Full exam for Orbital Mechanics in the Aerospace Engineering degree programme at Politecnico di Milano. The document covers: Orbital Mechanics Academic Year 2024-2025 Exam – 27 January 2025 Orbital Mechanics Academic Year 2024-2025 Lecturer: Camilla Colombo Exam 27 January 2025 Duration: 2 h 45 min (plus 15 minutes of time for loading the scanned solution) PAPER SOLUTION SHEET ▪ All solution sheets
Full exam for Orbital Mechanics in the Aerospace Engineering degree programme at Politecnico di Milano. The document covers: Orbital Mechanics Academic Year 2024-2025 Exam – 27 January 2025 Orbital Mechanics Academic Year 2024-2025 Lecturer: Camilla Colombo Exam 27 January 2025 Duration: 2 h 45 min (plus 15 minutes of time for loading the scanned solution) PAPER SOLUTION SHEET ▪ All solution sheets
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Orbital Mechanics Academic Year 2024-2025 Exam – 27 January 2025 Orbital Mechanics Academic Year 2024-2025 Lecturer: Camilla Colombo Exam 27 January 2025 Duration: 2 h 45 min (plus 15 minutes of time for loading the scanned solution) PAPER SOLUTION SHEET ▪ All solution sheets must be written in clear form in pen. You can use coloured pen except red colour. Do not write in pencil. ▪ On top of each sheet, please write in capital letters your surname, name, and student matriculation number. ▪ In the solution of the exercise all the analytical and numerical procedures must be reported. Please clearly report the number of the question you are answering to. Numerical results need to have proper units of measure. ▪ The written solution needs to be submitted in Forms in PDF form at the end of the exam. ONLINE FORM ▪ You must insert in Forms the answers to the questions in the provided text boxes following the instruction given in the text. ▪ For numerical results, you must introduce only the number with sign, using the units indicated in the text but without writing the units in the result box. ▪ The point to each question will be given only if the numerical result is correct (within reasonable thresholds), the written solution will be used for validation and check purposes. SUBMISSION ▪ At the end of the exam, you will be asked if you intend to Submit your solution or Withdraw from the exam. If you select Withdraw no upload will be needed. Instead, if you select Submit, you must upload the PDF scans of the procedure and computations you performed. A failure to do so will invalidate the exam. Please put your name and matriculation number on each sheet, number each sheet, and clearly identify the questions with their number. ▪ Name the PDF file you will upload as…
Orbital Mechanics Academic Year 2024-2025 Exam – 27 January 2025 Exercise 1 – Navigation by stars A fishing ship is sailing on the Northern hemisphere, to the East of Greenwich meridian. The GPS of the ship unexpectedly breaks, and the crew tries to determine their position and local time using the stars. The only other instrument available on board is a clock that gives the 𝐺𝑀𝑇 and date at Greenwich. On December 21 (date at Greenwich), at 𝐺𝑀𝑇 [see DATA], the crew observes a star just rising on the horizon (that is, with zero elevation), with Azimuth 𝐴𝑧∗ [see DATA]. From ephemerides, the right ascension and declination of the star on that day are 𝛼∗ = 235 deg, 𝛿∗ = 20 deg. Note: The date and time (𝐺𝑀𝑇) given in the text data correspond to Greenwich meridian. The date at the position of the ship may be different from the date at Greenwich. Because the ship is located East of Greenwich, the longitude of the ship must be 𝜆 ∈ [0,180]deg. Find: Q1. The latitude of the ship’s position, 𝜑. (Insert the NUMBER without units, expressed in degrees with 2 decimal digits) (+1.5) Q2. The hour angle of the star at the