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Orbital Mechanics Overview 2 MAE 155B G. Nacouzi GN/MAE155B 1 Orbital Mechanics Overview 2 • Summary of first quarter overview – Keplerian motion – Classical orbit parameters • Orbital perturbations • Central body observation – Coverage examples using Excel • Project workshop GN/MAE155B 2 Introduction: Orbital Mechanics • Motion of satellite is influenced by the gravity field of multiple bodies, however, two body assumption is usually sufficient. Earth orbiting satellite Two Body approach: – Central body is earth, assume it has only gravitational influence on S/C, assume M >> m (M, m ~ mass of earth & S/C) • Gravity effects of secondary bodies including sun, moon and other planets in solar system are ignored • Gravitational potential function is given by: = GM/r – Solution assumes bodies are spherically symmetric, point sources (Earth oblateness not accounted for) – Only gravity and centrifugal forces are present GN/MAE155B 3 Two Body Motion (or Keplerian Motion) • Closed form solution for 2 body exists, no explicit soltn exists for N >2, numerical approach needed • Gravitational field on body is given by: Fg = M m G/R2 where, M~ Mass of central body; m~ Mass of Satellite G~ Universal gravity constant R~ distance between centers of bodies For a S/C in Low Earth Orbit (LEO), the gravity forces are: Earth: 0.9 g Sun: 6E-4 g Moon: 3E-6 g Jupiter: 3E-8 g GN/MAE155B 4 Elliptical Orbit Geometry & Nomenclature V a c Periapsis R Line of Apsides Rp Apoapsis b S/C position defined by R & , R = [Rp (1+e)]/[1+ e cos()] is called true anomaly • Line of Apsides connects Apoapsis, central body & Periapsis • Apogee~ Apoapsis; Perigee~ Periapsis (earth nomenclature) GN/MAE155B 5 Elliptical Orbit Definition • Orbit is defined using the 6 classical orbital elements: – Eccentricity, i – semi-major axis, – true anomaly: position of SC on the orbit Vernal – inclination, i, is the Equinox angle between orbit plane and equatorial Ascending plane Node – Argument of Periapsis (). Angle from Ascending Node (AN) - Longitude of Ascending Node ()~Angle from to Periapsis. AN: Pt Vernal Equinox (vector from center of earth to sun on where S/C crosses first day of spring) and ascending node equatorial plane South to North GN/MAE155B 6 Sources of Orbital Perturbations • Several external forces cause perturbation to spacecraft orbit – 3rd body effects, e.g., sun, moon, other planets – Unsymmetrical central bodies (‘oblateness’ caused by rotation rate of body): • Earth: Requator = 6378 km, Rpolar = 6357 km – Space Environment: Solar Pressure, drag from rarefied atmosphere Reference: C. Brown, ‘Elements of SC Design’ GN/MAE155B 7 Relative Importance of Orbit Perturbations Reference: Spacecraft • J2 term accounts for effect from oblate earth Systems Engineering, Fortescue & Stark •Principal effect above 100 km altitude • Other terms may also be important depending on application, mission, etc... GN/MAE155B 8 Principal Orbital Perturbations • Earth ‘oblateness’ results in an unsymmetric gravity potential given by: a n GM e Jn Pn( w) Note: 1 r r J2~1E-3, n 2 J3~1E-6 where ae = equatorial radius, Pn ~ Legendre Polynomial Jn ~ zonal harmonics, w ~ sin (SC declination) • J2 term causes measurable perturbation which must be accounted for. Main effects: – Regression of nodes – Rotation of apsides GN/MAE155B 9 Orbital Perturbation Effects: Regression of Nodes Regression of Nodes: Equatorial bulge causes component of gravity vector acting on SC to be slightly out of orbit plane This out of orbit plane component causes a slight precession of the orbit plane. The resulting orbital rotation is called regression of nodes and is approximated using the dominant gravity harmonics term, J2 GN/MAE155B 10 Regression of Nodes • Regression of nodes is approximated by: 2 3 n J R cos ( i) d 2 dt 2a 2 1 e 2 2 Where, ~ Longitude of the ascending node; R~ Mean equatorial radius J2 ~ Zonal coeff.(for earth = 0.001082) n ~ mean motion (sqrt(GM/a3)), a~ semimajor axis Note: Although regression rate is small for Geo., it is cumulative and must be accounted for GN/MAE155B 11 Orbital Perturbation: Rotation of Apsides Rotation of apsides caused by earth oblateness is similar to regression of nodes. The phenomenon is caused by a higher acceleration near the equator and a resulting overshoot at periapsis. This only occurs in elliptical orbits. The rate of rotation is given by: d 3n J R 2 4 5 sin ( i) 2 dt 2 2 2 2 4a 1 e GN/MAE155B 12 Ground Track • Defined as the trace of nadir positions, as a function of time, on the central body. Ground track is influenced by: – S/C orbit – Rotation of central body – Orbit perturbations Trace is calculated using spherical trigonometry (no perturbances) sin (La) = sin (i) sin ALa Lo = + asin(tan (La)/tan(i))+Re where: Ala ~ (ascending node to SC) ~ Longitude of ascending node I ~ Inclination Re~Earth rotation rate= 0.0042t (add to west. longitudes, subtract for eastern longitude) GN/MAE155B 13 Example Ground Trace Ground trace from i= 45 deg GN/MAE155B 14 Spacecraft Horizon • SC horizon forms a circle on the spherical surface of the central body, within circle: – SC can be seen from central body – Line of sight communication can be established – SC can observe the central body GN/MAE155B 15 Central Body Observation From simple trigonometry: sin(h) = Rs/(Rs+hs) Dh = (Rs+hs) cos(h) Sw~ Swath width = 2 h Rs GN/MAE155B 16

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Orbital Mechanics, Space Technology, Cambridge University Press, Orion Books, The Cambridge Encyclopedia of Space, Space Directory, Andrew Wilson, Information Group, Encyclopedia Astronautica, Michael Rycroft

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posted: | 6/4/2011 |

language: | English |

pages: | 16 |

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