Limitations of the force field equation to describe cosmic ray modulation

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 109,, doi: /2003ja010098, 2004 Limitations of the force field equation to describe cosmic ray modulation R. A. Caballero-Lopez 1 Institute for Physical Science and Technology, University of Maryland, College Park, Maryland, USA H. Moraal School of Physics, Potchefstroom University, Potchefstroom, South Africa Received 20 June 2003; revised 7 October 2003; accepted 30 October 2003; published 2 January [1] The force field approximation to the transport equation which describes cosmic ray modulation in the heliosphere is a widely used tool. It is popular because it provides an easy to use, quasi-analytical method to describe the level of modulation with a single parameter. A simple numerical solution of the one-dimensional cosmic ray transport equation is used to show that this is a good approximation for galactic cosmic rays in the inner heliosphere but that its accuracy decreases toward the outer heliosphere. On the other hand, the even simpler convection-diffusion approximation improves with radial distance. The reason for the complementary behavior of these two approximations is that energy losses are relatively important in the inner heliosphere but not in the outer heliosphere. The force field approximation is worse for anomalous cosmic rays at all radial distances due to the exponential cutoff of these spectra at high energies. The ranges of validity are quantified. Since both approximations have their limitations, a simple numerical solution of the one-dimensional transport equation is provided for general use. INDEX TERMS: 2104 Interplanetary Physics: Cosmic rays; 2194 Interplanetary Physics: Instruments and techniques; 7807 Space Plasma Physics: Charged particle motion and acceleration; 7843 Space Plasma Physics: Numerical simulation studies; 7859 Space Plasma Physics: Transport processes; KEYWORDS: limitations of the force field equation Citation: Caballero-Lopez, R. A., and H. Moraal (2004), Limitations of the force field equation to describe cosmic ray modulation, J. Geophys. Res., 109,, doi: /2003ja Introduction [2] Cosmic ray modulation studies in the heliosphere are done with various approximations of the cosmic ray transport equation, first derived by Parker [1965]. The transport processes causing the modulation include: (a) pitch angle scattering of the particles along the field lines; (b) diffusion perpendicular to the field lines; (c) gradient, curvature, neutral sheet and shock drift effects; (d) convection in the solar wind; (e) adiabatic energy gains, typically in the compressions associated with shocks; and (f ) adiabatic losses in the expanding solar wind. [3] The full description of the modulation problem requires the determination of the cosmic ray intensity as a function of the three spatial coordinates, time and energy (alternatively momentum or rigidity). Because of the complexity, the full numerical solution of the transport equation, developed by Kota and Jokipii [1991] is seldom used. In order of progressing simplicity, the various levels of approximation are: (1) the assumption of azimuthal symmetry which eliminates cross-terms in the numerical solution 1 On leave from Institute of Geophysics, National Autonomous University of Mexico, Mexico City, Mexico. Copyright 2004 by the American Geophysical Union /04/2003JA that are difficult to handle, but which reduces its validity to time scales of one solar rotation or longer [e.g., Le Roux and Potgieter, 1991; Steenkamp, 1995]; (2) a quasi-steady state which reduces the time variable to a parameter (for technical reasons it becomes impossible to calculate shock acceleration effects in this case) [e.g., Jokipii and Kopriva, 1979; Moraal et al., 1979]; (3) spherical symmetry, which also eliminates the latitudinal coordinate, and in which reference to the heliospheric magnetic field and drift effects are lost [Fisk, 1971]; (4) a replacement of adiabatic energy losses by a simulated energy loss, which is called the force field solution; and (5) the neglect of energy losses, so that only the processes of inward radial diffusion and outward convection remain. [4] This paper reexamines to what extent the widely used force field solution and the convection-diffusion solution are valid approximations to the next level (level 3) of approximation, that includes adiabatic energy losses. 2. Derivation of the Force Field Solution [5] The force field formalism was originally derived by Gleeson and Axford [1968a, 1968b], but the following derivation closely follows that of Gleeson and Urch [1973]. Most of the calculations presented here were also done by Garrard [1972]. 1of7

