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402 Chapter 8 These approximations are quite good if the axial velocity is much greater than both the initial radial velocities and the ion sound speed. With the assumption that the ion beam profile starts out and remains a Gaussian profile, the set of equations can be solved analytically. The beam profile is written as (r)= o exp r2 , m h ( t ) 2 2 R 2 h ( t ) 2 where the initial ion beam mass density, o , is o = M Ib , (8.3-10) parameter R is chosen to best represent initial beam width, and the initial value of the expansion parameter h(0) is unity. The density spreads out as the beam moves axially, but the beam profile remains Gaussian, as shown in Fig. 8-8. An ion that starts out at a radial position ro will move radially outward proportionallyto h(t): r(ro ,t) = ro h(t). (8.3-12) This implies that the radial velocity, vr , is proportional to the time derivative of h(t): v r ( r , t ) = r o h ( t ) . ( 8 . 3 - 1 3 ) Equations (8.3-12) and (8.3-13) can be combined to obtain an expression for the local velocity that doesn’t use the initial radial position: vr (r,t) = r h ̇(t) . (8.3-14) h(t) The solution obtained below is valid for a beam with no initial radial velocity, or for an initial radial velocity distribution that is proportional to the radius: vr(r,0)=vr0 r. (8.3-15) (8.3-11) and the function h(t) describes how the beam expands radially. The 2 vb R2PDF Image | Fundamentals of Electric Propulsion: Ion and Hall Thrusters
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