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Where does the heliospheric modulation of galactic cosmic rays start?
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ScienceDirect Advances in Space Research 53 (2014) 1015–1023 www.elsevier.com/locate/asr
Where does the heliospheric modulation of galactic cosmic rays start? R.D. Strauss ⇑, M.S. Potgieter Centre for Space Research, North-West University, 2520 Potchefstroom, South Africa Received 24 October 2013; received in revised form 12 December 2013; accepted 6 January 2014 Available online 14 January 2014
Abstract The long outstanding question of where the heliospheric (solar) modulation of galactic cosmic rays actually begins, in terms of spatial position, as well as at what high kinetic energy, can now be answered. Both answers are possible by using the results of an advanced numerical model, together with appropriate observations. Voyager 1 has been exploring the outskirts of the heliosphere and is presently entering what can be called the very local interstellar medium. It has been generally expected, and accepted, that once the heliopause is crossed, the local interstellar spectrum (LIS) should be measured in situ by the Voyager spacecraft. However, we show that this may not be the case and that modulation effects on galactic cosmic rays can persist well beyond the heliopause. For example, proton observations at 100 MeV close to the heliopause can be lower by 25% to 40% than the LIS, depending on solar modulation conditions. It is also illustrated quantitatively that significant solar modulation diminishes above 50 GeV at Earth. It is found that cosmic ray observations above this energy contain less that 5% solar modulation effects and should therefore reflect the LIS for galactic cosmic rays. Input spectra, in other words the very LIS, for solar modulation models are now constrained by in situ observations and can therefore not any longer be treated arbitrarily. It is also possible for the first time to determine the lower limit of the very LIS from a few MeV/nuc to very high energies. Ó 2014 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Cosmic rays; Solar modulation; Heliopause; Local interstellar spectra; Positron excess
1. Introduction A recent report by Gurnett et al. (2013) indicates that the spacecraft Voyager 1 has entered the very local interstellar medium. Corresponding cosmic ray (CR) observations (Stone et al., 2013; Krimigis et al., 2013) can thus be considered as indicative of what the very local interstellar spectra (LIS) could be (see also Potgieter et al., 2013a; Potgieter, 2013a). These in situ observations by Voyager 1 have led to a renewed interest in the long outstanding question of where exactly does the solar modulation of galactic CRs commence in terms of spatial position. At Earth, the PAMELA space detector (e.g. Adriani et al., 2013a) has been observing CR spectra since mid-2006, down to kinetic energies of E 100 MeV, in particular also
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an excess of CR positrons (Adriani et al., 2013b), evident from 200 GeV down in energy to the point where solar modulation dominates, thus obscuring the very LIS. Where the solar modulation of CRs commences in terms of energy (or rigidity) has thus also become of interest. This report is focused on answering these questions quantitatively. In the next section, our investigation of the energy dependence of the solar modulation of CRs, from 100 MeV up to 350 GeV will be given in order to find the energy range at which solar modulation becomes significant. In the section that follows, the spatial dependence of the solar modulation of CRs is investigated in order to establish where the modulation process begins. Does it occur from the heliopause (HP) of the heliosphere, usually assumed as the modulation boundary, or is it beginning beyond the HP? Discussions are given in the context of the results of numerical modelling and the subsequent insight gained
E-mail address: [email protected] (R.D. Strauss). 0273-1177/$36.00 Ó 2014 COSPAR. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.asr.2014.01.004
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by doing such modelling. The emphasis is on the implications for the interpretation of in situ measurements of the very LIS of galactic CRs inside and outside of the heliosphere, including low energies E < 1 GeV which is of special interest to solar modulation studies. A preliminary presentation on this topic was given by Potgieter and Strauss (2013). 2. Energy dependence of the solar modulation of CRs It is not straightforward to establish at what energy/ rigidity the modulation of galactic CRs actually begins in the heliosphere. This should also change as a function of time and position, e.g. if the modulation conditions would be significantly different in the nose direction of the heliosphere compared to the tail-direction. It is not yet possible to determine observationally at what high energy (rigidity) the solar modulation of galactic CRS becomes insignificant. It can only be done precisely if the relevant very LIS for CRs were known (observed) at the appropriate rigidities, and reliable measurements are made simultaneously inside and outside the heliosphere over a relatively large rigidity range, e.g. between 10 GV and 200 GV. Neutron monitors, observing mainly galactic CR protons indirectly (inside the Earth’s atmosphere and magnetosphere) at rigidities larger than 40 GV do observe solar related changes in the CR intensity over relatively long periods at meaningful levels of modulation, even during solar minimum activity conditions. This indicates that the modulation of CRs is occurring at higher rigidities than what is usually assumed. Also from a numerical modelling side, it is not as easy as it may seem because a so-called nomodulation limit at high energies is difficult to determine so that this required initial condition in numerical modelling is simply assumed. In the past, this assumption was many times adjusted to lower energies to save on the computational time it took to do these computations. Most numerical models applied to solar modulation are based on what can be called standard finite difference schemes which require boundary conditions at all phasespace domains, one of which is a high energy initial condition where the LIS is specified as the unmodulated spectrum, usually between 30 to 50 GeV. This artificial condition therefore assumes no modulation above this selected energy, usually without considering what the values of the CR transport coefficients are for this assumed energy. Fundamentally, the energy at which modulation is no longer present, should be determined entirely by the values of transport coefficients at these high energies (in other words, by the physics), and should be independent of the adopted numerical scheme. Unfortunately, despite significant progress in turbulence and diffusion theory (see e.g. Shalchi, 2013), we also do not know what exactly the values of these diffusion coefficients everywhere are, inside and outside the heliosphere. Recently, progressively more CR transport and modulation models are based on solutions of stochastic differential
equations (SDEs). A reason is that the numerical scheme and process are perfectly parallelizable and is therefore very appropriate in utilizing the advantage given by large computer clusters. This approach offers the opportunity to determine the level of modulation for any given rigidity (or kinetic energy/nucleon), from very high to low rigidities. With these types of models it is possible to determine the energy at which CR modulation fades, because no energy domain boundary conditions need to be prescribed. In this section, the SDE based galactic CR transport model of Strauss et al. (2011a,b) is used to study the modulation of galactic CR protons, in particular its energy dependence. For details of this numerical method, see the work of Zhang (1999), Pei et al. (2010), Kopp et al. (2012) and references therein. The left panel of Fig. 1 shows energy spectra at Earth for galactic CR protons, for the two modulation drift cycles, indicated as A < 0 and A > 0. For a detailed discussion of this charge-sign dependent modulation and the 22-year cycle, see also e.g. Potgieter (2013c) and Strauss et al. (2012b). These spectra were computed with the mentioned SDE model and are shown with respect to the computed LIS of Moskalenko et al. (2002), as unmodulated input to the model, specified at a distance of r ¼ 120 AU from the Sun. The drift solutions are compared to observations of von Rosenvinge et al. (1979) and Balasubrahmanya et al. (1965) to validate the choice of transport coefficients as implemented, illustrating that realistic levels of modulation are obtained in this manner. For these simulations, a wavy heliospheric current sheet (HCS) with a tilt angle of a ¼ 10 is assumed, as implemented by Strauss et al. (2012a), along with the diffusion coefficients as discussed by Strauss et al. (2012c). This figure illustrates the general features and drift characteristics of modulated proton spectra at Earth. Of particular importance is that the LIS, according to this drift modulation model, gives two modulated spectra at Earth, produced with one set of modulation parameters but different drift cycles caused by flipping only the solar magnetic field direction every 11 years, thus creating a 22-year CR modulation cycle. The right panel of Fig. 1 shows the corresponding modulation ratio as a function of kinetic energy, obtained by normalizing the spectra at Earth to the LIS levels; unity therefore indicates that the LIS is unmodulated (no reduction in intensity). The shaded regions respectively indicate energy regimes where less than 30% (light grey; E > 10 GeV) and less than 5% (dark grey; E > 50 GeV) modulation are computed. This modulation ratio approaches unity at high energies as expected, but does not quite reach this value at 350 GeV, where the computations were stopped, with a small fraction of residual modulation still present. Clearly, the drift modulation cycles produce somewhat different modulation ratios, becoming quite evident between 10 to 20 GeV, progressively so with decreasing energy. Note that the drift effect seems to fade at low energies but this is simply caused by the use of a
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Fig. 1. The left panel shows computed galactic CR proton spectra at Earth, for the two drift cycles, indicated as A < 0 and A > 0, with respect to the assumed LIS of Moskalenko et al. (2002), specified at an assumed modulation boundary of r ¼ 120 AU. The two drift solutions are compared to the observations of von Rosenvinge et al. (1979) and Balasubrahmanya et al. (1965), respectively observed during the A > 0 solar minimum period before and after 1977 and the A < 0 period around 1965. The right panel shows the resulting modulation ratio (the energy spectra a Earth, normalized to LIS levels), while the grey bands indicate energy regions where different levels of modulation are present.
