GPS SATELLITE ORBIT COMPUTATION
GPS SATELLITE ORBIT COMPUTATION
INTRODUCTION OF KEPLER’S LAWS
INTRODUCTION OF KEPLER’S LAWS
The orbit of each planet is an ellipse with the Sn at one focs! The line "oinin# the planet The orbit of each planet is an ellipse with the Sn at one focs! The line "oinin# the planet to the sn sweeps ot e$al
to the sn sweeps ot e$al areas in e$al ti%es an&' the s$are areas in e$al ti%es an&' the s$are of the perio& of a planet of the perio& of a planet isis proportional to the cbe of its %ean &istance to the sn!
proportional to the cbe of its %ean &istance to the sn! Orbit of the satellite is (nown as
Orbit of the satellite is (nown as Keplerian, Keplerian, i!e! orbit is an ellipse with earth at one of thei!e! orbit is an ellipse with earth at one of the foci' where)
foci' where) Ass%ptions are) Ass%ptions are)
* Earth is a point %ass or e$i+alentl,
* Earth is a point %ass or e$i+alentl, a sphere with nifor% &ensit, so the attraction isa sphere with nifor% &ensit, so the attraction is towar& center
towar& center
* Mass of the satellite is ne#li#ible * Mass of the satellite is ne#li#ible * Satellite %o+es in a +ac% * Satellite %o+es in a +ac% E-ceptions)
E-ceptions)
* Gra+it, fiel& +ariation' attraction of sn' %oon an&
* Gra+it, fiel& +ariation' attraction of sn' %oon an& other planetsother planets * At%ospheric &ra# an& solar ra&iation pressre
* At%ospheric &ra# an& solar ra&iation pressre * Relati+istic effects etc!
* Relati+istic effects etc!
.i#re / ) Coor&inate s,ste% in the orbital plane! .i#re / ) Coor&inate s,ste% in the orbital plane!
1
KEPLERIAN ELEMENT
There are 0 t,pes of ele%ents in &eter%inin# the orbit! It is calle& the satellite orbital ele%ent (nown as 1eplerian Ele%ent 2na%e& after 3ohann 1epler 2/45/6/0789! In the 1eplerian' satellite orbitin# the ellipse is the shape an& orientation of the e$ip%ent! The Earth is at one focs of the ellipse an& not in the %i&&le 2bt when the elliptical orbit is a co%plete circle9!
A normal orbit is co%pletel, #i+en b, the followin# (eplerian orbital ele%ents' see fi#res /9 Se%i6%a"or a-is (a) = size
:9 Inclination 2ί 9 ; tilt 79 Eccentricit, 2e9 ; shape
<9 Lon#it&e of ascen&in# no&e 2Ω 9 ; pin 49 Ar#%ent of periapsis 2 ω 9 ; twist 09 Mean ano%al, ( M 9or 2v9 ; an#le
The first for ele%ents &escribe the si=e' shape' an& orientation of the orbit! The fifth ele%ent alon# with an an#lar $antit, calle& anomaly &eter%ines the position of the satellite in its orbit at an, instant! 1eplerian ele%ent can be obtaine& fro% orbital state +ectors sin# software or calclation &irectl,!
/!/ Si- 1eplerins Ele%ent E-planation
/! Semi major axis (a
The &istance between the centers of the bo&ies' not the &istance of the bo&ies fro% the center of the %ass
:! I!"#i!a$io! (i
Inclination is the an#lar &istance between the satellite>s orbit an& e$ator! Elliptical orbit is a flat area (nown as the orbital plane! This habit orbital plane thro#h &irectl, into the center of the earth bt will sli#htl, tilt at an, an#le relati+e to the e$ator! Inclination is the an#le between the orbital plane an& the e$atorial plane!
Nor%all,' the inclination is a n%ber between 8 an& /?8 &e#rees! .or orbit with inclination close to 8 is calle& the e$atorial orbit 2this is becase the satellite passe& o+er the north an& soth poles9! Crossin# the e$atorial plane an& the orbital plane is a line calle& the line of no&es!
