Orbital Mechanics - Chapter 08

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Chapter 8

ORBITAL MECHANICS

Knowledge of orbital motion is essential for a full understanding of space operations.The vantage point of space can be visualized through the motion Kepler described and bycomprehending the reasons for that motion as described by Newton. Thus, the objectiveshere are to gain a conceptual understanding of orbital motion and become familiar withcommon terms describing that motion.

A HISTORY OFTHE LAWS OF MOTION1

Early Cosmology

This generation is far too

knowledgeable to perceive the universe asearly man saw it. Each generation usesthe knowledge of the previous generationas a foundation to build upon in the ever-continuing search for comprehension.When the foundation is faulty, the towerof understanding eventually crumbles anda new building proceeds in a differentdirection. Such was the case during thedark ages in medieval Europe and theRenaissance.

The Babylonians, Egyptians andHebrews each had various ingeniousexplanations for the movements of theheavenly bodies. According to theBabylonians, the Sun, Moon and starsdanced across the heavenly dome enteringthrough doors in the East and vanishingthrough doors in the West. The Egyptiansexplained heavenly movement with riversin a suspended gallery upon which theSun, Moon and planets sailed, enteringthrough stage doors in the East andexiting through stage doors in the West.

Though one may view these ancient

cosmologies with a certain arrogance andmarvel at the incredible creativity bywhich they devised such a picture of theuniverse, their observations wereamazingly precise. They computed the

length of the year with a deviation of lessthan 0.001% from the correct value, andtheir observations were accurate, enablingthem to precisely predict astronomicalevents. Although based on mythologicalassumptions, these cosmological theories

worked.Greece took over from Babylon and

Egypt, creating a more colorful universe.However, the 6th century BC (the centuryof Buddha, Confucius and Lo Tse, theIonian philosophers and Pythagoras) wasa turning point for the human species. Inthe Ionian school of philosophy, rationalthought was emerging from themythological dream world. It was thebeginning of the great adventure in whichthe Promethean quest for naturalexplanations and rational causes wouldtransform humanity more radically than inthe previous two hundred thousand years.

Astronomy

Many early civilizations recognized thepattern and regularity of the stars andplanets motion and made efforts to trackand predict celestial events. Theinvention and upkeep of a calendarrequired at least some knowledge ofastronomy. The Chinese had a working

calendar at least by the 13th or 14thcentury BC. They also kept accuraterecords for things such as comets, meteorshowers, fallen meteorites and otherheavenly phenomena. The Egyptianswere able to roughly predict the floodingof the Nile every year: near the time whenthe star Sirius could be seen in the dawn

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1Much of this information comes from Arthur

KoestlersThe Sleepwalkers.


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The shape of the world must be aperfect sphere, and that all motion mustbe in perfect circles at uniform speed.

sky, rising just before the Sun. TheBronze Age peoples in northwesternEurope left many monuments indicatingtheir ability to understand the movementof celestial bodies. The best known isStonehenge, which was used as a crude

calendar.The early Greeks initiated the orbitaltheories, postulating the Earth was fixedwith the planets and other celestial bodiesmoving around it; a geocentric universe.About 300 BC, Aristarchus of Samossuggested that the Sun was fixed and theplanets, including the Earth, were incircular orbits around the Sun; aheliocentric universe. AlthoughAristarchus was more correct (at leastabout a heliocentric solar system), hisideas were too revolutionary for the time.

Other prominent astronomers/philosopherswere held in higher esteem and, since theyfavored the geocentric theory,Aristarchus heliocentric theory wasrejected and the geocentric theorycontinued to be predominately accepted.

This circular motion was soaesthetically appealing that Aristotle

promoted this circular motion into adogma of astronomy. Themathematicians task was now to design asystem reducing the apparent irregularitiesof planetary motion to regular motions inperfectly fixed circles. This task wouldkeep them busy for the next two thousandyears.

Perhaps the most elaborate and fancifulsystem was one Aristotle constructedusing fifty-four spheres to account for themotions of the seven planets.2 DespiteAristotles enormous prestige, this system

was so contrived that it was quicklyforgotten. In the 2nd century AD,Ptolemy modified and amplified thegeocentric theory explaining the apparentmotion of the planets by replacing thesphere inside a sphere concept with awheel inside a wheel arrangement.According to his theory, the planetsrevolve about imaginary planets, which inturn revolve around the Earth. Thus, thistheory employed forty wheels: thirty-nineto represent the seven planets and one forthe fixed stars.

Aristotle, one of the more famousGreek philosophers, wrote encyclopedictreatises on nearly every field of humanendeavor. Aristotle was accepted as theultimate authority during the medievalperiod and his views were upheld by theRoman Catholic Church, even to the timeof Galileo. However, his expositions inthe physical sciences in general, andastronomy in particular, were less soundthan some of his other works.Nevertheless, his writings indicate theGreeks understood such phenomena asphases of the Moon and eclipses at least inthe 4th century BC. Other early Greekastronomers, such as Eratosthenes andHipparchus, studied the problemsconfronting astronomers, such as: Howfar away are the heavenly bodies? How

large is the Earth? What kind of geometrybest explains the observations of theplanets motions and their relationships?

Even though Ptolemys system wasgeocentric, this complex system more orless described the observable universe andsuccessfully accounted for celestialobservations. With some latermodifications, his theory was acceptedwith absolute authority throughout theMiddle Ages until it finally gave way tothe heliocentric theory in the 17th century.

Modern Astronomy

Copernicus

In the year 1543, some 1,800 yearsafter Aristarchus proposed a heliocentricsystem, a Polish monk named Nicolas

The Greeks were under the influence ofPlatos metaphysical understanding of theuniverse, which stated:

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2In this instance the seven planets include the

Sun, Moon, Mercury, Venus, Mars, Jupiter, and

Saturn.


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Koppernias (better known by his Latinname, Copernicus) revived theheliocentric theory when he publishedDeRevolutionibus Orbium Coelestium (Onthe Revolutions of the Celestial Spheres).This work represented an advance, but

there were still some inaccuracies. Forexample, Copernicus thought that theorbital paths of all planets were circleswith their centers displaced from thecenter of the sun.

Tycho and Keplers relationship wasfar from a great friendship. It was short(eighteen months) and fraught withcontroversy. This brief relationship endedwhen Tycho De Brahe, the meticulousobserver who introduced precision into

astronomical measurement andtransformed the science, becameterminally ill and died in 1601.

KeplerCopernicus did not prove that the Earth

revolves about the sun; the Ptolemicsystem, with some adjustments, couldhave accounted just as well for theobserved planetary motion. However, theCopernican system had more asceticvalue. Unlike the Ptolemic system, it waselegant and simple without having to

resort to artful wheel upon wheelstructures. Although it upset the churchand other ruling authorities, Copernicusmade the Earth an astronomical body,which brought unity to the universe.

Johannes Kepler was born inWurttemberg, Germany, in 1571. Heexperienced an unstable childhood that,by his own accounts, was unhappy andridden with sickness. However, Keplersgenius propelled him through school andguaranteed his continued education.

Kepler studied theology and learnedthe principles of the Copernican system.He became an early convert to theheliocentric hypothesis, defending it inarguments with fellow students.

