RELATIVISTIC TRANSFORMATIONS FOR TIME SYNCHRONIZATION AND DISSEMINATION IN THE SOLAR SYSTEM

Transcription

1 RELATIVISTIC TRANSFORMATIONS FOR TIME SYNCHRONIZATION AND DISSEMINATION IN THE SOLAR SYSTEM Rober A. Nelson Saellie Engineering Researh Corporaion 77 Woodmon Avenue, Suie 8, Behesda, MD 84, USA Todd A. Ely Je Propulsion Laboraory 48 Oak Groove Drive, MS 3-5L, Pasadena, CA, 99, USA Absra The measuremen of ime is an essenial aspe of navigaion. The Global Posiioning Sysem (GPS) is omprised of a onsellaion of saellies ha ransmi one-way pseudorandom noise (PRN) oded signals used for range and ime measuremens. The signals are referened o onboard aomi loks. The GPS provides a model for posiion deerminaion wih a preision of a few meers and ime disseminaion wih a preision of abou nanoseonds. The mahemaial algorihms used in he GPS reeiver require he appliaion of he priniples of general relaiviy. Similar models will be needed for high-preision navigaion in he solar sysem. The adopion of an appropriae oordinae sysem and ime sale is required. This paper oulines he fundamenal oneps of relaivisi ime ransfer and desribes he deails of he mahemaial model. The approximae magniudes of various relaivisi effes for loks onboard he GPS saellies, oher saellies in Earh orbi, and a lok on he surfae of or on he Moon are derived. INTRODUCTION Relaiviy has beome an imporan aspe of modern preise imekeeping sysems. Thus, far from being simply a exbook problem or merely of heoreial sienifi ineres, he analysis of relaivisi effes on ime measuremen is an essenial praial onsideraion. The Global Posiioning Sysem (GPS) is an example of an engineering sysem in whih he reogniion of appropriae relaivisi orreions are neessary for is suessful operaion. Cloks onboard GPS saellies run fas by 38 μs/d due o heir aliude and veloiy and have a periodi omponen wih ampliude of abou 46 ns due o he small orbi eenriiy ompared o loks on he geoid. The onsan drif is ompensaed by a rae offse prior o launh. Negle of he periodi effe would resul in a radial posiion error of abou 5 m. Similarly, relaivisi ransformaions beween loks hroughou he solar sysem will be required in fuure spae missions. The purpose of his paper is o desribe he fundamenal heoreial priniples for relaivisi ime ransfer. The prinipal relaivisi effes are derived for ime ransfer beween loks on he Earh s surfae o loks on Earh-orbiing saellies,, and he Moon. 35

2 PROPER TIME AND COORDINATE TIME In he heory of general relaiviy, here are wo kinds of ime. Proper ime is he aual reading of a lok. The proper imes are differen for loks in differen saes of moion and in differen graviaional poenials. The proper ime measured by a lok may be ompared o he proper ime measured by anoher lok hrough he inermediae variable alled oordinae ime, whih, by definiion, has he same value everywhere for a given even. The relaionship beween oordinae ime and proper ime depends on he veloiy of he lok and he graviaional poenial a he loaion of he lok. I is esablished hrough he invariane of he four-dimensional spae-ime inerval. The heory of spae, ime, and graviaion aording o he general heory of relaiviy is founded upon he noion of an invarian Riemannian spae-ime inerval of he form μ ν j i j = μν = + j + ij μ= ν= j= i= j= ds g dx dx g d g d dx g dx dx where x α (, x i ). The fundamenal mahemaial obje is he meri ensor g μν, whose omponens are funions of he oordinaes and are symmeri in he indies μ, ν (ha is, g μν = g νμ ). The meri ensor plays he role of he graviaional poenials. For a ranspored lok, he spae-ime inerval is 3 3 = μν μ ν μ= ν= ds g dx dx d where is he proper ime reorded by he lok. For a given oordinae sysem, his equaion esablishes a well-defined ransformaion beween oordinae ime and proper ime. The oordinae ime is arbirary, as he omparison is made beween wo proper imes. In an inerial oordinae sysem wih no graviaion, he meri omponens are g =, g = g = g 33 =, g μν = for μ ν. Then ds = ( v / ) d = d, where v is he lok veloiy as in speial relaiviy, whih implies he phenomenon of ime dilaion of a moving lok relaive o a saionary lok. For an eleromagnei signal, he spae-ime inerval saisfies he ondiion 3 3 μ ν ds gμν dx dx μ= ν= = = In an inerial oordinae sysem wih no graviaion, his redues o ds = d + dx + dy + dz = as in speial relaiviy, whose invariane implies ha he speed of ligh is in all inerial sysems. EARTH-ORBITING SATELLITE CLOCKS To a suffiien approximaion in he analysis of lok ranspor, he omponens of he meri ensor in an Earh-Cenered Inerial (ECI) oordinae sysem are g U /, g j =, and g i j δ i j, where U is he Newonian graviaional poenial and δ i j is he Kroneker dela. For a lok onboard a saellie, he elapsed oordinae ime is given in erms of he proper ime by he inegral U v d Δ =