2 [6] The equation for the evolution of the omnidirectional part of the cosmic ray distribution function, f, in terms of particle momentum, p, is a continuity equation of the þrsþ p p2 h _pif ¼ Q; ð1þ S ¼ 4pp 2 ðcvf K rfþ: ð2þ The momentum variable may be changed to rigidity, P, or kinetic pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi energy per nucleon, T, with P = pc/q = A/Z TTþ ð 2E 0 Þ, where A and Z are mass and charge numbers respectively. S is the differential current density, V the solar wind velocity, and K (r, P, t) the diffusion tensor which contains elements k k (r, P, t) and k? (r, P, t). It also contains an antisymmetric coefficient k T = bp/(3b) which describes gradient, curvature, neutral sheet, and shock drift effects [e.g., Isenberg and Jokipii, 1979; Moraal, 2001]. C is the Compton-Getting coefficient, C = (1/3)@ ln f/@ ln p, i.e., one third of the spectral index of a power law spectrum in momentum space. The adiabatic rate of momentum loss of cosmic rays in the expanding solar wind is h _pi =(p/3)v rf /f (while it is ( p/3)r V in the frame of the solar wind, as was shown by Gleeson and Webb [1978]). [7] When the flux and momentum loss terms are inserted into the continuity equation, three equivalent forms of the transport equation =@t þrðcvf K rfþþ 3p p3 V rf ¼ Q; =@t þrðvf K rfþ 1 3p p3 f ¼ Q; =@t þ V rf rðkrf Þ 1 ð 3 rv Þ@f =@ ln p ¼ Q: ð5þ convective and diffusive flows nearly cancel. The steadystate, no-source form of (3) then also requires that the adiabatic loss term must be negligible compared to the diffusive and the convective terms. The first condition implies that Vr/k 1, which is valid at high energy (or rigidity) where k is large. The second condition requires that g =(1/f )@f/@r C/r, which is easier satisfied for small r. Thus the force field approximation is a weak-modulation approximation that is most accurate in the inner heliosphere. [9] When C is introduced explicitly, (7) becomes a first order partial differential equation ¼ 0; with solution f (r, p) = constant = f b (r b, p b ) along contours of the characteristic equation dp/dr = Vp/3k in (r, p) space. The subscript b designates values on the outer boundary of the modulation region. The heliosphere has two distinct boundaries, namely the solar wind termination shock at an estimated 90 AU, and the heliosheath beyond that. The current non-shock model can not distinguish between these boundaries, but this is not important in the current context. [10] The name force field originates from the fact that (8) can be written in þ ¼ 0; where we have changed the variable from momentum, p, to rigidity, P. Particle rigidity has the dimensions of potential and the coefficient of the second term has the dimensions of potential per unit length, or a field - hence the name force field. [11] The rigidity P b = P b (r, P) is obtained by integrating the characteristic equation from the initial phase space point (r, P) to the point (r b, P b ) at the outer boundary r b.ifthe diffusion coefficient is separable in the form kðr; PÞ ¼ bk 1 ðþk r 2 ðpþ; ð8þ ð9þ ð10þ The force field approximation assumes that (a) there are no sources of cosmic rays, Q = 0, (b) there is a steady = 0, and (c) that the adiabatic energy loss rate h _pi = ( p/3)v rf /f = 0. In this case (3) reduces to the solution is Z Pb ðr;p P bðp 0 Þk 2 ðp 0 Z Þ rb P 0 dp 0 ¼ r Þ Vðr 0 Þ 3k 1 ðr 0 Þ dr0 fðþ; r ð11þ CVf K rf ¼ constant ¼ 0: The integration constant must be zero to avoid a singularity at the origin. If there is also spherical symmetry, this equation reduces to CVf k@f =@r ¼ 0; where the diffusion tensor K has contracted into a single, effective radial coefficient k. With this assumption of spherical symmetry the structure of the heliospheric magnetic field is lost. [8] Gleeson and Axford [1968a] discussed the validity of the zero streaming condition (7) for the first time. From an approximate analytical solution of the one-dimensional version of (3), they showed that the oppositely directed ð6þ ð7þ where f is called the force field parameter. When k 2 / P and b 1 the solution reduces to the widely used form P b P ¼ f; ð12þ so that the force field parameter becomes a rigidity loss or, alternatively, an energy or a momentum loss. Since it has the dimensions of potential, it is often called the force field potential. Typical values of f vary from 300 to 1000 MV, from solar minimum to solar maximum conditions. It is often forgotten that the force field rigidity (or momentum) loss in the form (12) applies only to the special case of relativistic particles, b = 1, and the rigidity dependence k / P. In all other cases P b P = f (f, P). A case that is often studied with the force field approach is the modulation of low energy electrons in the range 1 to 100 MeV. These 2of7

