Orbital Mechanics Overview 2

1. Orbital Mechanics Overview 2 MAE 155B G. Nacouzi GN/MAE155B

2. 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

3. 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

4. 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

5. Elliptical Orbit Geometry & Nomenclature V Periapsis a c R  Line of Apsides Rp b Apoapsis S/C position defined by R & ,  is called true anomaly R = [Rp (1+e)]/[1+ e cos()] • Line of Apsides connects Apoapsis, central body & Periapsis • Apogee~ Apoapsis; Perigee~ Periapsis (earth nomenclature) GN/MAE155B

6. Orbit is defined using the 6 classical orbital elements: Eccentricity, semi-major axis, true anomaly: position of SC on the orbit inclination, i, is the angle between orbit plane and equatorial plane Argument of Periapsis (). Angle from Ascending Node (AN) to Periapsis. AN: Pt where S/C crosses equatorial plane South to North Periapsis i  Vernal Equinox  Ascending Node Elliptical Orbit Definition - Longitude of Ascending Node ()~Angle from Vernal Equinox (vector from center of earth to sun on first day of spring) and ascending node GN/MAE155B

7. 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

8. Relative Importance of Orbit Perturbations Reference: SpacecraftSystems Engineering, Fortescue & Stark • J2 term accounts for effect from oblate earth • Principal effect above 100 km altitude • Other terms may also be important depending on application, mission, etc... GN/MAE155B

9. Principal Orbital Perturbations • Earth ‘oblateness’ results in an unsymmetric gravity potential given by: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 Note: J2~1E-3, J3~1E-6 GN/MAE155B

10. 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

11. Regression of Nodes • Regression of nodes is approximated by: 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

12. 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: GN/MAE155B

13. 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 ALaLo =  + 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

14. Example Ground Trace GN/MAE155B

15. 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

16. Central Body Observation From simple trigonometry: sin(h) = Rs/(Rs+hs) Dh = (Rs+hs) cos(h) Sw~ Swath width = 2 h Rs GN/MAE155B