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
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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
Kepler's three laws of planetary motion can be described as follows:
The path of the planets about the sun are elliptical in shape, with the center of the sun being located at one focus. (The Law of Ellipses)
An imaginary line drawn from the center of the sun to the center of the planet will sweep out equal areas in equal intervals of time. (The Law of Equal Areas)
The ratio of the squares of the periods of any two planets is equal to the ratio of the cubes of their average distances from the sun. (The Law of Harmonies)
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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
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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
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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
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7. Relative Importance of Orbit Perturbations
Reference: Spacecraft Systems 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...
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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
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9. Two Body Motion (Derivation)h r M m j
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 = (1)
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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
H= r2 ’
 is a unit vector normal to the unit vector r along the r vector
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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 = 
B is the constant of integration
P (semilatus rectum) is the the normal distance from the focus point to the ellipse p
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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< 
Potential energy is considered to be 0 at infinity and negative at distances less than infinity
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)
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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
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14. General Two Body Motion Trajectories
Hyperbola, a< 0 a
Circle, a=r
Parabola,
a = 
Ellipse, a > 0
Central Body
Parabolic orbits provide minimum escape velocity
Hyperbolic orbits used for interplanetary travel
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15. Elliptical Orbit Geometry & NomenclatureV
Periapsis a c R 
Line of Apsides Rp
Apoapsis b
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)
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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
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
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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 ()
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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()]
E is called the eccentric anomaly, n is the mean motion these are circular equivalent parameters to elliptical motion
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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 < km
Highly Elliptical Orbit (HEO)
Geosynchronous (GEO) Orbit (circular): Period = time it takes earth to rotate once wrt stars, R = km
Polar orbit => inclination = 90 degree
Molniya ~ Highly eccentric orbit with 12 hr period (developed by Soviet Union to optimize coverage of Northern hemisphere)
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20. Sample Orbits LEO at 0 & 45 degree inclination
Elliptical, e~0.46, I~65deg
Lat =..
Ground trace
from i= 45 deg
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21. 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)
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22. Orbital Maneuvers Discussion
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)
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23. Hohmann Transfer Description
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
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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
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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, aero forces are used to change the vehicle velocity
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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
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27. Examples & Announcements
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