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THE ASTROPHYSICAL JOURNAL, 528 : 1015È1025, 2000 January 10 ( 2000. The American Astronomical Society. All rights reserved. Printed in U.S.A.

INTERPLANETARY AND INTERACTING PROTONS ACCELERATED IN A PARALLEL SHOCK WAVE R. VAINIO, L. KOCHAROV, AND T. LAITINEN Space Research Laboratory, Department of Physics, FIN-20014, Turku University, Finland Received 1999 April 9 ; accepted 1999 August 24

ABSTRACT We present a test-particle model of di†usive shock acceleration on open coronal Ðeld lines based on one-dimensional di†usion-convection equation with Ðnite upstream and downstream di†usion regions. We calculate the energy spectrum of protons escaping into the interplanetary space and that of protons interacting with the subcoronal material producing observable secondary emissions. Our model can account for the observed power-law and broken power-law energy spectra as well as the values of the order of unity for the ratio of the interplanetary to interacting protons. We compare our model to Monte Carlo simulations of parallel shock acceleration including the e†ects of the diverging magnetic Ðeld. A good agreement between the models is found if (i) the upstream di†usion length is much smaller than the scale length L of the large-scale magnetic Ðeld, i /U > L , where U is the upstream scatB 1 1 B 1 tering center speed and i (p) is the momentum dependent upstream di†usion coefficent ; (ii) the down1 stream di†usion length is much smaller than the length of the downstream di†usive region L , for which 2 L > L has to be satisÐed ; and (iii) most of the particles are injected to the acceleration process within 2 B a couple of L Ïs above the solar surface. We emphasize that concurrently produced interplanetary and B interacting protons can be used as probes of turbulence in the vicinity of the shock ; our model has two turbulence parameters, the scattering-center compression ratio at the shock and the number of di†usion lengths in the upstream region, that may be experimentally determined if the interplanetary and interacting proton spectra are measured. Subject headings : acceleration of particles È shock waves È Sun : Ñares È Sun : particle emission È turbulence 1.

INTRODUCTION

and power-law energy spectra of the two proton populations ? Since shock acceleration involves solar-frame bulk motion of the plasma, it may be questioned whether particles leaving the shock in the downstream region ever reach the subcoronal regions where they should interact. Also, since there has to be turbulence in front of the shock to hinder particle escape if any acceleration of particles is to occur, we need to study how the spectrum of the escaping particles is related to the spectrum at the shock. To answer these questions, we perform an analytical study of onedimensional parallel shock acceleration to provide a wellestablished background for numerical work on more involved geometries. As a second part of the study, we perform Monte Carlo simulations of particle acceleration in a diverging Ðeld line geometry to verify the analytical results, to check the limits for their applications, and to provide a tool for further studies in more involved and realistic models of coronal/interplanetary shocks. In this stage, both the analytical and the numerical calculations are done using test-particle approximation : accelerated-particle e†ects on the medium (shock-structure modiÐcation, generation of waves) are probably also important, but their selfconsistent modeling is far beyond the scope of the present study. Our aim is to apply and further develop the ideas in the studies of Ellison & Ramaty (1985), Lee & Fisk (1982), and Lee & Ryan (1986). The Ðrst was a study of di†usive shock acceleration in Ñare site conditions, and the second and the third were studies of coronal shock acceleration in a global blast wave of inÐnite strength propagating from solar corona to interplanetary space. Thus, the studies were aimed at explaining observations in impulsive-phase and gradual-phase timescales, respectively. Alternatively, we may state that the studies developed theories for shock

Protons accelerated on/near the Sun can be directly measured in the interplanetary medium, or alternatively their properties may be deduced from measurements of secondary emissions, neutrons and c-rays, produced during nuclear interactions at the Sun (see Ramaty et al. 1993 for a review). The ratio (!) of the interplanetary proton number to the number of protons producing secondary emission at the Sun (interacting protons) is an important result of recent solar observations. This ratio varies in a very wide range, from about D0.01 to Z1, being typically in order of unity if a postÈimpulsive-phase acceleration is present. Recently, Kocharov et al. (1999) have studied the spectra of interacting and interplanetary protons in a model, where the particles are accelerated stochastically on open, diverging coronal magnetic Ðeld lines with no bulk motion of the plasma. They found out that the ratio of interplanetary to interacting protons is quite generally between 1 and 4.6 and that it varies with energy so that the spectrum of interacting particles is steeper than the spectrum in interplanetary space. In this paper, we study the spectrum of interplanetary ions and the parameter ! in an alternative acceleration model, parallel shock acceleration. Di†usive shock acceleration (Axford, Leer, & Skadron 1977 ; Bell 1978 ; Blandford & Ostriker 1978 ; Krymsky 1977) in coronal/interplanetary shocks provides the majority of interplanetary protons in large solar energetic particle (SEP) events according to many authors (Kahler 1993, 1994 ; Reames 1993). It is therefore important to study whether this acceleration mechanism can really account for the SEP spectrum, but equally important is to study the spectrum of protons precipitating in subcoronal regions behind the shock wave. Under which conditions (if any) can we produce the observed ratios ! D 1 (Ramaty et al. 1993) 1015

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VAINIO, KOCHAROV, & LAITINEN

acceleration in systems with length scales much smaller and much larger than the solar radius, respectively. We, on the other hand, consider a system, that has length scales just of the order of solar radius, so our study attempts to bridge the two extremes. None of these studies attempted to calculate spectra of interacting protons accelerated concurrently with the interplanetary protons at the shock. Hence, our second aim is to emphasize that interacting particle spectrum should be computed along with the interplanetary one, to give a possibility to use a richer variety of observational data to judge for the correctness of the models. In addition, we present a Ñexible numerical method that may be used in complicated situations that are beyond the scope of analytical modeling. 2.

THE MODEL

We assume that the acceleration of energetic particles is due to Ðrst-order Fermi acceleration at a parallel shock. The shock is propagating with a constant speed V into a s medium at rest with an exponentially decreasing magnetic Ðeld, B(f) \ B e~f@LB , where f is the coordinate measured 0 along the magnetic Ðeld lines (Fig. 1) and L \ [B/(LB/Lf) is the (constant) scale length of the magnetic BÐeld. We take the particles to be scattered by circularly polarized Alfven waves propagating along the magnetic Ðeld lines on both sides of the shock in a region that has spatial extent of L upstream [downstream] of the shock. In the 1*2+ the waves are taken to propagate outward upstream region, from the Sun in the plasma frame. This is consistent with waves generated at the Sun and with waves generated by the accelerated particles themselves through the streaming instability. The super-Alfvenic Ñow then carries the waves to and through the shock to the downstream region. As a

