Orbital Mechanics Overview

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Orbital Mechanics Overview

MAE 155A

Dr. George Nacouzi

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James Webb Space Telescope, Launch Date 2011

Primary mirror: 6.5-meter aperture Orbit: 930,000 miles from Earth , 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., area coverage

• 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, Two body assumption is usually sufficient. Earth orbiting satellite Two Body approach:

– Central body is earth, assume it has only gravitational influence on S/C, assume M >> m (M, m ~ mass of earth & S/C)

• 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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Two Body Motion (or Keplerian Motion)

• Closed form solution for 2 body exists, no explicit soltn 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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General Two Body Motion Equations

2t

rd

d

2 r

R 0 GMwhere

Solution is in form of conical section, i.e., circle, ellipse, parabola & hyperbola.2

trd

d

2 r

R 0

V

2

2

R

KE + PE, PE = 0 at R= ∞ ∞

V 2R

a

a~ semi major axis of ellipse

H = R x V = R V cos (), where H~ angular momentum & ~ flight path angle (between V & local horizontal)

& r ~Position vector

V

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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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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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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

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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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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

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Sample GEO Orbit

Figure ‘8’ trace due to inclination, zero inclination in 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 - Vf

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 DV2 = Vi2 + Vf2 - 2 Vi Vf cos (a)

Initial orbit

Final orbit

a Vf

DV

Vi

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Other Orbital Transfers...

• 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)

• 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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Basics of Rocket Equation

F = Ve dm/dt

F = M dV/dt

Thrust

dV/dtVe ~ Exhaust Vel.m ~ propellant massF = Thrust = Force

M ~S/C MassV ~ S/C Velocitygc~ gravitational constant

S/C

M dV/dt = Ve dm/dt = - Ve dM/dt

=> DV = Ve ln (Mi/Mf) where, Mi ~ Initial Mass; Mf~ Final Mass

Isp = Thrust/(gc dm/dt) => Ve = Isp x gc

Calculate mass of propellant needed for rocket to provide a velocitygain (DV)

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Basics of Rocket Equation (cont’d)M dV/dt = Ve dm/dt = - Ve dM/dt

=> DV = Ve ln (Mi/Mf) where, Mi ~ Initial Mass; Mf~ Final Mass

Isp = Thrust/(gc dm/dt) => Ve = Isp x gc

Substituting we get:Mi/Mf = exp (DV/ (gc Isp))but Mp = Mf - Mi =>Mp = Mi[1-exp(-DV/ gc Isp)]

Where, DV ~ Delta Velocity, Mp ~ Mass of Propellant

Mass of propellant calculated from Delta Velocity and propellant Isp.For Launch Vehicles: Isp ~ 260 - 300 sec for solid propellant Isp ~ 300 - 500 sec for liquid bipropellant

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Example& Announcements