In 1594, Kepler was offered a positionteaching mathematics and astronomy atthe high school in Gratz. One of hisduties included preparing almanacsproviding astronomical and astrologicaldata. Although he thought astrology, aspracticed, was essentially quackery, hebelieved the stars affected earthly events.

Tycho De Brahe

Three years after the publication of DeRevolutionibus, Tyge De Brahe was bornto a family of Danish nobility. Tycho, ashe came to be known, developed an earlyinterest in astronomy and made significantastronomical observations as a youngman. His reputation gained him royalpatronage and he was able to establish anastronomical observatory on the island ofHveen in 1576. For 20 years, he and hisassistants carried out the most completeand accurate astronomical observationsyet made.

During a lecture having no relation toastronomy, Kepler had a flash of insight;he felt with certainty that it was to guidehis thoughts throughout his cosmicjourney. Kepler had wondered why therewere only six planets and whatdetermined their separation. This flash ofinsight provided the basis for hisrevolutionary discoveries. Keplerbelieved that each orbit was inscribedwithin a sphere that enclosed a perfectsolid3 within which existed the next

orbital sphere and so on for all the planets.

Tycho was a despotic ruler of Hveen,which the king could not sanction. Thus,Tycho fell from favor, leaving Hveen in1597 free to travel. He ended his travels

in Prague in 1599 and became EmperorRudolph IIs Imperial Mathematicus. Itwas during this time that a youngmathematician, who would also becomean exile from his native land, begancorrespondence with Tycho. JohannesKepler joined Tycho in 1600 and, with nomeans of self-support, relied on Tycho formaterial well being.


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3A perfect solid is a three dimensional geometric

figure whose faces are identical and are regular

polygons. These solids are: (1) tetrahedron

bounded by four equilateral triangles, (2) cube, (3)

octahedron (eight equilateral triangles), (4) do-

decahedron (twelve pentagons), and (5) icosahe-

dron (twenty equilateral triangles).


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Keplers LawsHe did not believe these solids actuallyexisted, but rather, God created theplanetary orbits in relation to these perfectsolids. However, Kepler made the errantconnection that this was the basis of thedivine plan, because there are only five

regular solids and there were only sixknown planets.

Keplers earth-shaking discoveriescame in anything but a straightforwardmanner. He struggled through tediouscalculations for years just to find that they

led to false conclusions. Kepler stumbledupon his second law (which is actually theone he discovered first) through asuccession of canceling errors. He wasaware of these errors and in hisexplanation of why they canceled he gothopelessly lost. In the struggle for thefirst law (discovered second), Keplerseemed determined not to see the solution.He wrote several times telling friends thatif the orbits were just an ellipse, then allwould be solved, but it wasnt until muchlater that he actually tried an ellipse. In

his frustrating machinations, he derived anequation for an ellipse in a form he did notrecognize4. He threw out his formula(which described an ellipse) because hewanted to try an entirely new orbit: anellipse5.

Kepler explained his pseudo-discoveries in his first book, theMysterium Cosmographicum (CosmicMystery). Although based on faultyreasoning, this book became the basis forKeplers later great discoveries. Thescientific and metaphysical communitiesat the time were divided as to the worth ofthis first work. Kepler continued workingtoward proving his theory and in doing so,found fault with his enthusiastic first

book. In his attempts at validation, hecame to realize he could only continuewith Tychos databut he did not havethe means to travel and begin theirrelationship. Fortunately for theadvancement of astronomy, the power ofthe Catholic Church in Gratz grew to apoint where Kepler, a Protestant, wasforced to quit his post. He then traveledto Prague where his short tumultuousrelationship with Tycho began. On 4February 1600, Kepler finally met TychoDe Brahe and became his assistant.

Keplers 1stLaw(Law of Ellipses)

The orbits of the planets areellipses with the Sun at onefocus.

Tycho originally set Kepler to work onthe motion of Mars, while he kept themajority of his astronomical data secret.This task was particularly difficultbecause Mars orbit is the second mosteccentric (of the then known planets) anddefied the circular explanation. Aftermany months and several violentoutbursts, Tycho sent Kepler on a missionto find a satisfactory theory of planetarymotion (the study of Mars continued to bedominant in this quest); one compatible

with the long series of observations madeat Hveen. 4In modern denotation, the formula is:After Tychos death in 1601, Kepler

became Emperor Rudolphs ImperialMathematicus. He finally obtainedpossession of the majority of Tychosrecords, which he studied for the nexttwenty-five years of his life.

R=1+ecos( )

where Ris the distance from the Sun, the longi-

tude referred to the center of the orbit, and e the

eccentricity.


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5After accepting the truth of his elliptical hypothe-

sis, Kepler eventually realized his first equation

was also an ellipse.


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Later Sir Isaac Newton found that

certain refinements had to be made toKeplers first law to account forperturbing influences. Neglecting suchinfluences (e.g., atmospheric drag, mass

asymmetry and third body effects), thelaw applies accurately to all orbitingbodies.

Figure 8-1shows an ellipse where Flisone focus and F2 is the other. Thisdepiction illustrates that, by definition, anellipse is constructed by joining all pointsthat have the same combined distance (D)between the foci.

The maximum diameter of an ellipse iscalled its major axis; the minimumdiameter is the minor axis. The size of anellipse depends in part upon the length ofits major axis. The shape of an ellipse isdenoted by eccentricity (e) which is the

ratio of the distance between the foci tothe length of the major axis (see OrbitGeometry section).

The path of ballistic missiles (not

including the powered and reentryportion) are also ellipses; however, they

happen to intersect the Earths surface(Fig. 8-2).

With Keplers second law, he was onthe trail of Newtons Law of UniversalGravitation. He was also hinting atcalculus, which was not yet invented.

Keplers 2ndLaw(Law of Equal Areas)

The line joining the planet tothe Sun sweeps out equalareas in equal times.

Based on his observation, Keplerreasoned that a planets speed dependedon its distance to the Sun. He drew theconnection that the Sun must be thesource of a planets motive force.

Fig. 8-1. Ellipse with axis

With circular orbits, Keplers secondlaw is easy to visualize (Fig. 8-3). In acircular orbit an objects speed and radiusboth remain constant, and therefore, in agiven interval of time it travels the same

distance along the circumference of thecircle. The areas swept out over these

intervals are equal.

Fig. 8-3. Keplers 2ndLaw

Ball ist ic Missile

Fig. 8-2. Ballistic Missile Path

However, closed orbits in general arenot circular but instead elliptical with.non-zero eccentricity (An ellipse with zeroeccentricity is a circle6 see pg. 8-11).


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6That is, naturally occurring orbits have some non-

zero eccentricity. A circle is a special form of an

ellipse where the eccentricity is zero. Most artifi-


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distances, in what paths ought they togo? To Halleys astonishment, Newtonreplied without hesitation: Why inellipses, of course. I have alreadycalculated it and have the proof among mypapers somewhere. Newton was

referring to his work during the plagueoutbreak 20 years earlier and in thiscasual way, his great discovery was madeknown to the world.

