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A self similar solution of expanding cylindrical flux ropes for any polytropic index value

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Figure

Fig. 1. Profiles of quantities in the r direction on the z = 0 plane obtained at an arbitrary time
Fig. 2.Relation between the temperature and density while a virtualspacecraft moves across an evolving flux rope for 20 hours
Fig. 3. New parameter z1 is the axial distance from the center of the flux rope to the location of the spacecraft encounter and z2 is the one from the centerto the location where the spacecraft detached from the flux rope.
Fig. 4.Flux rope observed by the WIND spacecraft on Feb. 8, 1995. nis the proton number density, and vth is the thermal velocity of protons.The GSE coordinates are used

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