Course Overview/Orbital Mechanics

Transcription

1 Course Overview/Orbital Mechanics Course Overview Challenges of launch and entry Course goals Web-based Content Syllabus Policies Project Content An overview of orbital mechanics at point five past lightspeed David L. Akin - All rights reserved

2 Space Launch - The Physics Minimum orbital altitude is ~200 km P otential Energy kg in orbit Circular orbital velocity there is 7784 m/sec Kinetic Energy kg in orbit = µ r orbit + µ r E = = 1 2 µ r 2 orbit = J kg J kg Total energy per kg in orbit T otal Energy kg in orbit = KE + P E = J kg 2

3 Theoretical Cost to Orbit Convert to usual energy units Total Energy kg in orbit = J kg =8.9 kwhrs kg Domestic energy costs are ~$0.05/kWhr eoretical cost to orbit $0.44/kg 3

4 Actual Cost to Orbit Delta IV Heavy 23,000 kg to LEO $250 M per flight $10,870/kg of payload Factor of 25,000x higher than theoretical energy costs! 4

5 What About Airplanes? For an aircra in level flight, Energy = force x distance, so Total Energy kg Weight Thrust = Lift mg, or Drag T = thrust distance mass = L D = Td m = For an airliner (L/D=25) to equal orbital energy, d=81,000 km (2 roundtrips NY-Sydney) gd L/D 5

6 Equivalent Airline Costs? Average economy ticket NY-Sydney round-roundtrip (Travelocity 9/3/09) ~$1300 Average passenger (+ luggage) ~100 kg Two round trips = $26/kg Factor of 60x more than electrical energy costs Factor of 420x less than current launch costs But you get to refuel at each stop! 6

7 Equivalence to Air Transport 81,000 km ~ twice around the world Voyager - one of two aircra to ever circle the world non-stop, non-refueled - once! 7

8 Orbital Entry - The Physics 32 MJ/kg dissipated by friction with atmosphere over ~8 min = 66kW/kg Pure graphite (carbon) high-temperature material: c p =709 J/kg K Orbital energy would cause temperature gain of 45,000 K! us proving the comment about space travel, It s utter bilge! (Sir Richard Wooley, Astronomer Royal of Great Britain, 1956) 8

9 The Vision Once you make it to low Earth orbit, you re halfway to anywhere! - Robert A. Heinlein 9

10 Goals of ENAE 791 Learn the underlying physics (orbital mechanics, flight mechanics, aerothermodynamics) which constrain and define launch and entry vehicles Develop the tools for preliminary design synthesis, including the fundamentals of systems analysis Provide an introduction to engineering economics, with a focus on the parameters affecting cost of launch and entry vehicles, such as reusability Examine specific challenges in the underlying design disciplines, such as thermal protection and structural dynamics 10

11 Contact Information Dr. Dave Akin Space Systems Laboratory Neutral Buoyancy Research Facility/Room 2100D

12 Web-based Course Content Data web site at Course information Syllabus Lecture notes Problems and solutions Interactive web site at Communications for team projects (forums, wiki, blogs) Surveys for course feedback Videos of lectures 12

13 Syllabus Overview (1) Fundamentals of Launch and Entry Design Orbital mechanics Basic rocket performance Entry flight mechanics Ballistic entry Li ing entry Aerothermodynamics ermal Protection System (TPS) analysis Entry, Descent, and Landing (EDL) systems 13

14 Syllabus Overview (2) Launch flight mechanics Gravity turn Targeted trajectories Optimal trajectories Airbreathing trajectories Launch vehicle systems Propulsion systems Structures and structural dynamics analysis Avionics Payload accommodations 14

15 Syllabus Overview (3) Systems Analysis Cost estimation Engineering economics Reliability issues Safety design concerns Fleet resiliency Case studies Design project 15

16 Policies Grade Distribution 25% Problems 20% Midterm Exam 25% Term Project 30% Final Exam Late Policy On time: Full credit Before solutions: 70% credit A er solutions: 20% credit 16

17 Term Project - Fix NASA Design a system to replace shuttle, by carrying humans to ISS and the moon and return them to Earth safely Launch vehicle(s) Spacecra launch abort and EDL systems Mission models 12 crew/year to ISS (max 6 crew/flight) ISS cargo: 20,000 kg/yr upmass; 5,000 kg/yr downmass 8 crew/year to moon Lunar cargo: 5000 kg/yr upmass/1000 kg/yr downmass 17

18 Term Project Form teams (~3-4/team) Design an architecture to accomplish NASA s human space flight aspirations in the most cost effective manner possible All vehicles will be conceptually designed from scratch (no catalog engineering!) Parametric design parameters will be provided for human spacecra systems other than abort and EDL Design process should proceed throughout the term Formal design presentations at end of term 18

19 Orbital Mechanics: 500 years in 40 min. Newton s Law of Universal Gravitation F = Gm 1m 2 r 2 Newton s First Law meets vector algebra F = m a 19

20 Relative Motion Between Two Bodies m 2 r 2 F 12 F 21 r 1 m 1 F 12 = force due to body 1 on body 2 20

21 Gravitational Motion Let µ = G(m 1 + m 2 ) d 2 r dt 2 + µ r r 3 = 0 Equation of Orbit - Orbital motion is simple harmonic motion 21

