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Effects of the heliospheric termination shock on possible local interstellar spectra for cosmic ray electrons and the associated heliospheric modulation
Effects of the heliospheric termination shock on possible local interstellar spectra for cosmic ray electrons and the associated heliospheric m o d u l a t i o n S. E. S. Ferreira, M. S. Potgieter and U. W. Langner School of Physics, Potchefstroom University for CHE, 2520 Potchefstroom. South Africa
The 'local' interstellar spectrum (IS) for cosmic ray electrons at energies of interest to heliospheric modulation studies is still basically unknown. Recently, new computations of the IS based on advanced modeling of cosmic-ray propagation in the Galaxy [1], and observations including diffuse galactic gamma rays, indicated that the electron IS may be considerably lower at energies below-100 MeV than previously assumed. For this work different scenarios for the electron 'local' IS, and their subsequent modulation in the heliosphere, are studied using a shock-drift modulation model. The effects of the heliospheric termination shock (TS) on each of these scenarios are illustrated, together with the subsequent effects on their modulation in the heliosphere. We find that the computed effect of the TS on galactic electron intensities at 16 MeV is relatively small, in general, but more pronounced if the TS is positioned at 80 AU, than at 90 or 100 AU. The larger the 'local' IS value is, the larger the effect of the TS on electron modulation at this energy becomes. 1. I N T R O D U C T I O N The study of the modulation of cosmic ray electrons in the heliosphere is an important and useful tool in understanding various aspects of heliospheric modulation. Modulated electron intensities in the lower-MeV range give a direct indication of the average parallel and perpendicular mean free paths in contrast to protons that experience adiabatic energy changes below-300 MeV (e.g. [2]). Gradient and curvature drifts become less important for electron modulation at lower energies, with almost no effect below 100 MeV (e.g. [3]). The Pioneer 10 radial-intensity-profiles f o r - 1 6 MeV electrons ([4],[6]) indicate almost no radial gradients out to -70 AU, which put serious constraints on the diffusion tensor. New computations of the interstellar spectra (IS) [1], indicate that the electron IS may be considerably lower at energies below -100 MeV than previously assumed (e.g. [11]). For this work, two different (extreme) scenarios for the 'local' IS for cosmic ray electrons ([1],[5]), and their subsequent modulation in the heliosphere are studied using a shock-drift-modulation model. Satisfying the constraints imposed on the diffusion tensor by the Pioneer 10 electron data in the outer heliosphere, the effects of the location of the heliospheric termination shock (TS) on each of the IS scenarios are illustrated, together with the subsequent effects on their modulation. 2. M O D U L A T I O N
MODEL AND PARAMETERS
The model is based on the numerical solution of the transport equation (TPE) [7]:
- 223 -
S.E.S. Ferreira, M.S. Potgieter and U W. Langner
3fot = - ( V
I(V.V) + (v i) ) ) . V f + V . ( K s . V f ) + ~-
3f +J 3 ln-~
,
(1)
source
where f (r~,t) is the CR distribution function; R is rigidity, r is position, and t is time, with V the solar wind velocity. Terms on the right-hand side represent convection, gradient and curvature drifts, diffusion, adiabatic energy changes and a source function, respectively. The symmetric part of the tensor Ks consists of a parallel diffusion coefficient (Kll) and a perpendicular diffusion coefficient (K• The anti-symmetric element KA describes gradient and curvature drifts in the large scale HMF. J~our~e can be any source, e.g. Jovian electrons, pick-up ions, and/or the local interstellar spectra (IS). Here, we concentrate on the IS for electrons, neglecting all other local sources. The TPE was solved using a two dimensional TSdrift model developed by le Roux et al. [8], and expanded by Haasbroek [9]. The outer modulation boundary is assumed at 120 AU. A TS with a compression ratio of 3.2 < s < 4.0, and scale length of L = 1.2 AU was assumed at rs = 80 AU. The solar wind speed V was assumed to change from 400 km.s 1 in the equatorial plane (0 = 90 ~ to a maximum of 800 km.s ~ when 0 < 60 ~ At the shock, V decreases from the upstream value of V1 = 400 km.s -1 in the equatorial plane according to the relationship given by le Roux et al. [8]: V(r)= VI(S+ 1)_ V I ( S - 1 ) t a n h ( r - r~'] 2s 2s [, L )
