Orbital Mechanics - Resumo.doc

8/10/2019 Orbital Mechanics - Resumo.doc

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ORBITAL MECHANICS

Conic Sections

Orbital Elements

Types of Orbits

Newton's Laws of Motion and Universal Gravitation

Uniform Circular Motion

Motions of lanets and Satellites

Launc! of a Space "e!icle

osition in an Elliptical Orbit

Orbit erturbations Orbit Maneuvers

Escape "elocity

Orbital mechanics# also called fli$!t mec!anics# is t!e study of t!e motions of artificial satellites and spaceve!icles movin$ under t!e influence of forces suc! as $ravity# atmosp!eric dra$# t!rust# etc% Orbital

mec!anics is a modern offs!oot of celestial mec!anics w!ic! is t!e study of t!e motions of naturalcelestial bodies suc! as t!e moon and planets% T!e root of orbital mec!anics can be traced bac& to t!e

(t! century w!en mat!ematician )saac Newton *+,-.(-(/ put forward !is laws of motion andformulated !is law of universal $ravitation% T!e en$ineerin$ applications of orbital mec!anics include

ascent tra0ectories# reentry and landin$# rende1vous computations# and lunar and interplanetary

tra0ectories%

Conic Sections

2 conic section# or 0ust conic# is a curve formed by passin$ a plane t!rou$! a ri$!t circular cone% 2s s!owin t!e fi$ure to t!e ri$!t# t!e an$ular orientation of t!e plane relative to t!e cone determines w!et!er t!e

conic section is a circle, ellipse,parabola,or hyerbola% T!e circle and

t!e ellipse arise w!en t!eintersection of cone and plane is a

bounded curve% T!e circle is aspecial case of t!e ellipse in w!ic!

t!e plane is perpendicular to t!e a3isof t!e cone% )f t!e plane is parallel to

a $enerator line of t!e cone# t!econic is called a parabola% 4inally# if

t!e intersection is an unboundedcurve and t!e plane is not parallel to

a $enerator line of t!e cone# t!e
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fi$ure is a !yperbola% )n t!e latter case t!e plane will intersect bot!

!alves of t!e cone# producin$ two separate curves%

5e can define all conic sections in terms of t!e eccentricity% T!e type ofconic section is also related to t!e semi.ma0or a3is and t!e ener$y% T!e

table below s!ows t!e relations!ips between eccentricity# semi.ma0or

a3is# and ener$y and t!e type of conic section%

Conic Section Eccentricity, e Semi-major axis Energy

Circle 0 = radius < 0

Ellipse 0 < e < 1 > 0 < 0

Parabla 1 i!"i!i#$ 0

H$perbla > 1 < 0 > 0

Satellite orbits can be any of t!e four conic sections% )n t!is section we will discuss bounded conic orbits#

i%e% circles and ellipses%

Orbital Elements

To mat!ematically describe an orbit one must define si3 6uantities# called orbital elements% T!ey are

Semi.Ma0or 23is# a

Eccentricity# e

)nclination# i

2r$ument of eriapsis#

Time of eriapsis assa$e# T

Lon$itude of 2scendin$ Node#

2n orbitin$ satellite follows an oval s!aped pat! &nown as an ellipse wit! t!e body bein$ orbited# calledt!e primary# located at one of two points called foci% 2n ellipse is defined to be a curve wit! t!e followin$

property7 for eac! point on an ellipse# t!e sum of its distances from two fi3ed points# called foci# isconstant *see fi$ure to ri$!t/% T!e lon$est and s!ortest lines t!at can be drawn t!rou$! t!e center of an

ellipse are called t!e ma0or a3is and minor a3is# respectively% T!e semi-major axisis one.!alf of t!e ma0oa3is and represents a satellite's mean distance from its primary% Eccentricityis t!e distance between t!e

foci divided by t!e len$t! of t!e ma0or a3is and is a number between 1ero and one% 2n eccentricity of 1erindicates a circle%

Inclinationis t!e an$ular distance between a satellite's orbital plane and t!e e6uator of its primary *or t!ecliptic plane in t!e case of !eliocentric# or sun centered# orbits/% 2n inclination of 1ero de$rees indicates

an orbit about t!e primary's e6uator in t!e same direction as t!e primary's rotation# a direction calledprograde*or direct/% 2n inclination of 89 de$rees indicates a polar orbit% 2n inclination of :9 de$rees

indicates a retro$rade e6uatorial orbit% 2 retrogradeorbit is one in w!ic! a satellite moves in a direction

opposite to t!e rotation of its primary%

Periapsisis t!e point in an orbit closest to t!e primary% T!e opposite of periapsis# t!e fart!est point in anorbit# is called apoapsis% eriapsis and apoapsis are usually modified to apply to t!e body bein$ orbited#

suc! as peri!elion and ap!elion for t!e Sun# peri$ee and apo$ee for Eart!# peri0ove and apo0ove for;upiter# perilune and apolune for t!e Moon# etc% T!e argument of periapsisis t!e an$ular distance betwee


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t!e ascendin$ node and t!e point of periapsis *see fi$ure below/% T!e time of periapsis passageis t!e

time in w!ic! a satellite moves t!rou$! its point of periapsis%

Nodes are t!e points w!ere an orbit crosses a plane# suc! as a satellite crossin$ t!e Eart!'s e6uatorialplane% )f t!e satellite crosses t!e plane $oin$ from sout! to nort!# t!e node is t!e ascending node< if

movin$ from nort! to sout!# it is t!e descending node% T!e longitude of the ascending nodeis t!e node's

celestial lon$itude% Celestial lon$itude is analo$ous to lon$itude on Eart! and is measured in de$reescounter.cloc&wise from 1ero wit! 1ero lon$itude bein$ in t!e direction of t!e vernal e6uino3%

)n $eneral# t!ree observations of an ob0ect in orbit are re6uired to calculate t!e si3 orbital elements% Twoot!er 6uantities often used to describe orbits are period and true anomaly% Period# P# is t!e len$t! of time

re6uired for a satellite to complete one orbit% True anomaly# v# is t!e an$ular distance of a point in anorbit past t!e point of periapsis# measured in de$rees%

Types Of Orbits

4or a spacecraft to ac!ieve Eart! orbit# it must be launc!ed to an elevation above t!e Eart!'s atmosp!ereand accelerated to orbital velocity% T!e most ener$y efficient orbit# t!at is one t!at re6uires t!e least

amount of propellant# is a direct low inclination orbit% To ac!ieve suc! an orbit# a spacecraft is launc!ed inan eastward direction from a site near t!e Eart!'s e6uator% T!e advanta$e bein$ t!at t!e rotational speed

of t!e Eart! contributes to t!e spacecraft's final orbital speed% 2t t!e United States' launc! site in Cape

Canaveral *-:%= de$rees nort! latitude/ a due east launc! results in a >free ride> of #,( &m?! *8,mp!/% Launc!in$ a spacecraft in a direction ot!er t!an east# or from a site far from t!e e6uator# results in

an orbit of !i$!er inclination% @i$! inclination orbits are less able to ta&e advanta$e of t!e initial speed

provided by t!e Eart!'s rotation# t!us t!e launc! ve!icle must provide a $reater part# or all# of t!e ener$yre6uired to attain orbital velocity% 2lt!ou$! !i$! inclination orbits are less ener$y efficient# t!ey do !ave

advanta$es over e6uatorial orbits for certain applications% Aelow we describe several types of orbits andt!e advanta$es of eac!7

Geosynchronous orbits*GEO/ are circular orbits around t!e Eart! !avin$ a period of -, !ours% 2$eosync!ronous orbit wit! an inclination of 1ero de$rees is called a geostationary orbit% 2 spacecraft in a

$eostationary orbit appears to !an$ motionless above one position on t!e Eart!'s e6uator% 4or t!is reasont!ey are ideal for some types of communication and meteorolo$ical satellites% 2 spacecraft in an inclined

$eosync!ronous orbit will appear to follow a re$ular fi$ure.: pattern in t!e s&y once every orbit% To attain$eosync!ronous orbit# a spacecraft is first launc!ed into an elliptical orbit wit! an apo$ee of B=#(:+ &m

*--#-B+ miles/ called a geosynchronous transfer orbit*GTO/% T!e orbit is t!en circulari1ed by firin$ t!espacecraft's en$ine at apo$ee%


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Polar orbits*O/ are orbits wit! an inclination of 89 de$rees% olar orbits are useful for satellites t!at

carry out mappin$ and?or surveillance operations because as t!e planet rotates t!e spacecraft !as accessto virtually every point on t!e planet's surface%

Walking orbits7 2n orbitin$ satellite is sub0ected to a $reat many $ravitational influences% 4irst# planets

are not perfectly sp!erical and t!ey !ave sli$!tly uneven mass distribution% T!ese fluctuations !ave an

effect on a spacecraft's tra0ectory% 2lso# t!e sun# moon# and planets contribute a $ravitational influence onan orbitin$ satellite% 5it! proper plannin$ it is possible to desi$n an orbit w!ic! ta&es advanta$e of t!ese

influences to induce a precession in t!e satellite's orbital plane% T!e resultin$ orbit is called a walingorbit# or precessin$ orbit%

Sun synchronous orbits*SSO/ are wal&in$ orbits w!ose orbital plane precesses wit! t!e same period at!e planet's solar orbit period% )n suc! an orbit# a satellite crosses periapsis at about t!e same local time

every orbit% T!is is useful if a satellite is carryin$ instruments w!ic! depend on a certain an$le of solarillumination on t!e planet's surface% )n order to maintain an e3act sync!ronous timin$# it may be

necessary to conduct occasional propulsive maneuvers to ad0ust t!e orbit%

Molniya orbitsare !i$!ly eccentric Eart! orbits wit! periods of appro3imately - !ours *- revolutions pe

day/% T!e orbital inclination is c!osen so t!e rate of c!an$e of peri$ee is 1ero# t!us bot! apo$ee andperi$ee can be maintained over fi3ed latitudes% T!is condition occurs at inclinations of +B%, de$rees and

+%+ de$rees% 4or t!ese orbits t!e ar$ument of peri$ee is typically placed in t!e sout!ern !emisp!ere# st!e satellite remains above t!e nort!ern !emisp!ere near apo$ee for appro3imately !ours per orbit%

