Aerospace Engineering - Orbital Mechanics

Full exam

1 of 2 Orbital Mechanics Academic Year 2020-2021 Lecturer: Camilla Colombo Exam 29 January 2021 Duration: 3 h All solution sheets have to be written in clear form in pen. Do not write in pencil. On top of each sheet, please report in capital letters your surname, name and student number. In the solution of the exercise all the analytical and numerical procedure must be reported. Please report the number of the answers from the text of the exercise. Numerical results need to have proper units of measure. PART 1 – EARTH TO SATURN MISSION EX 1 – Orbital Manoeuvre Part 1/2 Consider a refueling mission around the Earth. The satellite (spacecraft A) is injected by the launcher on a parking orbit (Orbit 1) with the following characteristics: Eccentricity: ������������ 1=0 Orbit Altitude: ℎ 1=763 km Inclination: ������������ 1=10° RAAN: Ω 1=120° The spacecraft A shall reach a new orbit (Orbit 2) to refuel a spacecraft B, part of a constellation mission. The orbit 2 has the following characteristics: Eccentricity: ������������ 2=0 Orbit Altitude: ℎ 2= 3642 km Inclination: ������������ 2=20° RAAN: Ω 2=120° The spacecraft A begins the transfer when it is at the ascending node of Orbit 1 and performs a Hohmann transfer with a change of plane to reach the Orbit 2. The transfer orbit lies on the plane of Orbit 2. Constants R_earth = 6378.15 km; 2 of 2 ������������_earth = 398600 km^3/s^2; Q1) Characterise the transfer orbit in terms of Keplerian elements { ������������,������������,������������,Ω,������������} ������������������������������������������������������������������������������������������������ Q2) Compute the modulus of the delta-v that the satellite A should provide to pass form Orbit 1 to the transfer orbit. Q3) Compute the components of the delta-v at question Q2) in the {������������,������������,ℎ} frame of Orbit 1. Q4) Compute the delta-v that the satellite A should provide to pass from the transfer orbit to the Orbit 2. Part 2/2 Now that the spacecraft A has reached Orbit 2, it should rendezvous with satellite B to perform the re-fuel. At the end of the Hohman transfer orbit, the satellite A is ahead of satellite B on Orbit 2 with an angular distance Δ������������=27.5°. The satellite A completes the phasing manoeuvre in 3 revolutions of the phasing and the initial orbit, to rendezvous with satellite B. Q5) Compute the period of the phasing orbit. Q6) Compute the modulus of the overall delta-v required by the phasing manoeuvres. Q7) Compute the total time of the transfer from the plane change manoeuvre in Q2) to the end of the phasing manoeuvre in Q6) Ex 2 – Time Measurements and Spherical Geometry An observatory in the Northern Hemisphere, LAT = 66.84° N and longitude LON = 62.60° W, observes Venus at a Local Civil Time LCT = 20 h 58 min 10 s. The declination of Venus at the observation moment is 28.9°. Assume the observation is done on May 13 th. Constants R_earth = 6378.15 km; ������������_earth = 398600 km^3/s^2; Right ascension