linear scale for the ratio. A selection of values based on this figure are shown in Table 1. 2.1. Implications for astrophysical processes Based on the results presented in the previous section, Fig. 2 summarizes the different energy dependent modulation zones that occur at Earth. Table 1 Modulation ratio in percentage (100% means no reduction in the LIS intensity) between Earth, at 1 AU, and the HP, in this case, at 120 AU for selected CR proton kinetic energies and for both drift modulation cycles, during ideal solar minimum conditions. The present modulation cycle, around the minimum in 2009, is an A < 0 drift cycle when positive particles drift to Earth mainly through the equatorial plane of the heliosphere. Kinetic energy (GeV)
A > 0 cycle (%)
A < 0 cycle (%)
300 200 100 50 25 12 6 3 1 0.5 0.1 0.05 0.01
99.6 99.3 98 95 86 75 60 42 22 15 5.0 2.9 1.3
99.6 99.3 98 95 86 74 56 31 13 8 2.6 1.8 0.6
Above 50 GeV, less than 5% solar modulation is present so that observations in this energy range should reflect astrophysical related processes and transport effects. A mixed zone between 10 to 50 GeV, where modulation is less than 30% so that measurements in this energy regime should reflect both solar modulation and astrophysical effects, with solar modulation becoming progressively dominant with decreasing energy. The solar modulation regime between 100 MeV to 10 GeV. In this regime, solar modulation dominates so that all interstellar or astrophysical information regarding e.g. the LIS of CRs, will be obscured, again progressively with decreasing energy. Below 100 MeV solar modulation is obviously very dominant at Earth but local heliospheric sources of energetic particles may also become important or even dominant. A very specific example of such domination is Jovian electrons so that the intensity of galactic electrons cannot be observed below 50 MeV at Earth (Potgieter and Nndanganeni, 2013a). In the outer heliosphere, especially in the heliosheath, the so-called anomalous component completely dominates the galactic particles in this energy range (see the review by e.g. Potgieter,
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Fig. 2. Different modulation zones (coloured panels), as indicated in the figure and identified based on the right panel of Fig. 1. See text for a full description. Also shown is the observed positron fraction, without error bars, at Earth (for full data references, see Adriani et al., 2013b). Dashed line is the theoretical calculations of Moskalenko and Strong (1998), shown for illustrative purposes.
2013b). This component should disappear beyond the HP and this was indeed observed to occur (Stone et al., 2013). Shown additionally in Fig. 2, is an overview of the observed positron fraction at Earth (taken from Adriani et al., 2013b), the cause of which is still hotly debated because it deviates so much from the long-standing computed fraction of Moskalenko and Strong (1998), shown here for illustrative purposes. Of interest in the context of the discussion topic of this publication, is the obvious trend in the variation of this fraction below 10 GeV over time, which indicates, based on the arguments given above, that these differences can be attributed to solar modulation effects (including drifts), as discussed by Potgieter (2013c). (These observations obviously also contain systematic errors, as reported when published, which should always be carefully considered when making conclusions). Above this energy, the high energy positron excess must be attributed to an astrophysical origin. In additional to this generally accepted interpretation, it can be added that up to 30 GeV, solar modulation should also contribute noticeably to a variation in this fraction over longer time periods, such as the 11-year cycle. The modelling results in this and the next section, depend on the value and spectral shape of the assumed LIS. In the energy range of particular interest, E > 1 GeV, most LIS models produce similar results as they are constrained by observations into a very narrow range. As such, the quantitative results presented here will change marginally, while the qualitative results and conclusions remain unchanged.
2.2. Effect of heliospheric drifts at high energies Recently, Roberts (2011) attributed the above-mentioned positron excess above 10 GeV to differences in the drift directions of oppositely charged galactic CRs in the heliosphere, in other words, that the total observed effect shown in Fig. 2 could simply be explained by the charge-sign dependent effect in solar modulation. During A < 0 drift cycles (the present epoch), positrons drift to Earth mainly through the equatorial regions of the heliosphere, encountering the wavy HCS in the process, while electrons drift mainly through the polar regions to Earth. Since the HCS changes significantly with solar activity, positrons will respond to it while electrons remain largely unaffected. For an illustration of these drift effects and its consequences for protons and electrons observed by PAMELA from mid-2006 to the end of 2009, see Fig. 3 of Vos et al. (2013). From a solar modulation point of view, Robert’s interpretation that the positron intensity could be enhanced during this polarity epoch, is highly controversial. It is argued in this section, where the effectiveness of modulation drifts is discussed and illustrated, that heliospheric drifts of CRs as illustrated in Figs. 1 and 2, cannot be the cause of the increasing trend in the observed positron to electron ratio at higher energies. The SDE model used for this study is based on utilizing a full diffusion and drift tensor in three dimensions, so that diffusion parallel to the average heliospheric magnetic field, also perpendicular to it in the radial and polar directions, as well as a drift coefficient are used in solving the appropriate transport equation. In Fig. 3 the transport coefficients as a function of rigidity are summarized as applied
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Fig. 3. The transport coefficients used in this work, as a function of rigidity at Earth (1 AU) and at the HP (120 AU). The grey coloured region indicates values smaller than the radius of the heliosphere, whereas the green box shows generally accepted values of the parallel mean free path at Earth, known as the Palmer consensus values (e.g. Bieber et al., 1994). (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this article.)
in this study. The corresponding parallel ðkjj Þ and perpendicular ðk? Þ mean free paths (MFPs) are shown as a function of rigidity at Earth and at the HP. The so-called Palmer consensus values (Bieber et al., 1994) of kjj at Earth are included to show that the assumed values of kjj are reasonable. Also shown on the figure is the Larmor radius of a galactic CR proton at the same positions. When the weak scattering limit of drifts is assumed, the Larmor radius may be interpreted as the drift scale ðkd Þ; the length scale on which modulation drifts occur (see e.g. Minnie et al., 2007). By comparing the different transport scales, as is done in Fig. 3, the most dominant transport processes may be identified qualitatively; a larger length scale indicates a more efficient global modulation process. For all energy and spatial regimes under investigations, kjj > kd , meaning that diffusion remains the dominant transport process of galactic CRs in the heliosphere. Moreover, the interplay between k? and kd gives insight into when drift effects may be significant because both of these processes operate perpendicular to the mean magnetic field: In the inner heliosphere, kjj > kd > k? for rigidities > 10 GV, so that drifts are non-negligible. In the outer heliosphere, kjj > k? > kd , and the efficiency of drifts, relative to diffusion, decreases.
