Galactic Positron Annihilation

The Galactic Positron Annihilation Radiation & the Propagation of Positrons
in the Interstellar Medium

J. C. Higdon W. M. Keck Science Center, Claremont Colleges, Claremont, CA 91711-5916
and California Institute of Technology, Pasadena, CA 91125111Sabbatical 2008-2009
jimh@lobach.JSD.claremont.edu
R. E. Lingenfelter and R. E. Rothschild Center for Astrophysics and Space Sciences, University of California San Diego, La Jolla, CA 92093 rlingenfelter@ucsd.edu, rrothschild@ucsd.edu
Abstract

The ratio of the luminosity of diffuse 511 keV positron annihilation radiation, measured by INTEGRAL in its four years, from a Galactic “positron bulge” ( 1.5 kpc) compared to that of the disk is 1.4. This ratio is roughly 4 times larger than that expected simply from the stellar bulge to disk ratio of 0.33 of the Galactic supernovae, which are thought to be the principal source of the annihilating positrons through the decay of radionuclei made by explosive nucleosynthesis in the supernovae. This large discrepancy has prompted a search for new sources.

Here, however, we show that the measured 511 keV luminosity ratio can be fully understood in the context of a Galactic supernova origin when the differential propagation of these MeV positrons in the various phases of the interstellar medium is taken into consideration, since these relativistic positrons must first slow down to energies 10 eV before they can annihilate. Moreover, without propagation, none of the proposed positron sources, new or old, can explain the two basic properties on the Galactic annihilation radiation: the fraction of the annihilation that occurs through positronium formation 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 the disk ( 3.5 kpc), the cascade of the magnetic turbulence, which scatters the positrons, is damped by ion-neutral friction, allowing positrons to stream along magnetic flux tubes. We find that nearly 1/2 of the positrons produced in the disk escape from it into the halo. On the other hand, we show that within the extended, or interstellar, bulge ( 3.5 kpc), essentially all of the positrons are born in the hot plasmas which fill that volume. We find that the diffusion mean free path is long enough that only a negligible fraction annihilate there and 80% of them escape down into the HII and HI envelopes of molecular clouds that lie within 1.5 kpc before they slow down and annihilate, while the remaining 20% escape out into the halo and the disk beyond. This propagation accounts for the low 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 the bulge and the disk are in the warm ionized phases of the interstellar medium. Such annihilation can also account for those two basic properties of the emission, the fraction (%) of annihilation via positronium and the ratio ( 0.5) of broad ( 5.4 keV) to narrow ( 1.3 keV) components of the bulge 511 keV line emission. Moreover, we expect that the bulk of this broad line emission comes from the tilted disk region ( kpc) with a very large broad/narrow flux ratio of 6, while much of the narrow line emission comes from the inner bulge ( kpc) with a negligible broad/narrow flux ratio. Separate spectral analyses of the 511 keV line emission from these two region should be able to test this prediction, and further probe the structure of the interstellar medium.

Lastly, we show that the asymmetry in the inner disk annihilation line flux, which has been suggested as added evidence for new sources, can also be fully understood from positron propagation and the asymmetry in the inner spiral arms as viewed from our solar perspective without any additional sources.

Subject headings:
(ISM:)cosmic rays — elementary particles — gamma rays: theory — ISM: general – nuclear reactions, nucleosynthesis — (stars:) supernovae: general

1. Introduction

The discovery (Johnson et al. 1972; Leventhal et al. 1979) of the diffuse Galactic positron annihilation radiation at 511 keV from the inner Galaxy has led to extensive studies of the possible origin of the annihilating positrons. These studies have shown (e.g. Ramaty & Lingenfelter 1979; Knödlseder et al. 2005) that, of all the various potential sources, the positrons from the decay of radioactive nuclei produced by explosive nucleosynthesis in supernovae are the largest and, hence, most likely source.

Recent measurements, however, have raised new questions about the source. Analyses of the measurements by the gamma-ray spectrometer (SPI) on ESA’s INTEGRAL spacecraft of the distribution of the diffuse Galactic positron annihilation radiation have shown that the luminosity of a Galactic positron bulge within 1.5 kpc of the Galactic Center is 1.4 times that of the Galactic disk (Knödlseder et al. 2005; Weidenspointner et al. 2007, 2008a). Their best fit models are either concentric shell sources, or Gaussian spheroidal sources, in a bulge extending out to 1.5 kpc combined with a thick disk with a half thickness of 0.4 kpc, that extends from the Galactic Center to well beyond the Solar distance of 8 kpc. They also find a large additional contribution from a spherical halo extending beyond the bulge.

In order to compare with these observations, we use these model fitting definitions of the positron bulge, thick disk, and halo, following Knödlseder et al. (2005) and Weidenspointner et al. (2007, 2008a) in all of the discussion that follows, although both have been defined in various other ways in the literature.

1.1. 511 keV Line Bulge/Disk Luminosity Ratio

The best-fit analysis of the SPI/INTEGRAL first year’s data (Knödlseder et al. 2005, Shells+D1) gave a 511 keV line luminosity of photons s in a spherical bulge from 0 to 1.5 kpc and an old, thick disk luminosity of photons s, scaled to a Solar distance of 8 kpc. This gave a bulge to disk 511 keV luminosity ratio of . The subsequent best-fit analyses of the first two years’ data, which included more coverage of the disk (Weidenspointner et al. 2007, for 4 Nested Shells and Old Disk), gave a bulge flux of photons cm s in spherical shells out to 1.5 kpc, a thick disk flux of photons cm s, and a marginally detected spherical halo flux of photons cm s from 1.5 to 8 kpc. To infer the corresponding 511 keV line luminosities, we use the flux to luminosity ratios from their earlier modeling (Knödlseder et al. 2005 Table 3), which are essentially just where for the bulge, about 0.57 for the thick disk, and about 0.93 for the halo. From the best-fit fluxes, we infer a 511 keV positron bulge luminosity photons s in the spherical shells out to 1.5 kpc, and a thick disk luminosity of photons s for a .

Most recently from more than four years of observation, Weidenspointner et al. (2008a, model BD) fit the 511 keV fluxes and luminosities to a two component narrow (FWHM of 3, or 0.42 kpc) and wide (FWHM of 11, or 1.5 kpc) spheroidal Gaussian distribution and thick disk from more extensive observations that increased the effective sampling distance to 0.75 for the disk. They found best-fit fluxes of photons cm s from the bulge and photons cm s from the disk. Scaled to 8 kpc, these fluxes give a best-fit 511 keV luminosity photons s in the spherical bulge 1.5 kpc, and a thick disk luminosity of photons s. This gives a bulge to disk 511 keV luminosity ratio of . They also found a best-fit combined halo and bulge luminosity of photons s for similar scaling, and subtracting the bulge luminosity, suggests a spherical halo luminosity of photons s beyond the bulge, for a total Galactic 511 keV luminosity photons s.

Best-fit SPI/INTEGRAL spectral analyses by Weidenspointner et al. (2008b) also show an asymmetry in the positron annihilation line flux from two opposing inner disk components at 0 to 50 to either side of the Galactic bulge. These show an % excess in the 511 keV line flux at the negative longitudes compared to the positive.

Assuming simply that the positron annihilation and production rates are in local equilibrium, these observations of the luminosities have been taken to imply a similar bulge/disk ratio and asymmetric spatial distribution for Galactic positron production. This suggested production ratio and distribution has challenged supernova source models, since it is much larger than that expected from the distribution and mean bulge/disk ratio of Galactic supernovae (e.g. Knödlseder et al. 2005; Weidenspointner et al. 2008b). As we show, however, these spatial properties can be fully explained by positron propagation.

1.2. 511 keV Line Width & Positronium Fraction

We also consider the other two fundamental spectral properties of the Galactic positron annihilation: the ratio of the broad to narrow 511 keV line emission and the fraction () of the annihilation that occurs via positronium (Ps) formation. As shown by Jean et al. (2006) neither of these can be explained by any of the proposed positron sources, either old or new, without extensive positron propagation, since all of these sources are expected to produce most of the positrons in the hot tenuous phases of the interstellar medium, which cannot be their primary annihilation site. For this reason alone a detailed treatment is required of the production, propagation and annihilation in each phase and region of the Galaxy.