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result of the interaction of the waves and the shock compression, the downstream turbulence always consists of waves propagating in both directions along the Ðeld. One can compute the intensities of both wave modes from given upstream conditions and it turns out that the outwardpropagating wave component is dominating in intensity so that stochastic acceleration in the downstream region may be neglected (Vainio & Schlickeiser 1998, 1999). The upstream Alfven speed V is taken to be constant, A1 which implies an exponentially decreasing plasma density, n (f) \ n e~2f@LB , where the subscript i \ 1[2] refers to the i 0i [downstream] region of the shock, and the comupstream pression ratio of the shock, o \ n /n , is also a constant. 02 01 We consider a system where the waves scattering the particles have their wave vectors directed along the mean magnetic Ðeld. Therefore, there is no resonant cross-Ðeld di†usion and particles remain on their initial magnetic Ðeld lines forever. A bundle of neighboring Ðeld lines, thus, deÐnes a Ñux tube. If the particle scattering o† the waves is strong enough to keep the particle distribution close to isotropic, we may describe the linear particle density f (f, p, t) \ d2N/dfd p in a Ñux tube, i.e., the number of particles per unit length of the magnetic Ðeld line and unit momentum, with a Fokker-Planck equation

A

B

A

i L V LV Lf L V] f[ ] ] L Lp L Lf Lt Lf B B

B

p L Lf f\ i ]s , 3 Lf Lf (1a)

where p is the particle momentum, s(f, p, t) \ Q(t)d(f [ V t)d(p [ p ) is the source function, Q(t) gives the inj number of sinjected particles to the considered Ñux tube at the shock per unit time,

04 V \ VA1 , f [ Vs t , (1b) V (f, t) \ 5 1 f\V t , 06 V2 , s is the phase speed of the waves scattering the particles, and 04 i (p) , Vs t ] L 1(p, t) [ f [ Vs t , (1c) i(f, p, t) \ 5 1 06 i2(p) , Vs t [ L 2(p, t) \ f \ Vs t , is the spatial di†usion coefficient of the particles related to the particle mean free path j by i \ jv/3. In the regions f [ V t ] L and f \ V t [ L the particles are taken to s adiabatically 1 s 2 stream without scattering. If the spatial coordinate is changed to x \ V t [ f, we may write down equas tion (1a) in the form

A

B A A B

B

i LU V [ U L p Lf L U[ f[ ] s f ] L Lx L Lp 3 Lt Lx B B Lf L i ]s , \ Lx Lx

(2a)

where s(x, p, t) \ d(x)d(p [ p )Q(t), inj 04 V [ V1 \ U1 , x \ 0 , (2b) U(x) \ V [ V (x) \ 5 s s 06 U2 \ U1/oc , x [ 0 , o \ U /U is the scattering-center compression ratio, c 1 2be calculated if the upstream plasma properties which can are known (see Vainio & Schlickeiser 1998, 1999), and FIG. 1.ÈGeometry of the acceleration region : a shock wave propagating parallel to the open magnetic Ðeld line emits accelerated particles both toward the Sun and toward the interplanetary medium.

04 i (p) , 0 [ x [ [L 1(p, t) , i(x, p, t) \ 5 1 06 i2(p) , 0 \ x \ L 2(p, t) .

(2c)

No. 2, 2000

INTERPLANETARY AND INTERACTING PROTONS

We require the particle distribution function, f (x, p, t)/ [4np2A(x, t)], where A(x, t) P 1/B(x, t) is the Ñux tube crosssectional area, to be continuous at x \ 0. For a parallel shock wave, this means that f (x, p, t) has to be continuous at the shock. In addition, equation (2a) has to be completed with appropriate boundary conditions at x \ [L , L . At 1 2 the outer boundary, we may assume that the particles escape, f ([L , p, t) \ 0, but the boundary condition at 1 x \ L is, in general, more involved because of the possible 2 mirroring of particles before they reach the solar surface. If the positions of the boundaries L (p) are time inde1,2 pendent, we may integrate equation (2a) and the boundary conditions over time to get

A

B A A B

i L U[ F[ L Lx B L i \ Lx

B

LU V [ U L p ] s F Lx L Lp 3 B LF ]S , Lx

(3)

where S(x, p) \ N d(x)d(p [ p ), N is the total number of 0 the considered inj Ñux 0 tube, and F(x, p) \ injected particles to /= fdt. Here we have assumed that f (x, p, t) ] 0 as t ] 0, 0 which is reasonable if the injection will tend to zero rapidly enough with time since the particles will escape at the outer boundary and be swept out of the system to the far downstream by the moving scattering centers.

RESULTS

3.

3.1. Analytical Solution In case of nondiverging magnetic Ðeld lines (L ] O), B adiaequation (3) can be solved analytically. In this case the batic motion of the particles outside the shock is free streaming, and the particles escape from both boundaries : F([L , p) \ F(]L , p) \ 0. Then, the solution is (see, e.g., 1 2 Ostrowski & Schlickeiser 1996)

0 0 4

exp (b x) [ exp ([b L ) 1 1 1 , 0 [ x [ [L , 1 1 [ exp ([b L ) 1 1 F(x, p) \ 5 exp (b L ) [ exp (b x) 2 2 2 , F (p) 0\x\L , 2 exp (b L ) [ 1 6 0 2 2 (4) F (p) 0

where b~1 \ i /U is the upstream [downstream] dif1*2+ and 1*2+F (p) 1*2+ fusion length \ F(0, p) satisÐes the equation 0 U U e~b2 L2 F p d F 1 0 0] [ 2 (U [ U ) 1 2 3 dp p2 1 [ e~b1 L1 1 [ e~b2 L2 p2

A

B

N d(p [ p ) inj \ 0 p2 that can be obtained by dividing equation (3) by p2 and integrating it from x \ 0 [ to 0 ] . This equation has the solution

A B G P C

F (p) \ CH(p [ p ) 0 inj ] exp [c

p c~2 inj p

p

pinj

D H

e~g1(p{) 1 e~g2(p{) dp@ [ , 1[e~g1(p{) o 1[e~g2(p{) p@ c (5a)

1017

where H(p) is the Heaviside function and 3N 0 , (5b) p (U [ U ) inj 1 2 3o c . c\ (5c) o [1 c Thus, if the numbers of di†usion lengths, g 4 b L , in the i i i up- and downstream regions are large, g ] O, the shock i spectrum approaches the canonical power law, where the spectral index c is fully determined by the scattering-center compression ratio at the shock. Also, since it is reasonable to assume that g ? g at all momenta, we may simplify 2 1 equation (5a) somewhat by neglecting the second term of the integrand to get C\

A B C P

F (p) \ CH(p [ p ) 0 inj

p c~2 inj p p

] exp [c

D

1 dp@ . eg1(p{) [ 1 p@

(6) pinj The di†erential energy spectrum of particles escaping at the boundaries is dN dp 1*2+ \ ][[] dE dE

KA BK LF Lx

. (7) x/~L1*`L2+ Using equations (4)È(7), the spectrum of interplanetary particles is obtained as dN dp U F (p) 1\ 1 0 dE dE eg1(p) [ 1

i

A B

p c~2 cN inj 0 p v(eg1(p) [ 1) p inj p 1 dp@ . (8) ] exp [ c { eg1(p ) [ 1 p@ pinj Note, however, that a fraction of particles escaping toward the Sun will not precipitate, because the shock is an outward moving source and, thus, the particle, although leaving the downstream region of the shock, may still propagate outward from the Sun. This fraction depends on the angular distribution of the particles, but for an isotropic distribution it would be of the order of V 2/v2, where v is the s fraction may be particle speed. Thus, at high energies, this neglected and the ratio of interplanetary to interacting protons is \

