1. GN/MAE155A1 Orbital Mechanics Overview MAE 155A Dr. George Nacouzi

2. GN/MAE155A2 James Webb Space Telescope, Launch Date 2011 Primary mirror: 6.5-meter aperture Orbit: 930,000 miles from Earth, Mission lifetime: 5 years (10-year goal) Telescope Operating temperature: ~45 Kelvin Weight: Approximately 6600kg

3. GN/MAE155A3 Overview: Orbital Mechanics Study of S/C (Spacecraft) motion influenced principally by gravity. Also considers perturbing forces, e.g., external pressures, on-board mass expulsions (e.g, thrust) Roots date back to 15th century (& earlier), e.g., Sir Isaac Newton, Copernicus, Galileo & Kepler In early 1600s, Kepler presented his 3 laws of planetary motion –Includes elliptical orbits of planets –Also developed Kepler’s eqtn which relates position & time of orbiting bodies

4. GN/MAE155A4 Overview: S/C Mission Design Involves the design of orbits/constellations for meeting Mission Objectives, e.g., area coverage Constellation design includes: number of S/C, number of orbital planes, inclination, phasing, as well as orbital parameters such as apogee, eccentricity and other key parameters Orbital mechanics provides the tools needed to develop the appropriate S/C constellations to meet the mission objectives

5. GN/MAE155A5 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 –Solution assumes bodies are spherically symmetric, point sources ( Earth oblateness can be important and is accounted for in J2 term of gravity field) –Only gravity and centrifugal forces are present

6. GN/MAE155A6 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

7. GN/MAE155A7 General Two Body Motion Equations where Solution is in form of conical section, i.e., circle, ellipse, parabola & hyperbola. KE + PE, PE = 0 at R=  ∞ ∞ a~ semi major axis of ellipse H = R x V = R V cos (  ), where H~ angular momentum &  ~ flight path angle (between V & local horizontal) & r ~Position vector V 

8. GN/MAE155A8 General Two Body Motion Trajectories Central Body Circle, a=r Ellipse, a> 0 Hyperbola, a< 0 Parabola, a =  a Parabolic orbits provide minimum escape velocity Hyperbolic orbits used for interplanetary travel

9. GN/MAE155A9 General Solution to Orbital Equation Velocity is given by: Eccentricity: e = c/a where, c = [Ra - Rp]/2 Ra~ Radius of Apoapsis, Rp~ Radius of Periapsis e is also obtained from the angular momentum H as: e =  [1 - (H 2 /  a)]; and H = R V cos (  )

10. GN/MAE155A10 Circular Orbits Equations Circular orbit solution offers insight into understanding of orbital mechanics and are easily derived Consider: F g = M m G/R 2 & F c = m V 2 /R (centrifugal F) V is solved for to get: V=  (MG/R) =  (  /R) Period is then: T=2  R/V => T = 2  (R 3 /  ) FcFc FgFg V R

11. GN/MAE155A11 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( )]

12. GN/MAE155A12 Elliptical Orbit Definition Orbit is defined using the 6 classical orbital elements including: –Eccentricity, semi- major axis, true anomaly and inclination, where Inclination, i, is the angle between orbit plane and equatorial plane i Other 2 parameters are: Argument of Periapsis (  ). Ascending Node: 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 Vernal Equinox   Ascending Node Periapsis

13. GN/MAE155A13 Some Orbit Types... Extensive number of orbit types, some common ones: –Low Earth Orbit (LEO), Ra < 2000 km –Mid Earth Orbit (MEO), 2000< Ra < 30000 km –Highly Elliptical Orbit (HEO) –Geosynchronous (GEO) Orbit (circular): Period = time it takes earth to rotate once wrt stars, R = 42164 km –Polar orbit => inclination = 90 degree –Molniya ~ Highly eccentric orbit with 12 hr period (developed by Soviet Union to optimize coverage of Northern hemisphere)

14. GN/MAE155A14 Sample Orbits LEO at 0 & 45 degree inclination Elliptical, e~0.46, I~65deg Ground trace from i= 45 deg

15. GN/MAE155A15 Sample GEO Orbit Figure ‘8’ trace due to inclination, zero inclination in no motion of nadir point (or satellite sub station) Nadir for GEO (equatorial, i=0) remain fixed over point 3 GEO satellites provide almost complete global coverage

16. GN/MAE155A16 Orbital Maneuvers Discussion Orbital Maneuver –S/C uses thrust to change orbital parameters, i.e., radius, e, inclination or longitude of ascending node –In-Plane Orbit Change Adjust velocity to convert a conic orbit into a different conic orbit. Orbit radius or eccentricity can be changed by adjusting velocity Hohmann transfer: Efficient approach to transfer between 2 Non-intersecting orbits. Consider a transfer between 2 circular orbits. Let Ri~ radius of initial orbit, Rf ~ radius of final orbit. Design transfer ellipse such that: Rp (periapsis of transfer orbit) = Ri (Initial R) Ra (apoapsis of transfer orbit) = Rf (Final R)

17. GN/MAE155A17 Hohmann Transfer Description DV1 DV2 Transfer Ellipse Final Orbit Initial Orbit Rp = Ri Ra = Rf DV1 = Vp - Vi DV2 = Va - Vf Note: ( )p = transfer periapsis ( )a = transfer apoapsis RpRa Ri Rf

18. GN/MAE155A18 General In-Plane Orbital Transfers... Change initial orbit velocity Vi to an intersecting coplanar orbit with velocity Vf DV 2 = Vi 2 + Vf 2 - 2 Vi Vf cos (a) Initial orbit Final orbit a Vf DV Vi

19. GN/MAE155A19 Other Orbital Transfers... Bielliptical Tranfer –When the transfer is from an initial orbit to a final orbit that has a much larger radius, a bielliptical transfer may be more efficient Involves three impulses (vs. 2 in Hohmann) Plane Changes –Can involve a change in inclination, longitude of ascending nodes or both –Plane changes are very expensive (energy wise) and are therefore avoided if possible

20. GN/MAE155A20 Basics of Rocket Equation F = Ve dm/dt F = M dV/dt Thrust dV/dt Ve ~ Exhaust Vel. m ~ propellant mass F = Thrust = Force M ~S/C Mass V ~ S/C Velocity g c ~ gravitational constant S/C M dV/dt = Ve dm/dt = - Ve dM/dt => DV = Ve ln (Mi/Mf) where, Mi ~ Initial Mass; Mf~ Final Mass Isp = Thrust/(g c dm/dt) => Ve = Isp x g c Calculate mass of propellant needed for rocket to provide a velocity gain (DV)

21. GN/MAE155A21 Basics of Rocket Equation (cont’d) M dV/dt = Ve dm/dt = - Ve dM/dt => DV = Ve ln (Mi/Mf) where, Mi ~ Initial Mass; Mf~ Final Mass Isp = Thrust/(g c dm/dt) => Ve = Isp x g c Substituting we get: Mi/Mf = exp (DV/ (g c Isp)) but Mp = Mf - Mi => Mp = Mi[1-exp(-DV/ g c Isp)] Where, DV ~ Delta Velocity, Mp ~ Mass of Propellant Mass of propellant calculated from Delta Velocity and propellant Isp. For Launch Vehicles: Isp ~ 260 - 300 sec for solid propellant Isp ~ 300 - 500 sec for liquid bipropellant

22. GN/MAE155A22 Example & Announcements