(2)
For the diffusion coefficients that describe diffusion parallel, Kjl, and perpendicular, K• to the average heliospheric magnetic field (HMF) as well as the asymmetric coefficient KA, which describes gradient and curvature drifts in the background HMF, we assumed: Be", Kii = K ofif ( R ) --ff-
K_l_r = a 1 + (k,KI]ll/rg
" )2 '
K 2.0
Kll )5," = b 1 + (All/rg
/3n KA = (KA)o ~
(3)
Here/3 is the ratio of the speed of the cosmic ray particles to the speed of light; f(R) gives the rigidity dependence (in GV) with f(R)=R when R > 0.4 GV, and f(R)=0.4 when R ~ 0.4 GV; B is the HMF magnitude modified [12] in qualitatively agreement with Ulysses observations [13]; K0 is a constant in units of 6.0 x l 0 2~ c m 2 S"1", a is a constant which determines the value of K• which contributes to perpendicular diffusion in the radial direction, and b is a constant that determines K• which contributes to perpendicular diffusion in the polar direction. Diffusion perpendicular to the HMF is therefore enhanced in the polar direction by assuming b > a. ([3],[10],[11]). The ratio of the scattering mean free path to the particle gyroradius, )~ll /rg, is larger than unity in view of the Bohm limit, ~'11= rg. The TPE was solved in a spherical coordinate system with the current sheet "tilt angle" (x = 10 ~ for so-called A > 0 epochs (-1990 to present) when electrons primarily drift inward through the equatorial regions of the heliosphere. 3. R E S U L T S
AND DISCUSSION
Figure l a shows the radial profiles of computed 16 MeV galactic electron intensities with an outer boundary at rB -- 120 AU, with the IS of Strong et al. [5] assumed to be the local electron spectrum. Three different radial profiles are shown; the solid line corresponding to a TS at rs = 80 AU, the dashed line to a TS at rs = 90 AU, and the dotted line to a TS at rs = 100 AU. The electron data from Pioneer 10 are presented as shaded areas for radial distances up to
- 224 -
Effects o f the heliospheric termination shock on ...
Figure l a" Computed radial profiles of 16 MeV galactic electron intensities with an outer boundary at rB = 120 AU, and with the IS of Strong et al.[5] assumed to be electron local spectra. The solid line corresponds to a TS at rs = 80 AU, the dashed line to a TS at rs = 90 AU, and the dotted line to a TS at rs - 100 AU. The electron data from Pioneer 10 are presented as shaded areas, lb" The same as in la, but with the IS from the recent calculations of Strong et al. [1]. --50 AU [4], and at --70 AU [6]. The height of the shaded areas incorporate the error bars present in the data, as well as a small time dependent effect due to the solar modulation of the electrons over the period --1972 to --1991. The data are assumed to be of galactic origin, with the Jovian component dominating only for r < 25 AU [4]. The computed intensities are compatible with both data sets. We assumed Ko = 0.5, a - 0.25 and b = 0.6. The effect of the shock is visible in all the radial profiles, indicating an increase in the radial gradient upstream of the shock but a decrease beyond the shock. For a TS at rs -- 80 AU, the effect of the shock on the radial gradients upstream and downstream of the shock is significantly larger than for the two other scenarios. Although the shock is more effective for larger radial distances due to the larger shock radius, the radial diffusion coefficient, due to its radial dependence oc r, is also larger, leading to a smaller effect for the more distant shocks. Figure l b shows the case when the more recently calculated IS of Strong et al. [1] is assumed as the local electron spectrum. Evidently, it is considerably lower than the IS used in Figure la. All three scenarios for the shock locations are again compatible with the observed Pioneer 10 data, but in order to obtain that, we had to assume Ko = 58, a=18 and b=24 in Equation 3. These larger diffusion coefficients are needed in order to produce less modulation between the outer boundary and the data a t - 7 0 AU. The effect of the different TS locations on the computed electron intensity profiles is much less pronounced, with almost no difference between the three scenarios, in contrast to Figure la. These larger diffusion coefficients clearly decrease the effectiveness of the shock.
