, Ramaty & Fisk 1971). POSITRON PROPAGATION & ANNIHILATION IN THEPOSITRON BULGE
Using these models of relativistic electron transport,we now investigate the propagation, slowing down, andannihilation of positrons produced in each of the dif-ferent phases of the interstellar medium in the positronbulge, Galactic disk and halo. The schematic model ofthe bulge, disk and halo is shown in Figure 1. Becausethe positrons, produced by SNIa and SNIp, are widelydistributed in the bulge and disk (equation 3), we assumethat the fraction produced in the various phases of theinterstellar medium are proportional to their appropri-ate filling factors. Although the supernovae themselvesdisturb the local medium, the ∼ MeV positrons from thedecay of radionuclei escape from the remnant into theundisturbed surroundings.We estimate the relative probabilities, P, of positronannihilation within each phase and positron escape intoneighboring phases by the following procedure. Fromthe properties of the medium in each phase, we deter-mine the propagation mode, diffusion or streaming, andcalculate the diffusion mean free path λ k (equation 6) inthe MHD scattering mode, or the mean velocity, ¯ βc , inthe unscattered streaming mode.Although we have also made Monte Carlo simulations,we estimate the propagation of positrons during theirslowing down in each phase of the interstellar medium,using a simple approximation. Since even the averageproperties and structure of the magnetic field especiallyare very poorly known in the various phases, no more so-phisticated treatment seems justified. For the 1-D diffu-sion in a uniform medium with a mean free path λ k alonga magnetic flux tube, N positrons produced at a point, l = 0 and t = 0, will be distributed in both directions alongthe flux tube with a density, n ( l ) = ( N/l sd ) e − ( l/l sd ) ,where l sd = (2 λ k βct sd / / , by the time t sd that theyslow down and annihilate. This corresponds to a meanpositron density ¯ n = N/ l sd over the total mean length2 l sd . Similarly for positrons streaming at their veloc-ity ¯ βc with an isotropic pitch angle distribution they willalso have a mean density N/ l sd over a total mean length2 l sd , where l sd ∼ ¯ βct sd /
2, when they slow down and an-nihilate.Thus, if the positrons are produced uniformly alongsome mean length x l B of the magnetic flux tubes thread-ing through some phase x , from the above we expect thatthe probability, P x : x , that such positrons will slow downand annihilate in that phase before they escape is crudely, P x : x ∼ x l B / x l sd for x l B < x l sd and P x : x ∼ x l B / ( x l B + x l sd ) for x l B > x l sd , (8)alactic Positron Annihilation 11where x l sd is the slowing down length in that phase. Theremaining positrons have a probability, 1 − P x : x , of es-caping from phase x into a neighboring phase, and weestimate the relative fractions of the escaping positronsthat go into each of the adjacent phases from simple geo-metric arguments, allowing large uncertainties of ± x will slow down and annihilate there is onlyslightly different from those born uniformly within it.These positrons are effectively born at their first scatterwithin the phase, roughly within a scattering mean freepath of the boundary. So if x l B < x l sd , the probability oftheir slowing down and annihilating in the phase, P x : x isthe same as for those born uniformly within, ∼ x l B / x l sd .But if x l B > x l sd , then P x : x remains at ∼ /
2, sinceafter the first scatter near the boundary half would beexpected to be spread over x l sd in both directions alongthe flux tube by the time they slow down and annihilate.Thus only about half of them will slow down and anni-hilate in phase x , while the other half will escape, or beeffectively reflected. As we show, however, in the denselabyrinth of clouds in the CMZ and tilted disk the halfof the positrons that diffuse into clouds from the VH andHM and are reflected back out rather than annihilatingwithin, quickly diffuse into another cloud where half areagain reflected, and by the time they have encountered ahalf dozen other clouds only a small fraction, ∼ (1 / , or <
1% remain. As we also show, however, in the diffusedisk beyond 1.5 kpc that is not the case.From these simple approximations, we estimate the dif-ferent propagation fractions P that relate the positronproduction rates Q and annihilation rates A , as definedin equation (4) for the expected bulge/disk annihila-tion ratio, expanding that equation to explicitly definethese fractions in the different phases of the interstellarmedium within the bulge, disk and halo.We consider here the Galactic positron bulge as itis defined by the spherical component ( < < < R < Bi , ( < Bm, (0.5 to 1.5 kpc);3) the hot, tenuous outer bulge, Bo between ∼ ∼ Do , and 5) the hot, tenuous halo, H , beyond 3.5 kpc and above the disk. The properties ofeach of these regions, which we discuss below, are sum-marized in Tables 1 and 2.Based on the distributions of the surface density of the three positron source components, given in equation (3):the bulge and disk distributions of SNIa and SNIp occur-rences, and the disk distribution of massive (OB) stars(McKee & Williams 1997), we estimated the fraction ofpositron production in each of these regions. The cu-mulative production as a function of Galactic radius isshown in Figure 2 for each component. We find that 45%,38% and 17% of the positrons from the stellar bulge B ∗ component of SNIa and SNIp are produced in the regions < Al component within 0.5kpc. Of the positrons from the stellar disk D ∗ compo-nent of SNIa and SNIp, 2%, 11% and 30% are producedin those same regions, and the remaining 57% are pro-duced beyond 3.5 kpc, together with 90% of the massivestar Al component.Thus the total positron production in the inner bulge Bi within 0.5 kpc, the middle bulge Bm between 0.5 and1.5 kpc and the outer bulge Bo between 1.5 and 3.5 kpc,including both stellar bulge and disk components, andthe production in the outer disk Do beyond 3.5 kpc, are Q Bi = [(4 . ± . f + (0 . ± . × e + s − Q Bm = [(5 . ± . f + (0 . ± . × e + s − ,Q Bo = [(8 . ± . f + (0 . ± . × e + s − ,Q Do = [(13 . ± . f + (0 . ± . × e + s − . (9)And using the best-fit positron survival fraction, f ∼ ±
2% from § Q Bi = (0 . ± . × e + s − Q Bm = (0 . ± . × e + s − ,Q Bo = (0 . ± . × e + s − ,Q Do = (1 . ± . × e + s − . (10)Within each of these regions, y , we determine the ex-pected positron production rates in each separate inter-stellar gas and plasma phase, x , by the relative fillingfactors of those phases, f yx , times the production ratein that region, Q y , such that Q yx ∼ f yx Q y . We thenestimate the final positron annihilation rates, A y ′ x ′ , ineach phase x ′ within each region y ′ , as the sums over x, y, x ′ and y ′ of products of the production rates, Q yx ,times the propagation fractions, P yx : y ′ x ′ , where each isthe fraction of the positrons born in yx that propagateto and annihilate in y ′ x ′ , as discussed above and givenfor x : x in equation (8).In the next three sections, we estimate these propaga-tion fractions from modelling the propagation, slowingdown and annihilation of positrons in each of these re-gions and phases starting with the Galactic bulge andmoving outward in the disk and ultimately into the halo,which is an important region of propagation and annihi-lation, despite its lack of local production. Inner Bulge (R < We model the complex interstellar phenomena of theinner region of the Galaxy (R < ∼ K), high-pressure plasma permeates theregion, pressure equilibrium exists among the interstellarphases (Spergel & Blitz 1992; Carral et al. 1994), and theproperties of the HII, HI, and H regions are related toeach other via the scenario of photodissociation regions(PDR) (Tielens & Hollenbach 1985).O-star radiation is the primary photo-ionization sourcein the interstellar medium (McKee & Williams 1997).Outside of galactic centers, ultraviolet radiation flux atan arbitrary location in galactic nuclear regions is gen-erated by nearby OB associations, or by single O starsdistributed randomly throughout the regions (Wolfire,Tielens, & Hollenbach 1990). However, in the immediatevicinity of a galactic center ionizing radiation is domi-nated by the radiation contribution of active galactic nu-clei. In the inner Galactic bulge the bulk of the ionizingradiation seems to be generated by randomly distributedO-stars which photo-ionize the outer layers of nearby in-terstellar clouds (Wolfire, Tielens, & Hollenbach 1990;Carral et al. 1991). However, in the outer ( > < r H , which contains the bulkof the cloud mass at a molecular hydrogen density n H .Surrounding the molecular cores are cold atomic HI en-velopes, ∆ r HI thick, which in turn generate infrared con-tinua and fine-structure line emissions (Wolfire, Tielens,& Hollenbach 1990). Finally, envelopes of photoionizedHII plasma form outer shells, ∆ r HII thick, surroundingthese HI shells (Carral et al. 1994), ionized by Lymancontinuum emission from randomly distributed hot, mas-sive stars within the central 0.5 kpc. Following Caral etal., we assume that these HII envelopes constitute theprimary photoionized gas component and a schematicmodel of the clouds is shown in Figure 3. Although thethe hot, massive O stars that generate the HII ionizingemission are thought to occur only in the CMZ, we ex-pect that clouds in the inner portion of the surroundingtilted disk are also likely to be irradiated with sufficientflux to maintain HII envelopes. The extent of such irra-diation is not known, however, and here we arbitrarilyassume that it extends only to 0.5 kpc.We also assume that the magnetic flux tubes nomi-nally pass through the cold clouds in a parallel array,but with large turbulent perturbations superimposed, assuggested by magnetic field in giant molecular clouds. Liet al. (2006), using the SPARO 450 µ m polarimetry ob-servations, have found that within some such clouds inthe disk, the direction of the magnetic field is roughlycorrelated with that of the local Galactic field. However,extensive Zeeman measurements of molecular clouds alsoindicate strong field perturbations from supersonic mo-tions driven by MHD turbulence with energies compara-ble to that of the magnetic fields within the clouds (seereviews by Crutcher 1999; Falceta-Goncalves, Lazarian& Kowal 2008). Thus, we assume a nominal field to calculate the basemean length of flux tubes through the clouds, and thenfrom an estimate of the scale length of turbulent motions,we estimate the increase of that mean length resultingfrom the effective random walk or meandering of the fluxtubes through the turbulent perturbations. The nominalmean flux tube length through cold cloud cores is l ′ B ∼ r/
3, where r is the core radius, while the nominal meanlength through an overlying shell of thickness ∆ r , is l ′ B ∼ (4 / r + ∆ r ) − r ] / ( r + ∆ r ) . Conservatively takingthe outer scale of turbulent motion l o as the step size,which minimizes the random walk, we expect a meanmeandering flux tube length, l B ∼ ( l ′ B /l o ) l ′ B .In the inner bulge the molecular clouds concentrateinto two nested disks, the Central Molecular Zone (CMZ)and the surrounding Tilted Disk, which extends well be-yond 0.5 kpc out to edge of the middle bulge at 1.5 kpc.The CMZ is a highly asymmetric region, which is an el-lipsoidal disk, ∼
40 pc thick vertically, with lateral axesof ∼
500 pc by ∼
250 pc and an interstellar H massof 1.9 × M ⊙ (Ferri`ere, Gillard, & Jean 2007). In thisregion we estimate a mean molecular cloud core mass, M H core ∼ × M ⊙ and a mean core radius, r H ∼ × M ⊙ ≤ M H core ≤ × M ⊙ ) and the size spectrum (3.3 pc(resolution limit) ≤ r H ≤ pc) of the CMZ molecu-lar cloud population determined by Miyazaki & Tsuboi(2000). This implies a total of ∼
400 such cloud coreswith a H density n H ∼ − . It also gives a totalvolume filling factor of these H clouds in the CMZ is f H ∼ < f H ∼ × − .The much larger Tilted Disk, which is described inmore detail in the next subsection (4.2), surrounds theCMZ, extending out to ∼ mass of 3.4 × M ⊙ (Ferri`ere, Gillard, &Jean 2007). About 20% of the Tilted Disk lies withinthe bulge source region < ∼ × M ⊙ of interstellar H .We assume that this H gas is also concentrated in sim-ilar molecular cloud cores, which would number ∼ < α He-like transition of ionized Fe indicate that this VH plasmahas a temperature of ∼ K, and fills most of thiscentral region (Koyama 1989; Yamauchi et al. 1990;Spergel & Blitz 1992; Koyama et al. 1996; Muno etal. 2004). Thus we assume that a hot ( ∼ × K),plasma ( n ∼ .
