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GN/MAE155A 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
• 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...
Reference: SpacecraftSystems Engineering,Fortescue & Stark
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
Two Body Motion (Derivation)
For m, we havem.h’’ = GMmr/(r^2 |r|)m.h’’ = GMmr/r^3 h’’ = r/r^3where h’’= d2h/dt2 & = GMFor M,
Mj’’ = -GMmr/(r^2 |r|) j’’ = -Gmr/r^3, but r = j-h => r’’ = -G(M+m) r/r^3for M>>m => r’’ + GM r/r^3= 0, or r’’ + r/r^3 = 0 (1)
h
rM
m
j
GN/MAE155A 10
Two Body Motion (Derivation)
From r’’ + r/r^3 = 0 => r x r’’ + r x r/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:(r’’ x H) + /r^3 (r x 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 sectionwhere, p ~ semilatus rectum
Specific Mechanical Energy Equation is obtained by taking the dotproduct of the 2 body ODE (with r’), and then integrating the resultr’.r’’ + r.r’/r^3 = 0, integrate to get:
r’2/2 - /r =
GN/MAE155A 12
General Two Body Motion Equations
Solution is in form of conical section, i.e., circle ~ e = 0, ellipse ~ e < 1 (parabola ~ e = 1 & hyperbola ~ e >1)
2t
rd
d
2 r
R 0
V
2
2
R
KE + PE, PE = 0 at R= & PE<0 for R<
V 2R
a
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)
d2r/dt2 + r/R3 = 0 (1) where, = GM, r ~Position vector, and R = |r|
V LocalHorizon
Specific mechanical energy is:
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=2R/V => T = 2(R3/)
Fc
Fg
V
R
* Period = time it takes SC to rotate once wrt earth
GN/MAE155A 14
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
GN/MAE155A 15
Elliptical Orbit Geometry & Nomenclature
Periapsis
ApoapsisLine of Apsides
R
a c
V
Rpb
• Line of Apsides connects Apoapsis, central body & Periapsis• Apogee~ Apoapsis; Perigee~ Periapsis (Earth nomenclature)
S/C position defined by R & , is called true anomalyR = [Rp (1+e)]/[1+ e cos()]
GN/MAE155A 16
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
AscendingNode
Periapsis
GN/MAE155A 17
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 - (H2/a)]; and H = R V cos ()
V 2R
a
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
Ground tracefrom i= 45 deg
Lat =..
GN/MAE155A 21
Sample GEO Orbit
Figure ‘8’ trace due to inclination, zero inclination has nomotion of nadir point(or satellite sub station)
• Nadir for GEO (equatorial, i=0) remain fixed over point• 3 GEO satellites provide almostcomplete global coverage
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
DV1
DV2
TransferEllipse
Final Orbit
Initial Orbit
Rp = RiRa = RfDV1 = Vp - ViDV2 = Va - VfDV = |DV1|+|DV2|
Note:( )p = transfer periapsis( )a = transfer apoapsis
RpRa
RiRf
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)
Initial orbit
Final orbit
a Vf
DV
Vi
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