C P

D

dN /dE o 1 c . (9) \ dN /dE eg1(p) [ 1 2 One notes immediately that the ratio of interplanetary to interacting protons is somewhat more involved in the case of shock acceleration than in the case of stochastic acceleration, where it is unity if L \ O. B validity of the obtained soluLet us try to estimate the tion, when L is Ðnite. First, one has to be able to neglect the streamingB speed caused by focusing, i /L , in comparison with the scattering-center speeds, iU .B Second, the i inverse adiabatic deceleration rate of the particle has to be !(E) \

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VAINIO, KOCHAROV, & LAITINEN

much greater than the acceleration timescale,

C

D

i (p) i (p) 3L 3 1 ] 2 B \q ?q B . AD acc U [ U U U V [U 1 2 s i 1 2 The approximate equality holds for particles relatively close to the shock. For these particlesÈif b ? b and if we 2 1 neglect all factors of the order of unityÈthe second condition, thus, reduces to the Ðrst one. Hence, we need j (p)v/(3U ) > L to hold for all momenta of interest to 1 1 B obtain the shock spectrum in correct form. In addition, particle propagation from the shock to the escape boundaries has to be rapid enough to neglect adiabatic deceleration during the propagation, also. In the upstream region, we may demand that the di†usion time L 2/i has to be 1 1 small in comparison with the adiabatic deceleration timescale 3L /V . This yields j v/(3U ) > 3L U /(V g2), which, B coronal 1 1 1larger 1 for typical shock 1speeds 1of a fewB times than V , reduces to j v/(3U ) \ L /g2. This imposes no further 1 1 1 B 1 restrictions on j as long as g is a number of the order of 1 1 unity. As the downstream di†usion coefficient is typically small, we estimate the downstream propagation time to be DL /U . For this to be much smaller than the respective 2 2 of adiabatic deceleration, 3L /V , we have to timescale B 2 demand L /L > 3U /V D 1. 2 B 2 2 One further condition, however, has to be satisÐed : the use of the zero boundary condition at the base of the acceleration region has to be justiÐed. As we noted above, magnetic mirroring reÑects a fraction of particles trying to precipitate. However, the mirrored particle then hits the turbulent downstream region of the shock from behind. If the downstream scattering is efficient enough, the mirrored particle has a negligible chance of di†using back to the shock. Instead it is rapidly scattered back for another trial of precipitation. This occurs if g (p) is a large number, which 2 deceleration in the downwe shall assume. Since adiabatic stream turbulent region could be neglected by the assumption of L > L , the particle has the same energy (in the 2 scattering B frame of the centers) when leaving the turbulent region as it had when entering it. Such a particle does not contribute to the time-integrated Ñux of particles at the boundary, so the correct solution for the downstream particle Ñux is indeed obtained using the zero boundary condition at both boundaries when solving equation (3). Thus, equation (8) for the interplanetary particle spectrum is correct as long as j (p)v/(3U ) > L , g (p) ? 1, and 1 1 B 2 L >L . 2Although B the diverging magnetic Ðeld might not have large e†ects on the spectrum of interplanetary particles, it may still have a large e†ect on the ratio of interplanetary to interacting protons. A particle emitted at the base of the acceleration region will precipitate, if its shock-frame pitchangle cosine

S

AB

B(V t [ L ) V s 2 ] O s B [J1 [ e~Vs t@LB , v B 0 yielding a probability of P(t) \ e~Vs t@LB for an emitted particle to precipitate for a Ñux-weighted isotropic particle distribution ( f P o k o , k \ 0), if terms of the order V /v and s L /L are neglected. A particle not precipitating will get 2 B mirrored back to the acceleration region, if its speed exceeds the shock speed (in the Ðxed frame). The time it takes for a particle to get mirrored (from k \ 0 to [ k) is t \ 2L v~1 ln [(1 [ k)/(1 ] k)], which B B k\[

1[

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yields an average mirroring time of St T \ 4L /v for a ÑuxB B weighted particle distribution. At high speeds, this may be used as the average time for the emitted particle to get back to the acceleration region provided that P(t) > 1. (If P(t) was not a small number, the average would have to be taken over the returning particles, only, yielding a smaller St T.) Then, the particle will simply di†use in the downB stream region, if g (p) is a very large number (as we assumed earlier). After an 2average time of Dj /U \ 3L /(g v) > 2 path,2it will 2 be 4L /v, where j is the downstream mean2 free B 2 emitted again for another chance-of-precipitation cycle. Thus, the rate of precipitation per emitted particle is about vP(t)/(4L ). Since it involves the particle speed, the rate has B to be completed with the rate of deceleration of the particles in the expanding region behind the shock. A mirroring particle loses a Ñux-weighted average (downstream) momentum S*pT \ 4pV /3v per cycle, and the deceleration rate is, 2 /3L . If one particle is emitted at the therefore, p5 /p \ [V 2 region B base of the acceleration with a nonrelativistic velocity v \ v at time t \ t , the number of particles in the 0 system at0t [ t is 0 3v 0 G(t ; v , t ) \ exp e~Vs t0@LB 0 0 4(3V ] V ) s 2

C

D

] (e~(3Vs`V2)(t~t0)@3LB [ 1) , and the fraction of precipitating particles is thus G (v , t ) \ 1 [ G(O ; v , t ) prec 0 0 0 0 3v P(t ) 0 0 . (10) \ 1 [ exp [ 4(3V ] V ) s 2 Those particles that do not precipitate are cooled down to thermal energies and become part of the downstream plasma Ñow. Note that if P(t ) is not too small, say, if 0 v P(t ) Z V , the rate of precipitation will exceed the rate of 0 0 2 deceleration, and most precipitating particles will have an energy close to their emission energy. Thus, if particles are not emitted too far from the Sun, our estimate for !(E) should be a good approximation at least at high energies. We shall show later how equation (10) can be used to give a valid analytical solution to this ratio even when particles are emitted relatively far from the Sun.

