lecture3 - 16.522, Space Propulsion Prof. Manuel...

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16.522, Space Propulsion Lecture 3 Prof. Manuel Martinez-Sanchez Page 1 of 9 16.522, Space Propulsion Prof. Manuel Martinez-Sanchez Lecture 3: Approximate V for Low-Thrust Spiral Climb Assume initial circular orbit, at co 0 v=v = r µ . Thrust is applied tangentially. Call F a= M . By conservation of energy, assuming the orbit remains near-circular v r ⎛⎞ µ ⎜⎟ ⎝⎠ ± , d -a dt 2r r µµ ± 2 dr a dt r 2r ± -3 2 1 2 r dr a dt 2 µ ± When we integrate, b t 0 a dt = V, and so b t 1- 1 22 0 -r =V µ∆ () 0b V= - rr t or final co c V=v -v (1) The result appears to be trivial, but it is not. Notice that the “velocity increment” V is actually equal to the decrease in orbital velocity. The rocket is pushing forward, but the velocity is decreasing. This is because in a r -2 force field, the kinetic energy is equal in magnitude but of the opposite sign as the total energy (potential = - 2 × kinetic). If thrust were applied opposite the velocity (negative a), the definition of V would be b t 0 (-a) dt , so the result in general is c V= v ∆∆ (2)
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16.522, Space Propulsion Lecture 3 Prof. Manuel Martinez-Sanchez Page 2 of 9 For simplicity, assume now F a = M = constant, which is actually optimal for many situations. Equation (1) can be recast as 0 -= a t rr µµ and solving for r, 00 22 co 0 r= = at at 1- v r ⎡⎤ ⎢⎥ ⎛⎞ ⎜⎟ ⎝⎠ µ ⎣⎦ (3) This shows how the radial distance “spirals out” in time. In principle, this says r → ∞ at co v t= a , a crude indication of “escape”. But of course, the orbit is no longer “near- circular” when approaching escape, so this result is not precise. One can get some improvement for the estimation of escape V as follows. The radial velocity r i can be calculated from (3) by differentiation. Notice that this is in the nature of an iteration, since r i was implicitly ignored in the energy balance which led to (3). We obtain 0 co 3 co 2a r v at v i (4) The tangential component v= ±r θ θ i is still approximated as the orbital velocity, i.e., co 0c o at r= = - a v 1 - v θ
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lecture3 - 16.522, Space Propulsion Prof. Manuel...

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