Halley encouraged his friend tocompletely develop and publish hisexplanation of planetary motion. Theresult appeared in 1687 as TheMathematical Principles of NaturalPhilosophy, or simply thePrincipia.

Newtons Laws

As weve seen, many great thinkerswere on the edge of discovery, but it wasNewton that took the pieces andformulated a grand view that wasconsistent and capable of describing andunifying the mundane motion of a fallingapple and the motion of the planets:9

Newtons 1stLaw(Inertia)

Every body continues in a stateof uniform motion in a straight

line, unless it is compelled tochange that state by a forceimposed upon it.

This concise statement encapsulates thegeneral relationship between objects andcausality. Newton combined Galileosidea of inertia with Descartes uniformmotion (motion in a straight line) to createhis first law. If an object deviates fromrest or motion in a straight line withconstant speed, then some force is beingapplied.

Newtons first law describesundisturbed motion; inertia, accordingly,

is the resistance of mass to changes in itsmotion. His second law describes howmotion changes. It is important to definemomentum before describing the secondlaw. Momentum is a measure of anobjects motion. Momentum (

vp) is a

vector quantity defined as the product ofan objects mass (m) and its relativevelocity (

vv )

10.

Newtons second law describes therelationship between the applied force, the

mass of the object and the resultingmotion:

vm=p vv

Newtons 2ndLaw(Momentum)

When a force is applied to abody, the time rate of changeof momentum is proportionalto, and in the direction of, theapplied force.

When we take the time rate of changeofan objects momentum (essentiallydifferentiate momentum with respect totime, dp dt

v), this second law becomes

Newtons famous equation:11

Newton continued his discoveries andwith his third law, completed his grandview of motion:

v vF = ma

10Velocity is an inertial quantity and, as such, is

relative to the observer. Momentum, as measured,

is also relative to the observer.


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11The differentiation of momentum with respect to

time actually givesv v v

F = mv + mv& & where m is

the rate of change of mass and

&v&v is the rate of

change of velocity which is accelerationva . In

simple cases we assume that the mass doesnt

change, so m = and the equation reduces to& 0v v v v

F = mv F = m&

a& . For an acceleratingbooster the mterm is not zero.

9We still essentially see the Universe in Newto-

nian terms; Einsteins general relativity and quan-

tum mechanics are a modification to Newtonian

mechanics, but have yet to be unified into a single

grand view.


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Newtons Derivation of Keplers Laws

Newtons 3rdLaw Keplers laws of planetary motion areempirical (found by comparing vastamounts of data in order to find thealgebraic relationship between them); and

describe the way the planets are observedto behave. Newton proposed his laws as abasis for all mechanics. Thus Newtonshould have been able to derive Keplerslaws from his own, and he did:

(Action-Reaction)

For every action there is a

reaction that is equal inmagnitude but opposite indirection to the action.

This law hints at conservation ofmomentum; if forces are always balanced,then the objects experiencing the opposedforces will change their momentum inopposite directions and equal amounts.

Keplers First Law: If twobodies interact gravitationally,each will describe an orbit thatcan be represented by a conicsection about the commoncenter of mass of the pair. In

particular, if the bodies arepermanently associated, theirorbits will be ellipses. If theyare not permanentlyassociated, their orbits will behyperbolas.

Newton combined ideas from varioussources in synthesizing his laws. Keplerslaws of planetary motion were among hissources and provided large scale

examples. Newton synthesized hisconcept of gravity, but thought that onemust be mad to believe in a force thatoperated across a vacuum with nomaterial means of transport.

Newton theorized gravity, which hebelieved to be responsible for the fallingapples and the planetary motion, eventhough he could not explain gravity orhow it was transmitted. In essence,Newton developed a system that describedmans experience with his environment.

Keplers Second Law: If twobodies revolve about eachother under the influence of acentral force (whether they arein a closed orbit or not), a line

joining them sweeps out equalareas in the orbit plane inequal intervals of time.

Universal Gravitation

Every particle in the universeattracts every other particle witha force that is proportional to the

product of the masses andinversely proportional to thesquare of the distance betweenthe particles.

Keplers Third Law: If twobodies revolve mutually abouteach other, the sum of theirmasses times the square oftheir period of mutual revolutionis in proportion to the cube oftheir semi-major axis of the

relative orbit of one about theother.

F GM m

Dg

1 2

2=

Where Fgis the force due to gravity, G isthe proportionality constant, M1 and m2the masses of the central and orbitingbodies, and D the distance between thetwo bodies.

ORBITAL MOTION


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Newtons laws of motion apply to allbodies, whether they are scurrying acrossthe face of the Earth or out in the vastness


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of space. By applying Newtons laws onecan predict macroscopic events with greataccuracy.

Motion

According to Newtons first law,bodies remain in uniform motion unlessacted upon by an external force; thatuniform motion is in a straight line. Thismotion is known as inertial motion,referring to the property of inertia, whichthe first law describes.

Velocity is a relative measure ofmotion. While standing on the surface ofthe Earth, it seems as though thebuildings, rocks, mountains and trees areall motionless; however, all of theseobjects are moving with respect to many

other objects (Sun, Moon, stars, planets,etc.). Objects at the equator are travelingaround the Earths axis at approximately1,000 mph; the Earth and Moon system istraveling around the Sun at 66,000 mph;the solar system is traveling around thegalactic center at approximately 250,000mph, and so on and so forth.

The only way motion can beexperienced is by seeing objects changeposition with respect to ones location.Change in motion may be experienced byfeeling the compression or tension withinthe body due to acceleration (sinking inthe seat or being held by seat belts). Insome cases, acceleration cannot be felt, asin free-fall. Acceleration is felt when theforces do not operate equally on everyparticle in the body; the compression ortension is sensed in the bodys tissues.With this feeling and other visual clues,any change in motion that has occurredmay be detected. Gravity is felt asopposing forces and the resultingcompression of bodily tissues. In free-

fall, acceleration is not felt because everyparticle in the body is experiencing thesame force and so there is no tissuecompression or tension; thus, no physicalsensation. What is felt is the suddenchange from tissue compression to a stateof no compression.

According to Newtons second law, fora body to change its motion there must be

a force imposed upon it. Everyone hasexperience with changing objects motionor compensating for forces that changetheir motion. An example is playingcatchwhen throwing or catching a ball,its motion is altered; thus, gravity is

compensated for by throwing the ballupward by some angle allowing gravity topull it down, resulting in an arc. Whenthe ball leaves the hand it startsaccelerating toward the ground accordingto Newtons laws (at sea level on theEarth the acceleration is approximately9.81 m/s or 32.2 ft/s). If the ball isinitially motionless, it will fall straightdown. However, if the ball has somehorizontal motion, it will continue in thatmotion while accelerating toward the

ground. Figure 8-5shows a ball releasedwith varying lateral (or horizontal)velocities.

Horizontal Velocity

Fig. 8-5. Newtons 2ndLaw

In Figure 8-5, if the initial height of the

ball is approximately 4.9 meters (16.1 ft)above the ground, then at sea level, itwould take 1 second for the ball to hit the

Table 8-1. Gravitational Effects

HorizontalVelocity

Distance (@ 1 sec)Vertical Horizontal

1248

16

4.94.94.94.94.9

1248

16All values are in meters and meters/second.

ground. How far the ball travels along theground in that one second depends on itshorizontal velocity (see Table 8-1).