- 225 -
S.E.S. Ferreira, M.S. Potgieter and U.W. Langner
4. SUMMARY AND CONCLUSIONS Two different scenarios for the electron IS ([1],[5]), and their subsequent modulation in the heliosphere have been studied using a shock-drift modulation model. The effects of the heliospheric termination shock (TS) on each of these scenarios were also studied. At 16 MeV the two electron IS differ by a factor of--100. They were assumed as the local IS at 120 AU, as an outer boundary of the heliosphere. Compatibility between the observed Pioneer 10 ([4],[6]) radial profiles and the model simulations were strictly required for the two IS cases, and the three different TS shock positions, as shown in Figure 1. For the highest IS of Strong et al. [5], very large radial gradients (-~10%/AU) were found between 70 AU, the shock position rs, and the outer boundary. The effect of the shock on the radial intensity profile is more pronounced with rs = 80 AU, than for rs = 90 AU or 100 AU. The radial gradients are found to be larger upstream of the shock than downstream. Although the TS should more effective for larger radial distances due to the larger shock radius, the diffusion coefficients are also larger, leading to a smaller modulation effect for the more distant TS positions. For the lowest IS of Strong et al. [1], the radial gradients beyond 70 AU were significantly less, with the effect of the different TS locations on the computed electron intensities far less pronounced than in the first case. The reason is that larger diffusion coefficients are needed to provide compatibility with the data. These larger diffusion coefficients not only decrease the total modulation in the outer heliosphere, but also the effectiveness of the shock. For the interstellar spectra given in Figure 1, the computed radial intensity profiles for--16 MeV electrons indicate that the detection of the crossing of the TS by a spacecraft registering only these low energy electrons would be ambiguous. Some other observations have to be utilized (e.g. magnetic field or solar wind speed changes) in order to provide information on when a spacecraft actually crosses the TS. REFERENCES ~
2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
A.W. Strong et al., Astrophys. J. 537 (2000) 763. L. J. Haasbroek et al., Proc. 24th ICRC 4 (1995) 706. S. E. S. Ferreira et al., J. Geophys. Res. 105 (2000) 18305 C. Lopate, Proc. 22nd ICRC (Dublin) 2 (1991) 149. A. W. Strong et al., A&A 292 (1994) 82. C. Lopate, private communication (2000). E. N. Parker, Planet. & Space Sci. 13 (1965) 9. J. A. le Roux et al., J. Geophys. Res., 101 (1996) 4791. L. J. Haasbroek, Ph.D. thesis, Potchefstroom University, South Africa (1997). J. K6ta and J.R. Jokipii, Proc. 24th ICRC (Rome) 4 (1995) 680. M. S. Potgieter, J. Geophys. Res. 101 (1996) 24411. J. R. Jokipii and J. K6ta, Geophys. Res. Lett. 16 (1989) 1. A. Balogh et al., Science. 268 (1995) 1007.
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The 'local' interstellar spectrum (IS) for cosmic ray electrons at energies of interest to heliospheric modulation studies is still basically unknown. Recently, new computations of the IS based on advanced modeling of cosmic-ray propagation in the Galaxy [1], and observations including diffuse galactic gamma rays, indicated that the electron IS may be considerably lower at energies below-100 MeV than previously assumed. For this work different scenarios for the electron 'local' IS, and their subsequent modulation in the heliosphere, are studied using a shock-drift modulation model. The effects of the heliospheric termination shock (TS) on each of these scenarios are illustrated, together with the subsequent effects on their modulation in the heliosphere. We find that the computed effect of the TS on galactic electron intensities at 16 MeV is relatively small, in general, but more pronounced if the TS is positioned at 80 AU, than at 90 or 100 AU. The larger the 'local' IS value is, the larger the effect of the TS on electron modulation at this energy becomes. 