Studies (e.g. Guessoum, et al. 1991; Guessoum, et al. 2005) show that positron annihilation in different phases of the interstellar medium occurs in differing ratios of direct annihilation on both free and bound electrons, to indirect annihilation via positronium formation, depending on the ionization fraction and temperature. Direct annihilation produces two photons at 511 keV, while positronium annihilation produces either two 511 keV photons or a three photon continuum depending on the spin state. Positronium is formed 25% of the time in the singlet state, parapositronium, which annihilates with a mean life of s into two 511 keV photons, while 75% of the time it forms in the triplet state, orthopositronium, 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 the Galactic 2 511 line flux to that of the 3 continuum in the bulge and disk shows that the bulk of the positrons annihilate via positronium (Ps) with % (Churazov et al. 2005), % (Weidenspointner et al. 2006), and % (Jean et al. 2006). The weighted mean positronium fraction of % also allows us to determine the total Galactic positron annihilation rates, , in the bulge and disk, since . From the above we see that , which equals 1.690.17. Thus the inferred best-fit positron bulge and Galactic disk annihilation rates are e s and e s using the previously derived L and L. In the hot tenuous plasma of the halo, however, the positronium fraction depends on the refractory grain abundance, ranging from only about 18% with narrow (FWHM 2 keV) 511 keV line emission, if all the refractories are in grains, to 42% with broad (FWHM 11 keV) if the grains all disintegrated (Jean et al. 2006). Thus we assume a possible range of there. From the best-fit halo flux, we thus infer a halo annihilation rate of e s, implying a total Galactic positron production rate of e s.

The width of the 511 keV line also depends on the temperature of the medium and its state of ionization which effects the fraction of positronium formed by charge exchange 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 this fast moving positronium produces a characteristic broad (FWHM 6 keV) component of the 511 keV line, while the subsequent thermal annihilation, either directly on free and bound electrons or via positronium formed by radiative combination or charge exchange give much narrower ( 2 keV) lines. Even the annihilation in the hot (K), tenuous ionized medium which would produce a broader (10 keV) line from the hot free elections, can instead produce a narrow line if most of the positrons annihilate on refractory dust grains (Guessoum et al. 1991; Guessoum et al. 2005). The dust, as it does with interstellar molecule formation, provides a 2-dimensional regime with much higher interaction rates for surface chemistry.

The width of the Galactic bulge 511 keV line measured by SPI/INTEGRAL from the first years’ data has been recently fitted (Churazov et al. 2005; Jean et al. 2006) by two components with about 6710% of the emission in a narrow line with a width of 1.30.4 keV and the remaining 3310% in a broad line with a width of 5.41.2 keV by positronium formation in flight, giving a broad/narrow 511 keV line flux ratio of 0.5. The combined positronium fraction and relative fluxes of the broad and narrow 511 keV line emission strongly constrain the bulge annihilation processes, as we discuss in detail below.

1.3. Positron Production & Propagation

The large difference between the ratio of the 511 keV annihilation line bulge/disk luminosity and that of the expected positron production by supernovae has led to suggestions there is some new, unrecognized source of positrons in the Galactic bulge. This seems unlikely, however, since various recent reviews (e.g. Dermer & Murphy 2001; Knödlseder et al. 2005; Guessoum, Jean & Prantzos 2006) of the potential Galactic positron sources all conclude that supernovae are still the most plausible source and that other suggested sources, including cosmic-ray interactions, novae, and various exotic processes, all seem to be weaker and much less certain.

However, as we have shown (e.g. Guessoum, Ramaty & Lingenfelter 1991), the 511 keV annihilation emission only illuminates the annihilation sites, not the sources, of the positrons, since these positrons must first slow down to energies 10 eV before they can annihilate. Thus, we examine in detail the propagation and annihilation of the -decay positrons in the bulge, disk, and halo, and we show here that the expected positron propagation and annihilation are very different in the bulge and disk, and the cloud and inter-cloud environments. When these differences are taken into consideration, we find that the spatial distribution, the 511 keV line widths and the positronium fraction can all in fact be clearly understood in the context of a Galactic supernova origin.

Although positron propagation has previously been discussed, the arguments range from the simple suggestion (Prantzos 2006) that most of the disk positrons might diffuse along dipolar field lines and annihilate in the bulge, to Monte Carlo transport simulations (Gillard et al. 2007) suggesting that they travel less than a few hundred parsecs in all but the hot phase. We therefore also carefully examine both the physics of MeV electron propagation and the observational evidence of such propagation from extensive measurements of Solar flare and Jovian electrons in the heliosphere.

Here, we examine in detail the mechanisms of the relativistic positron propagation in the various phases of the interstellar medium.

Overall, we assume that the Galactic stellar and interstellar distributions are defined by two superimposed systems: a stellar bulge-disk-and-halo system of stars and an interstellar bulge-disk-and-halo system of gas and plasma, as shown in Figure 1. The stellar bulge and disk populations essentially determine the distribution of supernovae from the decay of whose radionuclei the positrons are born. Following Ferriére et al. (2007), the interstellar bulge, disk and halo define the distribution of gas and plasma where the positrons die by annihilation.

As we discuss in more detail later, the stellar bulge is confined within the interstellar bulge, which is blown out by a bulge wind to about 3.5 kpc in the Galactic plane and above the disk it feeds and merges into the halo. The stellar disk extends not only throughout the interstellar disk out to at least about 15 kpc, but also all the way into the interstellar bulge. The interstellar disk, however, extends in only to the outer edge of the interstellar bulge, and starts in the so-called “molecular ring” at around 3.5 kpc, which is defined primarily by the two opposing innermost spiral arms.

Threading the combined stellar and interstellar systems, is the Galactic magnetic field, along whose flux tubes 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 the interstellar bulge and are blown out by the bulge wind (e.g. Bregman 1980; Blitz et al. 1993) roughly radially in the outer bulge, out to the molecular ring, where they begin to be tightly wound into the spiral arms of the interstellar disk (e.g. Beck 2001). There, in giant star formation regions along these spiral arms, hot, massive OB star associations are formed, whose collective supernovae generate superbubbles, that blow out plasma as well as flux tubes into the overlying halo (e.g. Parker 1979).

Within this framework we model the various aspects of the production, propagation and annihilation of Galactic positrons.

We show that roughly half of the Galactic positrons are born in the extended interstellar bulge (e.g. Ferriére, Gillard & Jean 2008) in the inner 3.5 kpc of the Galaxy, and nearly all result from SNIa occurring in the hot, tenuous high-pressure plasma fills nearly all of that volume. We show that MeV positrons created in these SNIa supernovae are expected to escape from the remnants into the surrounding tenuous plasma. There they propagate along magnetic flux tubes by 1-dimensional diffusion, resonantly scattered in pitch angle by turbulently generated magnetohydrodynamic (MHD) waves that cascade down to the thermal electron gyroradius. In these plasmas the diffusion mean free path is long enough that only a negligible fraction annihilate there and 80% of the positrons escape down into the dense HII and HI shells of neighboring molecular clouds within 1.5 kpc, where they quickly slow down and annihilate, while the remaining 20% escape out into the halo and disk beyond.

In the Galactic disk beyond 3.5 kpc, the filling factor of hot ( K) plasma is more modest ( 20%). Here the plasma is mostly in superbubbles, created by the correlated core-collapse supernovae of massive stars. Consequently, the bulk of positron production via Al decay, synthesized by these stars, occurs preferentially in the superbubbles. The larger bubbles blow out into the halo, sweeping the magnetic flux tubes up with them. Again the diffusion mean free paths of the positrons are such that the positrons escape along the flux tubes, either up into the halo, or down into the warm ionized envelopes, surrounding the base of the superbubble generated chimneys, where they slow down and annihilate.

In the warm ( K), but essentially neutral HI gas, which fills the largest fraction of the disk 3.5 kpc, we show that the cascade of magnetic turbulence, created at parsec scales, is damped by friction between the ions and neutral gas long before the cascading turbulence reaches the positron resonant gyroradii. Thus the positrons suffer much less scattering, and stream along the flux tubes with an isotropic pitch angle distribution. Therefore the positrons also mostly stream out into adjoining phases, either into the halo or into those warm ionized envelopes, separating the ubiquitous warm HI gas from the hot superbubbles, where they slow down and annihilate.

Therefore we find that roughly 1/2 of the positrons, produced anywhere in the disk 3.5 kpc, slow down and annihilate there, primarily in the warm ionized gas, while the rest escape from the disk into the halo. This escape thus accounts for the low observed annihilation radiation luminosity of the disk and explains the observed ratio of annihilation rates in the bulge and disk.

