Caballero-Lopez, R A

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29th International Cosmic Ray Conference Pune (2005) 2, 125128Termination shock and heliosheath effects on cosmic ray energychanges in the heliosphereR.A. Caballero-Lopez  , H. Moraal and F.B. McDonald(a) Instituto de Geof·sica, UNAM, 04510, M·exico(b) School of Physics, North-West University, Potchefstroom 2520, South Africa(c) Institute for Physical Science and Technology, University of Maryland, College Park, MD, 20742, USAPresenter: R.A. Caballero-Lopez (rogelioc@geofisica.unam.mx), mex-caballero-lopez-RA-abs3-sh33-posterAdiabatic energy losses form a well-understood part of cosmic ray modulation. However, shock acceleration atthe termination shock and the modulation in the heliosheath beyond this shock alter the net cosmic ray energychanges from that in the standard models. This paper compares cosmic ray energy changes in several versionsof the modulation process, from the simplest Force-Field model to one with a shock and heliosheath included.1. IntroductionThe energy loss of cosmic rays in the radially expanding solar wind in the heliosphere is an integral part ofthe cosmic ray modulation, the other three processes being diffusion, convection and drift in the heliosphericmagnetic field (HMF) in the solar wind. In the Force-Field approximation of the theory by [1], all threephysical processes are, in fact, combined into a single modulation potential (and consequent energy loss).These energy losses are used for the interpretation of observations, and on the whole are well understood.However, the acceleration and modulation of cosmic rays at the termination shock of the solar wind and inthe heliosheath downstream of this shock, has become increasingly important in our understanding of theoverall modulation. Specifically, the shock accelerates cosmic rays, while in the divergence-free downstreammedium of the heliosheath there are no adiabatic losses. In view of this, we update a set of standard energyloss calculations done by [1], [2] and [3] to estimate the net energy change of galactic cosmic rays in this morerealistic model of the heliosphere.In an accompanying paper we discuss applications of these energy change mechanisms and apply them to theexample of K-capture cosmic ray secondaries, where the ratio of these secondaries to their primary progenitorsis quite sensitive to energy loss processes, as was shown by [4] and [5].2. CalculationsWe calculate energy losses in the heliospheric modulation process from numerical solutions of the cosmic raytransport equation for the evolution of the cosmic ray distribution function  in terms of particle momentum, : 	 V  K ﬀﬁ V ﬃﬂ "!$#&%(1)Here V is the solar wind velocity and K(r ')(*' the diffusion tensor which contains elements +-, r ')(*' and+	./ r ')(*' for scattering along and perpendicular to the HMF, 0 , together with an antisymmetric coefficient+21!43(ﬃ5607 which describes gradient, curvature, neutral sheet, and shock drift effects . The HMF structureis described by the standard Parker spiral magnetic field, given in spherical polar coordinates 98';:<'>=? byB!0A@98B@C5B8D;EF e GHI>JLKe MN , with I>J/K!PO98QR8CSB6T;U:V5W , whereOis the angular frequency of solarrotation, and 8XS is the Alfven radius of several solar radii. The values of the field at Earth, 0Y@ , vary from 5 to126 R.A. Caballero-Lopez, H. Moraal and F.B. McDonald10-310-210-1100101102101 102 103 104Intensity (particles/m2 -s-sr-MeV/n) Force-FieldALISIntensity (particles/m2 -s-sr-MeV/n)Intensity (particles/m2 -s-sr-MeV/n)Intensity (particles/m2 -s-sr-MeV/n)Intensity (particles/m2 -s-sr-MeV/n)Intensity (particles/m2 -s-sr-MeV/n)Intensity (particles/m2 -s-sr-MeV/n)Intensity (particles/m2 -s-sr-MeV/n)102 103 104Kinetic Energy (MeV/n)One Dim.B102 103 10410-310-210-1100101102  ShockC       Figure 1. Modulation examples for protons, showing assumed LIS in dashed, and modulated spectra at Earth in full lines.Modulation is shown for full LIS and three Gaussians. (a) Force-Field solution. (b) One-dimensional steady-state solutionof the transport equation. (c) Two-dimensional time-dependent shock-heliosheath solution (with no drifts).10 nT from solar minimum to maximum. The diffusion coefficients can be written in terms of mean free pathsby +!FZ5B[ , and the radial and latitudinal components areZG)G!Z,	\^]VT;EKZ.T;UEK,Z	_>_!Z. .All