1. I N T R O D U C T I O N The study of the modulation of cosmic ray electrons in the heliosphere is an important and useful tool in understanding various aspects of heliospheric modulation. Modulated electron intensities in the lower-MeV range give a direct indication of the average parallel and perpendicular mean free paths in contrast to protons that experience adiabatic energy changes below-300 MeV (e.g. [2]). Gradient and curvature drifts become less important for electron modulation at lower energies, with almost no effect below 100 MeV (e.g. [3]). The Pioneer 10 radial-intensity-profiles f o r - 1 6 MeV electrons ([4],[6]) indicate almost no radial gradients out to -70 AU, which put serious constraints on the diffusion tensor. New computations of the interstellar spectra (IS) [1], indicate that the electron IS may be considerably lower at energies below -100 MeV than previously assumed (e.g. [11]). For this work, two different (extreme) scenarios for the 'local' IS for cosmic ray electrons ([1],[5]), and their subsequent modulation in the heliosphere are studied using a shock-drift-modulation model. Satisfying the constraints imposed on the diffusion tensor by the Pioneer 10 electron data in the outer heliosphere, the effects of the location of the heliospheric termination shock (TS) on each of the IS scenarios are illustrated, together with the subsequent effects on their modulation. 2. M O D U L A T I O N
MODEL AND PARAMETERS
The model is based on the numerical solution of the transport equation (TPE) [7]:
- 223 -
S.E.S. Ferreira, M.S. Potgieter and U W. Langner
3fot = - ( V
I(V.V) + (v i) ) ) . V f + V . ( K s . V f ) + ~-
3f +J 3 ln-~
,
(1)
source
where f (r~,t) is the CR distribution function; R is rigidity, r is position, and t is time, with V the solar wind velocity. Terms on the right-hand side represent convection, gradient and curvature drifts, diffusion, adiabatic energy changes and a source function, respectively. The symmetric part of the tensor Ks consists of a parallel diffusion coefficient (Kll) and a perpendicular diffusion coefficient (K• The anti-symmetric element KA describes gradient and curvature drifts in the large scale HMF. J~our~e can be any source, e.g. Jovian electrons, pick-up ions, and/or the local interstellar spectra (IS). Here, we concentrate on the IS for electrons, neglecting all other local sources. The TPE was solved using a two dimensional TSdrift model developed by le Roux et al. [8], and expanded by Haasbroek [9]. The outer modulation boundary is assumed at 120 AU. A TS with a compression ratio of 3.2 < s < 4.0, and scale length of L = 1.2 AU was assumed at rs = 80 AU. The solar wind speed V was assumed to change from 400 km.s 1 in the equatorial plane (0 = 90 ~ to a maximum of 800 km.s ~ when 0 < 60 ~ At the shock, V decreases from the upstream value of V1 = 400 km.s -1 in the equatorial plane according to the relationship given by le Roux et al. [8]: V(r)= VI(S+ 1)_ V I ( S - 1 ) t a n h ( r - r~'] 2s 2s [, L )
(2)
For the diffusion coefficients that describe diffusion parallel, Kjl, and perpendicular, K• to the average heliospheric magnetic field (HMF) as well as the asymmetric coefficient KA, which describes gradient and curvature drifts in the background HMF, we assumed: Be", Kii = K ofif ( R ) --ff-
K_l_r = a 1 + (k,KI]ll/rg
" )2 '
K 2.0
Kll )5," = b 1 + (All/rg
/3n KA = (KA)o ~
(3)
Here/3 is the ratio of the speed of the cosmic ray particles to the speed of light; f(R) gives the rigidity dependence (in GV) with f(R)=R when R > 0.4 GV, and f(R)=0.4 when R ~ 0.4 GV; B is the HMF magnitude modified [12] in qualitatively agreement with Ulysses observations [13]; K0 is a constant in units of 6.0 x l 0 2~ c m 2 S"1", a is a constant which determines the value of K• which contributes to perpendicular diffusion in the radial direction, and b is a constant that determines K• which contributes to perpendicular diffusion in the polar direction. Diffusion perpendicular to the HMF is therefore enhanced in the polar direction by assuming b > a. ([3],[10],[11]). The ratio of the scattering mean free path to the particle gyroradius, )~ll /rg, is larger than unity in view of the Bohm limit, ~'11= rg. The TPE was solved in a spherical coordinate system with the current sheet "tilt angle" (x = 10 ~ for so-called A > 0 epochs (-1990 to present) when electrons primarily drift inward through the equatorial regions of the heliosphere. 3. R E S U L T S
AND DISCUSSION
Figure l a shows the radial profiles of computed 16 MeV galactic electron intensities with an outer boundary at rB -- 120 AU, with the IS of Strong et al. [5] assumed to be the local electron spectrum. Three different radial profiles are shown; the solid line corresponding to a TS at rs = 80 AU, the dashed line to a TS at rs = 90 AU, and the dotted line to a TS at rs = 100 AU. The electron data from Pioneer 10 are presented as shaded areas for radial distances up to
- 224 -
Effects o f the heliospheric termination shock on ...
