MIT OpenCourseWare http://ocw.mit.edu 16.346 Astrodynamics Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.Exercises 18 1. Derive the Lagrange Interpolation Formulas for a sequence of four measurements. 2. Derive the differential equation used in Gibbs Method to determine the parameter p: d2r = (p − r)dt2 3. Derive the relation �dr �2 �2 p 1� dt = µr − r2 − a� 4. Derive the differential equation used in Gibbs Method to determine the semimajor axis a: d2 �11� dt2 (r 2)=2µr − a 5. Given a pair of range and range-rate measurements for a spacecraft in interplanetary space: r1 =0.600762027 a.u. r2 =0.603053915 a.u. dr1 dr2=0.288618834 a.u./year =0.575041077 a.u./year dt dt Determine the elements a, e, and p of the orbit. Problem 3–35 Courtesy of AIAA. Used with permission. Answer: a =1.19999993 a.u. and p =0.89999998 a.u. 6. Three observations of a satellite are made from the earth t1 =0.005274926 year t2 =0.010576712 year t3 =0.021370777 year ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎣ −0.830593168 ⎣ −0.851865512 ⎣ −0.890539844 iρ1 = 0.554953271 ⎦ iρ2 = 0.521052142 ⎦ iρ3 = 0.450207593 ⎦ 0.046280201 0.053196007 0.065206661 when the earth is positioned at ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 0.999450810 0.997792649 0.990998441 d1 = ⎣ 0.033137270 ⎦ a.u. d2 = ⎣ 0.066406537 ⎦ a.u. d3 = ⎣ 0.133873410 ⎦ a.u. 0 0 0 Find the position vector of the satellite at time t2 . Note: The earth’s orbit is assumed to be a circle of radius one astronomical unit and the earth crossed the reference x-axis at time zero. Answer: ⎡ ⎤ ⎡ ⎤ 0.1489 ... 0.159321004 ρ1 =0.952633214 a.u. r2 ≈ ⎣ 0.5856 ... ⎦ Exact answer r2 = ⎣ 0.579266185 ⎦ ρ2 =0.984277017 a.u. 0.0530 ... 0.052359607 ρ3 =1.048132894 a.u. ⎡ ⎤ ⎡ ⎤ diρ �� ⎣ −3.870867561 d2iρ �� 53.35190114 dt �� = −6.449952154 ⎦ dt2 �� = ⎣ −20.99951742 ⎦ t2 1.241279013 t2 −23.82234043 D1 = −144.0082104 D2 =1.044242145 D3 = −12.42980519 dρ� r(t2)=0.606571899 ρ(t2)=0.996437289 � =5.930383272 dt �
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