04 cm − ) fills the inner ∼
500 pc of thebulge (e.g. Yamauchi et al. 1990). The thermal pressure, p th /k = n V H T V H /µ ∼ K cm − , where µ representsthe mean particle mass for fully ionized H, and k is Boltz-mann’s constant. This value agrees well with previousestimates of the CMZ thermal pressure of ∼ × Kcm − by Spergel & Blitz (1992).Analyzing the diffuse nonthermal radio emission of theGalactic center region, Spergel & Blitz (1992) found thatthe magnetic field pressure is in approximate equilibriumwith the thermal gas pressure. Consequently, we assumealactic Positron Annihilation 13that both the intercloud and cloud field strengths are ∼ µ G, the value expected from such pressure argu-ments, 5 × − erg cm − .Between the VH plasma and the H core of each cloudare the two surrounding envelopes, the upper warm, ion-ized HII gas and the lower, cold neutral HI gas. TheHII density is estimated from pressure balance betweenHII gas and the VH (Carral et al. 1994). Although thethermal pressure of the overlying VH is approximatelymatched by the total pressure of the cold medium (H ),the latter pressure is dominated by turbulence (Tielens &Hollenbach 1985; Spergel & Blitz 1992; Oka et al. 1998).Similarly, the large line widths observed in the extendedHII gases in the Galactic center region (e.g. Rodriguez-Fernandez & Martin-Pintado 2005) imply that turbu-lence also dominates the HII pressure. We assume thatthe magnetic field strengths are the same in the VHplasma and the HII gas. Thus we find that a turbulentHII pressure, p turb = ρ HII ( u HII ) (Tielens & Hollen-bach 1985), constituting ∼
85% of the total HII pressureequal to the total VH pressure, possesses a root meansquare turbulent velocity, ( u HII ) / , of ≥
25 km s − , fora mass density, ρ HII , corresponding to n HII ∼
100 cm − .Note that the HII thermal pressure, 2 n HII T HII , ∼ K cm − , is small compared to that of the VH, since T HII is ∼ n HI , of ∼ − with T HI ∼
150 K. For R < × M ⊙ , with aspace averaged density, < n HI > ∼
25 cm − in the CMZand a filling factor there of f HI ∼ r HI ∼ f HII ∼ r HII ∼ r c ∼ f c = f H + f HII + f HI is ∼ d cc ∼ r c f − / c ofonly ∼
23 pc and that between cloud surfaces, d cc − r c of just ∼
11 pc.As we will show, the annihilation of the positrons takesplace almost entirely in the cloud shells. However, onlya negligible fraction ( < Q BiV H ∼ Q Bi ∼ (0 . ± . × e + s − , Q BiHII ∼ Q BiHI ∼ Q BiH ∼ . The subsequent positron annihilation in these phasesof the bulge is then defined by the propagation fractions,which we estimate below using this general model of theCMZ. Because, as we show, the positron slowing downand annihilation in the HII phase of CMZ is so efficient,the probability of positrons, formed in the VH, pene-trating through the HII into the underlying HI and H is negligible, so we only show here the terms for diffusionbetween the other adjacent phases with significant fillingfactors. We also find that the probability of positronssuccessfully escaping out of the inner disk is quite small.Thus the significant terms in each phase are, A BiV H ∼ P V H : V H ( Q BiV H + Q Bm P Bm : BiV H + Q Bo P Bo : BiV H ) ,A BiHII ∼ P HII : HII ( Q BiHII + Q BiV H P V H : HII + Q Bm P Bm : BiHII + Q Bo P Bo : BiHII ) ,A BiHI ∼ A BiH ∼ , (11)where, like those discussed above, the propagation frac-tions P , subscripted VH:VH is the fraction of thepositrons produced in the VH phase that slow down andannihilate in that phase, VH:HII is the fraction that es-caped from it into the adjacent HII, and other fractionsare similar. The propagation fractions and the resultantannihilation rates are listed in Table 2. Very Hot Medium (VH)
The bulk of the positrons from the inner bulge SNIaand SNIp are expected to be born in the VH, which in-cludes both the high density and temperature plasma inthe CMZ and all of the lower density and temperaturehot plasma that fills nearly ∼ < d ∼
10 pc, suggests that in a fairly regular magnetic field themean flux tube length l B would also be ∼
10 pc in theCMZ. On the other hand, for the bulge component withthe production roughly uniform throughout the spheri-cal volume, the positrons would be distributed along amean distance d ∼ (4 / πr / πr ∼ (2 / r ∼
300 pcboth above and below the CMZ with similar mean fluxtube lengths, l B , along the roughly vertical magnetic fieldin that region (e.g. Beck 2001).In order to determine what fraction of the positrons,born in the VH, can slow down and annihilate there andwhat fraction escape into the adjacent HII envelopes, wealso need to estimate the diffusion mean free path of thesepositrons in the VH plasma.It is difficult to quantify the properties of hypotheti-cal turbulent flows in the VH, since the VH origin andage are unknown. However, a plausible constraint onsuch turbulence is that the dissipation rate of the tur-bulent energy must be less than the cooling rate of theVH plasma. The rate, at which turbulent energy is dis-sipated is ∼ ρ V H ( u V H ) / /l o , in units of erg cm − s − ,where ρ V H is the mass density of the VH, ( u V H ) / is4 Higdon, Lingenfelter and Rothschildthe root mean square turbulent velocity, and l o is theouter scale of turbulence (Townsend 1976).From the X-ray observations Muno et al. (2004) esti-mated that the VH cooling time, t cool ∼ yr. Since theVH thermal energy density, ǫ th ≈ p th , the cooling con-straint becomes ( u V H ) / ∼ ( l o ǫ th / ( t cool ρ V H )) / . Tak-ing the maximum estimate of l o ∼
10 pc, the meanseparation between cloud surfaces, this relation limits( u V H ) / to less than 50 km s − , which is much less thanthe VH Alfv´en speed, V a ∼
920 km s − . This cooling re-lation places a severe constraint on VH turbulence, if theVH is long lived on the time scale of t cool . However, ifthe VH is a transient phenomenon, this constraint couldbe weakened and the turbulence would be stronger.Thus, if the mechanism generating the VH turbulenceoperated over a duration, t m , less than t cool , the dis-sipation rate of the turbulent energy can be greater, ∼ E th /t m . Thus ( u V H ) / ∼ ( l o ǫ th / ( t m ρ V H )) / . If,very conservatively, t m ∼
25 Myr, then ( u V H ) / isless than 90 km s − . Since in strong turbulent MHDflows, where energy is equipartitioned between magneticand velocity fields < δB V H > ∼ πρ V H < u V H > s(e.g. Iroshnikov 1964; Kraichnan 1965; Biskamp 2003),constraining < u V H > limits < δB V H > , and then < δB k > / , since < δB k > / ∼ . < δB V H > / .Thus ( B o /δB k ) ∼ λ k ∼
25 pc in the CMZ from equation (6).In the VH the positron slowing down time is t sd ∼ × /n (1 + x e ) ∼ . × yr from equation (5) at n ∼ .
04 cm − . In the slowing down time, t sd , the positronsdiffusing along a flux tube in the VH will be distributedover a mean length l sd ∼ (2 λ k ¯ βct sd / / ∼ ∼ . c .Roughly 61% of the inner bulge positrons are expectedto occur in the CMZ, predominantly from the SNIa andSNIp in the star burst and the inner disk, along with10% of the massive OB stars producing Al. The re-maining 39%, all from SNIa and SNIp, are expected tooccur throughout the nuclear bulge < l B , for positrons escaping into the HII. For thoseproduced by the star burst and disk component of SNIaand SNIp in the CMZ and inner disk, where the mean dis-tance between the HII envelopes of neighboring clouds, l B is only ∼
10 pc, we would expect from equation (8)that P V H : V H ∼ l B / l sd < ∼ − and the temperatureis much lower, ∼ × K (Almy et al. 2000). Assum-ing a mean B o ∼ µ G in equilibrium with the thermalpressure, and taking the same t m of 25 Myr, and a longer l o ∼
50 pc, we expect a mean free path λ k ∼
30 pc fromequation (6). Thus, with a slowing down time from equa-tion (5) of t sd ∼ × yr, the slowing down length l sd ∼ l B ∼ P V H : V H ∼ Al) production fractions, wewould expect the overall P V H : V H ∼ > P V H : V H ∼ .
02 and P V H : HII ∼ . , while P V H : Be ∼ . . Thus, only 2% of the positrons born in the VH anddiffusing along meandering magnetic flux tubes, wouldbe expected to slow down and annihilate there beforethey moved into the HII shells surrounding the molecularclouds.
Photoionized Medium (HII)
From the source-weighted filling factors we expect thatonly ∼ n HII ∼
100 cm − and athickness, ∆ r HII ∼ x e ∼
1, in view of the expected presence of the small-scale, ionized density fluctuations. Thus, a positronslowing down time, t sd , is ∼
450 yr from equation(5). We model the spatial scale of the HII turbulence, l o ∼ . r HII ∼ r HII wide (Pope 2000).Further, we find that ( u HII ) / ∼
25 km s − from our as-sumption that turbulent pressure is about 6 times that ofthe thermal pressure. From equation (6) we expect that λ k is very small, ∼ . × − pc. In the slowing downtime, t sd , these positrons diffusing along a flux tube inthe HII plasma will be distributed over a mean length l sd ∼ (2 λ k ¯ βct sd / / ∼ l ′ B ∼ (4 / r + ∆ r ) − r ] / ( r + ∆ r ) ∼ l o . For a tur-bulent scale equal to a quarter of the HII shell thicknessor ∼ l B ∼ ( l ′ B /l o ) l ′ B ) ∼ ∼ l sd , which is much lessthan the mean flux tube length l B through the HII. Sincethese positrons are effectively born close to the boundary,the other half are expected to escape back into the VH.But there, because of the close proximity of the clouds,the escaping positrons quickly diffuse into another HIIenvelope, where the process is repeated, and soon virtu-ally all are slowed down and annihilated in the HII. Thusthe net propagation fractions are effectively P HII : HII ∼ P HII : V H ∼ P HII : HI ∼ Thus we expect essentially all of the positrons that areeither born in the inner bulge or diffuse into it from be-yond, slow down and annihilate in the HII envelopes,making them the annihilation trap of the region.