3 The firs erm under he inegral is he elapsed proper ime, he seond erm is he orreion due o he graviaional poenial U (graviaional redshif), and he hird erm is he orreion due o he veloiy v of he saellie (ime dilaion). In he roaing Earh-Cenered Earh-Fixed (ECEF) oordinae sysem, i is onvenien o apply a hange of sale o define a new oordinae ime W ( U W ) v d Δ = Δ = + + where W = m /s is he geopoenial over he surfae of he Earh, whih is a onsan. Then he oordinae ime Δ orresponds o he proper ime regisered by a lok a res on he geoid. Thus, he lok beomes a oordinae lok. Upon inegraion, he elapsed oordinae ime for an Earh-orbiing saellie lok is 3GM Δ = + W Δ + GM a esine a where a is he orbial semimajor axis, e is he orbial eenriiy, and E is he eenri anomaly. The firs erm represens a onsan rae offse beween he saellie lok and a lok on he geoid. The seond erm is a small relaivisi periodi orreion due o he orbial eenriiy. This erm may be expressed wihou approximaion as Δ rel = GM a esin E = r v where r and v are he posiion and veloiy veors of he saellie. As r v is a salar, i may be evaluaed in eiher he ECI or ECEF oordinae sysem. ELECTROMAGNETIC SIGNALS The oordinae ime of propagaion of an eleromagnei signal is 3 ρ g j i Δ = + dx j= g pah where ρ is he propagaion pah lengh. The inegral erm is alled he Sagna effe. In he roaing ECEF oordinae sysem, he meri omponens are g, g j = (ω r) j /, and g i j δ i j, where ω is he Earh s roaional angular veloiy. Therefore, he Sagna effe beween wo poins A and B is B Sagna ( ) B ( ) B ω A Δ d d d A A A ω r r = = = ω r r ω A where A is he perpendiular projeion of he area formed by he ener of roaion and he endpoins of he ligh pah. For endpoins a (x A, y A ) and (x B, y B ), he Sagna effe may be expressed ω A ω Δ Sagna = = ( xa yb ya xb). In he ase of a reeiver a res on he Earh, an observer in he ECEF frame regards he reeiver as saionary and applies he Sagna orreion. However, an observer in he ECI frame sees ha he reeiver has moved due o he Earh s roaion during he signal ime of fligh and insead applies a 37