Eventually one would come to thepoint where the Earths surface dropsaway as fast as the ball drops toward it.As Fig. 8-6 depicts, the Earths surface


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curves down about 5 meters for every 8km.

At the Earths surface (withoutcontending for the atmosphere, mountainsor other structures), a satellite would have

to travel at approximately 8 km/sec (orabout 17,900 mph) to fall around theEarth without hitting the surface; in otherwords, to orbit.12

Figure 8-7 shows how differingvelocity affects a satellites trajectory ororbital path. The Figure depicts a satelliteat an altitude of one Earth radius (6378km above the Earths surface). At thisdistance, a satellite would have to travel at

5.59 km/sec to maintain a circular orbitand this speed is known as its circularspeed for this altitude. As the satellitesspeed increases, it falls farther and farther

away from the Earth and its trajectorybecomes an elongating ellipse until thespeed reaches 7.91 km/sec. At this speedand altitude the satellite has enoughenergy to leave the Earths gravity andnever return; its trajectory has now

become a parabola, and this speed isknown as its escape speed for thisaltitude. As the satellites speed continuesto increase beyond escape speed itstrajectory becomes a flattened hyperbola.From a low Earth orbit of about 100miles, the escape velocity becomes 11.2km/sec. In the above description, the twospecific speeds (5.59 km/sec and 7.91km/sec) correspond to the circular andescape speeds for the specific altitude ofone Earth radius.

Fig. 8-6. Earths Curvature

The satellites motion is described by

Newtons three laws and his Law ofUniversal Gravitation. The Law ofUniversal Gravitation describes how theforce between objects decreases with thesquare of the distance between theobjects. As the altitude increases, theforce of gravity rapidly decreases, andtherefore the satellite can travel slowerand still maintain a circular orbit. For theobject to escape the Earth, it has to haveenough kinetic energy (kinetic energy isproportional to the square of velocity) toovercome the gravitational potentialenergy of its position. Since gravitationalpotential energy is proportional to thedistance between the objects, the fartherthe object is from the Earth, the lesspotential energy the satellite mustovercome, which also means the lesskinetic energy is needed.

5.59 km\sec

5.59


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overcome the gravitational attraction thenthe object will follow an open path(parabola or hyperbola) and escape fromthe central force.

FocusApogee

Perigee

a

c c Fig. 8-9. Elliptical Geometry

Figure 8-8 shows the basic geometryfor the various possible conic sections.

The parameters that describe the size andshape of the conics are itssemi-major axisa (half of the large axis) and eccentricitye(the ratio between the separation of thefocilinear eccentricity cand the semi-major axis).

Figure 8-9 depicts a satellite orbitwith additional parameters whose conicsection is an ellipse.Semi-major Axis (a)half of the distance

between perigee and apogee, ameasure of the orbits size, also theaverage distance from the attractingbody.

Linear Eccentricity (c)half of thedistance between the foci.

Eccentricity (e)ratio of the distancebetween the foci (c) to the size of theellipse (a); describes the orbits shape.

Perigeethe closest point in an orbit to

the attracting body.Apogeethe farthest point in an orbit tothe attracting body.

These parameters apply to alltrajectories. A circular orbit is a specialcase of the elliptical orbit where the focicoincide (c = 0). A parabolic path is a

transition between an elliptical and ahyperbolic trajectory. The parabolic pathrepresents the minimum energy escapetrajectory. The hyperbolic is also anescape trajectory; and represents atrajectory with excess escape velocity.

Fig. 8-8. Conic Section Geometry

Table 8-2 shows the values for the

eccentricity (discussed later) for thevarious types of orbits. Eccentricity isassociated with the shape of the orbit.Energy is associated with the orbits size(for closed orbits).

Table 8-2. Eccentricity ValuesConic Section Eccentricity(e)

circle

ellipse

parabola

hyperbola

e= 0

0 < e< 1

e= 1

e> 1

CONSTANTS OF ORBITALMOTION: MOMENTUM AND

ENERGY

For some systems, there are basicproperties which remain constant or fixed.Energy and momentum are two such

properties required for a conservativesystem.

Momentum


8 - 11

Momentum is the product of masstimes velocity (

v vp mv= ). This is the term

for linear momentum and remainsconstant internal to the system in every


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direction. In some instances it is moreadvantageous to describe motion inangular terms. For example, whendealing with spinning or rotating objects,it is simpler to describe them in angularterms An important angular property in

orbital mechanics is angular momentum.Angular momentum is the product oflinear momentum times the radius ofrevolution.13 This property, like linearmomentum, remains constant internal tothe system for such things as orbitingobjects.

A simple experiment can be performedillustrating conservation of angularmomentum. Starting with some object onthe end of a string, the object may be spunto impart angular momentum to thesystem. The amount of angular

momentum depends on the objects massand velocity, and the length of the string(radius):

v v vh m . Now, if the string

is shortened, the object will speed up (spinfaster). From the above equation, mass(m) remains constant; angular momentum(

(r v= )

vh ) must remain constant as the radius (

vr)

decreases, so the objects velocity (vv )

increases. This same principle holds truefor orbiting systems. In an elliptical orbit,the radius is constantly varying and so isthe orbital speed, but the angularmomentum remains constant. Hence,there is greater velocity at perigee than atapogee.

Energy

A systems mechanical energy can alsobe conserved. Mechanical energy (denotedby E) is the sum of kinetic energy (KE)and potential energy (PE): E KE PE= + .Kinetic energy is the energy associatedwith an objects motion and potentialenergy is the energy associated with an

objects position. Every orbit has acertain amount of mechanical energy. Acircular orbits radius and speed remainconstant, so both potential and kinetic

energy remain constant. In all other orbits(elliptical, parabolic and hyperbolic) theradius and speed both change, andtherefore, so do both the potential andkinetic energy in such a way that the totalmechanical energy of the system remains

constant. Again, for an elliptical orbit,this results in greater velocity at perigeethan apogee. If a satellites position andvelocity is known, a satellites orbit maybe ascertained. Position determinespotential energy while velocity determineskinetic energy.

COORDINATE REFERENCESYSTEMS AND ORBITAL

ELEMENTS

Reference systems are used everyday.

Once an agreed upon reference has beendetermined, spatial information can betraded. The same must be done whenconsidering orbits and satellite positionsThe reference system used depends on thesituation, or the nature of the knowledgeto be retrieved.

How does one know where satellitesare, were or will be? Coordinatereference systems allow measurements tobe defined, resulting in specificparameters which describe orbits. A setof these parameters is a satellites orbitalelement set. Two elements are needed todefine an orbit: a satellites position andvelocity. Given these two parameters, asatellites past and future position andvelocity may be predicted.

