Orbital Mechanics 2

8/19/2019 Orbital Mechanics 2

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• Orbit transfer

• Change of orbital plane• Orbit Rendezvous• Orbit maintenance• De-orbit

Orbitmaneuvering

• Low-altitude Earth orbit• Medium-altitude Earth oribt• High-altitude Earth orbit

Type of orbits

• The orbit desing process• Orbits during the space mission lifetime• Earth coverage

Orbit design



ORBIT MANEUVERING

• Most satellites must change their OE during their lifetime:• Examples:

1. Transfer from initial parking orbit to final orbit2. Correct orbits from perturbation

3. Intercept a target4. De-orbit

• To change the orbit of a satellite: to change the satellitesvelocity vector in magnitude or direction using a thruster

V=Vneed -Vcurrent• The position does not change significantly during

impulsive burns.



ORBIT MANEUVERING

• The most common type of orbit maneuvering: change sizeand energy of the orbit remaining in the same orbital plane

• There exists different transfer

orbits depending of the missionrequirements:

• Smallest amount of energy•

Minimum transfer time

Coplanar orbit transfers



ORBIT MANEUVERING

• Walter Hohmann (1880-1945)german engineer.

• Hohmann transfer orbit consists in

a elliptical orbit tangent to both theinitial and final circular orbits at thetransfer’s orbit perigee and apogee,respectively.

• Velocity vectors are collinear at theintersecction points

• Represents the most fuel-efficienttransfer between two coplanarcircular orbits

Hohmann transfer orbit



ORBIT MANEUVERING




ORBIT MANEUVERING

Hohmann transfer orbit formulation

Parking orbit

r A= 6567 kmGeostationary orbit

r B= 42160 km

Orbital velocity of the parking orbitV A= 7.79 km/s

Orbital velocity of the GOV B= 3.08 km/s

Semimajor axis of the transfer orbita T= 0.5( r A+r B)=24364 km

Btransf B A Atransf B A V V V V V V V ,,



ORBIT MANEUVERING

Hohmann transfer orbit formulation

Transfer velocity at perigee V TA= 10.25 km/s

Transfer velocity at apogee V TB= 1.49 km/s

Change velocity at perigee V A=2.46 km/s

Change velocity at apogee V B=1.49 km/s

Total velocity change V =3.95 km/s

Time of transfer, P/2 T=5 hr 15 min



ORBIT MANEUVERING

• Hohmann transfer is efficient in energy terms, but notin transfer time

• The time of transfer can be calculated with the 3rdKepler’s law

• Paradox: we have increase the velocity in two burns,but the final velocity is slower than originally

Hohmann transfer orbit discussion

1/ 23transf

flight

aT

GM



ORBIT MANEUVERING

• Sometimes we need totransfer a satellite in lesstime than that required for

a Hohmann transfer• In the one-tangent burn

the transfer orbit istangential to the initial

orbit• The transfer orbit

intersects the final orbitwith an angle equal to theflight-path-angle

One-tangent-burn transfer



ORBIT MANEUVERING

Variable Hohmann Transfer One-tangent-burn

rA 6570 km 6570 km

rB 42200 km 42200 km

a tx 24385 km 28633 km

VT 3.935 km/s 4.699 km/s

TOF 5.256 h 3.457 h

• Comparison of coplanar orbit transfers from LEO toGeosynchronous orbit

Comparison



ORBIT MANEUVERING

• Another option for changing the size of the orbit is to usea constant low-thrust burn (ionic engines)

• The resulting transfer orbit is a spiral orbit.• We can approximate the velocity change for this type of

orbits by

• For the previous example, the total change of velocitywould be 4.71 km/s

Constant low-thrust burn

A B V V V



ORBIT MANEUVERING

Smart-1



ORBIT MANEUVERING

• To change the orbital plane we must change the direction of the velocity vector

• For a simple plane change, the required change in velocity is

• Plane changes are very expensive and requires an important

fuel consumption

Orbit plane changes

2sin2 iV V



ORBIT MANEUVERING

• An example:• Change in velocity required to transfer from a low-

altitude (h=185 km) inclined (i=28 deg) orbit to an

equatorial orbit (i=0) at the same altitute:r=6563 km V=7.79 km/s V=3.77 km/s

• For a i=60 deg, the required change in velocity equalsthe current velocity.

Orbit plane changes



ORBIT MANEUVERING

Orbit plane changes

Describe a 3-burn orbitalchange from an

inclined low-altitude

circular orbit to a high-altitude equatorial

circular orbit.