3 particles are still relativistic (b = 1), and in order to fit the observed spectra it is found that k must be independent of rigidity. In this case f is not a rigidity loss, but (11) gives a fractional loss of the form 4P=P ¼ e f 1: ð13þ In all such cases f alone is insufficient to describe the modulation and it does not have the dimensions of potential. Gleeson and Urch [1973] explicitly emphasized that the full force field parameter is k 2 /f. It follows from (11) that this quantity is given by f ¼ 1 Z rb V bk 2 3 r k dr; ð14þ and that it is dimensionless. [12] Notice that the force field formalism ends up with a modulation potential that causes energy, rigidity, or momentum changes while the original assumption was that the adiabatic energy loss term is negligible in comparison with the two spatial streaming terms. This is due to the interpretation of the coefficient VP/3k as a field in (9). There is no physical reason why this field, force, or energy loss, is related to the true adiabatic loss. The force field energy loss originates from the introduction of the Compton- Getting spectral term C in (8). It seems fortuitous that this energy loss is a reasonable approximation of the adiabatic energy loss in certain circumstances. Gleeson and Urch [1971] showed, in fact, that the force field energy loss in the form (12) is an upper limit of the true, adiabatic loss. [13] An almost equivalent approximation of the transport equation follows from the second form, (4), of the equation if, once again, and the third term are zero (although this term is not the adiabatic energy loss in the stationary frame). In the spherical symmetric approximation this results in the so-called convection-diffusion equation Vf k@f =@r ¼ 0; ð15þ Figure 1. Graphical representation of the description of the modulation with the force field solution (horizontal line) and the convection-diffusion solution (vertical line). The sloped line represents the actual modulation as a combination of intensity reduction and energy loss. intensity reduction and adiabatic energy loss, as represented by the inclined line in Figure 1. [15] For completeness we mention that the measured cosmic ray intensity, j T, with respect to kinetic energy per nucleon, T, is related to the omnidirectional distribution function f through with solution j T ¼ c A Z P2 fðpþ ¼ A c 4p 2 fðpþ: Z ð18þ Z rb f ¼ f b e M Vdr ; where M ¼ r k : ð16þ The integral M is called the modulation function, and it follows from (11) that it is related to the force field parameter f through M ¼ 3f=bk 2 : ð17þ Notice that M is a dimensionless quantity, which is natural (while f is a dimensional parameter). [14] The relationship between the force field and convection-diffusion formalisms provides two ways to describe the same process. Figure 1 shows a boundary spectrum f b that is modulated to a spectrum f. The force field approach describes the modulation along a horizontal line as an energy/momentum/rigidity loss which is a function of f (not necessarily equal to f), while the convection-diffusion approach describes it as a vertical reduction in intensity, given by M. In reality, the modulation consists of both Thus, in terms of intensities the force field relationship f (r, P) =f b (r b, P b ) becomes j T ðr; P Þ ¼ ðp=p b Þ 2 j Tb ðr b ; P b Þ: ð19þ 3. Validity of the Two Approximations [16] In this section we compare the force field and convection-diffusion solutions to the full numerical solution of the steady state, spherically symmetric (or one-dimensional) transport equation, i.e., the equation that includes adiabatic energy losses correctly. This equation follows from (5) 1 r @f 3r r2 ln p ¼ 0: ð20þ The solid lines in the left panel of Figure 2 are the intensity spectra, j T, calculated from the solution of this equation, 3of7