22 Orbital Angular Momentum h is angular momentum vector (constant) = r and v are in a constant plane 22

23 Fun and Games with Algebra 0 23

24 More Algebra, More Fun 24

25 Orientation of the Orbit e eccentricity vector, in orbital plane e points in the direction of periapsis 25

26 Position in Orbit θ = true anomaly: angular travel from perigee passage 26

27 Relating Velocity and Orbital Elements 27

28 Vis-Viva Equation p a(1 e 2 )= 1 e2 a = v 2 = µ v 2 2 r v2 µ 2 r v2 µ 1 2 r 1 a 2 µ r = µ 2a 28

29 Energy in Orbit Kinetic Energy K.E. = 1 2 mν 2 K.E. m Potential Energy P.E. = mµ r P.E. m Total Energy = v2 2 = µ r Const. = v2 2 µ r = µ 2a <--Vis-Viva Equation 29

30 Suborbital Tourism - Spaceship Two 30

31 How Close are we to Space Tourism? Energy for 100 km vertical climb µ r E km + µ = km2 MJ = r E sec2 kg Energy for 200 km circular orbit µ 2(r E km) + µ = 32.2 km2 MJ = 32.2 r E sec2 kg Energy difference is a factor of 33! 31

32 Implications of Vis-Viva Circular orbit (r=a) v circular = Parabolic escape orbit (a tends to infinity) v escape = µ r 2µ Relationship between circular and parabolic orbits r v escape = 2v circular 32

33 Some Useful Constants Gravitation constant µ = GM Earth: 398,604 km 3 /sec 2 Moon: km 3 /sec 2 Mars: 42,970 km 3 /sec 2 Sun: 1.327x10 11 km 3 /sec 2 Planetary radii r Earth = 6378 km r Moon = 1738 km r Mars = 3393 km 33

34 Classical Parameters of Elliptical Orbits 34

35 Basic Orbital Parameters Semi-latus rectum (or parameter) Radial distance as function of orbital position Periapse and apoapse distances Angular momentum h = r v p = a(1 e 2 ) r = p 1+e cos θ r p = a(1 e) r a = a(1 + e) h = µp 35

36 The Classical Orbital Elements Ref: J. E. Prussing and B. A. Conway, Orbital Mechanics Oxford University Press,

37 The Hohmann Transfer r 2 r 1 v 2 v apogee v 1 v perigee 37

38 First Maneuver Velocities Initial vehicle velocity Needed final velocity Required ΔV v perigee = v 1 = µ v 1 = r 1 µ µ r 1 r 1 2r2 r 1 + r 2 ( ) 2r2 1 r 1 + r 2 38

39 Second Maneuver Velocities Initial vehicle velocity Needed final velocity Required ΔV v apogee = v 2 = µ v 2 = r 2 µ r 2 ( 1 µ r 2 2r1 r 1 + r 2 2r1 r 1 + r 2 ) 39

40 Limitations on Launch Inclinations Equator 40

41 Simple Plane Change v 1 v perigee v apogee Δv 2 v 2 41

42 Optimal Plane Change v perigee v 1 Δv v apogee 1 Δv 2 v 2 42

43 First Maneuver with Plane Change Δi 1 Initial vehicle velocity Needed final velocity Required ΔV v 1 = v p = v 1 = µ r 1 µ r 1 2r2 r 1 + r 2 v v2 p 2v 1 v p cos i 1 43

44 Second Maneuver with Plane Change Δi 2 Initial vehicle velocity v a = µ Needed final velocity r 2 v 2 = 2r1 r 1 + r 2 µ r 2 Required ΔV v 2 = v v2 a 2v 2 v a cos i 2 44

45 Sample Plane Change Maneuver Optimum initial plane change =

46 Calculating Time in Orbit 46

47 Time in Orbit Period of an orbit P = 2π Mean motion (average angular velocity) n = µ a 3 Time since pericenter passage a 3 µ M = nt = E e sin E M=mean anomaly 47

48 Dealing with the Eccentric Anomaly Relationship to orbit Relationship to true anomaly Calculating M from time interval: iterate until it converges r = a (1 e cos E) tan θ 2 = 1 + e 1 e tan E 2 E i+1 = nt + e sin E i 48

49 Example: Time in Orbit Hohmann transfer from LEO to GEO h 1 =300 km --> r 1 = =6678 km r 2 =42240 km Time of transit (1/2 orbital period) a = 1 2 (r 1 + r 2 ) = 24, 459 km t transit = P 2 = π a 3 µ = 19, 034 sec = 5h17m14s 49

50 Example: Time-based Position Find the spacecra position 3 hours a er perigee µ rad n = = 1.650x10 4 a3 sec e = 1 r p a = E j+1 = nt + e sin E j = sin E j E=0; 1.783; 2.494; 2.222; 2.361; 2.294; 2.328; 2.311; 2.320; 2.316; 2.318; 2.317; 2.317;

51 Example: Time-based Position (cont.) E =2.317 r = a(1 e cos E) = 12, 387 km tan θ 1+e 2 = 1 e tan E = θ = 160 deg 2 Have to be sure to get the position in the proper quadrant - since the time is less than 1/2 the period, the spacecra has yet to reach apogee --> 0 <θ<180 51

52 Velocity Components in Orbit v r = dr dt = d dt p r = 1+ecos θ p = 1+ecos θ v r = pe sin θ dθ (1 + e cos θ) 2 dt 1+e cos θ = p r v r = h = r v p( e sin θ dθ dt ) (1 + e cos θ) 2 r2 dθ dt e sin θ p 52

53 Velocity Components in Orbit (cont.) 53