Aerospace Engineering - Orbital Mechanics

Full exam

1 of 5 Orbital Mechanics Academic Year 20 20 -20 21 Lecturer: Camilla Colombo Exam 30 June 20 21 Duration: 1.5 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. Second to last digit of Person Code (Codice Persona) 0 or 1 2 or 3 4 or 5 6 or 7 8 or 9 Exam Version 1 2 3 4 5 PART 1 (16 PT) E x e r c i s e 1 – O r b i t a l m a n o e u v r e Earth Data : • �������= 398600 km3/s2 • ������������= 6378 .16 km • ������2= 0.0010826 Part 1/2) A satellite, named servicer, is designed to perform maintenance and repairs of two different satellites in Low Earth Orbit (LEO), namely SAT -1 and SAT -2. To perform the repair and servicing tasks, the servicer shall reach SAT -1 and SAT -2 orbits, in that or der. The servicer is injected into a circular orbit with position and velocity equal Version 1 2 3 4 5 Δθ 40° 50° 60° 70° 80° �0 [ 4478 .9 1608 .3 5638 .2 ] ������������ ������0 [ −5.5240 −1.1389 4.7131 ] ������������ /� ℎ0 [ 14001 −52255 3783 ] ������������2/� ΔΩ 10° 20° 30° 40° 50° 2 of 5 to �0, ������0 and ������0 (see the table). The first satellite to be serviced, SAT -1, lies on the same orbit as the servicer injection orbit, but with an argument of latitude (anomaly of perigee + true anomaly) of ������1= ������������+ Δθ , with Δθ (see the table) . The servicer is required to perform an in -plan e phasing manoeuvre sequence to reach SAT -1 in two revolutions of both satellites. 1. Compute the Keplerian orbital elements of the servicer injection orbit. a. The semi -major axis of the servicer injection orbit [Output in km with 1 decimal digits] b. The incli nation of the servicer injection orbit [Output in degrees with 2 decimal digits] c. The right ascension of the ascending node of the servicer injection orbit [Output in degrees with 2 decimal digits] d. The argument of latitude of the servicer injection orbit me asured from the right ascension of the ascending node (as the orbit is circular the argument of latitude is used in place of the true anomaly) [Output in degrees with 2 decimal digits] 2. Compute the semi -major axis of the phasing orbit to reach SAT -1 with th e servicer. [Output in km with 1 decimal digits] 3. Compute the eccentricity of the phasing orbit to reach SAT -1 with the servicer. Consider four significant digits for the eccentricity. [Output with 4 decimal digits] 4. Compute the magnitude of the total Δ������ to reach SAT -1 with the phasing orbit manoeuvre sequence. [Output in km/s with 2 decimal digits] Part 2/2) After the first repair tasks, the servicer shall move to the orbit of SAT -2, which is characterised by the same orbital elements of SAT -1 orbit, apart from for the Right Ascension of the Ascending Node ( RAAN ). Particularly, the RAAN of SAT -2 is given by Ω2= Ω1+ ΔΩ , with ΔΩ (see the table). The servicer is required to use an intermediate orbit to change the RAAN of the satellite, exploiting the node regression provided by the J 2 perturbation. The waiting time on the intermediate orbit lasts 100 days. ( HINT : the intermediate orbit shall provide a natural RAAN evolution, Ω̇= ������( a,e,i,J2 , μE), to reach SAT -2 in the fixed time). Assume the analytical ex pression of the secular effect of J2 perturbation. 5. Compute the intermediate orbit inclination to perform the RAAN phasing between the orbit of SAT -1 and the orbit of SAT -2. NOTE: the intermediate orbit must have same energy and eccentricity of SAT -1 and SA T-2 orbits. [Output in degrees with 2 decimal digits] 6. Compute the total Δ������ to inject the servicer from the SAT -1 orbit (excluding initial in -orbit phasing of Q4) to SAT -2 orbit exploiting the intermediate orbit of Q5. [Output in km/s with 2 decimal digits] 3 of 5 E x e r c i s e 2 – T i m e M e a s u r e m e n t s a n d S p h e r i c a l G e o m e t r y During the operations of servicing of satellite 2, a dangerous operation of refueling