JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 114, A02103, doi: /2008ja013689, 2009

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1 Click Here for Full Article JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 114,, doi: /2008ja013689, 2009 Galactic propagation of cosmic ray nuclei in a model with an increasing diffusion coefficient at low rigidities: A comparison of the new interstellar spectra with Voyager data in the outer heliosphere W. R. Webber 1 and P. R. Higbie 2 Received 19 August 2008; revised 14 October 2008; accepted 26 November 2008; published 17 February [1] Using both a Monte Carlo Diffusion Model and a Leaky Box Model for propagation in the galaxy, we have determined the interstellar spectra of H, He, and heavier nuclei with emphasis on energies below 1 2 GeV nucleon 1. This calculation uses diffusion coefficients that are increasing at rigidities below 2 3 GV and which are based on those used to fit the interstellar electron spectrum to the galactic nonthermal radio synchrotron spectrum by Webber and Higbie (2008). The resulting interstellar spectra of the cosmic ray nuclei are thus reduced at low energies in a charge-dependent way from those derived using a diffusion coefficient = constant as was the case in some earlier calculations. The resulting interstellar intensities at 200 MeV nucleon 1 near the peak in the differential spectrum have been reduced by factors of 2.00 and 1.46 for H and He nuclei, respectively. These new interstellar intensities are then compared with recent Voyager measurements beyond the heliospheric termination shock. Solar modulation effects, corresponding to values of f between 60 and 80 MV in simple force field models, are still evident in the Voyager 1 H and He spectra at 107 AU. In 2009, when the 11-year solar modulation effects are expected to be a minimum in the outer heliosphere, a comparison of future Voyager measurements at 110 AU and beyond and these new interstellar spectra along with the corresponding electron spectrum measured by Voyager 1 will provide an important test of both modulation and interstellar propagation models. Citation: Webber, W. R., and P. R. Higbie (2009), Galactic propagation of cosmic ray nuclei in a model with an increasing diffusion coefficient at low rigidities: A comparison of the new interstellar spectra with Voyager data in the outer heliosphere, J. Geophys. Res., 114,, doi: /2008ja Introduction [2] In diffusive models for the propagation of cosmic rays in the galaxy the resulting spectra of those particles depends mainly on (1) the accelerated spectrum of cosmic rays and (2) the diffusion coefficient whose spectrum is determined by the spectrum of scattering irregularities. At rigidities less than 1 GV ionization energy loss becomes important. In the range of rigidities from 2 3 GV up to perhaps GV this diffusion coefficient appears to increase P 0.5 as determined, for example, by the ratios of secondary/ primary particles such as B/C which imply a total matter traversal which is P 0.5 in this rigidity range [Webber, 1997; Strong and Moskalenko, 1998]. This means that if the source or accelerated spectrum is taken to be P 2.3 then the observed spectrum will have an exponent = = 2.8 at higher energies, with the steepened 1 Department of Astronomy, New Mexico State University, Las Cruces, New Mexico, USA. 2 Physics Department, New Mexico State University, Las Cruces, New Mexico, USA. Copyright 2009 by the American Geophysical Union /09/2008JA013689$09.00 spectrum being essentially due to the more rapid escape of the higher-rigidity particles from the galaxy. The most precise energy spectra in this energy region, those for H and He nuclei from the BESS-TeV experiment, give spectra between 50 and 10 3 GV with exponents in the range = 2.75 ± 0.06 [Haino et al., 2004]. [3] At3 GV(1 GeV nucleon 1 )) it is well known that the secondary/primary ratios of nuclei such as B/C reach their maximum and begin to decrease at lower rigidities. This changing ratio is generally interpreted as being due to the fact that the rigidity dependence of the diffusion coefficient begins to change at this rigidity [e.g., Strong and Moskalenko, 1998]. Other effects such as reacceleration [e.g., Seo and Ptuskin, 1994] or a convective outflow from the disk [e.g., Jones, 1979; Soutoul and Ptuskin, 1999] have also been shown to produce peaks in the secondary/primary ratios at 1 GeV nucleon 1. Until recently no detailed model for a changing diffusion coefficient below 1 GeV nucleon 1 had been proposed so various simple assumptions have been made for the behavior of the diffusion coefficient at lower rigidity. One is to assume that D const below 2 3 GV [e.g., Stephens and Streitmatter, 1998]. The fact that the cross sections for the production of secondary nuclei such as B, generally peak at 1 GeV 1of6

