Orbital Mechanics Overview

1. Orbital Mechanics Overview MAE 155A Dr. George Nacouzi GN/MAE155A

2. 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 GN/MAE155A

3. 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 GN/MAE155A

4. 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 GN/MAE155A

5. 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 GN/MAE155A

6. 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/MAE155A

7. General Two Body Motion Equations where & r ~Position vector Solution is in form of conical section, i.e., circle, ellipse, parabola & hyperbola. V  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) GN/MAE155A

8. General Two Body Motion Trajectories Hyperbola, a< 0 Circle, a=r a Parabola, a =  Ellipse, a> 0 Central Body • Parabolic orbits provide minimum escape velocity • Hyperbolic orbits used for interplanetary travel GN/MAE155A

9. General Solution to Orbital Equation • Velocity is given by: • Eccentricity: e = c/a where, c = [Ra - Rp]/2Ra~ Radius of Apoapsis, Rp~ Radius of Periapsis • e is also obtained from the angular momentum H as: e = [1 - (H2/a)]; and H = R V cos () GN/MAE155A

10. Circular Orbits Equations • Circular orbit solution offers insight into understanding of orbital mechanics and are easily derived • Consider: Fg = M m G/R2 & Fc = m V2 /R (centrifugal F) V is solved for to get: V= (MG/R) = (/R) • Period is then: T=2R/V => T = 2(R3/) V Fc R Fg GN/MAE155A

11. 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/MAE155A

12. 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 Elliptical Orbit Definition Periapsis i  Vernal Equinox  Ascending Node • 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 GN/MAE155A

13. 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) GN/MAE155A

14. Sample Orbits LEO at 0 & 45 degree inclination Elliptical, e~0.46, I~65deg Ground trace from i= 45 deg GN/MAE155A

15. Sample GEO Orbit • Nadir for GEO (equatorial, i=0) • remain fixed over point • 3 GEO satellites provide almostcomplete global coverage Figure ‘8’ trace due to inclination, zero inclination in no motion of nadir point (or satellite sub station) GN/MAE155A

16. 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) GN/MAE155A

17. Hohmann Transfer Description Transfer Ellipse Rp = Ri Ra = Rf DV1 = Vp - Vi DV2 = Va - Vf Note: ( )p = transfer periapsis ( )a = transfer apoapsis DV1 Ra Rp Ri Rf Initial Orbit DV2 Final Orbit GN/MAE155A

18. General In-Plane Orbital Transfers... • Change initial orbit velocity Vi to an intersecting coplanar orbit with velocity Vf DV2 = Vi2 + Vf2 - 2 Vi Vf cos (a) Final orbit DV Initial orbit Vi Vf a GN/MAE155A

19. 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 GN/MAE155A

20. Basics of Rocket Equation Calculate mass of propellant needed for rocket to provide a velocity gain (DV) Thrust Ve ~ Exhaust Vel. m ~ propellant mass F = Thrust = Force F = Ve dm/dt S/C dV/dt F = M dV/dt M dV/dt = Ve dm/dt = - Ve dM/dt => DV = Ve ln (Mi/Mf) where, Mi ~ Initial Mass; Mf~ Final Mass Isp = Thrust/(gc dm/dt) => Ve = Isp x gc M ~S/C Mass V ~ S/C Velocity gc~ gravitational constant GN/MAE155A

21. 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/(gc dm/dt) => Ve = Isp x gc Substituting we get: Mi/Mf = exp (DV/ (gc Isp)) but Mp = Mf - Mi => Mp = Mi[1-exp(-DV/ gc 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 GN/MAE155A

22. Example& Announcements GN/MAE155A