' .; t' '". I'

,,t \''!

{\ {'lrtr-Y F 5FF /-arnnn^r ri9rX. I &l,t- L I I L t.r f-ttl/Cl-rr ?

T-k-

i. ifrTRilSE-rC'Fi#H

Sai*liiies (nair.rral *r ariifi*iali, iiaivc inade gici:n3 suivclis a realistii;abieclive.

* flcr,iur$ can he sp*."n;re,J ttt *E:rrlrrcl c*nlinenls l

* c**dr:tic;:r:siti*n *a;l b* lirii:siiig*d &irv*ily *ri gi*i:ili daiurii

hiiliuiui saieiiitss; [in]r]ri, sr-ui, gilal-icis auri siarss Path trf-the natural sat*lJit*s are functions of the oirs*n'ers positi*n

on *arf,l:s ]-{ence. if *tati*n re**rit siiffi*ient cliseiviiiisir alang the satelliies

path, then their position are reiated

Artiti* ial {man-rnad*) sateilites\- & Ohsen'atiofis arff made electr*rricatly *r optically& h4easure range, range ra.te s;r clir:estiern ti*n: the *hsen er tc th*

sateliite* Fr*duccrl, *$ a gl*bal datuln:

s Relativr: positi*n anlilng oirserving sens$rso Sensors pasiticno lJ*finiti*n of sarellire r:r'bits

\11r.'J

Ariifiuiai sateiiites are differerri lrtlrn ordinruy planet duc t*:* lts *rbit is mr"rch closer to the earth ti:len tire orbit cf any planet lvirtr

rrrspecf to its prinraryr h rr^+ion cf artifi*ial satellite is *igpifi*atfl_v af1bcted by th** lvisJl

earth's gravitaii*nai {ield md by ifs ieinp*rill valj*tir:n (c,g.rides)

*' flnabie ddtemiinatioll r,rf pararneters rhat defines this fiefd-t'"hich will y-ietd infarnratiorl orl tlre shape, fiia$s distributionemci dymamic hehaviour *f'the eefilr

* lt filoEjss. at least pnrtiall,r,'. ful an *tmosph*re ia'hij* thc nalri::*trsatellite in*l,;€ii practicall3'in a vacrium

u Enable thc deter:ninati*n of tlle ahr*spheric structur* aud itsbehaviour

o Affecreci lry ofirer phenon:etra, such as attraction cf the sun and themoon. solar radiation pressure, lunar anel solar tidal dist*rtion,ef1bct of the nragnetic field of the earth. *t*.

* T'lt* e e!ftcts arE: relzezlively n*nor, but crumot brl neglect*:d ,

t,i

2. F ['ie-P#SE U S' S.rtT?- LLi ]'n SSSERVAT-I$H

? g*nernl purposc *f sateilite *bservationa) Scientiii* purp*seh) {}perati*n*l pilry*se

?"l Scieniifir purp*se

Scientific purpo$e cf satellite *Lrseffi'atic:n is f'*rther break dow.'n int* 2categories

a lltarnic purpfiso€ {ier_rntetric purpose

a) Symamie purp*s*: 7is obserr;e p*silittns t*uf xtoli*rus r$'u suteilitu ':t.s

a Juitttion af tiwe, vsifh sffitient {ttcurttw for Ceveloping * t:heory *.f:nafiott cap*hle *f predirtingJuture S:asilicns rq'tlze sale.lliles" at lesstcis dtcctttdttt$t as tizej,tan be ohsercetl

Requires:a Pre*ise knowledge of ph3,sical paranreters that de{ines fh* forcsr

f,eld in li,hich the sateliites moves qorbit)& Accurate ge*cerdric positi*n af *bservers6 F,nables.

* Irositi*n of salellit* to be computecl at any time, J* Predicted satellitc p*siticn t* he estimated using least squarefr clj r;stment pr*cedure

* C*rrection to the assumed i:aralneters and statjonsc<xrrdfurates due to tJre diffbrences between actual andpredicted orbit" cnlculated using lcast squere adjus*nentprocedure

( r,t ;

