keplerian element

GPS SATELLITE ORBIT COMPUTATION INTRODUCTION OF KEPLER’S LAWS The orbit of each planet is an ellipse with the Sn at one focs! The line "oinin# the planet to the sn sweeps ot e$al areas in e$al ti%es an&' the s$are of the perio& of a planet is  proportional to the cbe of its %ean &istance to the sn! Orbit of the satellite is (nown as Keplerian, as Keplerian, i!e! orbit is an ellipse with earth at one of the foci' where) Ass%ptions are) * Earth is a point %ass or e$i+alentl, a sphere with nifor% &ensit, so the attraction is towar& center * Mass of the satellite is ne#li#ible * Satellite %o+es in a +ac% E-ceptions) * Gra+it, fiel& +ariation' attraction of sn' %oon an& other planets * At%ospheric &ra# an& solar ra&iation pressre * Relati+istic effects etc!

.i#re / ) Coor&inate s,ste% in the orbital plane!

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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 focs of the ellipse an& not in the %i&&le 2bt 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 :9 79 s orbit an& e$ator! Elliptical orbit is a flat area (nown as the orbital plane! This habit orbital plane thro#h &irectl, into the center of the earth bt will sli#htl, tilt at an, an#le relati+e to the e$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 becase the satellite passe& o+er  the north an& soth 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 abot the shape of an ellipse! @hen the +ale of e ; 8' the ellipse is a circle! @hen the +ale of e approaches /' the ellipse will be a +er, lon# an& thin!

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.i#re < ) eccentricit, an& %a"or a-is

Line of no&es> can co%e at an, point alon# the e$ator! If it is fon& at the e$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$ator fro% soth to north! .or the other' it is calle& as >&escen&in# no&e> where the satellite will cross the e$ator fro% north to soth! Nor%all,' the >ascen&in# no&es> will be specifie&! If the earth in a state of spin' this %eans we cannot se the sal %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' %easre& fro% the %i&point of the earth' fro% the >+ernal e$ino-> to the ascen&in# no&e!

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.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!

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.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 pre %athe%atical $antit, #i+en below b, 1eplers e$ation)

M ; E 6 e sin E

If the satellite is in a circlar orbit 2%o+in# at a constant spee&9 an& +iewe& on the center  of the earth an& %easrin# 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,! Usall, the peri#ee will appear on the Mean Ano%al, ; 8' an& apo#ee also appeare& in Mean Ano%al, ; /?8 &e#rees!

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COMPUTATION OF SATELLITE POSITIONIN+ IN OR,IT USIN+ KEPLERIAN ELEMENT

The position of the satellite %st be correcte& for the e-traneos effects state& earlier! Satellite ephe%eris "st as solar or star ephe%eris li(e an Al%anac' it contain infor%ation abot the location of a satellite at an, #i+en ti%e! As the orbit of a satellite is not $ite 1eplerian' its location can onl, be pre&icte& fro% the infor%ation collecte& b, trac(in# its orbit constantl,! In the case of positionin# satellites' ephe%eris infor%ation consists of 1eplerian  para%eters at a certain epoch' rate of chan#e of these ele%ents' cloc( infor%ation an& cloc(  correction ter%s! Accrate ephe%eris infor%ation is re$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%ptation!

Table / pro+i&es the GPS or Galileo broa&cast ephe%eris para%eters to co%pte their satellite coor&inates at an, obser+ation epoch! These para%eters are perio&icall, renewe& 2t,picall, e+er,

hors for GPS an&

hors for Galileo9 an& %st not be se& ot of the prescribe& ti%e

2abot for hors9' becase the e-trapolation error #rows e-ponentiall, be,on& its +ali&it,  perio&! The al#orith% pro+i&e& is fro% the GPSSPS6SS! The Galileo satellites follow a si%ilar sche%e

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Table /) Ephe%eris Para%eters

In or&er to co%pte satellite coor&inates fro% na+i#ation %essa#e' the al#orith% pro+i&e& as follows %st be se&! An accrac, of abot

%eters 2RMS9 is achie+e& for GPS satellites with

SA;8ff an& se+eral tens of %eters with SA;on!

Co%pte 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' sbtract

sec fro% ! If

sec' a&&

sec!

•

Co%pte the %ean ano%al, for

' 8

•

Sol+e 2iterati+el,9 the 1epler e$ation for the eccentricit, ano%al,

•

Co%pte the tre ano%al,

•

Co%pte the ar#%ent of latit&e an& corrections

an&

)

Co%pte the ra&ial &istance

•

Co%pte the inclination

•

fro% the ar#%ent of peri#ee

' tre ano%al,

)

•

' an& corrections

)

' consi&erin# corrections

an&

)

of the orbital plane fro% the inclination an&

at reference ti%e

)

Co%pte the lon#it&e of the ascen&in# no&e

2with respect to Greenwich9! This

calclation ses the ri#ht ascension at the be#innin# of the crrent wee( 2

9' the

9

correction fro% the apparent si&ereal ti%e +ariation in Greenwich between the be#innin# of the wee( an& reference ti%e ascen&in# no&e fro% the reference ti%e

' an& the chan#e in lon#it&e of the )

•

Co%pte the coor&inates in TRS fra%e' appl,in# three rotations 2aron&

•

@here

an&

'

an&

9)

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 consecti+e rotations aron& the coor&inate a-es 2 '

' three '

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)

where)

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•

A&optin# the con+ention se& b, IERS' the pre+ios e$ation 2/9 can be written as follows)

•

@here an&

'

'

are three translation para%eters'

is a scale factor an&

'

are three rotation an#les!

Referrin# to Transfor%ation para%eters fro% ITR.:888 to past ITR.s are liste& in table