Miyake, S

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29th International Cosmic Ray Conference Pune (2005) 2, 203206Effects of the tilted and wavy current sheet on the solar modulation ofgalactic cosmic raysShoko Miyake and Shohei YanagitaFaculty of Science, Ibaraki University, Mito 310-8512, JapanPresenter: Shoko Miyake (miyakesk@cc.nao.ac.jp), jap-miyake-S-abs1-sh34-posterTransport equation of the galactic cosmic ray (GCR) is numerically solved for   and   basedon the stochastic differential equation (SDE) method. We have developed a fully time-dependent and three-dimensional code adapted for the wavy heliospheric current sheet (HCS). Results anticipated by the drift patternare obtained for sample trajectories and distributions of arrival points at the heliospheric boundary for GCRprotons. Our simulation reproduced a 22-year cycle of solar modulation which is qualitatively consistent withobservations. Energy spectra of protons at 1 AU are calculated and compared with the observation by BESS.1. IntroductionTransport of the galactic cosmic rays in the heliosphere is described by Parker’s transport equation. TheParker’s equation is equivalent to a coupled stochastic differential equation (SDE) [1, 2]. This SDE methodallows us to get some information about solar modulation phenomena of galactic cosmic ray (GCR) not ob-tained by other numerical methods, such as distributions of arrival time, energy lost, and trajectory. We havedeveloped a fully time-dependent and three-dimensional code based on the SDE method. We present somenumerical results on solar modulation effects on GCR obtained by this new code.2. The ModelThe SDE equivalent to Parker’s transport equation is written as [2]	  ﬀﬁﬃﬂ "!$#ﬀ%#'&#(ﬂ)+*,  -.,"/0123ﬂ4*(1)where	and,are the position and the momentum of the particle,is the solar wind velocity,is thegradient-curvature drift velocity,is the diffusion coefficient tensor,!$#ﬀ576#598#;:3<=6>8, and&#is a Wienerprocess given by the Gaussian distribution. We adopted ?A@ﬃ km/s,<CBDFEﬃGIH7JKE3L-NMPORQ2SETVU?QWXN0ZY[U>QY\]X^LQ>_,<ﬀ`$GH3< B,Yis the magnetic field, andYaUbcH>d=eisYat the earth. We assume the structure ofheliospheric magnetic field (HMF) as standard Parker spiral. We have used the “drift velocity field method”[3] for the calculation of drift in the heliospheric current sheet (HCS). In our simulation, particles start at 1 AUon the equatorial plane and run backward in time until they exit the heliospheric boundary, 80 AU.We assume the structure of the HCS is asfhg:_Nijd7k-\lmmmmm_)ijdnog:gﬂprqkqtsuEﬃEwvﬃxy>z{_}|mmmmm_Ni~d==Kpﬂ9Ł2? K*(2)where r,f,are spherical polar coordinates relative to the solar rotation axis,9is a constant,is the radiusof the Sun, andis the rotational angular velocity of the Sun. Similar expression for HCS had been adopted204 Shoko Miyake and Shohei YanagitaFigure 1. (a) A view of theHCS projected onto a merid-ional plane as a function ofradial distance from the Sun.The tilt angle near the Sunis { , and the solar activ-ity heading for the maximum;(b) Same as (a), but the solaractivity heading for the mini-mum;(c) Same as (b), but thetilt angle near the Sun is w  .by Jokipii and Thomas [4], but with constant tilt angles contrary to our model. In this simple HCS model,the tilt angle changes continuously from  3 to ﬃ to  > over a period of 11 year. The solar polarity of theHMF reverses when the tilt angle of HCS near the sun reaches   . Figure 1 shows views of HCS projectedonto a meridional plane for three phases in the solar cycle. We have not taken into account in our model suchphenomena as the global merged interaction regions, and the dynamically changing HCS with Fisk-like field[5].3. ResultsFigure 2. Sample trajectory for two protons.The tilt angle near the Sun is   , and the solaractivity heading for the maximum. The energyat 1 AU on the equatorial plane is 1 GeV.Figure 2 shows a sample trajectory for two protons which propa-gate from a point at 1 AU on the equatorial plane to 10 AU. Thecoordinate system of the figure is corotating with the sun. Thedrift pattern of the GCR depends on the polarity   , where   is acharge sign of the sample particle and  is a constant character-izing HMF. For the positive polarity   , the particles driftfrom the polar region towards the HCS, and outward along theHCS. In contrast, for the negative polarity  c$ , the directionof the drift is opposite to the case for   .Figure 3 shows the heliolatitude-longitude distribution of simu-lated particles when they exit the heliospheric boundary, wherethe longitude is shown in the corotating coordinate system. Thethick line indicates the HCS at 1 AU when the particles start offfrom a point (indicated by “star”) at 1 AU on the equatorial plane.Latitudes of the arrival points for  c distribute around theHCS as expected, because particles arrive at 1 AU along the HCS. The thickness of the distribution of latitude( Ł ﬃ ) reflects the structure of spatiotemporal variation of the HCS. The structure of the HCS seen by particlesis different from particle to particle depending on the arrival time. On the contrary, the latitude for   dis-tribute around the polar region, either northern (a) or southern (b) depending on whether the observer is locatedat a point northern (a) the HCS or southern (b), respectively. This tendency is