1 / 27

Orbital Mechanics Overview

Orbital Mechanics Overview. MAE 155A G. Nacouzi. James Webb Space Telescope, Launch Date 2011. Primary mirror: 6.5-meter aperture Orbit: 930,000 miles from Earth , L3 point Mission lifetime: 5 years (10-year goal) Telescope Operating temperature: ~45 Kelvin Weight: Approximately 6600kg.

dane
Download Presentation

Orbital Mechanics Overview

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


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

  2. James Webb Space Telescope, Launch Date 2011 Primary mirror: 6.5-meter aperture Orbit: 930,000 miles from Earth , L3 point 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., coverage area • 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, 2 body assumption is usually used for initial studies. Earth orbiting satellite 2 Body assumptions: • Central body is Earth, assume it has only gravitational influence on S/C, MEarth >> mSC • 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. 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: Radius at equator = 6378 km, Radius at polar = 6357 km • Space Environment: Solar Pressure, drag from rarefied atmosphere GN/MAE155A

  7. 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/MAE155A

  8. Two Body Motion (or Keplerian Motion) • Closed form solution for 2 body exists, no explicit solution 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

  9. M r m j h Two Body Motion (Derivation) For m, we have m.h’’ = GMmr/(r^2 |r|) m.h’’ = GMmr/r^3 h’’ = r/r^3 where h’’= d2h/dt2 &  = GM For M, Mj’’ = -GMmr/(r^2 |r|) j’’ = -Gmr/r^3, but r = j-h => r’’ = -G(M+m) r/r^3 for M>>m => r’’ + GM r/r^3= 0, or r’’ + r/r^3 = 0 (1) GN/MAE155A

  10. Two Body Motion (Derivation) From r’’ + r/r^3 = 0 => r xr’’ + r x r/r^3 = 0 => r xr’’ = 0, but r xr’’ = d/dt ( r x r’) = d/dt (H),  d/dt (H) =0, where H is angular momentum vector, i.e. r and r’ are in same plane. Taking the cross product of equation 1with H, we get: (r’’x H) + /r^3 (rx H) = 0 (r’’x H) = /r^3 (H x r), but d/dt (r’ x H) = (r’’x H) + (r’ x H’) => d/dt (r’ x H) = /r^3 (H x r) => d/dt (r’ x H) = /r^3 (r2 ’) r  =  ’  =  r’ ( r is unit vector)  d/dt (r’ x H) =  r’ ; integrate => r’ x H =  r + B =0 GN/MAE155A

  11. Two Body Motion (Derivation) r . (r’ x H) = r . ( r + B) = (r x r’) . H = H.H = H2 => H2 = r + r B cos () => r = (H2 / )/[1 + B/ cos()] p = H2 / ; e = B /  ~ eccentricity;  ~ True Anomally => r = p/[1+e cos()] ~ Equation for a conic section where, p ~ semilatus rectum Specific Mechanical Energy Equation is obtained by taking the dot product of the 2 body ODE (with r’), and then integrating the result r’.r’’ + r.r’/r^3 = 0, integrate to get: r’2/2 - /r =  GN/MAE155A

  12. General Two Body Motion Equations d2r/dt2 +  r/R3 = 0 (1) where,  = GM,r ~Position vector, and R = |r| Solution is in form of conical section, i.e., circle ~ e = 0, ellipse ~ e < 1 (parabola ~ e = 1 & hyperbola ~ e >1) V Specific mechanical energy is:  Local Horizon KE + PE, PE = 0 at R=  & PE<0 for R<  a~ semi major axis of ellipse H = R x V = R V cos (), where H~ angular momentum &  ~ flight path angle (FPA, between V & local horizontal) GN/MAE155A

  13. 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 * Period = time it takes SC to rotate once wrt earth GN/MAE155A

  14. 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

  15. 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

  16. 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

  17. 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

  18. More Solutions to Orbital Equation • FPA is given by: tan() = e sin()/ ( 1+ e cos()) • True anomaly is given by, cos() = (Rp * (1+e)/R*e) - 1/e • Time since periapsis is calculated as: t = (E- e sin(E))/n, where, n = /a3; E = acos[ (e+cos())/ ( 1+ e cos()] GN/MAE155A

  19. 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

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

  21. 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 has no motion of nadir point (or satellite sub station) GN/MAE155A

  22. 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

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

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

  25. Aerobraking • Aerobraking uses aerodynamic forces to change the velocity of the SC therefore its trajectory (especially useful in interplanetary missions) • Instead of retro burns, aeroforces are used to change the vehicle velocity GN/MAE155A

  26. Other Orbital Transfers... • Hohman transfers are not always the most efficient • 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) • Low Thrust Transfers • When thrust level is small compared to gravitational forces, the orbit transfer is a very slow outward spiral • Gravity assists - Used in interplanetary missions • 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

  27. Examples& Announcements GN/MAE155A

More Related