2 nucleon 1 and then decrease at lower energies means that the B/C ratio and other secondary/primary ratios will also peak and then decrease at lower energies so that, even if the diffusion coefficient is constant, the secondary to primary ratios will still decrease. However, more precise measurements of these secondary/primary ratios at energies down to less than 100 MeV nucleon 1 (ffi880 MVrigidity) [e.g., Davis et al., 2000] and at lower solar modulation levels [Webber et al., 2003a] have shown that the observed decrease in these secondary/primary ratios is more than can be explained by changing cross sections or solar modulation levels and that changes (increases) in the diffusion coefficient itself (or convection and/or reacceleration) may be required. All three of these effects produce similar results at low rigidities, i.e., a reduction of cosmic ray intensity. In some cases this decrease in the diffusion coefficient has been taken to be rapid, e.g., P 3.0 [Davis et al., 2000; Yanasak et al., 2001; Strong and Moskalenko, 1998; Ptuskin et al., 2006] resulting in a rapidly decreasing lifetime (and intensity) at lower rigidity; however, this type of behavior of the diffusion coefficient has not been well understood until recently. [4] Ptuskin et al. [2006] have recognized this problem and suggested an explanation for a changing diffusion coefficient as follows: The magneto-hydrodynamic waves upon which the cosmic rays are scattered may themselves be dissipated by resonant interactions with the cosmic rays. This dissipation sets in rapidly and the diffusion coefficient therefore increases rapidly between 1 and 1.5 GV in their specific model. This causes a more rapid escape of cosmic rays from the galaxy and therefore a rapid reduction of intensity that is largest for electrons but still very significant for protons and heavier nuclei. In essence, Ptuskin et al. argue that a Kraichnan scale of turbulence with a wave number spectrum K 3/2 which determines the cosmic ray diffusion to be P 0.5 at higher energies [e.g., Jokipii, 1971] changes abruptly at 1 2 GV as a result of these resonant interactions. This abrupt change would be manifested, for example, in a change in the wave number power spectrum at cm since this is the radius of curvature of a 1 GV particle in a 5mG interstellar field. [5] Webber and Higbie [2008] have shown that the abrupt changes suggested by Ptuskin et al. [2006] modify the IS electron spectrum so severely that it is inconsistent with (lower than) the electron spectra measured near the Earth by the AMS experiment [Alcaraz et al., 2000] as well producing radio emission that is much lower than the polar galactic radio emission spectrum produced by synchrotron emission from these same electrons moving in the galactic magnetic fields. However, Webber and Higbie [2008] also find that to provide a match to the galactic polar radio spectrum and to the electron intensities observed by Voyager 1 (V1) at the beginning of 2008 (when it was at 105 AU and already 20 AU beyond the heliospheric termination shock) [McDonald et al., 2007], did require changes in the lowrigidity diffusion coefficient. Specifically this new diffusion coefficient does begin to increase below 2 3 GV as suggested by Ptuskin et al. [2006] and earlier by Strong and Moskalenko [1998], but much more slowly reaching a P 1.0 dependence at 0.25 GV (see Figure 1, spectrum (2) of this paper). The effects of this changing diffusion coefficient, in addition to producing IS electron spectra more compatible with Voyager observations, would be to produce IS electron spectra that result in polar radio synchrotron spectra that provide a much better fit to the observed galactic radio spectrum. [6] It is our objective in this paper to use these new estimates of the diffusion coefficient at low rigidities based on electrons by Webber and Higbie [2008] to derive new IS cosmic ray spectra for protons, He and heavier nuclei. These new IS spectra for nuclei are then compared with measurements by the Voyager spacecraft as they approach the assumed outer modulation boundary, the heliopause, to provide a better understanding of both the modulation in the outer heliosphere and the local IS cosmic ray spectra. Earlier propagation calculations of Webber and Lockwood [2001] and Moskalenko et al. [2002] of IS proton and Helium nuclei spectra (as used by Langner and Potgieter [2004] and Langner et al. [2006], for example, to calculate modulated proton and Helium spectra in the heliosphere near the termination shock), have given modulated intensities of these nuclei that are much larger than those measured by Voyager [e.g., Webber et al., 2008] and thus provide an impetus for these new propagation studies. 