Document Sample

Orbital Mechanics Overview MAE 155A G. Nacouzi GN/MAE155A 1 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 2 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 3 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 4 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 5 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 6 Relative Importance of Orbit Perturbations Reference: Spacecraft Systems Engineering, • J2 term accounts for effect from oblate earth Fortescue & Stark •Principal effect above 100 km altitude • Other terms may also be important depending on application, mission, etc... GN/MAE155A 7 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 8 Two Body Motion (Derivation) M r m j For m, we have h m.h’’ = GMmr/(r^2 |r|) m.h’’ = GMmr/r^3 h’’ = mr/r^3 where h’’= d2h/dt2 & For M, m = GM 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’’ + mr/r^3 = 0 (1) GN/MAE155A 9 Two Body Motion (Derivation) From r’’ + mr/r^3 = 0 => r x r’’ + r x mr/r^3 = 0 => r x r’’ = 0, but r x r’’ = 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: =0 (r’’ x H) + m/r^3 (r x H) = 0 (r’’ x H) = m/r^3 (H x r), but d/dt (r’ x H) = (r’’x H) + (r’ x H’) => d/dt (r’ x H) = m/r^3 (H x r) => d/dt (r’ x H) = m/r^3 (r2 q’) r q = m q’ q = m r’ ( r is unit vector) \ d/dt (r’ x H) = m r’ ; integrate => r’ x H = m r + B GN/MAE155A 10 Two Body Motion (Derivation) r . (r’ x H) = r . (m r + B) = (r x r’) . H = H.H = H2 => H2 = mr + r B cos (q) => r = (H2 / m)/[1 + B/m cos(q)] p = H2 / m; e = B / m ~ eccentricity; q ~ True Anomally => r = p/[1+e cos(q)] ~ 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’’ + mr.r’/r^3 = 0, integrate to get: r’2/2 - m/r = e GN/MAE155A 11 General Two Body Motion Equations d2r/dt2 + m r/R3 = 0 (1) where, m = 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 g 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 (g), where H~ angular momentum & g ~ flight path angle (FPA, between V & local horizontal) GN/MAE155A 12 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 V= Ö(MG/R) = Ö(m/R) • Period is then: T=2pR/V Fc => T = 2pÖ(R3/m) R Fg * Period = time it takes SC to rotate once wrt earth GN/MAE155A 13 General Two Body Motion Trajectories Hyperbola, a< 0 a Circle, a=r Parabola, Ellipse, a > 0 a=¥ Central Body • Parabolic orbits provide minimum escape velocity • Hyperbolic orbits used for interplanetary travel GN/MAE155A 14 Elliptical Orbit Geometry & Nomenclature V a c Periapsis R J Line of Apsides Rp Apoapsis b S/C position defined by R & J, R = [Rp (1+e)]/[1+ e cos(J)] J is called true anomaly • Line of Apsides connects Apoapsis, central body & Periapsis • Apogee~ Apoapsis; Perigee~ Periapsis (Earth nomenclature) GN/MAE155A 15 Elliptical Orbit Definition ps is • Orbit is defined using Peria the 6 classical orbital elements including: i – Eccentricity, semi- w major axis, true Vernal anomaly and Equinox W inclination, where Ascending • Inclination, i, is the Node angle between orbit Other 2 parameters are: plane and equatorial • Argument of Periapsis (w). Ascending Node: Pt where plane S/C crosses equatorial plane South to North • Longitude of Ascending Node (W)~Angle from Vernal Equinox (vector from center of earth to sun on first day of spring) and ascending node GN/MAE155A 16 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 = Ö[1 - (H2/ma)]; and H = R V cos (g) GN/MAE155A 17 More Solutions to Orbital Equation • FPA is given by: tan(g) = e sin(J)/ ( 1+ e cos(J)) • True anomaly is given by, cos(J) = (Rp * (1+e)/R*e) - 1/e • Time since periapsis is calculated as: t = (E- e sin(E))/n, where, n = Öm/a3; E = acos[ (e+cos(J))/ ( 1+ e cos(J)] GN/MAE155A 18 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 19 Sample Orbits LEO at 0 & 45 degree inclination Elliptical, e~0.46, I~65deg Lat =.. Ground trace from i= 45 deg GN/MAE155A 20 Sample GEO Orbit • Nadir for GEO (equatorial, i=0) remain fixed over point • 3 GEO satellites provide almost complete global coverage Figure ‘8’ trace due to inclination, zero inclination has no motion of nadir point (or satellite sub station) GN/MAE155A 21 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 22 Hohmann Transfer Description Transfer Ellipse Rp = Ri Ra = Rf DV1 DV1 = Vp - Vi Ra Rp DV2 = Va - Vf DV = |DV1|+|DV2| Ri Rf Initial Orbit DV2 Note: ( )p = transfer periapsis ( )a = transfer apoapsis Final Orbit GN/MAE155A 23 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 a Vf GN/MAE155A 24 Aerobraking • Aerobraking uses aerodynamic forces to change the velocity of the SC therefore its trajectory (especially useful in interplanetary missions) – Instead of retro burns, aero forces are used to change the vehicle velocity GN/MAE155A 25 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 26 Examples & Announcements GN/MAE155A 27

DOCUMENT INFO

Shared By:

Categories:

Tags:

Stats:

views: | 4 |

posted: | 1/25/2014 |

language: | English |

pages: | 27 |

OTHER DOCS BY hcj

How are you planning on using Docstoc?
BUSINESS
PERSONAL

By registering with docstoc.com you agree to our
privacy policy and
terms of service, and to receive content and offer notifications.

Docstoc is the premier online destination to start and grow small businesses. It hosts the best quality and widest selection of professional documents (over 20 million) and resources including expert videos, articles and productivity tools to make every small business better.

Search or Browse for any specific document or resource you need for your business. Or explore our curated resources for Starting a Business, Growing a Business or for Professional Development.

Feel free to Contact Us with any questions you might have.