4 Figure 2. Left panel: Full numerical solution of one-dimensional transport equation (full lines), force field solution (dashed lines) and convection-diffusion solution (dotted lines) for GCR protons in a heliosphere with r b = 90 AU, V = 400 pffiffiffiffiffi km/s, l = 0.29P(GV) AU, k = lv/3, and f (1 AU) = 407 MV. Intensities are multiplied by factors 10 to enhance visibility. Right panel: Average radial intensity gradients calculated from g =ln(j 2 /j 1 )/(r 2 r 1 ), with the radial distance corresponding to subscripts 1 and 2 shown in the figure. IMP data are from Goddard Medium Energy Detector (MED) (P.I.: R. E. McGuire). using a local interstellar spectrum at r b = 90 AU, given by Webber and Lockwood [2001] as j Tb = 21.1T 2.8 /( T T 2.54 ). The intensities are calculated pffiffiffiffiffi at 1, 20, 40, 60, and 80 AU, and are offset by factors of 10 for better visibility. The radial solar wind speed has a value of 400 km/s, while k = bp (GV) cm 2 /s, which is equivalent to a diffusion mean free path l = 0.29P(GV) AU. The data points at 1 AU are IMP8 observations during the solar minimum periods of 1987 (open symbols) and 1997 (solid symbols). They show that the parameters of this nondrift, spherically symmetric solution give a reasonable compromise fit between the observed intensities during solar minimum periods with opposite drift states. The dashed and dotted lines are the force field and convectiondiffusion solutions respectively, using the same transport parameters. [17] At 1 AU this comparison between the solutions confirms the well-known behavior that the force field solution is a much better approximation to the full solution than the convection-diffusion solution, because the former includes energy loss effects. With increasing radial distance the force field approximation gets progressively worse, but the convection-diffusion solution improves. The reason is that the force field solution progressively over-estimates the true, adiabatic energy loss. For the same reason, the convection-diffusion approximation improves because energy losses become progressively smaller with increasing radial distance. [18] The radial intensity gradient is defined as g r =(1/j T T /@r. The right panel of Figure 2 shows average values of this gradient, calculated from g r ¼ ln ð j T2=j T1 Þ ; ð21þ r 2 r 1 for four sets of values of r 1 and r 2. The full line solutions indicate the well-known behavior that the gradients peak between 20 and 200 MeV, and that they recede to zero at low energies due to adiabatic energy losses. The dashed lines are the force field gradients, and the dotted line is the convection-diffusion gradient, which, from (15), is given by g r = V/k. Since k was chosen as independent of r, all these convection-diffusion gradients are the same, and they are a poor approximation to the true gradient at all radial distances and at all energies. At high energies the force field gradient is a good approximation to the true gradient, but from 100 MeV downward it becomes progressively worse. Also notice from (7) that the force field gradient is g r = CV/k, so for low energies where C goes to zero, it should go to zero. This does not occur because V/k increases as fast with decreasing energy as C decreases, forcing CV/k into an 4of7

5 Figure 3. The same as in Figure p 2ffiffiffiffiffi but for ACR protons. The intensities in the left panel are absolute, i.e., not multiplied by factors of 10. The shock spectrum used is given in the text. asymptotic value. If we had chosen k independent of rigidity below a certain rigidity (which is quite a common choice to fit spectra) this contradiction would go away, and the force field gradient would go to zero at low energies. This sensitive parameter dependence demonstrates a limitation of the force field formalism. [19] Figure 3 is a repetition of Figure 2, but for anomalous cosmic ray (ACR) Hydrogen. Steenberg and Moraal [1999] showed how to replace ACR acceleration in the solar wind termination shock by an effective spectrum on a passive outer boundary. The appropriate scaling factors to other ACR species are also discussed there. For the present parameters this spectrum is given by j Tb = (0.062/T)exp[ 0.189(T/0.062) ]. In this case the intensities in the left panel are the true p ffiffiffiffiffiintensities, i.e., they are not multiplied by factors of 10. The right hand panel shows that the force field gradients in this case are substantially larger