2. Calculation of Propagated Proton and Nuclei Spectra in the Galaxy [7] We have used two models to calculate the propagated IS spectra of cosmic ray H, He, C and Fe nuclei. The first is a Monte Carlo Diffusion Model (MCDM), the same model as used for the electron calculations described by Webber and Higbie [2008]. The second model used for comparison is the simple but widely used Leaky Box Model (LBM) similar to the one used to interpret recent Voyager spectral data on He and heavier nuclei [Webber et al., 2003a]. [8] For the Monte Carlo calculation the basic parameters are the same as those described in an earlier calculation which includes secondary and radioactive nuclei [e.g., Webber, 2000], except that the diffusion coefficient below 3 GV which was taken to be cm 2 s 1 in the earlier calculation, curve (1) in Figure 1, is replaced by a rigidity-dependent one, curve (2), at low rigidities. Above 3 GV the diffusion coefficient is (P/3.0) 0.5 cm 2 s 1 as before. The propagation region is assumed as before to consist of a matter disk with a density n at Z = 0 of 1.2 cm 3 and a Z dependence exp Z/Z m where Z m = 0.2 kpc, giving a matter density integral for the disk: Z I m ¼ n dz ¼ 8: cm 2 This disk is embedded in a mass-less diffusing halo of thickness = ±3 kpc. This calculation reproduces very well the observed B/C ratio and 10 Be/Be ratio after a suitable solar modulation [Webber, 2000]. This model may be viewed as a cylindrically symmetric one-dimensional model. [9] For the LBM the important parameter is the escape length, l, which we assume to be 15.1 b (P/3.0) 0.5 above 3 GV and with a dependence below 3 GV shown by curve (2) in Figure 1. The IS medium is assumed to be 90% H and 10% He in both models and the average density in the LBM is taken to be 0.4 cm 3. Added ionization energy loss due to 15% ionized hydrogen is also included. Again except for the diffusion coefficient below 3 GV, these parameters are the 2of6

3 Figure 1. Rigidity dependence of the diffusion coefficients used in the Monte Carlo Diffusion Model and Leaky Box Model in this paper. Also shown are the diffusion coefficients used in the DRD and PD models described by Ptuskin et al. [2006]. same as used in the earlier Leaky Box calculation [Webber et al., 2003a], as well as being similar to other calculations [see, e.g., Stephens and Streitmatter, 1998]. [10] In both calculations the cross sections used for interactions with IS matter are from our latest cross section formula [Webber et al., 2003b]. Also in both calculations the cosmic ray source spectra are assumed to be: dj/dp P 2.3 /b where the source spectral index, s, = 2.30 and the factor b 1 is appropriable for shock acceleration models (V. S. Ptuskin, private communication, 2000). [11] The two calculations, MCDM and Leaky Box Model, give almost the same spectra (for primary nuclei) for what are essentially the same diffusion coefficient or equivalent path length. Therefore the two models may be used interchangeably, with the MCDM providing the most direct examples of the effects of the changing diffusion coefficients. [12] In Figure 1 the rigidity dependence of the diffusion coefficient used in our two models is shown. These new diffusion coefficients (2) are determined from a recent comparison of propagated cosmic ray electron spectra and the galactic polar radio spectrum by Webber and Higbie, 2008 and are estimated to have an accuracy ±10% above 0.5 GV. For comparison we also show the diffusion coefficient used in the DRD and PD, 3-D propagation models as described by Ptuskin et al. [2006]. [13] In Figure 2 the calculated IS H, He, C and Fe spectra are shown for the MCDM using the new diffusion coefficient at low rigidities along with earlier calculations using a LBM with a constant value of the diffusion coefficient below 3 GV. These spectra may be directly compared. [14] In Figure 3 we show in more detail the IS H and He spectra calculated using the MCDM for the new parameters. We also show calculations of the IS spectra using the MCDM made using the DRD and PD diffusion coefficients of Ptuskin et al. [2006]. The new IS spectra calculated using the MCDM are given in tabular form