Middle Bulge (0.5 < R < As Ferri`ere, Gillard, & Jean (2007) have pointed out,“reliable observational information” in this region of theGalaxy is very sketchy and the gas distributions are veryuncertain. The middle bulge from 0.5 to 1.5 kpc is dom-inated by a tilted elliptical interstellar gas disk, observedprimarily in CO and HI lines, with a hole in the middleof sufficient size to envelop the CMZ (Ferri`ere, Gillard,& Jean 2007). They suggest that the most plausible diskmodel is that of Liszt & Burton (1980) with a semi-majoraxis of 1600 pc and a semi-minor axis of 600 pc, and acentral hole are 800 pc and 260 pc, all tilted by 29 o rel-ative to the Galactic plane. Thus, as can be seen inour Figure 1, this disk appears in a more face-on pro-jection on the plane of the sky and extends ∼
800 pcabove and below that plane. This is twice the nomi-nal 400 pc scale height of the overall Galactic thick diskmodeled in the SPI/INTEGRAL analyses of the 511 keVline emission (Weidenpointner et al. 2007, 2008). Thus,we might expect that much of emission from positronannihilation in the Tilted Disk would not have beencounted as part of the modeled thick planar disk fittedfrom the SPI/INTEGRAL data and would instead havebeen counted as part of their modeled extended sphericalbulge from which it is essentially indistinguishable.Here we model the interstellar phenomena of this outerportion of the tilted interstellar gas disk, again assumingthat a hot tenuous plasma permeates the region, pressureequilibrium exists among all the interstellar phases, mag-netic pressure is in equilibrium with gas thermal pressure,and the properties of the HI, and H regions are relatedto each other via the PDR scenario.This disk contains an estimated ∼ . × M ⊙ ofH and the corresponding disk-averaged gas density is < n H > ∼ − (Ferri`ere, Gillard, & Jean 2007).In the 80% of the disk beyond 0.5 kpc, we assume thatthe H also resides in compact molecular clouds whichare roughly similar to those in the CMZ with a nominal molecular core radius of 5 pc, but because of the lowerexternal pressure, these clouds have a lower mean H density n H ∼ − . Thus, the typical core mass, M H core is ∼ . × M ⊙ , and the number of suchcloud cores, N H core , is ∼ componentwe expect a cloud H filling factor in the outer part ofthe tilted disk, f H ∼ . / ∼ ∼ v ) to shield the underlying H gas from dissociation via FUV absorption (Wolfire, Hol-lenbach & Tielens 1993). A visual extinction, A v ∼ N ( HI ) ∼ × cm − (Tielens 2005). Thus to shield the tilted diskmolecular cloud cores from dissociation, n HI ∆ HI of theoverlying HI shells, ∼ × cm − . If we assume thatthe nominal value of the HI density n HI ∼ − ,half the density of the H cores, then ∆ r HI ∼ f HI ∼ [(1 + ∆ r HI /r H ) − × (4 . / ∼ ROSAT ≤ K), tenuous ( ≤ − cm − ) plasma. However, this thermal HM pressure issignificantly greater than the thermal pressure at which both cold HI and warm HI can coexist, based on Wolfireet al. (2003). Thus, we expect that the cold HI is theonly stable HI phase in the tilted disk regionUnlike the CMZ and inner tilted disk region, the outertilted disk beyond around 0.5 kpc does not appear tohave any significant HII component. Although a popu-lation of molecular clouds exists in the Tilted Disk, nogiant HII regions, standard tracers of massive star forma-tion, have been resolved in this region (Smith, Biermann,& Mezger 1978). Moreover, no stars younger than 200Myr have been found here by van Loon et al. (2003) inthe mid-IR survey data from DENIS and ISOGAL. Sowe conclude that the stellar Lyman continuum flux andits resulting HII component are negligible in this region.Therefore the filling factors in the tilted disk are f H ∼ f HI ∼ f HM ∼ f c = f HI + f H ) implies a mean distance betweenneutral cloud centers d cc ∼ r c /f / c of ∼
60 pc, and amean distance from a random point in the disk intercloudmedium to a cloud surface of ( d cc / − ( r c + ∆ r HI ) ∼ ∼
800 pc above the Galactic plane. For as we saw fromequation (3) and Figure 2, 55% of these supernovae arein the stellar bulge population, distributed throughoutthe spherical shell from 0.5 to 1.5 kpc, and the other 45%are in the disk population, concentrated in the Galacticplane with a scale height of ∼ ∼ l B ∼ f HM ∼
1, and the other filling factors in the full middlebulge volume are f H ∼ × − and f HI ∼ × − .Thus Q HM ∼ Q Bm ∼ (0 . ± . × e + s − , and Q H ∼ Q HI ∼ cloud cores, where theyslow down and annihilate. A HM ∼ Q HM P HM : HM ,A HI ∼ Q HM P HM : HI P HI : HI ,A H ∼ Q HM P HM : HI P HI : H P H : H , (12)where, like those discussed above, the propagation frac-tions P , subscripted HM:HM, HI:HI and H :H , are thefractions of the positron that slow down and annihilatein that phase, independent of whether they are producedin that phase or escaped into it from an adjacent phase,as given by the other P combinations of escape from onephase to another adjacent phase. These fractions andannihilation rates are also listed in Table 2. Hot Medium (HM)
Since the hot medium fills effectively all of the middlebulge, we expect that all of the positrons produced be-tween 0.5 and 1.5 kpc are born in the HM. Based onthe analysis of Almy et al. (2000) at a Galactic pla-nar radius of ∼ T HM ∼ × K, the den-sity n HM ∼ × − cm − . Thus the positron slowingdown time t sd ∼ × yr from equation (5).As with the VH, we estimate the diffusion mean freepath of the positrons in the HM as follows. We assume,as did Spergel & Blitz (1992) for the CMZ, that the mag-netic field pressure is in equilibrium with the gas pres-sure. Thus we expect B o ∼ √ πp th ∼ µ G. As withthe CMZ a plausible constraint on the nature of such tur-bulent flows is that the rate at which turbulent energy isdissipated is less than the rate at which the hot plasmacools. Almy et al. (2000) estimate the cooling time ofthis hot plasma, t c ∼ × yr, but we again considera conservative value of ∼ × yr. Approximating anouter scale, l o ∼
50 pc for a hypothetical turbulent HMflow, this cooling time limits ( u ) / to be less than 45 kms − for a thermal energy density, ǫ th ≈ . th ≈ − erg cm − . Since the estimated Alfv´en speed, V a ∼ − , this limit on ( u ) / corresponds to λ k ∼ t sd of9 × yr, HM positrons can diffuse a mean distance l sd ∼ (2 λ k ¯ βct sd / / ∼ l B ∼ P HM : HM ∼ l B / l sd ∼ ∼ P HM : HII ∼ . ∼ P HM : HI ∼ P HM : H ∼ Cold Medium, (HI)
As with the molecular clouds in the CMZ, we assumethat only a fraction of the pressure is thermal in thetilted disk molecular cloud cores, so the mean temper-ature T H ∼
20 K and density n H ∼ cm − ,although we assume they also have nominal radii of ∼ cores.In the overlying HI envelopes between the cold H cloud cores and the hot HM plasma, we assume a meandensity of n HI ∼ − at a temperature T HI ∼
100 K, giving it thickness ∆ r HI ∼ t sd is ∼
90 yr from equation (5), and since theMHD cascade is also damped by ion-neutral friction, theisotropic streaming positrons are distributed along a fluxtube over a mean length l sd ∼ ¯ βct sd / ∼ l ′ B ∼ (2 / r + ∆ r ) − r ] / ( r + ∆ r ) ∼ l B ∼ ( l ′ B /l o ) l ′ B ) ∼
12 pc, expected for an turbulent scale l o ∼ . r HI ∼ ∼ ∼
69% of all these positrons are expectedto diffuse into the HI from the HM. As with the positronsthat diffuse into the HII envelopes of CMZ clouds, thesepositrons are all effectively born within a scattering meanfree path of the surface. Thus, since l sd < l B , we expectalactic Positron Annihilation 17also that ∼ cores below, and theremainder escape back into the HM above. There, againbecause of the close proximity of the clouds, the escapingpositrons quickly diffuse into another HI envelope, wherethe process is repeated, and soon virtually all are sloweddown and annihilated in the HI. Thus the net propaga-tion fractions are effectively P HI : HI ∼ P HI : V H ∼
0. Although the density of the HI envelopes are quite un-certain, the expectation that they are thick enough toslow down and annihilate the positrons is rather robust.That results from the fact that the thickness of the HIand the slowing down length are both inversely propor-tional to the density, because the column depth of theHI, which shields the H from dissociation, is fixed andso is the required column depth for the slowing down.Therefore, l ′ B , l sd , l o , l B and l B /l sd all nominally scaleproportionally. Thus, we expect most ( ∼ As we discuss in the summary ( § Outer Bulge (1.5 < R < Based on the stellar disk distribution of SNIa andSNIp, as discussed above, we see from Figure 2 that ∼
30% of all the stellar disk positrons and ∼
22% of allthe stellar bulge positrons produced by them should bemade in the outer bulge between 1.5 kpc and 3.5 kpc,and, as we now show, they become a major contributorto the modeled positron bulge component of the observedannihilation radiation within 1.5 kpc.This region has been known for some time (Faber &Gallagher 1976; G¨usten 1989; Dehnen & Binney 1998) tocontain little HI, as traced by 21 cm radio line emission(Lockman 1984), and little H , as traced by 2.6 mm lineof CO (Sanders, Solomon & Scoville 1984; Robinson etal. 1988). The lack of neutral gas results from a “highpressure galactic wind” (e.g. Blitz et al. 2007, Breg-man 1980), driven primarily by stellar bulge supernovaewithin ∼ n ∼ × − cm − av-eraged over the scale height of ∼ B o ∼ µ G, an outer scalefor MHD turbulence, l o ∼
75 pc, based on simulations ofsupernova-driven MHD turbulence (de Avillez and Bre-itschwerdt 2004), and, finally, we assume that the rootmean square strength of the turbulent magnetic field δB/B o ∼ λ k ∼
14 pc. Assuming that the bulge wind blows the magnetic field out radially, the mean length of a fluxtube through the region is l B ∼ t sd ∼
23 Myr from equation (5). In that time the positronsdiffuse both directions along a flux tube a mean length l sd ∼ (2 λ k ¯ βct sd / / ∼ ∼ l B / l sd ∼ l B < P Bo : Bo ∼ P Bo : Bm ∼ ∼
57% of the outer bulgepositrons going down into the Bm from the Bo will alsomeet the same fates in the same proportions, mostly dif-fusing into the labyrinths of the tilted disk and the CMZ,annihilating in the HI and HII envelopes of the molecularclouds. Thus, of all the positrons produced in the Be weexpect P Bo : HI ∼ .
40 and P Bo : HM ∼ .
04 in the middlebulge and P Bo : HII ∼ .
