Relativity for Global Navigation Satellite Systems

Transcription

1 Relativity fo Global Navigation Satellite Systems Notes by Anna Heffenan based on the Living eviews atile, Relativity in the Global Positioning Systems, Neil Ashby, Living Rev. Relativity 6, (003),1 whih will be efeed to as [1] in the below notes. A good aompaniment is Setion 10.3 of Gavity by Ei Poisson and Cliffod Will. This notebook seves as a supplement to [1] hene equations and setions have been labelled aodingly - it in no means eplaes the moe omplete mateial povided in [1]. Inteested students an avail themselves of a moe thoough undestanding by eading the Living eveiws atile (as well as Gavity S10.3). S1 Intodution The system 3 segments GPS has 3 piniple segments: - Spae segment - Contol segment - Use segment Spae segment atomi lok aying satellites (GPS:4 but spaes exist) - oganised in seveal tilted obital planes (GPS: 4 satellites in 6 planes inlines at 55 to eaths equato) - timing signals alulated fom loks ae tansmitted fom eah satellite, - i.e., event: satellites emit signal with spaetime oodinates of the emission. - additional infomation inludes infomation on: the entie satellite onstellation, satellite health and univesal oodinated time Contol segment - gound-based monitoing stations (keep tak of infomation fom the satellites) - satellite infomation is analyized and pojets the natual positions and lok behaviou of the satellites fo next few hous - this is uploaded to the satellites fo etansmission to uses.

2 Letue_GPS.nb Use segment - eeive signals tansmitted fom the satellites and use them to detemine position, veloity and time on thei loal loks. Tehnology The main tehonology fo GPS is the extemely auate, stable loks. Fig. 1 (taken fom [1]) plots the Allen deviation, fo both a high-pefomane Cessium lok and a Quatz osillato (typial fo GPS eeive), as a funtion of sample time τ. We an see that Quatz osillatos have bette shot tem stability than Cesium loks but afte 100 seonds o so, Cesium has fa bette pefomane. FIg. 1 is fo guidane - in eality thee is vaiation aoundt the nominal values. Fig.1. Allen deviations of Cesium loks and quatz osillatos, plotted as a funtion of aveaging time τ. Taken fom [1] The plot fo Cesium haateizes the best obiting loks in the GPS system afte initializing a Cesium lok, and leaving it alone fo a day ~10 6 s, it should be oet to within 5 pats in o 4 ns. Relativisti effets ae huge ompaed to this. Relativisti Effets Despite a weak gavitational field and small lok veloities, signifiant elativisti effets ae pesent in the GPS system, these inlude, - fist and seond ode Dopple fequeny shifts of loks due to thei elative motion - gavitational fequeny shifts - Sagna effet due to Eath s otation. If unaounted fo lage eos in GPS navigation and time tansfe. To detemine position and time tansfe with GPS, one onsides 4 synhonized atomi loks tansmitting thei time and position t j, j fo j = 1,,3,4. These fou signals ae eeived at position t, (t - t j ) = - j, j = 1,,3,4 ;(1)

3 Letue_GPS.nb 3 whee is the speed of light 4 unknowns t, j with 4 equations solvable. Timing eos of 1 ns 30 m position eo (~3 10 8, Δ t~10-9 ). We will see howeve that one must be vey aeful in speifying the efeene fame in whih the tansmitte loks ae synhonized to ensue (1) emains valid. S Refeene Fames and the Sagna Effet Refeene Fames ECEF: Eath-Centeed, Eath-Fixed efeene fame - Eath-Centeed, Eat-fixed in whih Eath otates aound a fixed axis with a defined otation, ω E. - Eath atually otates about a slightly diffeent axis with vaiable otation but this is of little onsequene fo elativity - As GPS uses ae fixed o slowly moving on the Eath s sufae ealy deision to desibe satellite s position in ECEF - ECEF satellite position use position will be outputted in ECEF also - Rotating not inetial to use (1) we need a diffeent (inetial) efeene fame Tanfoming fom inetial to otating fames Conside tansfomation fom an inetial fame, in whih spaetime is MInkowskian, to a otating fame of efeene (ignoing gavitational potential fo now), the meti in an inetial fame in ylindial oodinates is, -d s = - d t + d + dϕ + d z, ;() and the tansfomation is given by, t = t, =, ϕ = ϕ' + ω E t ', z = z ;(3) d s = ω E ' d t ' + ω E ' d ϕ' d t ' + d σ ', ;(4) whee d σ ' = d ' + ' d ϕ' + d z'. and t is oodinate time pope time as measued by an inetial obseve at infinity. Synhonization