of Venus: ������������ ������������ Correction factors: E_13/05 = 32 min 12 s E_14/05 = 32 min 54 s E_15/05 = 33 min 36 s 3 of 2 Q1) Compute the Greenwich Mean Time at the observation moment. Q2) Compute the Local Apparent Time at the observation moment. Q3) Compute the hour angle of the Sun. Q4) Compute the hour angle of Venus at the observation moment. Q5) Compute the elevation of Venus. Q6) Compute the Azimuth of Venus. PART 2 – EARTH TO SATURN MISSION Note that the DATA section reports the numerical data for the whole exercise. Read the whole exercise till the end before proceeding into its solution. DATA Assume that all planets follow circular and coplanar orbits (contained in the ecliptic) ������������ ������������ Versions V_sc_1 [km/s] ΔV mag nitude [km/s] eSat 1 38 .80 2.89 0.8 2 39.10 3.51 0.7 3 39.40 4.12 0.6 4 39.70 4.71 0.5 5 40.00 5.28 0.4 4 of 2 A Cassini-like mission is launched into a first heliocentric transfer from the Earth on the Winter solstice to target the planet Jupiter for a fly-by that will redirect the spacecraft in a second heliocentric leg towards the planet Saturn. The heliocentric velocity of the spacecraft after the Earth escape is parallel and in the same direction to the Earth�s velocity around the Sun and equal to V_sc_1 [see Data]. The spacecraft thus enters onto a heliocentric transfer leg (Orbit 1) which intersect the planet Jupiter on its orbit at the first intersection point and performs a fly-by with it. After the fly-by with Jupiter the spacecraft continues its travel on a second heliocentric leg (Orbit 2) which has its apogee in correspondence of the Saturn�s orbit where the spacecraft intersect Saturn and rendezvous with it. The magnitude of the relative velocity between the spacecraft and Saturn at the encounter is equal to ΔV [see Data]. Earth-Jupiter leg 1) Find the ecliptic longitude of the Earth at the spacecraft escape moment. (1 point) 2) Find the components of the eccentricity vector of Orbit 1 in the Sun-centred inertial heliocentric reference frame. (1 point) 3) Find the true anomaly of the point on Orbit 1 where the spacecraft meets Jupiter. (1 point) 4) Find the magnitude and the flight-path angle of the spacecraft velocity before the Jupiter fly-by. (2 point) Jupiter-Saturn leg 5) Characterise Orbit 2 (semi-major axis and eccentricity). (1 point) 6) Find the true anomaly of the point on Orbit 2 (Jupiter-Saturn leg) of the spacecraft after the fly-by with Jupiter. (1 point) 7) Find the magnitude and the flight-path angle of the spacecraft velocity after the Jupiter fly-by. (2 point) Jupiter fly-by 8) Draw the triangle of velocities at the fly-by showing clearly in the same drawing: the heliocentric velocities of the spacecraft before and after the fly-by, the relative velocity of the spacecraft at the entry and exit of sphere of influence and the planet velocity. Place the triangle in radial-transversal reference frame with respect to Jupiter’s orbit about the Sun. (2 point) 9) Draw the fly by hyperbola clearly showing the entry and