T!is orientation can provide $ood $round covera$e at !i$! nort!ern latitudes%

Hohmann transfer orbitsare interplanetary tra0ectories w!ose advanta$e is t!at t!ey consume t!e

least possible amount of propellant% 2 @o!mann transfer orbit to an outer planet# suc! as Mars# isac!ieved by launc!in$ a spacecraft and acceleratin$ it in t!e direction of Eart!'s revolution around t!e su

until it brea&s free of t!e Eart!'s $ravity and reac!es a velocity w!ic! places it in a sun orbit wit! anap!elion e6ual to t!e orbit of t!e outer planet% Upon reac!in$ its destination# t!e spacecraft must

decelerate so t!at t!e planet's $ravity can capture it into a planetary orbit%

To send a spacecraft to an inner planet# suc! as "enus# t!e spacecraft is launc!ed and accelerated in t!e

direction opposite of Eart!'s revolution around t!e sun *i%e% decelerated/ until it ac!ieves a sun orbit wit!a peri!elion e6ual to t!e orbit of t!e inner planet% )t s!ould be noted t!at t!e spacecraft continues to

move in t!e same direction as Eart!# only more slowly%

To reac! a planet re6uires t!at t!e spacecraft be inserted into an interplanetary tra0ectory at t!e correct

time so t!at t!e spacecraft arrives at t!e planet's orbit w!en t!e planet will be at t!e point w!ere t!e

spacecraft will intercept it% T!is tas& is comparable to a 6uarterbac& >leadin$> !is receiver so t!at t!efootball and receiver arrive at t!e same point at t!e same time% T!e interval of time in w!ic! a spacecraft

must be launc!ed in order to complete its mission is called a launch window%

Newton's Laws of Motion and Universal Gravitation

!ewton"s laws of motiondescribe t!e relations!ip between t!e motion of a particle and t!e forces actin$on it%

T!e first law states t!at if no forces are actin$# a body at rest will remain at rest# and a body in motionwill remain in motion in a strai$!t line% T!us# if no forces are actin$# t!e velocity *bot! ma$nitude and

direction/ will remain constant%


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T!e second law tells us t!at if a force is applied t!ere will be a c!an$e in velocity# i%e% an acceleration#

proportional to t!e ma$nitude of t!e force and in t!e direction in w!ic! t!e force is applied% T!is law maybe summari1ed by t!e e6uation

w!ere #is t!e force# mis t!e mass of t!e particle# and ais t!e acceleration%

T!e t!ird law states t!at if body e3erts a force on body -# t!en body - will e3ert a force of e6ualstren$t!# but opposite in direction# on body % T!is law is commonly stated# >for every action t!ere is an

e6ual and opposite reaction>%

)n !is law of universal gravitation# Newton states t!at two particles !avin$ masses m$and m%and

separated by a distance rare attracted to eac! ot!er wit! e6ual and opposite forces directed alon$ t!eline 0oinin$ t!e particles% T!e common ma$nitude #of t!e two forces is

w!ere &is an universal constant# called t!e constant of gravitation# and !as t!e value +%+(-=839.N.

m-?&$-*B%,B:839.:lb.ft-?slu$-/%

Let's now loo& at t!e force t!at t!e Eart! e3erts on an ob0ect% )f t!e ob0ect !as a mass m# and t!e Eart!

!as mass '# and t!e ob0ect's distance from t!e center of t!e Eart! is r# t!en t!e force t!at t!e Eart!e3erts on t!e ob0ect is &m' (r%% )f we drop t!e ob0ect# t!e Eart!'s $ravity will cause it to accelerate

toward t!e center of t!e Eart!% Ay Newton's second law *4 ma/# t!is acceleration gmust e6ual)&m' (r%*(m# or

2t t!e surface of t!e Eart! t!is acceleration !as t!e valve 8%:9++= m?s-*B-%(, ft?s-/%

Many of t!e upcomin$ computations will be somew!at simplified if we e3press t!e product &'as a

constant# w!ic! for Eart! !as t!e value B%8:+99=39,mB?s-*%,9:39+ftB?s-/% T!e product &'is often

represented by t!e Gree& letter %

4or additional useful constants please see t!e appendi3 Aasic Constants%

4or a refres!er on S) versus U%S% units see t!e appendi3 5ei$!ts D Measures%

Uniform Circular Motion

)n t!e simple case of free fall# a particle accelerates toward t!e center of t!e Eart! w!ile movin$ in astrai$!t line% T!e velocity of t!e particle c!an$es in ma$nitude# but not in direction% )n t!e case of uniform

circular motion a particle moves in a circle wit! constant speed% T!e velocity of t!e particle c!an$escontinuously in direction# but not in ma$nitude% 4rom Newton's laws we see t!at since t!e direction of t!e

velocity is c!an$in$# t!ere is an acceleration% T!is acceleration# called centripetal accelerationis directedinward toward t!e center of t!e circle and is $iven by
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w!ere vis t!e speed of t!e particle and ris t!e radius of t!e circle% Every acceleratin$ particle must !avea force actin$ on it# defined by Newton's second law *4 ma/% T!us# a particle under$oin$ uniform

circular motion is under t!e influence of a force# called centripetal force# w!ose ma$nitude is $iven by

T!e direction of #at any instant must be in t!e direction of aat t!e same instant# t!at is radially inward%

2 satellite in orbit is acted on only by t!e forces of $ravity% T!e inward

acceleration w!ic! causes t!e satellite to move in a circular orbit is t!e$ravitational acceleration caused by t!e body around w!ic! t!e satellite orbits%

@ence# t!e satellite's centripetal acceleration is g# t!at is g + v%(r% 4rom Newton'slaw of universal $ravitation we &now t!at g + &' (r%% T!erefore# by settin$ t!ese

e6uations e6ual to one anot!er we find t!at# for a circular orbit#

Clic& !erefor e3ample problem B%

*use your browser's >bac&> function to return/

Motions of lanets and Satellites

T!rou$! a lifelon$ study of t!e motions of bodies in t!e solar system# ;o!annes Fepler *=(.+B9/ wasable to derive t!ree basic laws &nown as epler"s laws of planetary motion% Usin$ t!e data compiled by !

mentor Tyc!o Ara!e *=,+.+9/# Fepler found t!e followin$ re$ularities after years of laboriouscalculations7

% 2ll planets move in elliptical orbits wit! t!e sun at one focus%-% 2 line 0oinin$ any planet to t!e sun sweeps out e6ual areas in e6ual times%

B% T!e s6uare of t!e period of any planet about t!e sun is proportional to t!e cube of t!e planet's meandistance from t!e sun%

T!ese laws can be deduced from Newton's laws of motion and law of universal $ravitation% )ndeed#

Newton used Fepler's wor& as basic information in t!e formulation of !is $ravitational t!eory%

2s Fepler pointed out# all planets move in elliptical orbits# !owever# we can learn muc! about planetarymotion by considerin$ t!e special case of circular orbits% 5e s!all ne$lect t!e forces between planets#

considerin$ only a planet's interaction wit! t!e sun% T!ese considerations apply e6ually well to t!e motionof a satellite about a planet%

Let's e3amine t!e case of two bodies of masses 'and mmovin$ in circular orbits under t!e influence ofeac! ot!er's $ravitational attraction% T!e center of mass of t!is system of two bodies lies alon$ t!e line

0oinin$ t!em at a point suc! t!at mr + '.% T!e lar$e body of mass 'moves in an orbit of constant
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radius .and t!e small body of mass min an orbit of constant radius r# bot! !avin$ t!e same an$ular

velocity % 4or t!is to !appen# t!e $ravitational force actin$ on eac! body must provide t!e necessarycentripetal acceleration% Since t!ese $ravitational forces are a simple action.reaction pair# t!e centripetal

forces must be e6ual but opposite in direction% T!at is# m %rmust e6ual ' %.% T!e specific re6uirement#t!en# is t!at t!e $ravitational force actin$ on eit!er body must e6ual t!e centripetal force needed to &eep

it movin$ in its circular orbit# t!at is

)f one body !as a muc! $reater mass t!an t!e ot!er# as is t!e case of t!e sunand a planet or t!e Eart! and a satellite# its distance from t!e center of mass is

muc! smaller t!an t!at of t!e ot!er body% )f we assume t!at mis ne$li$iblecompared to '# t!en .is ne$li$ible compared to r% T!us# e6uation *B%(/ t!en

becomes

)f we e3press t!e an$ular velocity in terms of t!e period of revolution# + % (Pwe obtain

w!ere Pis t!e period of revolution% T!is is a basic e6uation of planetary and satellite motion% )t also !oldsfor elliptical orbits if we define rto be t!e semi.ma0or a3is *a/ of t!e orbit%

2 si$nificant conse6uence of t!is e6uation is t!at it predicts Fepler's t!ird law of planetary motion# t!at isP%/r0%

Clic& !erefor e3ample problem B%-Clic& !erefor e3ample problem B%B

Fepler's second law of planetary motion must# of course# !old true for circular orbits% )n suc! orbits bot!

and rare constant so t!at e6ual areas are swept out in e6ual times by t!e line 0oinin$ a planet and t!esun% 4or elliptical orbits# !owever# bot! and rwill vary wit! time% Let's now consider t!is case%

T!e fi$ure at ri$!t s!ows a particle revolvin$ around alon$ some arbitrary pat!% T!e area swept out by

t!e radius vector in a s!ort time interval tis s!own s!aded% T!is area# ne$lectin$ t!e small trian$ularre$ion at t!e end# is one.!alf t!e base times t!e !ei$!t or appro3imately r)r t*(%% T!is e3pressionbecomes more e3act as tapproac!es 1ero# i%e% t!e small trian$le $oes to 1ero more rapidly t!an t!e

lar$e one% T!e rate at w!ic! area is bein$ swept out instantaneously is t!erefore
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4or any $iven body movin$ under t!e influence of a central force# t!e value r%is

constant%

Let's now consider two points P$and P%in an orbit wit! radii r$and r%# and velocitiesv$and v%% Since t!e velocity is always tan$ent to t!e pat!# it can be seen t!at if is

t!e an$le between rand v# t!en

w!ere vsin is t!e transverse component of v% Multiplyin$ t!rou$! by r# we !ave

or# for two points P$and P%on t!e orbital pat!