In three-dimensional spaces, it takesthree parameters each to describe positionand velocity. Therefore, any element setdefining a satellites orbital motionrequires at least six parameters to fullydescribe that motion. There are differenttypes of element sets, depending on the

use. The Keplerian, or classical, elementset is useful for space operations andtellsus four parameters about orbits, namely:

Orbit size

l

Orbit shape13Angular momentum is actually the vector cross

product of inear momentum and the radius of

revolution:v v v v vh r .p m( r v=

Orientation- orbit plane in space


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) - orbit within plane


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Location of the satellite

Ascending Node

Inclination

Angle

Equatorial Orbit

Inclined Orbit

Fig. 8-10. Inclination Tilt

Semi-Major Axis (a)

. Thesemi-major axis (a) describes anorbits size and is half of the distance

between apogee and perigee on theellipse. This is a significant measurementsince it also equals the average radius, andthus is a measure of the mechanicalenergy of the orbiting object.

Eccentricity (e)

Eccentricity (e) measures the shape ofan orbit and determines the positionalrelationship to the central body whichoccupies one of the foci. Recall from the

orbit geometry section that eccentricity isa ratio of the foci separation (lineareccentricity, c) to the size (semi-majoraxis, a) of the orbit:

e = c/a

Size and shape relate to orbit geometry,and tell what the orbit looks like. Theother orbital elements deal withorientation of the orbit relative to a fixedpoint in space.

Inclination (i)

The first angle used to orient the orbitalplane isinclination(i): a measurement ofthe orbital planes tilt. This is an angularmeasurement from the equatorial plane tothe orbital plane (0 i 180), measuredcounter-clockwise at the ascending nodewhile looking toward Earth (Fig. 8-10).

Inclination is utilized to define severalgeneral classes of orbits. Orbits with

inclinations equal to 0 or 180 areequatorial orbits, because the orbitalplane is contained within the equatorialplane. If an orbit has an inclination of90, it is a polar orbit, because it travelsover the poles. If 0 i< 90,the satelliteorbits in the same general direction as theEarth (orbiting eastward around the Earth)

and is in a prograde orbit. If 90< i180, the satellite orbits in the oppositedirection of the Earths rotation (orbitingwestward about the Earth) and is in aretrograde orbit. Inclination orients theorbital plane with respect to the equatorialplane (fundamental plane).

Right Ascension of the AscendingNode ()

Right Ascension of the AscendingNode, (upper case Greek letter omega),is a measurement of the orbital planesrotation around the Earth. It is an angularmeasurement within the equatorial planefrom the First point of Aries eastward tothe ascending node (0 360) (Fig.8-11).

The First Point of Aries is simply afixed point in space. The Vernal Equinoxis the first day of spring (in the northernhemisphere). However, for theastronomer, it has added importancebecause it is a convenient way of fixingthis principle direction. The Earths

Line of Nodes

Inclined Orbit

Perigee

Argument of Perigee,

Approximately 270 deg

Ascending Node

Fig. 8-13. Argument of Perigee


8 - 13


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The ancient astronomers called this theFirst Point of Aries because, at the time,this line pointed at the constellation Aries.The Earth is spinning like a top, and like atop, it wobbles on its axis. It takesapproximately 25,800 years for the axis to

complete one revolution. With the axischanging over time, so does the equatorialplanes orientation. The intersectionbetween the ecliptic and the equatorialplane is rotating westward around theecliptic. With this rotation, the principledirection points to different constellations.Presently, it is pointing towards Pisces.The orbital elements for Earth satellitesare referenced to inertial space (a non-rotating principle direction) so the orbitalelements must be referenced to where theprinciple direction was pointing at a

specific time. The orbital analyst doesthis by reporting the orbital elements asreferenced to the mean of 1950 (a popularepoch year reference). Most analystshave updated their systems and are nowreporting the elements with respect to themean of 2000.

Ascending Node

Line of Nodes

0 degrees = First Point of Aries

0

45

180

225315

135

Inclined Orbit

Fig. 8-11. Right Ascension of the Ascending Node

equatorial plane and its orbit about theSun provide the principle direction. TheEarth orbits about the Sun in the eclipticplane, and this plane passes through thecenters of both the Sun and Earth; theEarths equatorial plane passes throughthe center of the Earth, which is tilted at

approximately 23 to the ecliptic. Theintersection of these two planes forms aline that passes though Earths center andpasses through the Suns center twice ayear: at the Vernal and Autumnal

Equinoxes. The ancient astronomerspicked the principle direction as that fromthe Suns center through the Earthscenter on the first day of Spring, theVernal Equinox (Fig. 8-12).

Fig. 8-12. Vernal Equinox

Argument of Perigee ()

Inclination and Right Ascension fix theorbital plane in inertial space. The orbitmust now be fixed within the orbitalplane. For elliptical orbits, the perigee isdescribed with respect to inertial space.

The Argument of Perigee, (lowercase Greek letter omega),orients the orbitwithin the orbital plane. It is an angularmeasurement within the orbital plane fromthe ascending node to perigee in thedirection of satellite motion (0 360)(seeFig. 8-13).


8 - 14


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Table 8-3. Classical Orbital ElementsElement Name Description Definition Remarks

a semi-major

axis

orbitsize half of the long axis of the

ellipse

orbital period and energy

depend on orbit size

e eccentricity orbitshape ratio of half the foci

separation (c) to the semi-

major axis

closed orbits: 0 e < 1open orbits: 1 e

i inclination orbital planes

tilt

angle between the orbital

plane and equatorial plane,measured counterclockwise at

the ascending node

equatorial: i = 0or 180prograde: 0i


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Geostationary Period =23hrs56mini= 0e= 0

AU 7/23/2003

8 - 16Space primer

GeosynchronousPeriod =23hrs56minMolniya Period =11hrs58min

i= 63.4

e = .72Sun-synchronousPeriod=1hr41mini= 98

Semi-synchronous-Period =11hr58min

GROUND TRACKS

The physics of two-body motiondictates the motions of the bodies will liewithin a plane (two-dimensional motion).The orbital plane intersects the Earths

surface forming a great circle. Asatellites ground track is the intersectionof the line between the Earths center andthe satellite, and the Earths surface; thepoint on the line at the surface of theEarth is called thesatellite subpoint. Theground track, then, is the path the satellitesubpoint traces on the Earths surface over

time (Fig. 8-14). However, the Earthdoes rotate on its axis at the rate of onerevolution per 24 hours. With the Earthrotating under the satellite, the

intersection of the orbital plane,14and theEarths surface is continually changing.

The ground track is the expression ofthe relative motion of the satellite in itsorbit to the Earths surface rotatingbeneath it. Because of this relative

motion, ground tracks come in almost anyform and shape imaginable. Ground trackshape depends on many factors:

Inclination iPeriod PEccentricity eArgument of Perigee

Inclination defines the tilt of the orbitalplane and therefore, defines the maximumlatitude, both North and South of theground track.

Fig. 8-14. Ground Track

The period defines the ground tracks

westward regression. With a non-rotatingEarth, the ground track would be a greatcircle. Because the Earth does rotate, bythe time the satellite returns to the sameplace in its orbit after one revolution, theEarth has rotated eastward by someamount, and the ground track looks like ithas moved westward on the Earthssurface (westward regression). Theorientation of the satellites orbital planedoes not change in inertial space, theEarth has just rotated beneath it. The timeit takes for the satellite to orbit (its orbitalperiod) determines the amount the Earthrotates eastward and hence its westwardregression.