13 in the inner bulge.Those positrons going beyond 3.5 kpc are also expectedto annihilate and not return, and within large uncertain-ties we assume that they have roughly equal likelihoodof going into the halo or the disk. Thus P Bo : Do ∼ P Bo : H ∼ Positron Bulge ( < We now estimate the total positron annihilation ratewithin the 1.5 kpc bulge, using the contributions of thevarious phases and the propagation fractions within andbetween them, summarized in Table 2.Using these fractions and combining equations (9)through (13), the total annihilation rate in the positronbulge < A HII ∼ (0 . ± . × e + s − ,A HI ∼ (0 . ± . × e + s − ,A V H + HM ∼ . × e + s − ,A Bi ∼ (0 . ± . × e + s − ,A Bm ∼ (0 . ± . × e + s − ,A B ∼ (1 . ± . × e + s − . (13)As we discuss in detail in the summary ( § ±
50% in the estimated propagation frac-tions between interstellar gas phases, which producecomparably large uncertainties in the individual anni-hilation rates in each, their sums are more tightly con-strained.We also see (Table 2) that although the hot phase ofthe interstellar medium fills most of the bulge, becauseof propagation it is not the site of most of the annihi-lation. We find (Table 2 and equation 13) from the ex-pected propagation and slowing down, that ∼ ±
8% ofthe bulge positrons annihilate in the HII shells surround-ing the molecular clouds in the highly ionizing radiationenvironment of the inner bulge < ∼ ∼ ±
12% of theannihilation is expected to occur in the middle bulge,0.5 to 1.5 kpc, in the neutral HI shells surrounding themolecular clouds which lack significant HII shells, via in-flight positronium formation resulting in the broad ( ∼ § ∼ P Bo : Bo ∼
14% of the positrons areexpected to slow down in the outer bulge, producingan annihilation rate A Bo ∼ . × e + s − in anessentially spherical shell contribution between 1.5 and3.5 kpc. This is consistent with the effective limit of < (0 . ± . × e + s − from that particular region, in-ferred from the best-fit modeling of the SPI/INTEGRALdata (Kn¨odlseder et al. 2005). But since the contributionfrom that region has subsequently been included as partof the model halo in the analyses of Weidenspointner etal. (2008a), as discussed above, we include A Bo in thehalo annihilation instead. Even though the annihilationfraction in the outer bulge is small, its contribution tothe observed 511 keV line flux is still significant, becauseof the high γ /e + yield of ∼ . § ∼ (0 . ± . × e + s − of the outer bulge positrons thatescape beyond 3.5 kpc into both the halo and the innerspiral arms that define the Galactic molecular ring of theinterstellar disk. As we also show below ( § POSITRON PROPAGATION & ANNIHILATION IN THEGALACTIC DISK ( > Having estimated the expected production, propaga-tion and annihilation of positrons in the positron bulge( < Do ) beyond 3.5 kpc.As we have seen, the total expected positron produc-tion rate in this region consists of those born there, Q Do ,and those that diffuse into it from the outer bulge be-tween 1.5 and 3.5 kpc, Q Bo P Bo : Do , we estimate Q Do ∼ (1 . ± . × e + s − ,Q Bo P Bo : Do ∼ (0 . ± . × e + s − . (14)We again focus on the major phases of the interstellarmedium, which fill essentially all of the outer disk vol-ume in which essentially all of the supernovae in this re-gion occur, namely: the warm predominantly neutral HImedium, (HI), the hot tenuous medium (HT), and thephotoionized medium produced by O stars: the densedense HII regions (HII) and the diffuse warm ionizedmedium (WI). The dense ionized phases together withthe molecular clouds, OB star associations and super-bubbles are all concentrated along the spiral arms (e.gDrimmel & Spergel 2001; Stark & Lee 2006), which dom-inate the structure of the outer disk. Their properties,discussed below, are summarized in Table 3.We assume nominal average values of filling factorsof these phases beyond 3.5 kpc of f HI ∼ f HT ∼ f HII ∼ f W I ∼ ∼
400 pc half thickness. There are large uncertainties inthese numbers, but since they provide only an estimateof where the positrons are born, the uncertainties do nothave a significant impact on where they eventually slowdown and annihilate, for as we have seen elsewhere, thatis dominated by their propagation. Although we do notconsider independently the other interstellar phases ofthe disk, namely the molecular H and HI clouds, be-cause of their very small filling factors ( <
1% Ferri`ere1998), we do consider their effects in the diffuse neutralmedium (HI).As we saw from the integration of equation (3) andFigure 2, 57% of the SNIa and SNIp in the disk occur > ∼ (0 . ± . × e + s − . Weassume that they occur randomly in the various phasesof the disk proportional to their filling fractions and that,as discussed above, ∼
90% of the ∼ (0 . ± . × e + s − , produced by Al decay from massive stars in thedisk, are born in the HT phase of hot superbubbles. Thuswe expect that the positron production rates in each ofthe major phases of the disk beyond 3.5 kpc are Q HT ∼ (0 . ± . × e + s − ,Q HI ∼ (0 . ± . × e + s − ,Q HII ∼ Q W I ∼ (0 . ± . × e + s − . (15)From the positron production in these phases, we es-timate the propagation fractions, listed in Table 4, andthe resulting positron annihilation in the disk beyond 3.5kpc as follows, A HT ∼ P HT : HT ( Q HT + Q HII P HII : HT + Q W I P W I : HT ) ,A HI ∼ P HI : HI ( Q HI + Q HII P HII : HI + Q W I P W I : HI ) , alactic Positron Annihilation 19 A HII ∼ P HII : HII ( Q HII + Q HI P HI : HII + Q W I P W I : HII + Q HT P HT : HII + Q Bo P Bo : HII , ) A W I ∼ P W I : W I ( Q W I + Q HI P HI : W I + Q HII P HII : W I + Q HT P HT : W I ) , (16)where, like those discussed above, the propagation frac-tions P , subscripted HT:HT, HI:HI, HII:HII, and WI:WI,are the fractions of the positrons that are either producedin that phase or escape into it from an adjacent phase,that then slow down in and annihilate in that phase,while the other P s are all combinations of escape fromone phase to another adjacent phase. That excludes onlydirect escape from the HT into the HI, or vise versa, be-cause they are separated from one another by the HIIand WI, and any return from the halo, which we assumeis negligible.Because of the large vertical scale height of the hottenuous plasma in the large blown out superbubbles, an-nihilation there is also effectively part of the halo, nota resolvable component of the ±
400 pc disk, so we add A HT to A H . Hot Tenuous Medium (HT)
Ferri`ere (2001) found that the mean HT filling fac-tor in the outer disk was a modest, f HT ∼ ∼ ∼ Al from the massive starWolf Rayet winds and SNII, which create them, and fromthe decay of Co and Ti from the random SNIa andSNIp. Thus, as discussed above, the production in theHT is expected to be Q HT ∼ (0 . ± . × e + s − ,with Q ∼ (0 . ± . × e + s − from clusteredmassive stars and Q Ia + Ip ∼ (0 . ± . × e + s − from the random SNIa and SNIp.The typical properties of the hot medium (e.g. Ferri`ere1998; Yan & Lazarian 2004) are: a nominal tempera-ture T HM ∼ × K, a scale height averaged density, n HM ∼ × − cm − , and a mean magnetic field, B o ∼ µ G. As discussed in § t cool ∼ × yr, the log average cooling time fora hot medium, whose evolution is dominated by time-dependent cloud evaporation following the prescriptionof McKee & Ostriker (1977), and a cooling time fromTielens (2005) for the above values of T HM and n HM .Using an outer scale, l o ∼
75 pc (de Avillez and Bre-itschwerdt 2004), the above cooling time limits ( u ) / to be less than 40 km s − . With V a ∼
80 km s − andthe root-mean-square δB/B o ≤ λ k ≥
10 pc. In this very tenuous plasma the positron slowing downtime is t sd ∼
23 Myr from equation (5) and the slow-ing down length l sd ∼ (2 λ k ¯ βct sd / / > l B < P HT : HT ∼ l B / l sd < . A HT ∼ P HT : HT Q HT < . × e + s − is includedin the halo A H .As with the bulge wind dominated hot medium be-tween 1.5 and 3.5 kpc, we again assume that the escap-ing positrons go into both the overlying halo and thephotoionized, WI and HII phases in the disk. For al-though the upper ends of these flux tubes are blown outinto the overlying halo, so that the positrons could allescape, their roots feed into the surrounding HII andWI envelopes that separate the HT from the HI. Thuswithin large uncertainties, we also assume the nominalcase where escape up into the halo or down into the sur-rounding photoionized gas is equally probable ( ∼ ∼
23% each).As we show below, however, we expect that only 66%of the positrons that enter either the WI or the HII,slow down and annihilate before there, while the remain-der escape into the halo, where they annihilate. Thus,we expect that the effective net propagation factors are P HT : W I ∼ . ± . P HT : HII ∼ . ± . , and P HT : H ∼ . ± . , including P HT : HT . Warm Neutral Medium (HI)
Studies by Wolfire et al. (2003) find that over mostof the Galactic disk > ∼
100 K) neutralmedium and a warm ( ≤ K) neutral medium, butthe filling factor of cold HI is negligible (Ferri`ere 1998).Further, Wolfire et al. found relative electron densitiesin the predominantly neutral HI, n HI /n i ∼ /x e ∼ f crit of equation (7), so turbulentAlfv´enic cascades are damped by ion-neutral friction (e.g.Kulsrud & Pearce 1969). Thus we expect the positronsto stream isotropically along magnetic flux tubes at ¯ βc/ − ,using the HI gas density distribution from Ferri`ere (1998)applicable at > t sd ∼ × yr from equation (5), and inthat time they are spread both directions along a fluxtube over a slowing down length l sd ∼ ¯ βc/ ∼
90 kpc.We estimate the flux tube lengths in the HI in theGalactic disk from the mean separation between super-bubbles, since are the primary agents for disrupting theinterstellar magnetic field, which otherwise tends to liein the plane of the disk. From an analysis of the Galac-tic distribution of OB associations by McKee & Williams(1997), we estimate that the mean distance between su-perbubbles in the Galactic disk, s ∼ . e − R/H R kpcas a function of galactocentric distance with H R ∼ l B ∼ s may range fromonly 1.4 kpc in the inner part of the warm disk at R l B of about 3 kpc. Since these distances are farless that the slowing down length, we would expect thatonly a very small fraction ( ∼ l B / l sd ∼ / ∼ Thus, we expect that nearly all ( ∼ . ) of the SNIapositrons, streaming along the magnetic flux tubes, escapefrom the HI into the overlying halo or the neighboringwarm ionized HII and WI, going into the latter phaseswith equal likelihood. As we noted above, we expect thatabout 66% of the positrons entering either the WI or theHI, slow down and annihilate there, while the remainderescape into the halo and annihilate. Therefore with largeuncertainties, we take for the net factors, P HI : HI ∼ . P HI : W I ∼ . ± . , P HI : HII ∼ . ± .
08, and P HI : H ∼ . ± . Photoionized Medium in the Galactic Disk
OB associations ionize the interstellar medium, andcreate localized, high-density (HII) regions as well as anextended, diffuse warm ionized medium (WIM) (e.g. Mc-Kee & Williams 1997). Ionizing radiation from such clus-tered association stars, which are embedded initially intheir parent molecular cloud, heat as well as ionize thesurrounding dense molecular gas. Ultimately such ion-ized gas expands, streams away from the molecular cloud,evacuating large cavities, finally destroying the molecu-lar cloud (Tenorio-Tagle 1979; Whitworth 1979). ThusOB associations create dense ( ≥ − ), compact( ≤ ∼ ≥
100 pc), more tenuous ( ≤
10 cm − ) envelopes of pho-toionized gas (Anantharamaiah 1985; Heiles, Reach, &Koo 1996; McKee & Williams 1997). Following McKee& Williams we model Galactic-disk HII regions as twocomponents: the dense central cores surrounding the par-ent OB associations and the extended HII envelopes. Asnoted by McKee & Williams the tenuous HII envelopescompletely dominate the HII filling factor, and also abouthalf the volume of an HII region is filled with a hot( geq K) tenuous, superbubble plasma.Analyses of pulsar dispersion measures and diffuse, op-tical line emissions require the existence of a widespread(volume filling factor ∼ ∼ ∼ − )distributed throughout the Galactic disk and lower halo(Reynolds 1984, 1991; Tielens 2005). The extended WIMhas a large scale height, ∼ ∼ ∼ Warm Ionized Medium (WI)
Based on thermal pressure balance and the assumptionthat the WI is highly ionized, the mean electron densityin the WI, n e ∼ − (Kulkarni & Heiles 1987). Theneutral fraction of the WI is not well constrained obser-vationally by optical-line diagnostics, but neutral frac-tions ∼ ∼ − , a primary WIM tracer (Cordes et al. 1991).Thus Cordes et al. ascertained that the thermal elec-trons, which for create the pulsar dispersion measures,also generate corresponding radio scatterings. Since suchradio scatterings are created by electron density fluctu-ations on spatial scales, ∼ − cm (Cordes et al.1991), significantly smaller than the mean free path forproton-HI collisions, ∼ × cm in unit-density media,means that efficient turbulent radio scattering exists inWI and, thus, HI fraction must be less than the criticalvalue, equation (7). Consequently, we assume that in WI x e ∼ n HI ¯ α HI ) − , where ¯ α HI ∼ × − cm (Tielens 2005)and the neutral HI density, n HI ∼ (1 − x e ) n e /x e . Tak-ing a nominal value of x e ∼ n ∼ n e /x e ∼ − gives a typical WI thickness, ∆ l W I ∼
40 pc, anda positron slowing down time t sd ∼ . × yr fromequation (5).Recent simulations (Esquivel et al. 2006) of the turbu-lent mixing generated by shear flow instabilities in theWI layers between the HT and HI suggest an outer tur-bulent scale l o ∼
10 pc and ( u ) / ∼
20 km s − . As-suming B o ∼ µ G, ( δB ⊥ ) / /B o ∼
1, and T W I ∼ λ k ∼ × yr we would expectthe positrons to be distributed both directions along aflux tube over a distance l sd ∼ (2 λ k ¯ βct sd / / ∼
65 pcin both directions along the flux tubes.Again assuming that the mean length l ′ B of nominallyparallel flux tubes through these thick turbulent phasesis equal to the typical thickness of the region, ∆ l W I ∼ ∼
10 pc,the mean meandering tube length l B ∼ l ′ B /l o ∼ / ∼
160 pc. The probability that those positrons initiallyborn in the WI, slow down and annihilate there beforethey escape into the neighboring HI and HT is ∼ l B / ( l B + l sd ) ∼ / ∼ P W I : W I ∼ . ± . P W I : HII ∼ . ± .