than the gradients of the full solution. Furthermore, they increase with energy. This is due to the exponential, rather than power law form of the high energy roll-over of the ACR spectra. [20] The ultimate evaluation of the force field approximation is to test the zero-streaming condition (7) from which it is derived. In Figure 4 we plot the ratio of the outward convective streaming term, CVf, to the inward diffusive term, k@f/@r. If k ff is the diffusion coefficient deduced from the force field condition (7), then the ratio k ff /k is the same as the ratio of the two streaming terms, where k is the true diffusion coefficient deduced from the numerical solution of the transport equation. Thus the vertical axis of Figure 4 also represents the ratio of the diffusion coefficients. The results confirm that for GCRs the condition, and the deduced values of k are valid to within ±20% down to 20 MeV at 1 AU and to 200 MeV at large radial distances. For ACRs the situation is much worse, because the ratio diverges at both low and high energies. The high energy divergence is due to the fact that the ACR spectrum has an exponential type cutoff because of the upper limit of the acceleration at the termination shock. For such a spectrum C increases with energy, whereas it is constant for a power law. Thus, in the ratio CVf/(k@f /@r) = CV/(kg), the numerator CV increases faster than the denominator kg. The effective ACR spectra may be different from this form, however. Mewaldt et al. [1996] and Jokipii [1996] showed, for instance, that ACR species other than H may become harder in the cutoff region due to multiple charge stripping [see also Cummings et al., 2002]. For this reason we repeat the ACR calculation for the case where the spectrum at T > 300 MeV is extended as a power law /T 13. The ratio at these energies then recedes back to 1. Thus the validity of the force field formalism is sensitive to the exact, species dependent, spectral form at high energies. 4. Usefulness of the Modulation Parameter M [21] The combination of the values r b = 90 AU, V = 400 km/s, and k = bp (GV) cm 2 /s, which fit the observed spectra at 1 AU, give the modulation parameter M, defined in (16), as Mð1AUÞ ¼ 1:22=bPðGVÞ: ð22þ 5of7

6 Figure 4. The ratio of convective to diffusive streaming (or anisotropy) as function of kinetic energy: left for GCRs, middle for ACRs, right for ACRs when the boundary (or equivalent shock) spectrum is modified to a power law j Tb / T 13 for T > 300 MeV. The ratio of the k deduced from the force field condition to the true k is given by the same number. This is equivalent to a force field parameter f(1au) = 407 MV. The question arises as to how the solutions will change if a different solar wind, boundary distance, or k are chosen. Equation (16) implies that the convectiondiffusion solution will remain the same for all combinations that give the same M, i.e., that only the integrated effect of the modulation can be detected. This is not true for the full numerical solution of (20), because the adiabatic loss term that it contains does not scale proportional to M. In Figure 5 we test to what extent the full numerical solution deviates from the scaling with M. A set of five diffusion coefficients was chosen, being proportional to r 1, r 0.5, r 0, r 0.5, and r 1. The magnitudes were chosen such that M (1 AU) is the same for all of these. The figure shows that the five numerical solutions at 1 AU are very similar (but the 80 AU solutions differ dramatically, because M (80 AU) is different for each of them). At 10 MeV, for instance, the modulation from the LIS to the 1 AU intensity is a factor of 470, while the solutions for k / r 1 to k / r +1 differ by only a factor of 2, i.e., by a relative amount of 0.4%. This shows that the modulation parameter M is a single expression that adequately describes the modulation process, including its rigidity dependence. The individual values of k, r b, and V can only be deduced from two or more spectra at different radial distances. 5. Conclusions [22] The lower limit of the energy range for which the force field approximation is valid for galactic cosmic rays increases with increasing radial distance, and it is a much weaker approximation for the steep, exponential type anomalous cosmic ray spectra in general. On the other hand, the full numerical solution of the one-dimensional transport Figure 5. Full numerical solution of the one-dimensional transport equation for GCR protons using different radial dependencies of the diffusion coefficient k, such that the modulation function is M(1 AU) = 1.22/bP (GV). 6of7

