# Orbital Mechanics Overview - MAELabs UCSD.ppt

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```					Orbital Mechanics Overview

MAE 155A
G. Nacouzi

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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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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

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Overview: S/C Mission Design
• Involves the design of orbits/constellations for meeting
Mission Objectives, e.g., coverage area

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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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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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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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...
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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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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)
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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
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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
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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)
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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

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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
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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)

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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
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General Solution to Orbital Equation
• Velocity is given by:

• Eccentricity: e = c/a where,
c = [Ra - Rp]/2

• e is also obtained from the angular momentum H as:
e = Ö[1 - (H2/ma)]; and H = R V cos (g)
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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)]

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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)
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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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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
(or satellite sub station)

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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
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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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

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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

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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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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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Examples
& Announcements

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