Moreover, the efficient diffusion of positrons out of the hot tenuous plasmas in both the bulge and disk, where nearly 70% are born, and into the warm ionized envelopes and cold neutral interiors of molecular clouds, where they annihilate, also explains both the observed positronium fraction and broad to narrow 511 keV line flux ratio.

The positrons escaping into the halo from the bulge and disk also lead to the annihilation of roughly 1/3 of all the positrons in the halo, even though there in no significant production there.

1.4. Outline

Although the processes of Galactic positron production, propagation and annihilation are rather straightforward, all the details still make the determination of the expected 511 keV line emission both lengthly and complicated.

We consider positron production from the decay of three separate long-lived radioisotopes, each synthesized by distinct classes of stars: the Ni and Ti, respectively, by SNIa and SNIp from thermonuclear explosions of Gyr-old accreting white dwarves in close binary systems, and the Al by young ( 40 Myr), massive ( 8 M) stars, most likely through Wolf Rayet (WR) winds and core collapse SNII and SNIbc. The supernovae, SNIa and SNIp, from older stellar populations are known to occur in galactic bulges and disks, while WR stars, SNII and SNIbc, formed in extremely young stellar populations, occur in active, or recent, galactic star-formation sites, such as spiral arms, molecular-cloud complexes, and galactic nuclei. Thus in section 2 we briefly re-examine the production of -decay positrons by Galactic supernovae and their spatial and temporal distribution in the light of recent observations and theory.

In section 3 we then explore in detail the expected energy loss, scattering, and propagation processes of the relativistic positrons from these supernovae in the different phases of the interstellar medium. We consider collisionless scattering of these positrons by small-scale fluctuations generated by magnetohydrodynamic turbulence in the ionized interstellar phases, based on extensive studies and observations of relativistic electrons in the analgous turbulence in the interplanetary medium. Within this context we adopt a self-consistent transport model tied to field-aligned turbulence dependent on the properties of the ambient interstellar phases.

In sections 4, 5, and 6 we determine the expected spatial distribution of positron production, propagation, slowing down and annihilation in the various phases of the positron bulge, disk and halo. In each of these we review the fundamental properties of the interstellar medium, which constrain positron transport as well as leave signatures in the positron annihilation line emission features. Here we explore the very large differences between the interstellar properties, and the resulting positron propagation and annihilation, in the well-studied local interstellar medium compared to those in the inner 1.5 kpc of our Galaxy, where the diffuse 511 keV emission is most intense.

Finally, in sections 7 & 8 we give a summary and conclusions, comparing the expected 511 keV line emission with the SPI/INTEGRAL observations. We show that the observed spatial and spectral features of the Galactic positron annihilation radiation can all be fully understood in the context of a supernova radionuclide origin, when the differential propagation of these MeV positrons in the various phases of the interstellar medium is taken into consideration.

2. Galactic Nucleosynthetic Positron Sources

Here we briefly re-examine the production of positrons by -decay radioactive nuclei produced by different types of Galactic supernovae. We also re-examine the rates of such supernovae in the Galaxy and the resulting spatial and temporal distribution of their production of positrons in both the Galactic bulge and disk in the light of recent observations and theory, in order to determine the expected positron bulge/disk ratio of the such positron production, .

2.1. Positrons from Supernovae

Positrons resulting from the decay chains of the longer-lived radioactive nuclei, Ni, Ti, Al, all produced by explosive nucleosynthesis in supernovae, have long been thought (Colgate 1970; Burger, Stephens & Swanenburg 1970; Clayton 1973; Ramaty & Lingenfelter 1979; Chan & Lingenfelter 1993; Milne, The & Leising 1999) to be the major source of Galactic positrons. The total Galactic positron production rates from these radionuclei depend on their supernova mass yields , the supernova occurrence rates and the survival fraction, , of their positrons in the supernova ejecta as it expands into the interstellar medium.

2.1.1 Ni

The decay chain of NiCoFe produces a positron only 19% of the time, since Ni decays solely by electron capture with a mean life of 8.8 days into Co, which also decays primarily by electron capture, and just 19% by emission into Fe with a mean life of 111.4 days. Thus the production rate of positrons from Co decay is , or e s, where is the time-integrated survival fraction of Co decay positrons in the expanding supernova ejecta, is the production rate of Ni atoms per s, is the Ni yield in M per SN, and is the Galactic supernova rate in SN per 100 yr. This Galactic rate alone would greatly exceed the observed rate of positron annihilation, if all of the positrons escaped from the supernova ejecta, but most do not.

The nucleosynthetic yields of Ni, whose decay energy powers the visible light from the exploding ejecta, have been extensively calculated in all different types of supernovae. This is particularly true of the cosmologically important Type Ia thermonuclear supernovae in accreting white dwarves, which are best described by the deflagration model W7 of Nomoto, Thielemann, & Yoko (1984) that gives a Ni yield of 0.58 M. The peculiar subclass of Type Ia(bg), also called Type Ip, supernovae in accreting sub-Chandrasakar white dwarves are expected to produce 0.44 M of Ni from the helium detonation model (Woosley, Taam & Weaver 1986).

Because of the relatively short decay mean life of Co, most of these positrons lose their energy and annihilate in the ejecta before it becomes either thin enough for them to escape or sufficiently rarefied for them to survive. Indeed, quantitative studies of the survival of positrons in the expanding supernova ejecta (Chan & Lingenfelter 1993; Milne, The & Leising 1999), for various supernova models, have shown that essentially all of the positrons produced by its decay in the most frequent, but massive, core collapse SNII supernovae slow down and annihilate unobserved in their much denser ejecta and do not escape into the interstellar medium.

In particular, Chan & Lingenfelter (1993) have calculated of positrons from Co decay expected for various supernova model distributions of the ejecta density, velocity and magnetic fields. They found that in the limit where the magnetic field is thoroughly tangled, the positron survival fraction for SNIa deflagration model, , ranged from 0.1% to 2.5% for unmixed versus uniformly mixed ejecta. Alternatively in the limit that the magnetic field is fully combed-out (e.g. Colgate, Petschek & Kriese 1980) the SNIa was much greater, ranging from 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 % for the deflagration model gave the best fits to SNIa light curves at late times (1 yr), when the positrons from Co are the dominant source of energy. Using the deflagration model (Nomoto, Thielemann, & Yoko 1984) Ni yield of 0.58 M, we calculate the expected positron production rate e s.

2.1.2 Ti

Chan & Lingenfelter (1993) have shown that the positrons from TiScCa decay, which produces a positron 95% of the time from emission of Sc to Ca, is a significant source of the annihilating positrons. The Solar system abundance ratio of Ca to Fe of 1.2310 (e.g. Lodders 2003) requires a similar relative nucleosynthetic yield ratio of Ti to Ni, since these radionuclei are the primary sources of Ca and Fe (Woosley & Pinto 1988). The much longer Ti decay mean life of 89 yr would allow essentially all ( 97%) of its positrons to survive in the expanding ejecta of even the most massive supernovae (Chan & Lingenfelter 1993). Therefore the total Galactic positron production rate from Ti decay can be scaled by the Solar system abundance ratio of Ca to Fe, assuming that half of the Galactic Fe is produced by SNIa (e.g. Timmes, Woosley & Weaver 1995) and , so that or e s.

The most likely sources of these Ti decay positrons are the peculiar SNIp supernovae, typified by SN 1991bg (Filippenko et al. 1992; Turatto et al. 1996), so that we assume the same spatial distribution as SNIa. In particular, the primary source of Galactic Ca appears to be He detonations in accreting sub-Chandrasakar mass white dwarves, whose calculated (Woosley, Taam & Weaver 1986; Woosley & Weaver 1994) Ti yields of are 16 times the Solar system ratio. And such models also give the best fits to the light curves of these peculiar supernovae (e.g. Milne, The & Leising 1999; Blinnikov & Sorokina 2004; The et al. 2006). The ratios calculated for other types of supernovae are much less than the Solar system ratio. Thus we assume the spatial distribution of the SNIp is the same as that of the SNIa and we also scale the Ti production to that of Ni in SNIa.