02, and P W I : H ∼ . ± . l sd < l B , we expect that ∼ P W I : W I ∼ . ± . P W I : HII ∼ . ± .
04, and P W I : H ∼ . ± . HII Regions
Although the direct positron production by supernovaein the HII regions is modest because of their small fillingfactor, a large fraction of positrons made in the neighbor-ing HT and HI escape into it. Thus, it becomes the ma-jor site of positron annihilation in the disk, because of itshigh density ( ∼
10 times that in the HI), which reducestheir slowing down time, so that most of the positronsslow down and annihilate before they can escape.McKee & Williams (1997) derived the Galactic volumefilling factor of the HII regions, as a function of galacto-centric radius, R , assuming that these HII regions repro-duced the observed Galactic plane dispersion measures.Further, they assumed the electron density is constantacross the Galactic disk, and all the R -dependence re-sides in the filling factor, f HII ( R ). They ascertainedthat f HII ( R ) = f HII (0) e − R/H R , where H R = 3.5 kpc,and in such HII regions n e = 2.6 cm − , and f HII (0) =0.13. In these HII regions, the positrons slow down in t sd ∼ . × yr from equation (5).The typical thickness an HII envelope can be deter-mined from the average dispersion measure contributedby the walls of a single large superbubble HII envelope,∆ DM ∼
70 cm − pc (McKee & Williams 1997). For amean density n e ∼ − , this corresponds to a typ-ical wall thickness, ∆ l HII ∼ . DM/n e ∼
13 pc. So,again assuming nominally parallel flux tubes, the meanlength through one of these highly turbulent walls l ′ B isalso ∼
13 pc. Although the nature of turbulent processesin the HII is poorly known, Haverkorn et al. (2004) havedetermined from the analysis of rotation measures thatthe outer scale of turbulent flows in disk HII regions, l o ,is relatively small, ∼ l B ∼ l ′ B /l o ∼ / ∼
85 pc.To compare that with the expected slowing downlength l sd , we assume B o ∼ µ G and T HII ∼ t cool ∼ . × yr. So tak-ing l o ∼ u ) / <
23 km s − , which is greaterthan the Alfv´en speed in this plasma, V a ∼ − , so( δB ) / ∼ B o , and from equation (6) we thus find that λ k ∼ ∼ . × yr, we wouldthus expect the positrons to be distributed over a length l sd ∼ (2 λ k ¯ βct sd / / ∼
15 pc in both directions alonga flux tube. Thus we expect from equation (8) that ∼ l B / ( l B + l sd ) ∼ / (85 + 15) ∼
85% of the positronsborn in one of these HII envelopes, would slow down and annihilate there before they escape into the neighboringHI and HT. We again assume that half of the escap-ing positrons go into each of these two phases, and thatfrom there some diffuse into other HII and WI regionsor into the halo and annihilate in the same proportionas those initially born in the HI and HT. Thus the ef-fective propagation fractions for those born in the HIIare P HII : HII ∼ . ± . P HII : W I ∼ . ± .
01, and P HII : H ∼ . ± . l sd < l B , we expect that ∼ P HII : HII ∼ . ± . P HII : W I ∼ . ± . P HII : H ∼ . ± . Disk Annihilation Flux Asymmetry
The asymmetry in the 511 keV line flux (Weidenspoint-ner et al. 2008b) from the inner disk provides further evi-dence of positron propagation and annihilation in the HIIand WI. We show that this flux asymmetry can be fullyaccounted for by the apparent asymmetry of positron an-nihilation in the inner spiral arms, as viewed from our So-lar perspective, without any asymmetry in the positronproduction.Modeling the inner Galactic disk in two compo-nents, Weidenspointner et al. (2008b) find a best-fitSPI/INTEGRAL 511 keV line flux of (4 . ± . × − photons cm − s − from the component between − o
40% in the 511 keV fluxfrom negative longitudes compared to that from positivelongitudes, and it amounts to an asymmetry of about20 ±
8% in the total observed disk flux (Weidenspointneret al. 2008a).The asymmetry in the Galactic spiral arms, as viewedfrom our solar perspective, can be seen in Figure 4, show-ing the current mapping of the four spiral arms deter-mined from the study by Vall´ee (2005a,b). As summa-rized in Table 4, we expect that the positron annihilationin the disk occurs almost entirely in the warm ionized HIIand WI phases, which are concentrated along the spiralarms (e.g. Drimmel & Spergel 2001). Thus, we assumethat equal fractions of the positrons produced at any par-ticular radius propagate to and annihilate in the ionizedgas in their nearest spiral.We estimate the positron annihilation rate per unitlength along each of the four arms, using the radial dis-tributions of the positron production surface density inthe disk as a function of Galacto-centric radius, R , givenin equation (3), multiplied by πRdR/ γ /e + ratio to deter-mine the differential 511 keV line luminosity and thendivide by 4 πd ⊙ , where d is the distance from the sun,to calculate the expected differential 511 keV line fluxat the earth. Finally, integrating along each arm out to ± o Galactic longitude within ± o latitude to comparewith the SPI/INTEGRAL model, we find asymmetric511 keV line disk fluxes of (3 . ± . × − photons cm − s − from the negative longitudes, and (2 . ± . × − photons cm − s − from the positive, and a negative-to-positive longitude flux ratio of 1.6 ± . ± . × − photons cm − s − and(2 . ± . × − photons cm − s − , and a negative-to-positive flux ratio of 1.8 ± Thus we show that the observed (Weidenspointner etal. 2008b) SPI/INTEGRAL asymmetry in the 511 keVline flux from the inner disk can be fully explained by theapparent asymmetry in the inner spiral arms as viewedfrom our Solar perspective, which leads to different line-of-sight intensities, without any difference in the positronannihilation or production rates.This asymmetry in theannihilation radiation from the Galactic disk is a productof the Norma arm dominating the negative longitudes dueto its proximity to the inner region of the Galaxy, andthe weakness of the Sagittarius-Carina arm due to itsgreater distance from the Galactic center from which theannihilation rate falls off exponentially from the Galacticcenter and its lower density/activity (Drimmel & Spergel2001) compared to the other 3 arms.
Disk Positron Annihilation > We now estimate the total outer disk annihilation ratefrom the contributions of the various phases and thepropagation fraction within and between them. Theseare summarized in Table 4.As we have discussed above, we see that more thanhalf of the positrons, produced in the dominant phasesof the interstellar medium in the disk, the HI and HT,escape into adjacent phases before they can slow downand annihilate. Only those produced in the HII and WIphases have a large chance of annihilating before theyescape, and the rest either end up in the HII, the WI, orthe halo.
This result is quite robust, even allowing thelarge uncertainties in the relative likelihood of positronsfollowing magnetic flux tubes that lead into these threeannihilation traps.
Thus, we expect that the total annihilation rate in thedisk beyond 1.5 kpc is A D ∼ (0 . ± . × e + s − . As we discuss in detail in the summary ( § POSITRON PROPAGATION & ANNIHILATION IN THEGALACTIC HALO
Lastly we consider the halo, which for comparison withthe SPI/INTEGRAL analyses we take to lie beyond 1.5kpc from the Galactic center and more than 0.4 kpcabove the disk. Although it has no direct positron pro-duction by supernovae, a large fraction of positrons madein all the underlying phases of the bulge and disk can es-cape into it. Thus, it too becomes a major site of positronannihilation, because its pervasive hot tenuous plasma,coming primarily out of the outer bulge and disk super-bubbles (e.g. Norman & Ikeuchi 1989), may be turbu-lent enough to produce a scattering mean free path muchsmaller than its size, so that most of the positrons thatenter it slow down and annihilate before they can escapeon out of the Galaxy.From the positron production in all of the regions andphases and their propagation fractions, discussed above,we estimate the positron escape into the halo and theirannihilation there to be, A H ∼ P H : H ( Q Bi P Bi : H + Q Bm P Bm : H + Q Bo P Bo : H + Q HT P HT : H + Q HI P HI : H + Q W I P W I : H + Q HII P HII : H ) , (17)Extensive observations of X-ray emission from theGalactic halo (e.g. Snowden et al. 1998; Pietz et al.1998) have shown that hot tenuous plasma filling it isessentially indistinguishable from that in the outer bulgeand superbubbles from which it is derived. Thus weassume a similar T of 1 × K, a density of 2 × − cm − , and magnetic field, B o ∼ µ G. This plasmahas a scale height, z XR , ∼ z > l o ∼
75 pc and ( B ⊥ ) / /B o ≤ λ k ∼
14 pc from equation (6).Assuming a minimum flux tube escape length l B ∼
10 kpc, roughly 2 times the scale height, then the ex-pected positron escape time t esc ∼
200 Myr. By com-parison the positron slowing down time, t sd ∼
23 Myrfrom equation(5), and in that time positrons travel l sd ∼ (2 ¯ βct sd λ k / / ∼ l sd < l B . Thus, for allof the positrons that enter the halo, we expect P H : H ∼ ∼ (0 . ± . × e + s − . As we show in Table 6 and dis-cuss further in the summary, this annihilation rate givesan average 511 keV line flux from the halo of ∼ (1 . ± . × − cm − s − , depending on how much refrac-tory dust survives in the halo. This flux together withthe bulge flux of ∼ (0 . ± . × − cm − s − givesa combined bulge and halo flux of ∼ (2 . ± . × − cm − s − (see Table 7), which is quite consistent withthat of ∼ (2 . ± . × − cm − s − , determinedfrom the SPI/INTEGRAL best-fit flux model (Weidens-pointner et al. 2008a).As we discuss further in the summary, not only the511 keV line intensity but also the Galactic averages ofboth its broad/narrow line flux ratio and the positron-ium fraction are very sensitive to the mean dust contentof the hot halo plasma. For as Jean et al. (2006) haveshown and we list in Table 6, if the refractory elementsare all in dust in the hot plasma, only ∼
18% of the re-sulting positron annihilation occurs via positronium for-mation, and fully ∼
90% of the 511 keV line emission isnarrow (FWHM ∼ ∼
11 keV). This gives a halo flux of ∼ (1 . ± . × − cm − s − in the narrow line alonewhich in turn gives a combined bulge and halo flux of ∼ (2 . ± . × − cm − s − , which is likewise quiteconsistent with that from the SPI/INTEGRAL best fitabove. Combined with disk flux, that would give a totalGalactic bulge, disk and halo flux of ∼ (3 . ± . × − cm − s − which is also quite consistent with the com-bined best-fit flux of ∼ (2 . ± . × − cm − s − ,from the SPI/INTEGRAL analyses (Weidenspointneret al. 2008a). This also gives Galactic average val-ues of the positronium fraction of only ∼ . ± . ∼ . ± .
04, for which the best-fit SPI/INTEGRALvalues have not yet been published.If on the other hand, the grains have all disintegrated,the halo flux would still be also as large, ∼ (1 . ± . × − cm − s − , but it would all be in the very wide ( ∼
11 keV) 511 keV line, so the Galactic average value ofthe broad/narrow 511 keV line flux ratio would jump to ∼ . ± .
29, and the positronium fraction would in-crease only slightly to ∼ . ± .
13. The actual dustcontent of the halo undoubtedly lies in between and theSPI/INTEGRAL observations can provide a unique mea-sure.This is also consistent with that implied by theSPI/INTEGRAL best-fit flux model of A H ∼ (0 . ± . × e + s − (Weidenspointner et al. 2008a), scaledto a galacto-centric distance of 8 kpc and ( e + /γ ) ∼ . ± .
07 expected (Jean et al. 2006) in the halo plasma,as discussed above. SUMMARY
The recent analyses by Weidenspointner et al. (2007,2008a) of the SPI/INTEGRAL measurements of theGalactic 511 keV positron annihilation radiation lumi-nosity suggest that the best fit to the observations con-sists of three basic components: 1) a spherical bulge of ∼ ±
400 pc) Galactic disk, and 3)a spherical halo extending beyond 1.5 kpc. Their most recent best-fit model (Weidenspointner et al. 2008a),scaled to a galacto-centric distance of 8 kpc, gives 511keV line luminosities of L B ∼ (0 . ± . × photonss − from the bulge, L D ∼ (0 . ± . × photons s − from the disk, and a net L H ∼ (1 . ± . × photonss − from the halo beyond the bulge, for a total Galactic511 keV luminosity L ∼ (2 . ± . × photons s − .This gives a positron annihilation luminositybulge/disk ratio of ∼ ± Here we have shown, however, that the measured 511keV luminosity ratio can be fully explained by positronsfrom the decay of radionuclei made by explosive nucle-osynthesis in supernovae, if the propagation of these rel-ativistic positrons in the various phases of the interstellarmedium is taken into account, since these positrons mustfirst slow down to energies ≤
10 eV before they can anni-hilate, and if the geometry of the sources is also consid-ered. Moreover, as Jean et al. (2006) have shown, with-out propagation none of the proposed positron sources,new or old, can explain the two fundamental propertiesof the Galactic annihilation radiation: the fraction of theannihilation that occurs through positronium formationand the ratio of the broad/narrow components of the 511keV line.
Positron Production
First, we have shown, using recent estimates (Cap-pellaro, Evans & Turatto 1999) of the SNIa and SNIprate in our Galaxy of ∼ ± β + -decay chains ofthe radioactive nuclei, Ni, Ti, and Al, produced inthese supernovae can fully account for the total Galacticpositron production rate of ∼ (2 . ± . × e + s − ,implied by the best-fit SPI/INTEGRAL analyses (Wei-denspointner et al. 2008a).The total Galactic production rate of positrons fromlong lived (1.04 Myr meanlife) decay of Al of ∼ (0 . ± . × e + s − , produced by massive stars, is deter-mined directly from the SPI/INTEGRAL measurements(Diehl et al. 2006) of the Galactic luminosity of the 1809keV line emission which accompanies the decay. TheGalactic positron production rate the Ti decay chain(89 yr meanlife) of ∼ (0 . ± . × e + s − can be de-termined from the relative Galactic abundances (Lodders2003) of Ca and Fe, which are primarily produced by Ti and Ni decays, using the calculated SNIa rate and Ni yield, assuming that these supernovae presently pro-duce about half of the Fe and core collapse supernovaeproduce the other half. The Galactic positron produc-tion from the much shorter lived (111.4 day meanlife)decay chain Ni, however, depends on an additional fac-tor, since most of these decay positrons slow down andannihilate in the dense ejecta and only a fraction, f survive into the interstellar medium. We showed thatall of the additional positron production, amounting to ∼ . × e + s − , can be accounted for by the Ni de-cay chain with a positron survival fraction in SNIa ejectaof f ∼ ± This is the time-integrated mean of ∼
5% calculated (Chan & Lingenfelter 1993 Fig. 3) from4 Higdon, Lingenfelter and Rothschildthe standard deflagration models of SNIa (Nomoto et al.1984, W7), assuming combed out magnetic fields, and itis also consistent with the mean fraction of ∼ . ± ∼
40 to 50 %,they are necessarily anticorrelated. Therefore their prod-uct yields an uncertainty of only ∼
25% in the productionrates.Using these positron production rates and the Galac-tic bulge and disk radial distributions of the SNIa, SNIp,and massive stars, gave an expected positron productionof (0 . ± . × e + s − in the inner bulge < . ± . × e + s − in the middle bulge from0.5 to 1.5 kpc, (0 . ± . × e + s − in the outerbulge from 1.5 to 3.5 kpc, and (1 . ± . × e + s − in the disk beyond 3.5 kpc. For a positron annihilationbulge within 1.5 kpc used in SPI/INTEGRAL observa-tion analyses, that gave a positron production bulge/diskratio of 0 . ± . Positron Propagation
We have modeled the properties of the various regionsand phases of the interstellar medium to determine thepropagation of the positrons as they slowed down and an-nihilated. In ionized plasmas we assumed that diffusivepropagation was controlled by resonant scattering of therelativistic positrons by MHD waves at their cyclotronradius. These waves are generated by the cascade ofthe larger scale magnetic turbulence down to sufficientlysmall scales which scatter the positrons and they diffusealong magnetic flux tubes. Such propagation has beenobserved in interplanetary plasmas with MeV electronsand we use a phenomenological model based on thesestudies. In neutral, or at least nearly neutral, phases ofthe interstellar medium, however, this cascade is dampedby ion-neutral collisions, and the positrons are expectedto stream along the magnetic flux tubes with an isotropicpitch angle distribution.From these propagation processes, we have calcu-lated the propagation, slowing down and annihilation ofpositrons formed in the various regions and phases of theinterstellar medium described in Tables 1 and 3. Fromthe properties of the medium in each phase, we deter-mined the propagation mode, and calculated the diffu-sion mean free path in the undamped cascade mode, orthe streaming velocity in the damped mode. From anestimate of the mean length of the magnetic flux tubeswithin a phase, compared to the tube length over whichthe positrons would propagate before they slowed down,we estimated the fraction of positrons expected to slowdown and annihilate in that phase before they escape.We then estimated the relative fractions of the escapingpositrons that go into each of the adjacent phases by sim-ple geometric arguments. We have assumed large uncer-tainties of ±
50% in the estimated propagation fractionsbetween phases, which produce comparably large uncer-tainties in the annihilation rates in each phase, althoughtheir sums within the bulge and disk are more tightlyconstrained. These propagation fractions between var- ious components and the resulting distribution of theirannihilation rates are presented in Tables 2 and 4.
Positron Annihilation Rate & Flux
The expected positron production and annihilationrates in the different regions and phases of the interstel-lar medium are summarized in Tables 5 and 6. Finally,the expected positron annihilation rates and 511 keV lineproperties are compared with SPI/INTEGRAL analysesin Table 7.We show that ∼ ±
7% of the all the Galacticpositrons are expected to be produced within the bulge < ∼ ±
6% are born within the positronbulge < ∼
80% of them diffuseinto the warm HII and cold HI envelopes of molecularclouds that lie within 1.5 kpc, where they slow downand annihilate, while the remaining ∼
20% escape intothe halo and disk beyond. This propagation thus re-sults in a positron bulge annihilation rate < A B ∼ (1 . ± . × e + s − .Of the positrons that are produced in the Galactic diskbeyond 3.5 kpc, we expect from the relative filling factorsthat that about 70% of the disk positrons are born eitherin the ubiquitous warm neutral HI medium of the disk,or in the hot tenuous plasmas of superbubbles, in bothof which the conditions are such that positrons rapidlystream or diffuse along magnetic flux tubes. Some seg-ments of these flux tubes thread through the warm ion-ized gas and HII shells that separate the warm neutralgas from the hot tenuous plasmas, while other segmentshave been blown up into the halo by the hot expand-ing superbubbles (e.g. Parker 1979), and, since there isno preferred direction, the positrons diffusing or stream-ing along them should have roughly equal likelihood ofeither going into the warm denser ionized shells, wherethey mostly slow down and annihilate, or out into thehalo. From the estimated propagation fractions, assum-ing large uncertainties, we find that nearly half of thepositrons enter the ionized shells and annihilate in thedisk, and that other half escape into the overlying halo,giving a total annihilation rate in the disk beyond 3.5kpc is A D ∼ (0 . ± . × e + s − . So even though roughly equal numbers of positrons areborn in the bulge within 3.5 kpc and in the disk beyond,their annihilation rate is higher in the bulge than in thedisk, because ∼
80% of those born in the interstellar bulgeslow down and annihilate in the cloud envelopes within1.5 kpc, while only ∼
51% of those born in the disk slowdown and annihilate before they escape into the halo.
