Astrodynamics (AERO0024)

Transcription

1 Astrodynamics (AERO0024) 5A. Orbital Maneuvers Gaëtan Kerschen Space Structures & Systems Lab (S3L)

2 Course Outline THEMATIC UNIT 1: ORBITAL DYNAMICS Lecture 02: The Two-Body Problem Lecture 03: The Orbit in Space and Time Lecture 04: Non-Keplerian Motion THEMATIC UNIT 2: ORBIT CONTROL Lecture 05: Orbital Maneuvers Lecture 06: Interplanetary Trajectories 2

3 Definition of Orbital Maneuvering It encompasses all orbital changes after insertion required to place a satellite in the desired orbit. This lecture focuses on satellites in Earth orbit. 3

4 Motivation Without maneuvers, satellites could not go beyond the close vicinity of Earth. For instance, a GEO spacecraft is usually placed on a transfer orbit (LEO or GTO). 4

5 STK: Astrogator Astrogator is a specialized analysis module for modeling and targeting a spacecraft s trajectory, including Impulsive and finite burns. Targeted maneuver sequences. High-fidelity orbit propagation. Fuel-usage calculations. Deep-space mission design and analysis. 5

6 STK: Astrogator 6

7 From LEO to GEO: Astrogator Video 7

8 5. Orbital Maneuvers 5.1 Introduction 5.2 Coplanar maneuvers 8

9 5. Orbital Maneuvers 5.1 Introduction Why? How? How much? When? 9

10 Orbital Maneuvers Why? (cite real-life examples) Why? 10

11 Orbit Circularization Ariane V is able to place heavy GEO satellites in GTO: perigee: km apogee: ~35786 km. GTO GEO Why? 11

12 Orbit Raising: Reboost ISS reboost due to atmospheric drag (ISS, Shuttle, Progress, ATV). The Space Shuttle is able to place heavy GEO satellites in near-circular LEO with a few hundred kilometers altitude Why? 12

13 Orbit Raising: Evasive Maneuvers Spot-2 raised its orbit by 400m to avoid a fragment of Thor Agena D orbital stage (July 1997) See also Why? 13

14 Orbit Raising: Deorbiting GEO Satellites Graveyard orbit: to eliminate collision risk, satellites should be moved out of the GEO ring at the end of their mission. Their orbit should be raised by about 300 km to avoid future interference with active GEO spacecraft Why? 14

15 Orbit Lowering Why? 15

16 Orbit Phasing Replacement of a failed satellite of a constellation by an existing on-orbit spare Why? 16

17 Final Rendezvous The crew of Gemini 6 took this photo of Gemini 7 when they were about 7 meters apart Why? 17

18 Plane Change A launch site location restricts the initial orbit inclination for a satellite. Which one is correct? For a direct launch 1. launch site latitude desired inclination. 2. launch site latitude desired inclination Why? 18

19 Hint ˆK Ĵ Î i ˆ 1h 1 ( ). cos z cos r v K h r v Why? 19

20 Plane Change The Space Shuttle cannot provide a direct launch to equatorial orbits Why? 20

21 And Launch Errors! Ariane V User s Manual Why? 21

22 And Launch Errors! Due to a malfunction in Ariane V s upper stage, Artemis was injected into an abnormally low transfer orbit. Artemis could still be placed, over a period of 18 months, into its intended operating position in GEO: 1. Several firings of the satellite s apogee kick motor raised the apogee and circularized the orbit at about km. 2. An unforeseen use of the ion engine was used to maneuver into GEO. 3. A final trim maneuver nudged Artemis into its originally intended trajectory Why? 22

23 Rocket Engines Maneuvers are performed using firings of onboard rocket motors. Chemical rocket engines: Assumption of impulsive thrust in this lecture: because the burn times are small compared with the time intervals between burns, the thrust can be idealized as having infinitely small duration (no thrust included in the equation of motion). Electric propulsion: Not covered herein (continuous and low thrust) How? 23

24 Rocket Engines: Monopropellant Astrium CHT 1 N: Hydrazine Burn life: 50h Length: 17cm Attitude and orbit control of small satellites and deep space probes. Herschel, Globalstar How? Astrium CHT 400 N: Hydrazine Burn life: 30m Length: 32cm Ariane V attitude control system 24

25 Rocket Engines: Bipropellant Astrium S 10 N: MMH (Fuel) N2O4-MON1-MON3 (Oxidizers) Attitude and orbit control of large satellites and deep space probes Venus Express, Arabsat How? Astrium S 400 N: MMH (Fuel), N2O4-MON1-MON3 (Oxidizers) For apogee orbit injection of GEO satellites and for planetary orbit maneuvers of of deep space probes Venus Express, Artemis 25

26 Rocket Engines: Solid ATK Star 27 (TE-M-616) 27 kn: Burn time: 34s Length: 1.3m Gross mass: 361 kg Apogee motor (GOES,GPS) How? 26

27 Rocket Engines: Low-Thrust Astrium RITA 150 mn: Xenon Beam voltage: 1200V Burn time: >20000h Gross mass: 154 kg Stationkeeping, orbit transfer, deep space trajectories RITA-10 (Artemis) How? 27