7 equation gives a result that can just as easily be presented as a single expression as the force field parameter. This is the modulation integral M, defined in (16), and expressed in a form such as (22). [23] Thus we recommend that the numerical solution of the full one-dimensional transport equation (20) should be used instead of the force field approximation. Gleeson and Urch [1971] published the solution algorithm, but it has seldom been used. To encourage its use, we publish short Fortran and C solutions of the equation at ac.za/physics/physics%20web/research/helio-phy.htm. [24] Acknowledgments. We acknowledge the valuable suggestions by F. B. McDonald and V. Ptuskin, and the advice of A. C. Cummings and E. C. Stone regarding the results of Figure 4. This work was supported by NASA SRT Grant NAG , the NASA JPL Voyager Program (contract ), and the South African National Research Foundation. [25] Shadia Rifai Habbal thanks Garry M. Webb and another referee for their assistance in evaluating this paper. References Cummings, A. C., E. C. Stone, and C. D. Steenberg (2002), Composition of anomalous cosmic rays and other heliospheric ions, Astrophys. J., 578, 194. Fisk, L. A. (1971), Solar modulation of galactic cosmic rays, 2, J. Geophys. Res., 76, 221. Garrard, T. L. (1972), A quantitative investigation of the solar modulation of cosmic-ray protons and helium nuclei, Ph.D. thesis, Calif. Inst. of Technol., Pasadena, Calif. Gleeson, L. J., and W. I. Axford (1968a), The solar radial gradient of galactic cosmic rays, Can. J. Phys., 46, S937. Gleeson, L. J., and W. I. Axford (1968b), Modulation of galactic cosmic rays, Astrophys. J., 154, Gleeson, L. J., and I. A. Urch (1971), Energy losses and modulation of galactic cosmic rays, Astrophys. Space Sci., 11, Gleeson, L. J., and I. A. Urch (1973), A study of the force-field equation for the propagation of galactic cosmic rays, Astrophys. Space Sci., 25, Gleeson, L. J., and G. M. Webb (1978), Energy changes of cosmic rays in the interplanetary region, Astrophys. Space Sci., 58, 21. Isenberg, P. A., and J. R. Jokipii (1979), Gradient and curvature drifts in magnetic fields with arbitrary spatial variation, Astrophys. J., 234, Jokipii, J. R. (1996), Theory of multiply charged anomalous cosmic rays, Astrophys. J., 466, L47 L50. Jokipii, J. R., and D. A. Kopriva (1979), Effects of particle drift on the transport of cosmic rays III. Numerical models of galactic cosmic ray modulation, Astrophys. J, 234, 384. Kota, J., and J. R. Jokipii (1991), The role of corotating interaction regions in cosmic ray modulation, Geophys. Res. Lett., 18, Le Roux, J. A., and M. S. Potgieter (1991), The simulation of Forbush decreases with time-dependent cosmic ray modulation models of varying complexity, Astron. Astrophys., 234, 531. Mewaldt, R. A., R. S. Selesnick, J. R. Cummings, E. C. Stone, and T. T. von Rosenvinge (1996), Evidence for multiply charged anomalous cosmic rays, Astrophys. J., 466, L43 L46. Moraal, H. (2001), The discovery and early development of anomalous cosmic rays, in Cospar Colloquia Series, vol. 11, edited by K. Scherer et al., pp , Pergamon, New York. Moraal, H., L. J. Gleeson, and G. M. Webb (1979), Effects of charged particle drift on the modulation of the intensity of galactic cosmic rays, Conf. Pap. Int. Cosmic Ray Conf. 16th, 3, 1. Parker, E. N. (1965), The passage of energetic particles through interplanetary space, Planet. Space Sci., 13, 9. Steenberg, C. D., and H. Moraal (1999), The form of the anomalous cosmic ray spectrum at the solar wind termination shock, J. Geophys. Res., 104, 24,879. Steenkamp, R. (1995), Shock acceleration as source of the anomalous component of cosmic rays in the heliosphere, Ph.D. thesis, Potchefstroom Univ., Potchefstroom, South Africa. Webber, W. R., and J. A. Lockwood (2001), Voyager and Pioneer spacecraft measurements of cosmic ray intensities in the outer heliosphere: Toward a new paradigm for understanding the global modulation process: 1. Minimum solar modulation (1987 and 1997), J. Geophys. Res., 106, 29,323. R. A. Caballero-Lopez, Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742, USA. (rogelio@ glue.umd.edu) H. Moraal, School of Physics, Potchefstroom University, Potchefstroom, 2520, South Africa. (fskhm@puknet.puk.ac.za) 7of7