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This illustrates that the transport of galactic CRs in the global heliosphere fundamentally remains a diffusion dominated process and that drifts only augment the process, sometimes significantly depending on solar activity levels and, of course, the polarity cycles of the solar magnetic field. Also shown on Fig. 3, as the shaded region, is the assumed radial distance to the HP. This assumption is based on the recent observation of the HP position at 122 AU by Voyager 1 (e.g. Stone et al., 2013). Comparing the different length scales with this value, adds further reason to the different modulation zones as identified in Fig. 2: Above 50 GV both diffusion MFPs at the HP become comparable to the radius of the heliosphere, and insignificant modulation effects are thus to be expected, even at Earth, as illustrated above. The interplay between diffusion and drifts, and their effectiveness as modulation processes, is illustrated in Fig. 4. Here, pseudo-particle traces of galactic protons, reaching Earth with kinetic energy of 100 MeV in the A < 0 drift cycle, are shown in the meridional plane of the heliosphere (for details see Strauss et al., 2012a). The shape of the wavy HCS, with a ¼ 45 , is shown as the dotted line, and the position of the HP as the dashed line. Keep in mind that in this picture 3D modulation effects are projected onto the meridional plane. Panel (a) corresponds to a scenario similar to that of Fig. 1, except for a larger tilt angle. This scenario thus produces the most realistic levels of modulation at Earth. Successive panels each time show the computations when both kjj and k? are decreased by an additional factor of 10, in order to illustrate what happens when diffusion is diminishing globally in such a way. From panel (a), considered as the most realistic modulation scenario, it is clearly evident that although protons drift primarily along the HCS to reach Earth, as expected for this drift cycle, their transport and consequent modulation remain diffusion dominated because CRs are scattered off the wavy HCS and throughout the heliosphere. The scenario shown in panel (d), gives a picture of drifts dominating the modulation process completely, with smooth, deterministic transport along the HCS. Because this scenario represents a clear picture of HCS drifts in the heliosphere, it is a popular but one-sided view and cannot be considered as a realistic modulation scenario. The CRs drifting to Earth via the equatorial regions of the heliosphere in such a way must of course again drift away from Earth via the polar regions during this polarity epoch to assure that the divergence of the drift velocity remains zero everywhere. The magnetic focussing effect described by Roberts (2011) thus requires that the positrons must drift away from Earth in the same perfect way that they drift in because in his approach there exists no other process that can take them away from the HCS which cannot happen in a realistic drift modulation model. These simulations reiterate that diffusion is a determining CR transport process in the heliosphere, even during solar minimum activity (in this context, see also Potgieter
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Fig. 4. Computed pseudo-particle traces of 100 MeV galactic protons with a 3D SDE-type modulation model, projected onto the meridional plane of the heliosphere. The dotted line shows a projection of the HCS, with a ¼ 45 , onto the same plane and the dashed line the HP position. For each consecutive panel, the diffusion coefficients are divided by an additional factor of 10 relative to those used in panel (a), which is considered as the most realistic modulation scenario. For additional trajectories e.g. with increasing tilt angles, see Strauss et al. (2012a).
et al., 2013b), and that studies neglecting diffusion, or handling it in a totally arbitrarily manner in a drift model (e.g. Loparco et al., 2013), should be taken with serious caution since a distorted view of solar modulation could be the result. 3. Spatial dependence of heliospheric modulation The long awaited entering of the very local interstellar medium by Voyager 1 is confirmed to have taken place on 25 August 2012 at a radial distance of 122 AU (e.g. Stone et al., 2013). Although there were originally some reservations whether the spacecraft actually crossed the
HP because of the seemingly incongruous magnetic field observations (Burlaga et al., 2013), the inferred plasma density measurements (Gurnett et al., 2013) confirmed that it is currently sampling the very local interstellar space in situ. A long held hypothesis is that once Voyager 1 crossed the HP, it would immediately sample the pristine LIS for galactic CRs. This paradigm means that the solar modulation of CRs should abruptly stop at the HP. Recently, it has been argued and now confirmed by sophisticated SDE modelling (Scherer et al., 2011; Strauss et al., 2013), that this may not be the case and that solar modulation can persist beyond the HP, perhaps up to where the
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Fig. 5. Computed spectra using a hybrid modulation model to investigate the occurrence of galactic CR modulation beyond the HP (taken from Strauss et al., 2013). Spectra are shown at the HP (corresponding to the Voyager 1 trajectory) and at Earth for solar minimum (red lines) and maximum (blue lines) modulation conditions. (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this article.)