in Table Discussion and Comparison of Calculated IS Spectra [15] First it is observed that the IS intensities calculated with the new larger diffusion coefficient at lower rigidities are systematically lower than the earlier calculations for H and He nuclei [e.g., Webber and Lockwood, 2001]. For H this decrease is a factor 2.00 at 200 MeV. For He the decrease is 1.46 x at 200 MeV nucleon 1. For C the decrease is 1.30, and for Fe the decrease is 1.16 (see Figure 2). Thus the decreases in the calculated IS intensities resulting from changes in the diffusion coefficient are charge dependent with the lowest Z exhibiting the largest decrease. This dependence is due to a combination of the ionization energy loss for each type of particle which is Z 2 /A, where A is the mass number, as well as to the particles rigidity which depends on Z/A and is lower at the same energy for H nuclei than for He and the other A/Z = 2 nuclei. For the higher Z nuclei the E loss process dominates at energies below a few hundred MeV nucleon 1 (2 GV) over the escape from the galaxy which is due to the increased diffusion coefficient which is rigidity dependent. For H nuclei, on the other hand, the increasing diffusion coefficient and the resulting more rapid escape from the galaxy dominate, at least above 100 MeV (0.5 GV). In the energy region where diffusion dominates, the relative spectral intensity dj/de scales as D 0.5. This scaling thus applies to H nuclei over the rigidity range from GV which corresponds to an energy range from 100 MeV to 1 GeV; however, for He and heavier nuclei, because of the higher mass/charge ratio, the rigidity range in which this diffusion scaling dominates is only from GVand at these higher rigidities the changes in D are smaller. 3of6

4 Figure 2. Calculated IS H, He, C, and Fe spectra using the LBM and MCDM with diffusion coefficients (1) and (2), respectively, in Figure 1. Intensities are in particles/m 2 sr s MeV/nucleon. [16] Particularly interesting are the H and He spectra shown in Figure 3 calculated using the MCDM but using the DRD and PD diffusion coefficients of Ptuskin et al. [2006]. For H, which is most sensitive to the value of D, the DRD spectrum with its rapid increase in the value of D below 1.5 GV, gives IS intensities that are already below the most recent V1 measurements of H nuclei [Webber et al., 2008] in the outer heliosphere. For He the intensity decrease resulting from the larger DRD diffusion coefficient is much less obvious. For both H and He nuclei the IS spectra calculated by us using the PD diffusion coefficient of Ptuskin et al. [2006] are similar to but slightly lower, below 1 2 GV, than those obtained using the diffusion coefficient (2) in Figure 1. This would be expected from the Figure 3. Calculated IS spectra for H and He nuclei using the MCDM compared with measurements of these spectra made by the Voyager 1 spacecraft at 105 AU at Also shown are the IS spectra calculated with DRD and PD model diffusion coefficients of Ptuskin et al. [2006] normalized at 1 GeV nucleon 1 (for clarity, the calculations using the PD model diffusion coefficients are not shown for He; they lie between by MCDM and DRD curves that are shown). The MCDM spectrum modulated by a force field potential of 60 MV is shown as a dotted line. Intensities are in particles/m 2 sr s MeV/nucleon. 4of6

5 Table 1. Interstellar Spectra a Energy (GeV nucleon 1 ) H He C O Fe b a Intensities are in particles/m 2 srsmev. b Intensities E2.5 for all energies >1. magnitudes of the two sets of diffusion coefficients as seen in Figure 1. [17] A close correspondence exists between our new IS spectra for nuclei with Z 6 and those published earlier by Davis et al. [2000] [see also Yanasak et al., 2001]. These earlier calculations were based on a somewhat different decreasing path length at low energies (equivalent to an increasing diffusion coefficient) originally suggested by Soutoul and Ptuskin [1999] as being the result of altitudedependent convection above the galactic plane. 