2.1.3 Al

The additional contribution of positrons, produced 82% of the time from long-lived ( yr) decays of Al to Mg, can be determined much more directly from the measured Galactic luminosity of the 1.809 MeV line, which accompanies that decay. This luminosity implies a steady-state Galactic mass 0.8 M of Al (Diehl et al. 2006), or a positron production e s. Thus they can contribute 154% of the total Galactic positron annihilation rate of e s inferred from the best-fit bulge, disk and halo annihilation luminosity discussed above.

Knödlseder (1999) has shown that the Al 1.809 MeV line luminosity is strongly correlated with the distribution of massive stars in the disk, which confirms that it is produced in Wolf Rayet winds and/or in core collapse supernovae, SNII and SNIb/c.

With these calculated yields of Ni, Ti and Al in various types of supernovae, we now estimate the total Galactic positron production rate from the decay of radionuclei synthesized in supernovae in the bulge and disk, assuming no significant production in the halo,

(1)

scaled to the mean Galactic occurrence rate, of SNIa supernovae per 100 yrs.

2.2. Mean Supernova Rates in Our Galaxy

Galactic supernova occurrence rates depend on the Hubble 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 , where is the Hubble constant in units of 75 km s Mpc and is the blue luminosity of the galaxy in units of . The Hubble classification of our Milky Way Galaxy is Sbc (Binney & Merrifield 1998; Kennicutt 2001). From a study of the extra-galactic observations Cappellaro, Evans & Turatto (1999) find that the SNIa rate is 0.210.08 SNu in Sbc galaxies. Thus from the Galactic blue luminosity, of (Freeman 1985), and this extragalactic rate for Sbc galaxies, we estimate the mean supernova rate in our Galaxy for SNIa to be 0.400.16 per 100 years.

However, recent studies of the secular evolution of the Galactic bulge suggest that episodic (Gyr) bursts of star formation could cause modest variations in the supernova bulge/disk ratio. From observations of oxygen-rich, cool giants, Sjouwerman et al. (1998) conclude that the most recent burst of star formation in the inner bulge occurred roughly 1 Gyr ago and produced 10 SNII and SNIb/c supernovae. Such a burst would then be followed by a much more extended period of enhanced SNIa and SNIp, occurring with a range of delay times between 0.1 and 1 Gyr, calculated from evolutionary models by Greggio (2005). Thus, although the SNII and SNIbc supernovae all occurred during the burst of star formation a Gyr ago, we would expect the SNIa and SNIp to still be occurring at the present time. Using the average ratio of SNIa to SNII for our Galaxy of 0.25 (e.g. van den Bergh & Tammann 1991; Cappellaro, Evans & Turatto 1999), we would expect 2.5 SNIa from this star burst spread over 1 Gyr, or an added contribution of 0.025 SNIa per 100 years. Thus we estimate the present supernova rate in our Galaxy for SNIa to be 0.430.16 per 100 years.

Using this SNIa rate we then estimate from equation (1) the expected mean Galactic positron production from the decay of supernova produced Co, Ti and Al,

(2)

Thus, of the Galactic positron production of e s, inferred from the observed bulge, disk and halo annihilation radiation as discussed above, Al decay accounts for e s, and Ti decay produces e s, nearly all from SNIp, leaving e s, which can come from Ni decay positrons, if they have a survival fraction in SNIa ejecta of %, including the uncertainty in the Galactic SNIa rate. A survival fraction of 5%, was in fact just what was calculated (Chan & Lingenfelter 1993 Fig. 3) for the standard deflagration model of SNIa (Nomoto et al. 1984 W7) with a combed out magnetic field in the ejecta. It is also quite consistent with the mean value of % inferred (Milne, The & Leising 1999) from SNIa light curves at late times, when the positrons become a major source of ejecta heating.

2.3. Spatial Distribution of Galactic Supernovae

Here we investigate both the stellar bulge/disk ratio () of Galactic supernovae and their radial and transverse distributions within the stellar bulge and disk. Of primary concern are the SNIa and SNIp, which appear to be the source of 85% of the annihilating positrons.

We first determine the relative contributions of SNIa occurring in the Galactic stellar bulge and disk. It is well known that the SNIa occurrence rate is essentially the same across the Hubble sequence of galaxy types from ellipticals (E) through late-type spirals (Sd), when such rates are defined in SNu, the number of supernovae per year per unit of blue luminosity of the parent galaxy (e.g. Cappellaro, Evans & Turatto 1999). Across this broad Hubble sequence, the fraction of galactic blue-band luminosities contributed by Galactic disks range from zero in ellipticals to 95% in late-type Scd galaxies (Simien & de Vaucouleurs 1986), yet the SNIa birth rate per unit blue luminosity of parent galaxies remains the same.

Since the SNIa occurrence rates per blue luminosity is independent of the relative contributions of galactic bulges and disks, we scale the ratio of SNIa birthrates in the stellar bulge and disk of our Galaxy by the corresponding ratio of stellar bulge to disk blue luminosities, . In their classic study Simien & de Vaucouleurs (1986) investigated the systematics of bulge-to-disk ratios in the blue-band. The mean value of their tabulations of for our Galaxy is 0.25. Thus, we expect that the Galactic stellar luminosity bulge/disk ratio implies a similar time-averaged, mean stellar bulge/disk ratio 0.25 for Galactic SNIa and SNIp. Adding the star burst contribution of 0.025 SNIa per 100 years in the bulge, which is a 30% enhancement over the mean bulge rate of 0.08 SNIa per 100 years, gives a current stellar Galactic bulge/disk ratio 0.33 for SNIa and SNIp supernovae.

To model the spatial distribution of SNIa in the Galaxy, we use the representations of Dehnen & Binney (1998) who parameterized the large-scale stellar distributions within the Galaxy by comparing models to the Galactic rotational and other observational constraints. Thus the number density of SNIa in the Galactic disk, , and the number density of SNIa in the Galactic bulge, , expressed in cylindrically symmetric Galactocentric coordinates, are

(3)

where , represents the planar distance from the Galactic center in kpc, represents the distance normal to the Galactic plane in kpc, and the constant = 1.9 kpc (Dehnen & Binney 1998). Here the normalization constants, and are found from the total Galactic SNIa birthrate and the ratio. Dehnen and Binney ascertained that the more than 90% of the stellar disk population could be represented by of 0.180 kpc. They concluded that the most important model parameter was the ratio of the disk scale length, , to , the distance of the Sun from the Galactic center. They found that the observational constraints were best satisfied by any of four models with = 8 kpc and of 0.25, 0.3, 0.35, & 0.4. Here we will use their second model with of 2.4 kpc. We also include a star burst contribution to the bulge located in the CMZ (Sjouwerman et al. 1998).

2.4. Spatial Distribution of Galactic Positron Production

In Figure 2 we show the relative spatial distributions, , as functions of distance, , from the Galactic center for SNIa residing in the stellar bulge and disk employing the relative number densities, equation (3) for and , expressed a functions of & in a Galactocentric cylindrical coordinate system,

(4)

is the relative spatial distribution of massive stars, the expected Wolf-Rayet and SNII & SNIbc progenitor sources of Al. Although the great majority of massive stars, which synthesize Al, are located in the outer ( 3.5 kpc) Galactic disk (McKee & Williams 1997), some massive stars found in the inner ( 0.2 kpc) stellar bulge. Since Galactic Lyman continuum radiation is also produced by a massive star population (e.g. McKee & Williams 1997), we use the ratio of Lyman continuum radiation emitted by the inner stellar bulge to that from the outer disk to estimate the corresponding Al ratio. Güsten (1989) estimated a luminosity of ionizing photons from massive stars in the inner bulge of photons s. Comparing this to the estimated (McKee & Williams 1997) total Galactic value of photons s, we expect that roughly 10% of the massive stars and their generated Al occur in the inner bulge, amounting to e s.

For the remaining disk portion we use the spatial distribution of OB associations, , of (McKee & Williams (1997, eq. 35) with = 0.15 kpc to represent that of the Al made by massive stars. These spatial distributions have been normalized at large to their expected relative contributions to the total Galactic positron production rate fractions: = 0.21, = 0.64, and = 0.15, based on the SNIa and SNIp bulge/disk ratio of 0.33 in the 85% of positrons produced by them.