heliosphere begins to disturb the very local interstellar medium by moving through it. This is discussed further in the next section. 3.1. Implications for in situ LIS measurements Fig. 5 shows modelling results from Strauss et al. (2013), who investigated the residual modulation that may be present beyond the HP. For these simulations, a hybrid modulation model was used, where the heliospheric plasma environment, as simulated by a 3D magneto-hydrodynamic model, is coupled to a SDE based galactic CR modulation model. This produces a realistic plasma flow (solar wind), with a subsequent magnetic field and a realistic heliospheric geometry in contrast to simply assuming these features in a particle transport model. Doing this is an important and necessary requirement for the realistic modelling of CR transport in the heliosheath, up to the HP and beyond. The very LIS, in this model, is not prescribed at the HP, but rather at the edge of the spatial computational domain, at least 400 AU from the Sun in order to compute what may happen to it beyond the HP. In Fig. 5, energy spectra for galactic protons are shown at Earth (the bottom two curves) and at the HP, with
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respect to the unmodulated LIS. It follows from these results that CR modulation persists beyond the HP, with the 100 MeV intensity 25% to 40% lower than the LIS value at the HP. This process is energy dependent and diminishes above a few GeV. The two sets of simulations represent solar minimum (red curves) and solar maximum (blue curves) modulation conditions, with an interesting finding that modulation beyond the HP even exhibits solar cycle related changes. This consequently gives lower proton intensities at the HP during times of solar maximum conditions. Modulation in this previously neglected modulation region is mostly the result of the re-entry (return) of galactic CRs which were already inside the heliosphere back into this region. These returning CRs had thus already been subjected to some modulation (for a detailed discussion see Scherer et al., 2011; Strauss et al., 2013). The computational results discussed in this section indicate that the galactic CR observations currently being made by Voyager 1 should still be influenced by the global heliospheric environment. These measurements should continue to exhibit a small, positive radial intensity gradient beyond the HP for the foreseeable future. Spatially, this should continue to the position where the heliosphere no longer disturbs the local interstellar medium. Simulations suggest that this effect should persist for at least 100 AU beyond the HP along the Voyager 1 trajectory. It should be kept in mind that for galactic CRs this is not a local effect: A 40% decrease in intensity occurs over 100 AU, so that an average radial gradient of 0.4%/ AU is expected. As such, Voyager 1 may take several years to observe this effect. In contrast, the anomalous cosmic ray particles which are created inside the heliosphere, and accelerated in the heliosheath, should exhibit a negative radial gradient that varies from a very large value close to the HP to essentially zero when far enough from the HP. The present Voyager 1 observations of proton, helium, carbon and oxygen spectra below a few hundred MeV/ nuc should thus be considered as the lower limits of the very LIS and should rather be referred to as the HP spectra of galactic CRs until proven otherwise. In this context, see also discussions by Strauss et al. (2012b), Potgieter and Nndanganeni (2013a) and Potgieter (2013a). 4. Conclusions The solar modulation of galactic CRs at Earth diminishes progressively with increasing energy (rigidity) primarily caused by the corresponding increasing values of mainly the diffusion coefficients. It was quantitatively shown that solar modulation of CRs becomes significant below 10 GeV, less so between 10 to 50 GeV, and progressively negligible above 50 GeV. Cosmic ray observations above this energy contain less that 5% solar modulation effects and should therefore reflect the very LIS for galactic CRs at Earth. The probability of particle drifts having significant modulation effects with increasing energy was investigated and
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it was confirmed that solar modulation remains fundamentally a diffusion dominated process with drifts augmenting the process depending on solar activity and subsequent modulation conditions but can be considered mostly of second-order importance at higher energies. Drifts, as the main charge-sign dependent process in the heliosphere, cannot be the main contributor of the increasing positron excess above 10 GeV but will significantly change the positron to electron ratio below this energy (for examples, see Potgieter, 2013b). It was shown that CR modulation may persist beyond the HP, and can be quite significant below 1 GeV. At present, Voyager 1 is probably not measuring the LIS of galactic CRs at these low energies, but rather a HP spectrum that should gradually unfold to LIS values during the coming years. Spectra of protons, helium, carbon and oxygen below a few hundred MeV/nuc should thus be considered as the lower limits of the very LIS and can thus be used to estimate the LIS. At the same time, these HP spectra can be used to check predictions of the galactic spectra as obtained with galactic propagation models. Combining the current low energy galactic CR observations of Voyager 1 at and beyond the HP with high energy observations at Earth can thus provide a full HP spectrum for the mentioned CR species (see also e.g. Potgieter and Nndanganeni, 2013b; Potgieter, 2013a). Input spectra for solar modulation models are now constrained by in situ observations and can therefore not any longer be treated arbitrarily, especially at these low energies. For galactic CR transport models, HP spectra together with observations at Earth provide further constraint for the choice of e.g. the transport parameters because for the first time the total modulation inside the heliosphere, between 1 MeV and 300 GeV, can be calculated accurately, of course limited by the accuracy of the corresponding CR observations. References Adriani, O.and the PAMELA collaboration, 2013a. Time dependence of the proton flux measured by PAMELA during the July 2006– December 2009 solar minimum. Astrophys. J. 765 (91), 1–8. Adriani, O.and the PAMELA collaboration, 2013b. Cosmic-ray positron energy spectrum measured by PAMELA. Phys. Rev. Lett. 765, 081102:1–081102:6. Balasubrahmanya, V.K., Hagge, D.E., Ludwig, G.H., McDonald, F.B., 1965. Galactic cosmic rays at solar minimum. In: Proceedings of the 9th International Cosmic Ray Conference, vol. 1, pp. 427–436. Bieber, J.W., Matthaeus, W.H., Smith, C.W., Wanner, W., Kallenrode, M.-B., Wibberenz, G., 1994. Proton and electron mean free paths: the Palmer consensus revisited. Astrophys. J. 420, 294–306. Burlaga, L.F., Ness, N.F., Stone, E.C., 2013. Magnetic field observations as Voyager 1 entered the heliosheath depletion region. Science 341 (6142), 147–150. Gurnett, D.A., Kurth, W.S., Burlaga, L.F., Ness, N.F., 2013. In situ observations of interstellar plasma with Voyager 1. Science 341 (6153), 1489–1492. Kopp, A., Busching, I., Strauss, R.D., Potgieter, M.S., 2012. A stochastic differential equation code for multidimensional Fokker–Planck type problems. Comput. Phys. Commun. 183, 530–542.
Krimigis, S.M., Decker, R.B., Roelof, E.C., et al., 2013. Search for the exit: voyager 1 at heliosphere’s border with the Galaxy. Science 341 (6142), 144–147. Loparco, F., Maccione, L., Mazziotta, M.N, 2013. Inferring the local interstellar spectrum of cosmic ray protons from PAMELA data. arXiv:1306.1354v1. Minnie, J., Bieber, J.W., Matthaeus, W.H., et al., 2007. Suppression of particle drifts by turbulence. Astrophys. J. 670, 1149–1158. Moskalenko, I.V., Strong, A.W., 1998. Production and propagation of cosmic-ray positrons and electrons. Astrophys. J. 493, 694–707. Moskalenko, I.V., Strong, A.W., Ormes, J.F., Potgieter, M.S., 2002. Secondary antiprotons and propagation of cosmic rays in the galaxy and heliosphere. Astrophys. J. 565 (1), 280–296. Pei, C., Bieber, J.W., Burger, R.A., Clem, L., 2010. A general timedependent stochastic method for solving Parker’s transport equation in spherical coordinates. J. Geophys. Res. 115, A12107. Potgieter, M.S., 2013a. Very local interstellar spectra for galactic electrons, protons and helium. In: Proc. 33rd Inter. Cosmic Ray Conf. (Highlight Talk), Brazil. J. Phys. (in press). Potgieter, M.S., 2013c. The charge-sign dependent effect in the solar modulation of cosmic rays. Adv. Space Res., JASR11331. Potgieter, M.S., 2013b. Solar modulation of cosmic rays. Liv. Rev. Solar Phys. 10 (1), 1–66. Potgieter, M.S., Nndanganeni, R.R., 2013a. The solar modulation of electrons in the heliosphere. Astrophys. Space Sci. 345, 33–40. Potgieter, M.S., Nndanganeni, R.R., 2013b. A local interstellar spectrum for galactic electrons. Astropart. Phys. 48, 25–29. Potgieter, M.S., Strauss, R.D., 2013. At what rigidity does the solar modulation of galactic cosmic rays begin? In: Proc. 33rd Inter. Cosmic Ray Conf. arXiv:1308.1614. Potgieter, M.S., Vos, E.E., Nndanganeni, R.R., Boezio, M., Munini, R., 2013a. A very local interstellar spectrum for galactic electrons. In: Proc. 33rd Inter. Cosmic Ray Conf. arXiv:1308.1666. Potgieter, M.S., Vos, E.E., Boezio, M., De Simone, N., Di Felice, V., Formato, V., 2013b. Modulation of galactic protons in the heliosphere during the unusual solar minimum of 2006 to 2009. Solar Phys. 281, 391–406. Roberts, J.P., 2011. PAMELA through a magnetic lens. J. Cosmol. Astropart. Phys. 2, 29–42. Scherer, K., Fichtner, H., Strauss, R.D., Ferreira, S.E.S., Potgieter, M.S., Fahr, H.-J., 2011. On cosmic ray modulation beyond the heliopause: where is the modulation boundary? Astrophys. J. 735 (128), 1–5. Shalchi, A., 2013. Simple analytical forms of the perpendicular diffusion coefficient for two-component turbulence I. Magnetostatic turbulence. Astrophys. J. 774 (7), 1–6. Stone, E.C., Cummings, A.C., McDonald, F.B., Heikkila, B.C., Lal, N., Webber, W.R., 2013. Voyager 1 observes low-energy galactic cosmic rays in a region depleted of heliospheric ions. Science 341 (6142), 150– 153. Strauss, R.D., Potgieter, M.S., Bu¨sching, I., Kopp, A., 2011a. Modelling the modulation of galactic and Jovian electrons by stochastic processes. Astrophys. J. 735 (83), 1–13. Strauss, R.D., Potgieter, M.S., Kopp, A., Bu¨sching, I., 2011b. On the propagation times and energy losses of cosmic rays in the heliosphere. J. Geophys. Res. 116 (A12105), 1–13. Strauss, R.D., Potgieter, M.S., Bu¨sching, I., Kopp, A., 2012a. Modelling heliospheric current sheet drift in stochastic cosmic ray transport models. Astrophys. Space Sci. 339, 223–236. Strauss, R.D., Potgieter, M.S., Ferreira, S.E.S., 2012b. Modeling ground and space based cosmic ray observations. Adv. Space Res. 49, 392– 407. Strauss, R.D., Potgieter, M.S., Boezio, M., et al., 2012c. The heliospheric transport of protons and anti-protons: a stochastic modelling approach to PAMELA observations. Proc. Conf. Astroparticle, Particle, Space Physics and Detectors for Physics Applications, vol. 7. World Scientific, pp. 288–294.