Note t!at at periapsis and apoapsis# 89 de$rees% T!us# lettin$ P$and P%be t!ese two points we $et

Let's now loo& at t!e ener$y of t!e above particle at points P$and P%% onservation of energystates t!at

t!e sum of t!e &inetic ener$y and t!e potential ener$y of a particle remains constant% T!e &inetic ener$yTof a particle is $iven by mv%(%w!ile t!e potential ener$y of $ravity 1is calculated by t!e e6uation

-&'m(r% 2pplyin$ conservation of ener$y we !ave

4rom e6uations *B%,/ and *B%=/ we obtain

earran$in$ terms we $et


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Multiplyin$ t!rou$! by -.p%()r$%v$%*and rearran$in$# we $et

Note t!at t!is is a simple 6uadratic e6uation in t!e ratio ).p(r$*and t!at %&' ()r$ x v$%*is a nondimensional parameter of t!e orbit%

Solvin$ for ).p(r$*$ives

Li&e any 6uadratic# t!e above e6uation yields two answers% T!e smaller of t!e two answers corresponds t.p# t!e periapsis radius% T!e ot!er root corresponds to t!e apoapsis radius# .a%

lease note t!at in practice spacecraft launc!es are usually terminated at eit!er peri$ee or apo$ee# i%e%89% T!is condition results in t!e minimum use of propellant%

Clic& !erefor e3ample problem B%:

E6uation *B%-+/ $ives t!e values of .pand .afrom w!ic! t!e eccentricity of t!e orbit can be calculated#

!owever# it may be simpler to calculate t!e eccentricity edirectly from t!e e6uation

Clic& !erefor e3ample problem B%8

To pin down a satellite's orbit in space# we need to &now t!e an$le from t!e periapsis point to t!e launcpoint% T!is an$le is $iven by


)n most calculations# t!e complement of t!e 1enit! an$le is used# denoted by % T!is an$le is called t!e

flight-path angle# and is positive w!en t!e velocity vector is directed away from t!e primary as s!own in
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t!e fi$ure to t!e ri$!t% 5!en fli$!t.pat! an$le is used# e6uations *B%-+/ t!rou$! *B%-:/ are rewritten as

follows7

osition in an Elliptical Orbit

;o!annes Fepler was able to solve t!e problem of relatin$ position in an orbit to t!e elapsed time# t-tO# orconversely# !ow lon$ it ta&es to $o from one point in an orbit to anot!er% To solve t!is# Fepler introduced

t!e 6uantity '# called t!e mean anomaly# w!ic! is t!e fraction of an orbit period t!at !as elapsed sinceperi$ee% T!e mean anomaly e6uals t!e true anomaly for a circular orbit% Ay definition#

w!ere 'Oin t!e mean anomaly at time tOand nis t!e mean motion# or t!e avera$e an$ular velocity#

determined from t!e semi.ma0or a3is of t!e orbit as follows7

T!is solution will $ive t!e avera$e position and velocity# but satellite orbits are elliptical wit! a radius

constantly varyin$ in orbit% Aecause t!e satellite's velocity depends on t!is varyin$ radius# it c!an$es aswell% To resolve t!is problem we can define an intermediate variable E# called t!e eccentric anomaly# for

elliptical orbits# w!ic! is $iven by

w!ere vis t!e true anomaly% Mean anomaly is a function of eccentric anomaly by t!e formula

4or small eccentricities a $ood appro3imation of true anomaly can be obtained by t!e followin$ formula*t!e error is of t!e order eB/7


12/43

T!e precedin$ five e6uations can be used to */ find t!e time it ta&es to $o from one position in an orbit

to anot!er# or *-/ find t!e position in an orbit after a specific period of time% 5!en solvin$ t!ese e6uationit is important to wor& in radians rat!er t!an de$rees# w!ere - radians e6uals B+9 de$rees%

Clic& !erefor e3ample problem B%

Clic& !erefor e3ample problem B%-

2t any time in its orbit# t!e ma$nitude of a spacecraft's position vector# i%e% its distance from t!e primarybody# and its fli$!t.pat! an$le can be calculated from t!e followin$ e6uations7

2nd t!e spacecraft's velocity is $iven by#

Clic& !erefor e3ample problem B%B

Orbit erturbations

T!e orbital elements discussed at t!e be$innin$ of t!is section provide an e3cellent reference for

describin$ orbits# !owever t!ere are ot!er forces actin$ on a satellite t!at perturb it away from t!enominal orbit% T!eseperturbations# or variations in t!e orbital elements# can be classified based on !ow

t!ey affect t!e Feplerian elements% 3ecular variationsrepresent a linear variation in t!e element# short-period variationsare periodic in t!e element wit! a period less t!an t!e orbital period# and long-period

variationsare t!ose wit! a period $reater t!an t!e orbital period% Aecause secular variations !ave lon$.term effects on orbit prediction *t!e orbital elements affected continue to increase or decrease/# t!ey will

be discussed !ere for Eart!.orbitin$ satellites% recise orbit determination re6uires t!at t!e periodicvariations be included as well%

Third-Body Perturbations

T!e $ravitational forces of t!e Sun and t!e Moon cause periodic variations in all of t!e orbital elements#

but only t!e lon$itude of t!e ascendin$ node# ar$ument of peri$ee# and mean anomaly e3perience seculavariations% T!ese secular variations arise from a $yroscopic precession of t!e orbit about t!e ecliptic pole%

T!e secular variation in mean anomaly is muc! smaller t!an t!e mean motion and !as little effect on t!eorbit# !owever t!e secular variations in lon$itude of t!e ascendin$ node and ar$ument of peri$ee are

important# especially for !i$!.altitude orbits%

4or nearly circular orbits t!e e6uations for t!e secular rates of c!an$e resultin$ from t!e Sun and Moon

are

Lon$itude of t!e ascendin$ node7
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13/43

2r$ument of peri$ee7

w!ere iis t!e orbit inclination# nis t!e number of orbit revolutions per day# and and are in de$rees peday% T!ese e6uations are only appro3imate< t!ey ne$lect t!e variation caused by t!e c!an$in$ orientation

of t!e orbital plane wit! respect to bot! t!e Moon's orbital plane and t!e ecliptic plane%

Clic& !erefor e3ample problem B%,

Perturbations due to Non-spherical Earth

5!en developin$ t!e two.body e6uations of motion# we assumed t!e Eart! was a sp!erically symmetrica!omo$eneous mass% )n fact# t!e Eart! is neit!er !omo$eneous nor sp!erical% T!e most dominant featureare a bul$e at t!e e6uator# a sli$!t pear s!ape# and flattenin$ at t!e poles% 4or a potential function of t!e

Eart!# we can find a satellite's acceleration by ta&in$ t!e $radient of t!e potential function% T!e mostwidely used form of t!e $eopotential function depends on latitude and $eopotential coefficients#4n# called

t!e2onal coefficients%

T!e potential $enerated by t!e non.sp!erical Eart! causes periodic variations in all t!e orbital elements%T!e dominant effects# !owever# are secular variations in lon$itude of t!e ascendin$ node and ar$ument o

peri$ee because of t!e Eart!'s oblateness# represented by t!e ;-term in t!e $eopotential e3pansion% T!e

rates of c!an$e of and due to ;-are

w!ere nis t!e mean motion in de$rees?day#4%!as t!e value 9%999:-+B# .Eis t!e Eart!'s e6uatorial

radius# ais t!e semi.ma0or a3is in &ilometers# iis t!e inclination# eis t!e eccentricity# and and are inde$rees?day% 4or satellites in GEO and below# t!e ;-perturbations dominate< for satellites above GEO t!e

Sun and Moon perturbations dominate%

'olniyaorbits are desi$ned so t!at t!e perturbations in ar$ument of peri$ee are 1ero% T!is conditionsoccurs w!en t!e term 5-6sin%iis e6ual to 1ero or# t!at is# w!en t!e inclination is eit!er +B%, or +%+de$rees%

Clic& !erefor e3ample problem B%=

Perturbations from Atmospheric Drag


14/43

Hra$ is t!e resistance offered by a $as or li6uid to a body movin$ t!rou$! it% 2 spacecraft is sub0ected to

dra$ forces w!en movin$ t!rou$! a planet's atmosp!ere% T!is dra$ is $reatest durin$ launc! and reentry#!owever# even a space ve!icle in low Eart! orbit e3periences some dra$ as it moves t!rou$! t!e Eart!'s

t!in upper atmosp!ere% )n time# t!e action of dra$ on a space ve!icle will cause it to spiral bac& into t!eatmosp!ere# eventually to disinte$rate or burn up% )f a space ve!icle comes wit!in -9 to +9 &m of t!e

Eart!'s surface# atmosp!eric dra$ will brin$ it down in a few days# wit! final disinte$ration occurrin$ at an

altitude of about :9 &m% 2bove appro3imately +99 &m# on t!e ot!er !and# dra$ is so wea& t!at orbitsusually last more t!an 9 years . beyond a satellite's operational lifetime% T!e deterioration of a

spacecraft's orbit due to dra$ is called decay%

T!e dra$ force #7on a body acts in t!e opposite direction of t!e velocity vector and is $iven by t!ee6uation

w!ere 7is t!e dra$ coefficient# is t!e air density# vis t!e body's velocity# and8is t!e area of t!e body

normal to t!e flow% T!e dra$ coefficient is dependent on t!e $eometric form of t!e body and is $enerallydetermined by e3periment% Eart! orbitin$ satellites typically !ave very !i$! dra$ coefficients in t!e ran$e

of about - to ,% 2ir density is $iven by t!e appendi3 2tmosp!ere roperties%

T!e re$ion above 89 &m is t!e Eart!'s thermospherew!ere t!e absorption of e3treme ultraviolet radiatio

from t!e Sun results in a very rapid increase in temperature wit! altitude% 2t appro3imately -99.-=9 &mt!is temperature approac!es a limitin$ value# t!e avera$e value of w!ic! ran$es between about +99 and

#-99 F over a typical solar cycle% Solar activity also !as a si$nificant affect on atmosp!eric density# wit!!i$! solar activity resultin$ in !i$! density% Aelow about =9 &m t!e density is not stron$ly affected by

solar activity< !owever# at satellite altitudes in t!e ran$e of =99 to :99 &m# t!e density variations betwee

solar ma3imum and solar minimum are appro3imately two orders of ma$nitude% T!e lar$e variations impt!at satellites will decay more rapidly durin$ periods of solar ma3ima and muc! more slowly durin$ solar

minima%

4or circular orbits we can appro3imate t!e c!an$es in semi.ma0or a3is# period# and velocity per revolutionusin$ t!e followin$ e6uations7

w!ere ais t!e semi.ma0or a3is# Pis t!e orbit period# and 1#8and mare t!e satellite's velocity# area# andmass respectively% T!e term m()78*# called t!e ballistic coefficient# is $iven as a constant for mostsatellites% Hra$ effects are stron$est for satellites wit! low ballistic coefficients# t!is is# li$!t ve!icles wit!

lar$e frontal areas%

2 rou$! estimate of a satellite's lifetime# 9# due to dra$ can be computed from
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15/43

w!ere :is t!e atmosp!eric density scale !ei$!t% 2 substantially more

accurate estimate *alt!ou$! still very appro3imate/ can be obtained binte$ratin$ e6uation *B%,(/# ta&in$ into account t!e c!an$es in

atmosp!eric density wit! bot! altitude and solar activity%

Clic& !erefor e3ample problem B%+

Perturbations from Solar Radiation

Solar radiation pressure causes periodic variations in all of t!e orbital elements% T!e ma$nitude of t!e

acceleration in m?s-arisin$ from solar radiation pressure is

w!ere8is t!e cross.sectional area of t!e satellite e3posed to t!e Sun and mis t!e mass of t!e satellite i

&ilo$rams% 4or satellites below :99 &m altitude# acceleration from atmosp!eric dra$ is $reater t!an t!atfrom solar radiation pressure< above :99 &m# acceleration from solar radiation pressure is $reater%