As we see from Figure 2, of the positrons produced from SNIa and SNIp in the disk, 13% are born within 1.5 kpc. Although the best fit disk model used in the SPI/INTEGRAL analyses (Weidenspointner et al. 2007, 2008a) of the annihilation radiation also includes 12% occurring within that radius, in order to treat the propagation of the positrons before the annihilate, we obviously need to include them in the total production within 1.5 kpc. Once we have determined the distribution of the subsequent annihilation of all of the positrons, we then adjust the estimated bulge ( 1.5 kpc) and disk ( 1.5 kpc) annihilation luminosities to compare with the SPI/INTEGRAL bulge and disk components, as we discuss in section 7.

Therefore, the positron bulge production ( 1.5 kpc), , and the disk production, including the OB contribution, . Thus the effective positron production ratio in the bulge and disk within and beyond 1.5 kpc is 0.45.

From the ratio of positron production rates in the bulge and disk we now determine the differential propagation of the positrons within and between those regions and the halo before they annihilate. In simplest terms we define the resulting positron bulge/disk annihilation ratio as, where and are respectively the fractions of the positrons, born in the bulge and disk beyond 1.5 kpc, which annihilate in the positron bulge kpc, while and are the corresponding fractions of those positrons which annihilate in the disk beyond the positron bulge kpc. As we discuss below, however, we need to investigate the positron propagation and annihilation in each of the interstellar gas phases within each of the regions.

3. Nature of Galactic Positron Propagation

We now look in detail at the nature of these positrons, and their propagation, slowing down and annihilation in the various phases of the interstellar medium in the bulge, disk and halo. In particular, we investigate the plasma processes that determine the rate of positron propagation, drawing upon the extensive studies and observations of relativistic electron propagation in the interplanetary medium.

3.1. Relativistic -Decay Positrons from Supernovae

Assuming that the dominant source of Galactic positrons is the decay of Co, Sc and Al from supernovae, the detailed studies of their survival in and escape from the ejecta show that the great bulk of the surviving and escaping positrons are relativistic.

Positrons emitted in the Co Fe decay are distributed in kinetic energy with a maximum of 1.459 MeV and a mean of 0.630 MeV, and those from Sc Ca decay have a very similar spectrum. The expected (Chan & Lingenfelter 1993; Milne, The & Leising 1999) mean energy 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 in the still dense ejecta, while the mean energy of those from the longer lived (89 yr Ti dominated meanlife) decay of Sc is essentially unchanged at about 0.6 MeV in the much less dense ejecta. The mean energy of positrons from the decay of the very long-lived (1.0410 yr) decay of Al is likewise about 0.5 MeV.

In order for these surviving relativistic positrons to annihilate in the interstellar medium they must first be decelerated to energies, 10 eV. In order to slow down to thermal energies, these 0.5 MeV positrons, must run through 0.12 g cm (Berger & Seltzer 1964), or H cm, in unionized interstellar gas, and this range is reduced by a factor of if the gas is ionized with an ionization fraction, , relying solely on collisional ionization losses. In the process of traveling this slowing down distance, kpc, the average velocity of these positrons, is 0.7 of the velocity of light, so their slowing down time,

(5)

where is the mean H density in cm.

Thus we see that since the slowing down distance of the radioactive decay positrons in the nominal phases of the interstellar medium is 0.5 kpc in HI clouds with typical 40 cm, 50 kpc in warm media with typical 0.4 cm, and 3 Mpc in hot phases with typical cm. Since all of these distances are far greater than the scales of their corresponding interstellar components, these fast positrons, streaming along the large-scale interstellar magnetic field, would all escape the Galactic disk before they even slow down, unless there is sufficient plasma turbulence to efficiently isotropize their trajectories. We thus investigate in detail under what conditions the interstellar turbulence can isotropize the trajectories of these relativistic positrons.

3.2. Nature of the Turbulence which Scatters Interstellar Positrons

In ionized interstellar phases we expect that the interstellar MeV positrons are scattered by resonant interactions with ambient turbulent magnetic fluctuations. Consequently, the efficiency of these scatterings in turn depends on the fundamental properties of the MHD turbulence at very small spatial scales of the electron gyroradius, . However, the basic properties of MHD turbulence is poorly understood. In a recent review of turbulence and magnetic fields in astrophysical plasmas, Schekochihin & Cowley (2007) state “despite over fifty years of research and many major advances a satisfactory theory of MHD turbulence remains elusive. Indeed, even the simplest (most idealized) cases are not fully understood.”

An important advancement was the realization that in many 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) as well as in increasingly more accurate, numerical simulations (Shebalin, Matthaeus, & Montgomery 1983; Maron & Goldreich 2001; Cho, Lazarian, & Vishniac 2002; Mason, Cattaneo, & Boldyrev 2006; Perez & Boldyrev 2008). In a pioneering study Goldreich & Sridhar (1995) developed a sophisticated theory of anisotropic incompressible MHD turbulence. They presented the concept of “critical balance” which predicted in inertial regime of turbulent MHD flows, first, a filamentary shape for turbulent fluctuations, via a relationship between the wave numbers parallel () and transverse () to the direction of the local mean magnetic field, k k and, second, the modeled energy spectra of transverse turbulent components possesses Kolmogorov (1941) scaling, , observed in low-speed, terrestrial flows.

At present, a puzzling dichotomy exists between these model predictions and a plethora of solar-wind observations and numerical simulations. Power spectra of small scale electromagnetic fluctuations in solar-wind plasmas have been found to have the expected 5/3 spectral index (e.g. Horbury et al. 1996; Leamon et al. 1998; Bale et al. 2005), but corresponding spectra in many precise numerical simulations have spectral indices much closer to 3/2 (Mason & Goldreich 2001; Beresnyak & Lazarian 2006; Mininni & Pouquet 2007; Perez & Boldyrev 2008). Yet the critical balance prediction about the wavenumber relationship of turbulent eddies seems to be consistent with many numerical simulations (Cho & Vishniac 2000; Maron & Goldreich 2001; Cho, Lazarian, Vishniac 2002; Beresnyak & Lazarian 2006). Unfortunately, analyses of single spacecraft measurements are unable to determine the full modal three-dimensional wavevector spectra of turbulent fluctuations (e.g. Horbury, Forman, & Oughton 2005) and thus cannot confirm or refute the critical balance prediction about filamentary-shaped fluctuations (Beresnyak & Lazarian 2008).

Although much progress has been made in the understanding and modeling of anisotropic MHD turbulence, such models, which assume that the frequency of the turbulent fluctuations is low compared to the proton gyrofrequency , still have difficulty in representing high-frequency collisionless turbulence evolving on frequencies, , which seem to be most efficient at resonantly scattering MeV positrons, where represents electron gyrofrequency. The application of such models of anisotropic turbulence seems to imply that collisionless scattering of MeV electrons would be modest and MeV electron transport should be dominated by streaming at . Yet a great variety of independent investigations of MeV electron transport in the interplanetary medium irrefutably demonstrate that such particles propagate diffusively in the collisionless solar wind plasma, although the nature of the plasma fluctuations which actually scatter such electrons remains yet unknown. Thus, we employ a simple phenomenological model of collisionless MeV electron scarttering by turbulent interstellar fluctuations tied heavily to the well determined properties of electron scattering in turbulent solar-wind plasmas.

The propagation of MeV electrons has been well studied in the interplanetary medium employing measurements of electrons accelerated by solar particle events as well as by the Jovian magnetosphere. The propagation of Jovian MeV electrons is, for the most part, dominated by diffusion (Ferreira et al. 2001a,b, 2003). Fits to intensity and anisotropy time profiles of solar-flare electrons also illustrate the dominance of diffusive transport at these electron energies, and, consequently, lead to the straightforward determination of scattering mean free paths via fits to intensity and anisotropy time profiles (Kallenrode 1993).

Bieber, Wanner, & Matthaeus (1996) related such phenomenologically-derived scattering mean free paths from solar flare particle measurements to the determination of simultaneous power spectra of magnetic fluctuations at MHD spatial scales. Since it is impossible to determine the full three-dimensional wavevector spectra of turbulent fluctuation from single spacecraft measurements (e.g. Fredricks & Coroniti 1976), Bieber, Wanner, & Matthaeus employed a simple two-component representation of the anisotropic interplanetary MHD turbulence. They found that their magnetic-field data consisted of two anisotropic populations, fluctuations with large correlation lengths transverse to the direction of the mean magnetic field (slab turbulence) and fluctuations with large correlation lengths parallel to the direction of the mean magnetic field (quasi-two-dimensional turbulence). In such slab models the wave vectors of the fluctuations are aligned parallel to the direction of the mean magnetic field, and a simple interpretation is that in the inertial range slab turbulence consists of Alfvén wave propagating along the mean magnetic field. Later investigations (Oughton, Dmitruk, & Matthaeus 2004, 2006; Oughton & Matthaeus 2005) have demonstrated that this simple two-component model of anisotropic turbulence seems to model the basic properties of solar wind MHD turbulence.