C

D

3.2. Numerical Solutions To verify our analytical model, we used a Monte Carlo simulation code developed for the study of test-particle transport and acceleration in the inner heliosphere. Instead of solving a di†usion-convection equation, the code employs a kinetic treatment, where particles are moved in small time steps conserving their energy and magnetic moment in a frame, where the large-scale magnetic Ðeld is static. After each such time step, an isotropic small-angle scattering is performed via a random generator. The scatterings are elastic in the (local) frame comoving with the scattering centers. After a large amount of such particles is simulated, we print out the results in the form of histograms over quantities of interest. Very similar model was used by Ellison, Jones, & Reynolds (1990) in a study of relativistic shock acceleration, but in their study the background magnetic Ðeld was strictly one-dimensional (no magnetic focusing) and their scattering centers were frozen-in into the

No. 2, 2000

INTERPLANETARY AND INTERACTING PROTONS

plasma. More details about the simulations and the kinetic model can be found in the Appendix. We performed a set of simulations to verify equations (8) and (9). We Ðxed the value of focusing length to L \ 0.3 B R (where R is the solar radius), and used a shock speed _ _ of V \ 1200 km s~1, an upstream Alfven speed of V \ s A1 300 km s~1, which give U \ 900 km s~1. We also used a 1 squared upstream sound speed of c2 \ 0.3V 2 correspondS1 ing to a coronal electron temperature of T A1D 2 ] 106 K. e According to Vainio & Schlickeiser (1998, 1999), the gas compression ratio of the shock should be o B 4/ (1 ] 3c2 /V 2) B 3.8 as long as the upstream wave amplitude S1 s is very small compared with the magnitude of the ordered Ðeld. In addition to o, the scattering-center compression ratio is dependent on the upstream wave spectrum but a characteristic value for incompressible upstream turbulence is near o \ 5.5 (see Fig. 5 of Vainio & Schlickeiser 1999). c We set j , L ] 0 while keeping g ] O (see Appendix for 2 2 2 how the downstream scattering is treated, explicitly) and inject low-energy particles to the acceleration process at the shock with an injection speed of v \ 0.03c measured in the shock frame. The shock-frame inj angular distribution of the injected particles was taken to be 2H(k)k corresponding to an isotropic particle population just crossing the shock from the downstream region to the upstream. The number of upstream di†usion lengths was set at g \ ln (1 ] o ) \ c 1.872, giving a value of ! \ 1 in equation 1(9). These choices, thus, give a theoretical prediction of

A B

dN 7.0 ] 10~3 mc 2.33 \ p mcv dE for the interplanetary and interacting proton spectra per one injected particle per Ñux tube, momentum being measured in the Ðxed frame. As a test for the simulation code, we veriÐed that it produces this solution when all the e†ects of diverging magnetic Ðeld were switched o†. All simulations with the e†ects of diverging magnetic Ðeld included were started when the shock was at f \ 0.3 R _ above the solar surface. The upstream mean free path of the particles, j, was varied to study the e†ect of focusing on the

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TABLE 1 PARAMETERS FOR THE PRESENTED MONTE CARLO SIMULATIONS Simulation Number 1 ..................... 2 ..................... 3 .....................

j/R

_ 5 ] 10~4 5 ] 10~4 2.5 ] 10~3

Injection, Q(t)1 d(t [ t@) H(t [ t@)V L ~1 e~Vs(t~t{)@LB s B d(t [ t@)

1 Time t@ \ 0.3 R /V is the start time of the injection. _ s

interplanetary spectrum. To analyze the e†ect of mirroring behind the acceleration region, we took the injection at the shock, s(x, p, t) \ d(x)d(p [ p )Q(t), to be either impulsive inj or proportional to the ambient plasma density. The values used in the simulations are given in Table 1. The simulated spectra of interplanetary and interacting protons are given in Figure 2. Simulation 1 gives a good agreement with the analytical form of the interplanetary spectrum, as it should, since the mean free path is so small that one does not expect large contributions from either the streaming term caused by focusing or the adiabatic deceleration term in equation (3) : for j \ 5 ] 10~4 R we have i /L \ 0.185(v/c)U . The _ spectrum at1 EB[ 600 MeV is 1due to slight softening of the the Ðnite, although small, value of jv/(3U L ) \ 0.185(v/c) B was comat particle speeds v D c. If the analytical 1model pletely valid, the chosen parameters would produce identical interacting and interplanetary proton spectra, as pointed above. For most part of the spectrum this is true, but below E D 20 MeV there is a deÐcit of interacting particles. This is because at low velocities the precipitating rate for the particles is not high enough for all of them to interact before adiabatically cooling. Therefore, the theoretical prediction we use here for the interacting spectrum is

A B

dN 7.0 ] 10~3 mc 2.33 2 \ N (v) . prec p dE mcv The multiplying factor N (v) is the fraction of precipitating particles at speed v prec obtained as a convolution inte-

FIG. 2.ÈSimulated (see text for details) interplanetary (crosses) and interacting (diamonds) proton energy spectra along with the analytically predicted spectra (curves).

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VAINIO, KOCHAROV, & LAITINEN

gral over t of the injection function Q(t) and the ““ GreenÏs 0 function ÏÏ for the fraction of precipitating particles, G (v , prec 0 t ) from equation (10) : 0 = Q[t [ T (v)]G (v, t )dt , (11) N (v) \ 0 prec 0 0 prec 0 where v has simply been replaced by the particle speed, 0 and

P

T (v) \

A

v ] c Jc2 [ v2 j o inj c ln v ]c U2 o [ 1 Jc2 [ v2 inj 1 c

B

(12)

is the mean acceleration time. It is obtained from the standard equation for the acceleration rate, p5 U [U U 2 1, \ 1 p 3 i 1 and it may be in error by D10% due to a Ðnite g D 2 (see 1 equaFig. 2 of Ostrowski & Schilckeiser 1996). Note that tion (12) is valid only for a constant mean free path ; for a momentum dependent mean free path, the form of T (v) should be reintegrated from the acceleration rate. In case of impulsive injection, Q(t) \ d(t [ t@), where t@ \ 0.3 R /V _ \s is the time of the injection, we have simply N (v) prec G [v, t@ ] T (v)]. Our treatment is not rigorous, and it prec not even take into account the fact that the particles does will have somewhat lower energies when precipitating than when leaving the shock. Nevertheless, it seems to give a very good Ðt to the spectrum of interacting particles at least in the considered case, where the spectrum is quite hard and the fraction of precipitating particles is still above 50% at energies above 1 MeV. In simulation 2 the injection is taken to be proportional to the linear density of the ambient plasma. At low energies, the mean injection time t@ ] L /V is much greater than the acceleration time T and twiceBas smuch as the start time of the injection. This means that the low-energy particles have much smaller probability to precipitate than in simulation 1 ; this is clearly seen in the simulated spectrum as a decrease of dN /dE at low energies. In addition, the precipi2 high-energy particles starts to be slow, as tation rate of the well, and we see a deÐcit of more than 20% of interacting particles at all energies. This shows that if a substantial fraction of particles is injected to the acceleration process at distances farther than D2L Ïs from the solar surface, the analytical model is no longerB valid for interacting particles. There is, however, no signiÐcant di†erence between the interplanetary spectra of the analytical model and the two simulations, which is also consistent with the model prediction. In simulation 3, we have increased the value of the mean free path by a factor of 5 relative to the value used in simulation 1. This has two e†ects : (i) the interplanetary spectrum is now considerably softer than the model prediction above E D 200 MeV, and (ii) the ratio of interplanetary to interacting protons is larger than in simulation 1. E†ect (i) is due to clearly non-negligible e†ects of focusing in equation (3) : streaming with velocity [i /L \ [0.925(v/c)U , giving a 1 EB\ 200 MeV, and1 adiabatic value of jv/(3U ) D 0.5L at 1 B deceleration at rate V /(3L ) B 0.11U /L , which is more B than one-third of the A1 acceleration rate1 atBhighest energies given by (U [ U )U /(jc) B 0.29U /L . E†ect (ii) is due to 1 2 1 1 B

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the acceleration time increased by factor of 5, which makes the probability of precipitation much smaller, especially at high energies where T now becomes dominant over t@. As can be judged from the given numerical solutions, the analytical model describes the interplanetary and interacting proton spectra rather well in all cases, where the mean free path is so small that particle acceleration to relativistic energies becomes possible. The only notable exception is the suppression of the interacting proton spectrum from the theoretical value, if a substantial fraction of the particle injection occurs at altitudes of a couple of L or B more above the solar surface. The analytical model is, however, quite adequate basis for the analysis of measured accelerated particle spectra.

