times(evi be the effective ionization energy per atom, and a, the degree of ionization at the exit we then have (26) from(25) m. 4H HoI 9(W (27) This indicates very rapid increase of the ionization fraction as the current increases or as the flow is reduced Instabilityonset For a given thruster, as I/m is increased a increases rapidly when it reaches unity the behavior of the plasma near the exit changes drastically. This is because any extra dissipation cannot be absorbed into ionization anymore, and goes instead directly into heating the plasma (or perhaps the electron component only ). This causes conductivity to increase whenever the current concentrates, which leads to further current concentration We have here the classical prescription for constriction into an arc, and one can expect heavy arcing(with the corresponding damage to electrodes) when a, approaches 1. This behavior has indeed been observed repeatedly and has been the focus of a lot of attention because it limits the practically achievable value of I2/m. Since(as we will see)efficiency increases with I /m, this is a major hurdle in the path towards efficient mPD operation It has been dubbed "the onset condition and we are now in a position to see what it implies lantitatively Setting(27)to unity, we get H HoP2-19v3 ev and from the exit velocity expression(18), 1e=098 4 The velocity at which the particle's kinetic energy would be capable of ionizing it is called the" Alfven critical speed". Many years ago Alfven used this conversion of kinetic to ionization energy to construct a model of the"condensation"of matter expanding from the proto-Sun to form the existing planets We see here that the exhaust speed of an MPD thruster (or a PPt, which works on the same principles )is 16.522, Space Propulsion Lecture 22 Prof. Manuel martinez-Sanchez Page 6 of 816.522, Space Propulsion Lecture 22 Prof. Manuel Martinez-Sanchez Page 6 of 8 times (eVi) be the effective ionization energy per atom, and αe the degree of ionization at the exit. We then have e i i ' m D eV m α i  (26) or, from (25), 2 2 i 0 e i ' m 4 H I eV 9 3 w m ⎛ ⎞ µ α ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ i  (27) This indicates very rapid increase of the ionization fraction as the current increases, or as the flow is reduced. Instability “onset” For a given thruster, as 2 I mi is increased, αe increases rapidly. When it reaches unity, the behavior of the plasma near the exit changes drastically. This is because any extra dissipation cannot be absorbed into ionization anymore, and goes instead directly into heating the plasma (or perhaps the electron component only). This causes conductivity to increase whenever the current concentrates, which leads to further current concentration. We have here the classical prescription for constriction into an arc, and one can expect heavy arcing (with the corresponding damage to electrodes) when αe approaches 1. This behavior has indeed been observed repeatedly, and has been the focus of a lot of attention, because it limits the practically achievable value of 2 I mi . Since (as we will see) efficiency increases with 2 I mi , this is a major hurdle in the path towards efficient MPD operation. It has been dubbed “the onset condition and we are now in a position to see what it implies quantitatively. Setting (27) to unity, we get 2 0 i i ' H 93 I eV = w 4m m µ i (28) and from the exit velocity expression (18), i i e i i ' ' 1 93 eV eV u = = 0.987 2 4m m (29) The velocity i i ' 2eV m at which the particle’s kinetic energy would be capable of ionizing it is called the “Alfvèn critical speed”. Many years ago Alfvèn used this conversion of kinetic to ionization energy to construct a model of the “condensation” of matter expanding from the proto-Sun to form the existing planets. We see here that the exhaust speed of an MPD thruster (or a PPT, which works on the same principles) is