lecture_35_36

lecture_35_36 - 16.512 Rocket Propulsion Prof Manuel...

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16.512, Rocket Propulsion Prof. Manuel Martinez-Sanchez Lecture 35-36: Impulsive and Low-Thrust Maneuvers in Space See Lectures 3-4 of 16.522 (Space Propulsion) for coverage of Low Thrust Maneuvers and Re-positing within an orbit . We add here material on Hyperbolic Orbits and Interplanetary Transfer . Hyperbolic trajectories : r = p 1e c o s e > 1 Asymptotes : = -1 ecos θ cos θ = - 1 e 1 1 cos e ⎛⎞ θ=θ =π− ⎜⎟ ⎝⎠ hP still valid Instead of “semimajor axis” a>0 distance from perigee to “center” is (-a), and we still have E0 2a µ = −> Also ( ) () 22 pa ae 1 = −= Similarly, distance |focus - center| is (-a)e (it is ae in ellipse) Turning angle : 11 2( ) 2cos 2 cos e2 e −− π δ=π− π−θ =π− = 1 1 2sin e δ= 16.512, Rocket Propulsion Lecture 35-36 Prof. Manuel Martinez-Sanchez Page 1 of 7
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Miss distance or Impact Parameter (ae )s in ( ) (ae ∆= − π−θ = − θ 2 1 )1 e or 2 P e e1 ∆= 2 2 e 2 P Excess hyperbolic velocity () 2 V2 E ap µ− µ == = used in many books, reports 2 3 "C " v = Orbit characterized by any pair of parameters, like (p, e) , , (v , (v p (v , ) θ , ) ∞∞ θ , ) , p (, v) γ , p (,) γ δ Example : Given (v , -a = , ) 2 v µ 2 2 v a −= = µ then 2 11 2 1 v 1 2sin 2cot e 1 e −− ⎛⎞ δ= = = ⎜⎟ µ ⎝⎠ Planetary “Spheres of Influence” (SOI) Locus of points for which the ratio of (the Sun’s disturbing acceleration of the relative vehicle-planet motion) to (the planet-induced vehicle acceleration) equals the ratio of (the planet-induced acceleration of the relative vehicle-Sun motion) (the Sun-induced vehicle acceleration).
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lecture_35_36 - 16.512 Rocket Propulsion Prof Manuel...

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