Bieber, Wanner, & Matthaeus found that interplanetary turbulence is dominated111Recently, Horbury, Forman & Oughton (2005) have investigated in more detail the properties of anisotropic solar wind turbulence, implementing a wavelet-based method to track the direction of the local mean magnetic field; such an approach reduces greatly the noise in the magnetic field measurements. Employing the two-component model, they ascertained that their results agreed “remarkably well” with the results of Bieber, Wanner, & Matthaeus. by quasi-two-dimensional turbulence, and the mean ratio of the power in slab turbulence to the power in two-dimensional turbulence is 0.15. Further, observing that the modeled two-dimensional fluctuations do not contribute to particle scatterings, they calculated scattering mean free paths expected by resonant interactions solely with turbulent slab fluctuations via quasi-linear theory (Jokipii 1969). They found that such scattering mean free paths agreed well with the values determined independently (of the analyses of the turbulent magnetic variations) from data fits to intensity and anisotropy time profiles.

Based on these discussions, we model interstellar MHD turbulence as composite slab/two-dimensional turbulence. We assume here that the ratio of the power of interstellar slab turbulence along the mean magnetic field to the power of 2-dimensional turbulence normal to the direction of the mean field is 0.15 following Bieber et al. (1996). Similarly, we assume that the spectrum of interstellar slab turbulence steepens from a spectral index of 5/3 to 3 at a wavenumber , where is the proton gyroradius, since Leamon et al. (1998) have found that at higher wave numbers the spectral index of solar-wind variations varied from 2 to 4.4 with an average index of 3.

To quantify such scattering mean free paths for interstellar MeV positrons we use an analytic approximation (Teufel & Schlickeiser 2002) calculated for slab MHD turbulence employing quasi-linear theory. Following Teufel & Schlickeiser we assume that energy is injected into the interstellar medium at an outer spatial scale, . For 1 and k, the modeled turbulence possesses a spectral index, . From the discussion above, we assume = 5/3 and at = 1/r, we assume the spectral shape steepens to = 3. Teufel & Schlickeiser (2002, their eq. 58), find that for such slab turbulence the electron scattering mean free path along the magnetic field is

(6)

where is the energy density of the turbulent magnetic fluctuation in the direction of the mean magnetic field, is the speed of light, is the gyro-frequency of the positron, the mean magnetic field is , is the Alfvén speed, , is the ion mass density, is the electron gyroradius; the average positron speed is , and is a dimensionless quantity involving a Gauss hypergeometric function (Table 3 of Teufel & Schlickeiser 2002).

Thus, by such one-dimensional (1-D) diffusion the positrons in their slowing down time would be distributed along a flux tube a mean length in either direction from their point of origin.

3.3. Ion-Neutral Damping

The bulk of the Galactic interstellar mass resides in primarily neutral phases (HI & H), concentrated in a cloud population (e.g. Tielens 2005). Moreover, a major fraction of the disk ( 3.5 kpc) is filled with warm neutral HI (Kulkarni & Heiles 1987). Consequently, the nature of MHD turbulence in predominantly neutral interstellar phases needs to be addressed.

In partially neutral plasmas, where magnetic forces act directly on charges, and indirectly on neutral atoms via ion-neutral collisions, the turbulent NHD fluctuations are dissipated into heat when ion and neutral velocities differ significantly (Higdon 1984; Goldreich & Sridhar 1995; Lithwick & Goldreich 2001). Turbulent MHD cascades are quenched at scales of roughly the collision mean free path of protons with neutral atoms, if the proton collision rate with neutral atoms exceeds the eddy cascade rate. The mean free path for proton-HI collisions is cm at 8000 K, or if hydrogen is fully ionized, and He is the dominant neutral species, the mean free path for proton-He collisions is cm where is the H number density, including both ions and neutrals (Lithwick & Goldreich 2001).

Consequently, they showed that MHD turbulence can cascade to small spatial scales, much less than these damping collision mean free paths, only if the HI fraction, , is less than a critical value,

(7)

Therefore in predominantly neutral interstellar phases turbulent MHD cascades are halted by ion-neutral collisions at , spatial scales far greater than the gyroradii of MeV positrons. Hence we expect collisionless scattering of positrons by such large-scale turbulent MHD fluctuations to be very inefficient. Similarly simple elastic collisions with ambient electrons, whose mean pitch angle scattering is 90 and mean energy loss is 50%, have a mean free path much longer than slowing down distance due to ionization losses (e.g. Berger & Seltzer 1964), so they too offer no significant scattering.

In the absence of such scatterings, MeV positrons might be expected to generate plasma waves by a resonant streaming instability, similar to the creation (Kulsrud 2005) of ion Alfvén waves by streaming relativistic cosmic ray nuclei. But these do not appear to be effective either. These MeV positrons generate resonant whistler waves (Schlickeiser 2002) at wavelengths significantly less than the scale of the thermal proton gyroradius, , at frequencies approaching those of thermal electrons, where is the thermal proton speed. The phase speed of whistler waves in the direction of the mean field is the electron Alfvén speed, If the positron streaming velocity, , and if the whistler waves weren’t damped so they grew to saturation, then the resulting magnetic field fluctuations in these waves would interact with the streaming positrons via quasi-linear wave-particle interactions that change their pitch angles, reducing to a level just less than . However, in predominantly neutral interstellar phases such whistler waves are also subject to severe ion-neutral damping (Kennel 2008 private communication).

Thus, we expect that positron streaming velocities along the magnetic flux tubes are comparable to their particle velocities, , similar to the very high drift speeds found (Felice & Kulsrud 2001) for cosmic-ray nuclei in warm HI regions. Nonetheless, in turbulent media the flux tubes themselves may essentially random walk, so that the propagation is described by so-called “compound diffusion” (Lingenfelter, Ramaty & Fisk 1971).

4. Positron Propagation & Annihilation in the Positron Bulge

Using these models of relativistic electron transport, we now investigate the propagation, slowing down, and annihilation of positrons produced in each of the different phases of the interstellar medium in the positron bulge, Galactic disk and halo. The schematic model of the bulge, disk and halo is shown in Figure 1. Because the positrons, produced by SNIa and SNIp, are widely distributed in the bulge and disk (equation 3), we assume that the fraction produced in the various phases of the interstellar medium are proportional to their appropriate filling factors. Although the supernovae themselves disturb the local medium, the MeV positrons from the decay of radionuclei escape from the remnant into the undisturbed surroundings.

We estimate the relative probabilities, P, of positron annihilation within each phase and positron escape into neighboring phases by the following procedure. From the properties of the medium in each phase, we determine the propagation mode, diffusion or streaming, and calculate the diffusion mean free path (equation 6) in the MHD scattering mode, or the mean velocity, , in the unscattered streaming mode.

Although we have also made Monte Carlo simulations, we estimate the propagation of positrons during their slowing down in each phase of the interstellar medium, using a simple approximation. Since even the average properties and structure of the magnetic field especially are very poorly known in the various phases, no more sophisticated treatment seems justified. For the 1-D diffusion in a uniform medium with a mean free path along a magnetic flux tube, positrons produced at a point, = 0 and = 0, will be distributed in both directions along the flux tube with a density, , where , by the time that they slow down and annihilate. This corresponds to a mean positron density over the total mean length . Similarly for positrons streaming at their velocity with an isotropic pitch angle distribution they will also have a mean density over a total mean length , where , when they slow down and annihilate.

Thus, if the positrons are produced uniformly along some mean length of the magnetic flux tubes threading through some phase , from the above we expect that the probability, , that such positrons will slow down and annihilate in that phase before they escape is crudely,

(8)

where is the slowing down length in that phase. The remaining positrons have a probability, , of escaping from phase into a neighboring phase, and we estimate the relative fractions of the escaping positrons that go into each of the adjacent phases from simple geometric arguments, allowing large uncertainties of 50%.