R.D. Strauss, M.S. Potgieter / Advances in Space Research 53 (2014) 1015–1023 Strauss, R.D., Potgieter, M.S., Ferreira, S.E.S., Fichtner, H., Scherer, K., 2013. Cosmic ray modulation beyond the heliopause: a hybrid modeling approach. Astrophys. J. Lett. 765 (L18), 1–6. von Rosenvinge, T.T., McDonald, F.B., Trainor, J.H., Webber, W.R., 1979. The modulation of galactic cosmic rays between 1 and 17 AU. In: Proceedings of the 16th International Cosmic Ray Conference, vol. 12, pp. 170–175.
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ScienceDirect Advances in Space Research 53 (2014) 1015–1023 www.elsevier.com/locate/asr
Where does the heliospheric modulation of galactic cosmic rays start? R.D. Strauss ⇑, M.S. Potgieter Centre for Space Research, North-West University, 2520 Potchefstroom, South Africa Received 24 October 2013; received in revised form 12 December 2013; accepted 6 January 2014 Available online 14 January 2014
Abstract The long outstanding question of where the heliospheric (solar) modulation of galactic cosmic rays actually begins, in terms of spatial position, as well as at what high kinetic energy, can now be answered. Both answers are possible by using the results of an advanced numerical model, together with appropriate observations. Voyager 1 has been exploring the outskirts of the heliosphere and is presently entering what can be called the very local interstellar medium. It has been generally expected, and accepted, that once the heliopause is crossed, the local interstellar spectrum (LIS) should be measured in situ by the Voyager spacecraft. However, we show that this may not be the case and that modulation effects on galactic cosmic rays can persist well beyond the heliopause. For example, proton observations at 100 MeV close to the heliopause can be lower by 25% to 40% than the LIS, depending on solar modulation conditions. It is also illustrated quantitatively that significant solar modulation diminishes above 50 GeV at Earth. It is found that cosmic ray observations above this energy contain less that 5% solar modulation effects and should therefore reflect the LIS for galactic cosmic rays. Input spectra, in other words the very LIS, for solar modulation models are now constrained by in situ observations and can therefore not any longer be treated arbitrarily. It is also possible for the first time to determine the lower limit of the very LIS from a few MeV/nuc to very high energies. Ó 2014 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Cosmic rays; Solar modulation; Heliopause; Local interstellar spectra; Positron excess
1. Introduction A recent report by Gurnett et al. (2013) indicates that the spacecraft Voyager 1 has entered the very local interstellar medium. Corresponding cosmic ray (CR) observations (Stone et al., 2013; Krimigis et al., 2013) can thus be considered as indicative of what the very local interstellar spectra (LIS) could be (see also Potgieter et al., 2013a; Potgieter, 2013a). These in situ observations by Voyager 1 have led to a renewed interest in the long outstanding question of where exactly does the solar modulation of galactic CRs commence in terms of spatial position. At Earth, the PAMELA space detector (e.g. Adriani et al., 2013a) has been observing CR spectra since mid-2006, down to kinetic energies of E 100 MeV, in particular also
⇑ Corresponding author. Tel.: +27182992404.