observation, 𝑡∗ . (Insert the NUMBER without units, expressed in degrees with 2 decimal digits) (+1.5) Q3. The Local Sidereal Time 𝐿𝑆𝑇 on the ship’s position, at the time of observation. (Insert the NUMBER without units, expressed in hours with 3 decimal digits) (+1) Q4. The hour angle of the Sun on the ship’s position at the time of observation, 𝑡𝑆 . (Insert the NUMBER without units, expressed in hours with 3 decimal digits) (+1) Q5. The Local Apparent Time 𝐿𝐴𝑇 on the ship’s position at the time of observation. (Insert the NUMBER without units, expressed in hours with 3 decimal digits) (+1) Q6. The mean-to-apparent time correction 𝐸 at the time…
Orbital Mechanics Academic Year 2024-2025 Exam – 27 January 2025 Exercise 3 – Orbital reconfiguration Scientists decide to make observations of the NEO–Earth close encounter using a satellite already in orbit around Earth. To this end, they need to modify the orbit of the satellite, performing a single-impulse apse line rotation manoeuvre with shape change but no inclination change. The initial orbit of the satellite has its pericentre in correspondence with its ascending node. Furthermore, at a given point, called point A, the satellite has orbital radius 𝑟𝐴 [see DATA], orbital velocity 𝑣𝐴 [see DATA], and flight path angle 𝛾𝐴 [see DATA]. The manoeuvre changes the apocentre of the orbit by Δℎ𝑎 = −100 km, its pericentre by Δℎ𝑝 = +50 km and its argument of pericentre by Δ𝜔 = 5 deg, and it is performed at the manoeuvre opportunity closest to point A. Find: Q25. The semimajor axis of the initial orbit of the satellite. (Insert the NUMBER without units, expressed in km with 2 decimal digits) (+1) Q26. The eccentricity of the initial orbit of the satellite. (Insert the NUMBER without units, with 4 decimal digits) (+1) Q27. The true anomaly of point A in the initial orbit of the satellite. (Insert the NUMBER without units, expressed in degrees with 2 decimal digits) (+1) Q28. The argument of pericentre of the initial orbit of the satellite. (Insert the NUMBER without units, expressed in degrees with 2 decimal digits) (+0.5) Q29. The semimajor axis of the satellite’s orbit after the manoeuvre. (Insert the NUMBER without units, expressed in km with 2 decimal digits) (+0.5) Q30. The eccentricity of the satellite’s orbit after the manoeuvre. (Insert the NUMBER without units, with 4 decimal digits) (+0.5) Q31. The argument of pericentre of the satellite’s orbit after the…
Orbital Mechanics Academic Year 2024-2025 Exam – 27 January 2025 NUMERICAL SOLUTIONS EXERCISE 2 Q11) Q12) Q13) Q14) Q15) Q16) Q17) Q18) Version [deg] [deg] [-] [km/s] [km/s] [km] [-] [km] [0.000; [0.000; 1 0.00 180.00 0.1000 31.226; 1.443; -144902.44 1.0579 2012.95 -0.818] -0.818] [0.000; [0.000; 2 0.00 180.00 0.2000 32.615; 2.831; -45571.34 1.4845 15709.32 -0.854] -0.854] [0.000; [0.000; 3 0.00 180.00 0.3000 33.947; 4.163; -21994.39 2.4835 26258.35 -0.889] -0.889] [0.000; [0.000; 4 0.00 180.00 0.1500 31.928; 2.145; -75220.38 1.2008 8730.85 -0.836] -0.836] [0.000; [0.000; 5 0.00 180.00 0.2500 33.287; 3.504; -30572.15 1.9170 21662.79 -0.872] -0.872] 14 of 16
Orbital Mechanics Academic Year 2024-2025 Exam – 27 January 2025 NUMERICAL SOLUTIONS EXERCISE 3 Q25) Q26) Q27) Q28) Q29) Q30) Q31) Q32) Q33) Version [km] [-] [deg] [deg] [km] [-] [deg] [deg] [m/s] [80.50; 1 7935.78 0.0834 209.05 0.000 7910.78 0.0741 5.00 243.94 -5.80; 0.00] [100.51; 2 8953.91 0.1485 162.36 0.000 8928.91 0.1405 5.00 216.31 -1.30; 0.00] [-76.11; 3 7491.76 0.0539 141.68 0.000 7466.76 0.0441 5.00 55.91 -8.93; 0.00] [127.63; 4 8117.17 0.1837 161.68 0.000 8092.17 0.1750 5.00 211.18 0.42; 0.00] [73.44; 5 9455.16 0.0995 214.33 0.000 9430.16 0.0919 5.00 230.62 -3.55; 0.00] 16 of 16
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