The probability that positrons which diffuse into a phase will slow down and annihilate there is only slightly different from those born uniformly within it. These positrons are effectively born at their first scatter within the phase, roughly within a scattering mean free path of the boundary. So if , the probability of their slowing down and annihilating in the phase, is the same as for those born uniformly within, . But if , then remains at , since after the first scatter near the boundary half would be expected to be spread over in both directions along the flux tube by the time they slow down and annihilate. Thus only about half of them will slow down and annihilate in phase , while the other half will escape, or be effectively reflected. As we show, however, in the dense labyrinth of clouds in the CMZ and tilted disk the half of the positrons that diffuse into clouds from the VH and HM and are reflected back out rather than annihilating within, quickly diffuse into another cloud where half are again reflected, and by the time they have encountered a half dozen other clouds only a small fraction, or 1% remain. As we also show, however, in the diffuse disk beyond 1.5 kpc that is not the case.

From these simple approximations, we estimate the different propagation fractions that relate the positron production rates and annihilation rates , as defined in equation (4) for the expected bulge/disk annihilation ratio, expanding that equation to explicitly define these fractions in the different phases of the interstellar medium within the bulge, disk and halo.

We consider here the Galactic positron bulge as it is defined by the spherical component ( 1.5 kpc) of the 511 keV luminosity in the SPI/INTEGRAL analyses (Knödlseder et al. 2005; Weidenspointner et al. 2007, 2008a). Also based on their model fitting we discuss it in two parts: the inner bulge region (R 0.5 kpc) and the surrounding outer, or tilted-disk, bulge region (0.5 kpc R 1.5 kpc).

In order to compare with the SPI/INTEGRAL measurements, we need to separate the positron production and annihilation in the bulge and disk into five regions, shown in Figure 1, defined primarily by their galacto-centic radius, although they also have very different mixes of interstellar phases, as recently reviewed by Ferrière, Gillard & Jean (2007). These are 1) the very hot, dense inner or nuclear bulge, , ( 0.5 kpc); 2) the hot, middle bulge and tilted disk, (0.5 to 1.5 kpc); 3) the hot, tenuous outer bulge, between 1.5 kpc and the Galactic molecular ring at around 3.5 kpc, dominated by the bulge wind (e.g. Bregman 1980; Blitz et al. 1993); 4) the predominantly warm neutral outer disk beyond 3.5 kpc, , and 5) the hot, tenuous halo, , beyond 3.5 kpc and above the disk. The properties of each of these regions, which we discuss below, are summarized 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 occurrences, and the disk distribution of massive (OB) stars (McKee & Williams 1997), we estimated the fraction of positron production in each of these regions. The cumulative production as a function of Galactic radius is shown in Figure 2 for each component. We find that 45%, 38% and 17% of the positrons from the stellar bulge component of SNIa and SNIp are produced in the regions 0.5 kpc, 0.5 to 1.5 kpc, and 1.5 to 3.5 kpc, respectively, plus 10% of the massive star Al component within 0.5 kpc. Of the positrons from the stellar disk component of SNIa and SNIp, 2%, 11% and 30% are produced in those same regions, and the remaining 57% are produced beyond 3.5 kpc, together with 90% of the massive star Al component.

Thus the total positron production in the inner bulge within 0.5 kpc, the middle bulge between 0.5 and 1.5 kpc and the outer bulge between 1.5 and 3.5 kpc, including both stellar bulge and disk components, and the production in the outer disk beyond 3.5 kpc, are

(9)

And using the best-fit positron survival fraction, % from 2.2 above, the total positron production rates are

(10)

Within each of these regions, , we determine the expected positron production rates in each separate interstellar gas and plasma phase, , by the relative filling factors of those phases, , times the production rate in that region, , such that We then estimate the final positron annihilation rates, , in each phase within each region , as the sums over and of products of the production rates, , times the propagation fractions, where each is the fraction of the positrons born in that propagate to and annihilate in , as discussed above and given for in equation (8).

In the next three sections, we estimate these propagation fractions from modelling the propagation, slowing down and annihilation of positrons in each of these regions and phases starting with the Galactic bulge and moving outward in the disk and ultimately into the halo, which is an important region of propagation and annihilation, despite its lack of local production.

4.1. Inner Bulge (R 0.5 kpc)

We model the complex interstellar phenomena of the inner region of the Galaxy (R 0.5 kpc), assuming that a very hot ( 10 K), high-pressure plasma permeates the region, pressure equilibrium exists among the interstellar phases (Spergel & Blitz 1992; Carral et al. 1994), and the properties of the HII, HI, and H regions are related to each other via the scenario of photodissociation regions (PDR) (Tielens & Hollenbach 1985).

O-star radiation is the primary photo-ionization source in the interstellar medium (McKee & Williams 1997). Outside of galactic centers, ultraviolet radiation flux at an arbitrary location in galactic nuclear regions is generated by nearby OB associations, or by single O stars distributed randomly throughout the regions (Wolfire, Tielens, & Hollenbach 1990). However, in the immediate vicinity of a galactic center ionizing radiation is dominated by the radiation contribution of active galactic nuclei. In the inner Galactic bulge the bulk of the ionizing radiation seems to be generated by randomly distributed O-stars which photo-ionize the outer layers of nearby interstellar clouds (Wolfire, Tielens, & Hollenbach 1990; Carral et al. 1991). However, in the outer ( 3.5 kpc) Galaxy the great majority of O stars are clumped in OB associations, which in turn create dense, compact HII regions, which surround such OB associations (McKee & Williams 1997). In the outer galaxy, only, small fraction ( 15%) of ionizing radiation generated in these OB associations escapes to maintain the diffuse HII.

Thus the photo-ionized, as well as the neutral, phases in the centers of galaxies have been modeled as ensembles of PDRs (Wolfire, Tielens, & Hollenbach 1990; Carral et al. 1991). Each cloud is viewed as having a spherical molecular core of radius, , which contains the bulk of the cloud mass at a molecular hydrogen density . Surrounding the molecular cores are cold atomic HI envelopes, thick, which in turn generate infrared continua and fine-structure line emissions (Wolfire, Tielens, & Hollenbach 1990). Finally, envelopes of photoionized HII plasma form outer shells, thick, surrounding these HI shells (Carral et al. 1994), ionized by Lyman continuum emission from randomly distributed hot, massive stars within the central 0.5 kpc. Following Caral et al., we assume that these HII envelopes constitute the primary photoionized gas component and a schematic model of the clouds is shown in Figure 3. Although the the hot, massive O stars that generate the HII ionizing emission are thought to occur only in the CMZ, we expect that clouds in the inner portion of the surrounding tilted disk are also likely to be irradiated with sufficient flux to maintain HII envelopes. The extent of such irradiation is not known, however, and here we arbitrarily assume that it extends only to 0.5 kpc.

We also assume that the magnetic flux tubes nominally pass through the cold clouds in a parallel array, but with large turbulent perturbations superimposed, as suggested by magnetic field in giant molecular clouds. Li et al. (2006), using the SPARO 450 m polarimetry observations, have found that within some such clouds in the disk, the direction of the magnetic field is roughly correlated with that of the local Galactic field. However, extensive Zeeman measurements of molecular clouds also indicate strong field perturbations from supersonic motions driven by MHD turbulence with energies comparable to that of the magnetic fields within the clouds (see reviews by Crutcher 1999; Falceta-Goncalves, Lazarian & Kowal 2008).

Thus, we assume a nominal field to calculate the base mean length of flux tubes through the clouds, and then from an estimate of the scale length of turbulent motions, we estimate the increase of that mean length resulting from the effective random walk or meandering of the flux tubes through the turbulent perturbations. The nominal mean flux tube length through cold cloud cores is , where is the core radius, while the nominal mean length through an overlying shell of thickness , is . Conservatively taking the outer scale of turbulent motion as the step size, which minimizes the random walk, we expect a mean meandering flux tube length, .

In the inner bulge the molecular clouds concentrate into two nested disks, the Central Molecular Zone (CMZ) and the surrounding Tilted Disk, which extends well beyond 0.5 kpc out to edge of the middle bulge at 1.5 kpc. The CMZ is a highly asymmetric region, which is an ellipsoidal disk, 40 pc thick vertically, with lateral axes of 500 pc by 250 pc and an interstellar H mass of 1.9(Ferrière, Gillard, & Jean 2007). In this region we estimate a mean molecular cloud core mass, M and a mean core radius, 5 pc, from integrations of the mass spectrum (2 M M 2 M) and the size spectrum (3.3 pc (resolution limit) pc) of the CMZ molecular cloud population determined by Miyazaki & Tsuboi (2000). This implies a total of 400 such cloud cores with a H density cm. It also gives a total volume filling factor of these H clouds in the CMZ is 0.08, but in the full inner bulge 0.5 kpc volume, the filling factor is only .