energy loss calculations are compared with the widely used Force-Field energy loss. The Force-Fieldformalism was derived by [6] and [7], recast in its generally used form by [1], and summarized by [8]. Itassumes that the radial cosmic ray streaming is zero in a spherically symmetric heliosphere, leading to theresult that a particle observed with rigidity ( somewhere inside the heliosphere had a rigidity (` on the outermodulation boundary given byaRbdcCeG)fbhgb3i(kjl;+Em(kjl(j n(j!aG;oGWp98jl+	qD98jn8j?r=dm8Ds' (2)where + G>G is assumed separable in the form + G)G 98')(Q!t3+?q98D)+Em(Q . This produces an energy loss u7v!wﬃx= , wherewﬃxis is the particle charge. The solution of the Force-Field equation then is y98'>(Q!ym8z^'>(Y`C ,where  is the omnidirectional distribution function, related to the kinetic energy intensity spectrum by R{|15(kE . Values of +?qm8D!~}x#E)E cm E /s (independent of 8 ) and +Em(Q!( , with ( in units of GV, are usedhere. The outer boundary of the modulation region is put at 8z = 150 AU. For W = 400 km/s these values give == 500 MV, which produces an amount of modulation observed roughly between solar minimum and maximum.3. ResultsCalculations were done for protons with an LIS on the outer modulation boundary given by|1z!%vQ	EC s5&~% vkq E>E%vk?E^ )^ . Other model details are given by [9]. Figure 1 showsintensity spectra calculated from (a) the Force-Field equation with the parameters above (b) the simple one-dimensional steady-state transport equation of [8] with the same parameters, and (c) the solution of the time-dependent transport equation of [9] which contains a termination shock at 90 AU. This shock has a compressionEnergy changes ... 127102103101 102 103 104Mean Energy Loss (MeV/n) Force-FieldOne Dim.One Dim., +σOne Dim., -σAMean Energy Loss (MeV/n)Mean Energy Loss (MeV/n)Mean Energy Loss (MeV/n)102 103 104Energy at 1 AU (MeV/n)No ShockNo DriftqA<0qA>0B102 103 104102103  ShockqA>0qA<0No DriftC   Figure 2. Mean energy losses, Ł4H/A~- , suffered by particles observed at Earth. (a) The Force-Field energyloss compared to that of a one-dimensional steady-state solution of the transport equation with the same parameters. Thedashed lines indicate the wide band from within which 68% of the particles orginate in the LIS. (b) The steady-state two-dimensional solution with no shock, for the no-drift and qA  0 and qA  0 drift cases. (c) The same as (b) but for thetime-dependent shock-heliosheath solution with a strong shock inserted at X = 90 AU.ratio  = 4, and outside the shock the solar wind falls off {8 	E up to 8Bz = 150 AU. As described in that paper,this means that cosmic rays in the heliosheath do not suffer adiabatic losses. The dashed upper curve in eachpanel is the LIS, while the full line is the modulated spectrum at Earth. In addition to these full-spectrumsolutions, we also show three Gaussian input spectra together with their modulated spectra, in the same formatas in [3]. These Gaussians demonstrate the spreading in energy due to energy changes. The modulated peaksfor the Force-Field case shift down the furthest, indicating that it gives the largest energy loss. The ones for theone-dimensional steady-state case also have long tails, while the shock/heliosheath solutions show the energygain due to acceleration by the shock. The qA  0 and qA #drift cases for both the no-shock and shock-heliosheath solutions produce qualitatively similar results and are not shown. Notice that particles with a givenenergy vA` on the boundary produce particles with a range of energies, v , at the point of observation inside theheliosphere. Conversely, particles observed with energy v at the point of observation have come from a rangeof energies vA` in the LIS. Average energy losses were calculated by dividing up the LIS in a large amount ofGaussian spectra, and then counting the contribution of each to the intensity at energy v . The average energyin the unmodulated spectrum, vk`k , is then that value above which half the contribution has come. Figure2 shows the average energy loss Pu7v!