Orbit Maneuvers

2t some point durin$ t!e lifetime of most space ve!icles or satellites# we must c!an$e one or more of t!eorbital elements% 4or e3ample# we may need to transfer from an initial par&in$ orbit to t!e final mission

orbit# rende1vous wit! or intercept anot!er spacecraft# or correct t!e orbital elements to ad0ust for t!eperturbations discussed in t!e previous section% Most fre6uently# we must c!an$e t!e orbit altitude# plane

or bot!% To c!an$e t!e orbit of a space ve!icle# we !ave to c!an$e its velocity vector in ma$nitude ordirection% Most propulsion systems operate for only a s!ort time compared to t!e orbital period# t!us we

can treat t!e maneuver as an impulsive c!an$e in velocity w!ile t!e position remains fi3ed% 4or t!isreason# any maneuver c!an$in$ t!e orbit of a space ve!icle must occur at a point w!ere t!e old orbit

intersects t!e new orbit% )f t!e orbits do not intersect# we must use an intermediate orbit t!at intersects

bot!% )n t!is case# t!e total maneuver will re6uire at least two propulsive burns%

Orbit Altitude hanges

T!e most common type of in.plane maneuver c!an$es t!e si1e and ener$y of an orbit# usually from a low

altitude par&in$ orbit to a !i$!er.altitude mission orbit suc! as a $eosync!ronous orbit% Aecause t!e initia

and final orbits do not intersect# t!e maneuver re6uires a transfer orbit% T!e fi$ure to t!e ri$!t representsa :ohmann transfer orbit% )n t!is case# t!e transfer orbit's ellipse is tan$ent to bot! t!e initial and final

orbits at t!e transfer orbit's peri$ee and apo$ee respectively% T!e orbits are tan$ential# so t!e velocityvectors are collinear# and t!e @o!mann transfer represents t!e most fuel.efficient transfer between two

circular# coplanar orbits% 5!en transferrin$ from a smaller orbit to a lar$er orbit# t!e c!an$e in velocity is

applied in t!e direction of motion< w!en transferrin$ from a lar$er orbit to a smaller# t!e c!an$e of

velocity is opposite to t!e direction of motion%

T!e total c!an$e in velocity re6uired for t!e orbit transfer is t!e sum of t!e velocity c!an$es at peri$eeand apo$ee of t!e transfer ellipse% Since t!e velocity vectors are collinear# t!e velocity c!an$es are 0ust

t!e differences in ma$nitudes of t!e velocities in eac! orbit% )f we &now t!e initial and final orbits# r2andrA# we can calculate t!e total velocity c!an$e usin$ t!e followin$ e6uations7


16/43

Note t!at e6uations *B%=B/ and *B%=,/ are t!e same as e6uation *B%+/# and e6uations *B%==/ and *B%=+/

are variations of e6uations *B%+/ and *B%(/ respectively%

Clic& !erefor e3ample problem B%(

Ordinarily we want to transfer a space ve!icle usin$ t!e smallest amount of ener$y# w!ic! usually leads tousin$ a @o!mann transfer orbit% @owever# sometimes we may need to transfer a satellite between orbits

in less time t!an t!at re6uired to complete t!e @o!mann transfer% T!e fi$ure to t!e ri$!t s!ows a fastertransfer called t!e One-Tangent ;urn% )n t!is instance t!e transfer orbit is tan$ential to t!e initial orbit% )t

intersects t!e final orbit at an an$le e6ual to t!e fli$!t pat! an$le of t!e transfer orbit at t!e point ofintersection% 2n infinite number of transfer orbits are tan$ential to t!e initial orbit and intersect t!e final

orbit at some an$le% T!us# we may c!oose t!e transfer orbit by specifyin$ t!e si1e of t!e transfer orbit#t!e an$ular c!an$e of t!e transfer# or t!e time re6uired to complete t!e transfer% 5e can t!en define t!e

transfer orbit and calculate t!e re6uired velocities%

4or e3ample# we may specify t!e si1e of t!e transfer orbit# c!oosin$ any semi.ma0or a3is t!at is $reater

t!an t!e semi.ma0or a3is of t!e @o!mann transfer ellipse% Once we &now t!e semi.ma0or a3is of t!e

ellipse *at3/# we can calculate t!e eccentricity# an$ular distance traveled in t!e transfer# t!e velocity c!an$re6uired for t!e transfer# and t!e time re6uired to complete t!e transfer% 5e do t!is usin$ e6uations

*B%=B/ t!rou$! *B%=(/ and *B%=8/ above# and t!e followin$ e6uations7


17/43

Clic& !erefor e3ample problem B%:

2not!er option for c!an$in$ t!e si1e of an orbit is to use electric propulsion to produce a constant low.

t!rust burn# w!ic! results in a spiral transfer% 5e can appro3imate t!e velocity c!an$e for t!is type oforbit transfer by

w!ere t!e velocities are t!e circular velocities of t!e two orbits%

Orbit Plane hanges

To c!an$e t!e orientation of a satellite's orbital plane# typically t!e inclination# we must c!an$e t!edirection of t!e velocity vector% T!is maneuver re6uires a component of 1to be perpendicular to t!e

orbital plane and# t!erefore# perpendicular to t!e initial velocity vector% )f t!e si1e of t!e orbit remainsconstant# t!e maneuver is called a simple plane change% 5e can find t!e re6uired c!an$e in velocity by

usin$ t!e law of cosines% 4or t!e case in w!ic! 1fis e6ual to 1i# t!is e3pression reduces to

w!ere 1iis t!e velocity before and after t!e burn# and is t!e an$le c!an$e re6uired%


4rom e6uation *B%+(/ we see t!at if t!e an$ular c!an$e is e6ual to +9 de$rees# t!e re6uired c!an$e invelocity is e6ual to t!e current velocity% lane c!an$es are very e3pensive in terms of t!e re6uired c!an$e

in velocity and resultin$ propellant consumption% To minimi1e t!is# we s!ould c!an$e t!e plane at a point


18/43

w!ere t!e velocity of t!e satellite is a minimum7 at apo$ee for an elliptical orbit% )n some cases# it may

even be c!eaper to boost t!e satellite into a !i$!er orbit# c!an$e t!e orbit plane at apo$ee# and return t!satellite to its ori$inal orbit%

Typically# orbital transfers re6uire c!an$es in bot! t!e si1e and t!e plane of t!e orbit# suc! as transferrin$

from an inclined par&in$ orbit at low altitude to a 1ero.inclination orbit at $eosync!ronous altitude% 5e ca

do t!is transfer in two steps7 a @o!mann transfer to c!an$e t!e si1e of t!e orbit and a simple planec!an$e to ma&e t!e orbit e6uatorial% 2 more efficient met!od *less total c!an$e in velocity/ would be to

combine t!e plane c!an$e wit! t!e tan$ential burn at apo$ee of t!e transfer orbit% 2s we must c!an$ebot! t!e ma$nitude and direction of t!e velocity vector# we can find t!e re6uired c!an$e in velocity usin$

t!e law of cosines#

w!ere 1iis t!e initial velocity# 1fis t!e final velocity# and is t!e an$le c!an$e re6uired% Note t!at

e6uation *B%+:/ is in t!e same form as e6uation *B%+B/%

Clic& !erefor e3ample problem B%-9

2s can be seen from e6uation *B%+:/# a small plane c!an$e can be combined wit! an altitude c!an$e foralmost no cost in 1or propellant% Conse6uently# in practice# $eosync!ronous transfer is done wit! a

small plane c!an$e at peri$ee and most of t!e plane c!an$e at apo$ee%

2not!er option is to complete t!e maneuver usin$ t!ree burns% T!e first burn is a coplanar maneuverplacin$ t!e satellite into a transfer orbit wit! an apo$ee muc! !i$!er t!an t!e final orbit% 5!en t!e

satellite reac!es apo$ee of t!e transfer orbit# a combined plane c!an$e maneuver is done% T!is places t!e

satellite in a second transfer orbit t!at is coplanar wit! t!e final orbit and !as a peri$ee altitude e6ual tot!e altitude of t!e final orbit% 4inally# w!en t!e satellite reac!es peri$ee of t!e second transfer orbit#

anot!er coplanar maneuver places t!e satellite into t!e final orbit% T!is t!ree.burn maneuver may savepropellant# but t!e propellant savin$s comes at t!e e3pense of t!e total time re6uired to complete t!e

maneuver%

Orbit Rende!"ous

Orbital transfer becomes more complicated w!en t!e ob0ect is to rende1vous wit! or intercept anot!erob0ect in space7 bot! t!e interceptor and t!e tar$et must arrive at t!e rende1vous point at t!e same time

T!is precision demands a p!asin$ orbit to accomplis! t!e maneuver% 2phasing orbitis any orbit t!atresults in t!e interceptor ac!ievin$ t!e desired $eometry relative to t!e tar$et to initiate a @o!mann

transfer% )f t!e initial and final orbits are circular# coplanar# and of different si1es# t!en t!e p!asin$ orbit issimply t!e initial interceptor orbit% T!e interceptor remains in t!e initial orbit until t!e relative motion

between t!e interceptor and tar$et results in t!e desired $eometry% 2t t!at point# we would in0ect t!einterceptor into a @o!mann transfer orbit%

#aunch $indo%s

Similar to t!e rende1vous problem is t!e launc!.window problem# or determinin$ t!e appropriate time to

launc! from t!e surface of t!e Eart! into t!e desired orbital plane% Aecause t!e orbital plane is fi3ed ininertial space# t!e launc! window is t!e time w!en t!e launc! site on t!e surface of t!e Eart! rotates

t!rou$! t!e orbital plane% T!e time of t!e launc! depends on t!e launc! site's latitude and lon$itude andt!e satellite orbit's inclination and lon$itude of ascendin$ node%


19/43

Orbit &aintenance

Once in t!eir mission orbits# many satellites need no additional orbit ad0ustment% On t!e ot!er !and#

mission re6uirements may demand t!at we maneuver t!e satellite to correct t!e orbital elements w!enperturbin$ forces !ave c!an$ed t!em% Two particular cases of note are satellites wit! repeatin$ $round

trac&s and $eostationary satellites%

2fter t!e mission of a satellite is complete# several options e3ist# dependin$ on t!e orbit% 5e may allowlow.altitude orbits to decay and reenter t!e atmosp!ere or use a velocity c!an$e to speed up t!e process

5e may also boost satellites at all altitudes into beni$n orbits to reduce t!e probability of collision wit!active payloads# especially at sync!ronous altitudes%

' Budget

To an orbit desi$ner# a space mission is a series of different orbits% 4or e3ample# a satellite mi$!t be

released in a low.Eart! par&in$ orbit# transferred to some mission orbit# $o t!rou$! a series ofresp!asin$s or alternate mission orbits# and t!en move to some final orbit at t!e end of its useful life%

Eac! of t!ese orbit c!an$es re6uires ener$y% T!e 1 budgetis traditionally used to account for t!is

ener$y% )t sums all t!e velocity c!an$es re6uired t!rou$!out t!e space mission life% )n a broad sense t!e" bud$et represents t!e cost for eac! mission orbit scenario%