Fig. 8-15. Earths Rotation Effects

2

3

1

Figure 8-15 shows the effect of theEarths rotation on the ground track. The

14Except for equatorial orbits, whose orbital plane

is contained within the equatorial plane.


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Fig. 8-16. Geostationary Orbit/Ground Track

Earth rotates through 360 in 24 hours,giving a rotation rate of 15/hr.15 With aperiod of 90 min., a satellites groundtrace regresses 22.5 westward perrevolution (15/hr 1.5 hrs= 22.5).Westward regression is the angle through

which the Earth has rotated underneaththe satellite during the time it takes thesatellite to complete one orbit.

Eccentricity affects the ground trackbecause the satellite spends differentamounts of time in different parts of itsorbit (its moving faster or slower). Thismeans it will spend more time over certainparts of the Earth than others. This hasthe effect of creating an unsymmetricalground track. It is difficult to determinehow long the satellite spends in eachhemisphere by simply looking at the

ground trace. The time depends on boththe length of the trace and the speed of thesatellite.

Argument of perigee skews the groundtrack. For a prograde orbit, at perigee thesatellite will be moving faster eastwardthan at apogee; in effect, tilting the groundtrack.


8 - 17

A general rule of thumb is that if theground track has any portion in theeastward direction, the satellite is in aprograde orbit. If the ground trace doesnot have a portion in the eastwarddirection, it is either a retrograde orbit orit could be a super-synchronous progradeorbit.

Relative Motions

Because the Earth is a rotating thevelocity of points on the surface isdifferent depending on their distance fromthe Earths axis of rotation. In otherwords, points on the equator have a greatereastward velocity than points north and

south of the equator.Figure 8-16 conceptualizes a geo-stationary orbit and its ground track. A

satellite in a ideal geostationary orbit hasthe same orbital period as the Earthsrotational period, its inclination is 0 andits eccentricity is 0. The ground track willremain in the equatorial plane, the

westward regression will be 360and thesatellites speed never changes.Therefore, from the earth, the groundtrack will be a point on the equator.

Now take the same orbit and give it aninclination of 45).

The period and eccentricity remain thesame. The westward regression will be

360so the ground trace will retrace itselfwith every orbit. The ground trace will

also vary between 45 North and 45South. The apparent ground trace lookslike a figure eight(Fig. 8-17 for thesimplest case. If the orbital parametersare varied (such as eccentricity andargument of perigee), the relative motionsof the satellite and the

Fig. 8-17. Ground Traces of Inclined, Circular,

Synchronous Satellites

15In reality, the 28-hour rotation rate is with re-

spect to the Sun and is called the solar day. The

rotation rate with respect to inertial space (the

fixed stars, or background stars) is actually 23 hrs

56 min and is called thesidereal day.


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HHiigghh

IInncclliinnaattiioonn

1122,,550000MMiilleessAAllttiittuuddee

Fig. 8-18. Semi-synchronous Orbit

HighlyElliptical

Inclination (63.4/116.6)Altitude: 200-23,800 Mile

Fig. 8-19. Molniya Orbit

Earths surface can become quitecomplicated. For orbits with smallinclinations, the eccentricity and argumentof perigee dominate the effect of theEarths surface speed at different latitudesand can cause the ground track to varysignificantly from a symmetric figureeight. These parameters can be combinedin various ways to produce practically any

ground track.


8 - 18

The semi-synchronous orbit (used bythe Global Positioning System) alsoprovides a unique ground track. Thisorbit, with its approximate 12-hourperiod, repeats twice a day. Since theEarth turns half way on its axis duringeach complete orbit, the points where thesinusoidal ground tracks cross the equatorcoincide pass after pass and the groundtracks repeat each day (Fig. 8-18).

Figure 8-19shows a typical Molniya

orbit that might be used for northernhemispheric communications. TheRussians are credited with the discoveryof this ingenious orbit. With the highdegree of eccentricity the satellite travelsslowly at apogee and can hang over theNorthern Hemisphere for about two thirdsof its period. Since the period is 12 hours,

the ground track retraces itself every day,much the same as the semi-synchronousorbit

Fig. 8-20. Sun-synchronous Orbit

5 4 3 2 1

.

Figure 8-20 shows a representativesun-synchronous orbit. In this case, theorbital elements represent a DMSP(Defense Meteorological SatelliteProgram) satellite. The satellite is in aslightly retrograde orbit; therefore, thesatellite travels east to west along thetrack..


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8 - 19

A sun-synchronous orbit is one inwhich the orbital plane rotates eastwardaround the Earth at the same rate that theEarth orbits the Sun. So, the orbit mustrotate eastward around the Earth at a littleless than 1/day {(360/year)/(365.25days/ year) = .986/day}. Thisphenomenon occurs naturally due to theoblateness of the Earth (see the section onperturbations).

Sun-synchronous orbits can beachieved at different altitudes andinclinations. However, all the inclinationsfor sun-synchronous satellites are greaterthan 90(retrograde orbits). Figure 8-21plots inclination versus altitude for sun-synchronous orbits.

LAUNCH CONSIDERATIONS

The problem of launching satellitescomes down to geometry and energy. Ifthere were enough energy, satellites couldbe launched from anywhere at any timeinto any orbit. However, energy is limitedand so is cost.

When a satellite is launched, it isintended to end up in a specific orbit, notonly with respect to the Earth, but oftenwith respect to an existing constellation.Also, the geometry of the planets must betaken into consideration when launching

an interplanetary probe. Meetingoperational constraints determines thelaunch window. The launch system isdesigned to accomplish the mission withthe minimum amount of energy requiredbecause it is usually less

expensive.16Keeping energy to aminimum restricts the launch trajectoriesand the launch location.

Fig. 8-21. Inclination versus Altitude

Launch site latitude and orbitinclination are two important factorsaffecting how much energy boosters have

to supply. Orbit inclination depends onthe satellites mission, while launch sitelatitude is, for the most part, fixed (to ourexisting launch facilities).17 Onlyminimum energy launches (direct launch)will be addressed. A minimum energy isone in which a satellite is launcheddirectly into the orbital plane (i.e., noplane change or inclination maneuver).Bylooking at the geometry, the launch sitemust pass through the orbital plane to becapable of directly launching into thatplane. Imagine a line drawn from the

center of the Earth through the launch siteand out into space. After a day, this lineproduces a conical configuration due tothe rotation of the Earth. A satellite canbe launched into any orbital plane that istangent to, or passes through, this cone.As a result of this geometry, the lowestinclination that can be achieved bydirectly launching is equal to the latitudeof the launch site.

If the orbital plane inclination is greaterthan the launch site latitude, the launchsite will pass through the orbital planetwice a day, producing two launchwindows per day. If the inclination of theorbital plane is equal to the launch sitelatitude, the launch site will be coincidentwith the orbital plane once a day,producing one launch window a day. Ifthe inclination is less than the launch sitelatitude, the launch site will not passthrough, or be coincident with the orbitalplane at any time, and so there will not beany launch windows for a direct launch.

16There are some situations when it is less expen-

sive to use an existing system with extra energy

because a lower class booster will not meet the

mission needs. In this case, there is an extra mar-

gin and the launch windows are larger.17There are various schemes to get around the

problem of fixed launch sites: air launch, sea

launch, and portable launch facilities, for instance.