3.3. Interplanetary and Interacting Particles as Probes of T urbulence In our analytical model, two parameters describing the turbulence in vicinity of the shock, g (p) and o , determine 1 both the spectrum of interplanetary protons andc the ratio of interplanetary to interacting protons (at least at high energies). This gives us a possibility to use the measurements of dN /dE and ! to deduce the turbulence param1 be done, e.g., by Ðtting. We note, however, eters. This could that it is possible to use a direct inversion method to obtain these parameters in case, when the interplanetary particle spectrum is close to a power law over momentum at high energies, i.e., vdN /dE P p~c1 above, say, p \ p and when 1 protons an average value1 of interplanetary to interacting ratio at high energies, ! , is also measured. In this case we = (9) to deduce the constant values may use equations (8) and of o and g at p º p : equation (8) gives c 1 1 z ]1 c \c = [2 1 z = and equation (9) combined with equation (5c) gives c ! \ , = z (c [ 3) = where z is the value of z(p) 4 exp (g ) [ 1 at p º p . Thus, = 1 1 c ] 2 ] 3! = c\ 1 1]! = and c ] 2 ] 3! =. z \ 1 = (c [ 1)! 1 = We have, thus, solved the value of c, which Ðxes the value of the scattering-center compression ratio through equation (5c). If the spectral shape of the interplanetary particles is di†erent from the power-law form at lower momenta, we may solve for z(p) as follows. Rearrange equation (8) in a form of di†erential equation for z(p). Multiply it Ðrst by vpc~2z(p), then take a logarithm of both sides and derivate with respect to p to get

C A

dN d dz 1 ln vpc~2 ] dE dp dp

BD

z\[

1 , p

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INTERPLANETARY AND INTERACTING PROTONS

1021

and Ðnally integrate from p to p to arrive at 1 p1 dN dN 1 1 dp ]c vpc~3 z(p) \ z vpc~2 = dE dE p/p1 p dN ~1 1 ] vpc~2 . (13) dE

C A

B

P

D

A

B

This holds at momenta p \ p , whatever the form of the 1 measured interplanetary spectrum is. We note also that the asymptotic form of the spectrum does not have to be exactly of the form proposed ; we may use also the values of ! and c measured at p \ p as long as o dc /d ln p o > c at 1 1 1 1 p\p . 1 Torsti et al. (1996) analyzed the interplanetary protons of the solar cosmic ray event on 1990 May 24. They concluded that the injection of protons into the interplanetary medium consisted of two components, Ðrst of which was released 10È40 minutes after the X-ray Ñare. The timescale of the this particle release is consistent with accelerationÈsite length scales of the order of solar radius. This Ñare was also a source of gamma rays and neutrons, which have been analyzed as well (Kocharov et al. 1994, 1996), and there were indications of a prolonged emission of neutrons that could be originating from shock-accelerated particles. These observational facts make the event acceptable for application of our model. The energy spectrum of the prompt-component interplanetary protons could be represented by

A

B C A

BD

E ~1.6 E 3 ~1 dN 1\N 1] , 0 160 MeV 360 MeV dE with N \ 1.4 ] 1030 protons MeV~1 in the range E \ 30È 0 assuming a solid angle of 2 sr for the size of the 1000 MeV acceleration region. The number of high-energy interacting protons was found to be equal to or several times less than the number of high-energy interplanetary protons. As we apply our model to these observations we take the proton momentum p \ J120m c corresponding to the p have c \ 5.0. We then kinetic energy of 10m 1c2, where we p adopt ! \ 1, which gives a value of c \1 5.0. This yields o \ 2.5,=a relatively small value when compared to the c theoretical value of about 5.5. As another case, we note that the theoretical compression ratio yields c \ 3.67 and ! \ 5, so it may marginally Ðt the observed data as well. =For these two cases, we then have z \ 2.5 and 1.1, respectively. The integral in equation (13) =performed numerically, we present the number of upstream di†usion lengths in Figure 3 and the inferred spectra of interacting protons in Figure 4 resulting for the two considered cases. Note, that the actual values should lie between the curves of these Ðgures, although at low energies the curves for dN /dE should be 2 taken as upper limits because of the mirroring e†ects discussed above. 3.4. A Brief Parameter Study Let us next try to estimate model parameters that could explain usual observations of interplanetary particle spectrum as well as interplanetary to interacting proton ratio. The above-adopted o \ 5.5 is actually an upper limit c MHD-turbulence component in because any compressible the upstream region can couple to fast MHD waves in the downstream region and these can only propagate away

FIG. 3.ÈNumber of upstream di†usion lengths as a function of particle energy deduced from the measured spectrum of interplanetary protons in the 1990 May 24 solar Ñare for the two limiting cases of the observed 1 [ ! [ 5. =

from the shock. Also a Ðnite value of the magnetic amplitude of the upstream waves will slightly reduce the scattering-center compression ratio from this ideal value (see Vainio & Schlickeiser 1999). Thus, we may estimate using equation (5c) connecting c and o that c º 3.7. In the c to o , we are left view of the analytical model, in addition c number of with one (momentum dependent) parameter, the di†usion lengths in the upstream region g (p) to determine 1 ratio of interthe interplanetary proton spectrum and the planetary to interacting protons. We must next ask, what values of these parameters are consistent with observations of interplanetary and interacting particle spectra. We note Ðrst that to obtain a single power law in momentum or energy in equation (8) for the energy spectrum of interplanetary particles at either nonrelativistic (v \ p/m, E \ p2/2m) or ultrarelativistic (v B c, E B pc) energies, g (p) must have a constant value. A broken power law can1 be produced, e.g., if 1/[eg1(p) [ 1] \ a(p/p )a/[1 ] (p/p )a], k where p is the momentum where the knee kof the spectrum k should be situated, and a and a are positive constants. This

FIG. 4.ÈMeasured interplanetary proton spectrum (thick curve) and the deduced spectra of the interacting protons (thin curves) in the 1990 May 24 solar Ñare for the two limiting cases of the observed 1 [ ! [ 5. =

