'

.; t

' '

'". I

,,t ''!

\

{\ {'lrtr-Y

i9rX.

I

&l,t-

F 5FF

/-arnnn^r

L I I L t.r f-ttl/Cl-rr

r

?

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

hiiliuiui saieiiitss; [in]r]ri,

s

s

sr-ui, gilal-icis auri siars

Path trf-the natural sat*lJit*s are functions of the oirs*n'ers positi*n on *arf,l: ]-{ence. if *tati*n re**rit siiffi*ient cliseiviiiisir alang the satelliies path, then their position are reiated

\11r.'J

\-

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: Relativr: positi*n anlilng oirserving sens$rs Sensors pasiticn lJ*finiti*n of sarellire r:r'bits

s o o

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 prinrary h rr^+ion * lvisJl cf artifi*ial satellite is *igpifi*atfl_v af1bcted by th* earth's gravitaii*nai {ield md by ifs ieinp*rill valj*tir:n (c,g. rides)

r

*'

* o

flnabie ddtemiinatioll r,rf pararneters rhat defines this fiefdt'"hich will y-ietd infarnratiorl orl tlre shape, fiia$s distribution emci dymamic hehaviour *f'the eefilr lt filoEjss. at least pnrtiall,r,'. ful an *tmosph*re ia'hij* thc nalri::*tr satellite in*l,;€ii practicall3'in a vacrium u Enable thc deter:ninati*n of tlle ahr*spheric structur* aud its behaviour Affecreci lry ofirer phenon:etra, such as attraction cf the sun and the moon. 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

,

,i

t

2.

F

['ie-P#SE U S' S.rtT?- LLi

]'n SSSERVAT-I$H

? g*nernl purposc *f sateilite *bservation a) Scientiii* purp*se

h) {}perati*n*l pilry*se ?"l Scieniifir purp*se

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

a lltarnic purpfiso € {ier_rntetric purpose

a) Symamie purp*s*:

p*silittns t*uf xtoli*rus r$'u suteilitu ':t.s a Juitttion af tiwe, vsifh sffitient {ttcurttw for Ceveloping * t:heory *.f :nafiott cap*hle *f predirtingJuture S:asilicns rq'tlze sale.lliles" at lesst cis dtcctttdttt$t as tizej,tan be ohsercetl 7is obserr;e

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 *bservers

6 F,nables. * Irositi*n of salellit* to be computecl at any time, * Predicted satellitc p*siticn t* he estimated using least square J

r;stment pr*cedure C*rrection to the assumed i:aralneters and statjons c
*

(

r,t

clj

;

"fi

h) 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;.{or determirzir?g tf?e re lt:rtive sso^ritirsrz a.f the stution,r {and sate{lites}

n i.

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

*rhitcl

meth.od', Safellite is qrbserved in its ortrit fr$m several stali*us ot'assumect pusiti*n" The ol-rservation results are then contpare
cuor:ciirtaies. Froin iht: riifferr:n{,*s, eurrsciioris

iri i"he a-csuins.i the e*ardiiiates, or b*th, niay be ,-lruiuir* simulraneo u s or leasr s q u nre adi usrnenl Dis tri to inrprove the force fiuicticm, the initial conclitions, oritlr" utution Fosition of the observ'er. paraffiltsfs *r

tr

-

ii'

iy crepil;;;-use

Stellsr {spsce) triangtlation i trilateration:Satellite is used as a triangrlation. mark in space, w.hich is sirnultanrouulv ou;;;;; $y tneasuring directions or distances or both) from station of knonrr and unkm*n positiorx. Frorn obsen'ation* ut l"oiitr ; tfr* position of the satellite is cletennined at the firslant of obseruation. Tlte observation at the unlcnonn station to the knou,n (detemrined) position of the satellite rviliyielcl the position oi-the

"*ii"* the

abserver.

o Determineel station position is useful for: n strengthening satellite-tracking netrvork establislrcd " * r

tbr d1'namic applicaticn Providing connectians between geodetic datr*n, separated by large distan*es Determining disiortions w-itrrin geodetic datum Detecting motions behveen stations clue to geodynamic phenomena

2.2 Operationa I pu rpos€

Operational puqpose of satellife observation is ta meet flre operational needs of the satellite, i.e. Fcr regular suveillance of the satellite and to keep the orbita data up to date Guidurg and controlling the clirection of the transmittirlg or telenretry antenna of the sfttellite

o

I

s

Before stut$,ing the dynuntic and geomefi,ic ntethods in scientific purpases satellite geodeq'can be achieved, it is necessar,uio struty the _of 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 orbit around a ielestial body, u,hich is fixed in space, is called 'closed satel{ite tlreot-v*t,

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

l: Start rvith a simplified case

i. ii. iii.

The earth is treated as a point rrrass, or, ecluirralently, as a sphere with constant densiry distribution. The gravitational field o1'such a hr:dy is radialiy slumretric: i.e. the plumb lines are all straight lines and point towald the cenfie of the sphere The mass of the satellite is negligible conrpared to the ntass of the earth l'he rnoticn takes place in vacuuni [i.e. there are no distubing effects present drrc to cther celestial bodies (sun, moon. ar other celestial body, etc.) ar t{r some physical phenomena (aUirospheric dnag solar radiation pressure, etc.)]