THINK



ORBIT MANEUVERING

• Objective: to rendezvous with or intercept another object inspace, eg:

• A probes send to a planet, a comet, an asteroid.• Shuttle send to the ISS• A communication satellite in a geosynchronous orbit

Orbit Rendezvous

Phasing orbit : Hohmann transfer orbit which the interceptormust arrive at the rendezvous point at the same time asthe target



ORBIT MANEUVERING

Orbit Rendezvous

f is the phase angle needed forrendezvous

i is the initial phase angle

k is the number of rendezvousopportunities

int is the angular velocity of theinterceptor

tgt is the angular velocity of thetarget



ORBIT MANEUVERING

• Wait time : the interceptor remains in the initial orbit untilinterceptor and target achive the desired geometry forrendezvous

• The lead angle is calculated by:

• And the phase angle

• The total time

Orbit Rendezvous

)/()2( int tgt f iwait k T

L tg flight T

180º f L

flight wait total T T T



ORBIT MANEUVERING

Orbit Rendezvous



ORBIT MANEUVERING

• After the mission is complete we must to think “what todo with the satellite ”

• Low-altitude orbits: allow to decay and reenter the

atmosphere• Boost satellites into benign orbits to reduce the

probability of collision• Sometimes the deorbit must be controlled: used for the

recovery of the payload (eg. Manned mission)

Deorbit



TYPE OF ORBITS

Classifications• Heliocentric orbit• Geocentric orbit

1 Centricclassifications

• Low Earth orbit• Medium Earth orbit• High Earth orbit

2 Altitudeclassifications

• Polar orbit• Inclined orbit• Equatorial orbit

3 Inclinationclassifications

• Circular orbit• Elliptic orbit• Parabolic orbit• Hyperbolic orbit

4 Eccentricityclassifications

• Geosynchronous orbit• Geostationary orbit• Semi-syncrhonousorbit• Heliosynchronous

5 Synchronousclassifications



TYPE OF ORBITS

• Earth observation satellites• Manned spaceflight• Aid remote sensing satellites

LEO, Low-altitudeEarth Orbit (160-

2000 km)

• Global positioning (GPS, GLONASS andGalileo)

MEO, Medium-altitude Earth

Orbit

• Geostationary orbit for communication• Geostationary orbit for meteorology

HEO, High-altitute Earth

Orbit

Altitude classification



TYPE OF ORBITS

PROS:• Below the inner Van Allen radiation belt

(reduction to exposed radiation in mannedspaceflight)

• Good resolution for Earth observation• Low energetic requirements for communication

CONS:• Bad Earth coverage. Several groundsation

needed.• Space debris. LEO orbit congested.

LEO



TYPE OF ORBITS

MEO

PROS:• Good Earth visibility and coverage• Low apparent motion of the

satellite

CONS:

• High energetic requirement forcommunication. (Earth laptop onlyin reception mode)



TYPE OF ORBITS

HEO

PROS:• Possibility that the satellite remains at rest in the

sky• Large Earth coverage

CONS:• Very high energy requirements for

communication. (Exclusive groundsatation, largesolar panels)

• Communication latency becomes important• Affected by cosmic radiation and solar wind



• The orbit selection process is complex

• The orbit typically defines:• Space mission lifetime•

Cost• Environment• Viewing geometry• Payload performance

ORBIT DESIGN

The orbit design process

Gold rule: to meet the largest number of mission

requirements at the least possible cost



ORBIT DESIGN

Orbits during the space mission lifetime

Parking orbit: for spacecraft checkoutor storage

Transfer orbit: to move thespacecraft between orbit

Injection orbit: where the spacecraftsepareted from the launch vehicle

Operational orbit: orbit/s for missionactivities

Disposal orbit: where the spacecraftwill do minimum damage



ORBIT DESIGN

Footprint Area (FOV): area that aspecific instrument or antenna

can see at any instant.

Instantaneous Access Area (IAA):all the area that the instrumentcould potentially see

Area Coverage Rate (ACR): therate at which the instrument issensing new land

Area Access Rate (AAR): the rateat which new land is coming intothe SC’s access area.

Earth coverage



ORBIT DESIGN

• The fraction of the orbitover which the point Pis in view is

• Where

• 2 max is called theswath width

oview F 180/

cos/coscos max

Earth coverage geometry



• Launch windows: to determine the appropiate time tolaunch from the Earth surface into the desired orbitaltime

ORBIT DESIGN

Launch windows

• No launch windows exist if L>i for direct orbit or L>180deg- i for retrograde orbit

• One launch window exists if L=i or L=180 deg –i • Two launch windows exist if L<i or L<180 deg - i



ORBIT DESIGN

• An example: interplanetary probe send to Mars

Launch windows

Orbital Mechanics 2

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