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A simplified model for determininginclination from launch site latitude andlaunch azimuth is:18

( ) ( ) ( )AzLi sincoscos = i= inclinationL = launch site latitude

Az = launch azimuth


8 - 20

The cosine of the latitude reduces therange of possible inclinations and the sineof the azimuth varies the inclinationwithin the reduced range. When viewingthe Earth and a launch site, it is possibleto launch a satellite in any direction(launch azimuth). The orbital plane mustpass through the launch site and the centerof the Earth.

For launches due east (no matter whatthe launch site latitude) the inclinationwill equal the launch site latitude. Forlaunches on any other azimuth, theinclination will always be greater than thelaunch site latitude.

Just as the launch site latitudedetermines the minimum inclination(launching due east), it also determinesthe maximum inclination by launchingdue west. The maximum inclination is180 minus the latitude.

The actual launch azimuths allowed (inmost countries) are limited due to the

safety considerations of not launchingover populated areas, which further limitsthe possible inclinations from any launchsite. However, the inclination can changeafter launch by performing an out of planemaneuver (see next section).

Launch Velocity

When a satellite is launched, energy isimparted to it. The two tasks ofincreasing the satellites potential andkinetic energies must be accomplished.

Potential energy is increased by raisingthe satellite above the Earth (increasing itsaltitude by at least 90-100 miles). Inorder to maintain a minimum circularorbit at that altitude, the satellite has to

travel about 17,500 mph. Due to theEarths rotation, additional kinetic energymay need to be supplied depending onlaunch azimuth to achieve this orbitalvelocity (17,500 mph). The startingvelocity at the launch sites vary with

latitude. It ranges from zero mph at thepoles to 1,037 mph at the equator.If a satellite is launched from the

equator prograde (in the same direction asthe earths rotation) starting with 1,037mph, only 16,463 mph must be supplied(17,500 mph- 1,037 mph). If launchedfrom the equator retrograde (against therotation of the earth), 18,537 mph must besupplied. Launching with the earthsrotation saves fuel and allows for largerpayloads for any given booster.

There are substantial energy savings

when locating launch sites close to theequator and launching in a progradedirection.

ORBITAL MANEUVERS

It is a rare case indeed to launchdirectly into the final orbit. In general, asatellites orbit must change at least onceto place it in its final mission orbit. Oncea satellite is in its mission orbit,perturbations must be counteracted, orperhaps the satellite must be moved intoanother orbit.

As was previously mentioned, asatellites velocity and position determineits orbit.19 Thus, one of these parametersmust be changed in order to change itsorbit. The only option is to change thevelocity, since position is relativelyconstant. By changing the velocity, thesatellite is now in a different orbit. Sincegravity is conservative, the satellite willalways return to the point where itperformed the maneuver (provided it

doesnt perform another maneuver beforereturning).

19The position and velocity correspond to the

force of gravity and the satellites momentum.

Knowing the forces on the satellite and its momen-

tum, we can apply Newtons second law and pre-

dict its future positions (in other words, we know

its orbit).

18This is a simplified model because it ignores the

Earths rotation, which has a small effect.


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8 - 21

Mission Considerations

Since both position and velocitydetermine a satellites orbit, and manydifferent orbits can pass through the same

point, the velocity vectors must differ20

toresult in a different orbit while passingthrough the same point.

When an orbit is changed through itsvelocity vector, a delta-v (v) isperformed. For any single v orbitalchange, the desired orbit must intersectthe current orbit, otherwise it will take atleast two vs to achieve the final orbit.

When the present and desired orbitsintersect, a v is employed to change thesatellites velocity vector. The vvectorcan be determined by subtracting the

present vector from the desired vector.The resultant velocity vector is the vrequired to get from one point to another.

PERTURBATIONS

Perturbations are forces which changethe motion (orbit) of the satellite. Theseforces have a variety of causes/origins andeffects. These forces are named andcategorized in an attempt to model theireffects. The major perturbations are:

Earths oblateness; Atmospheric drag; Third-body effects; Solar wind/radiation pressure; Electromagnetic drag.

Earths Oblateness

The Earth is not a perfect sphere. It issomewhat misshapen at the poles andbulges at the equator. This squashed

shape is referred to as oblateness. TheNorth polar region is more pointed thanthe flatter South polar region, producing aslight pear shape. The equator is not aperfect circle; it is slightly elliptical. Theeffects of Earths oblateness are

20 Remember that velocity is a vector it has

both a magnitude and a direction.

gravitational variations or perturbations,which have a greater influence the closera satellite is to the Earth. This bulge isoften modeled with complex mathematicsand is frequently referred to as the J2effect.21 For low to medium orbits, these

influences are significant.One effect of Earths oblateness isnodal regression. Westward regressiondue to Earths rotation under the satellitewas discussed in the ground trackssection. Nodal regression is an actualrotation of the orbital plane, relative to theFirst Point of Aries, about the Earth (theright ascension changes). If the orbit isprograde, the orbital plane rotateswestward around the Earth (rightascension decreases); if the orbit isretrograde, the orbital plane rotates

eastward around the Earth (rightascension increases).

In most cases, perturbations must becounteracted. However, in the case ofsun-synchronous orbits, perturbations canbe advantageous. In the slightlyretrograde sun-synchronous orbit, theangle between the orbital plane and a linebetween the Earth and the Sun needs toremain constant. As the Earth orbitseastward around the Sun, the orbital planemust rotate eastward around the Earth atthe same rate. Since it takes 365 days forthe Earth to orbit the Sun, the sun-synchronous orbit must rotate about theEarth at just under one degree per day.The oblateness of the Earth perturbs theorbital plane by nearly this amount.

A sun-synchronous orbit is beneficialbecause it allows a satellite to view thesame place on Earth with the same sunangle (or shadow pattern). This is veryvaluable for remote sensing missionsbecause they use shadows to measureobject height.22

21J2 is a constant describing the size of the bulge

in the mathematical formulas used to model the

oblate Earth.22With a constant sun angle, the shadow lengths

give away any changes in height, or any shadow

changes give clues to exterior configuration

changes.


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8 - 22

Another significant effect of Earthsasymmetry is apsidal line rotation. Onlyelliptical orbits have a line of apsides andso this effect only affects elliptical orbits.This effect appears as a rotation of theorbit within the orbital plane; the

argument of perigee changes. At aninclination of 63.4 (and its retrogradecompliment, 116.6), this rotation is zero.The Molniya orbit was specificallydesigned with an inclination of 63.4 totake advantage of this perturbation. Withthe zero effect at 63.4 inclination, thestability of the Molniya orbit improveslimiting the need for considerable on-board fuel to counteract this rotation. At asmaller inclination (but larger than116.6), the argument of perigee rotateseastward in the orbital plane; at

inclinations between 63.4 and 116.6, theargument of perigee rotates westward inthe orbital plane. This could present aproblem for non-Molniya communicationssatellites providing polar coverage. If theapogee point rotated away from thedesired communications (rotated from theNorthern to Southern Hemisphere), thesatellite would be useless.

