△Mc2 here the definition of Vch(Equation(6) is now made using no instead of n. Once gain,only approximate expressions for Av/vch are feasible for the optimum c and mass fractions Normalizing all velocities by vch e obtain xn=√1+82-y (1 224√1+82 =1-21+82v+v2- 121+82 (13) M M-五+8224(1+ For 8=0, and neglecting the last term included in each case, we recover the simple expressions of Equation (7)and Figure 2. The main effect of the losses(8 can be seen to be: (a)An increase of the optimum C, seeking to take advantage of the higher efficiency thus obtained (b)a reduction of the maximum payload c)A reduction of the fuel fraction Both these last effects indicate a higher structural fraction due to the need to raise rated power to compensate for the efficiency loss. It is worth noting also that the losses are felt least in the trucking"mode(high Vch, i. e light engine or long duration). 2. The Optimum: Thrust Profile As was mentioned there is no a priori reason to operate an electric thruster at a constant thrust or specific impulse even if the power is indeed fixed We examine here a simple case to illustrate this point, namely, one with a constant efficiency as in the classical Stuhlinger optimization but allowing f, m and c to vary in time if this is advantageous. Of course these variations are linked by the constancy of the 16.522, Space Propulsion ecture 2 Prof. manuel martinez-Sanchez Page 6 of 1916.522, Space Propulsion Lecture 2 Prof. Manuel Martinez-Sanchez Page 6 of 19 V 22 V - - L L c c so 2 o o ch M c +v M =e - - 1-e M M v ∆ ∆ ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ (10) where the definition of vch (Equation (6)) is now made using η0 instead of η. Once again, only approximate expressions for ∆V vch are feasible for the optimum c and mass fractions. Normalizing all velocities by vch : L ch ch ch c v v x ; v = ; = vvv ∆ ≡ δ (11) we obtain 2 2 OPT 2 v v x = 1+ - - + ... 2 24 1+ δ δ (12) 3 L so 2 2 2 o o MAX M M v + = 1 - 2 1+ v + v - + ... M M 12 1+ ⎛ ⎞ ⎜ ⎟ δ ⎝ ⎠ δ (13) 3 P 2 2 o M v 1v = - + ... M 24 1+ 1+ ⎛ ⎞ ⎜ ⎟ δ δ ⎝ ⎠ (14) For δ = 0,and neglecting the last term included in each case, we recover the simple expressions of Equation (7) and Figure 2. The main effect of the losses ( ) δ can be seen to be: (a) An increase of the optimum c, seeking to take advantage of the higher efficiency thus obtained. (b) A reduction of the maximum payload, (c) A reduction of the fuel fraction. Both these last effects indicate a higher structural fraction, due to the need to raise rated power to compensate for the efficiency loss. It is worth noting also that the losses are felt least in the “trucking” mode (high vch , i.e. light engine or long duration). 2. The Optimum: Thrust Profile As was mentioned, there is no a priori reason to operate an electric thruster at a constant thrust or specific impulse, even if the power is indeed fixed. We examine here a simple case to illustrate this point, namely, one with a constant efficiency as in the classical Stuhlinger optimization, but allowing F, m i and c to vary in time if this is advantageous. Of course these variations are linked by the constancy of the power: