PHYS Dynamics of Space Vehicles

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PHYS 4110 - Dynamics of Space Vehicles Chapte 3: Two Body Poblem Eath, Moon, Mas, and Beyond D.

1 PHYS Dynamics of Space Vehicles Chapte 3: Two Body Poblem Eath, Moon, Mas, and Beyond D. Jinjun Shan, Pofesso of Space Engineeing Depatment of Eath and Space Science and Engineeing Room 55, Petie Science and Engineeing Building Tel: ext jjshan@yoku.ca Homepage:

2 Keple s Thee Laws Two-Body Poblem -

3 Newton s Law of Gavity Two-Body Poblem - 3

4 Equations of Motion in an Inetial Fame Two-Body Poblem - 4

5 Two-Body Equation of Motion F " " F 1 1! m R " 1 0 ( m + m ) 0 1! m1r " " F 1 + m 1 1! R "! " G R G F " 1 m1m m "! R " Two-Body Poblem - 5

6 Angula Momentum and Obit Fomulas Two-Body Poblem - 6

7 Two-Body Poblem - 7 Angula Momentum and Obit Fomulas ( ) e h h h h H c dt d dt d m µ µ µ µ!!!!!!!!!!!! o /1

8 Two-Body Poblem - 8

9 Solution to Two-Body Poblem Two-Body Poblem - 9

10 The Enegy Law Two-Body Poblem - 10

11 Cicula Obit (e0) Two-Body Poblem - 11

12 Example - Calculate the speed and peiod of satellite in cicula obit as a function of altitude z. Two-Body Poblem - 1

13 Example - 3 Detemine the speed of a geostationay eath satellite. Two-Body Poblem - 13

14 Elliptical Obit (0<e<1) β Two-Body Poblem - 14

15 Example - 4 Obits 1 and ae two cicula obits with altitudes of 300 km and 35,786 km, espectively. An elliptical obit is tangent to both cicles at its apse line. Detemine the velocities of spacecaft at point A and B on both elliptical and cicula obits. Two-Body Poblem - 15

16 Example - 5 [Example.5 in textbook]: An eath satellite is in an obit with peigee altitude z p 400 km and e 0.6. Find (a) v (d) (h) v when (i) γ when (j) γ θ p ; max (b) ; (e) v (g) θ when a θ a θ θ and θ when γ ; (c) a ; (f) T max Two-Body Poblem - 16

17 Example - 6 [Example.6 in textbook]: At two points on a geocentic obit the altitude and tue anomaly ae Find (a) e; z z km; 85 km; (b) z p ; θ 16 θ 1 58 (c) a;!! (d) T Two-Body Poblem - 17

18 Paabolic Obit (e1) Two-Body Poblem - 18

19 Example - 7 [Example.7 in textbook]: The peigee of a satellite in a paabolic geocentic tajectoy is 7000 km. Find the distance d between points P1 and P on the obit which ae 8000 km and km fom the cente of the eath. Two-Body Poblem - 19

20 Hypebolic Obits (e>1) The system consists of two symmetic cuves. One of them is occupied by the obiting body, the othe one is its empty, mathematical image. Two-Body Poblem - 0

21 Vaious Obits Obits of vaious eccenticities, having a common focus F and peiapsis P Two-Body Poblem - 1

22 Example - 8 [Example.8 in textbook]: At a given point, km; v1 8.6 km/s; γ1! 50 show the tajectoy is a hypebola and calculate the following: C ; h; θ; e; ; δ ; a; 3 p and Δ Two-Body Poblem -

23 Obital Position as a Function of Time n Given a tue anomaly, calculate the time equied by spacecaft to aive this point. n Given a time instant, calculate the position of spacecaft. Two-Body Poblem - 3

24 Two-Body Poblem - 4 Time Since Peiapsis ( ) ( ) ( ) θ θ ϑ ϑ µ ϑ ϑ µ θ θ µ θ θ µ d cos 1 1 d cos 1 1 d cos 1 1 d cos 1 1 e h t e t h e t h h e h!! h

25 Cicula Obit (e0) 3 t h 3 1 θ h 3 θ dϑ µ 0 1+ ecosϑ µ 0 µ θ ( ) dϑ Two-Body Poblem - 5

26 Elliptical Obit (0<e<1) µ θ h 3 t 1 dϑ 1 tan θ e 1 e sinθ tan! " ( 1+ ecosϑ) e ( e ) 3 $! cosθ 1!!!!!! #!!!!!! e e µ h t 1 3 ( 1 e ) 3 M e M e µ h 3 M e 1 e ( ) 3 t π T t nt t M e π T Two-Body Poblem - 6

27 Example - 9 [Example 3.1 in textbook]: A geocentic elliptical obit with a km and p km. Calculate the time to fly fom peigee P to a tue anomaly of 10 degees. Two-Body Poblem - 7

28 Paabolic Obit (e1) Two-Body Poblem - 8

29 Example - 10 [Example 3.4 in textbook]: Geocentic paabola has a peigee velocity of 10 km/s. How fa is the satellite fom the cente of the eath six hous afte peigee passage. Two-Body Poblem - 9

30 Hypebolic Obits (e>1) Two-Body Poblem - 30

31 Example - 11 [Example 3.5 in textbook]: A geocentic tajectoy has a peigee velocity of 15 km/s and a peigee altitude of 300 km. Find (a) the adius when the tue anomaly is 100 degees; and (b) the coesponding time t at that point. Two-Body Poblem - 31

32 Resticted Thee-Body Poblem Two-Body Poblem - 3

33 Example - 9 n Find the Lagange points fo eath-moon system. Two-Body Poblem - 33

34 Example - 10: Sun-Eath System Two-Body Poblem - 34

HW6 Physics 311 Mechanics

HW6 Physics 311 Mechanics Fall 015 Physics depatment Univesity of Wisconsin, Madison Instucto: Pofesso Stefan Westehoff By Nasse M. Abbasi June 1, 016 Contents 0.1 Poblem 1.........................................