4. Comparison of Newly Calculated IS Spectra With Voyager Data [18] In Figure 3 we compare the newly calculated IS spectra for H and He using the MCDM with the most recent measured V1 spectra for these nuclei (2008.5) [Webber et al., 2008]. At this time V1 is at 105 AU, 20 AU beyond the HTS which is now estimated to be at 85 AU [Webber, 2005]. The ratio between the newly calculated IS spectra for these nuclei and the measured spectra at V1 at 200 MeV is 1.55 times for H and 1.30 times for He nuclei. This difference can be attributed to solar modulation effects in the outer heliosheath region of the heliosphere, plus the uncertainties in the derived IS spectra. The uncertainties in the IS spectra listed in Table 1 are estimated to be ±5 10% above 1 GeV nucleon 1, increasing to ±10 20% at 100 MeV nucleon 1, owing mainly to uncertainties in the shape of the low-rigidity diffusion coefficient and in the normalization of the spectra for the different charges. The uncertainties in the data are ±5% or less. [19] Another way to compare the new IS spectra and those measured by V1 is to determine, using a simple force field modulation model [e.g., Gleeson and Axford, 1968], what value of the modulation parameter, f, will reproduce the V1 H and He spectra observed at by starting from the calculated IS spectra using the MCDM. The value of f is determined to be (60 ± 20 MV) for both H and He as shown by the dotted lines in Figure 3. Starting from these new IS H and He spectra the value of f required to reproduce the observed H and He spectra at the Earth at a time of minimum modulation in 1998 [e.g., McDonald, 1998; Sanuki et al., 2000] is 320 MV. Thus in these very simple terms there still appears to be a small but significant solar modulation beyond V1 at 105 AU which requires a modulation potential f in a force field model that is 0.20 of that required to explain the spectra observed at a time of minimum solar modulation at the Earth. This modulation is sufficient to reduce the IS H and He spectra to those observed at V1 at [20] We should note, however, that the intensities being observed at V1 at are probably not yet at their maximum for the current 11-year modulation cycle. At the H and He intensities at V1 are still increasing at a rate 10 20%/year [Webber et al., 2008]. Since the maximum cosmic ray intensity at the Earth at these energies was observed at about using data from the ACE spacecraft ( and since the solar wind propagation time from the Earth to V1 is 1 year it is likely that V1 will observe a maximum intensity in 2010 when it is at 114 AU. Therefore some of the differences between the IS spectrum predictions and the current V1 measurements can be explained in terms of this continued intensity recovery. 5. Summary and Conclusions [21] In this paper we utilize new diffusion coefficients for galactic propagation of cosmic ray nuclei that have recently been derived for cosmic ray electrons [Webber and Higbie, 2008]. These modified diffusion coefficients, which increase at rigidities below 1 2 GV, are used to determine new IS H, He, C and Fe spectra using a Monte Carlo Diffusion Model for galactic propagation. The resulting energy spectra of these cosmic ray nuclei are reduced at low energies in a charge-dependent way from those derived using a constant diffusion coefficient at low rigidities as was used in earlier calculations. This charge-dependent intensity reduction is a result of a combination of the loss mechanisms due to diffusion, ionization energy loss and nuclear interactions during propagation. [22] The IS spectra for H and He nuclei are reduced by factors of 2.00 and 1.46 at 200 MeV nucleon 1 from earlier values [e.g., Webber and Lockwood, 2001]. The amount of solar modulation in simple force field models required to explain the H and He nuclei spectra observed by V1 in 2008 is 60 MV or about 0.20 of the 320 MV required to explain the solar modulation of H and He observed at the Earth. The new IS intensities calculated for both H and He are still larger by 10 30% at 200 MeV nucleon 1 than those projected to be observed at the time of the next 11-year maximum intensity at V1, expected to occur in 2010 when V1 will be at 114 AU. Our understanding of the solar modulation process in the outermost heliosphere along with the propagation and acceleration of galactic cosmic rays in the galaxy at energies MeV nucleon 1 and below, therefore depends on a 5of6

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Lal (2008), Galactic cosmic ray H and He nuclei energy spectra measured by Voyagers 1 and 2 near the heliospheric termination shock in positive and negative solar magnetic polarity cycles, J. Geophys. Res., 113, A10108, doi: /2008ja Yanasak, N. E., et al. (2001), Measurements of the secondary radionuclides 10 Be, 26 Al, 36 Cl, 54 Mn and 14 C and implications for the cosmic ray age, Astrophys. J., 563, , doi: / P. R. Higbie, Physics Department, New Mexico State University, Las Cruces, NM 88003, USA. W. R. Webber, Department of Astronomy, New Mexico State University, P. O. Box 30001, 1320 Frenger Street, Las Cruces, NM 88003, USA. (bwebber@nmsu.edu) 6of6