The ellipticity of the equator has aneffect that shows up most notably ingeostationary satellites (also in inclinedgeosynchronous satellites). Because theequator is elliptical, most satellites arecloser to one of the lobes and experience aslight gravitational misalignment. Thismisalignment affects geostationarysatellites more because they view thesame part of the earths surface all thetime, resulting in a cumulative effect.

The elliptical force causes the subpointof the geostationary satellite to move eastor west with the direction depends on itslocation. There are two stable points at 75East and 105 West, and two unstable

stable points 90 out (165 East and 5West).23

23A stable point is like a marble in the bottom of a

bowl; an unstable stable point is like a marble per-

fectly balanced on the top of a hill.

Atmospheric Drag

The Earths atmosphere does notsuddenly cease; rather it trails off intospace. However, after about 1,000 km(620 miles), its effects become minuscule.

Generally speaking, atmospheric drag canbe modeled in predictions of satelliteposition. The current atmospheric modelis not perfect because of the many factorsaffecting the upper atmosphere, such asthe earths day-night cycle, seasonal tilt,variable solar distance, fluctuation in theearths magnetic field, the suns 27-dayrotation and the 11-year sun spot cycle.The drag force also depends on thesatellites coefficient of drag and frontalarea, which varies widely betweensatellites.

The uncertainty in these variablescause predictions of satellite decay to beaccurate only for the short term. Anexample of changing atmosphericconditions causing premature satellitedecay occurred in 1978-1979, when theatmosphere received an increased amountof energy during a period of extreme solaractivity. The extra solar energy expandedthe atmosphere, causing several satellitesto decay prematurely, most notably theU.S. space station SKYLAB.

The highest drag occurs when thesatellite is closest to the earth (at perigee),and has a similar effect in performing adelta-V at perigee; it decreases the apogeeheight, circularizing the orbit. On everyperigee pass, the satellite looses morekinetic energy (negative delta-V),circularizing the orbit more and more untilthe whole orbit is experiencing significantdrag, and the satellite spirals in.

Third Body Effects

According to Newtons law ofUniversal Gravitation, every object in theuniverse attracts every other object in theuniverse. The greatest third body effectscome from those bodies that are verymassive and/or close such as the Sun,Jupiter and the Moon. These forces ffectsatellites in orbits as well. The farther asatellite is from the Earth, the greater the


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8 - 23

third body forces are in proportion toEarths gravitational force, and therefore,the greater the effect on the high altitudeorbits.

Radiation pressure

The Sun is constantly expelling atomicmatter (electrons, protons, and Heliumnuclei). This ionized gas moves with highvelocity through interplanetary space andis known as the solar wind. The satellitesare like sails in this solar wind, alternatelybeing speeded up and slowed down,producing orbital perturbations.

Electromagnetic Drag

Satellites are continually traveling

through the Earths magnetic field. Withall their electronics, satellites producetheir own localized magnetic fields whichinteract with the earths, causing torqueon the satellite. In some instances, thistorque is advantageous for stabilization.More specifically, satellites are basically amass of conductors. Passing a conductorthrough a magnetic field causes a currentin the conductor, producing electricalenergy. Some recent experiments used along tether from the satellite to generateelectrical power from the earths magneticfield (the tether also provided otherbenefits).

The electrical energy generated by theinteraction of the satellite and the earthsmagnetic field comes from the satelliteskinetic energy about the earth. Thesatellite looses orbital energy, just as itdoes with atmospheric drag, which resultsin orbital changes. The magnetic field isstrongest close to the Earth where

satellites travel the fastest. Thus, thiseffect is largest on low orbiting satellites.However, the overall effect due toelectromagnetic forces is quite small.

DEORBIT AND DECAY

So far the concern has been withplacing and maintaining satellites in orbit.When no longer useful, satellites must beremoved from their operational orbit.Sometimes natural perturbations such asatmospheric drag take care of disposal,but not always.

For satellites passing close to the earth(low orbit or highly elliptical orbits),satellites can be programmed to re-enter,or they may re-enter autonomously.Deliberate re-entry of a satellite with the

purpose of recovering the vehicle intact iscalled deorbiting. This is usually done torecover something of value: people,experiments, film, or the vehicle itself.The natural process of spacecraft (or anydebris rocket body, payload, or piece)eventually re-entering Earths atmosphereis called decay.

In some situations, the satellites are insuch stable orbits that naturalperturbations will not do the disposal job .In these situations, the satellite must beremoved from the desirable orbit. Toreturn a satellite to earth withoutdestroying it takes a considerable amountof energy. Obviously, it is impractical toreturn old satellites to earth from a highorbit. The satellite is usually boosted intoa slightly higher orbit to get it out of theway, and there it will sit for thousands ofyears.


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REFERENCES

Bates, Roger R., Jerry E. White, and Donald D. Mueller. Fundamentals of

Astrodynamics. New York: Dover Publications, Inc., 1971.

Koestler, Arthur. The Sleepwalkers: A History of Mans Changing Vision of the

Universe. New York: Universal Library, Grosset & Dunlab, 1976.

An Introduction to Orbital Mechanics, CBT Module, Prepared for AFSPC SWC/DOT

http://imagine.gsfc.nasa.gov/docs/features/movies/kepler.html

Contains Video clips (AVI) on Keplers Laws.

www.hq.nasa.gov/office/pao/History/conghand/traject.htm

Orbits & Escape velocities.

http://astro-2.msfc.nasa.gov/academy/rocket_sci/launch/mir_window.html

Launch windows-click onLaunch Windowfor more information.

http://topex-www.jpl.nasa.gov/discover/mission/maintain.htm

Limited information on perturbationsuses TOPEX as an example.

www.allstar.fiu.edu/aero/rocket1.htm

Newtons Laws.

http://www.usafa.af.mil/

Go to: Dean, Engineering Division-Astronautics, Instructional Wares, then browse.

http://www.jpl.nasa.gov/basics/

Browse.

http://www.jpl.nasa.gov/basics/bsf-toc.htm

Select topic of interest

TOC
http://imagine.gsfc.nasa.gov/docs/features/movies/kepler.htmlhttp://www.hq.nasa.gov/office/pao/History/conghand/traject.htmhttp://astro-2.msfc.nasa.gov/academy/rocket_sci/launch/mir_window.htmlhttp://topex-www.jpl.nasa.gov/discover/mission/maintain.htmhttp://www.allstar.fiu.edu/aero/rocket1.htmhttp://www.usafa.af.mil/http://www.jpl.nasa.gov/basics/http://www.jpl.nasa.gov/basics/bsf-toc.htmhttp://www.jpl.nasa.gov/basics/bsf-toc.htmhttp://www.jpl.nasa.gov/basics/http://www.usafa.af.mil/http://www.allstar.fiu.edu/aero/rocket1.htmhttp://topex-www.jpl.nasa.gov/discover/mission/maintain.htmhttp://astro-2.msfc.nasa.gov/academy/rocket_sci/launch/mir_window.htmlhttp://www.hq.nasa.gov/office/pao/History/conghand/traject.htmhttp://imagine.gsfc.nasa.gov/docs/features/movies/kepler.html

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Orbital Mechanics - Chapter 08