Orbits Supplement GEOSTATIONARY ORBIT PERTURBATIONS INFLUENCE OF ASPHERICITY OF THE EARTH: The gravitational potential of the Earth is no longer µ/r, but varies with longitude. A tangential acceleration is created, depending on the longitudinal location of the satellite, with four points of stable equilibrium: two stable equilibrium points (L 75° E, 105° W) two unstable equilibrium points ( 15° W, 162° E) This tangential acceleration causes a drift of the satellite longitude. Longitudinal drift d'/dt in terms of the longitude about a point of stable equilibrium expresses as: (d/dt)2 - k cos 2 = constant Orbits Supplement GEO PERTURBATIONS (CONT'D) INFLUENCE OF EARTH ASPHERICITY VARIATION IN THE LONGITUDINAL ACCELERATION OF A GEOSTATIONARY SATELLITE: Orbits Supplement GEO PERTURBATIONS (CONT'D) INFLUENCE OF SUN & MOON ATTRACTION Gravitational attraction by the sun and moon causes the satellite orbital inclination to change with time. The evolution of the inclination vector is mainly a combination of variations: period 13.66 days with 0.0035° amplitude period 182.65 days with 0.023° amplitude long term drift The long term drift is given by: -4 dix/dt = H = (-3.6 sin M) 10 ° /day -4 diy/dt = K = (23.4 +.2.7 cos M) 10 °/day where M is the moon ascending node longitude: M = 12.111 -0.052954 T (T: days from 1/1/1950) 2 2 2 2 cos d = H / (H + K ); i/t = (H + K ) Depending on time within the 18 year period of M d varies from 81.1° to 98.9° i/t varies from 0.75°/year to 0.95°/year Orbits Supplement GEO PERTURBATIONS (CONT'D) INFLUENCE OF SUN RADIATION PRESSURE Due to sun radiation pressure, eccentricity arises: EFFECT OF NON-ZERO ECCENTRICITY L = difference between longitude of geostationary satellite and geosynchronous satellite (24 hour period orbit with e0) With non-zero eccentricity the satellite track undergoes a periodic motion about the subsatellite point at perigee. The MAXIMUM LONGITUDINAL VARIATION Lmax is given by: Lmax = 114 e = 0.1° (assuming e = 0.001) and is reached 6 HOURS BEFORE/AFTER PERIGEE Orbits Supplement EFFECT OF NON-ZERO ORBIT INCLINATION The subsatellite point describes a 'figure of eight' around the desired station. Given residual inclination i (degrees): peak longitude variation (degrees): -3 2 Lmax =4.36 10 i peak latitude variation (degrees): lmax = i -5 typical i = 0.05°, hence Lmax = 10 ° ( negligible compared to other drifts) Orbits Supplement STATION-KEEPING REQUIREMENTS The combined non-zero inclination and eccentricity keeps the satellite from being perfectly geostationary (daily movement). The East-West drift moves the satellite away from its nominal longitude (long term movement) The mission requires that the satellite which displays daily+long term movements should remain within a specific solid angle as seen from the center of the earth (window). Example of a typical window: longitude variation L= 0.1° latitude variation l = 0.1° The necessary compensation of forces which drive the satellite outside the window is obtained by accurate firing of reaction equipment (thrusters) on-board the satellite in order to keep it on if station. Orbits Supplement SATELLITE DRIFT EXAMPLE SIRIO satellite subpoint trace on June 1 and 2, 1978: Orbits Supplement ORBIT CORRECTIONS NORTH/SOUTH STATION KEEPING (CORRECTION OF ORBIT INCLINATION): Forces acting perpendicular to the orbit plane are delivered by thrusters at the nodes of the orbit. Cost is of the order of 50 m/s per year. EAST/WEST STATION KEEPING (CORRECTION OF SATELLITE DRIFT): Forces acting in the orbital plane, in the direction of satellite velocity or in opposite direction depending on the drift direction, are delivered by thrusters when the satellite is expected to reach the boundaries of the station keeping window. Note that part of the satellite drift can be induced while North/South station keeping maneuvers are performed, due to thrusters misalignment relative to the perpendicular to the orbital plane. Cost is of the order of 2-10 m/s per year depending on the satellite location with respect to the points of equilibrium. Orbits Supplement ORBIT CORRECTIONS INCLINATION CONTROL iX = VN cos S / VS ; iy = VN sin S / VS where S = satellite right ascension = + +M Required Velocity Increment to Compensate for Orbit Inclination Year Velocity increment (m/s) 1980 41.24 1981 42.58 1982 44.19 1983 45.99 1984 47.70 1985 49.16 1986 50.20 1987 50.72 1988 50.62 1989 49.96 1990 48.77 1991 47.21 1992 45.48 1993 43.71 1994 42.15 Orbits Supplement EAST-WEST STATION KEEPING Orbits Supplement EAST-WEST STATION KEEPING Longitude drift and east-west corrections (a) when the satellite is located far away from the equilibrium point, and (b) when the satellite is located near the equilibrium point. Annual velocity increment for east-west station keeping versus station longitude. Orbits Supplement EXERCISE: ORBITAL MECHANICS, LAUNCHING & POSITIONING DATA: Universal Gravitational Constant G = 6.673 x 10-11 N m2 kg-2 Mass of Earth M = 5.977 x 1024 kg 6 Equatorial Radius of Earth RE = 6.3782 x 10 m Obliquity of the Ecliptic = 23.4° 1. A communications satellite is placed in a geostationary Earth-orbit at an altitude of 35,788 km. Calculate the following: (i) the orbital period; (ii) the maximum eclipse period (from geometry alone). 