Büsching, I

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29th International Cosmic Ray Conference Pune (2005) 3, 205208A method to calculate the Galactic cosmic ray density with high spatialresolution, new developmentsI. Bu¨sching and M. S. PotgieterSchool of Physics, Potchefstroom Campus, North-West University, Venter Street, Potchefstroom 2520, South AfricaPresenter: I. Bu¨sching (fskib@puk.ac.za), saf-buesching-I-abs2-og13-posterThe recent discovery of direct evidences for the acceleration of high energetic particles at the shell supernovaremnant RXJ1713.7-3946 underlined the need to calculate the cosmic ray (CR) distribution in the Galaxy on aspatial grid fine enough to resolve the changes in the CR density due to these kind of objects. This is a progressreport on the further development of a method to solve the time dependent CR propagation equation with highspatial resolution for arbitrary source distributions.1. IntroductionThe recent discovery of direct evidences for the acceleration of high energetic particles at the shell supernovaremnant RXJ1713.7-3946 [1] underlined the need to calculate the cosmic ray (CR) distribution in the Galaxyon a spatial grid fine enough to resolve the changes in the CR density due to these kind of objects, what resultsin a huge numerical grid.As described in [2, 3], we developed a method to solve the time-dependent CR propagation equation:  	        ﬀﬁ(1)for the CR densitywith high spatial and temporal resolution avoiding the need of a huge numerical grid.Eq. (1) takes into account diffusion and catastrophic losses, which is suitable for CR nuclei with ﬂﬃ .!	"$#&%('*)is the spatial diffusion coefficient, ﬁ+	+,(-/.10325476987:5; 476 describes the rate of catastrophic lossesand <	=>#''''?%)represents the sources. These quantities also depend on particle momentum % whichis treated as a a free parameter. For our calculations, the Galactic disc is assumed to be filled with gas, witha distribution depending ononly. The disk also contains the sources. The Galactic disc is embedded in theGalactic halo, a region where particles may still be confined in the system. The diffusive volume is treatedas a cylinder with radius @ and height  5A . Free escape boundary conditions are imposed at the boundaries.Cylindrical coordinates are used, which is suggested by the geometry of the Galaxy.Under the condition that neither the diffusion coefficient  nor the loss term ﬁ depend on galactocentric radiusor azimuth, we apply a series ansatz for the CR density:	BCDCFE#?GD5HIJLK#?,M)OND5HKQPSR#?,M)Q)TD#VUDWH5)#TXD#YUDWH@)Q) (2)UD5His the Z th the solutions of TD#?U@)	\[ , and TDthe Bessel function of order , which transforms the three-dimensional spatial problem into a system of one-dimensional equations for the coefficients of this expansion,which then can be solved efficiently with standard numerical methods. The resolution in,only depends onthe number of coefficients in Eq. (2) and is limited only by the available CPU time.206 I. B¤usching and M. S. Potgieter2. New DevelopmentsDuring the last year, the method was improved significantly. The calculation time has been reduced drasticallyby changing the numerical code from a DuFort Frankel to a Crank Nicholson scheme [4], which allowed toincrease the time step by a factor of 100 and also proved to be numerically stable for a much larger parameterspace.On the technical side, we optimized the code for computer clusters by implementing the MPI framework [6].Alternatively the already implemented lock-file mechanism can be chosen, which is suitable for inhomoge-neous computer grids.3. Extending the methodEq. (1) describes the propagation of CR nuclei heavier than helium. Considering also electrons, protons andlight nuclei, continuous losses have to be taken into account. In this new model, we further include effects ofthe Galactic wind and reacceleration. So we have instead of Eq. (1), using cylindrical coordinates   	    ]  ^       >  _#Y`bac def)  %g  %  %!hjikmlﬁ(3)where `bac de is the speed of the Galactic wind,ikdescribes continuous losses.gis the diffusion coefficient fordiffusion in momentum space. Again, we can assume that `ac de ,ik,gare independent of the galactocentricradiusand azimuthso we apply ansatz Eq. (2) to Eq. (3), which leaves us with equations for the expansioncoefficients GD5H  	 MnDWHbUD5HGDWH   GD5H   o#Y`pac deqGDWH)  %g GD5H %  %hrikGD5HlﬁsGD5H(4)where ut n$vDWHis given bytn$vD5H	xw/yzw|{z}#'''V%('Q)I~JK#,M)TD#VUDWH5)bLL(5)and similar equations forND5H. Note that GD5H,NDWHnow depend on,and % where the latter is no more afree parameter. Eq. (4) are solved with a ADI scheme [5], which is in the process of testing.4. SummaryWe presented a method that solves CR-propagation type PDEs for cylindrical geometries. The code is capableof dealing with all three spatial dimensions and time, where the solution may be obtained with high resolu-tion (easy adjustable) in space and time. The change to the Crank Nicholson scheme drastically reduced thecomputation time. The method allows distributed computation and is usable on PC-style hardware or clusters(using MPI). We are currently expanding this method to also be usable for CR electrons and protons.A method to calculate the Galactic cosmic ray density with high spatial resolution 2075. AcknowledgementsWe thank the National Research Foundation for partial financial support, GUN number 2049412.References[1] F.A. Aharonian et al., Nature 432, 75 (2004)[2] I. Bu¨sching et al., ApJ 619, 314 (2005)[3] I. Bu¨sching et al., Proc. 28 ? ICRC, 1981 (2003)[4] J. Crank, The mathematics of diffusion, Clarendon, Oxford (1956)[5] J.Jr. Douglas and J.E. Gunn, Numer. Math. 6, 428 (1964)[6] W. Gropp et al., Parallel Computing 22, 789 (1996)208 I. B¤usching and M. S. Potgieter

A method to calculate the Galactic cosmic ray density with high spatial resolution, new developments

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