Bat 6(△t+t)=-2 (8) and after the period t of reversed thrust we return to the initial orbit with O, and with 89 as in( 8), plus a further 89(t,) This is then the total 49 3at At -t (9) Clearly, the mission(given A9 and At )can be accomplished with different choices of thrusting time ti(but notice that t, At/ 2 in any case). The required thrust/mass ratio(a)and AV=2 at depend on this choice (10 3t1(△t-t1) 4v=2 3 4『o△6 2 We can see here that low thrust ends up as a penalty on Av, so that the thrusting time should be selected as short as possible within the available on-board power. 16.522, Space P pessan Lecture 4 Prof. Manuel martinez3 ⎞ δϑ (∆tt ) = - 3at1 ⎜ ⎛ ∆t - t1 ⎟ - (8) 1 r0 ⎝ 2 ⎠ and, after the period t1 of reversed thrust, we return to the initial orbit with 0 d (δϑ ) =, and with δϑ as in (8), plus a further δϑ (t1 ) . dt This is then the total ∆ϑ : 3at1 ⎜ ⎛ ∆t - 3 ⎞ 3 a 2 ∆ϑ = - r0 ⎝ 2 t1 ⎠ ⎟ - 2 r0 t1 ∆ϑ = - 3at1 (∆t - t1 ) (9) r0 Clearly, the mission (given ∆ϑ and ∆t ) can be accomplished with different choices of thrusting time t1 (but notice that t1 < ∆t 2 in any case). The required thrust/mass ratio (a) and V=2 ∆ t a 1depend on this choice: 1 r0∆ϑ -a = 3 t1 (∆t - t1 ) (10) 2 r0∆ϑ ∆V= 3 (∆t t1 ) (11) - We can see here that low thrust ends up as a penalty on ∆V , so that the thrusting time should be selected as short as possible within the available on-board power. 16.522, Space Propulsion Lecture 4 Prof. Manuel Martinez-Sanchez Page 3 of 10