We now compare the spatial distribution of the Galac-tic positron annihilation radiation, expected from the β + -decay chains of the radioactive nuclei, Ni, Ti,and Al, produced in supernovae, with that observedby SPI/INTEGRAL.The most direct comparison is through the 511 keV linefluxes. From more than four years of SPI/INTEGRALobservations Weidenspointner et al. (2008a, model BD)fit the 511 keV fluxes and luminosities to a combinedbulge component within a spheroidal Gaussian (FWHMalactic Positron Annihilation 25of 1.5 kpc) distribution and thick disk component. Theyfound best-fit fluxes of ∼ (0 . ± . × − photonscm − s − from the bulge and ∼ (0 . ± . × − photons cm − s − from the disk. They also found a best-fit combined halo and bulge flux of ∼ (2 . ± . × − photons cm − s − .For comparison we calculate in Table 5 the expectedfluxes from the annihilation rates in the different re-gions and phases listed in Tables 2 and 4, using the511 keV photons yields per positron expected in thosephases (Guessoum, Jean & Gillard 2005; Jean et al.2006). Thus, we expect 511 keV line fluxes of ∼ (0 . ± . × − photons cm − s − from the bulge, ∼ (1 . ± . × − photons cm − s − from the disk,and ∼ (2 . ± . × − photons cm − s − from thecombined halo and bulge. These expected fluxes, as sum-marized in Table 7, are all in excellent agreement withthe best-fit fluxes observed by SPI/INTEGRAL (Wei-denspointner et al. 2008a).We also calculated the expected 511 keV fluxes fromthe inner bulge within 0.5 kpc and the middle bulgebetween 0.5 and 1.5 kpc, ∼ (0 . ± . × − and ∼ (0 . ± . × − photons cm − s − , respec-tively. As mentioned above, these boundaries were orig-inally chosen for our calculations because these were thetwo regions modeled in the previous analysis by Wei-denspointner et al. (2007) of the first two years ofSPI/INTEGRAL, where they found best-fit fluxes of ∼ (0 . ± . × − and ∼ (0 . ± . × − pho-tons cm − s − from the same regions. These expectedand observed values are also consistent within the 1 − σ uncertainties.We have compared with the earlier analysis, becausethe fluxes in the later Gaussian component analysis by(Weidenspointner et al. 2008a) overlapped in the innervolume which was also smaller than 0.5 kpc. The best-fit total fluxes from the analyses of (Weidenspointner etal. 2007) and (Weidenspointner et al. 2008a) within 1.5kpc are essentially the same, ∼ (0 . ± . × − and ∼ (0 . ± . × − photons cm − s − , respectively.We also compare the expected positron annihila-tion rates in the bulge, disk and halo, and the cor-responding bulge/disk ratio, with those determinedby Weidenspointner et al. (2008a) from their best-fit SPI/INTEGRAL observations. Since they used agalacto-centric distance of 8.5 kpc rather than the 8 kpc,which we assumed, we scale our annihilation rates by(8 . / for comparison. In addition, since ∼
12% oftheir disk annihilation component lies within 1.5 kpc,we further modify our expected annihilation rates tocompare with their model components. Thus, to com-pare with the Weidenspointner et al. (2008a) values,we calculate an equivalent SPI disk annihilation com-ponent
SP I A D ∼ A D (8 . / / (1 − . ∼ (0 . ± . × e + s − . This is in very good agreementwith the SPI/INTEGRAL value of (0 . ± . × e + s − . Similarly, we calculate an equivalent SPI bulgeannihilation component SP I A B ∼ ( A B − A D [0 . / (1 − . . / ∼ (1 . ± . × e + s − , which isalso in very good agreement with the SPI/INTEGRALvalue of (1 . ± . × e + s − . Thus, we expect an equivalent bulge/disk ratio of theannihilation rate
SP I A B / SP I A D ∼ (1 . ± . / (0 . ± . ∼ . ± . , which is also in excellent agreement with the measured value of (1 . ± . / (0 . ± . ∼ . ± . from the SPI/INTEGRAL data analyses ofthe best-fit bulge and disk components (Weidenspointneret al. 2008a). By comparison we also note, however,that the ratio of the total bulge annihilation with 1.5kpc compared to that in the disk beyond is A B /A D ∼ (1 . ± . / (0 . ± . ∼ . ± .
45. This is nonethe-less consistent with the SPI/INTEGRAL best-fit modelwhen the 12% disk component within 1.5 kpc is sub-tracted from their disk component and added to that ofthe bulge, ∼ (0 .
94 + 0 . / (0 . − . ∼ . ± . ± o that show a best-fit excess of 80 ±
40% in the 511keV line flux from the negative longitudes compared tothat from the positive longitudes. As we have shown ( § < > SP I A H + B ∼ ( A B + A H )(8 . / ∼ (1 . ± . × e + s − . From their combined haloand bulge flux, discussed above, Weidenspointner et al.(2008a) estimate a significantly larger combined annihi-lation rate of ∼ (3 . ± . × e + s − .But they use a uniform e + /γ ∼ .
82 to convert fromobserved the 511 keV photons luminosity to an assumedpositron annihilation rate. This conversion factor, how-ever, was based on a positronium fraction of ∼ e + /γ ∼ . ± .
07, as discussed above. Thuswe use this much more appropriate conversion factor forflux contribution from the halo beyond the bulge, whichfrom the best-fit fluxes amounts to roughly 65% of thecombined halo and bulge flux. We estimate an annihila-tion rate of
SP I A H ∼ (0 . / . . − . × ∼ (0 . ± . × e + s − just for the halo beyondthe bulge, which, adding back the bulge contribution,gives an adjusted best-fit SPI/INTEGRAL model of A H + B ∼ (1 . ± . × e + s − . Those values wouldbe in good agreement with the expected equivalent SPIannihilation components, SP I A H ∼ (0 . ± . × and SP I A H + B ∼ (1 . ± . × e + s − . Positronium Formation ± ±
6% (Churazov et al.2005), 95 ±
3% (Jean et al. 2006) and 92 ±
9% (Weidens-pointner et al. 2006), and 2) the observed ratio of broad( ∼ ∼ ∼ ± < . ± . × e + s − annihilate in the warm ionized and HII phases of thedisk, (0 . ± . × e + s − annihilate in the warmHII phase of the bulge, and (0 . ± . × e + s − annihilate in the cold neutral phase of the bulge. In thesephases positronium is formed ∼
89% and ∼
95% of thetime, respectively (Guessoum, Jean & Gillard 2005; Chu-razov et al. 2005; Jean et al. 2006), so we would expect(as outlined in Table 6) a combined positronium annihi-lation rate of (1 . ± . × e + s − out of a totalpositron annihilation rate of (1 . ± . × e + s − for a total Galactic positronium fraction f P s sim ± ∼
56% isin the disk and ∼
44% is in the bulge, while only ∼ ∼
98% is in the bulge. This gives an expected positroniumfraction of ∼ ±
2% in the bulge and ∼ ±
2% in thedisk (Table 6).
The mean positronium fraction of f P s ∼ ±
2% of the Galactic positron bulge within 1.5 kpc is invery good agreement with the mean of 94 ±
6% measuredby SPI/INTEGRAL (Churazov et al. 2005; Jean et al.2006; Weidenspointner et al. 2006) from the inner ± o of the Galaxy. Broad/Narrow 511 keV Line Ratio
We also consider the ratio of the broad and narrowcomponents of the 511 kev line fluxes, since the bulgeand disk values are expected to be quite different (Table6), and compare the expected values with those measuredby SPI/INTEGRAL (Table 7).In the warm ionized and HII phases, essentially all( ∼ ∼ γ
511 keVemission 25% of the time and 3 γ continuum emission therest of the time. In the cold neutral phases essentially allof the positronium is formed by charge exchange in flightproducing the broad ( ∼ ∼
2% of the neutral phase positronium for- mation, the fraction of in-flight formation is ∼
70% forthe typical weakly ionized ( x e ∼ . ± . × e + s − frompositronium formed in-flight and an annihilation rate of(0 . ± . × e + s − from that formed thermally byradiative combination. With positronium producing two511 keV photons in 25% of its annihilations, this givesa broad 511 keV line luminosity of (0 . ± . × γ s − from in-flight annihilation and a narrow line lumi-nosity of (0 . ± . × γ s − from thermal annihi-lation. However, the residual direct thermal annihilationof (0 . ± . × e + s − from non-positronium pro-cesses also produces 2 narrow 511 keV photons from ev-ery annihilation with a luminosity of (0 . ± . × γ s − , which gives a total narrow 511 keV line luminosityof luminosity of (0 . ± . × γ s − .We expect (Tables 6 & 7) that of the bulge and diskannihilation that 68% occurs in the warm ionized gasversus 32% in the neutral gas. Nearly all ( ∼ ±
12% is in the ionized gasversus 51 ±
14% in the neutral. The latter values are quiteconsistent with the SPI/INTEGRAL measurements fromthe bulge region, where Jean et al. (2006) find 51 ± ±
3% in the neutral.We further expect (Table 6) that of the narrow linepositronium formed in the warm ionized and HII phases ∼
56% is in the disk and ∼
44% is in the bulge, whileagain only ∼
2% of that formed in the neutral phases isin the disk and ∼
98% is in the bulge. In the bulge thisyields a broad 511 keV line luminosity of (0 . ± . × γ s − from in-flight annihilation and a narrow lineluminosity of (0 . ± . × γ s − from thermalannihilation, for a broad/narrow line luminosity ratio of0.68 ± ∼ . × γ s − and the narrow line luminosity is(0 . ± . × γ s − , yielding a broad/narrow lineluminosity ratio of only ∼ ± πR B ∼ . × cm , where R B isthe distance to the Galactic center ∼ × πR B (Weiden-spointner et al. 2008a). Thus the broad line flux is(0 . ± . × − γ cm − s − from the bulge and(0 . ± . × − γ cm − s − from the disk, while thenarrow line flux is (0 . ± . × − γ cm − s − fromthe bulge and (0 . ± . × − γ cm − s − from thedisk. This gives a total observed broad 511 keV line fluxof (0 . ± . × − γ cm − s − and a narrow lineflux of (1 . ± . × − γ cm − s − , which gives anapparent broad/narrow line ratio of 0.24 ± ∼ Thus,we expect the apparent ratio to be quite sensitive to theeffective sky coverage, and we suggest that this angular alactic Positron Annihilation 27 dependence can test that prediction and further probe thenature and structure of the interstellar medium.
Indeed, analyses of SPI/INTEGRAL observations ofthe Galactic bulge by Jean et al. (2006) give a broadline flux of (0 . ± . × − γ cm − s − and a narrowline flux of (0 . ± . × − γ cm − s − , which givesa broad/narrow line flux ratio of 0.49 ± . ± . × − γ cm − s − and a narrow lineflux of (0 . ± . × − γ cm − s − , which gives abroad/narrow line flux ratio of 0.47 ± ∼ ± − σ . Moreover, within the positron bulge ( < Thenarrow line emission is expected to come effectively allfrom central molecular zone and inner tilted disk within0.5 kpc, arising from thermal positronium and direct an-nihilation in the warm HII shells surrounding molecularclouds, ionized by FUV from central O stars. Essen-tially all of the broad line emission on the other handis expected to come from the surrounding outer tilteddisk between 0.5 and 1.5 kpc, arising from annihilationof positronium formed in flight in the cold neutral HIenvelope of molecular clouds, where there is insufficientFUV to generate surrounding HII shells.In particular, from the positron annihilation rates inTables 2 & 6, we expect from the HII shells within 0.5kpc of the Galactic Center a narrow line flux of (0 . ± . × − γ cm − s − and no significant broad lineflux, giving an expected broad/narrow line ratio of just ∼ within 0.5 kpc. Whereas from the HI gas in thesurrounding tilted disk, we expect a broad line flux of(0 . ± . × − γ cm − s − , and a narrow line fluxof (0 . ± . × − γ cm − s − from direct thermalannihilation, giving an expected broad/narrow line ratioof ∼ ± between 0.5 kpc and 1.5 kpc, which is nearly 10times the combined full bulge average. Spectral analysisof the emission from these two regions should be able totest this prediction. Halo Contributions
The 511 keV line flux from the halo makes up nearlyhalf of the total Galactic annihilation line flux measuredby SPI/INTEGRAL (Weidenspointner et al. 2008a) andfrom our estimates of positron propagation we also ex-pect that it accounts for a very similar ( ∼ ∼
18% of the positron annihilation occursvia positronium formation, and fully ∼
90% of the 511keV line emission is narrow with a FWHM ∼ ∼ ∼ (1 . ± . × − cm − s − in the narrow line alone. That in turn gives a combined bulge and halo flux of ∼ (2 . ± . × − cm − s − , which is quite consistent with that from theSPI/INTEGRAL best fit flux of ∼ (2 . ± . × − cm − s − (Weidenspointner et al. 2008a), see Table 7.Combined with disk flux, that would give a total Galac-tic bulge, disk and halo flux of ∼ (3 . ± . × − cm − s − which is also quite consistent with the com-bined best-fit flux of ∼ (2 . ± . × − cm − s − ,from the SPI/INTEGRAL analyses. This also givesGalactic average values of the positronium fraction ofonly ∼ . ± .
12 and the broad/narrow 511 keV lineflux ratio of only ∼ . ± .
