Aerospace Engineering - Orbital Mechanics

Full exam

1 of 4 Orbital Mechanics Academic Year 2020-2021 Lecturer: Camilla Colombo Exam 01 September 2021 Duration: 1.5 h All solution sheets must 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. Last digit of Person Code (Codice Persona) 0 or 2 or 4 or 6 or 8 1 or 3 or 5 or 7 or 9 Exam Version 1 2 PART 1 Exercise 1 – Orbital manoeuvre A satellite performs a manoeuvre at the orbital position r around the Earth characterised by the orbital elements given in DATA. A manoeuvre is performed to exclusively rotate of an angle α [see DATA] the transversal component of the spacecraft velocity at the orbital position r. The rotation is performed positively around the +������������� direction. Data Earth data: ������������ ������������= 398600 km 3/s 2 ������������ ������������= 6378.16 km Version Semi - major axis [km ] Eccentricity Inclination [deg] RAAN [deg] Anomaly of perigee [deg] True anomaly [deg] α [deg] 1 -13 0000 1.1 40 30 50 10 3 2 -120000 1.2 30 40 35 20 4 1. Compute the transversal component of the velocity at point r before the manoeuvre. [Output in km/s with 2 decimal digits] 2. Compute the radial component of the velocity at point r before the manoeuvre. [Output in km/s with 2 decimal digits] 2 of 4 3. Is the shape of the orbit changed? 4. Compute the inclination of the orbit after the manoeuvre. [Output in degrees with 2 decimal digits] 5. Compute the right ascension of the ascending node after the manoeuvre. [Output in degrees with 2 decimal digits] 6. Compute the anomaly of perigee of the orbit after the manoeuvre. [Output in degrees with 2 decimal digits] 7. Find the semi-major axis of the orbit after the manoeuvre. [Output in km with 0 decimal digit] 8. Find the eccentricity of the orbit after the manoeuvre. [Output with 4 decimal digits] 9. Find the angle of which the angular momentum vector is rotated. [Output in degrees with 2 decimal digits] 10. Compute the time interval from the perigee to the manoeuvre point on the initial orbit. [Output in seconds with 2 decimal digits] Exercise 2 – Time measurements and spherical geometry The landing of the NASA Perseverance rover at Mars is observed from a ground station located at the following latitude ������������ and longitude ������������ (see DATA). The declination of Mars at the time of the observation is δ [see DATA]. Version Latitude ������������ [deg] Longitude ������������ [deg] Declination of Mars δ [deg] 1 46.52 S 168.37 E - 21.58 2 34.23 S 143.25 E - 22 .20 11. Compute the maximum elevation of Mars above the horizon as seen from the control centre. [Output in degrees with 2 decimal digits] 12. Find the azimuth of Mars when it rises with respect to the ground station horizon. [Output in degrees with 2 decimal digits] 13. Find the time angle of Mars when it rises with respect to the ground station horizon. [Output in degrees with 2 decimal digits] 14. Find the azimuth of Mars when it sets with respect to the ground station horizon. [Output in degrees with 2 decimal digits] 15. Find the time angle of Mars when it sets with respect to the ground station horizon. [Output in degrees with 2 decimal digits] 3 of 4 Orbital Mechanics Academic Year 2020-2021 Lecturer: Camilla Colombo Exam 01 September 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 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. Last digit of Person Code (Codice Persona) 0 or 2 or 4 or 6 or 8 1 or 3 or 5 or 7 or 9 Exam Version 1 2 PART 2 – INTERPLA NETARY TRAJECTORY WITH 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 km 3/s 2 gravitational constant of the Sun ������������ ������������������������������������������������ℎ =398600 km 3/s 2 gravitational constant of Earth ������������ ������������������������������������������������ℎ =149.6∙10 6 km Earth orbital radius (around the Sun) ������������ ������������ Version v1 [km/s] 1 [ 1.2715; 2.7503; 0 ] 2 [1.9753 ; 1.7151; 0] A spacecraft performs a fly-by with the Moon, after which its relative velocity with respect to the Moon is ������������ 1 [see DATA] given in the radial-transversal-out of plane reference frame of the Moon. The Moon fly-by puts the spacecraft on a hyperbolic orbit with respect to the Earth. The subsequent escape from Earth takes place at the autumn equinox, and the escape velocity is parallel to the Earth’s velocity with respect to the Sun and with the same direction and orientation. The resulting heliocentric orbit intersects with the orbit of Mars. Consider the Moon’s orbit to be circular and to lie on the same plane as the Earth’s orbit around the Sun. Consider Earth and Mars orbits to be circular and ecliptic too. 4 of 4 1- Find the absolute velocity with respect to the Earth after the Moon fly-by and express it in the radial- transversal-out of plane reference frame of the Moon at the fly-by point. [Output three components in km/s with 2 decimal digits in the format [������������ Part 1 Version Last digit a e i Om om2 f alpha Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 1 0,2,4,6,8 -130000 1.1 40 50 30 10 3 7.960434 0.729878 'NO' 41.56754 33.91696 47.03338 -130000 1.1 2 1,3,5,7,9 -120000 1.2 30 35 40 20 4 5.845855 1.127677 'NO' 32.44461 46.1142 29.76885 -120000 1.2 Version Last digit Q9 Q10 lat lon dec Q11 Q12 Q13 Q14 Q15 1 0,2,4,6,8 3 284.2699 -46.52 168.37 -21.58 65.06 122.311 245.3492 237.689 114.6508 2 1,3,5,7,9 4 1417.391 -34.23 143.25 -22.2 77.97 117.1937 253.8799 242.8063 106.1201 Part 2 Version Q1) Velocity with respect to Earth after flyby, in RTH [km/s] Q2) Semimajor axis of Earth- centred hyperbola [km] Q3) Eccentricity of Earth- centred hyperbola Q4) Hyperbolic excess velocity at Earth escape [km/s] Q5) True anomaly at Earth escape [deg] Q6) Velocity wrt Sun at escape, in HECI [km/s] Q8) Semimajor axis of heliocentric orbit [km] Q9) Eccentricity of heliocentric orbit Q10) Initial angular position of Mars [deg] Q11) Dv to rendezvous with Mars [km/s] 1 [1.27;3.77;0] -28999 13.5112 3.71 94.24 [0;3 3.49;0] 203384879 0.2644 42.63 5.78 2 [1.98;2.73;0] -42863 8.1005 3.05 97.09 [0;32.83;0] 190635316 0.2153 45.18 3.26