Escape "elocity

5e &now t!at if we t!row a ball up from t!e surface of t!e Eart!# it will rise for a w!ile and t!en return% )

we $ive it a lar$er initial velocity# it will rise !i$!er and t!en return% T!ere is a velocity# called t!e escapevelocity, 1esc# suc! t!at if t!e ball is launc!ed wit! an initial velocity $reater t!an 1esc# it will rise and

never return% 5e must $ive t!e particle enou$! &inetic ener$y to overcome all of t!e ne$ative$ravitational potential ener$y% T!us# if mis t!e mass of t!e ball# 'is t!e mass of t!e Eart!# and .is t!e

radius of t!e Eart!# t!e potential ener$y is -&m' (.% T!e &inetic ener$y of t!e ball# w!en it is launc!ed# imv%(%% 5e t!us !ave

w!ic! is independent of t!e mass of t!e ball%

4or a spacecraft launc!ed to escape velocity from a par&in$ orbit# .is t!e radius of t!e orbit%

Clic& !erefor e3ample problem B%--

%##p&''((()braeu!i*)us'space'

BASIC CONSTANTS

&athematical onstants
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20/43

+)1,1-./-+-.2.+

e /)21/1/,-.0,-

Physical onstants

Speed " li*%# 3c4 /..52./5,- 6's

C!s#a!# " *ra7i#a#i! 384 )2/-.910:11 N:6/';*/

Accelera#i! " *ra7i#$ 3*4 .)0- 6's/

!i7ersal *as c!s#a!# 3R4 5+1,)-10 N:6';*:

S#a!dard a#6sp%ere5 sea le7el 1015+/- Pa

Astronomical onstants

As#r!6ical u!i# 3A4 1,.5-.2520 ;6

Li*%# $ear 3l$4 .),0-+09101/ ;6

Parsec 3pc4 +)/1++ l$

Sidereal $ear +-)/-+ da$s

Mass " Su! 1)..1910+0 ;*

Radius " Su! .5000 ;6

Mass " Ear#% -).2+2910/, ;*

Eua#rial radius " Ear#% 5+2)1,0 ;6

Ear#% bla#e!ess 1'/.)/-2

Obliui#$ " #%e eclip#ic5 epc% /000 /+),+./.11 de*rees

Mea! lu!ar dis#a!ce +,5,0+ ;6

Radius " M! 152+ ;6

Mass " M! 2)+,910// ;*

Lu6i!si#$ " Su! +)/2910/ ?

Slar c!s#a!#5 a# 1 A 15+- ?'6/

Slar 6a9i6a 1..0 @ 11! 3da#e4

Spaceflight onstants

8M 3Su!4 1)+/21/,+910/0 6+'s/

8M 3Ear#%4 +).00-9101, 6+'s/

8M 3M!4 ,).0/2.,9101/ 6+'s/

/3Ear#%4 0)0010/+

/3M!4 0)000/0/2


21/43

Planet Data

Na6e

Eua#rial

Radius3;64

Mass

310/,;*4

Sur"ace

8ra7i#$36's/4

8M

3101-6+'s+4

Se6i6ar

A9is310;64

Ecce!#rici#$

Orbi#

I!cli!a#i!3de*rees4

Sidereal

Perid3da$s4

Sidereal

R#a#i!

Perid

3%urs4

Mercur$ /5,+.)2 0)++0/ +)20 0)0//0+ -2).1 0)/0- 2)00 2).0 15,02)

e!us 50-1) ,)- )2 0)+/,. 10)/1 0)002 +)+. //,)201 :-5+/)-

Ear#% 5+2)1 -).2,/ .)0 0)+. 1,.)0 0)012 0)000 +-)/- /+).+,-

Mars +5+.2)0 0),1- +)21 0)0,/+ //2)./ 0)0.+- 1)-0 ).0 /,)//.

upi#er 215,./ 15.) /,)2. 1/) 22)-2 0)0,. 1)+0, ,5++/)-. .)./-0

Sa#ur! 05/ -), 10),, +2).+1 15,++)-+ 0)0-- /),- 1052-.)/ 10)-

ra!us /-5--. )+/ )2 -)2., /52/), 0)0,-2 0)22/ +05-), :12)/,

Nep#u!e /,52, 10/),+ 11)1- )+- ,5,.-)0 0)011+ 1)2. 051. 1)11

Plu# 151.- 0)01/- 0)- 0)000+ -5.) 0)/,,, 12)1 .05,- :1-+)/.

?EI8HTS D MEASRES

Two systems of wei$!ts and measures coe3ist in t!e United States today7 t!e U%S% Customary System an

t!e )nternational System of Units *S)# after t!e initials Systeme )nternational/% T!e S) System# commonlyidentified wit! t!e metric system# is actually a more complete and co!erent version of it% T!rou$!out U%S

!istory# t!e Customary System *in!erited from# but now different from# t!e Aritis! )mperial System/ !asbeen $enerally used%

T!e use of t!e S) System !as slowly and steadily increased in t!e United States# particularly in t!e

scientific community< !owever# t!e $eneral public still uses# almost e3clusively# t!e U%S% Customary

System% Aecause t!is 5eb site !as a lar$e international audience# all calculations and e3ample problemsma&e e3clusive use of S) units# !owever t!e formulae provided will wor& in eit!er system of units%

5!enever a basic constant is $iven# its U%S% Customary e6uivalent is also $iven in parent!eses%

)n t!e S) System# t!e basic units are t!e units of len$t!# mass# and time# and are called respectively# t!e

meter *m/# t!e &ilo$ram *&$/# and t!e second *s/% T!e unit of force# called t!e Newton *N/# is a derived

unit and is defined as t!e force t!at $ives an acceleration of m?s-to a mass of &$%

)n t!e U%S% Customary System t!e base units are t!e units of len$t!# force# and time# and are calledrespectively# t!e foot *ft/# t!e pound *lb/# and t!e second *s/% T!e unit of mass# called t!e slu$# is a

derived unit and is defined as t!e mass t!at receives an acceleration of ft?s-w!en a force of lb isapplied to it%

5!en wor&in$ in t!e U%S% Customary System# it is commonplace to e3press >mass> in pounds< !owever#

w!en doin$ so# it is necessary to reco$ni1e t!at we are actually e3pressin$ >wei$!t># w!ic! is t!e measur


22/43

of t!e $ravitational force on a body% 5!en used in t!is way# t!e wei$!t is t!at of a mass w!en it is

sub0ected to an acceleration of one g% )n t!e study of dynamics# w!ere forces# masses# and accelerationsare involved# it is important t!at we e3press t!e mass min slu$s of a body of w!ic! t!e wei$!t


23/43

t 6 90 s

E5"ation 81.19:&

7 6 7e ! L2@ M / 8M = 5t: 7 6 3&100 ! L2@ 30&000 / 830&000 = 830 ! 90:: 7 6 1>, m/s

PROBLEM 1.3

A spacecraft's *r% mass is +)&000 kg an* te effecti#e e!a"st gas #e$ocit% of itsmain engine is 3&100 m/s. ow m"c prope$$ant m"st ?e carrie* if te prop"$sion s%stemis to pro*"ce a tota$ # of +00 m/s

OLO2& i#en4 Mf 6 +)&000 kg C 6 3&100 m/s 7 6 +00 m/s

E5"ation 81.,0:&

Mo 6 Mf ! eD8 7 / C: Mo 6 +)&000 ! eD8+00 / 3&100: Mo 6 >&000 kg

Prope$$ant mass&

Mp 6 Mo = Mf Mp 6 >&000 = +)&000 Mp 6 1>&000 kg

PROBLEM 1.

A )&000 kg spacecraft is in Eart or?it tra#e$ing at a #e$ocit% of +&+>0 m/s. tsengine is ?"rne* to acce$erate it to a #e$ocit% of 1,&000 m/s p$acing it on an escape

trajector%. e engine e!pe$s mass at a rate of 10 kg/s an* an effecti#e #e$ocit% of3&000 m/s. Ca$c"$ate te *"ration of te ?"rn.

OLO2&

i#en4 M 6 )&000 kg 5 6 10 kg/s C 6 3&000 m/s


24/43

7 6 1,&000 = +&+>0 6 &,10 m/s

E5"ation 81.,1:&

t 6 M / 5 ! @ 1 = 1 / eD8 7 / C: t 6 )&000 / 10 ! @ 1 = 1 / eD8&,10 / 3&000: t 6 3++ s

PROBLEM 1.)

A rocket engine ?"rning $i5"i* o!%gen an* kerosene operates at a mi!t"re ratio of,.,9 an* a com?"stion cam?er press"re of )0 atmosperes. f te no(($e is e!pan*e*to operate at sea $e#e$& ca$c"$ate te e!a"st gas #e$ocit% re$ati#e to te rocket.

OLO2& i#en4 O/; 6 ,.,9 Pc 6 )0 atm Pe 6 Pa 6 1 atm

;rom LOF/Gerosene Cartswe estimate&

c 6 3&+0 G M 6 ,1.0

k 6 1.,,1

E5"ation 81.,,:&

7e 6 HR@ 8, ! k / 8k = 1:: ! 8R' ! c / M: ! 81 = 8Pe / Pc:8k=1:/k: 7e 6 HR@ 8, ! 1.,,1 / 81.,,1 = 1:: ! 8I&31.)1 ! 3&+0 / ,1.0: ! 81 = 81 / )0:81.,,1=1:/1.,,1: 7e 6 ,&+> m/s

PROBLEM 1.9

A rocket engine pro*"ces a tr"st of 1&000 k2 at sea $e#e$ wit a prope$$ant f$ow

rate of 00 kg/s. Ca$c"$ate te specific imp"$se.

OLO2&

i#en4 ; 6 1&000&000 2 5 6 00 kg/s

E5"ation 81.,3:&
http://www.braeunig.us/space/comb-OK.htmhttp://www.braeunig.us/space/comb-OK.htm


25/43

sp 6 ; / 85 ! g: sp 6 1&000&000 / 800 ! >.I099): sp 6 ,)) s 8sea $e#e$:

PROBLEM 1.+

A rocket engine "ses te same prope$$ant& mi!t"re ratio& an* com?"stion cam?erpress"re as tat in pro?$em 1.). f te prope$$ant f$ow rate is )00 kg/s& ca$c"$atete area of te e!a"st no(($e troat.

OLO2&

i#en4 Pc 6 )0 ! 0.1013,) 6 ).099 MPa c 6 3&+0 G M 6 ,1.0 k 6 1.,,1 5 6 )00 kg/s

E5"ation 81.,9:&

Pt 6 Pc ! @1 < 8k = 1: / ,=k/8k=1:

Pt 6 ).099 ! @1 < 81.,,1 = 1: / ,=1.,,1/81.,,1=1:

Pt 6 ,.I3> MPa 6 ,.I3>!1092/m,

E5"ation 81.,+:&

t 6 c / 81 < 8k = 1: / ,: t 6 3&+0 / 81 < 81.,,1 = 1: / ,: t 6 3&1,) G

E5"ation 81.,):&

At 6 85 / Pt: ! HR@ 8R' ! t: / 8M ! k: At 6 8)00 / ,.I3>!109: ! HR@ 8I&31.)1 ! 3&1,): / 8,1.0 ! 1.,,1: At 6 0.1+)9 m,

PROBLEM 1.I

e rocket engine in pro?$em 1.+ is optimi(e* to operate at an e$e#ation of ,000 meters.Ca$c"$ate te area of te no(($e e!it an* te section ratio.