4 propagaion ime orreion due o he addiional pah lengh. The erm Sagna effe is par of he voabulary of only he observer in he roaing referene frame. The orresponding orreion applied by he inerial observer migh be alled a veloiy orreion. While he inerpreaion of he orreion is differen in he wo frames, he numerial value is he same in eiher frame. RELATIVISTIC EFFECTS ON SATELLITE CLOCKS AND SIGNALS THE GLOBAL POSITIONING SYSTEM The GPS has served as a laboraory for doing relaiviy physis. The onsisen appliaion of relaiviy o ime and posiion measuremens has been demonsraed by he operaional preision of he sysem and by numerous experimens designed o es he individual effes over a wide range of ondiions. The GPS provides a model for he appliaion of relaiviy algorihms o similar appliaions aross a broad sperum of imekeeping sysems. The relaivisi effes enounered in he GPS illusrae ha he effes ha mus be onsidered are no negligible. The saellies ransmi one-way pseudorandom noise (PRN) oded signals ha are used for range and ime measuremens. The signals are referened o onboard aomi loks. For measuremens wih a preision a he -o- nanoseond level, here are hree relaivisi effes ha mus be onsidered. Firs, here is he effe of ime dilaion. The veloiy of a moving lok auses i o appear o run slow relaive o a lok on he Earh. GPS saellies revolve around he Earh wih an orbial period of.967 hours and a veloiy of km/s. Thus, on aoun of is veloiy, a GPS saellie lok appears o run slow by 7 μs per day. Seond, here is he effe of he graviaional redshif, a frequeny shif aused by he differene in graviaional poenial. (The erm redshif is generi, regardless of sign, bu for a saellie lok he frequeny shif is aually a blueshif. ) The differene in graviaional poenial beween he aliude of he orbi and he surfae of he Earh auses he saellie lok o run fas. A an aliude of,84 km, he lok runs fas by 45 μs per day. The ne effe of ime dilaion and graviaional redshif is ha he saellie lok runs fas by approximaely 38 μs per day when ompared o a similar lok a res on he geoid, inluding he effes of he Earh s roaion and he graviaional poenial a he Earh s surfae. This is an enormous rae differene for a lok ha mainains ime wih a preision of a few nanoseonds over a day. To ompensae for his large seular effe, he lok is given a fraional rae offse prior o launh of from is nominal frequeny of exaly.3 MHz, so ha when in orbi is average rae is he same as he rae of a lok on he ground. The aual frequeny of he saellie lok prior o launh is hus MHz. Alhough GPS saellie orbis are nominally irular, here is always some residual eenriiy. The eenriiy auses he orbi o be slighly ellipial. Thus, he veloiy and graviaional poenial vary slighly over one revoluion and, alhough he prinipal seular effe is ompensaed by a rae offse, here remains a small residual variaion ha is proporional o he eenriiy. For example, wih an orbial eenriiy of., here is a relaivisi sinusoidal variaion in he apparen lok ime having an ampliude of 46 ns a he orbial period. This orreion mus be alulaed and aken ino aoun in he user s reeiver. The hird relaivisi effe is he Sagna effe. For a saionary erresrial reeiver on he geoid, he Sagna orreion an be as large as 33 ns (orresponding o a GPS signal propagaion ime of 86 ms and a veloiy of 465 m/s a he equaor in he ECI frame). This orreion is also applied in he reeiver. Higher-order effes no presenly modeled in he GPS inlude he Earh oblaeness onribuion o he graviaional redshif, he idal poenials of he Moon and Sun, and he effe of he graviaional poenial 38

5 on he speed of signal propagaion. When saellie ross links are implemened in he fuure, he orbial eenriiy effe will have o be aken ino aoun a boh he ransmier and reeiver. OTHER SATELLITES IN EARTH ORBIT To illusrae heir orders of magniude, he relaivisi effes on loks and signal propagaion for a variey of Earh orbiing saellies are ompared in Table. Consans Table. Relaivisi effes on loks and signals for saellies in Earh orbi. Veloiy of ligh m/s Graviaional onsan of Earh km 3 /s Radius of Earh km J oblaeness oeffiien.86 Angular veloiy of Earh roaion rad/s Geopoenial on geoid U m /s U / Saellie orbial properies Saellie ISS GLONASS GPS Galileo Molniya GEO Semimajor axis km Eenriiy Inlinaion deg Argumen of perigee deg 5 Apogee aliude km Perigee aliude km Asending node aliude km Period of revoluion s Mean moion rev/d Mean veloiy km/s Clok effes Seular ime dilaion μs/d Seular redshif μs/d Ne seular effe μs/d Ampliude of periodi effe due o eenriiy ns Seular oblaeness onribuion o redshif ns/d Ampliude of periodi effe due o oblaeness ps Ampliude of periodi idal effe of Moon ps Ampliude of periodi idal effe of Sun ps Signal propagaion Maximum Sagna effe ns Graviaional propagaion delay along radius ps Ampliude of periodi fraional Doppler shif