28 Specific Impulse, Isp It is a measure of the performance of a propulsion system. Astrium CHT 1N: Astrium CHT 400N: 210s 220s Monopropellant Astrium S 10N: 291s Astrium S 400N: 318s. Bipropellant ATK STAR 27: 288s Solid Astrium RITA-150: s Electric [Cold gas: ~50s Liquid oxygen/liquid hydrogen 455s ] How? 28

29 Rocket Engines: Isp RITA, Astrium The Ion Propulsion System for the Future How? 29

30 STK: Maneuver How? 30

31 STK: Engine How? 31

32 STK: View\Astrogator Browser How? 32

33 Further Reading on the Web Site How? 33

34 Goal: Efficiency Use a minimum amount of fuel. Do not take too much time How much? 34

35 Delta-V Each impulsive maneuver results in a change v, an indicator of how much propellant will be required. t1 Tt () t1 Ispg0 m t1 dm v dt dt I t spg 0 o m() t to m to m I g sp ln m I g ln m sp 0 m0 m0 m m m 0 v I g sp 1 e How much? 35

36 m/m Delta-V (1000,0.288) How much? v (m/s) Isp=300s 36

37 STK: Delta-V How much? 37

38 Delta-V: Examples Maneuver Average v per year [m/s] Drag compensation ( km) <25 Drag compensation ( km) < 5 Stationkeeping GEO GTO GEO 1460 (~90% N/S, ~10% E/W) Attitude control (3-axis) 2 6 First cosmic velocity 7900 Second cosmic velocity Space Ship One How much? 38

39 Delta-V: Can a CubeSat Go to the Moon? GTO to lunar orbit: 1700 m/s. Lunar descent and landing: 2100 m/s. 725 grams of propellant (Isp=300s) How much? 39

40 Delta-V Budget It is the sum of the velocity changes required throughout the space mission life. It is a good starting point for early design decisions. As important as power and mass budgets. In some cases, it may become a principal design driver and impose complex trajectories to deep space probes (see Lecture 6)! How much? 40

41 Delta-V Budget: GEO How much? 41

42 Time Time is another key parameter, especially for manned missions. Rendez-vous between the Space Shuttle and ISS cannot take more than a few days How much? 42

43 First Orbital Maneuvers January 2, 1959, Luna 1: The spacecraft missed the Moon by about 6000 km. But coming even this close required several maneuvers, including circularizing the initial launch orbit and doing midcourse corrections. September 12, 1959, Luna 2: Intentional crash into the lunar surface When? 43

44 First Maneuvers for Manned Spacecraft March 23, 1965, Gemini 3: A 74s burn gave a V of 15.5 meters per second. The orbit was changed from km x km to an orbit of 158 km x 169 km. December 12, 1965: Gemini 6 and 7: First rendezvous. The two Gemini capsules flew around each other, coming within a foot (0.3 meter) of each other but never touching When? 44

45 Gemini Program When? 45

46 Gemini Capsule When? 46

47 5. Orbital Maneuvers 5.2 Coplanar maneuvers One-impulse transfer Two-impulse transfer Three-impulse transfer Nontangential burns Phasing maneuvers 47

48 Perturbation Equations (Gauss) a 2 (1 e ) N sin2 sin i 1 ecos 3 a a 2 Resin T 1 ecos 2 1 e i a 2 (1 e ) N cos2 1 ecos 2 a(1 e ) e Rsin T cos cos E 2 1 a(1 e ) Tsin 2 ecos cos i Rcos e 1 ecos M nt, with a 2 2 (1 e ) R 2e cos ecos T sin 2 ecos e 1 ecos J.E. Prussing, B.A. Conway, Orbital Mechanics, Oxford University Press 48

49 Different Types of Maneuvers Coplanar / noncoplanar: coplanar maneuvers can change a, e, ω, θ. WIDE APPLICABILITY! Tangential / nontangential: tangential burns occur only at apoapsis and periapsis or on circular orbit. Impulsive / continuous: an impulsive maneuver corresponds to an instantaneous burn. One-, two-, and three-impulse transfers: different purposes and efficiency. 49

50 Two-impulse burn (nontangential, coplanar, impulsive) One-impulse burn (tangential, coplanar, impulsive) Two-impulse burn (tangential, coplanar, impulsive)

51 One-impulse burn (nontangential, noncoplanar, impulsive) Continuous burn (lowthrust orbit transfer)

52 Modifying the Semi-Major Axis 2 2 v1 v2, 2 r 2a 2 r 2a r r, v v Δv, v v v 2v v cos v v 1 v v1 v2 1 2 v 2v1v cos 2a 2a Fixed quantity v minimum if =0,v 1 =max(v 1 ). To get the most efficient burn, do the maneuver as close to perigee as possible in a direction collinear to the velocity One-impulse transfer 52

53 From GTO to GEO The impulse is necessarily applied at the apogee of the GTP, because we want to circularize the orbit. GTO The maneuver at apogee is in fact a combination of two maneuvers. Why? GEO One-impulse transfer 53