1022

VAINIO, KOCHAROV, & LAITINEN

choice corresponds to g (p) \ ln M1 ] a~1[(p /p)a ] 1]N, so 1 k at p > p the number of di†usion lengths increases as ln p~a k as p decreases. In this case, the spectrum in equation (8) can be analytically integrated to give

A B

A

B

pa ] pa ac@a dN caN p c~2 pa k 0 1\ 0 inj . pa ] pa pa ] pa dE p v p k k inj Here, the nonrelativistic energy-spectral index is (c [ a [ 1)/2 below the knee and [c(1 ] a) [ 1]/2 above the knee. At ultrarelativistic energies these indices are c [ a [ 2 and c(1 ] a) [ 2, respectively. Since the value of a is controlling the rate of change of the spectral index, one cannot choose it to be very small if a broken power law is to be produced ; in fact, a \ 0 produces a single power law with nonrelativistic and ultrarelativistic spectral indices of [c(1 ] a/2) [ 1]/2 and c(1 ] a/2) [ 2. This requirement implies, if the natural condition of g (p) ] O as p ] 0 is 1 imposed, that the interplanetary spectrum at low energies should be very Ñat for broken power laws generated by parallel shock acceleration, if c close to its limiting value of 3.7. We note, Ðnally, that a law increasing faster than g D ln p~a as p tends to zero, e.g., a power law g D p~a, will 1not 1 but to a speclead to a broken power law energy spectrum, trum with a low-energy cuto†. Even these spectra, however, may well present themselves as broken power laws over a limited range in energy. As stated above, the choice of g (p) Ðxes also the interplanetary to interacting proton 1ratio. For the broken power-law example, it is given by

AB

p a o a c , p 1 ] (p/p )a k k which has the asymptotic value of !(O) \ o a at large energies ; at low energies, !(E) B o a(p/p )a cif we may k assume that all particles leaving thec acceleration region really precipitate. Otherwise the interacting particle spectrum will be suppressed at low energies and the above expression will be a lower limit estimation. For the single power-law case, a \ 0, we have ! \ o a/2, which should c also be taken as a lower limit at low energies. !(E) \

4.

DISCUSSION AND CONCLUSIONS

We have presented an analytical model to describe the interplanetary and interacting proton spectra resulting from parallel shock acceleration in solar corona. We have shown that a very simple analytical model of di†usive shock acceleration leading to equations (8) and (9) can be used to give the accelerated particle spectra if (i) the upstream di†usion length is much smaller than the scale length of the background Ðeld, j (p)v/(3U ) > L , at all particle speeds of 1 B interest ; (ii) the 1downstream di†usion length, the length of the downstream di†usion region and the background-Ðeld scale length are ordered as j (p)v/(3U ) > L > L ; and (iii) 2 2 B close to the injection of low-energy particles is2concentrated the Sun, within a couple of L Ïs. Since j > j in any reaB turbulence 2 and 1 since the sonable description of shocked waves propagate toward the downstream in the shock frame, we feel it is extremely unlikely that the condition j (p)v/(3U ) > L is violated. If either condition (iii) or 2 L 2 > L is2 violated, one may not be able to describe the 2 B interacting spectrum analytically, although the analytical interplanetary spectrum should still be valid. Note that con-

Vol. 528

dition (i) is much more stringent than condition j > L , 1 B which is commonly thought to be enough to justify the use of one-dimensional di†usion without the e†ects of focusing. Similar conclusion was drawn by Kocharov et al. (1999) ; in their stochastic acceleration model there was no bulk motion of the plasma at all, which led to an even stronger version of this rule : focusing could not be neglected at all, no matter how small the mean free path may be. We have compared our analytical model to a numerical one, which uses a Monte Carlo method to trace particles in a predescribed shock system under a law of scattering resulting to a di†usion-convectionÈlike motion of particles. A good agreement of the models is found, when the abovestated conditions for its validity hold. We were also able to give an analytical correction factor, equations (10) and (11), for the interacting spectrum in case, where the particle spectrum emitted by the shock toward the downstream region is not entirely precipitated due to mirroring between the shock and the solar surface. Our correction neglects (partly) the change in the particle energy caused by adiabatic deceleration of particles in the expanding region between the shock and the receding solar surface. In our simulated cases this was a minor error since the precipitation rate dominated over the deceleration rate making most of the particles precipitate with speeds close to their emission speeds. Care has to be taken, however, if particles emitted from outer corona are considered since then the rate of deceleration will become comparable to or even exceed the rate of precipitation, and the resulting shift in the spectrum has to be taken into account. Also, our treatment of the precipitation process was nonrelativistic, so if N (v) will be small at relativistic speeds we advice against its prec use. In principle, the theory of di†usive shock acceleration could also be used for oblique shocks. We, however, limited the use of the model for nearly parallel shocks since in the geometry we have in mind (Fig. 1), oblique shocks would naturally have more involved time dependence than we proposed in our model. Obliquity would also lead to more complicated description of the downstream turbulence with waves propagating o† axis, and with time-dependent largescale magnetic Ðelds that would lead to complications and invalidate our simple analytical solution. If a proper description of the magnetic Ðelds as a function of time and position may be found in such a case, our numerical model is quite capable with simple updates to solve for those equations, too. We used constant values of the upstream mean free path in our simulations and concluded that the mean free path has to be at most of the order of j D 5 ] 10~4 R if relativistic energies are to be produced efficiently. It_is interesting to compare this to the Larmor radius of the particles. In a Ðeld of 1 G, the Larmor radius of 1 GeV proton is R \ 8 ] 10~5 R . Since the Larmor radius represents the L limit of the _ lower mean free path (the so called Bohm limit) in parallel shock acceleration, at which the magnetic Ðeld should already be very disordered, acceleration to such high energies would probably have to take place in regions of high magnetic Ðeld of at least 1 G in order to be described by our model. If we apply this value at, say, one solar radius above the solar surface, it implies a Ðeld of 30 G at the base of the corona with the adopted value of L \ 0.3 R . Thus, our Alfven speed of V \ 3 ] 107 cm s~1B yields a _ coronalA 1010 cm~3 for the electron density base value of n D 4 ] suggesting that ethe adopted Alfven speed may probably be

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INTERPLANETARY AND INTERACTING PROTONS

somewhat low. A larger value implies a more important role for adiabatic deceleration and, if as a result the shock speed is increased, shortens the time available for the acceleration. If we apply a mean free path scenario with j \ 10R in the L magnetic Ðeld discussed above, we have j(p, f)/L \ 5 B ] 10~5 exp (3.33f/R )p/(GeV c~1). This has to be much _ smaller than 3U /v D 0.01c/v for di†usive shock acceler1 ation to work efficiently. We thus conclude that shock acceleration to relativistic energies, if it occurs, should take place relatively close to the Sun in very turbulent conditions to avoid problems with the e†ects of diverging Ðeld that tend to produce softer spectra than those observed. In our model, the ratio of interplanetary to interacting particles is very sensitive to the value of g . Very large 1 values of this parameter cannot be proposed without obtaining extremely small values of ! never observed in gradual Ñares. Large values of g would also lead to another 1 difficulty ; the timescale of the adiabatic deceleration in the upstream region would get smaller than the di†usive escape time of particles. Interplanetary mean free paths at 1 AU for E \ 10È100 MeV protons are often observed to be of the order of 0.1 AU. WKB theory predicts that the mean free path near the Sun should be a decreasing function of heliocentric distance for typical wave spectra, if Sun is the only source for turbulence (Jokipii 1973 ; Ng & Reames 1994) ; in contrast, we need values that are 5 orders of magnitude smaller than this. All these facts both point toward the interpretation that the upstream waves are self-generated by the accelerated particles through the streaming insta-