6 RELATIVISTIC TRANSFORMATION FROM MARS TO EARTH For loks in ommuniaion and navigaion sysems used for spae exploraion, analogous orreions are required. Thus, appropriae relaivisi ransformaions are neessary in ransferring ime from one frame of referene o anoher, for example beween a lok on o a lok on Earh. Referene daa used in hese alulaions are summarized in Table. Table. Referene daa for he Sun, Earh, and. Mass Sun kg Earh kg kg Planeary radius Earh 6378 km 3397 km Orbial semimajor axis Earh. AU = km.54 AU =.79 8 km Orbial period Earh d d Average orbial veloiy Earh 9.8 km/s 4. km/s Orbial eenriiy Earh Speed of ligh 99,79,458 m/s (exa) Graviaional onsan m 3 / kg s The analysis of ime ransfer mus be arried ou in a ommon oordinae sysem. A onvenien oordinae sysem is one whose origin is a he solar sysem baryener. The orresponding oordinae ime is alled Baryenri Coordinae Time (TCB). For ime ransfer beween and Earh, wo ransformaions are required. The firs ransformaion is from Terresrial Time (TT) o Baryenri Coordinae Time (TCB). The seond is from Baryenri Coordinae Time (TCB) o Time (MT). The oordinae ime TCB is an inermediae variable ha ulimaely anels ou. These ransformaions are illusraed shemaially in Figure. BARYCENTRIC COORDINATE TIME - TERRESTRIAL TIME The elapsed oordinae ime Δ in a baryenri oordinae sysem orresponding o he proper ime Δ mainained by a lok, having an arbirary posiion and veloiy in his oordinae sysem, is 3

7 U ( ) v r d Δ = + + where r and v are he baryenri posiion and veloiy of he lok and U(r) is he graviaional poenial of all he bodies in he solar sysem (inluding he Earh) evaluaed a he lok. The inegral depends on he posiion and veloiy of he lok in he baryenri oordinae sysem. The oordinae ime Δ is idenified wih TCB. Proper ime as measured by loks on surfae Proper ime as measured by lok on spaeraf Spaeraf Baryenri Coordinae Time (TCB) Earh GPS Saellie Proper ime as measured by lok on GPS saellie Sun Proper ime as measured by loks on Earh s surfae Terresrial Time (TT) Inernaional Aomi Time (TAI) Coordinaed Universal Time (UTC) GPS Time Figure. Relaivisi ime ransformaions. I is desirable o separae he lok dependen par from he lok independen par. In his approximaion, one may express r and v as r = re + R and v = ve + R &, where r E and v E are he baryenri posiion and veloiy of he Earh s ener of mass, and R and R & are he geoenri posiion and veloiy of he lok, as illusraed in Figure. The oal poenial a posiion r is U(r) = U E (r) + U ex (r) where U E is he Newonian poenial of he Earh and U ex is he exernal Newonian poenial of all of he solar sysem bodies apar from he Earh. The exernal poenial may be expressed Also, U () r U ( r ) + U R ex ex E ex 3

8 and where v = v E + R& v E + R& d R& v = ( R v ) R a d E E E a dv d E E = U ex is he Earh s aeleraion in he baryenri oordinae sysem. Subsiuing hese expressions ino he inegral, one obains d + [ UE( r) + Uex( re) + Uex R] + v E + ( E) E + d R v R a R & d. R Clok h Earh R E r r E Baryener Figure. Geomery of lok, Earh, and solar sysem baryener. The graviaional aeleraion erms Uex R and Ra E anel ou. Thus, as U E (r) = U E (R) (hey refer o he same poin), he elapsed oordinae ime is ex ( re) E E( R) R& R v E Δ Δ + U + v d + U + d + 3