ﬁvk`/v as function of kinetic energy v at Earth, for eightdifferent scenarios. The horizontal line is the solution of the Force-Field equation (2) with the parametersstated there, which gives an energy loss of 500 MeV for all energies. All other energy losses are comparedto this value. The second solution in Figure 2a is the steady-state one-dimensional solution of (1) with thesame parameters. The energy loss is fairly independent of energy, but  20% smaller than the Force-Field loss,which confirms that the Force-Field overestimates the amount of energy loss, as was pointed out by [8]. Thetwo dashed lines on Figure 2a indicate the spread in energy loss. They define the one-  limits, i.e. 68% ofthe particles observed at v have suffered energy losses between these limits. All other solutions have similarenergy spreads, and they will therefore not be shown.128 R.A. Caballero-Lopez, H. Moraal and F.B. McDonaldFigure 2b shows solutions of the widely applied two-dimensional (radial distance, polar angle) transport equa-tion for no-drift and qA  0 and qA  0 drift states. In this case W and +G)G in the ecliptic plane are the same asfor the one-dimensional and Force-Field cases. W increases from 400 to 800 km/s between latitudes 10 and30  . +<G>G has no latitudinal dependence. Latitudinal diffusion +__= 10% of +2G)G is added. In the drift casesa wavy sheet tilt angle of 10  is assumed. These parameters provide a best fit to solar minimum observations(see [9]). The drift solutions have significantly less energy loss than the no-drift, the one-dimensional and theForce-Field solution. The general rule is that as more and easier spatial routes of access are opened up, the lessmigration there is in energy.Finally, Figure 2c shows the same cases of Figure 2b, but with a solar wind termination shock ( !) insertedat 8B = 90 AU. Here + G>G , +__and W all drop by a factor of  , and W decreases further {86	E in the heliosheath.Since the transport parameters inside the shock are identical to those of Figure 2b, the differences between thesetwo figures are due to the acceleration on the shock and the turn-off of the adiabatic losses in the heliosheath.Once again, the qA  0 and qA  0 solutions offer the best fits to observations inside the shock at solar minimumconditions (see [9]). In this case the energy losses are much less, particularly at high energies where theacceleration at the termination shock reduces the net loss.All results are shown for protons. Conversion to other species is readily done by noting that two specieshave the same modulation if they have the same + , which is {3( . Using the relation (!2 5D¡!¢5wY£v79v¤S !¢5w39v¤S  , between momentum  , rigidity, ( , kinetic energy per nucleon,v , and speed3R![&5  , where¢andware mass and charge number, and¤S!¥MeV, one can transformfrom one species to another. [10] showed that, while this is complicated for all energies, both the relativisticand non-relativistic limits are that two species have the same + (i.e.3( ) if ¦§¢5w¨v/©ﬀq!¦ ¢5w¨v/©E.4. ConclusionsThe two primary conclusions from this analysis are, first, that for a given set of modulation parameters theForce-Field formalism produces up to twice as much energy loss than more realistic no-shock models. Second,the acceleration at the solar wind termination shock is so strong that at high energies it offsets much of theadiabatic losses in the supersonic solar wind. These results have important implications for the interpretationof certain observations, and in an accompanying paper we investigate the example of the modulated secondaryto primary ratio in the cosmic ray intensity.Acknowledgements. This work was supported by NSF Grant ATM 0107181 and the South African NationalResearch Foundation. RCL was supported by UNAM-DGAPA grant IN106105.References[1] L.J. Gleeson and I.A. Urch, Astrophys. Space Sci., 25, 387–404, (1973).[2] L.J. Gleeson and I.A. Urch, Astrophys. Space Sci., 11, 288–308, (1971).[3] M.L. Goldstein et al., Phys. Rev. Lett. 25, 832, (1970).[4] S.M. Niebur et al., J. G. R., 108 (A10), 8033, doi:10.1029/2003JA009876, (2003).[5] R.A. Mewaldt et al., AIP Conf. Proc. 719, 127-132, (2004).[6] L.J. Gleeson and W.I. Axford, Can. J. Phys., 46, S937, (1968).[7] L.J. Gleeson and W.I. Axford, Astrophys. J., 154, 1011, (1968).[8] R.A. Caballero-Lopez and H. Moraal, J. G. R., 109, A01101, doi:10.1029/2003JA10098, (2004).[9] R.A. Caballero-Lopez et al., J.G.R., 109, A05105, doi:10.1029/2003JA010358, (2004).[10] C.D. Steenberg, Ph.D. thesis, Potchefstroom University, (1998).

Termination shock and heliosheath effects on cosmic ray energy changes in the heliosphere

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