1. GN/MAE155B1 Orbital Mechanics Overview 2 MAE 155B G. Nacouzi

2. GN/MAE155B2 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

3. GN/MAE155B3 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

4. GN/MAE155B4 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: F g = M m G/R 2 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

5. GN/MAE155B5 Elliptical Orbit Geometry & Nomenclature Periapsis Apoapsis Line of Apsides R ac V Rp b Line of Apsides connects Apoapsis, central body & Periapsis Apogee~ Apoapsis; Perigee~ Periapsis (earth nomenclature) S/C position defined by R &, is called true anomaly R = [Rp (1+e)]/[1+ e cos( )]

6. GN/MAE155B6 Elliptical Orbit Definition 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 - Longitude of Ascending Node (  )~Angle from Vernal Equinox (vector from center of earth to sun on first day of spring) and ascending node i Vernal Equinox   Ascending Node Periapsis

7. GN/MAE155B7 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’

8. GN/MAE155B8 Relative Importance of Orbit Perturbations 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... Reference: Spacecraft Systems Engineering, Fortescue & Stark

9. GN/MAE155B9 Principal Orbital Perturbations Earth ‘oblateness’ results in an unsymmetric gravity potential given by: where a e = equatorial radius, P n ~ Legendre Polynomial J n ~ 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

10. GN/MAE155B10 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

11. GN/MAE155B11 Regression of Nodes Regression of nodes is approximated by: Where,  ~ Longitude of the ascending node; R~ Mean equatorial radius J 2 ~ Zonal coeff.(for earth = 0.001082) n ~ mean motion (sqrt(GM/a 3 )), a~ semimajor axis Note: Although regression rate is small for Geo., it is cumulative and must be accounted for

12. GN/MAE155B12 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:

13. GN/MAE155B13 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 A La Lo =  + asin(tan (La)/tan(i))+Re where: A la ~  (ascending node to SC)  ~ Longitude of ascending node I ~ Inclination Re~Earth rotation rate= 0.0042t (add to west. longitudes, subtract for eastern longitude)

14. GN/MAE155B14 Example Ground Trace

15. GN/MAE155B15 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

16. GN/MAE155B16 Central Body Observation From simple trigonometry: sin(  h ) = Rs/(Rs+hs) D h = (Rs+hs) cos(  h ) Sw~ Swath width = 2  h Rs