of the SAT2 tank is performed. This operation is required to be closely monitored from ground, from the ground station located at the following latitude ������ and longitude �: Version V1 V2 V3 V4 V5 ������ 45.47° N 48.47° N 51.47° N 54.47° N 57.47° N � 10° E 15° E 20° E 25° E 30° E Consider that the two satellites, servicer and SAT2, are in the same exact position during the event. The event takes place at the following orbital position in the Earth Inertial Equatorial frame: V1 V2 V3 V4 V5 �0 [km] �0 [km] �0 [km] �0 [km] �0 [km] 4681.950 900.198 5630.828 4775.370 167.589 5621.832 4756.366 -572.970 5611.232 4624.523 -304.609 5599.020 4381.891 -2010.470 5585.184 Consider that the event takes place the following Greenwich Mean Time (GMT): V1 V2 V3 V4 V5 24/06 14:00:00 25/06 15:00:00 26/06 16:00:00 27/06 17:00:00 28/06 18:00:00 Earth data: Equatorial radius of the Earth: ������������ = 6378.1 6 km Earth's gravitational constant: ������� = 398600 km^3/s^2 1. Compute right ascension and declination of the event. [Output in degrees with 2 decimal digits] 2. Compute the Local Civil Time (LCT) at the ground station to understand the work shifts of the ground operators. [Output in h:min:sec] 3. Compute the Local Apparent Time (LAT). Assume the apparent and the mean time to coincide. [Output in h:min:sec] 4. Compute th e hour angle at the monitoring time. [Output in degrees with 2 decimal digits] 5. Compute the Azimuth and Elevation of the event. [Output in degrees with 2 decimal digits] 4 of 5 Orbital Mechanics Academic Year 2020 -2021 Lecturer: Camilla Colombo Exam 30 June 2021 Duration: 1.5 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 ana lytical 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. Second to last digit of Person Code (Codice Persona) 0 or 1 2 or 3 4 or 5 6 or 7 8 or 9 Exam Version 1 2 3 4 5 PART 2 – INTERPLA NETARY TRAJECTORY WITH PLANET FLYBY 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 ������������������� = 1.3271 ∙10 11 km3/s2 gravitational constant of the Sun Table 1 Input data table for Part 2 (flyby) Version h1 [km ^2 /s] e1 Vr [km/s] β [deg] ������ [deg] SOI radius [km] ������������ [������������ � �� ] 1 1.5357e+10 [0.1331; -0.2305; 0] 1.00 -34.23 85 54759700 37931000 2 1.5474e+10 [0.1396; -0.2417; 0] 0.9 0 -37.92 75 54759700 37931000 3 1.6268e+10 [0.2069; -0.3583; 0] 1.1 0 -40.98 65 54759700 37931000 4 2.1719e+10 [0.1318; -0.2283; 0] 0.7 0 -34.24 85 51705000 5794000 5 2.1579e+10 [0.1210; -0.2096; 0] 0.6 0 -36.23 80 51705000 5794000 A spacecraft is orbiting on a n ecliptic heliocentric orbit characterised by a specific angular momentum of magnitude h1 [see DATA] and the eccentricity vector e1 [see DATA] expressed in the Sun -centred heliocentric reference frame. At the orbital position characterised by having a radial spacecraft velocity of Vr [see DATA ] BEFORE passing the point of maximum radial velocity , the spacecraft meets with a planet an d performs a fly -by with it. Consider the orbit of the planet to be circular and belonging to the ecliptic plane. Heliocentric incoming orbit 1) Find the ecliptic celestial longitude b of the peri centre of the spacecraft orbit [Output in deg with 2 decimal digit s]. 2) Find the maximum radial velocity of the spacecraft on its orbit, and the corresponding true anomaly [Output in km/s and deg with 2 decimal digits]. 5 of 5 3) Find the true anomaly of the spacecraft at t he fly -by [Output in deg with 2 decimal digit s]. 4) Find the radius of the planet’s orbit around the Sun [Output in km with 0 decimal digit s]. 5) Find the component s of the spacecraft velocity in the radial -transversal -out of plane reference frame at the fly -by position [Output in km/s with 2 decimal digits]. Fly -by characterisation The fly -by hyperbola is such that the ecliptic celestial longitude (centred in the planet) of its pericentre is β [see DATA] and the entry relative velocity is rotated of a positive angle [see DATA] around the planet pole direction. Assume the planet pole and the pole of the ecliptic directions to coincide. 