2. What conditions are necessary for an orbit to be geostationary (GEO)? Explain why in practice GEO satellites depart from these conditions. 3. Sketch the trajectory of a practical GEO satellite as seen from a ground-station and explain its shape. 4. By means of a labelled diagram, describe five independent Keplerian Elements which can be used to describe the orbital trajectory of a satellite. 5. A NOAA meteorological satellite is launched into an 870 km altitude, circular, Sun- synchronous orbit, with an ascending node at 14.30 local solar time. (i) Define what is meant by a Sun-synchronous orbit, and explain how the effect occurs. (ii) Calculate the satellite's orbital velocity (iii) Calculate the orbital inclination angle (iv) Find the eclipse period at the Vernal Equinox NOTE: the period of time which an orbit spends in eclipse is given by: T x (1/180°) x cos-1 [(1 - n2)1/2 / sin β] Where cos-1 (x) is in degrees; n = R / (R + h); R = Earth's radius, h= satellite height; β = the angle between the Sun's rays and the orbit normal; T = orbital period. Provided cos β < n, otherwise the orbit is not eclipsed at all. 6. A mobile communications satellite is placed in a 'Molniya' orbit with a semi-major axis of 26,556 km and an eccentricity of 0.71. Calculate the following for the orbit: (i) perigee height; (ii) apogee radius; (iii) orbital speed at apogee. 7. Briefly describe the properties of a Molniya orbit and explain why it is most suitable for communications into high latitudes. CONT/D OVER Orbits Supplement 8. An orbit has an apogee height of 37,500 km, and inclination of 20° and an eccentricity of 0.7. Calculate: (i) perigee radius; (ii) orbital period; (iii) nodal regression rate; (iv) rate of change of the argument of perigee. 9. By means of a sketch diagram, explain what is meant by a Hohmann Transfer Ellipse. 10. Give one advantage and one disadvantage of the Hohmann Transfer over direct ascent from one orbit to another. 11. A satellite is to be deployed from the cargo bay of the STS (Space Shuttle), which is in a low-Earth circular parking orbit of 300 km altitude and 28.5° inclination. The satellite has a boost-motor which gives it an initial delta-V of 1.942 km s-1 to put the satellite into a transfer orbit. Calculate the following: (i) the orbital velocity of the satellite in the parking orbit; (ii) the maximum apogee height attainable by the transfer orbit; (iii) the additional delta-V required to circularise the orbit at apogee; (iv) the minimum delta-V required to change the inclination of the orbit to 90°. 12. Briefly explain the advantages of possessing a launch site close to the Equator. 13. A communications satellite is to be placed into a geostationary Earth orbit from the Kourou launch site with the following characteristics: Launch: Dual launch configuration with the ARIANE 3 launch vehicle Geostationary Transfer Orbit (GTO): perigee height 200 km apogee height 35,586 km inclination 7° Launch weight capability: 1140 kg into GTO Spacecraft Configuration: Body-fixed stabilisation with deployable solar arrays Station keeping accuracy: +/- 0.05° N-S and E-W End-of-Life satellite mass: 510 kg Propulsion: Apogee Kick motor (Solid Propellant) specific impulse (Isp): 295 seconds Attitude and Orbit control (monopropellant Hydrazine): Isp : 216 seconds Propellant provision for drift orbit control: 13 kg Propellant provision for attitude control: 2.3 kg Propellant provision for E-W station-keeping: 3.2 kg (i) Briefly describe the typical orbit transfer sequence from launch to final orbit. Calculate the velocity increment (delta-V) required for the injection of the satellite into GEO at the apogee of the GTO. (ii) Determine the apogee motor propellant mass. (iii) Estimate the required propellant mass required for station-keeping during the lifetime of the satellite (assume 10 years). State the beginning-of-life satellite mass (dry mass). Orbits Supplement EXERCISE: ORBITAL MECHANICS, LAUNCHING & POSITIONING -ANSWERS 1. (i) Orbital period of a GEO satellite: T2 = (4 2 / GM) a3 => T2 = (4 2 / 3.988 X 1014) ( [35788 + 6378] x 103)3 => T = 86148 s = 23 hrs 56 mins (ii) Maximum Eclipse period: = = R E 6378 / (35788 + 6378) RE +h => θ = 8.7° Teclipse = Torbit x (2θ / 360°) => 86164.1 x 17.4 / 360 = 69 mins (72 mins when the apparent motion of the Sun and penumbra is taken into account) 2.