R,(T R=Ooe/i* kte).e (6) E KT and,forx,o=3.6×1039m2,E1=eV,V=12.1V From(4), two first integrals ;-I。=IA= constant (8) T+rn=rm=constant (9) Conservation a 2. Ion Momentum Equation The force per unit volume on the ion gas is eEn(from the axial electric field E) There is also a"pick-up"drag due to ionization For each ionization event, a new ion is"incorporated"to the ion population(of velocity vi), jumping from the neutral velocity vn; this gives a drag-ne onm(v, -vn). We then have mv, dv=eE-mvion (v, -vn) (11) 3. Electron Momentum Equation Consider first only classical electron collisions(say, with neutrals). The vector equation of motion of electrons, including electric force, magnetic force and collisional"drag"(and neglecting inertia)is 16.522, Space P pessan Lecture 18 Prof. Manuel martinez Page 6 of 20
16.522, Space Propulsion Lecture 18 Prof. Manuel Martinez-Sanchez Page 6 of 20 Where νion n i e =nR T( ) (5) i e E - e kT e i 0 i kT R = c 1+2 e E ⎛ ⎞ σ ⎜ ⎟ ⎝ ⎠ (6) e e i 8 kT c = π m (7) and, for Xe, -20 2 σ0 i ii 3 .6 × 10 m , E = eV , V = 12 .1V -25 m = 2.2 × 10 kg i From (4), two first integrals : ie d ΓΓ Γ - = = constant (8) ΓΓ Γ in m + = = constant (9) Conservation Equations and a d m i I m = , = Ae Am Γ Γ i (10) 2. Ion Momentum Equation The force per unit volume on the ion gas is eEne (from the axial electric field E). There is also a “pick-up” drag due to ionization. For each ionization event, a new ion is “incorporated” to the ion population (of velocity vi ), jumping from the neutral velocity vn ; this gives a drag-n m v - v e ion i i n ν ( ) . We then have ( ) i i i i ion i n dv m v = eE - m v - v dx ν (11) 3. Electron Momentum Equation Consider first only classical electron collisions (say, with neutrals). The vector equation of motion of electrons, including electric force, magnetic force and collisional “drag” (and neglecting inertia) is
ⅴPQ=enE+vxB)- meVenneve (12) nn ce Project on x, y: dx =en(E+v B)-m v v (13) E=0, 0=-en(0-Vex B)-mevennev (14) From(14), Substitute in(13) ene-enb-cv-mv. nv en, - Ve In the Hall thruster plasma, ven <<o(low collisionality), so second term in parenthesis is neglected. The quantity -acts then as an effective collision frequency accounting for the magnetic effect 16.522, Space P pessan Lecture 18 Prof. Manuel martinez Page 7 of 20
16.522, Space Propulsion Lecture 18 Prof. Manuel Martinez-Sanchez Page 7 of 20 ( ) e e ∇ ν P = -en E + v ×B - m n v e e e en e G JG JG JG (12) ( ) e ν σ en n en =n c Project on x, y: ( ) e e x ey e en e ex dP = -en E + v B - m n v dx ν (13) e y P E = 0, 0 x ⎛ ⎞ ∂ ⎜ ⎟ = ⎝ ⎠ ∂ 0 = -en 0 - v B - m n v e ex e en e e ( ) y ν (14) From (14), c ey ex ex e en en eB v= v= v m ω ν ν (15) Substitute in (13): e c e x e ex e en e ex en dP = -en E - en B v - m n v dx ω ν ν 2 c e x e e ex en en = -en E - m n v + ⎛ ⎞ ω ⎜ ⎟ ν ν⎝ ⎠ (16) In the Hall thruster plasma, ν ω en c << (low collisionality), so second term in parenthesis is neglected. The quantity 2 c en ω ν acts then as an “effective collision frequency”, accounting for the magnetic effect 2 c e en " "= ω ν ν (17) y z B x vey vex