an excess of CR positrons (Adriani et al., 2013b), evident from 200 GeV down in energy to the point where solar modulation dominates, thus obscuring the very LIS. Where the solar modulation of CRs commences in terms of energy (or rigidity) has thus also become of interest. This report is focused on answering these questions quantitatively. In the next section, our investigation of the energy dependence of the solar modulation of CRs, from 100 MeV up to 350 GeV will be given in order to find the energy range at which solar modulation becomes significant. In the section that follows, the spatial dependence of the solar modulation of CRs is investigated in order to establish where the modulation process begins. Does it occur from the heliopause (HP) of the heliosphere, usually assumed as the modulation boundary, or is it beginning beyond the HP? Discussions are given in the context of the results of numerical modelling and the subsequent insight gained
E-mail address: [email protected] (R.D. Strauss). 0273-1177/$36.00 Ó 2014 COSPAR. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.asr.2014.01.004
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by doing such modelling. The emphasis is on the implications for the interpretation of in situ measurements of the very LIS of galactic CRs inside and outside of the heliosphere, including low energies E < 1 GeV which is of special interest to solar modulation studies. A preliminary presentation on this topic was given by Potgieter and Strauss (2013). 2. Energy dependence of the solar modulation of CRs It is not straightforward to establish at what energy/ rigidity the modulation of galactic CRs actually begins in the heliosphere. This should also change as a function of time and position, e.g. if the modulation conditions would be significantly different in the nose direction of the heliosphere compared to the tail-direction. It is not yet possible to determine observationally at what high energy (rigidity) the solar modulation of galactic CRS becomes insignificant. It can only be done precisely if the relevant very LIS for CRs were known (observed) at the appropriate rigidities, and reliable measurements are made simultaneously inside and outside the heliosphere over a relatively large rigidity range, e.g. between 10 GV and 200 GV. Neutron monitors, observing mainly galactic CR protons indirectly (inside the Earth’s atmosphere and magnetosphere) at rigidities larger than 40 GV do observe solar related changes in the CR intensity over relatively long periods at meaningful levels of modulation, even during solar minimum activity conditions. This indicates that the modulation of CRs is occurring at higher rigidities than what is usually assumed. Also from a numerical modelling side, it is not as easy as it may seem because a so-called nomodulation limit at high energies is difficult to determine so that this required initial condition in numerical modelling is simply assumed. In the past, this assumption was many times adjusted to lower energies to save on the computational time it took to do these computations. Most numerical models applied to solar modulation are based on what can be called standard finite difference schemes which require boundary conditions at all phasespace domains, one of which is a high energy initial condition where the LIS is specified as the unmodulated spectrum, usually between 30 to 50 GeV. This artificial condition therefore assumes no modulation above this selected energy, usually without considering what the values of the CR transport coefficients are for this assumed energy. Fundamentally, the energy at which modulation is no longer present, should be determined entirely by the values of transport coefficients at these high energies (in other words, by the physics), and should be independent of the adopted numerical scheme. Unfortunately, despite significant progress in turbulence and diffusion theory (see e.g. Shalchi, 2013), we also do not know what exactly the values of these diffusion coefficients everywhere are, inside and outside the heliosphere. Recently, progressively more CR transport and modulation models are based on solutions of stochastic differential
equations (SDEs). A reason is that the numerical scheme and process are perfectly parallelizable and is therefore very appropriate in utilizing the advantage given by large computer clusters. This approach offers the opportunity to determine the level of modulation for any given rigidity (or kinetic energy/nucleon), from very high to low rigidities. With these types of models it is possible to determine the energy at which CR modulation fades, because no energy domain boundary conditions need to be prescribed. In this section, the SDE based galactic CR transport model of Strauss et al. (2011a,b) is used to study the modulation of galactic CR protons, in particular its energy dependence. For details of this numerical method, see the work of Zhang (1999), Pei et al. (2010), Kopp et al. (2012) and references therein. The left panel of Fig. 1 shows energy spectra at Earth for galactic CR protons, for the two modulation drift cycles, indicated as A < 0 and A > 0. For a detailed discussion of this charge-sign dependent modulation and the 22-year cycle, see also e.g. Potgieter (2013c) and Strauss et al. (2012b). These spectra were computed with the mentioned SDE model and are shown with respect to the computed LIS of Moskalenko et al. (2002), as unmodulated input to the model, specified at a distance of r ¼ 120 AU from the Sun. The drift solutions are compared to observations of von Rosenvinge et al. (1979) and Balasubrahmanya et al. (1965) to validate the choice of transport coefficients as implemented, illustrating that realistic levels of modulation are obtained in this manner. For these simulations, a wavy heliospheric current sheet (HCS) with a tilt angle of a ¼ 10 is assumed, as implemented by Strauss et al. (2012a), along with the diffusion coefficients as discussed by Strauss et al. (2012c). This figure illustrates the general features and drift characteristics of modulated proton spectra at Earth. Of particular importance is that the LIS, according to this drift modulation model, gives two modulated spectra at Earth, produced with one set of modulation parameters but different drift cycles caused by flipping only the solar magnetic field direction every 11 years, thus creating a 22-year CR modulation cycle. The right panel of Fig. 1 shows the corresponding modulation ratio as a function of kinetic energy, obtained by normalizing the spectra at Earth to the LIS levels; unity therefore indicates that the LIS is unmodulated (no reduction in intensity). The shaded regions respectively indicate energy regimes where less than 30% (light grey; E > 10 GeV) and less than 5% (dark grey; E > 50 GeV) modulation are computed. This modulation ratio approaches unity at high energies as expected, but does not quite reach this value at 350 GeV, where the computations were stopped, with a small fraction of residual modulation still present. Clearly, the drift modulation cycles produce somewhat different modulation ratios, becoming quite evident between 10 to 20 GeV, progressively so with decreasing energy. Note that the drift effect seems to fade at low energies but this is simply caused by the use of a
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Fig. 1. The left panel shows computed galactic CR proton spectra at Earth, for the two drift cycles, indicated as A < 0 and A > 0, with respect to the assumed LIS of Moskalenko et al. (2002), specified at an assumed modulation boundary of r ¼ 120 AU. The two drift solutions are compared to the observations of von Rosenvinge et al. (1979) and Balasubrahmanya et al. (1965), respectively observed during the A > 0 solar minimum period before and after 1977 and the A < 0 period around 1965. The right panel shows the resulting modulation ratio (the energy spectra a Earth, normalized to LIS levels), while the grey bands indicate energy regions where different levels of modulation are present.
linear scale for the ratio. A selection of values based on this figure are shown in Table 1. 2.1. Implications for astrophysical processes Based on the results presented in the previous section, Fig. 2 summarizes the different energy dependent modulation zones that occur at Earth. Table 1 Modulation ratio in percentage (100% means no reduction in the LIS intensity) between Earth, at 1 AU, and the HP, in this case, at 120 AU for selected CR proton kinetic energies and for both drift modulation cycles, during ideal solar minimum conditions. The present modulation cycle, around the minimum in 2009, is an A < 0 drift cycle when positive particles drift to Earth mainly through the equatorial plane of the heliosphere. Kinetic energy (GeV)
A > 0 cycle (%)
A < 0 cycle (%)
300 200 100 50 25 12 6 3 1 0.5 0.1 0.05 0.01
99.6 99.3 98 95 86 75 60 42 22 15 5.0 2.9 1.3
99.6 99.3 98 95 86 74 56 31 13 8 2.6 1.8 0.6
Above 50 GeV, less than 5% solar modulation is present so that observations in this energy range should reflect astrophysical related processes and transport effects. A mixed zone between 10 to 50 GeV, where modulation is less than 30% so that measurements in this energy regime should reflect both solar modulation and astrophysical effects, with solar modulation becoming progressively dominant with decreasing energy. The solar modulation regime between 100 MeV to 10 GeV. In this regime, solar modulation dominates so that all interstellar or astrophysical information regarding e.g. the LIS of CRs, will be obscured, again progressively with decreasing energy. Below 100 MeV solar modulation is obviously very dominant at Earth but local heliospheric sources of energetic particles may also become important or even dominant. A very specific example of such domination is Jovian electrons so that the intensity of galactic electrons cannot be observed below 50 MeV at Earth (Potgieter and Nndanganeni, 2013a). In the outer heliosphere, especially in the heliosheath, the so-called anomalous component completely dominates the galactic particles in this energy range (see the review by e.g. Potgieter,
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Fig. 2. Different modulation zones (coloured panels), as indicated in the figure and identified based on the right panel of Fig. 1. See text for a full description. Also shown is the observed positron fraction, without error bars, at Earth (for full data references, see Adriani et al., 2013b). Dashed line is the theoretical calculations of Moskalenko and Strong (1998), shown for illustrative purposes.
2013b). This component should disappear beyond the HP and this was indeed observed to occur (Stone et al., 2013). Shown additionally in Fig. 2, is an overview of the observed positron fraction at Earth (taken from Adriani et al., 2013b), the cause of which is still hotly debated because it deviates so much from the long-standing computed fraction of Moskalenko and Strong (1998), shown here for illustrative purposes. Of interest in the context of the discussion topic of this publication, is the obvious trend in the variation of this fraction below 10 GeV over time, which indicates, based on the arguments given above, that these differences can be attributed to solar modulation effects (including drifts), as discussed by Potgieter (2013c). (These observations obviously also contain systematic errors, as reported when published, which should always be carefully considered when making conclusions). Above this energy, the high energy positron excess must be attributed to an astrophysical origin. In additional to this generally accepted interpretation, it can be added that up to 30 GeV, solar modulation should also contribute noticeably to a variation in this fraction over longer time periods, such as the 11-year cycle. The modelling results in this and the next section, depend on the value and spectral shape of the assumed LIS. In the energy range of particular interest, E > 1 GeV, most LIS models produce similar results as they are constrained by observations into a very narrow range. As such, the quantitative results presented here will change marginally, while the qualitative results and conclusions remain unchanged.
