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Orbital Mechanics Overview - MAELabs UCSD.ppt

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					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

				
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posted:1/25/2014
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