4 4 Letue_GPS.nb Sagna Effet To solve (1) fo the use s position, we must be able to ompae and synhonize times, one an use light to tanslate this infomation as the speed of light is onstant in evey fame. Anothe option is a slowly potable lok. Regadless of the method, (1) will not hold in a otating fame due to the Sagna effet we must ay out ou alulations in a loal inetial fame o ECI oodinate system. Synhonization via light A light ay is emitted at A and eeived at B: Light ds = 0 in (4), and keeping only tems of fist ode ωe ' in, (4) beomes, dt' - dσ ' - ' dϕ' dt' ωe ωe + O = 0, ;(5) dt' = dσ ' 1 + ' dϕ' dt' dσ' dt' = dσ ' 1 - ' dϕ' dσ' ωe ωe + O, ωe -1 + O ωe, = dσ ' + ' dϕ' ω E + O ω E. ;(6) z' ω E A B ' y' x' dϕ' da ' z Fig.. da ' z = π ' dϕ' is the aea swept out by a position veto, as it otates aound the z'-axis an infinitissimal angle of dϕ', π pojeted onto the equatoial plane of the sphee. The quantity ' dϕ' = π ' dϕ' is simply the infinitissimal aea da' π z as illustated in Fig.. The total time equied fo light to tavel between these two points is B A B dt' = 1 A dσ ' + ωe B d A' z. ;(7) A

5 Letue_GPS.nb 5 Obseves fixed on the eath, who wee unawae of the Eath s otation, would just use 1 dσ ' to synhonize thei lok netwok. Obseves at est in the undeling inetial fame would see that this leads to signifiant path-dependent inonsistenies. Fom the undelying inetial fame view: the additional tavel time equied by light to ath up to the moving efeene point. Synhonisation via a potable lok A lok slowly moves fom point A to point B: A timelike geodesi ds = - dτ whee τ is the pope time (measued by a lok), (4) now dτ = dt' 1 - ωe ' - ωe ' d ϕ' dt' - 1 dσ'. ;(9) dt' A slowly moving lok dσ' << 1, allowing us to disegad this tem, also keeping only fist ode of ωe ' yields, dt' dτ = dt' 1 - ωe ' d ϕ' = dt' - ωe ' d ϕ', ;(10) dt' B B dt' = dτ + B ωe d A' z. ;(11) A A A the time diffeene eoded on the lok (dτ) is diffeent to the oodinate time diffeene (dt) between events A (lok depated) and B (lok aived). Compaing with (7), egadless of whethe we use light o potable loks to disseminate time, pathdependent disepanies in the otating fame ae inesapable must use undelying inetial fame fo synhonization. S3 GPS Coodinate Time and TAI Side note: Legende polynomial geneating funtions By a Taylo expansion, povided u<<1, we have (1 - u) -1/ = n=0 f (n) (1) n! = n=0 (-u) n ( n)! ( n n!) u n Now if we onside, u = xt - t, we have,, whee f (x) = x -1/ f (n) (x) = (-1)n ( n - 1)!! x -( n+1) n. f (n) (1) = (-1)n n ( n - 1)!! = (-1)n ( n)! = (-1)n ( n)! n ( n)!! n n n!