.i#re 7) inclination
7! E""e!$ri"i$% (e
In the 1eplerian orbit %o&el' the satellite orbit is elliptical! Eccentricit, is %ore abot the shape of an ellipse! @hen the +ale of e ; 8' the ellipse is a circle! @hen the +ale of e approaches /' the ellipse will be a +er, lon# an& thin!
.i#re < ) eccentricit, an& %a"or a-is
<! Lo!&i$'e o) as"e!i!& !oe (Ω
After the inclination &escribe& in %ore &etail' there is still no orbital plane can be calclate&! >Line of no&es> can co%e at an, point alon# the e$ator! If it is fon& at the e$ator where the >line of no&es> co%e off' then the orbital plane can be specifie&!
Ar#abl,' it #oin# off at : locations' so we onl, nee& to specif, one onl,! One of the calle& as >ascen&in# no&es> where the satellite will cross the e$ator fro% soth to north! .or the other' it is calle& as >&escen&in# no&e> where the satellite will cross the e$ator fro% north to soth! Nor%all,' the >ascen&in# no&es> will be specifie&!
If the earth in a state of spin' this %eans we cannot se the sal %etho& of latit&e or lon#it&e coor&inate s,ste% to specif, the location of the >line of no&es! Con+ersel,' we can also se an astrono%ical coor&inate s,ste%' (nown also as the ri#ht ascension &eclination where it &oes not rotates with the earth! >Ri#ht ascension of ascen&in# no&e> is the an#le' %easre& fro% the %i&point of the earth' fro% the >+ernal e$ino-> to the ascen&in# no&e!
.i#re 4 ) Lon#it&e of ascen&in# no&e
4! Ar&'me!$ o) Perigee ( ω
Also (nown as the Ar#%ent of Periapsis! In a&&ition' it is preferable to specif, a corner! The point where the satellite is closer to the earth is calle& peri#ee' or also calle& periapsis! @hile the farthest point fro% the earth is calle& apo#ee! @hen we &raw a line
fro% peri#ee to apo#ee' this line is calle& the line of apsi&es! Line of apsi&es thro#h the center of the earth an& another line thro#h the center of the earth' as it is (nown the line of no&es!
The an#le between two lines is (nown as ar#%ent of peri#ee! @hen two lines intersect' the, for% two a&&itional an#les! To %ore clearl,' we can sa, that ar#%ent of peri#ee is the an#le of the ascen&in# no&e to the peri#ee!
.i#re 0 ) Ar#%ent of Perigee
0! Mea! a!oma#% ( M or ( v
Mean ano%al, is an an#le that %o+es nifor%l, in ti%e fro% 8 to 708 &e#rees in the rotation! This &efines that 8 &e#rees is at peri#ee an& apo#ee of /?8 &e#rees! Mean no%al, is a pre %athe%atical $antit, #i+en below b, 1eplers e$ation)
M ; E 6 e sin E
If the satellite is in a circlar orbit 2%o+in# at a constant spee&9 an& +iewe& on the center of the earth an& %easrin# the an#le fro% peri#ee' this will be reflecte& towar&s the satellite! Satellite in orbit is not a circle that %o+es at the spee& of ne+en' then the relationship will not last! This relationship will re%ain for : %ain point in orbit' howe+er' re#ar&less of eccentricit,! Usall, the peri#ee will appear on the Mean Ano%al, ; 8' an& apo#ee also appeare& in Mean Ano%al, ; /?8 &e#rees!
*
COMPUTATION OF SATELLITE POSITIONIN+ IN OR,IT
USIN+
KEPLERIAN ELEMENT
The position of the satellite %st be correcte& for the e-traneos effects state& earlier! Satellite ephe%eris "st as solar or star ephe%eris li(e an Al%anac' it contain infor%ation abot the location of a satellite at an, #i+en ti%e! As the orbit of a satellite is not $ite 1eplerian' its location can onl, be pre&icte& fro% the infor%ation collecte& b, trac(in# its orbit constantl,!
