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lecture4 - 16.522 Space Propulsion Prof Manuel...

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16.522, Space Propulsion Prof. Manuel Martinez-Sanchez Lecture 4: Re-positioning in Orbits Suppose we want to move a satellite in a circular orbit to a position ϑ apart in the same orbit, in a time t (assumed to be several orbital times at least). The general approach is to transfer to a lower (for positive ϑ ) or higher (for ϑ < 0 ) nearby orbit, then drift in this faster (or slower) orbit for a certain time, then return to the original orbit. The analysis is similar for low and high thrust, because we have radius ratios very close to 1, so that, in either case the satellite is nearly “in orbit” even during thrusting periods, and V's for orbit transfer amount (in magnitude) to the difference of the beginning and ending orbital speeds. In detail, of course, if done at high thrust the maneuver involves a two-impulse Hohmann transfer to the drift orbit and one other two-impulse Hohmann transfer back to the original orbit. For the low-thrust case, continuous thrusting is used during both legs, with some guidance required to JG remove the very slight radial component of v picked up during spiral flight (although ignored here). We will do the analysis for the low-thrust case only, then adapt the result for high- thrust. Let δϑ be the advance angle relative to a hypothetical satellite remaining in the original orbit and left undisturbed. The general shape of the maneuver is sketched below: Since the orbital angular velocity is = µ 3 , r its variation with orbit radius is 16.522, Space Propulsion Lecture 4 Prof. Manuel Martinez-Sanchez Page 1 of 10
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3 δ r d ( δϑ ) δ = - = (1) 2 r dt During thrusting, δ r is varying according to d µ F r a= M (2) dt - 2 r M v a µ F or µ dr a µ 1 dr = 2 ar 1 2 (3) 2 r 2 dt r rdt µ and so d ( δ ) d 2 ( δϑ ) 3 2 ar 1 2 3a = = - = − (4) dt dt 2 2 µ r For integration, we will regard r r 0 as a constant (small variations): d ( δϑ ) 3a + = - t constant (5) dt r 0 Starting from t = 0, δϑ = 0, d ( δϑ ) = 0 , dt we obtain d ( δϑ ) = - 3 a t ; δϑ = - 3 a t 2 ( t < t 1 ) (6) dt r 0 2 r 0 After ( t = t 1 ), we continue to drift at a constant rate d ( δϑ ) 3 a = - t 1 dt r 0 and, since we start from δϑ ( ) 3 a 2 t = - t 1 , 1 2 r 0 the δϑ during the coasting phase is t 1 δϑ = - 3 a t 1 2 - 3 a t 1 ( t - t 1 ) = - 3 at 1 t - (7) 2 r 0 r 0 2 r 0 - At the end of coasting ( t = t t ) , we have then 1 16.522, Space Propulsion Lecture 4 Prof. Manuel Martinez-Sanchez Page 2 of 10
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3 δϑ ( t t ) = - 3 at 1 t - t 1 (8) - 1 2 r 0 and, after the period t 1
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