*

The resulting orbits is called 'ndrmal

orbit'

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

i. ii.

Without the assumptions nrade in step t. the motion of the satellite r"'ill deviate front dle nonnal orbit. This disturbed normal orbit is called a'perturbed orbit' The deviation befiveen the perturbed orbit and normal orbit is cal led'p ertur bation'g

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

\

=gnPlare r' )

acting on the satellite i.e. the earth's gravitational field,

H/, equal to it's potential, (n, zero, then the orbit

=Y.rJ unO disturbing potential r is

will obey 'Kepler's lqw of planetary motion'.

4

i{epicr's three iaws are ricdur:ed (iiom obscrvations oipiallets or"niiing tlrc sun) in 1609 {law I &,2\ and i619 (lar.r: 3). !---- ^4 ?'!E'^-!^--t^ --^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

',ilil:'lr

Periot-e t

Asceneiing nocie

Where:

P

AN,(

"!:!"

r i"

a h L}

Apogee, points at rvhich the sateiiite is fufihest away frcnr the earth Perigee, point at whieh the satellife is nearest to iirc earth Ascending Norie, point u'here the satellite crosses the equator, travelling from south to north (of'lut pr:int af-Aries) liue aronraly iirstai rtaneous saiellite radiris Radius of apogee Radius of perigee

Orbital serni major axis Orbital semi minor axis Eccentricity

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

a ; i '. {i :

Semi-maj*r axis hlclirration Al"gum**?

*f

perigee

e.

g?: t,

Ilc,centricity Right nscensi*n Tin":e of pi+ssage at perige*

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

Line ef Apsides

[rluatorial plan* Ascerr{iillg Fk:iie af

/\

l',

1n!

-\ uff-)tiitl i-ieile

N*diri [.ine

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

hiailai line isthe inter,secliun o{'the orbiral plaule rviLh thu cquat*rirrl ptrane, wlrich ccnilects rhe ascending and riescending nades. Rigkt *sc*n.q;iatt. (n) is the angle h*fw*en ghs lines of- nodcr and the directi*r"r t* r'emal ecluin*x {zero nreridian)" {ncfinrsti*n nngle {i} is th* angle betrvcen the equatar:iai anql the ,-lrbital plaiies.

o{perigee

is ths angl* befr,veen the nodal 1i*e and th* orhital scmi-n'laj*r axis.

,!1rg;tdtrent

True *nrsrnulv i:*rigee.

./

(,ru'}

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

-ffire equati*rr

-' -,i}2 "'iccs

#{1

1--' '-

1I

/'

e;sntrs ,rro*r.

ii.

*f the *ri:itai eilipse , wilerf,, r is

i{*nce.

Lhe

r ilnci i'

baseci on

Kepier's i't iaw is:

distance clf the sateliites rrrl*l iire eru.ilh's

ftlirns a paii'ilf p*ii* c**r*jjriats.

Kepl*r's se*und lalv: i,ai.v *iareas tr(epler Znd iaw sfates titilf. llre aren *f tlre etliptical sect*r srnept by rh* radir.r* rzestor r betr,'u'**n rur},' trv* ptsiti*ns clf the sat*llite is

preipcrti*nal t* the tisn* it takes ttie slifellite t* pass frcinr ope pcsition 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. t" iu = c{Jn5tanl

= 4! i,t{{ | -

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

.'li:'.

'ii{t.

tlt"

Kepler's lhirci traiv: Hanluonic

laiq,

Ii

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

'tu

:

'I,

c*fiste*t

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

(-ii,l L_-J ( 4*t,.,t ','' -* i sl-___l

{,n _*;;__-_ '1',;

4rt

I

r;.,tr

j

3.LGeometry of elliptic orbit

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

a Satellite orbit

O-rP

ae

Orbital plane coordinates

i.

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

axis, d, and it' eccentricity, e, "^'-'- -2 where "'

ii. iii.

=1' a2=b.'

oJil,

From geometry of the ellipse, semlminor axis o = and the distance from the focus Fto the center O is equal to ae The point on semi major axis at the nearest approach is the pericenter, ffid farthest approach is apocenter (perigee and apogee)

When satellite is at arbitrary point, m, its position in orbit is defined by the angle mFP, or TRUE ANOMALY, which is measured in the direction of motion and denoted,,asf. The angle m'OP, which associated with the auxiliary circle of radius a, is catled ECCENTRIC ANQMALY and denoted by angle E. Ttre MEAN AN2MALY is the true anomaly corresponding to the motion of an imaginary satellite of uniform angular velocity. It can be visualized as an angle that is zerc atperigee and increases uniformly at a rate of

360'per revolution. It is denoted by M.

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 orbit plane to make right handed coordinate system. In normal orbit, zo is of course zero, but for real (perturbed) satellite orbit z, wrll assume nonzero value.