exit asymptote the delta-v provided by the fly- by, the hyperbola eccentricity vector in the radial-transversal reference frame with respect to Jupiter’s orbit about the Sun. (1 point) 10) Determine the turn angle, the eccentricity and the pericentre radius of the fly-by hyperbola. [HINT: due to numerical errors in your computation you may have some SMALL mismatch between the entry and exit relative velocity magnitude] (2 point) 11) Find the angle between the velocity of Jupiter around the Sun and the eccentricity vector of the hyperbola. (2 point) Saturn rendezvous 12) Find the minimum delta-velocity to be imparted at the perigee of the capture hyperbola for an optimal capture on an elliptical orbit with e Sat [see Data] around Saturn. (1 point) 5 of 2 Solutions Version Q1) lon_Earth_A [deg] Q2) e_x [-] Q2) e_y [ -] Q2) e_z [-] Q3) theta_1_B [deg] Q4) V_1_B [km/s] Q4) gamma_1_B [deg] 1 90 0 0.6970 0 165.20 8.49 28.63 2 90 0 0.723 4 0 157.61 9.77 39.76 3 90 0 0.7499 0 152.26 10.91 46.0 7 4 90 0 0.776 7 0 147.99 11.9 5 50.33 5 90 0 0.8036 0 144.40 12.91 53.4 7 Version Q5) a [km] Q5) e [ -] Q6) theta_2_B Q7) V_2_B [km/s] Q7) gamma_2_B [deg] 1 949019385.5 0.51050655 101.15 14.1 8 29.06 2 897890285.3 0.59652022 115.53 13. 90 35.9 3 3 856822468.3 0.67304203 126.25 13.6 4 42.0 4 4 824131308.8 0.73940728 134.71 13.41 47.6 0 5 797993409.4 0.79638075 141.71 13.21 52.77 Version Q10) delta [deg] Q10) e [ -] Q10) rp [ -] Q11) ang(VJ,e) Q12) DV_capture [km/s] 1 48.5 1 2.4344 3793289.1 2 150. 30 0.91 2 29. 10 3.981 2 5413806.8 6 153.0 4 1.3 6 3 17.15 6.7066 7866379.2 7 153.5 3 1.84 4 8.4 9 13.5104 13887285.27 153.49 2.35 5 1.67 68.476 8 62674849.09 154.5 6 2.89 Part 1 Exercise 1 -- Orbital Manoeuvre Branch 1 Q1) Kep = [8580.65, 0.16776, 20.000, 120.000, 0] Q2) DV_A = 1.482 km/s Q3) DV_A = [ 0; 0.480; 1.402] km/s Q4) DV_B = 0.553 km/s Q5) T_ph = 2.843 h Q6) DV_ph = 0.104 km/s Q7) DT_tot = 9.629 h Branch 2 Q1) Kep = [8830.27, 0.18509, 28.750, 130.000, 0] Q2) DV_A = 1.973 km/s Q3) DV_A = [ 0; 0.427; 1.926] km/s Q4) DV_B = 0.600 km/s Q5) T_ph = 3.092 h Q6) DV_ph = 0.176 km/s Q7) DT_tot = 13.514 h Branch 3 Q1) Kep = [9079.90, 0.20146, 37.500, 140.000, 0] Q2) DV_A = 2.467 km/s Q3) DV_A = [ 0; 0.336; 2.444] km/s Q4) DV_B = 0.643 km/s Q5) T_ph = 3.327 h Q6) DV_ph = 0.215 km/s Q7) DT_tot = 17.832 h Branch 4 Q1) Kep = [9329.52, 0.21696, 46.250, 150.000, 0] Q2) DV_A = 2.961 km/s Q3) DV_A = [ 0; 0.208; 2.953] km/s Q4) DV_B = 0.682 km/s Q5) T_ph = 3.558 h Q6) DV_ph = 0.238 km/s Q7) DT_tot = 22.596 h Branch 5 Q1) Kep = [9579.15, 0.23165, 55.000, 160.000, 0] Q2) DV_A = 3.452 km/s Q3) DV_A = [ 0; 0.043; 3.452] km/s Q4) DV_B = 0.718 km/s Q5) T_ph = 3.789 h Q6) DV_ph = 0.252 km/s Q7) DT_tot = 27.819 h Exercise 2 -- Time Measurements and Spherical Geometry Branch 1 Q1) GMT = 0.969 h (DAY + 1) Q2) LAT = 21.345 h (DAY 0) Q3) t_sun = 9.345 h Q4) t_Venus = 10.345 h Q5) h_v = 7.576° Q6) Az = 338.234° Q7) el_max = 52.060° Branch 2 Q1) GMT = 1.969 h (DAY + 1) Q2) LAT = 21.327 h (DAY 0) Q3) t_sun = 9.327 h Q4) t_Venus = 10.327 h