Figure l a" Computed radial profiles of 16 MeV galactic electron intensities with an outer boundary at rB = 120 AU, and with the IS of Strong et al.[5] assumed to be electron local spectra. The solid line corresponds to a TS at rs = 80 AU, the dashed line to a TS at rs = 90 AU, and the dotted line to a TS at rs - 100 AU. The electron data from Pioneer 10 are presented as shaded areas, lb" The same as in la, but with the IS from the recent calculations of Strong et al. [1]. --50 AU [4], and at --70 AU [6]. The height of the shaded areas incorporate the error bars present in the data, as well as a small time dependent effect due to the solar modulation of the electrons over the period --1972 to --1991. The data are assumed to be of galactic origin, with the Jovian component dominating only for r < 25 AU [4]. The computed intensities are compatible with both data sets. We assumed Ko = 0.5, a - 0.25 and b = 0.6. The effect of the shock is visible in all the radial profiles, indicating an increase in the radial gradient upstream of the shock but a decrease beyond the shock. For a TS at rs -- 80 AU, the effect of the shock on the radial gradients upstream and downstream of the shock is significantly larger than for the two other scenarios. Although the shock is more effective for larger radial distances due to the larger shock radius, the radial diffusion coefficient, due to its radial dependence oc r, is also larger, leading to a smaller effect for the more distant shocks. Figure l b shows the case when the more recently calculated IS of Strong et al. [1] is assumed as the local electron spectrum. Evidently, it is considerably lower than the IS used in Figure la. All three scenarios for the shock locations are again compatible with the observed Pioneer 10 data, but in order to obtain that, we had to assume Ko = 58, a=18 and b=24 in Equation 3. These larger diffusion coefficients are needed in order to produce less modulation between the outer boundary and the data a t - 7 0 AU. The effect of the different TS locations on the computed electron intensity profiles is much less pronounced, with almost no difference between the three scenarios, in contrast to Figure la. These larger diffusion coefficients clearly decrease the effectiveness of the shock.
- 225 -
S.E.S. Ferreira, M.S. Potgieter and U.W. Langner
4. SUMMARY AND CONCLUSIONS Two different scenarios for the electron IS ([1],[5]), and their subsequent modulation in the heliosphere have been studied using a shock-drift modulation model. The effects of the heliospheric termination shock (TS) on each of these scenarios were also studied. At 16 MeV the two electron IS differ by a factor of--100. They were assumed as the local IS at 120 AU, as an outer boundary of the heliosphere. Compatibility between the observed Pioneer 10 ([4],[6]) radial profiles and the model simulations were strictly required for the two IS cases, and the three different TS shock positions, as shown in Figure 1. For the highest IS of Strong et al. [5], very large radial gradients (-~10%/AU) were found between 70 AU, the shock position rs, and the outer boundary. The effect of the shock on the radial intensity profile is more pronounced with rs = 80 AU, than for rs = 90 AU or 100 AU. The radial gradients are found to be larger upstream of the shock than downstream. Although the TS should more effective for larger radial distances due to the larger shock radius, the diffusion coefficients are also larger, leading to a smaller modulation effect for the more distant TS positions. For the lowest IS of Strong et al. [1], the radial gradients beyond 70 AU were significantly less, with the effect of the different TS locations on the computed electron intensities far less pronounced than in the first case. The reason is that larger diffusion coefficients are needed to provide compatibility with the data. These larger diffusion coefficients not only decrease the total modulation in the outer heliosphere, but also the effectiveness of the shock. For the interstellar spectra given in Figure 1, the computed radial intensity profiles for--16 MeV electrons indicate that the detection of the crossing of the TS by a spacecraft registering only these low energy electrons would be ambiguous. Some other observations have to be utilized (e.g. magnetic field or solar wind speed changes) in order to provide information on when a spacecraft actually crosses the TS. REFERENCES ~
2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
A.W. Strong et al., Astrophys. J. 537 (2000) 763. L. J. Haasbroek et al., Proc. 24th ICRC 4 (1995) 706. S. E. S. Ferreira et al., J. Geophys. Res. 105 (2000) 18305 C. Lopate, Proc. 22nd ICRC (Dublin) 2 (1991) 149. A. W. Strong et al., A&A 292 (1994) 82. C. Lopate, private communication (2000). E. N. Parker, Planet. & Space Sci. 13 (1965) 9. J. A. le Roux et al., J. Geophys. Res., 101 (1996) 4791. L. J. Haasbroek, Ph.D. thesis, Potchefstroom University, South Africa (1997). J. K6ta and J.R. Jokipii, Proc. 24th ICRC (Rome) 4 (1995) 680. M. S. Potgieter, J. Geophys. Res. 101 (1996) 24411. J. R. Jokipii and J. K6ta, Geophys. Res. Lett. 16 (1989) 1. A. Balogh et al., Science. 268 (1995) 1007.
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