6 6 Letue_GPS.nb 1 - x t + t -1/ ( n)! = n=0 xt - t n ( n n!) = n=0 = n=0 [s/] = s=0 k=0 = s=0 = s=0 ( n)! n ( n n!) k=0 ( n)! n ( n n!) k=0 n k ( s- k)! s-k (s-k)! t s s s! [s/] k=0 t s s s! = s=0 t s P s (x) whee s = n + k n - k 0 k s ( x t)n-k -t k n! k! (n-k)! (-1)k n-k x n-k t n+k (s-k)! k! (s- k)! (-1)k s- k x s- k t s s! k! (s-k)! (-1)k d s dx s x (s-k), d s dx s x - 1 s,, Coodinate time Newtonian gavitational potential multipole moment deomposition The Newtonian gavitational potential fo a point mass is given by, U = G M. If we ae to apply this to a mass distibution, we get U t, x = G i x m i -x i, whee extenal field > and x x ' = ' Cosγ, ρ t,x' = G d3 x', ;P l (Cosγ) = = G 1 x -x' ' ' 1 - Cos γ + -1/ ρ t, x' d 3 x', ' = G n=0 n P n+1 n (Cosθ) ρ t, x' d 3 x', = G M = G M + G n=1 + n= G n+1 l+1 m=-l l Y * lm(θ', ϕ') Y lm (θ, ϕ), ;Cosγ = Cosθ Cosθ ' + Sinθ Sinθ' Cos(ϕ - ϕ'), n+1 ' n n m=-n Y * nm(θ', ϕ') Y nm (θ, ϕ) ρ t, x' d 3 x', ;P 0 = 1 n+1 n m=-n I nm Y nm (θ, ϕ), whee I lm (t) = ρ t, x ' ' l Y * lm(θ', ϕ') d 3 x' ae the multipole moments of the Eath s mass distibution, and we have used the fat that the dipole moments vanish (I 10 = I 1-1 = I 11 = 0) when we plae the oigin of the oodinate system at the body s ente of mass ρ x d 3 x = 0. Given Y * 10 = 3 Cosθ and Y * 1±1 = 3 8 π Sinθ e±iθ, this an be seen by, I 10 = 3 ρ t, x ' ' l Cosθ d 3 x' = 3 ρ t, x ' z d 3 x' = 0,

7 Letue_GPS.nb 7 I 1±1 = 3 8 π ρ t, x ' Sinθ e ±iθ d 3 x' = 3 8 π ρ t, x ' (x ± i y) d 3 x' = 0. Fo an axially symmeti body (mass density invaiant unde a otation about a symmety axis), Taking the z-dietion to be aligned with the symmety axis, the mass density is independent of the azimuthal angle ϕ the only non-vanishing multipole moments ae m=0. U t, x = G M + n= G n+1 n+1 I n0 Y n0 (θ, ϕ), using J l = - l+1 I l0 M R l, = G M - n= G M n+1 = G M R n 1 - n= J n n P n (Cosθ), R n n+1 J n Y n0 (θ, ϕ), P l (Cosθ) = l+1 Y l0(θ), whee the haateisti length sale R is hosen to be the body s equatoial adius and J is the Eath s quadupole moment oeffiient. Fo ou uent pupose, keeping up to n= will be suffiient, highe multipole moment ontibutions have a vey small effet fo elativity in GPS, U t, x = G M 1 - R J P (Cosθ). (13) Meti in Isotopi Coodinates In istopi oodinates ' 1 + m ', the Shwazshild meti tansfoms to, ds = - 1- m 1+ m dt m 4 d + dω, m + 1 m +... dt m m +... d + dω whee dω = dθ + sin θ dϕ. Theefoe to wite an appoximate solution of Einstein s field equations in istopi oodinates with the assumptions of a stati mass distibution and a loal inetial, non-otating, feely falling oodinate system with the oigin at the Eath s ente of mass, we have, ds = U dt U d + dω, (1) whee U is the Newtonian gavitational potential. Rotating into ECEF oodinate system To tansfe this (1) to a otating, ECEF oodinate system via spheial oodinates, we have, t = t ', = ', θ = θ', ϕ = ϕ' + ω E t ' (14) ds = U ωe ' Sinθ' - dt' + ωe ' Sin θ dϕ' dt ' U d + dω, (15) whee we have disgaded tems of ode 1 3 and highe. The meti oeffiient g' 00 in the otating fame is, g' 00 = U - ω E ' Sinθ' Φ, (16) whee Φ is the effetive gavitational potential in the otating fame, whih inludes the stati gavitational potential of the Eath, and a entipetal potential tem.