The much larger Tilted Disk, which is described in more detail in the next subsection (4.2), surrounds the CMZ, extending out to 1.5 kpc with an estimated interstellar H mass of 3.4 M (Ferrière, Gillard, & Jean 2007). About 20% of the Tilted Disk lies within the bulge source region 0.5 kpc. This inner portion of the Titled Disk contains M of interstellar H. We assume that this H gas is also concentrated in similar molecular cloud cores, which would number 140, if they have the same mean density and mass as those in the CMZ, and they would have an even smaller filling factor in the bulge source volume 0.5 kpc.

Within and surrounding the CMZ, containing the molecular clouds, is a very hot tenuous plasma (VH). Observations of 6.7 keV line emission from the K He-like transition of ionized Fe indicate that this VH plasma has a temperature of K, and fills most of this central region (Koyama 1989; Yamauchi et al. 1990; Spergel & Blitz 1992; Koyama et al. 1996; Muno et al. 2004). Thus we assume that a hot ( K), plasma ( cm) fills the inner 500 pc of the bulge (e.g. Yamauchi et al. 1990). The thermal pressure, K cm, where represents the mean particle mass for fully ionized H, and is Boltzmann’s constant. This value agrees well with previous estimates of the CMZ thermal pressure of K cm by Spergel & Blitz (1992).

Analyzing the diffuse nonthermal radio emission of the Galactic center region, Spergel & Blitz (1992) found that the magnetic field pressure is in approximate equilibrium with the thermal gas pressure. Consequently, we assume that both the intercloud and cloud field strengths are G, the value expected from such pressure arguments, 5 erg cm.

Between the VH plasma and the H core of each cloud are the two surrounding envelopes, the upper warm, ionized HII gas and the lower, cold neutral HI gas. The HII density is estimated from pressure balance between HII gas and the VH (Carral et al. 1994). Although the thermal pressure of the overlying VH is approximately matched 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 extended HII gases in the Galactic center region (e.g. Rodriguez-Fernandez & Martin-Pintado 2005) imply that turbulence also dominates the HII pressure. We assume that the magnetic field strengths are the same in the VH plasma and the HII gas. Thus we find that a turbulent HII pressure, (Tielens & Hollenbach 1985), constituting 85% of the total HII pressure equal to the total VH pressure, possesses a root mean square turbulent velocity, , of 25 km s, for a mass density, , corresponding to 100 cm. Note that the HII thermal pressure, , K cm, is small compared to that of the VH, since is 5000 K (Mezger & Pauls 1979).

Based on Wolfire et al. (2003), we expect that at the high thermal pressures found in the Galactic nuclear region, warm HI is unstable and only cold HI is thermally stable. From analyses of fine structure emissions from PDRs of CMZ molecular clouds located far from thermal radio sources Rodríguez-Fernández et al. (2004) found typical HI densities, , of cm with 150 K. For R 0.5 kpc, Ferrière, Gillard, & Jean estimated an interstellar HI mass of 1.7 M, with a space averaged density, cm in the CMZ and a filling factor there of 0.025. At such a mean density, the typical thickness of cold atomic shells is expected to be 0.5 pc. The typical thickness of HII shells, surrounding such HI shells with a filling factor 0.045, is expected to be 0.7 pc. Thus, the outer radius of a typical, double-shelled CMZ cloud, 6.2 pc. The total cloud filling factor in the CMZ, is 0.15 and it further implies a mean distance between cloud centers of only 23 pc and that between cloud surfaces, of just 11 pc.

As we will show, the annihilation of the positrons takes place almost entirely in the cloud shells. However, only a negligible fraction ( 1%) of them are produced there, since the filling factors of the cloud phases are all negligible compared to the inner bulge volume. Thus we assume that all of the inner bulge positrons are born in the pervasive, very hot medium, e s,

The subsequent positron annihilation in these phases of the bulge is then defined by the propagation fractions, which we estimate below using this general model of the CMZ. Because, as we show, the positron slowing down and annihilation in the HII phase of CMZ is so efficient, the probability of positrons, formed in the VH, penetrating through the HII into the underlying HI and H is negligible, so we only show here the terms for diffusion between the other adjacent phases with significant filling factors. We also find that the probability of positrons successfully escaping out of the inner disk is quite small.

Thus the significant terms in each phase are,

(11)

where, like those discussed above, the propagation fractions , subscripted VH:VH is the fraction of the positrons produced in the VH phase that slow down and annihilate in that phase, VH:HII is the fraction that escaped from it into the adjacent HII, and other fractions are similar. The propagation fractions and the resultant annihilation rates are listed in Table 2.

4.1.1 Very Hot Medium (VH)

The bulk of the positrons from the inner bulge SNIa and SNIp are expected to be born in the VH, which includes both the high density and temperature plasma in the CMZ and all of the lower density and temperature hot plasma that fills nearly 100% of the inner bulge 0.5 kpc. To determine the propagation fractions within the VH, and between it and the HII regions, we estimate the median distance from the positron production site to the HII envelopes around clouds in the CMZ and inner Tilted Disk. We consider separately the two distributed source components: the compact 57% of the inner bulge SNIa and SNIp, resulting from the star burst and inner disk population, and the diffuse 43%, resulting from the supernovae in the general bulge population that are expected to occur throughout the inner bulge.

For the star burst and disk component with positron production uniformly distributed sources in the CMZ with a mean distance between HII shells would be 10 pc, suggests that in a fairly regular magnetic field the mean flux tube length would also be 10 pc in the CMZ. On the other hand, for the bulge component with the production roughly uniform throughout the spherical volume, the positrons would be distributed along a mean distance 300 pc both above and below the CMZ with similar mean flux tube lengths, , along the roughly vertical magnetic field in 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 and what fraction escape into the adjacent HII envelopes, we also need to estimate the diffusion mean free path of these positrons in the VH plasma.

It is difficult to quantify the properties of hypothetical turbulent flows in the VH, since the VH origin and age are unknown. However, a plausible constraint on such turbulence is that the dissipation rate of the turbulent energy must be less than the cooling rate of the VH plasma. The rate, at which turbulent energy is dissipated is , in units of erg cm s, where is the mass density of the VH, is the root mean square turbulent velocity, and is the outer scale of turbulence (Townsend 1976).

From the X-ray observations Muno et al. (2004) estimated that the VH cooling time, yr. Since the VH thermal energy density, , the cooling constraint becomes . Taking the maximum estimate of 10 pc, the mean separation between cloud surfaces, this relation limits to less than 50 km s, which is much less than the VH Alfvén speed, 920 km s. This cooling relation places a severe constraint on VH turbulence, if the VH is long lived on the time scale of . However, if the VH is a transient phenomenon, this constraint could be weakened and the turbulence would be stronger.

Thus, if the mechanism generating the VH turbulence operated over a duration, , less than , the dissipation rate of the turbulent energy can be greater, . Thus . If, very conservatively, 25 Myr, then is less than 90 km s. Since in strong turbulent MHD flows, where energy is equipartitioned between magnetic and velocity fields s (e.g. Iroshnikov 1964; Kraichnan 1965; Biskamp 2003), constraining limits , and then , since . Thus , and, consequently a rather large 25 pc in the CMZ from equation (6).

In the VH the positron slowing down time is yr from equation (5) at cm. In the slowing down time, , the positrons diffusing along a flux tube in the VH will be distributed over a mean length 2000 pc, since the average positron velocity during deceleration is .

Roughly 61% of the inner bulge positrons are expected to occur in the CMZ, predominantly from the SNIa and SNIp in the star burst and the inner disk, along with 10% of the massive OB stars producing Al. The remaining 39%, all from SNIa and SNIp, are expected to occur throughout the nuclear bulge 0.5 kpc. This mean diffusion length is much larger than the mean flux tube length, , for positrons escaping into the HII. For those produced by the star burst and disk component of SNIa and SNIp in the CMZ and inner disk, where the mean distance between the HII envelopes of neighboring clouds, is only </