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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

• 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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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

• 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

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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)

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

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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

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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 =

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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:

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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= (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

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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

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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()]

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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

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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

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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()]

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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

Ground tracefrom i= 45 deg

Lat =..

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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

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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 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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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

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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)

Initial orbit

Final orbit

a Vf

DV

Vi

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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, aeroforces 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