OLO2&

i#en4 Pc 6 ).099 MPa At 6 0.1+)9 m,

k 6 1.,,1


26/43

;rom Atmospere Properties&

Pa 6 0.0+>) MPa

E5"ation 81.,I:&

2m,6 8, / 8k = 1:: ! @8Pc / Pa:8k=1:/k= 1 2m,6 8, / 81.,,1 = 1:: ! @8).099 / 0.0+>):81.,,1=1:/1.,,1= 1 2m,6 10.1) 2m 6 810.1):1/,6 3.1I)

E5"ation 81.,>:&

Ae 6 8At / 2m: ! @81 < 8k = 1: / , ! 2m,:/88k < 1: / ,:8k


27/43

PROBLEM 1.10

A two=stage rocket as te fo$$owing masses4 1st=stage prope$$ant mass 1,0&000 kg&1st=stage *r% mass >&000 kg& ,n*=stage prope$$ant mass 30&000 kg& ,n*=stage *r% mass

3&000 kg& an* pa%$oa* mass 3&000 kg. e specific imp"$ses of te 1st an* ,n* stagesare ,90 s an* 3,0 s respecti#e$%. Ca$c"$ate te rocket's tota$ 7.

OLO2&

i#en4 Mo16 1,0&000 < >&000 < 30&000 < 3&000 < 3&000 6 19)&000 kg Mf16 >&000 < 30&000 < 3&000 < 3&000 6 )&000 kg sp16 ,90 s Mo,6 30&000 < 3&000 < 3&000 6 39&000 kg Mf,6 3&000 < 3&000 6 9&000 kg sp,6 3,0 s

E5"ation 81.,:&

C16 sp1g C16 ,90 ! >.I099) 6 ,&))0 m/s

C,6 sp,gC,6 3,0 ! >.I099) 6 3&13I m/s

E5"ation 81.33:&

716 C1! L2@ Mo1/ Mf1 716 ,&))0 ! L2@ 19)&000 / )&000 716 3&313 m/s

7,6 C,! L2@ Mo,/ Mf, 7,6 3&13I ! L2@ 39&000 / 9&000 7,6 )&9,3 m/s

E5"ation 81.3:&

7ota$6 71< 7,7ota$6 3&313 < )&9,3

7ota$6 I&>39 m/s

PROBLEM 3.1

Ca$c"$ate te #e$ocit% of an artificia$ sate$$ite or?iting te Eart in a circ"$ar or?itat an a$tit"*e of ,00 km a?o#e te Eart's s"rface.

OLO2&


28/43

;rom Basics Constants&

Ra*i"s of Eart 6 9&3+I.10 kmM of Eart 6 3.>I900)!101m3/s,

i#en4 r 6 89&3+I.1 < ,00: ! 1&000 6 9&)+I&10 m

E5"ation 83.9:&

# 6 HR@ M / r # 6 HR@ 3.>I900)!101/ 9&)+I&10 # 6 +&+I m/s

PROBLEM 3.,

Ca$c"$ate te perio* of re#o$"tion for te sate$$ite in pro?$em 3.1.

OLO2&

i#en4 r 6 9&)+I&10 m

E5"ation 83.>:&

P,6 ! ,! r3/ M

P 6 HR@ ! ,! r3/ M

P 6 HR@ ! ,! 9&)+I&103/ 3.>I900)!101 P 6 )&310 s

PROBLEM 3.3

Ca$c"$ate te ra*i"s of or?it for a Eart sate$$ite in a geos%ncrono"s or?it& were teEart's rotationa$ perio* is I9&19.1 secon*s.

OLO2&

i#en4 P 6 I9&19.1 s

E5"ation 83.>:&

P,6 ! ,! r3/ M

r 6 @ P,! M / 8 ! ,: 1/3
http://www.braeunig.us/space/constant.htmhttp://www.braeunig.us/space/constant.htm


29/43

r 6 @ I9&19.1,! 3.>I900)!101/ 8 ! ,: 1/3

r 6 ,&19&1+0 m

PROBLEM 3.

An artificia$ Eart sate$$ite is in an e$$iptica$ or?it wic ?rings it to an a$tit"*e of,)0 km at perigee an* o"t to an a$tit"*e of )00 km at apogee. Ca$c"$ate te #e$ocit% ofte sate$$ite at ?ot perigee an* apogee.

OLO2&

i#en4 Rp 6 89&3+I.1 < ,)0: ! 1&000 6 9&9,I&10 m Ra 6 89&3+I.1 < )00: ! 1&000 6 9&I+I&10 m

E5"ations 83.19: an* 83.1+:&

7p 6 HR@ , ! M ! Ra / 8Rp ! 8Ra < Rp:: 7p 6 HR@ , ! 3.>I900)!101! 9&I+I&10 / 89&9,I&10 ! 89&I+I&10 < 9&9,I&10:: 7p 6 +&I,9 m/s

7a 6 HR@ , ! M ! Rp / 8Ra ! 8Ra < Rp:: 7a 6 HR@ , ! 3.>I900)!101! 9&9,I&10 / 89&I+I&10 ! 89&I+I&10 < 9&9,I&10:: 7a 6 +&), m/s

PROBLEM 3.)

A sate$$ite in Eart or?it passes tro"g its perigee point at an a$tit"*e of ,00 kma?o#e te Eart's s"rface an* at a #e$ocit% of +&I)0 m/s. Ca$c"$ate te apogee a$tit"*eof te sate$$ite.

OLO2&

i#en4 Rp 6 89&3+I.1 < ,00: ! 1&000 6 9&)+I&10 m 7p 6 +&I)0 m/s

E5"ation 83.1I:&

Ra 6 Rp / @, ! M / 8Rp ! 7p,: = 1 Ra 6 9&)+I&10 / @, ! 3.>I900)!101/ 89&)+I&10 ! +&I)0,: = 1 Ra 6 9&I0)&10 m

A$tit"*e J apogee 6 9&I0)&10 / 1&000 = 9&3+I.1 6 ,+.0 km


30/43

PROBLEM 3.9

Ca$c"$ate te eccentricit% of te or?it for te sate$$ite in pro?$em 3.).

OLO2&

i#en4 Rp 6 9&)+I&10 m 7p 6 +&I)0 m/s

E5"ation 83.,0:&

e 6 Rp ! 7p,/ M = 1 e 6 9&)+I&10 ! +&I)0,/ 3.>I900)!101= 1 e 6 0.019>9

PROBLEM 3.+

A sate$$ite in Eart or?it as a semi=major a!is of 9&+00 km an* an eccentricit% of 0.01.Ca$c"$ate te sate$$ite's a$tit"*e at ?ot perigee an* apogee.

OLO2&

i#en4 a 6 9&+00 km e 6 0.01

E5"ation 83.,1: an* 83.,,:&

Rp 6 a ! 81 = e: Rp 6 9&+00 ! 81 = .01: Rp 6 9&933 km

A$tit"*e J perigee 6 9&933 = 9&3+I.1 6 ,).> km

Ra 6 a ! 81 < e: Ra 6 9&+00 ! 81 < .01:

Ra 6 9&+9+ km

A$tit"*e J apogee 6 9&+9+ = 9&3+I.1 6 3II.> km

PROBLEM 3.I


31/43

A sate$$ite is $a"nce* into Eart or?it were its $a"nc #eic$e ?"rns o"t at ana$tit"*e of ,)0 km. At ?"rno"t te sate$$ite's #e$ocit% is +&>00 m/s wit te (enitang$e e5"a$ to I> *egrees. Ca$c"$ate te sate$$ite's a$tit"*e at perigee an* apogee.

OLO2& i#en4 r1 6 89&3+I.1 < ,)0: ! 1&000 6 9&9,I&10 m #1 6 +&>00 m/s 6 I>o

E5"ation 83.,9:&

8Rp / r1:1&,6 8 =C I900)!101/ 89&9,I&10 ! +&>00,: C 6 1.>,+1I

8Rp / r1:1&,6 8 =1.>,+1I ,+1I,= ! =0.>,+1I ! =sin,8I>: : / 8, !

=0.>,+1I: 8Rp / r1:1&,6 0.>>901> an* 1.0I,),

Perigee Ra*i"s& Rp 6 Rp16 r1 ! 8Rp / r1:1

Rp 6 9&9,I&10 ! 0.>>901> Rp 6 9&901&+)0 m

A$tit"*e J perigee 6 9&901&+)0 / 1&000 = 9&3+I.1 6 ,,3.9 km

Apogee Ra*i"s& Ra 6 Rp,6 r1 ! 8Rp / r1:,

Ra 6 9&9,I&10 ! 1.0I,), Ra 6 +&1+)&0>0 m

A$tit"*e J agogee 6 +&1+)&0>0 / 1&000 = 9&3+I.1 6 +>+.0 km

PROBLEM 3.>

Ca$c"$ate te eccentricit% of te or?it for te sate$$ite in pro?$em 3.I.

OLO2&

i#en4 r1 6 9&9,I&10 m #1 6 +&>00 m/s 6 I>o

E5"ation 83.,+:&


32/43

e 6 HR@ 8r1 ! #1,/ M = 1:,! sin, < cos, e 6 HR@ 89&9,I&10 ! +&>00,/ 3.>I900)!101= 1:,! sin,8I>: < cos,8I>: e 6 0.019,

PROBLEM 3.10

Ca$c"$ate te ang$e from perigee point to $a"nc point for te sate$$itein pro?$em 3.I.

OLO2&

i#en4 r1 6 9&9,I&10 m #1 6 +&>00 m/s 6 I>o

E5"ation 83.,I:&

tan 6 8r1 ! #1,/ M: ! sin ! cos / @8r1 ! #1,/ M: ! sin, = 1 tan 6 89&9,I&10 ! +&>00,/ 3.>I900)!101: ! sin8I>: ! cos8I>:

/ @89&9,I&10 ! +&>00,/ 3.>I900)!101: ! sin,8I>: = 1 tan 6 0.I3,>

6 arctan80.I3,>:6 ,).+>o

PROBLEM 3.11

A sate$$ite is in an or?it wit a semi=major a!is of +&)00 km an* an eccentricit%of 0.1. Ca$c"$ate te time it takes to mo#e from a position 30 *egrees past perigeeto >0 *egrees past perigee.