6) Find the magnitude of the velocity at infinity [Output in km /s with 2 decimal digits]. 7) Find the pericentre radius of the fly -by hyperbola [Output in km with 0 decimal digit s]. 8) Find the magnitude of the delta -velocity provided by the unpowered fly -by [Output in km/s with 2 decimal digits]. 9) Find the component s of the delta -velocity provided by the fly -by in the Sun -centred heliocentric reference frame [Output in km/s with 2 decimal digits]. 10) Find the component s of the delta -velocity provided by the fly -by in the radial -transversal -out of plane reference frame of the heliocentric spacecraft [Output in km/s with 2 decimal digits]. 11) Draw the velocities triangle of the fly -by in the radial -transversal -out of plane reference frame of the heliocentric spacecraft orbit . Clearly indicate [Answer on sheet]: a. The reference f rame with the direction of the three axes b. Direction and magnitude of the incoming and outgoing heliocentric velocity on Orbit 1 and Orbit 2 c. Direction and magnitude of the incoming and outgoing relative velocity with respect to the planet d. The direction and magnitude of the planet velocity 12) Compute the change of orbital elements caused by the fly -by: a. Change of eccentricity magnitude [Output with 4 decimal digit s]. b. Change of semi -major axis [Output in km with 0 decimal digit s]. c. Change of argument of peri centre [Output in deg with 2 decimal digits]. 13) Compute the time the spacecraft passes within the planet’s SOI [see DATA] [Output in days with 4 decimal digits ]. Part 1 Exe 1 Versions Δf ΔΩ Q1a Q1b Q1c Q1d Q2 Q3 Q4 Q5 Q6 [deg] [deg] [km] [deg] [deg] [deg] [km] [-] [km/s] [deg] [km/s] 1 40 10 7378.138826 86.00 15.00 50.00 7102.28 0.0388 0.2883 86.96 0.2461 2 50 20 7378.138826 86.00 15.00 50.00 7032.48 0.0492 0.3658 87.92 0.4919 3 60 30 7378.138826 86.00 15.00 50.00 6962.33 0.0597 0.4457 88.87 0.7376 4 70 40 7378.138826 86.00 15.00 50.00 6891.82 0.0706 0.5281 89.83 0.9832 5 80 50 7378.138826 86.00 15.00 50.00 6820.95 0.0817 0.6132 90.79 1.2286 Exe 2 Column1 Q1 alpha Q1 delta Q2 day Q2 m Q2 hour Q3 day Q3 m Q3 hour Q4 Q5 el Q5 az [deg] [deg] d m h:m:s d m h:m:s deg deg deg 1 10.88343824 49.74491844 24 6 15:00:00 24 6 14:40:00 40 63.02415759 293.6981435 2 2.009939175 49.63692717 25 6 16:00:00 25 6 16:00:00 60 51.72989892 295.1040337 3 353.1310306 49.50999185 26 6 17:00:00 26 6 17:20:00 80 41.69589086 301.086877 4 356.231474 50.38401342 27 6 19:00:00 27 6 18:40:00 100 34.23317869 310.5765156 5 335.3537119 49.19944956 28 6 20:00:00 28 6 20:00:00 120 27.55095551 320.3374757 Part 2 Version Ecliptic celestial longitude of pericentre of orbit 1 [deg] Max. radial velocity orbit 1 [km/s] True anomaly at max. vr point [deg] True anomaly in orbit 1 at flyby [deg] Radius of planet's orbit [km] Velocity of s/c before flyby, in RTH frame [km/s] |v_inf| [km/s] r_p flyby [km] magnitude of the delta - velocity [km/s] DV in HECI [km/s] 1 -60.00 2.30 90 25.77 1433485311 [1.00, 10.71, 0.00 ] 1.48 8313542 2.00 [-1.65, 1.13, -0.00 ] 2 -59.99 2.39 90 22.08 1433507672 [ 0.90, 10.79, 0.00 ] 1.48 11154383 1.80 [ -1.42, 1.11, -0.00 ] 3 -60.00 3.38 90 19.02 1433469089 [ 1.10, 11.35, 0.00 ] 2.05 7792029 2.20 [ -1.66, 1.44, -0.00 ] 4 -60.00 1.61 90 25.76 2872493226 [ 0.70, 7.56, 0.00 ] 1.04 2591431 1.40 [ -1.16, 0.79, -0.00 ] 5 -60.00 1.49 90 23.77 287257524 5 [ 0.60, 7.51, 0.00 ] 0.93 3695226 1.20 [ -0.97, 0.71, -0.00 ] Version DV in RTH [km/s] Delta e Delta a [km] Delta AoP [deg] Time in SOI [day] 1 [-2.00, 0.00, -0.00 ] 0.0000 73078 51.54 594.8267 2 [ -1.80, 0.00, -0.00 ] 0.0001 267859 44.16 607.9454 3 [ -2.20, 0.00, -0.00 ] 0.0000 190844 38.04 490.4744 4 [ -1.40, -0.00, -0.00 ] 0.0000 -131003 51.52 943.0216 5 [ -1.20, 0.00, -0.00 ] 0.0000 21731 47.55 1025.8995