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Flight and Orbital Mechanics Flight and Orbital Mechanics Lecture slides Challenge the future 1 Flight and Orbital Mechanics AE2-104, lecture hours 17+18: Perturbations Ron Noomen October 25, 2012 AE2104 Flight and Orbital Mechanics 1 | Example: solar sail spacecraft Questions: • what is the purpose of this mission? • where is the satellite located? • why does it use a solar sail? • …. [Wikipedia, 2010] AE2104 Flight and Orbital Mechanics 2 | Overview • Orbital mechanics (recap) • Irregularities gravity field • Third-body perturbations • Atmospheric drag • Solar radiation pressure • Thrust • Relativistic effects • Tidal forces • Thermal forces Special missions only; not treated here • Impacts debris/micrometeoroids AE2104 Flight and Orbital Mechanics 3 | Learning goals The student should be able to: • mention and describe the various perturbing forces that may act on an arbitrary spacecraft; • quantify the resulting accelerations; • make an assessment of the importance of the different perturbing forces, depending on the specifics of the mission (phase) at hand. Lecture material: • these slides (incl. footnotes) AE2104 Flight and Orbital Mechanics 4 | 2-dimensional Kepler orbits [Seligman, 2010] AE2104 Flight and Orbital Mechanics 5 | 2-dimensional Kepler orbits a: semi-major axis [m] e: eccentricity [-] [Cornellise, Schöyer and Wakker, 1979] θ: true anomaly [deg] E: eccentric anomaly [deg] AE2104 Flight and Orbital Mechanics 6 | 2-dimensional Kepler orbits: general equations 2 r a (1 e ) p ; r a (1 e) ; r a (1 e) 1 e cos 1 e cos p a 2 E E E V tot kin pot 2 r 2a 2 2 1 2 V ; Vcirc ; Vesc r a r a r 3 T 2 a AE2104 Flight and Orbital Mechanics 7 | 2-dimensional Kepler orbits: equations (cnt’d) ellips: 0 ≤ e < 1 a > 0 Etot < 0 n 3 a satellite altitude specific energy 1 e E [km] [km2/s2/kg] tan tan 2 1 e 2 launch 0 -62.4 M E esin E platform M n(t t0) SpaceShipOne 100+ -61.4 (culmination) M Ei esin Ei Ei1 Ei imaginary sat 100 -30.8 1ecosEi Kepler’s Equation Kepler’s r a(1ecosE) Envisat 800 -27.8 LAGEOS 5900 -16.2 AE2104GEO Flight and Orbital35900 Mechanics -4.78 | 2-dimensional Kepler orbits: equations (cnt’d) ellips: 0 ≤ e < 1 a > 0 Etot < 0 AE2104 Flight and Orbital Mechanics 9 | 2-dimensional Kepler orbits: equations (cnt’d) parabola: e = 1 a = ∞ Etot = 0 p r 1 cos 1 1 M tan tan3 2 2 6 2 M n (t t0) n p3 2 2 2 V Vesc r AE2104 Flight and Orbital Mechanics 10 | 2-dimensional Kepler orbits: equations (cnt’d) hyperbola: e > 1 a < 0 Etot > 0 e 1 F tan tanh 2 e 1 2 M e sinh F F M n (t t0) n (a)3 r a (1 e cosh F) 2 2 2 2 2 V Vesc V V r AE2104 Flight and Orbital Mechanics 11 | 3-dimensional Kepler orbits i: inclination [deg] [Cornelisse, Schöyer and Wakker, 1979] Ω: right ascension of the ascending node, or longitude of the ascending node [deg] ω: argument of pericenter [deg] u = ω + θ : argument of latitude [deg] AE2104 Flight and Orbital Mechanics 12 | coordinates transformations 1) from spherical (r,λ,δ) (λ in X-Y plane; δ w.r.t. X-Y plane) to cartesian (x,y,z): x = r cos δ cos λ y = r cos δ sin λ z = r sin δ 2) from cartesian (x,y,z) to spherical (r,λ,δ): r = sqrt ( x2 + y2 + z2 ) 2 2 rxy = sqrt ( x + y ) λ = atan2 ( y / rxy , x / rxy ) δ = asin ( z / r ) AE2104 Flight and Orbital Mechanics 13 | Orbital perturbations: introduction Questions: • what forces? • magnitude of forces? • how to model/compute in satellite orbit? • analytically? numerically? • accuracy? • efficiency? AE2104 Flight and Orbital Mechanics 14 | Inclusion in orbit modeling Option 1: include directly in equation of motion d x2 amain agrav adrag asolrad a3rdbody etcetera d t2 Option 2: express as variation of orbital elements da 2a2 p e.g. Sesin N dt p r (not further treated here; cf. ae4-874 and ae4-878) AE2104 Flight and Orbital Mechanics 15 | Perturbations: • Irregularities gravity field • Third body • Atmospheric drag • Solar radiation pressure • Thrust AE2104 Flight and Orbital Mechanics 16 | Perturbations: • Irregularities gravity field • Third body • Atmospheric drag • Solar radiation pressure • Thrust AE2104 Flight and Orbital Mechanics 17 | Irregularities gravity field Already treated in lecture hours 15+16 e.g., North-South acceleration due to J2: 2 4 acc 3 J2 Re r sin( )cos( ) e.g., East-West acceleration due to J2,2: 2 4 acc 6 J2,2 Re r 3cos( )sin(2( 2,2)) AE2104 Flight and Orbital Mechanics 18 | Perturbations: • Irregularities gravity field • Third body • Atmospheric drag • Solar radiation pressure • Thrust AE2104 Flight and Orbital Mechanics 19 | Third-body perturbations Reference frame M rsd rd rs G rs G Md 3 3 3 rs rsd rd Attractional forces (by definition): • satellite <-> Earth • satellite <-> perturbing body • Earth <-> perturbing body AE2104 Flight and Orbital Mechanics 20 | Third-body perturbations (cnt’d) Attractional forces (practice): • Earth attracts satellite • perturbing body attracts satellite M << M , M • perturbing body attracts Earth sat E S • net effect counts M rr rr GGMmain ps p s3 s p 3 3 rs r ps r p AE2104 Flight and Orbital Mechanics 21 | Third-body perturbations (cnt’d) perturbing acceleration ap: rps rp a p p 3 3 r r ps p maximum when on straight line: main sat 3rd body 3 amppr 2 s a m r mainmax main p AE2104 Flight and Orbital Mechanics 22 | Third-body perturbations (cnt’d) 2 situations: 1. heliocentric (i.e., orbits around Sun) 2. planetocentric (i.e., orbits around Earth) AE2104 Flight and Orbital Mechanics 23 | Third-body perturbations (cnt’d) Conclusions: • influence increases with distance to Sun • perturbation O(10-7) m/s2 AE2104 Flight and Orbital Mechanics 24 | Third-body perturbations (cnt’d) Conclusions: • influence Sun decreases with distance to Sun • influence planets increases with distance to Sun AE2104 Flight and Orbital Mechanics 25 | • acceleration from Sun O(10-2) m/s2; dominant (central body!!) • near planet: 3rd body becomes dominant Third-body perturbations (cnt’d) Conclusions: • in Solar System: Sun dominant • near Earth: Earth itself dominant AE2104 Flight and Orbital Mechanics 26 | • central body vs. 3rd body perturbation ?? Third-body perturbations (cnt’d) Sphere of Influence: • area around planet where gravity from planet is dominant (compared with gravity of other celestial bodies) • 3-dimensional shape • boundary • 1st-order approximation: sphere with constant radius • definition for determination location: relative acceleration w.r.t. system 1 = relative acceleration w.r.t. system 2 Sun Earth sat AE2104 Flight and Orbital Mechanics 27 | Third-body perturbations (cnt’d) accSun,3rd accEarth,3rd accEarth,main accSun,main without derivation : 0.4 Mmain rSoI r3rd M3rd example : Earth main body, Sun 3rd body : 0.4 M Earth rSoI,Earth distEarthSun 930,000 km MSun AE2104 Flight and Orbital Mechanics 28 | Moon is at 384,000 km…. Third-body perturbations (cnt’d) planet Sphere of Influence [km] [% AU] [% distance planet-Sun] Mercury 1.1 × 105 0.08 0.2 Venus 6.2 × 105 0.4 0.6 Earth 9.3 × 105 0.6 0.6 Mars 5.8 × 105 0.4 0.3 Jupiter 4.8 × 107 32.2 6.2 Saturn 5.5 × 107 36.5 3.8 Uranus 5.2 × 107 34.6 1.8 Neptune 8.7 × 107 57.9 1.9 Pluto 3.2 × 106 2.1 0.1 AE2104 Flight and Orbital Mechanics 29 | Third-body perturbations (cnt’d) 2 situations: 1. heliocentric (i.e., orbits around Sun) 2. planetocentric (i.e., orbits around Earth) AE2104 Flight and Orbital Mechanics 30 | Third-body perturbations (cnt’d) center of ref. frame AE2104 Flight and Orbital Mechanics 31 | Third-body perturbations (cnt’d) Conclusions: • influence of Sun as 3rd body increases with distance from Earth • effective 3rd body acceleration by Sun O(10-6) m/s2 AE2104 Flight and Orbital Mechanics 32 | Third-body perturbations (cnt’d) Question 1: a) Compute the dimension of the Sphere of Influence of the Earth (when the Sun is considered as the perturbing body). The SoI is given by the following general equation: 0.4 Mmain rSoI r3rd M3rd b) What is the value of the radial attraction exerted by the Earth at this distance? (if you were unable to make the question a, use a value of 1 × 106 km for this position). c) What is the effective gravitational acceleration by the Sun at this position? Assume that Earth, Sun and satellite are on a straight line. d) What is the relative perturbation of the solar attraction, compared to that of the main attraction of the Earth? 6 30 24 11 Data: 1 AU = 149.6 × 10 km, mSun = 2.0 × 10 kg, mEarth = 6.0 × 10 kg, μSun = 1.3271 × 10 3 2 3 2 km /s , μEarth = 398600 km /s . Answers: see footnotes below (BUT TRY YOURSELF FIRST!!) AE2104 Flight and Orbital Mechanics 33 | Third-body perturbations (cnt’d) Conclusions: • influence of Moon as 3rd body increases with distance from Earth • effective 3rd body acceleration by Moon O(10-5) m/s2 at GEO AE2104 Flight and Orbital Mechanics 34 | Third-body perturbations (cnt’d) Conclusions: • Earth dominant (central body; within SoI) • Moon most important 3rd body; Sun directly after AE2104 Flight and Orbital Mechanics 35 | • effect of planets about 4 orders of magnitude smaller Third-body perturbations (cnt’d) [Wikimedia, 2010] celestial ΔV-budget body for GEO sat [m/s/yr] Moon 36.93 Sun 14.45 AE2104 Flight and Orbital Mechanics 36 | Third-body perturbations (cnt’d) Question 2: Consider a hypothetical planet X with mass 5 × 1025 kg, orbiting the Sun in a circular orbit with radius 3 AU.
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