54 Hohmann Transfer The transfer between two coplanar circular orbits requires at least two impulses v 1 and v 2. In 1925, Walter Hohmann conjectured that The minimum-fuel impulsive transfer orbit is the elliptic orbit that is tangent to both orbits at its apse line. The rigorous demonstration came some 40 years later! Two-impulse transfer 54

55 Governing Equations v 2 v circ r v ellip 2 1 r a r 2 r 1 2r 2 v1 r 1 r1 r2 r1 v 1 2r 1 v2 r 2 r1 r2 r Two-impulse transfer 55

56 Initial circular orbit: 322 km v 1 = km/s Transfer orbit v 2 = km/s Final circular orbit: GEO

57 No Need to Compute the Vs in STK! v 1 = km/s v 2 = km/s Reproduce this during the exercise session Two-impulse transfer 57

58 Hohmann Transfer Elliptical Orbits The transfer orbit between elliptic orbits with the same apse line must be tangent to both ellipses. But there are two such transfer orbits. Which one should we favor? H. Curtis, Orbital Mechanics for Engineering Students, Elsevier Two-impulse transfer 58

59 Graphs of v 3 / v 3 The most efficient transfer is 3: it begins at the perigee on the inner orbit 1, where the kinetic energy is greatest, regardless of the shape of the outer target orbit. Inner elliptic orbit (A is the perigee) outer elliptic orbit Two-impulse transfer 59

60 Graphs of v 3 / v 3 The most efficient transfer terminates at the apogee of the outer ellipse, where the speed is the lowest. Inner circular orbit, outer elliptic orbit Two-impulse transfer 60

61 Bi-Elliptic Transfer Why? It is composed of two ellipses, separated by a midcourse tangential impulse (i.e., two Hohmann transfers in series). A limiting case is the biparabolic transfer (r B ). H. Curtis, Orbital Mechanics for Engineering Students, Elsevier Three-impulse transfer 61

62 Two or Three-Impulse Transfer? H. Curtis, Orbital Mechanics for Engineering Students, Elsevier Three-impulse transfer 62

63 Two or Three-Impulse Transfer? H. Curtis, Orbital Mechanics for Engineering Students, Elsevier Three-impulse transfer 63

64 Two- or Three-Impulse Transfer? It depends on the ratio of the radii of the inner and outer orbits (threshold: r C / r A = 11.94). For many practical applications (LEO to GEO), the twoimpulse transfer is more economical. It is also the case for interplanetary transfers from Earth to all planets except the outermost three. What is another important parameter to choose between two- or three- impulse transfer? Time of flight! For instance, the bi-parabolic transfer requires an infinite transfer time Three-impulse transfer 64

65 Exercise Session Find the total delta-v requirement for a bi-elliptic transfer from a geocentric circular orbit of 7000 km radius to one of km radius. Let the apogee of the first ellipse be km. Compare the delta-v schedule and total time of fliht time with that of a single Hohmann transfer ellipse. Verify using STK Three-impulse transfer 65

66 Tangential Burns or Not? The major drawback to the Hohmann transfer is the long flight time. Time of flight can be reduced at the expense of an acceptable increase in v. A possible solution is a one-tangent burn. It comprises one tangential burn and one nontangential burn Nontangential burns 66

67 Tangential Burns or Not? Vallado, Fundamental of Astrodynamics and Applications, Kluwer, Nontangential burns 67

68 Two Nontangential Burns? Solve Lambert s problem: it gives a relationship between two positions of a spacecraft in an elliptical orbit and the time taken to traverse them: The time required to traverse an elliptic arc between specified endpoints depends only on the semimajor axis, the chord length and the sum of the radii from the focus to the two points. It does not depend on eccentricity. If two position vectors and the time of flight are known, then the orbit can be fully determined. P 2 t P Nontangential burns 68

69 Lambert s Problem: Matlab Example Nontangential burns 69

70 Phasing Maneuvers Can we apply a tangential burn to intercept a target? Target Interceptor Phasing maneuvers 70

71 No! Imagine that you take a bend with your car and that you want to catch the car in front of you Phasing maneuvers 71

72 Yes! Can we exploit Hohmann transfer in a clever manner? Phasing maneuvers 72

73 GEO Repositioning: Astrogator Video Phasing orbit v=0.2 km/s for a longitude shift of 32º in one revolution. Target Interceptor Phasing maneuvers 73

74 Phasing Maneuver It can take the form of a two-impulse Hohmann transfer from and back to the same orbit. The target can be ahead or behind the chase vehicle. Usefulness: 1. Constellation (deployment or replacement of a failed satellite) 2. GEO 3. First phase of a rendezvous procedure Phasing maneuvers 74

75 Phasing Maneuver Design v Initial orbit Phasing Phasing maneuvers 75

76 Phasing Maneuver Design t 3 Tphasing orbit 2 a Lecture 2 a phasing orbit a r p r 2 a, r is known p r a e hv, P r p e r r a a r r 2 h 1, v 1 ecos p p p h r p v Phasing maneuvers 76

77 Astrodynamics (AERO0024) 5A. Orbital Maneuvers Gaëtan Kerschen Space Structures & Systems Lab (S3L)