9 This equaion is ompleely general, regardless of he posiion of he lok. The firs erm is he proper ime measured by he lok. The seond erm is due o he ombined redshif and ime dilaion effes a he geoener wih respe o he baryener and is independen of he lok. The hird erm is he ime differene beween a lok a he geoener and a lok a posiion R wih respe o he geoener. The fourh erm depends on he lok s veloiy and posiion. In he limi of fla spae-ime, i represens he speial relaiviy lok synhronizaion orreion in he moving geoenri frame when observed from he baryenri frame. The anellaion of he wo aeleraion erms is a manifesaion of he Priniple of Equivalene for a freely falling frame of referene. Tha is, he Earh onsiues a freely falling frame in is orbi abou he Sun. The oordinae ime sale of Terresrial Time (TT) is equivalen o he proper ime kep by a hypoheial lok on he geoid. This imesale is relaed o Inernaional Aomi Time (TAI) by he equaion The onsan offse represens he differene beween Ephemeris Time (an obsolee Newonian imesale used for asronomial ephemerides whih has been superseded by TT) and TAI a he defining epoh of TAI on January 958. For an aual lok a res a an elevaion h above he geoid where he loal aeleraion of graviy is g, he relaion beween TT and he proper ime reading Δ of he lok is =Δ = gh Δ TT ( / ) The ransformaion from TT o Geoenri Coordinae Time (TCG) is where W E is he Earh s geopoenial, L G W E / = μs/d, and ΔT is he ime elapsed sine January 977 h TAI (JD ). In he general equaion above, he firs inegral may be alulaed by numerial inegraion or i may be represened by an analyial formula. I is expressed as he sum of a seular erm L C ΔT and periodi erms P. For a lok on he geoid he seond inegral is simply W E Δ. Thus for a lok on he geoid TT = TAI s TCG TT ( E / ) = W Δ T = L ΔT TCB = TT + Uex + v d+ L Δ T + = TT + LC Δ T + P+ LG Δ T + R E ve ( re) E G RE ve G where L C = ms/d. The diurnal erm has a maximum ampliude of. μs (for a lok on he equaor). The leading erms in he evaluaion of he inegral are GM S UE ex ( re ) + ve d Δ T + GM sin S ae ee EE E 3 a where GM S is he graviaional onsan of he Sun, and where a E and e E are he Earh s orbial semimajor axis and eenriiy. The firs erm is an approximaion o L C ΔT. The seond erm is he prinipal periodi erm in P, whih has ampliude of.7 ms. 33

10 BARYCENTRIC COORDINATE TIME - MARS TIME By an analogous derivaion, one finds ha he ransformaion from Baryenri Coordinae Time (TCB) o Time (MT) is given by TCB = MT + Uex + v d+ L Δ T + = MT + LCM Δ T + P+ LM Δ T + R M vm ( rm ) M M RM vm where L CM = ms/d, L M W M / =.43. μs/d, W M is he areopoenial ( geopoenial on ), P represens periodi erms, and R M is he areoenri posiion of he lok on he surfae of. The diurnal erm has a maximum ampliude of.9 μs. The leading erms in he inegral are GM S UM ex ( rm ) + vm d Δ T + GM sin S am em EM M 3 a where GM S is he graviaional onsan of he Sun, and where a M and e M are he orbial semimajor axis and eenriiy. The firs erm is an approximaion o L CM ΔT. The seond erm is he prinipal periodi erm in P, whih has ampliude of.4 ms. NET EFFECTS: MARS TIME - TERRESTRIAL TIME The resuls of hese alulaions are summarized in Table 3. The differene in he readings of a lok on he surfae of and a lok on he surfae of he Earh has boh seular and periodi erms. The differene beween Time (MT) and Terresrial Time (TT) is MT TT = (TCB TT) (TCB MT) The ne seular drif is (.8 ms/d +.6 ms/d) (.84 ms/d +. ms/d) =.49 ms/d. The ampliudes of he periodi variaions are: (a).7 ms a he Earh orbial period (365.4 d); (b).4 ms a he orbial period (687 d). Therefore, in he ransfer of ime beween a lok on and a lok on he Earh, here are boh seular and periodi effes ha are on he order of o milliseonds. For a navigaion ranging sysem referened o a lok on and an ephemeris referened o loks on Earh, he radial posiion error ould be as muh as 3, km if he relaivisi effes were no modeled. 34

