05-02-2013 Astrodynamics 1 05-02-2013 2 05-02-2013 Orbit Perturbations • Secular variations – langsomt, lineært • Short-period variations • Long-period variations Element Short Period Long Period Orbit Time Third-Body Perturbations Right ascension of the ascending node Moon 0.00338 ( cos i ) / n Sun 0.00154 ( cos i ) / n Argument of perigee 2 Moon 0.00169 ( 4 - 5 sin i ) / n 2 Sun 0.00077 ( 4 - 5 sin i ) / n 3 05-02-2013 Perturbations Because of a Non-spherical Earth Right ascension of the ascending node -2.064741014 a-7/2 ( cos i ) (1- e2 )2 J2 cos(90) = 0,0 Argument of perigee 1.032371014 a-7/2 ( 4 -5 sin 2 i ) (1- e2 )2 J2 sin²(63,43495) = 0,8 4 05-02-2013 Perturbations From Atmospheric Drag Perturbations from Solar Radiation 6 aR 4.510 (1 r)A/m r: Reflection factor (1: specular reflection, 0: absorption) A: Satellite cross-section (exposed to the Sun) – m² m: Satellite mass –kg a: m/s² Perturbations From Atmospheric Drag 5 05-02-2013 Perturbations From Atmospheric Drag Perturbations From Atmospheric Drag 6 05-02-2013 Perturbations From Atmospheric Drag Ørsted H = 700 km P = 100 min T = 30-40 years 7 05-02-2013 http://www.heavens-above.com/ Astrodynamics II • Orbital elements • Orbit Maneuvering • Launch Windows • Orbit Maintenance • Earth Coverage 8 05-02-2013 a Den halve storakse e Ekscentriciteten i Inklinationen Den opstigende knudes længde (i forhold til forårspunktet Perifokus’ vinkelafstand fra den opstigende knude T Perifokustiden Low Earth orbit (LEO) Mir: 350km altitude, near circular a: 6730 km, e: 0.00 P: 91 minutes n: 15.725 omløb/døgn i: 51.6° Baikonur Launch site Change in right ascension of the ascending node: -5.1 deg/day 9 05-02-2013 10 05-02-2013 Landsat 7 orbit (LEO) Landsat 7 is an Earth resources spacecraft which images the Earth's surface in visible and infrared light. Near polar orbit of 700km altitude a: 7080 km P: 98 min. i: 98.8° Change in right ascension of the ascending node: +1 deg/day Perturbations Because of a Non-spherical Earth Right ascension of the ascending node -2.064741014 a-7/2 ( cos i ) (1- e2 )2 J2 Argument of perigee 1.032371014 a-7/2 ( 4 -5 sin 2 i ) (1- e2 )2 J2 11 05-02-2013 12 05-02-2013 13 05-02-2013 14 05-02-2013 15 05-02-2013 Geosynchronous Orbits (GEO) A geosynchronous orbit is an orbit which has an orbital period close to that of the Earths rotation. A geostationary orbit is a special case of the geosynchronous orbit where inclination = 0° and the period is equal to the rotation period of the earth (approx 1436 minutes). a: 42160 km e: 0 i: 0 P: 1436 min 16 05-02-2013 TDRS (USA) 17 05-02-2013 Geosynchronous Transfer Orbits (GTO) This is a 600 x 35,700km 28° inclination a = 24530 km i = 28 e=0.71 Change in right ascension of the ascending node: -0.31 deg/day Change in argument of perigee: +0.53 deg/day 18 05-02-2013 Molniya orbits This is a 400 x 40,000km 63.4° inclination a = 26600 km i = 63.4 e=0.75 P=11.967 hrs. Change in right ascension of the ascending node: -0.030 deg/day Change in argument of perigee: 0.000 deg/day 19 05-02-2013 Mid Earth Orbit (MEO or GPS) Circular orbit 60.0° inclination a = 26600 km i = 60 e=0.00 P=11.967 hrs Change in right ascension of the ascending node: -0.033 deg/day Change in argument of perigee: +0.008 deg/day 20 05-02-2013 21 05-02-2013 Highly Eccentric Orbits (HEO) A number of scientific satellites, particularly orbiting observatories, use highly eccentric orbits with apogee's of over 100,000km. The reason for using these orbits is generally to permit continues observations of celestial objects without the earth blocking the view every 30 to 40 minutes. The orbit of the Chandra X-ray observatory is a 9600 x 139000km, 28.4° inclination orbit of 3809 minute period (63.5h orbit). a = 80700 km i = 28.4 e=0.80 P=63.5 hrs Change in right ascension of the ascending node: -0.009 deg/day Change in argument of perigee: +0.015 deg/day 22 05-02-2013 Orbit Transfer Coplanar transfer Modify the velocity vector V 2 / 2 / r / 2a 23 05-02-2013 r r V 2 / 2 / r / 2a r r r V 2 / 2 / r / 2a 24 05-02-2013 r r r V 2 / 2 / r / 2a 25.