8 8 Letue_GPS.nb The Eath s Geoid TAI: Temps Atomique Intenational - The Eath s geoid is defined as the sufae of onstant gavitational equipotential at mean sea level in ECEF. - TAI is the pimay intenational standad fo atomi time - defined on the Eath s geoid. - In (1) and (15), the ate of oodinate time is detemined by atomi loks at est at infinity - not possible in eality - We an elate this oodinate time to that eoded on the Eath s geoid. Cloks on the Eath s geoid Fo a lok at est on Eath, (15) edues to, ds = U - ω E ' Sinθ' dt ' = Φ dt'. (17) This detemines the adius of the model geoid as a funtion of pola angle θ. The numeial value of Φ 0 an be deteined at the equato whee θ' = π and =R, Φ0 = G M [1 - J R P (0)] + ωe R, whee P (0) = - 1, G M = m 3 s -, = G M 1 + J R, R ;J = , R = = ;ω E = ad s -1 = (18) thee distint ontibutions to this effetive potential: - 1/ ontibution due to the Eath s mass, - a moe ompliated ontibution fom the quadupole potential, - a entipetal tem due to the Eath s otation. Fom (17) and ds = - dτ, fo loks on the geoid we have, dτ = dt' 1 - Φ0. (19) loks at est on the geoid un slow ompaed to loks at est at infinity by about These effets ae about 10 5 times lage than the fational fequeny stability of a high-pefomane Cesium lok. The shape of the geoid in this model an be obtained by solving (16) with Φ 0, i.e., Φ 0 = G M 1 - R J 3 ' ' Cos θ' ωe Sin θ', to obtain = (Cosθ ). If we assume =R(1+Δ) whee Δ is small, we have, Φ 0 = G M R J (1 - Δ) 1 - (1 - Δ) 3 Cos θ' ωe R (1+ Δ) 1-Cos θ' = G M R 1 - Δ - J (1 - Δ) 3 Cos θ' J Δ 3 Cos θ' ωe, R 3 (1+ Δ) 1-Cos θ' G M,

9 Letue_GPS.nb 9 = G M R = G M R J 1 - Δ - 3 Cos θ' ωe J ωe R 3 G M = Φ 0 - G M R Δ + 3 J Δ = - 3 J + ωe R 3 - Δ - 3 J + ω E R 3 G M Cos θ' R 3 1-Cos θ' G M + ωe R 3 G M Cos θ',, as J << 1, ωe R 3 G M << 1 G M Cos θ'. (0) This teatment of the gavitational field of the oblate Eath is limited by the simple model of the gavitational field. Hee, we have estimated the shape of the so-alled efeene ellipsoid, fom whih the atual geoid is onventionally measued. Geoid loks as efeene loks In (19), Φ 0 is a onstant and dτ epesents the inement of pope time as eoded by a lok at est. ideal loks at est on the geoid of the otating Eath all tik at the same ate. In (1), the ate of oodiante time is defined by standad loks at infinity, to swith to loks on the geoid, we define a new oodinate time, t : t '' = 1 - Φ0 t ' = 1 - Φ0 t. o t = t' = 1 + Φ0 t '' + O -4 (1) If we eall (15) ds = Φ dt ' + ωe ' Sin θ' dϕ' dt' U d' + ' dω', (15) we now have ds = (Φ-Φ0) dt'' + ωe ' Sin θ dϕ'' dt '' U d'' + '' dω'', () whee we have again only etained an auay to -. Caying out the same time sale hange in the non-otating meti (1), gives ds = (U-Φ0) dt'' U d'' + '' dω''. (3) () and (3) pope time elapsed on loks at est on the geoid is idential with the oodinate time t. This is the oet way to expess the fat that ideal loks at est on the geoid povide all of ou standad efeene loks. S4 Realisation of Coodinate Time We now dop the t and just use t with the undestanding that unit of time is efeened to TAI on the otating geoid with synhonisation established in an undelying, loal inetial, efeene fame. We theefoe wite (3), whih is in the ECI fame, as, ds = (U-Φ 0) dt U d + dω. (4) ds = - dτ dτ = 1 - (U-Φ0) U v dt, (5) whee v = d + dω dt (6) dτ = 1 - (U-Φ0) - v dt. keeping odes up to - (7) Solving (7) we an elate the inement of oodinate time (TAI) to the pope time inement given by