In the case of positionin# satellites' ephe%eris infor%ation consists of 1eplerian para%eters at a certain epoch' rate of chan#e of these ele%ents' cloc( infor%ation an& cloc(
correction ter%s! Accrate ephe%eris infor%ation is re$ire& for locatin# precise positions on earth sin# the si#nals fro% satellites
.or this reason' all positionin# satellites incl&e ephe%eris infor%ation in its si#nal sent to earth
Satellite Coor&inates Co%ptation!
Table / pro+i&es the GPS or Galileo broa&cast ephe%eris para%eters to co%pte their satellite coor&inates at an, obser+ation epoch! These para%eters are perio&icall, renewe& 2t,picall, e+er, hors for GPS an& hors for Galileo9 an& %st not be se& ot of the prescribe& ti%e 2abot for hors9' becase the e-trapolation error #rows e-ponentiall, be,on& its +ali&it, perio&! The al#orith% pro+i&e& is fro% the GPSSPS6SS! The Galileo satellites follow a si%ilar
Table /) Ephe%eris Para%eters
In or&er to co%pte satellite coor&inates fro% na+i#ation %essa#e' the al#orith% pro+i&e& as follows %st be se&! An accrac, of abot %eters 2RMS9 is achie+e& for GPS satellites with SA;8ff an& se+eral tens of %eters with SA;on!
Co%pte the ti%e fro% the ephe%eri&es reference epoch 2 an& are e-presse& in secon&s in the GPS wee(9)
If sec' sbtract sec fro% ! If sec' a&& sec!
• Sol+e 2iterati+el,9 the 1epler e$ation for the eccentricit, ano%al, )
• Co%pte the tre ano%al, )
• Co%pte the ar#%ent of latit&e fro% the ar#%ent of peri#ee ' tre ano%al,
an& corrections an& )
• Co%pte the ra&ial &istance ' consi&erin# corrections an& )
• Co%pte the inclination of the orbital plane fro% the inclination at reference ti%e
correction fro% the apparent si&ereal ti%e +ariation in Greenwich between the be#innin# of the wee( an& reference ti%e ' an& the chan#e in lon#it&e of the
ascen&in# no&e fro% the reference ti%e )
• Co%pte the coor&inates in TRS fra%e' appl,in# three rotations 2aron& ' an& 9)
• @here an& are the rotation %atrices &efine& in Transfor%ation between
Terrestrial .ra%es!
Transfor%ation between Terrestrial .ra%es
• .ro% ele%ental linear al#ebra' all transfor%ations between two Cartesian coor&inate
s,ste%s can be &eco%pose& in a shift +ector ' three
consecti+e rotations aron& the coor&inate a-es 2 ' ' 9' an& a scale factor 2 9! That is' the, can be &escribe& b, the followin# e$ation' which in+ol+es 5 para%eters)
• A&optin# the con+ention se& b, IERS' the pre+ios e$ation 2/9 can be written as
follows)
• @here ' ' are three translation para%eters' is a scale factor an& '
an& are three rotation an#les!
Referrin# to Transfor%ation para%eters fro% ITR.:888 to past ITR.s are liste& in table <!/ of IERS Con+entions 2:8879 Denis et al!' :88<F
1eplerian %otion
/! Broa&cast ephe%eri&es 2in GPS si#nal9 containor-i$a# .arame$ers M, e, a, , i, H ' n 2;an#lar +elocit,9 :! Orbital para%eters se& to co%pte orbit in i!er$ia# )rame
7! Orbit in inertial fra%e 2Earth rotatin#9 isro$a$e $o $erres$ria# )rame 2Earth fi-e&9
<! In or&er to &o this accratel, o+er lon# ti%e perio&s' one nee& to (now UT/' notations' an& polar %otion! 4! An& ta(e into accont orbit