also as expected, because it isdifficult for positive particles to penetrate through the HCS during the journey from the outer heliosphere. Themean arrival time of 1 GeV protons is 4.5 days and 80 days for  r and  r , respectively. The shortEffects of the tilted and wavy HCS on the solar modulation of GCRs 205-80-60-40-20020406080-100 0 100lmqA>0qA<0H(a)Longitude [deg]Heliolatitude [deg]-80-60-40-20020406080-100 0 100lmqA>0qA<0H(b)Longitude [deg]Heliolatitude [deg]Figure 3. (a) Heliolatitude-longitude distribution by simulation for protons with energy 1 GeV at 1 AU when they exit theheliospheric boundary, 80 AU. The observer at 1 AU on the equatorial plane is located at a point northern the HCS; (b)Same as (a), but the observer is located at a point southern the HCS. The tilt angle near the sun is  , and the solar activityheading for the maximum.time of only 4.5 days for   suggests that particles are less modulated. This relatively little modulationeffect is reflected in the fail to reproduce the observed spectrum for  " as we mention later.Figure 4a shows simulated variation of the proton intensities over the 22-year modulation cycle. The solid lineindicates the result for the model of the HCS by eq.2 ( (*)ﬂ)model). The dashed line shows the result forthe HCS model ( (ﬂ)model) where the tilt angle varies with time but does not depend on radius from the sun.In our *)ﬂ)model, the variation of the HMF near the sun propagates with the solar wind velocity. It takesabout 1 year for the signal of polarity change and change in the tilt angle reach the heliospheric boundary. In(ﬂ)model, on the contrary, these changes reach instantaneously the boundary. This difference between thetwo models in signal propagation is reflected in a time lag of the proton intensity changes by *)ﬂ)model asseen in Figure 4a. Our model succeeded in reproducing a 22-year modulation cycle and in reproducing, thoughqualitatively, rather flat maximum for   and a sharp maximum for   [6].Figure 4b shows proton energy spectra by our simulation (solid line) and the spectra observed by BESS ex-periment (open symbol). The open triangles refer to proton fluxes observed in 1998 [7] when the tilt anglenear the sun is aboutEﬃ and the solar polarity is positive. The open squares refer to proton fluxes observedin 2000 [8] when the tilt angle near the sun is about    and the solar polarity is negative. The symbols forsimulated energy spectra refer to the phases marked by the same symbol in Figure 4a. The view of the HCScorresponding to these phases are shown in Figure 1a and 1c. The local interstellar spectrum (LIS) of proton isassumed as ¡¢£M(¤GIH¥¦WLkLX§ ¨ , where¤is kinetic energy and¥is the mass of proton. The simulatedenergy spectra are normalized at 3.6 GeV to the observation in 2000 [8]. Our model spectrum for  ©hagrees quite well with BESS result, however our model fails to reproduce the observed spectrum for  c$ .The failure of our model for  ªA may come from inadequacies of the heliospheric structure in our model.As we mentioned before, positive particles drift from polar region within short times modulated rather weakly.If we take into account the existence of the random transverse component of the HMF in the polar regions [9],we may be able to reproduce the observed spectrum because it is expected that the random transverse compo-206 Shoko Miyake and Shohei Yanagita10-210-110 5 10 15 20α(r,t) Modelα(t) ModelSolar PolarityPositiveSolar PolarityNegative↓ ↓(a)Time [years]Differential Flux(arbitrary)10 210 310 410-11 10LIS (80AU)qA>0, α=10degqA<0, α=70deg (b)Kinetic Energy [GeV]Differential Flux [m-2s-1sr-1GeV-1]Figure 4. (a) Simulated variation of the 1 GeV proton intensities over the 22-year modulation cycle; (b) Simulated protonenergy spectra and observed proton energy spectra in 1998 and 2000.nent suppresses drifts in the polar regions. The actual variation of the HCS is much irregular in contrast to ourmodel. We have to consider the actual dynamically changing HCS.4. ConclusionsWe have developed a time-dependent and three-dimensional code based on the SDE method. The sampletrajectories and distributions of arrival points at the heliospheric boundary agree with the physical expectation.Our simulation reproduced a 22-year modulation cycle which is qualitatively consistent with observations.Our model spectrum for  «A agrees with BESS result, however our model fails to reproduce the observedspectrum for   .5. AcknowledgementsWe would like to thank Prof. J. Kota and Prof. P. Evenson for their comments and suggestions on this work.References[1] Yamada Y., S. Yanagita, and T. Yoshida, Geophys. Res. Lett., 25, 2353-2356, 1998.[2] Zhang M., Astrophys. J., 513, 409-420, 1999.[3] Burger R. A. and M. S. Potgieter, Astrophys. J., 339, 501-511, 1989.[4] Jokipii J. R. and B. Thomas, Astrophys. J., 243, 1115-1122, 1981.[5] Kota J. and J. R. Jokipii, 28th ICRC (Tsukuba), 3791-3794, 2003.[6] Webber W. R. and J. A. Lockwood, J. Geophys. Res., 93, 8735-8740, 1988.[7] Maeno T. et al., Astropart. Phys., 16, 121-128, 2001.[8] Asaoka Y. et al., Phys. Rev. Lett., 88, 051101, 2002.[9] Jokipii J. R. and J. Kota, Geophys. Res. Lett., 16, 1-4, 1989.

Effects of the tilted and wavy current sheet on the solar modulation of galactic cosmic rays

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