"fih) Ger:mbirir pur;los€: To ohserte pcsi{ions oJ's*teliites from ser;erctl

st*ti*ns o{ knot*'i cutd rsnkn*wn positksns stmu{t*rceousl1;.{ordetermirzir?g tf?e re lt:rtive sso^ritirsrz a.f the stution,r {and sate{lites}

n F*r this plxpo$e, satsllite is regard*rl as *hserv atian'd,talget *nrl*trservation is *ften carried *ut in tn'o lvays:

i. *rhitcl meth.od', Safellite is qrbserved in its ortrit fr$m severalstali*us ot'assumect pusiti*n" The ol-rservation results are thencontpare<l rvith tlrc computed {pr*dictedi satellite positian, rvhichare hasetl $n il$suirleil *r l.:nq:rtvn *rbitai parameters and station

cuor:ciirtaies. Froin iht: riifferr:n{,*s, eurrsciioris iri i"he a-csuins.iparaffiltsfs *r tr the e*ardiiiates, or b*th, niay be ,-lruiuir* iysimulraneo u s or leasr s q u nre adi usrnenl

-

Dis crepil;;;-use trito inrprove the force fiuicticm, the initial conclitions, oritlr" ututionFosition of the observ'er.

ii' Stellsr {spsce) triangtlation i trilateration:Satellite is used as atriangrlation. mark in space, w.hich is sirnultanrouulv ou;;;;; $ytneasuring directions or distances or both) from station of knonrrand unkm*n positiorx. Frorn obsen'ation* ut l"oiitr

"*ii"* tfr*position of the satellite is cletennined at the firslant of the

;

obseruation. Tlte observation at the unlcnonn station to the knou,n(detemrined) position of the satellite rviliyielcl the position oi-theabserver.

o Determineel station position is useful for:n strengthening satellite-tracking netrvork establislrcdtbr d1'namic applicaticn

" Providing connectians between geodetic datr*n,separated by large distan*es* Determining disiortions w-itrrin geodetic datumr Detecting motions behveen stations clue togeodynamic phenomena

2.2 Operationa I pu rpos€

Operational puqpose of satellife observation is ta meet flre operationalneeds of the satellite, i.e.

o Fcr regular suveillance of the satellite and to keep the orbita I dataup to date

s Guidurg and controlling the clirection of the transmittirlg ortelenretry antenna of the sfttellite

Before stut$,ing the dynuntic and geomefi,ic ntethods in scientificpurpases

_of satellite geodeq'can be achieved, it is necessar,uio struty the

orbit 0f the satellite.

3. i]L{iSE SATELLITE TI{flORY

'I'he tbeory, rvhich deals r,vitlr the nrotion of artiticial satellites in its orbitaround a ielestial body, u,hich is fixed in space, is called 'closed satel{itetlreot-v*t,

To understand the motion of a satellite easilv, lets proceed irr tr,vo steps:

Step l: Start rvith a simplified case

i. The earth is treated as a point rrrass, or, ecluirralently, as a spherewith constant densiry distribution. The gravitational field o1'such a

hr:dy is radialiy slumretric: i.e. the plumb lines are all straight linesand point towald the cenfie of the sphere

ii. The mass of the satellite is negligible conrpared to the ntass of theearth

iii. l'he rnoticn takes place in vacuuni [i.e. there are no distubingeffects present drrc to cther celestial bodies (sun, moon. ar othercelestial body, etc.) ar t{r some physical phenomena (aUirosphericdnag solar radiation pressure, etc.)]

* The resulting orbits is called 'ndrmal orbit'

Step 2: Reviert those factors tlut deviate from the assumptions above andtheir effect on the nonnal orbit

i. Without the assumptions nrade in step t. the motion of the satelliter"'ill deviate front dle nonnal orbit. This disturbed normal orbit iscalled a'perturbed orbit'

ii. The deviation befiveen the perturbed orbit and normal orbit iscal led' p ertur bation' g

S.lNormal orbit

In normal orbit, we arssume that the satellite is travellin--e in a perfectlyspherical grflvrty field, and il'\.ve assume that only gravitational forces

(,'. =gnPlare acting on the satellite i.e. the earth's gravitational field,\ r' )

H/, equal to it's potential, (n, =Y.rJ unO disturbing potential r is

zero, then the orbit will obey 'Kepler's lqw of planetary motion'.

4

',ilil:'lr

i{epicr's three iaws are ricdur:ed (iiom obscrvations oipiallets or"niiingtlrc sun) in 1609 {law I &,2\ and i619 (lar.r: 3).

?'!E'^-!^--t^ !---- ^4 --^r:^-J,.&r\epttsr $ ;aYl ui ilI{rai{it-t

i. Kepier"s iirsi iaw: Lalv of'urbii

Tire orbit a; a saieiiire v;iii be an eiiipse wiiit ihe eariiz ar ane af its.fuci.

satellrte

Periot-e

t

Asceneiing nocie

Where:Apogee, points at rvhich the sateiiite is fufihest away frcnrthe earthPerigee, point at whieh the satellife is nearest to iirc earthAscending Norie, point u'here the satellite crosses theequator, travelling from south to north (of'lut pr:int af-Aries)liue aronralyiirstai rtaneous saiellite radirisRadius of apogee

Radius of perigee

Orbital serni major axisOrbital semi minor axisEccentricity

"!:!"