The position of satellite at any given time, l, is defined by its polar coordinates, r andf. The radius vector r can be computed from the rectangular coordinate Xoand ro: X o = r cosf= acos-E

-

= rstrtf= Dsing = Zo=o Yo

ae =

a(cosE - e)

oJlllr*

g

The polar radius is: f=

* Yo'

=

o

= ol-ecosE)

Expression for true anomaly,f can also be calculated from simple geomefrical considerations : tan1^=

rsin

f

Yo

rcos

I

xo

=0-

u'Y ,i,.s cosB

And in a more convsnient form:

-

e

o"f;r:(#)'tanla

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 of an 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 now be computed from M,us,tng classical equation of celestial mechanics (Kepler's equation).

Mean anomaly: M = E - esinE

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

i.

'

a-

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 compute M1 using: M, = E, - esinE,

iv. v.

Calculate the difference'. LA/ = Mr-

Determine the correction

M

fo: rEt aE=_M ecos. 1

vi.

Then compute

2od

approximation E2: Ez=

vii. With E2,ftpe&tthis

sequence

until LM

=

Er+

M

0 (or nearly so)

r

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

10

3.2 C OORDINATES

TRANSFORMATION

For most application, we require the satellite position to be in 3D cartesian system, namely Geocentric cartesian Coordinates and Topocentric Cartesian Coordinates with its cente at the cenfte of mass of 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 being parallel to the earth's mean spin axis, X6 is towards the zero longitude and 16 towards 90oE. The cenfie of this system is near the cenffe of the mass of the earth. This coordinates system is often referred to as "earth fixed" because it rotates with the earth. \'ti:'l

')-rzo

ql;!:i'

4.,r

Yc

Yo

Orbital Plane

}D

irr.J lr

Geocentric Coordinates System

To transform\om orbital coordinate system to earth fixed geocentriQ coordinates system requires simple rnatter of 3 rotations (each involving Keplerian elements)

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

[";l r.f

I

R,(G,qsr I=

I't)

-o)R*

then:

["ll rJ

Fi)a,(-r)

I

lrt ) l1

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

follows: XE =rcosdcoscr

Yi ='cosdsina ZE =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(")

Rotation matrix about

[r o

R,(B)=lo

Z

ol

are as follows:

axis (axis 1) is:

,tn, Lt - sin a*or r-l cosct)

I

o Rotation matrix about X axis (axis 2) is: lcosi 0 -sinll tt n,(B)=l o

o1

fsinl 0

r

I

cosiJ

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 is sometimes used. Such system, referred to as an inertial coordinates system (i.e. fixed in space), is useful when relative star and satellite positions are important. Positions of the satellite in this coordinate system 1Xf ,Y,s,Zl )are derived from:

lxil t'

[x;'l l= R, (- o)R" (- i)n, F rl r,'

I

I't

I

I

Lr: )

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

I rd

["*

I cos GrsI

l=l-sinGlsr

lrt)[ o I

sin

GrsI ol["] I

cosG,4sl 0ll

o

ti

I

,)lti ]

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

..:,: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 of origin, observer). Zyis the direction of the vertical (up) at that point, Yyis north and ZTis

.l

X" I I

----------r-

lt

tl

r

east.

Now, lets inffoduce the Topocenfric coordinate system Xy,Y7,Zy parallel to the geocentric coordinate system X6,Y6,26. Satellite, S

Centre mass, o

Yi Topocentric Coordinate system

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

t4

In Topocentric Coordinate System: sind'=

4

1i p ^A p=sind'

Therefore, the satellite coordinates in this system can be expressed as in geocentric 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 coordinates of satellites in this system is:

+;

ly,l[f]

l,;l

These coordinates may also be expressed as a function of Topocentric Spherical Coordinates as follows:

d' d' p

... The Topocenffic right ascension ... The Topocenffic declination ... The Topocentric radius

And these values can be derived as follows:

.;lir:::

ii

\ir.

d,=tan-tf-r,''l

lxi

d'=

J

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 coordinates of the observing station P from the given latitudg longitude and height (0r,1r,hr), using: X [ = (v + H r) cos Q, cos X., ;

Where:

YI =(v + 1/" ) co s {,

sin }",

; ZI =(v(1 - e2 1 + H r)sin /"

v-, =a= -e2 sin' prlrtz

.,=,

(1

Then the satellite to ground distance

o=Wtr-xtrY*ki

-

p

is simply given by:

r[f +(z'.-rIYY

And direction cosines from:

,=ryr*=Wrr=ry

To determine the direction to the satellite, it is necessary to transform the satellite position in Earth Fixed to Topocentric coordinate system, given by the following:

lxil [t o o l[-sin/, cost", ol lx|-xt1 t' l=lo sin4, cos4,ll-'orz" sin)", o l-l t; -y: I o lJ lz|-z| ) lzi ) [o -cos4, sin/,][ o I

t6

Geodetic Coordin ates Sv stem For practical application, it is unlikely that such coordinates will be direct$ useful. They will normally have to be fansformed into the local national coordinate system, or perhaps to a particular pu{pose system that may have been adopted for a particular project in hand.

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

ItAW.n.'{;? \-

:r?t[i]

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

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

t7