2.2. Effect of heliospheric drifts at high energies Recently, Roberts (2011) attributed the above-mentioned positron excess above 10 GeV to differences in the drift directions of oppositely charged galactic CRs in the heliosphere, in other words, that the total observed effect shown in Fig. 2 could simply be explained by the charge-sign dependent effect in solar modulation. During A < 0 drift cycles (the present epoch), positrons drift to Earth mainly through the equatorial regions of the heliosphere, encountering the wavy HCS in the process, while electrons drift mainly through the polar regions to Earth. Since the HCS changes significantly with solar activity, positrons will respond to it while electrons remain largely unaffected. For an illustration of these drift effects and its consequences for protons and electrons observed by PAMELA from mid-2006 to the end of 2009, see Fig. 3 of Vos et al. (2013). From a solar modulation point of view, Robert’s interpretation that the positron intensity could be enhanced during this polarity epoch, is highly controversial. It is argued in this section, where the effectiveness of modulation drifts is discussed and illustrated, that heliospheric drifts of CRs as illustrated in Figs. 1 and 2, cannot be the cause of the increasing trend in the observed positron to electron ratio at higher energies. The SDE model used for this study is based on utilizing a full diffusion and drift tensor in three dimensions, so that diffusion parallel to the average heliospheric magnetic field, also perpendicular to it in the radial and polar directions, as well as a drift coefficient are used in solving the appropriate transport equation. In Fig. 3 the transport coefficients as a function of rigidity are summarized as applied
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Fig. 3. The transport coefficients used in this work, as a function of rigidity at Earth (1 AU) and at the HP (120 AU). The grey coloured region indicates values smaller than the radius of the heliosphere, whereas the green box shows generally accepted values of the parallel mean free path at Earth, known as the Palmer consensus values (e.g. Bieber et al., 1994). (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this article.)
in this study. The corresponding parallel ðkjj Þ and perpendicular ðk? Þ mean free paths (MFPs) are shown as a function of rigidity at Earth and at the HP. The so-called Palmer consensus values (Bieber et al., 1994) of kjj at Earth are included to show that the assumed values of kjj are reasonable. Also shown on the figure is the Larmor radius of a galactic CR proton at the same positions. When the weak scattering limit of drifts is assumed, the Larmor radius may be interpreted as the drift scale ðkd Þ; the length scale on which modulation drifts occur (see e.g. Minnie et al., 2007). By comparing the different transport scales, as is done in Fig. 3, the most dominant transport processes may be identified qualitatively; a larger length scale indicates a more efficient global modulation process. For all energy and spatial regimes under investigations, kjj > kd , meaning that diffusion remains the dominant transport process of galactic CRs in the heliosphere. Moreover, the interplay between k? and kd gives insight into when drift effects may be significant because both of these processes operate perpendicular to the mean magnetic field: In the inner heliosphere, kjj > kd > k? for rigidities > 10 GV, so that drifts are non-negligible. In the outer heliosphere, kjj > k? > kd , and the efficiency of drifts, relative to diffusion, decreases.
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This illustrates that the transport of galactic CRs in the global heliosphere fundamentally remains a diffusion dominated process and that drifts only augment the process, sometimes significantly depending on solar activity levels and, of course, the polarity cycles of the solar magnetic field. Also shown on Fig. 3, as the shaded region, is the assumed radial distance to the HP. This assumption is based on the recent observation of the HP position at 122 AU by Voyager 1 (e.g. Stone et al., 2013). Comparing the different length scales with this value, adds further reason to the different modulation zones as identified in Fig. 2: Above 50 GV both diffusion MFPs at the HP become comparable to the radius of the heliosphere, and insignificant modulation effects are thus to be expected, even at Earth, as illustrated above. The interplay between diffusion and drifts, and their effectiveness as modulation processes, is illustrated in Fig. 4. Here, pseudo-particle traces of galactic protons, reaching Earth with kinetic energy of 100 MeV in the A < 0 drift cycle, are shown in the meridional plane of the heliosphere (for details see Strauss et al., 2012a). The shape of the wavy HCS, with a ¼ 45 , is shown as the dotted line, and the position of the HP as the dashed line. Keep in mind that in this picture 3D modulation effects are projected onto the meridional plane. Panel (a) corresponds to a scenario similar to that of Fig. 1, except for a larger tilt angle. This scenario thus produces the most realistic levels of modulation at Earth. Successive panels each time show the computations when both kjj and k? are decreased by an additional factor of 10, in order to illustrate what happens when diffusion is diminishing globally in such a way. From panel (a), considered as the most realistic modulation scenario, it is clearly evident that although protons drift primarily along the HCS to reach Earth, as expected for this drift cycle, their transport and consequent modulation remain diffusion dominated because CRs are scattered off the wavy HCS and throughout the heliosphere. The scenario shown in panel (d), gives a picture of drifts dominating the modulation process completely, with smooth, deterministic transport along the HCS. Because this scenario represents a clear picture of HCS drifts in the heliosphere, it is a popular but one-sided view and cannot be considered as a realistic modulation scenario. The CRs drifting to Earth via the equatorial regions of the heliosphere in such a way must of course again drift away from Earth via the polar regions during this polarity epoch to assure that the divergence of the drift velocity remains zero everywhere. The magnetic focussing effect described by Roberts (2011) thus requires that the positrons must drift away from Earth in the same perfect way that they drift in because in his approach there exists no other process that can take them away from the HCS which cannot happen in a realistic drift modulation model. These simulations reiterate that diffusion is a determining CR transport process in the heliosphere, even during solar minimum activity (in this context, see also Potgieter
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Fig. 4. Computed pseudo-particle traces of 100 MeV galactic protons with a 3D SDE-type modulation model, projected onto the meridional plane of the heliosphere. The dotted line shows a projection of the HCS, with a ¼ 45 , onto the same plane and the dashed line the HP position. For each consecutive panel, the diffusion coefficients are divided by an additional factor of 10 relative to those used in panel (a), which is considered as the most realistic modulation scenario. For additional trajectories e.g. with increasing tilt angles, see Strauss et al. (2012a).