1023

bility : even if an external source could have produced the waves necessary for the intense scattering, it would be impossible for the accelerated to ever escape upstream to be detected near the Earth. To answer, whether it is possible for the particles to generate the waves self-consistently, we need a model that is time dependent and can address also the basic question of injection of particles to the acceleration process. This is beyond any di†usion theory and needs numerical kinetic modeling. Our aim is to study this possibility in future using a Monte Carlo model that keeps track of the energy density in waves in addition to an ion population starting from thermal energies. The sensitivity of the model to the value of g gives also a 1 good possibility to test it experimentally : we have o†ered our model as a natural way to explain the broken powerlaw spectra of the interplanetary ions observed in some SEP events (e.g., Torsti et al. 1996). In this case, the change in the spectral slope should also be seen as a change in the interplanetary to interacting proton ratio at the same energies so it should be, at least in principle, detectable even with the currently available gamma-ray and particle detectors. L. K. gratefully acknowledges a discussion with Jim Ryan in early 1996, when an idea to calculate spectrum of shock accelerated interacting protons was proposed. R. V. thanks Reinhard Schlickeiser for fruitful discussions about shock acceleration in low beta plasmas. The Academy of Finland is thanked for Ðnancial support.

APPENDIX KINETIC EQUATION AND MONTE CARLO SIMULATIONS To obtain numerical solutions to equations (1a) and (3), we have used a Monte Carlo simulation code developed for the analysis of energetic particle transport and acceleration in the inner heliosphere (Vainio 1998). The model is a kinetic one. It follows individual particles in a predescribed system of large-scale magnetic Ðelds and superposed waves that scatter the particles in pitch angle. Particle speed and magnetic moment are constants of motion in the large-scale Ðeld when the velocity is measured in the frame where the large-scale magnetic Ðeld is static. We call this frame the Ðxed frame. Scatterings, on the other hand, conserve the particle speed in the frame, where the magnetic Ðeld of the (nondispersive) waves is static. This is the frame moving with the phase speed V of the waves relative to the Ðxed frame, and we call it the wave frame. Let us assume, for simplicity, that for the considered Ñux tube the phase speed of the waves is directed along the mean magnetic Ðeld. We will study how the momentum and pitch angle of the particles change due to the adiabatic motion in the large-scale Ðeld when they are measured in the wave frame. We do not require this phase speed to be constant but allow it to change spatially and temporally. In what follows, we shall assume that terms of the order (V /c)2 may be neglected. In the Ðxed frame, we may write k5 \ (1 [ k2)v/(2L ) and p5 \ 0. Thus, a particle moving from event A (t , f ) to B (t , f ) along the magnetic Ðeld will su†er a B A A B B change in its wave-frame parallel momentum p@ \ p@k@ \ p [ pV (f)/v of A A *p@ \ p@ [ p@ \ p(k [ k ) [ (p/v)(V [ V ) , A AB AA B A B A which gives, when divided by *t \ *f/(vk), a rate of change for the wave-frame parallel momentum as seen from the Ðxed frame,

A

LV 1 LV dp@ p [ p5 @ \ A \ pk5 [ V0 \ p k5 [ k A Lf v Lt dt v

B

.

Since p \ p@ , we have M M 2p@p5 @ \

d d d p@2 \ p@2 ] p@2 \ 2p@ p5 @ [ 2kp2k5 , A A dt dt A dt M

1024

VAINIO, KOCHAROV, & LAITINEN

which yields a rate of acceleration/deceleration of p5 @ \

A

LV 1 LV p@k@ [ kp ] pk5 [ k@p k Lf v Lt p@

A A

Vol. 528

B

B

p2 V LV p LV \[ M [ k@ pk ] p@ 2L Lf v Lt B p@ p@2 V p@V LV k@v@V LV [ k@ \[ M [ k@ p@k@ ] 1] v@ Lt p@ 2L v@ Lf c2 B 1 [ k@2 V LV LV k@ LV LV V ] k@2 ] ] ]V . (A1) \[p@ 2L c2 Lt Lf v@ Lt Lf B In addition to the time derivatives, this expression di†ers from the deceleration rate given by Ru†olo (1995) in his equation (7) by the last term. This is because in Ru†oloÏs model, the increase of V (that was the solar wind speed in his paper) with f came entirely from the centrifugal acceleration of the medium in the Ðxed frame corotating with the Sun. In this case, one should also take into account the centrifugal force in the deceleration rate of the particles, which cancels the last term in equation (A1) exactly. We, however, consider phase speeds that have much larger gradients than those predicted by corotation (which are completely negligible at coronal distances from the Sun), so in our case the centrifugal force may be neglected and equation (A1) should be used with the last term included. Similarly, we may write for the rate of change in the wave-frame pitch-angle cosine

C

B

A

C A v@ 2L

1]

B

B A

p5 @ p5 @ k5@ \ A [ k@ p@ p@ \(1 [ k@2)

A

B

BD

A

k@V k@v@V LV 1 k@V [ [ k@ [ ] v@ c2 Lf v@ c2

B

LV V LV [ Lt v@ Lf

D

,

(A2)

B which also di†ers by the inclusion of the last-line terms from the expression (10) of Ru†olo (1995) for the same reason as the deceleration rate discussed above. Finally, particle speed along the magnetic Ðeld is given by

A

B

k@v@V ]V . f5 \ kv \ k@v@ 1 [ c2

(A3)

We may now write down a Fokker-Planck equation for the number of particles per unit length of the magnetic Ðeld, unit of pitch-angle cosine, and unit momentum, n(f, p@, k@, t) \ d3N/(dfdp@dk@), as

C A A B

v@ Ln L L \ [ (k@v@ ] V )n [ (1 [ k@2) 2L Lt Lf Lk@

C

1]