P

AN-

,(

ri"

ahL}

'i'iie p*siriorr of the *-rrbil iu sprir;* is defirted b5' six *rhital(Keplerian) eleurcnts:

a ; Semi-maj*r axisi '. hlclirration{i : Al"gum**? *f perigee

Ilc,centricityRight nscensi*nTin":e of pi+ssage at perige*

e.g?:t,

Tlrese riumbers de.ftne rni eiltstse, *rienl it ahout ihr e{trfh, ciwlploe:* tlre s#teili!.e *re the eltitrsst: at * pnrticular tinte.

Line ef Apsides

af -\ l', 1n!/\ uff-)tiitl i-ieile N*diri [.ine

*li1ial-elg11g$S_egd part o{thn qri}ii pryiectecl itnto_a seotelrricur$lsphsrs

hiailai line isthe inter,secliun o{'the orbiral plaule rviLh thu cquat*rirrlptrane, wlrich ccnilects rhe ascending and riescending nades.

Rigkt *sc*n.q;iatt. (n) is the angle h*fw*en ghs lines of- nodcr and thedirecti*r"r t* r'emal ecluin*x {zero nreridian)"{ncfinrsti*n nngle {i} is th* angle betrvcen the equatar:iai anql the,-lrbital plaiies.

,!1rg;tdtrent o{perigee (,ru'} is ths angl* befr,veen the nodal 1i*e andth* orhital scmi-n'laj*r axis.

True *nrsrnulv ./ is t!:e;rngula:'digtance *:f'th* satellite, S, ftorn

i:*rigee.

[rluatorial plan*

Ascerr{iillg Fk:iie

-ffire equati*rr *f the *ri:itai eilipse baseci on Kepier's i't iaw is:#{1 -,i}21--' '- "' -'

, wilerf,, r is Lhe distance clf the sateliites rrrl*l iire eru.ilh's1 I iccs /'e;sntrs ,rro*r. i{*nce. r ilnci i' ftlirns a paii'ilf p*ii* c**r*jjriats.

ii. Kepl*r's se*und lalv: i,ai.v *iareas

tr(epler Znd iaw sfates titilf. llre aren *f tlre etliptical sect*r srnept byrh* radir.r* rzestor r betr,'u'**n rur},' trv* ptsiti*ns clf the sat*llite ispreipcrti*nal t* the tisn* it takes ttie slifellite t* pass frcinr opepcsition tc) another, i.e. tlie time ratt *J'*rfifige af the $r"c& sw'upt bt;the r#di'ils vcctCIr is t*nsf.ant.

t t'!{

tlt;:*::---T.

= 4! i,t{{ | - t" iu = c{Jn5tanl

.'li:'.

'ii{t.

tlt"

Ar:AzSatcllite larlves faster neal perigee {gr*atast resistanc*)5 alellite lnoves stror+'er ne?rt' apogee {lorr est rcsistance}

Kepler's lhirci traiv: Hanluonic laiq,

Ii

Kepler 3d law statss that" thr: square of the perioc{ of revolurir}tl, 'I,is prop*lti*nnl f"o fhe cuhe *f its senri*lrrai*r axis clf'ths crbit, a.

,J''tu : c*fiste*t

Later in th* l?th century. hl*rvton proves all thr above lni,vs licmrhis universal law o{ g;lavitation and in fact derived Kepler's 3'd lawrelaiionship to he:

{,n (-ii,l -*

( 4*t,.,t ',''_*;;__-_ L_-J i sl-___l'1',; 4rt I r;.,tr j

3.LGeometry of elliptic orbit

First$, let us rotate our cartesian coordinate system to make Xaxiscoincide with major axis of the ellipse.

Satellite orbit

O-rPae

Orbital plane coordinates

i. Size and shape of the orbital ellipse are defined by it's semi major

axis, d, and it' eccentricity, e, where "' =1' =b.'-2 "^'-'- - a2

ii. From geometry of the ellipse, semlminor axis o = oJil, and

the distance from the focus Fto the center O is equal to ae

iii. The point on semi major axis at the nearest approach is thepericenter, ffid farthest approach is apocenter (perigee and apogee)

When satellite is at arbitrary point, m, its position in orbit is defined bythe angle mFP, or TRUE ANOMALY, which is measured in the directionof motion and denoted,,asf.

The angle m'OP, which associated with the auxiliary circle of radius a, iscatled ECCENTRIC ANQMALY and denoted by angle E.