04, for which the best-fitSPI/INTEGRAL values have not yet been published.If on the other hand, the grains have all disintegrated,the halo flux would still be almost as large, ∼ (1 . ± . × − cm − s − , but it would all be in the very wide( ∼
11 keV) 511 keV line, so the Galactic average valueof the broad/narrow 511 keV line flux ratio would jumpto ∼ . ± .
29, and the positronium fraction wouldincrease only slightly to ∼ . ± .
13. The actual dustcontent of the halo undoubtedly lies in between and sincewe expect the broad/narrow 511 keV line ratio from thehalo beyond 1.5 kpc to be extremely sensitive, rangingfrom ∼ ∞ , SPI/INTEGRAL measurements of thatratio should provide a unique measure of the halo dust. CONCLUSION
In conclusion, we find that roughly half of the Galac-tic positrons from supernova generated radionuclei areproduced in he bulge and inner disk within the stellarbulge of
R < ∼ − ) warm outer shells of the molecular clouds thatlie within R < ∼ − )neutral HI at close to c , or diffuse through the hot super-bubble plasma, escape into the overlying halo before theycan stop and annihilate in the low density (0.3-3 H cm − )photoionized outer shells of clouds and superbubbles inthe disk. Thus, positron propagation easily explains thelarger annihilation flux from the bulge. Moreover, it ex-plains the high observed postronium fraction and narrowobserved 511 keV line width, that both require annihila-tion predominantly in the warm ionized gas and not inthe hot plasma in which nearly all of the positrons areproduced, either in this or in other suggested sources.We expect these conclusions to be quite robust, sincethey are based on 1) extensive observations of galactic su-pernova rates and distributions, 2) observationally con-firmed models of supernova yields of radionuclei and theirdecay-positron survival fractions, 3) wide ranging obser-vations of the properties and distributions of the variousphases of the interstellar medium and magnetic fields,and 4) the widely used photodissociation region (PDR)models of molecular cloud shells. From these we havecalculated the expected propagation of positrons in the8 Higdon, Lingenfelter and Rothschildvarious phases, assuming streaming in the neutral phasesand diffusion in the ionized gas and plasma phases, us-ing diffusion mean free paths in the latter phases calcu-lated from a two-component, phenomenological model ofanisotropic turbulence, which is consistent with observa-tions of particle propagation in our best natural labora-tory, the interplanetary medium.Thus, as we show in Table 7, when positron propa-gation is considered, the positrons from the β + -decaychains of the radioactive nuclei, Ni, Ti, and Al,produced in Galactic supernovae, can fully account for all of the features of the diffuse Galactic 511 keVand 3- γ continuum annihilation radiation observed bySPI/INTEGRAL. We have also predicted additionalmeasurable features that can not only further test suchan origin of the positrons but provide new informationon the nature of the interstellar medium.
This work was supported by NASA’s
InternationalGamma-Ray Astrophysics Laboratory
Science Program,grant NNG05GE70G.
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Fig. 1.—
A schematic view of the Galactic Bulge, Disk and Halo assumed in these calculations, showing the stellar bulge and diskdistributions of stars (+) and the assumed boundaries of the interstellar gas and plasma subdivisions, chosen for comparison with theSPI/INTEGRAL observational analyses. alactic Positron Annihilation 31 F ( < r) Fig. 2.—
The cummulative fraction of total (thick solid curve) Galactic positron production F ( < r ) in the bulge and disk as a functionof Galactic radius r from equations (3) and (4). The assumed distribution of positrons from decay of Ni and Ti, produced by SNIa andSNIp, are shown for the stellar disk (dot-dashed curve), bulge (thin solid curve) and a star burst near the Galactic Center (dotted curve),while those from decay of Al, produced by massive Wolf Rayet stars and core collapse supernovae are shown in the short dashed curve. (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)
Central Molecular Zone (CMZ)Very Hot Medium UVn ~ 0.04~10 K Plasma
10 pccmcm −3−3
100 cm −3 cm −3 Fig. 3.—
A schematic view of molecular clouds in the Central Molecular Zone of the Inner Bulge within 0.5 kpc of the Galactic Center,together with a cross section of the HII and HI envelopes surrounding the H core of a model cloud. alactic Positron Annihilation 33 −10 −5 0 5 10−10−50510 S kpc k p c −10 o +10 o −50 o +50 o NormaScutum−CruxSagittarius−CarinaPerseus
Fig. 4.—
The Galactic spiral arms, determined from the study by Vall´ee (2005), showing the asymmetry, as viewed from our solarperspective, of the inner arms within 50 ◦ to either side of the positron bulge ( ± ◦ ), measured by SPI/INTEGRAL.The asymmetry in theannihilation radiation from the Galactic disk is a product of the Norma arm dominating the negative longitudes due to its proximity to theinner region of the Galaxy, and the weakness of the Sagittarius-Carina arm due to its greater distance from the Galactic center from whichthe annihilation rate falls off exponentially from the Galactic center and its lower density/activity (Drimmel & Spergel 2001) compared tothe other 3 arms. We find 511 keV line disk fluxes of (3.8 ± × − photons cm − s − from the negative longitudes, and (2.3 ± × − photons cm − s − from the positive longitudes, and a negative-to-positive longitude flux ratio of 1.6 ± TABLE 1Assumed Properties of the Galactic Interstellar Bulge ( < Inner Bulge Middle Bulge Outer Bulge
Galactic
R < f T K (90 − ×
106 5000 150 30 5 ×
106 20 20 1 × n H cm − x × − − × − − tsd yr (1 − ×
106 450 90 20 9 ×
106 900 150 2 × B µ G ∼ ∼ ∼ ∼ ∼ ∼ ∼
12 4Fluctuation δB/B ∼ ∼ ∼ lB pc 10-300 ∼ ∼ > ∼ ∼ ∼ lo pc 10-50 – – – 50 – – 75 λ k pc 25-30 0.0017 – – 45 – – 14Slowing Down lsd pc 2000-5000 0.3 12 3 7000 12 6 7000 lB/ ( lB + lsd ) 0.01-0.03 0.96 0.30 > QX e +/s 0.31 ± ∼ ∼ ∼ ± ∼ ∼ ± alactic Positron Annihilation 35 TABLE 2Positron Propagation & Annihilation in the Galactic Interstellar Bulge ( < Inner Bulge Middle Bulge Outer Bulge
Galactic
R < X VH HII HM HI BoFilling Factor f ∼ ∼ QX e +/s 0.31 ± ∼ ± ∼ ± PV H : X ∼ ∼ ∼ PHM : X – ∼ ∼ ∼ ∼ PBo : X – ∼ ∼ ∼ ∼ ]AnnihilationInner Bulge QPV H : X e +/s ∼ ± ∼ QPHM : X – 0.08 ± ± ± ∼ QPBe : X – 0.08 ± ± ± ∼ ] AX e +/s ∼ ± ± ± ∼ TABLE 3Assumed Properties of the Outer Bulge, Disk & Halo
Outer Bulge Disk Halo
Galactic R > > f T K 1 ×
106 1 ×
106 8000 8000 8000 1 . × n H cm − x tsd Myr 23 23 0.7 0.16 0.018 23Magnetic Field
B µ
G 4 4 4 4 4 4Fluctuation δB/B ∼ ∼ ∼ ∼ ∼ lB pc 1000&2000 < ∼ > lo pc 75 75 – 10 2 75 λ || pc 14 14 – 0.2 0.1 14Slowing Down lsd pc 7000 7000 90,000 120 15 7000 lB/ ( lB + lsd ) 0.13&0.22 < > QX e +/s 0.56 ± ± ± ± ± alactic Positron Annihilation 37 TABLE 4Positron Propagation & Annihilation in the Outer Bulge, Disk & Halo
Outer Bulge Disk Halo
Galactic R > > X Bo HT HI WI HII HFilling Factor fX ± ± ± ± QX e +/s 0.56 ± ± ± ± ± PBi/Bm : X [ ∼ ∼ PBo : X [ ∼ ] – – – 0.14 ± ± PHT : X – [ < ] – 0.15 ± ± ± PHI : X – – ∼ ± ± ± PWI : X – ∼ ∼ ± ± ± PHII : X ∼ ∼ ∼ ± ± ± PH : X ∼ ∼ ∼ ∼ ∼ ∼ Annihilation
QPBi/Bm : X e +/s [ ∼ ∼ QPBo : X [ ∼ – – – 0.08 ± ± QPHT : X – [ < ] – 0.07 ± ± ± QPHI : X – – ∼ ± ± ± QPWI : X – ∼ ∼ ± ± ± QPHII : X ∼ ∼ ∼ ∼ ± ± QPH : X ∼ ∼ ∼ ∼ ∼ ∼ A e +/s [ ∼ ∼ ∼ ± ± ± TABLE 5Summary of Positron Production & Annihilation in Positron Bulge, Galactic Disk & Halo
Production Rate Annihilation Rate Production Annihilation Q A Q/QTot % A/ATot %Bulge < ± ± ± ± ± ± > ± ± ±
15 25 ± > ± ± ± ± ±
13 37 ± ± ± ± ± ± ∼ ± ∼ ∼ ± ∼ ± ∼ ∼ ∼ ∼ < ± ± ± ± ± ± > ± ± > ± ± < ∼ ± ∼ ± > ± ± ± ± > ± ∼ ± ∼ < ∼ ± ∼ ±
7* Included in Halo alactic Positron Annihilation 39
TABLE 6Galactic Positronium (Ps) Fraction & 511 keV Line Luminosities & Fluxes
Bulge Disk Halo
HII HI HII & WI HI Grains No Grains < > > > > ∼ γ
511 1.50 1.74 1.50 1.94 0.58 0.73Broad γ
511 % 0 83 0 68 10 100Narrow γ
511 % 100 17 100 32 90 0
Annihilation Rate − ± ± ± ∼ ± ± ± ± ± ∼ ± ± ± ∼ ± ∼ ± ∼ ± ± ± ± ± ∼ ± ± s Fraction 0.93 ± ± ± Luminosity γ s − ± ± ± ∼ ± ± ± ± ± ∼ ± ± ± ± ± ∼ ± ± γ Continuum 0.88 ± ± ± ± ± ± ∼ ± ∼ ∼ ± ± ± ± ± ∼ ± ± ∼ < . ± . > Narrow 511 keV Line 0.34 ± ± < . ± . > Broad/Narrow Line Ratio 0.68 ± ∼ < . ± . > Broad/Narrow Line Ratio 0.33 ± Flux − γ cm2 s − ± ± ± ∼ ± ± γ Continuum 1.15 ± ± ± ± ± ± ∼ ± ∼ ∼ ± ± ± ± ± ∼ ± ∼ ± ∼ ∞ Total 511 keV Line 0.74 ± ± < . ± . > Broad 511 keV Line 0.30 ± ∼ < . ± . > Narrow 511 keV Line 0.44 ± ± < . ± . > Broad/Narrow Line Ratio 0.68 ± ∼ < . ± . > Broad 511 keV Line 0.32 ± ± ± TABLE 7Expected & Observed Galactic Positron Annihilation Radiation*
Expected Observed Reference511 keV Line Flux − γ /cm2 sBulge Component 0.74 ± ± ± ± ± ± < ± ± ± ± < ± ± − o < l < o ∼ ± o > l > o ∼ ± − o − o )/(+50 o − o ) ∼ ± Annihilation Rate ± ± ± ± ± ± /γ ± ± Positronium Fraction
Inner Galaxy 0.92 ± ± ± ± > ± > ± Broad & Narrow 511 keV Line
Bulge Warm Ionized Gas % 49 ±
12 51 ± ±
14 49 ± ± ± ± < ∼ ± > ∼0.02 – ? – –