OLO2&

i#en4 a 6 +&)00 ! 1&000 6 +&)00&000 m

e 6 0.1 tO6 0 #O6 30 *eg ! /1I0 6 0.),390 ra*ians

# 6 >0 *eg ! /1I0 6 1.)+0I0 ra*ians

E5"ation 83.3:&

cos E 6 8e < cos #: / 81 < e cos #:

Eo 6 [email protected] < cos80.),390:: / 81 < 0.1 ! cos80.),390::


33/43

Eo 6 0.+))+ ra*ians

E 6 [email protected] < cos81.)+0I0:: / 81 < 0.1 ! cos81.)+0I0:: E 6 1.+093 ra*ians

E5"ation 83.3):&

M 6 E = e ! sin E

Mo 6 0.+))+ = 0.1 ! sin80.+))+: Mo 6 0.,>+I ra*ians

M 6 1.+093 = 0.1 ! sin81.+093: M 6 1.3+113 ra*ians

E5"ation 83.33:&

n 6 HR@ M / a3 n 6 HR@ 3.>I900)!101/ +&)00&0003 n 6 0.000>+,0, ra*/s

E5"ation 83.3,:&

M = Mo 6 n ! 8t = tO:

t 6 tO< 8M = Mo: / n t 6 0 < 81.3+113 = 0.,>+I: / 0.000>+,0, t 6 >9I. s

PROBLEM 3.1,e sate$$ite in pro?$em 3.11 as a tr"e anoma$% of >0 *egrees. -at wi$$ ?e tesate$$ite's position& i.e. it's tr"e anoma$%& ,0 min"tes $ater

OLO2&

i#en4 a 6 +&)00&000 m e 6 0.1 tO6 0 t 6 ,0 ! 90 6 1&,00 s #O6 >0 ! /1I0 6 1.)+0I0 ra*

;rom pro?$em 3.11&

Mo 6 1.3+113 ra* n 6 0.000>+,0, ra*/s

E5"ation 83.3,:&

M = Mo 6 n ! 8t = tO:


34/43

M 6 Mo < n ! 8t = tO: M 6 1.3+113 < 0.000>+,0, ! 81&,00 = 0: M 6 ,.)3+))

METHOD #1, Low Accuracy:

E5"ation 83.39:&

# K M < , ! e ! sin M < 1.,) ! e,! sin ,M # K ,.)3+)) < , ! 0.1 ! sin8,.)3+)): < 1.,) ! 0.1,! sin8, ! ,.)3+)):

# K ,.93>9 6 1)1., *egrees

METHOD #2, High Accuracy:

E5"ation 83.3):&

M 6 E = e ! sin E ,.)3+)) 6 E = 0.1 ! sin E

B% iteration& E 6 ,.)I>>9 ra*ians

E5"ation 83.3:&

cos E 6 8e < cos #: / 81 < e cos #:

Rearranging #aria?$es gi#es&

cos # 6 8cos E = e: / 81 = e cos E:

# 6 arccos@8cos8,.)I>>9: = 0.1: / 81 = 0.1 ! cos8,.)I>>9: # 6 ,.903 6 1)1.3 *egrees

PROBLEM 3.13

;or te sate$$ite in pro?$ems 3.11 an* 3.1,& ca$c"$ate te $engt of its position#ector& its f$igt=pat ang$e& an* its #e$ocit% wen te sate$$ite's tr"e anoma$%is ,,) *egrees.

OLO2&

i#en4 a 6 +&)00&000 m e 6 0.1 # 6 ,,) *egrees

E5"ations 83.3+: an* 83.3I:&

r 6 a ! 81 = e,: / 81 < e ! cos #:r 6 +&)00&000 ! 81 = 0.1,: / 81 < 0.1 ! cos8,,)::

r 6 +&>I>&>++ m


35/43

6 arctan@ e ! sin # / 81 < e ! cos #:

6 arctan@ 0.1 ! sin8,,): / 81 < 0.1 ! cos8,,)::

6 =.3)1 *egrees

E5"ation 83.3>:&

# 6 HR@ M ! a ! 81 = e,: / 8r ! cos : # 6 HR@ 3.>I900)!101! +&)00&000 ! 81 = 0.1,: / 8+&>I>&>++ ! cos8=.3)1:: # 6 9&I,I m/s

PROBLEM 3.1

Ca$c"$ate te pert"r?ations in $ongit"*e of te ascen*ing no*e an* arg"ment ofperigee ca"se* ?% te Moon an* "n for te nternationa$ pace tation or?iting

at an a$tit"*e of 00 km& an inc$ination of )1.9 *egrees& an* wit an or?ita$perio* of >,.9 min"tes.

OLO2&

i#en4 i 6 )1.9 *egrees n 6 139 / >,.9 6 1).) re#o$"tions/*a%

E5"ations 83.0: tro"g 83.3:&

moon 6 =0.0033I ! cos8i: / nmoon 6 =0.0033I ! cos8)1.9: / 1).)moon 6 =0.00013) *eg/*a%

s"n 6 =0.001) ! cos8i: / n s"n 6 =0.001) ! cos8)1.9: / 1).)

s"n 6 =0.000091+ *eg/*a%

moon 6 0.0019> ! 8 = ) ! sin,i: / n moon 6 0.0019> ! 8 = ) ! sin,)1.9: / 1).)

moon 6 0.000101 *eg/*a%

s"n 6 0.000++ ! 8 = ) ! sin,i: / n s"n 6 0.000++ ! 8 = ) ! sin,)1.9: / 1).)

s"n 6 0.00009 *eg/*a%

PROBLEM 3.1)

A sate$$ite is in an or?it wit a semi=major a!is of +&)00 km& an inc$inationof ,I.) *egrees& an* an eccentricit% of 0.1. Ca$c"$ate te ,pert"r?ations in$ongit"*e of te ascen*ing no*e an* arg"ment of perigee.


36/43

OLO2&

i#en4 a 6 +&)00 km i 6 ,I.) *egrees e 6 0.1

E5"ations 83.: an* 83.):&

, 6 =,.09+!101! a=+/,! 8cos i: ! 81 = e,:=,

, 6 =,.09+!101! 8+&)00:=+/,! 8cos ,I.): ! 81 = 80.1:,:=,

, 6 =).09+ *eg/*a%

, 6 1.03,3+!101! a=+/,! 8 = ) ! sin,i: ! 81 = e,:=,

, 6 1.03,3+!101! 8+&)00:=+/,! 8 = ) ! sin,,I.): ! 81 = 80.1:,:=,

, 6 I.,)0 *eg/*a%

PROBLEM 3.19

A sate$$ite is in a circ"$ar Eart or?it at an a$tit"*e of 00 km. e sate$$iteas a c%$in*rica$ sape , m in *iameter ?% m $ong an* as a mass of 1&000 kg. esate$$ite is tra#e$ing wit its $ong a!is perpen*ic"$ar to te #e$ocit% #ector an*it's *rag coefficient is ,.9+. Ca$c"$ate te pert"r?ations *"e to atmosperic *ragan* estimate te sate$$ite's $ifetime.

OLO2&

i#en4 a 6 89&3+I.1 < 00: ! 1&000 6 9&++I&10 m A 6 , ! 6 I m,

m 6 1&000 kg CN6 ,.9+

;rom Atmospere Properties&

6 ,.9,!10=1,kg/m3

6 )I., km

E5"ation 83.9:&

7 6 HR@ M / a 7 6 HR@ 3.>I900)!101/ 9&++I&10

7 6 +&99> m/s

E5"ations 83.+: tro"g 83.>:&

are#6 8=, ! ! CN! A ! ! a,: / m are#6 8=, ! ! ,.9+ ! I ! ,.9,!10=1,! 9&++I&10,: / 1&000 are#6 =19., m

Pre#6 8=9 !,! CN! A ! ! a

,: / 8m ! 7:
http://www.braeunig.us/space/atmos.htm#scalehthttp://www.braeunig.us/space/atmos.htm#scaleht


37/43

Pre#6 8=9 !,! ,.9+ ! I ! ,.9,!10=1,! 9&++I&10,: / 81&000 ! +&99>:

Pre#6 =0.01>> s

7re#6 8 ! CN! A ! ! a ! 7: / m 7re#6 8 ! ,.9+ ! I ! ,.9,!10=1,! 9&++I&10 ! +&99>: / 1&000 7re#6 0.00>1 m/s

E5"ation 83.)0:&

L K = / are#L K =8)I., ! 1&000: / =19.,

L K 3&900 re#o$"tions

PROBLEM 3.1+

A spacecraft is in a circ"$ar parking or?it wit an a$tit"*e of ,00 km. Ca$c"$atete #e$ocit% cange re5"ire* to perform a omann transfer to a circ"$ar or?it atgeos%ncrono"s a$tit"*e.

OLO2&

i#en4 rA6 89&3+I.1 < ,00: ! 1&000 6 9&)+I&10 m

;rom pro?$em 3.3&

rB6 ,&19&1+0 m

E5"ations 83.),: tro"g 83.)>:&

at!6 8rA< rB: / , at!6 89&)+I&10 < ,&19&1+0: / , at!6 ,&3+1&1)) m

7iA6 HR@ M / rA 7iA6 HR@ 3.>I900)!101/ 9&)+I&10 7iA6 +&+I m/s

7fB6 HR@ M / rB 7fB6 HR@ 3.>I900)!101/ ,&19&1+0 7fB6 3&0+) m/s

7t!A6 HR@ M ! 8, / rA= 1 / at!: 7t!A6 HR@ 3.>I900)!10

1! 8, / 9&)+I&10 = 1 / ,&3+1&1)): 7t!A6 10&,3> m/s

7t!B6 HR@ M ! 8, / rB= 1 / at!: 7t!B6 HR@ 3.>I900)!101! 8, / ,&19&1+0 = 1 / ,&3+1&1)): 7t!B6 1&)>+ m/s

7A6 7t!A= 7iA 7A6 10&,3> = +&+I


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7A6 ,&)) m/s

7B6 7fB= 7t!B 7B6 3&0+) = 1&)>+ 7B6 1&+I m/s

76 7A< 7B 76 ,&)) < 1&+I 76 3&>33 m/s

PROBLEM 3.1I

A sate$$ite is in a circ"$ar parking or?it wit an a$tit"*e of ,00 km. sing a one=tangent ?"rn& it is to ?e transferre* to geos%ncrono"s a$tit"*e "sing an* a transfere$$ipse wit a semi=major a!is of 30&000 km. Ca$c"$ate te tota$ re5"ire* #e$ocit%

cange an* te time re5"ire* to comp$ete te transfer.