Ttre MEAN AN2MALY is the true anomaly corresponding to the motionof an imaginary satellite of uniform angular velocity. It can be visualizedas an angle that is zerc atperigee and increases uniformly at a rate of360'per revolution. It is denoted by M.

a

ORBITAL COORDINATES

orbital coordinate system (xo,yo,zo) is defined with xo towards perigee,

Io towards true anomaly of 90o and Zo pointing perpendicular to the orbitplane to make right handed coordinate system. In normal orbit, zo is ofcourse zero, but for real (perturbed) satellite orbit z, wrll assume non-zero value.

The position of satellite at any given time, l, is defined by its polarcoordinates, r andf. The radius vector r can be computed from therectangular coordinate Xoand ro:

X o = r cos f= acos-E - ae = a(cos E - e)

Yo = rstrt f= Dsing = oJlllr* g

Zo=o

The polar radius is:

= o = ol-ecosE)f=

Expression for true anomaly,f can also be calculated from simplegeomefrical considerations :

^ rsin ftan1 =

rcos I=0- u'Y ,i,.s

cosB - e

Yo

xo

And in a more convsnient form:

The polar coordinate r andf are given as a function of eccenfric anomaly,E. To find E, we first define the mean anomaly, M, as a true anomqly ofan imaginary satellite, m ', moving with uniform angular velocity and the

same period as ffi. Then, at time t, M is computed from: M =+Q 4 r)Where, T,theorbit period, is computed from Kepler's 3'd law. E can nowbe computed from M,us,tng classical equation of celestial mechanics(Kepler's equation).

o"f;r:(#)'tanla

* Yo'

Mean anomaly: M = E - esinE

If satellite position is to be determined as a function of time, this equationis solved by iteration, ffid outlined as follows:

' a-

i. First, compute M for a time I using: M =+Q -t r)

ii. Then calculate approximation of E using:

Et=M+esinMl,JU*sin2M

iii. Then substitute value into mean anomaly equation, and computeM1 using: M, = E, - esinE,

iv. Calculate the difference'. LA/ = Mr- M

rEt aE= Mv. Determine the correction fo: 1 _ ecos.

vi. Then compute 2od approximation E2: Ez= Er+ M

vii. With E2,ftpe&tthis sequence until LM = 0 (or nearly so)

* Knowin g E, r andf cannow be determined. These define thesatellite motion and position in the plane of the ellipse, with respectto polar and rectangular Cartesian coordinates system.

10

\'ti:'lql;!:i'

3.2 C OORDINATES TRANSFORMATION

For most application, we require the satellite position to be in 3Dcartesian system, namely Geocentric cartesian Coordinates andTopocentric Cartesian Coordinates with its cente at the cenfte of massof the earth.

G eo c entri c C artesian C o o rdin st es

Here we used right-handed coordinate system. 26is defined as beingparallel to the earth's mean spin axis, X6 is towards the zero longitudeand 16 towards 90oE. The cenfie of this system is near the cenffe of themass of the earth. This coordinates system is often referred to as "earthfixed" because it rotates with the earth.

)-r' zo 4.,r

Yc Yo

}D

Geocentric Coordinates Systemirr.J lr

To transform\om orbital coordinate system to earth fixed geocentriQcoordinates system requires simple rnatter of 3 rotations (each involvingKeplerian elements)

IfXo is a vector (xd,yJ ,ZEY and xo is (xf ,fo'

[";l ["llI r.f I = R,(G,qsr -o)R* Fi)a,(-r) rJ II't) lrt )

Orbital Plane

,ziY, then:

l1

The Cartesian Coordinates of the satellites in the equatorial system are as

follows:

XE =rcosdcoscrYi ='cosdsinaZE =rsin6

Where, geocentric declination, d, and right ascension , d , are:

d = sin-r[si"(7 + ar)sin i]

a = GAST- b = tan-r[costt an(y + d)]

With the rotation matrices &(r)q(ilRr(") are as follows:

. Rotation matrix about Z axis (axis 1) is:

[r o olR,(B)=lo cosct) ,tn,

I

Lt - sin a*or r-l

o Rotation matrix about X axis (axis 2) is:lcosi 0 -sinlltt

n,(B)=l o 1 o I

fsinl 0 cosiJ

r Rotation maffix about Z ans(axis 3) is:I cosa sina 0l

&(e)=l_;;;;;;l[o o rJ

L2

An alternative coordinates system withXtoward l't point of Aries issometimes used. Such system, referred to as an inertial coordinatessystem (i.e. fixed in space), is useful when relative star and satellitepositions are important. Positions of the satellite in this coordinate system1Xf ,Y,s,Zl )are derived from:

The relationship between Earth Fixed and Inertial coordinate systems isexpress by an angle known as Greenwich Apparent Sidereal rime,denoted as GAST (a), and the required transformation can be carried outusing:

The value for GAST at any particular time depends on number of factorsand exffemely complex to compute. Here, we simply mention that itdepends on the motion of the earth with respect to the sun (for definitionof vemal equinox), and hence the precession and nutation of the earth,and on rotation of the earth.

lxil [x;'lI t' l= R, (- o)R" (- i)n, F rl r,' I

I't I Lr: )

["* I I cos GrsI sin GrsI ol["] II rd l=l-sinGlsr cosG,4sl 0ll ti

Ilrt)[ o o ,)lti ]

..:,:r

l3

T?p o centric C artesian Co ordinates

Topocentric Cartesian Coordinates system is

also known as Local Coordinates System. The

system is defined at one point (called point oforigin, observer). Zyis the direction of thevertical (up) at that point, Yyis north and ZTiseast.

Now, lets inffoduce the Topocenfric coordinate system Xy,Y7,Zy parallel

to the geocentric coordinate system X6,Y6,26.

Centremass, o

Topocentric Coordinate system

Note that, q and a are known satellite celestial coordinates that would beviewed by an observer at the earth's centre; a' and 5' ate the celestialcoordinates actually observed (Topocenhic); p is the range from theobserver to the satellite; r is the range from the satellite to the earth'scentre; and X*,Y14,21a are the rectangular coordinates of the observer inthe celestial coordinate system at the moment of observation.

Yi

X".l

IIlt

----------r- rtl

Satellite, S

t4

In Topocentric Coordinate System: sind'= 4 ^A

p= 1ip sind'

Therefore, the satellite coordinates in this system can be expressed as ingeocentric coordinate system:

Xi = pcosd'cosa'= Zl cotd'cosa'

Yf = p cos d'sin a' : Zl cot 6' sin a'Zl = psinS'

Now, as both coordinate systems are parallel, the Topocentric coordinatesof satellites in this system is:+;

ly,l[f] l,;lThese coordinates may also be expressed as a function of TopocentricSpherical Coordinates as follows:

d' ... The Topocenffic right ascension

.;lir::: ii\ir.

d' ... The Topocenffic declinationp ... The Topocentric radius

And these values can be derived as follows:

d,=tan-tf-r,''llxi J

d'= cot-l[4r.ro']\zi )o=ffi=m

15

":l

In real situation, the Geocenffic and Topocentric coordinate systems are

not parallel (earth fxed but with cenffe at P and with xr,Yr,Z, are

directed east, north and up respectively).

It is necessary to initially compute the earth fixed Cartesian coordinatesof the observing station P from the given latitudg longitude and height(0r,1r,hr), using:

X [ = (v + H r) cos Q, cos X., ; YI =(v + 1/" ) co s {, sin }", ; ZI =(v(1 - e2 1 + H r)sin /"Where: v-, =a= .,=,

(1 -e2 sin' prlrtz

Then the satellite to ground distance p is simply given by:

o=Wtr-xtrY*ki - r[f +(z'.-rIYY

And direction cosines from: ,=ryr*=Wrr=ryTo determine the direction to the satellite, it is necessary to transform thesatellite position in Earth Fixed to Topocentric coordinate system, givenby the following:

lxil [t o o l[-sin/, cost", ol lx|-xt1I t' l=lo sin4, cos4,ll-'orz" sin)", o l-l t; -y: I

lzi ) [o -cos4, sin/,][ o o lJ lz|-z| )

t6

Geodetic Coordin ates Sv stem

For practical application, it is unlikely that such coordinates will bedirect$ useful. They will normally have to be fansformed into the localnational coordinate system, or perhaps to a particular pu{pose system thatmay have been adopted for a particular project in hand.

Nominally the transformation between geodetic coordinate system isexpressed by seven parameters transformation (Flelmert fransforrnation),with the required geodetic coordinates (xG ,yG,zcl betng related to thosegiven by satellite positioning (xty,t ,zs) by:

\- Where: dX,dY,dZ are translation parameters, 0y,0y,0, are small rotationparameters, s dscale parameter.

In areas where the seven transformation parameters are not known, then itis necessary to carry out satellite position fixing at stations with knownlocal coordinates and solving the seven parameters using least squares, bytreating qxc,yG,zo) and (xty,t,zt ) asobserved quantities.

ItAW.n.'{;? :r?t[i]

t7