B A D

B

D

1 LV k@V V LV [ [ k@ ] n v@ v@ Lf v@ Lt

B 1 [ k@2 V LV k@ LV L Ln L V ] k@ k@ ] ] n] (1 [ k@2)l ] s(f, p@, k@, t) , (A4) ] p@ v@ Lf Lk@ Lk@ 2L v@ Lt Lp@ B where time and distance are measured in the Ðxed frame and momentum and pitch angle in the wave frame, and where we have neglected all terms of the order (V /c) when applying equations (A1)È(A3). The second to last term describes the e†ect of scatterings, and it involves the scattering frequency l(f, p@, k@, t), while the last term is the source function, now written with dependence on k@ allowed. To our knowledge, equation (A4) is new in the sense that it allows for time dependence in the wave speed. Note, however, that time dependence will only be important in cases, where particle anisotropies are large, e.g., when particles of low speeds are considered. Our simulations e†ectively solve equation (A4). Equation (1a) for f \ d2N/(dfdp) results from equation (A4) through the use of well-known di†usion approximation scheme, where scattering is assumed to be intense enough to keep the particle distribution close to isotropic. We shall not reproduce the rigorous derivation here but refer the reader to Webb & Gleeson (1979) for a detailed description. To keep our simulations computationally efficient, we have made a couple of simpliÐcations : we assume nonrelativistic phase speeds for the waves in the sense that we use a Galilean transformation for time, t \ t@, between the Ðxed frame and the wave frame. This choice is also consistent with equation (A4), which assumed the relativistic e†ects due to the wave speed to be negligible. We also choose to work with the simplest form of isotropic small-angle scattering, where the scattering frequency is not allowed to depend on pitch angle. In this case, the spatial di†usion coefficient and the scattering frequency have a simple connection, i \ j(p)v/3 \ v2/[6l(p)]. For isotropic scattering, we have developed an accurate numerical method that e†ectively solves the pitch-angle di†usion part of the equation (A4). We consider the scattering process relative to the direction of the unperturbed particle motion in the wave frame. Relative to this rotating scattering axis, we may write an equation for the angular distribution g@ of particles of constant speed in the wave frame, Lg@ l L Lg@ l L Lg@ \ sin Ë B Ë , Lt sin Ë LË LË Ë LË LË

No. 2, 2000

INTERPLANETARY AND INTERACTING PROTONS

1025

with the initial condition g@(Ë, 0) \ d(Ë), where Ë ½ [0, n] is the angle between the unperturbed and the scattered velocity vectors. If small times compared to the inverse scattering frequency are considered, one may use the small-angle approximation sin Ë B Ë. Then, the number of particles in a di†erential solid angle d) is g@(Ë, t)d) \

A B

Ë2 1 Ë dË dr , exp [ 4lt 4nlt

(A5)

where r ½ [0, 2n) is the angle measured around the scattering axis. Making use of the above, our simulations thus work with the following scheme : Ðrst, the particle is injected at the shock so that it propagates into the upstream region. A time step is chosen as a small fraction of the inverse scattering time, *t \ al~1 with a D 0.01. The particle is then moved in the Ðxed frame according to f ^ f ] kv *t and k ^ k ] (1 [ k2)v *t/(2L ) keeping the Ðxed-frame particle speed constant. After this, a scattering is performed : Ðrst, the velocity vector of the particle Bis Lorentz-transformed to the wave frame. Then, in accordance with equation (A5), Ë2 and r are picked via a random generator from exponential and uniform distributions, respectively, and the new wave-frame pitch-angle cosine is computed by the use of spherical trigonometry, k@ ^ k@ cos Ë ] (1 [ k@2)1@2 sin Ë cos r. The new velocity vector is then transformed back to the Ðxed frame. Such time steps are taken until the particle either escapes at the upstream boundary or it hits the shock again. If it escapes, the particle is removed from the simulation, the interplanetary spectrum is updated, and a new particle is injected at the shock. If the particle hits the shock, it is transmitted into the downstream region and followed there accordingly until it either returns to the upstream region or goes through a boundary at a distance of 2j downstream of the shock, where we decide its 2 Ellison et al. 1990). This formula is valid if fate through its probability of return given by P \ (v@ [ U )2/(v@ ] U )2 (e.g., ret 2 2 j ] 0, which we have assumed in our simulations. It also implies that if the particle returns, it does so immediately, so no 2 spent in the downstream region needs to be taken into account. The returning particles are reinjected into the simulation time at the 2j distance downstream of the shock so that their pitch-angle cosines are picked from isotropic distributions measured 2 in the downstream wave frame and then Ñux-weighted with the shock-frame cosine factor. They are followed until they escape either through the upstream boundary or by failing to return when they hit the downstream 2j distance again ; these particles 2 solar surface into the region get convected away from the shock to the far downstream region and will be injected toward the with no turbulence. There they are followed until they precipitate or cool down to speeds v \ 2V . Particles mirroring and catching up with the outward propagating downstream turbulence of the shock are reinjected backs toward the solar surface. This back-scattering is done elastically in the downstream wave frame, which leads to the cooling as described in ° 3. A precipitating particle is removed from the simulation, the spectrum of interacting particles is updated, and a new particle is injected at the shock. REFERENCES Axford, W. I., Leer, E., & Skadron, G. 1977, Proc. 15th Internat. CosmicKrymsky, G. F. 1977, Dokl. Akad. Nauk. SSSR, 243, 1306 Ray Conf. 11 (Plovdiv), 132 Lee, M. A., & Fisk, L. A. 1982, Space Sci. Rev., 32, 205 Bell, A. R. 1978, MNRAS, 182, 147 Lee, M. A., & Ryan, J. M. 1986, ApJ, 303, 829 Blandford, R. D., & Ostriker, J. P. 1978, ApJ, 221, L29 Ng, C. K., & Reames, D. V. 1994, ApJ, 424, 1032 Ellison, D. C., Jones, F. C., & Reynolds, S. P. 1990, ApJ, 360, 702 Ostrowski, M., & Shlickeiser, R. 1996, Sol. Phys., 167, 381 Ellison, D. C., & Ramaty, R. 1985, ApJ, 298, 400 Ramaty, Mandzhavidze, N., Kozlovsky, B., & Skibo, J. G. 1993, Adv. Jokipii, J. R. 1973, ApJ, 182, 585 Space Res., 13, (9)275 Kahler, S. 1993, J. Geophys. Res., 98, 5607 Reames, D. V. 1993, Adv. Space Res., 13, (9)331 ÈÈÈ. 1994, ApJ, 428, 837 Ru†olo, D. 1995, ApJ, 442, 861 Kocharov, L. G., Lee, J. W., Zirin, H., Kovaltsov, G. A., Usoskin, I. G., Torsti, J., Kocharov, L. G., Vainio, R., Anttila, A., & Kovaltsov, G. A. 1996, Pyle, K. R., Shea, M. A., & Smart, D. F. 1994, Sol. Phys., 155, 149 Sol. Phys., 166, 135 Kocharov, L. G., Torsti, J., Vainio, R., Kovaltsov, G. A., & Usoskin, I. G. Vainio, R. 1998, Ph.D. thesis, Univ. Turku 1996, Sol. Phys., 169, 181 Vainio, R., & Schlickeiser, R. 1998, A&A, 331, 793 Kocharov, L., Kovaltsov, G. A., Laitinen, T., MaŽkelaŽ, P., & Torsti, J. 1999, ÈÈÈ. 1999, A&A, 343, 303 ApJ, 521, 898 Webb, G. M., & Gleeson, L. J. 1979, Ap&SS, 60, 335