10 10 Letue_GPS.nb the slowly moving lok. path dt = path 1 + (U-Φ0) + v dτ. (8) S5 Relativisti Effets on Satellite Cloks Kepleian obits, we an desibe the satellites with Keplaian obits, fo whih we have the standad equa- Fo U G M tions, = a 1-e Cos u-e, (9) whee Cos λ = 1+e Cos λ 1-e Cos u = a (1 - e Cosu) (31) whee a and e ae the semi-majo axis and eentiity espetively while u and λ ae the eenti and tue anomalies espetively. To alulate the eenti anomaly, one must solve, u - e sin u = (30) G M a 3 (t - t p ), (3) whee t p is the oodinate time of peigee passage. If we apply onsevation of enegy and momentum at the apogee and and peigee, we also have, E m = va - G M v a a = vp - G M, and v p p = v a a p = G M a a p -1, whee a = a (1 + e), p = a (1 - e). p = G M - e a a 1-e (1-e) -4 e, = G M (1-e) a (1+e) E m = G M 1-e a (1+e) - 1 a (1+e) = -G M (1+e) a (1+e) = - G M a. v - G M = - G M a. (33) Relativisti Effets Quadupole Contibutions - Fo atomi loks, we use ECI Sagna effet is ielevent (still equied when dealing with goundbased eeives). - Gavitational fequeny shifts and seond ode Dopple shifts must be taken into aount togethe. - In (8) Φ 0 inludes the oetion fo using loks at est on the geoid (instead of at infinity). - Fom (18) we an see that the quadupole ontibutes to Φ 0 in the tem G M J R ~ this must be aounted fo in GPS. - Fom (13) we also see that in U, the quadupole omponent falling off patiulaly fast, U quad = G M R 3 J P (Cosθ)~10-14 fo GPS, whih we an neglet U G M Keplaian obits ae

11 Letue_GPS.nb 11 appliable. Clok ates fo iula obits Reall (8) Δt = path 1 + (U-Φ0) = path 1 + G M = path 1 - Φ0 + v dτ, (8) - Φ0 + G M 1-1 dτ, a + G M 1-1 dτ, 4 a = path G M - Φ0 + G M 1 a - 1 dτ, (34) a whee if we put in numeial values fo the fist two omponents, we have 3 G M - Φ0 = = (35a) a fo iula obits the diffeene in tiks of a lok on the geoid and a lok in obit is a fato of , i.e., t = G M - Φ0 τ + τ a 0, whee τ 0 is a onstant of integation set by synhonization of the obiting lok. (35b) Time-dilation (seond tem in 35b) and gavitational blueshift (whih dominates - thid tem in (35b)) Negative sign standad lok in obit (dτ in ECI) is beating two fast beause it s fequeny is gavitationally blueshifted. fo satellite lok to appea to an obseve on the geoid to beat at the hosen fequeny of 10.3MHz, the satellite loks ae adjusted lowe in fequeny so that the pope fequeny is M Hz = M Hz, this is done on the gound befoe the lok is plaed in obit.

12 1 Letue_GPS.nb Fig.3. Taken fom [1]. Net fational shift of a lok in a iula obit (35b) plotted as a funtion of obit adius a. Seveal soues of elativisti effets ae pesent, with the dominant effet hanging with the adii. Fo a low eath obite (e.g., spae shuttle), high veloity time dilation dominant, while fo GPS, the gavitational blueshift dominates. These effets lealy anel at a 8545 km. Clok ates fo eenti obits Fo eenti obits (so when the last tem in (34) does not vanish) we an use (3) to alulate, d u = d t path G M G M a 3 (1 - e Cos u) -1, 1-1 a dτ G M path 1-1 a dt, as (8) dt = 1 + O - dτ, = G M a path e Cos u dt, as (31) = a (1 - e Cos u), 1-e Cos u = G M a path a 3 G M du e Cos u dt, dt = a G M e (Sin u - Sin u 0 ), = a G M e Sin u + onst, (38a) t = G M a - Φ 0 + a G M e Sin u τ + τ 1 (38b) As u = u(t), (38) gives us a elativisti oetion fo eenti obits that vays in time, whih numeially gives, Δt = e a Sin u s m. (39) - Fo a satellite of e = 0.001, this an eah up to 3 ns (taking a~ m) must be aounted fo. - Histoially due to limited omputational powe, this oetion is added by the eeive in GPS wheeas GLONASS oets fo it befoe boadast. - Looking at (38b), we an see a time dilation effet podued by obital motion (seond tem), a gavita-

13 Letue_GPS.nb 13 tional blue(ed)shift assoiated with the tansfe of the time signal fom high to low altitude (thid tem), and a time-dependent oeton assoiated with the obital eentiity (final tem).