OLO2&

i#en4 rA6 89&3+I.1 < ,00: ! 1&000 6 9&)+I&10 m rB6 ,&19&1+0 m at!6 30&000 ! 1&000 6 30&000&000 m

E5"ations 83.90: tro"g 83.9,:&

e 6 1 = rA/ at! e 6 1 = 9&)+I&10 / 30&000&000

e 6 0.+I0+,>

# 6 arccos@8at!! 81 = e,: / rB= 1: / e # 6 arccos@830&000&000 ! 81 = 0.+I0+,>,: / ,&19&1+0 = 1: / 0.+I0+,>

# 6 1)+.9+0 *egrees

6 arctan@ e ! sin # / 81 < e ! cos #:

6 arctan@ 0.+I0+,> ! sin81)+.9+0: / 81 < 0.+I0+,> ! cos81)+.9+0::

6 9.I+9 *egrees

E5"ations 83.)3: tro"g 83.)+:&

7iA6 HR@ M / rA

7iA6 HR@ 3.>I900)!101

/ 9&)+I&10 7iA6 +&+I m/s

7fB6 HR@ M / rB 7fB6 HR@ 3.>I900)!101/ ,&19&1+0 7fB6 3&0+) m/s

7t!A6 HR@ M ! 8, / rA= 1 / at!: 7t!A6 HR@ 3.>I900)!101! 8, / 9&)+I&10 = 1 / 30&000&000: 7t!A6 10&3II m/s


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7t!B6 HR@ M ! 8, / rB= 1 / at!: 7t!B6 HR@ 3.>I900)!101! 8, / ,&19&1+0 = 1 / 30&000&000: 7t!B6 ,&3+1 m/s

7A6 7t!A= 7iA 7A6 10&3II = +&+I

7A6 ,&90 m/s

E5"ation 83.93:&

7B6 HR@ 7t!B,< 7fB

,= , ! 7t!B! 7fB! cos 7B6 HR@ ,&3+1

,< 3&0+),= , ! ,&3+1 ! 3&0+) ! cos89.I+9: 7B6 ,&,90 m/s

E5"ation 83.)>:&

76 7A< 7B 76 ,&90 < ,&,90 76 &I9 m/s

E5"ations 83.9: an* 83.9):&

E 6 arctan@81 = e,:1/,! sin # / 8e < cos #: E 6 arctan@81 = 0.+I0+,>,:1/,! sin81)+.9+0: / 80.+I0+,> < cos81)+.9+0:: E 6 ,.119II ra*ians

O; 6 8E = e ! sin E: ! HR@ at!3/ M O; 6 8,.119II = 0.+I0+,> ! sin8,.119II:: ! HR@ 30&000&0003/ 3.>I900)!101

O; 6 11&>31 s 6 3.31 o"rs

PROBLEM 3.1>

Ca$c"$ate te #e$ocit% cange re5"ire* to transfer a sate$$ite from a circ"$ar900 km or?it wit an inc$ination of ,I *egrees to an or?it of e5"a$ si(e wit aninc$ination of ,0 *egrees.

OLO2&

i#en4 r 6 89&3+I.1 < 900: ! 1&000 6 9&>+I&10 m 6 ,I = ,0 6 I *egrees

E5"ation 83.9:&

7i 6 HR@ M / r 7i 6 HR@ 3.>I900)!101/ 9&>+I&10 7i 6 +&))I m/s

E5"ation 83.9+:&

7 6 , ! 7i ! sin8 /,:


40/43

7 6 , ! +&))I ! sin8I/,: 7 6 1&0) m/s

PROBLEM 3.,0

A sate$$ite is in a parking or?it wit an a$tit"*e of ,00 km an* an inc$ination of,I *egrees. Ca$c"$ate te tota$ #e$ocit% cange re5"ire* to transfer te sate$$iteto a (ero=inc$ination geos%ncrono"s or?it "sing a omann transfer wit a com?ine*p$ane cange at apogee.

i#en4 rA6 89&3+I.1 < ,00: ! 1&000 6 9&)+I&10 m rB6 ,&19&1+0 m 6 ,I *egrees

;rom pro?$em 3.1+&

7fB6 3&0+) m/s 7t!B6 1&)>+ m/s

7A6 ,&)) m/s

E5"ation 83.9I:&

7B6 HR@ 7t!B,< 7fB,= , ! 7t!B! 7fB! cos 7B6 HR@ 1&)>+

,< 3&0+),= , ! 1&)>+ ! 3&0+) ! cos8,I: 7B6 1&I,9 m/s

E5"ation 83.)>:&

76 7A< 7B

76 ,&)) < 1&I,9 76 &,I1 m/s

PROBLEM 3.,,

Ca$c"$ate te escape #e$ocit% of a spacecraft $a"nce* from an Eart or?it wit ana$tit"*e of 300 km.

OLO2& i#en4 r 6 89&3+I.1 < 300: ! 1&000 6 9&9+I&10 m

E5"ation 83.+0:&

7EC6 HR@ , ! M / r 7EC6 HR@ , ! 3.>I900)!10

1/ 9&9+I&10 7EC6 10&>,9 m/s


41/43

ATMOSPHERE PROPERTIES

Physical Properties of Standard Atmosphere in SI Units

Altitudemeters!

"emperature#!

PressurePa!

Density$g%m&!

'iscosity(-s%m)!

:-5000 +/0)2 1)22E@- 1).+1 1).,/E:-

:,5000 +1,)/ 1)-.E@- 1)220 1).1/E:-

:+5000 +02)2 1),+0E@- 1)1. 1)/E:-

:/5000 +01)/ 1)/2E@- 1),2 1)-/E:-

:15000 /.,)2 1)1+.E@- 1)+,2 1)/1E:-

0 /)/ 1)01+E@- 1)//- 1)2.E:-

15000 /1)2 ).E@, 1)11/ 1)2-E:-

/5000 /2-)/ 2).-0E@, 1)002 1)2/E:-+5000 /)2 2)01/E@, .)0.+E:1 1).,E:-

,5000 //)/ )1E@, )1.,E:1 1)1E:-

-5000 /--)2 -),0-E@, 2)+,E:1 1)/E:-

5000 /,.)/ ,)2//E@, )01E:1 1)-.-E:-

25000 /,/)2 ,)111E@, -).00E:1 1)-1E:-

5000 /+)/ +)--E@, -)/-E:1 1)-/2E:-

.5000 //.)2 +)00E@, ,)21E:1 1),.+E:-

105000 //+)+ /)-0E@, ,)1+-E:1 1),-E:-

1-5000 /1)2 1)/11E@, 1).,E:1 1),//E:-

/05000 /1)2 -)-/.E@+ ).1E:/ 1),//E:-

+05000 //)- 1)1.2E@+ 1),1E:/ 1),2-E:-

,05000 /-0), /)21E@/ +)..E:+ 1)01E:-

-05000 /20)2 2).2E@1 1)0/2E:+ 1)20,E:-

05000 /--) /)/,E@1 +)0-.E:, 1)/.E:-

205000 /1.)2 -)-/0 )2-,E:- 1),+E:-

05000 10)2 1)0+2 1)...E:- 1)/1E:-

.05000 10)2 1),,E:1 +)120E: 1)/1E:-

Atmospheric Scale *eight + Density, to &,./ $m

Altitude

$m!

Scale *eight

$m!

Atmospheric Density

0ean

$g%m&!

0aximum

$g%m&!

0 ), 1)//- 1)//-


42/43

100 -). -)/-E:2 -)2-E:2

1-0 /-)- 1)2+E:. 1)..E:.

/00 +2)- /),1E:10 +)-E:10

/-0 ,,) -).2E:11 1)/0E:10

+00 -0)+ 1)2E:11 ,),E:11

+-0 -,) )E:1/ /)1E:11

,00 -)/ /)/E:1/ 1)0-E:11

,-0 1)+ 1)0.E:1/ -)+-E:1/

-00 ,)- ,)2E:1+ /)/E:1/

--0 )2 /)1,E:1+ 1)-+E:1/

00 2,) .).E:1, ),E:1+

-0 ,), ,)2+E:1, ,)22E:1+

200 ..)+ /)+E:1, /)2+E:1+

2-0 1/1 1)/,E:1, 1)-.E:1+

00 1-1 ).-E:1- .),1E:1,-0 1 ,)//E:1- -)2E:1,

.00 // /)2E:1- +),.E:1,

.-0 /+ 1).E:1- /)/1E:1,

15000 /. 1),.E:1- 1),+E:1,

15/-0 ,0 -)20E:1 /)/E:1-

15-00 -1 /)2.E:1 1)1E:1-

/5000 /. .)0.E:12 +)0E:1

/5-00 1//0 ,)/+E:12 1)-,E:1

+5000 1-.0 /)-,E:12 2)0.E:12

+5-00 1.00 1)22E:12 +)2E:12

,5000 /10 1)+,E:12 /)11E:12

,5-00 /,+0 1)0E:12 1)+,E:12

-5000 /.0 )/E:1 .)+0E:1

5000 +/00 )0.E:1 -),1E:1

25000 +2-0 ,)-E:1 +)2,E:1

5000 ,+,0 +)-E:1 /)2E:1

.5000 ,.20 /)2E:1 /)+,E:1

105000 -+0 /)+2E:1 1).E:1

1-5000 .00 1)/1E:1 1)1E:1/05000 1,00 2)./E:1. ),/E:1.

/-5000 /0200 -).-E:1. )1E:1.

+05000 /200 ,)+E:1. -),E:1.

+-5000 +000 ,)1+E:1. -)/1E:1.

+-52 +2+00 ,)0,E:1. -)1/E:1.


43/43

Physical Properties of Standard Atmosphere in U1S1 Units

Altitude

feet!

"emperature

degrees 2!

Pressure

psia!

Density

slug%ft&!

'iscosity

l3-s%ft)!

:1-5000 -2/)/ /,)/ +)10E:+ ,)0+1E:2

:105000 --,)+ /0),2 +)1--E:+ +).+-E:2

:-5000 -+)- 12)--, /)2,-E:+ +)+-E:2

0 -1)2 1,). /)+22E:+ +)2+E:2

-5000 -00) 1/)0-, /)0,E:+ +)+E:2

105000 ,+)0 10)10 1)2-E:+ +)-+,E:2

1-5000 ,-)/ )/.2 1),.E:+ +),+1E:2

/05000 ,,2), )2-. 1)/2E:+ +)+/E:2

/-5000 ,/.) -),1 1)0E:+ +)/12E:2

+05000 ,11) ,)+2+ ).02E:, +)102E:2+-5000 +.,)1 +), 2)+/E:, /)..-E:2

,05000 +.0)0 /)2+0 -)2+E:, /)..E:2

,-5000 +.0)0 /)1,. ,)/+E:, /)..E:2

-05000 +.0)0 1)./ +)+.E:, /)..E:2

--5000 +.0)0 1)++/ /)-E:, /)..E:2

05000 +.0)0 1)0,. /)/-E:, /)..E:2

-5000 +.0)0 0)/ 1)222E:, /)..E:2

205000 +./)/ 0)-1 1)+./E:, /).+E:2

2-5000 +.-)0 0)-1, 1)0.1E:, +)001E:2

05000 +.2)2 0),0 )-21E:- +)01E:2

-5000 ,00), 0)+// )2,+E:- +)0+-E:2

.05000 ,0+)1 0)/-- -)+1-E:- +)0-/E:2

.-5000 ,0-) 0)/0+ ,)1.E:- +)020E:2

1005000 ,0) 0)1/ +)+1E:- +)02E:2

1-05000 ,2.)1 0)0/0 +),-E: +)-1/E:2

/005000 ,-2)0 0)00+ -)/20E:2 +)+/E:2

/-05000 +-1) 0)000 2)0+,E: /)2/1E:2

+005000 ++/). 0)000 ,)/-E:. /)-.+E:2

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