et al., 2013b), and that studies neglecting diffusion, or handling it in a totally arbitrarily manner in a drift model (e.g. Loparco et al., 2013), should be taken with serious caution since a distorted view of solar modulation could be the result. 3. Spatial dependence of heliospheric modulation The long awaited entering of the very local interstellar medium by Voyager 1 is confirmed to have taken place on 25 August 2012 at a radial distance of 122 AU (e.g. Stone et al., 2013). Although there were originally some reservations whether the spacecraft actually crossed the
HP because of the seemingly incongruous magnetic field observations (Burlaga et al., 2013), the inferred plasma density measurements (Gurnett et al., 2013) confirmed that it is currently sampling the very local interstellar space in situ. A long held hypothesis is that once Voyager 1 crossed the HP, it would immediately sample the pristine LIS for galactic CRs. This paradigm means that the solar modulation of CRs should abruptly stop at the HP. Recently, it has been argued and now confirmed by sophisticated SDE modelling (Scherer et al., 2011; Strauss et al., 2013), that this may not be the case and that solar modulation can persist beyond the HP, perhaps up to where the
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Fig. 5. Computed spectra using a hybrid modulation model to investigate the occurrence of galactic CR modulation beyond the HP (taken from Strauss et al., 2013). Spectra are shown at the HP (corresponding to the Voyager 1 trajectory) and at Earth for solar minimum (red lines) and maximum (blue lines) modulation conditions. (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this article.)
heliosphere begins to disturb the very local interstellar medium by moving through it. This is discussed further in the next section. 3.1. Implications for in situ LIS measurements Fig. 5 shows modelling results from Strauss et al. (2013), who investigated the residual modulation that may be present beyond the HP. For these simulations, a hybrid modulation model was used, where the heliospheric plasma environment, as simulated by a 3D magneto-hydrodynamic model, is coupled to a SDE based galactic CR modulation model. This produces a realistic plasma flow (solar wind), with a subsequent magnetic field and a realistic heliospheric geometry in contrast to simply assuming these features in a particle transport model. Doing this is an important and necessary requirement for the realistic modelling of CR transport in the heliosheath, up to the HP and beyond. The very LIS, in this model, is not prescribed at the HP, but rather at the edge of the spatial computational domain, at least 400 AU from the Sun in order to compute what may happen to it beyond the HP. In Fig. 5, energy spectra for galactic protons are shown at Earth (the bottom two curves) and at the HP, with
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respect to the unmodulated LIS. It follows from these results that CR modulation persists beyond the HP, with the 100 MeV intensity 25% to 40% lower than the LIS value at the HP. This process is energy dependent and diminishes above a few GeV. The two sets of simulations represent solar minimum (red curves) and solar maximum (blue curves) modulation conditions, with an interesting finding that modulation beyond the HP even exhibits solar cycle related changes. This consequently gives lower proton intensities at the HP during times of solar maximum conditions. Modulation in this previously neglected modulation region is mostly the result of the re-entry (return) of galactic CRs which were already inside the heliosphere back into this region. These returning CRs had thus already been subjected to some modulation (for a detailed discussion see Scherer et al., 2011; Strauss et al., 2013). The computational results discussed in this section indicate that the galactic CR observations currently being made by Voyager 1 should still be influenced by the global heliospheric environment. These measurements should continue to exhibit a small, positive radial intensity gradient beyond the HP for the foreseeable future. Spatially, this should continue to the position where the heliosphere no longer disturbs the local interstellar medium. Simulations suggest that this effect should persist for at least 100 AU beyond the HP along the Voyager 1 trajectory. It should be kept in mind that for galactic CRs this is not a local effect: A 40% decrease in intensity occurs over 100 AU, so that an average radial gradient of 0.4%/ AU is expected. As such, Voyager 1 may take several years to observe this effect. In contrast, the anomalous cosmic ray particles which are created inside the heliosphere, and accelerated in the heliosheath, should exhibit a negative radial gradient that varies from a very large value close to the HP to essentially zero when far enough from the HP. The present Voyager 1 observations of proton, helium, carbon and oxygen spectra below a few hundred MeV/ nuc should thus be considered as the lower limits of the very LIS and should rather be referred to as the HP spectra of galactic CRs until proven otherwise. In this context, see also discussions by Strauss et al. (2012b), Potgieter and Nndanganeni (2013a) and Potgieter (2013a). 4. Conclusions The solar modulation of galactic CRs at Earth diminishes progressively with increasing energy (rigidity) primarily caused by the corresponding increasing values of mainly the diffusion coefficients. It was quantitatively shown that solar modulation of CRs becomes significant below 10 GeV, less so between 10 to 50 GeV, and progressively negligible above 50 GeV. Cosmic ray observations above this energy contain less that 5% solar modulation effects and should therefore reflect the very LIS for galactic CRs at Earth. The probability of particle drifts having significant modulation effects with increasing energy was investigated and
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it was confirmed that solar modulation remains fundamentally a diffusion dominated process with drifts augmenting the process depending on solar activity and subsequent modulation conditions but can be considered mostly of second-order importance at higher energies. Drifts, as the main charge-sign dependent process in the heliosphere, cannot be the main contributor of the increasing positron excess above 10 GeV but will significantly change the positron to electron ratio below this energy (for examples, see Potgieter, 2013b). It was shown that CR modulation may persist beyond the HP, and can be quite significant below 1 GeV. At present, Voyager 1 is probably not measuring the LIS of galactic CRs at these low energies, but rather a HP spectrum that should gradually unfold to LIS values during the coming years. Spectra of protons, helium, carbon and oxygen below a few hundred MeV/nuc should thus be considered as the lower limits of the very LIS and can thus be used to estimate the LIS. At the same time, these HP spectra can be used to check predictions of the galactic spectra as obtained with galactic propagation models. Combining the current low energy galactic CR observations of Voyager 1 at and beyond the HP with high energy observations at Earth can thus provide a full HP spectrum for the mentioned CR species (see also e.g. Potgieter and Nndanganeni, 2013b; Potgieter, 2013a). Input spectra for solar modulation models are now constrained by in situ observations and can therefore not any longer be treated arbitrarily, especially at these low energies. For galactic CR transport models, HP spectra together with observations at Earth provide further constraint for the choice of e.g. the transport parameters because for the first time the total modulation inside the heliosphere, between 1 MeV and 300 GeV, can be calculated accurately, of course limited by the accuracy of the corresponding CR observations. References Adriani, O.and the PAMELA collaboration, 2013a. Time dependence of the proton flux measured by PAMELA during the July 2006– December 2009 solar minimum. Astrophys. J. 765 (91), 1–8. Adriani, O.and the PAMELA collaboration, 2013b. Cosmic-ray positron energy spectrum measured by PAMELA. Phys. Rev. Lett. 765, 081102:1–081102:6. Balasubrahmanya, V.K., Hagge, D.E., Ludwig, G.H., McDonald, F.B., 1965. Galactic cosmic rays at solar minimum. In: Proceedings of the 9th International Cosmic Ray Conference, vol. 1, pp. 427–436. Bieber, J.W., Matthaeus, W.H., Smith, C.W., Wanner, W., Kallenrode, M.-B., Wibberenz, G., 1994. Proton and electron mean free paths: the Palmer consensus revisited. Astrophys. J. 420, 294–306. Burlaga, L.F., Ness, N.F., Stone, E.C., 2013. Magnetic field observations as Voyager 1 entered the heliosheath depletion region. Science 341 (6142), 147–150. Gurnett, D.A., Kurth, W.S., Burlaga, L.F., Ness, N.F., 2013. In situ observations of interstellar plasma with Voyager 1. Science 341 (6153), 1489–1492. Kopp, A., Busching, I., Strauss, R.D., Potgieter, M.S., 2012. A stochastic differential equation code for multidimensional Fokker–Planck type problems. Comput. Phys. Commun. 183, 530–542.
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