Q5) h_v = 10.360° Q6) Az = 338.078° Q7) el_max = 51.422° Branch 3 Q1) GMT = 2.969 h (DAY + 1) Q2) LAT = 21.309 h (DAY 0) Q3) t_sun = 9.309 h Q4) t_Venus = 10.309 h Q5) h_v = 13.142° Q6) Az = 337.879° Q7) el_max = 50.785° Branch 4 Q1) GMT = 3.969 h (DAY + 1) Q2) LAT = 21.290 h (DAY 0) Q3) t_sun = 9.290 h Q4) t_Venus = 10.290 h Q5) h_v = 15.920° Q6) Az = 337.635° Q7) el_max = 50.147° Branch 5 Q1) GMT = 4.969 h (DAY + 1) Q2) LAT = 21.272 h (DAY 0) Q3) t_sun = 9.272 h Q4) t_Venus = 10.272 h Q5) h_v = 18.694° Q6) Az = 337.343° Q7) el_max = 49.510° Part 2 Codic e Perso na (last digit) Versi on Colum n3 Colum n2 Colum n1 Q1) lon_Eart h_A [deg] Q 2) e_ x [-] Q2) e_y [- ] Q 2) e_ z [-] Q3) theta_1_ B [deg] Q4) V_1_B [km/s] Q4) gamma_1 _B [deg] Q5) a [km] Q5) e [ -] Q6) theta_2_ B 0 1 38.8 -2.89 0.8 90 0 0.697037 33 0 165.2022 962 8.494067 728 28.63292 479 94901938 5.5 0.51050 655 101.1529 454 1 1 38.8 -2.89 0.8 90 0 0.697037 33 0 165.2022 962 8.494067 728 28.63292 479 94901938 5.5 0.51050 655 101.1529 454 2 2 39.1 -3.51 0.7 90 0 0.723381 629 0 157.6119 607 9.773391 764 39.76135 968 89789028 5.3 0.59652 022 115.5325 742 3 2 39.1 -3.51 0.7 90 0 0.723381 629 0 157.6119 607 9.773391 764 39.76135 968 89789028 5.3 0.59652 022 115.5325 742 4 3 39.4 -4.12 0.6 90 0 0.749928 837 0 152.2628 715 10.91188 282 46.06881 823 85682246 8.3 0.67304 203 126.2531 703 5 3 39.4 -4.12 0.6 90 0 0.749928 837 0 152.2628 715 10.91188 282 46.06881 823 85682246 8.3 0.67304 203 126.2531 703 6 4 39.7 -4.71 0.5 90 0 0.776678 954 0 147.9937 513 11.94986 136 50.33081 11 82413130 8.8 0.73940 728 134.7111 812 7 4 39.7 -4.71 0.5 90 0 0.776678 954 0 147.9937 513 11.94986 136 50.33081 11 82413130 8.8 0.73940 728 134.7111 812 8 5 40 -5.28 0.4 90 0 0.803631 98 0 144.4004 282 12.91159 117 53.46810 244 79799340 9.4 0.79638 075 141.7134 965 9 5 40 -5.28 0.4 90 0 0.803631 98 0 144.4004 282 12.91159 117 53.46810 244 79799340 9.4 0.79638 075 141.7134 965 Codice Persona (last digit) Version Q7) V_2_B [km/s] Q7) gamma_2_B [deg] Q10) delta [deg] Q10) e [ -] Q10) rp [ -] Q11) ang(VJ,e) Q12) DV_capture [km/s] Q11) supplementare 0 1 14.17984362 29.06281185 48.50660893 2.434443855 3793289.116 209.704967 0.913898244 150.295 1 1 14.17984362 29.06281185 48.50660893 2.434443855 3793289.116 209.704967 0.913898244 150.295 2 2 13.89622342 35.92554689 29.09508943 3.98115642 5413806.862 206.961205 1.359417155 153.0388 3 2 13.89622342 35.92554689 29.09508943 3.98115642 5413806.862 206.961205 1.359417155 153.0388 4 3 13.63894496 42.03737678 17.15029456 6.70661823 7866379.266 206.472127 1.842520013 153.5279 5 3 13.63894496 42.03737678 17.15029456 6.70661823 7866379.266 206.472127 1.842520013 153.5279 6 4 13.41181879 47.60099131 8.489459794 13.51045318 13887285.27 206.508665 2.355 153.4913 7 4 13.41181879 47.60099131 8.489459794 13.51045318 13887285.27 206.508665 2.355 153.4913 8 5 13.21372074 52.77297001 1.673496706 68.47676007 62674849.09 205.441192 2.891975104 154.5588 9 5 13.21372074 52.77297001 1.673496706 68.47676007 62674849.09 205.441192 2.891975104 154.5588