11 Table 3. Prinipal relaiviy effes for ime ransfer beween and Earh. Geoid o Geoener Seular drif Maximum ampliude of diurnal erm Geoener o Baryener Seular drif Ampliude of prinipal periodi erm surfae o ener Seular drif Maximum ampliude of diurnal erm ener o Baryener Seular drif Ampliude of prinipal periodi erm 6. μs/d. μs.8 ms/d.7 ms. μs/d.9 μs.84 ms/d.4 ms RELATIVISTIC TRANSFORMATION FROM THE MOON TO EARTH For ime ransfer from he surfae of he Moon o he surfae of he Earh, he proedure is similar, bu he relaive magniudes of he erms are differen. A onvenien oordinae sysem is one whose origin is a he ener of he Earh. (The moion of he Earh s ener abou he ener of mass of he Earh-Moon sysem will be negleed.) As above, he differene beween Geoenri Coordinae Time and Terresrial Time is TCG TT = LG ΔT where L G = 6. μs/d. Bu in addiion, TCG is relaed o Lunar Time (LT), he proper ime measured by loks on he Moon s surfae, by he equaion TCG = LT + L Δ T + P+ L Δ T + Cm m R m v m where he subsrip m refers o he Moon and L Cm 3 GM E a, P GM sin E am em Em, L m m GM. m Rm Referene daa for he Moon are summarized in Table 4. 35

12 Table 4. Referene daa for he Moon. Mass kg Radius 738. km Orbial semimajor axis 384,4 km Orbial eenriiy.549 Average orbial veloiy.3 km/s Disane of geoener from baryener 467 km Thus L Cm =.73 =.5 μs/d, ( / ) GME am e m =.48 μs, and L m = 3.4 =.7 μs/d. The differene beween Lunar Time (LT) and Terresrial Time (TT) is LT TT = (TCG TT) (TCG LT) The ne seular drif rae is 6. μs/d (.5 μs/d +.7 μs/d) = 56. μs/d and he ampliude of he periodi effe is.48 μs a he Moon s orbial period (7.3 d). TIME EPHEMERIS AND THE PLANETARY EPHEMERIDES Anoher oordinae ime of ineres is Time Ephemeris eph, whih is a solar sysem baryener oordinae ime ha has been resaled o have he same seular rae as TT. This relaivisi oordinae ime is used in he planeary and saellie ephemerides published by he Je Propulsion Laboraory. (Alhough i has a similar name, i is no relaed o he Newonian sale of Ephemeris Time.) This definiion of ime hanges he seular raes of oordinae ime by subraing he quaniy L G + L C =.55 8 =.34 ms/day from he given seular raes. Using he reen DE4 ephemeris published by JPL, numerial resuls of he differene beween proper ime and Time Ephemeris for a lander is shown in Figure 3. The simulaions are for 6 years and begin on he J epoh of January. The resuls learly show a large.4 ms periodi erm for a lok on. The resuls also show ha relaivisi ime alulaions a mus aoun for he seular rae differene beween in-siu loks and Earh loks. 36

13 Figure 3. Seular and periodi relaivisi effes, proper ime oordinae ime (seonds), for a lander using Time Ephemeris as he oordinae ime. CONCLUSION Transformaions beween loks operaing on he Earh and loks a or on he Moon will beome an essenial aiviy for fuure spae missions. The analysis of his paper has shown ha he relaivisi effes a are omprised of a seular rae differene of abou.49 ms/d and periodi variaions wih ampliudes of.7 ms and.4 ms relaive o Earh-based loks. The seular effe for he Moon is abou an order of magniude less. Aurae ime ransfer in he solar sysem for ommuniaions and navigaion sysems requires he onsideraion of hese relaivisi effes. ACKNOWLEDGMENT This work was arried ou a Saellie Engineering Researh Corporaion and a he Je Propulsion Laboraory, California Insiue of Tehnology, under onras wih he Naional Aeronauis and Spae Adminisraion. 37