Explicit formulae for the two way time-transfer in the T2L2 experiment including the J2 contribution to the Earth potential in a relativistic framework
Olivier Minazzoli, Bertrand Chauvineau, Etienne Samain, Pierre Exertier, Patrick Vrancken, Philippe Guillemot
EExplicit formulae for the two way time-transfer in the T2L2 experiment including the J contribution to the Earth potential in a relativistic framework O. Minazzoli* ∗ , B. Chauvineau* , E. Samain † , P. Exertier † , P. Vrancken † and P. Guillemot ‡ * UNS, OCA-ARTEMIS UMR 6162, Observatoire de la Cte d’Azur, Avenue Copernic, 06130 Grasse, France † UNS, OCA-Geosciences Azur UMR 6526, Observatoire de la Cte d’Azur, Avenue Copernic, 06130 Grasse, France ‡ CNES, Avenue Edouard Belin, 31401 Toulouse cedex 9, France
The topic of this paper is to study the two way time-transfer problem between a ground basedstation and a low orbit Earth’s satellite, in the aim of an application to the T2L2 experiment.The sudy is driven in a fully relativistic framework. Because of the rapid increase in clock’s preci-sion/measurements, the first term beyond the Earth’s potential monopolar term is explicitly takeninto account. Explicit formulae, for both the distance and offset problems (definitions in the text)are proposed for the relevant applications.
Keywords:
I. INTRODUCTION
The idea of optical space-based time transfer had firstbeen proposed to ESA in the seventies through theproject which was named LASSO [1] (Laser Synchroniza-tion from stationary Orbit). Even though it was quitedifficult to get some data with LASSO, the mission wasa great success with some time transfers obtained withina time stability in the range of 100 ps. In 1994 OCAproposed to build a new generation of optical transferbased on some new technology: it was the beginning ofthe T2L2 project. As compared to LASSO, the objec-tive was to improve the performances by at least 2 or-ders of magnitude. After several proposals on the MirSpace Station, ISS, GIOVE, and Myriade, T2L2 was fi-nally accepted in 2005 as a passenger instrument on theAltimetry Jason-2 satellite ([2],[3]).The full exploitation of the T2L2 experiment requires arelativistic description of the Earth gravitational field forboth the definition of the proper timescales relevant forthe (ground-based and on board) clocks and the photons’trajectories involved by laser links.The aim of the current paper is to propose explicituseful formulae dealing with two-way experiments, likeT2L2. We derive explicit formulae giving the ratio ν + G /ν − G of the ground-based frequencies at reception and emissionrespectively as a function of the required quantities (po-sitions, velocities,...) at a ground-based time. We alsogive explicit expressions for the time dependance of (1)the ground proper time needed for a photon to go to thesatellite and to come back and (2) the offset between theon board and the ground based clocks. All the quantitiesare explicitely written as a function of the time attachedto a unique event (emission or reception). To anticipatethe performance increase of future clocks, these formulaetake the J term in the Earth gravitational field effect onthe photons’ orbits into account.Blanchet et al [4] derive explicit formulae in both the ∗ email : [email protected] one-way and the two-way problems. However, in the two-way case, the ratio ν + G /ν − G of the ground-based frequen-cies at reception and emission respectively and other use-ful expressions are not explicitly given as a function ofthe required quantities at a unique time. The current pa-per proposes such formulae, which can be directly used inpractical applications. Besides, the photon’s orbit is de-termined in [4] using the monopolar Earth gravitationalterm only. (A more complete gravitational modelizationis used in [4], but for the link between proper and coor-dinate timescales only.) In the present paper, we includethe J gravitational contribution to light propagation,which will be useful as soon as the 10 − s level of preci-sion is required, as it should be in the future ACES exper-iment [5], for instance. Using the Synge world functionapproach [6], Linet and Teyssandier [7] have obtainedexplicit expressions of laser link in the one-way case forgeneral axisymetric gravitational field, but no formulaein the two-way case is given. This problem has been re-considered by Le Poncin-Lafitte et al [8], but they giveexplicit formulae in the spherical case and for the one-wayproblem. Let us remark that, from a theoretical point ofview, the applicability of the Synge world function ap-proach can be limited to regions of spatial extension ofthe order of 2 R Sun ( R Sun /R Schw ) ∼ A.U. ∼ . pc in the worse case (light rays grazzing the Sun and consid-ering points at typically ∼ A.U. ). The last remarkis obviously not relevant for solar system applications,but is relevant for interstellar scales.
II. EXPERIMENTAL DESCRIPTIONA. Principe
The experiment principle is issued from satellite laserranging techniques. T2L2 permits to synchronize remoteground clocks and compare their frequency stabilitieswith a performance never reached before. T2L2 allowssynchronization of a ground and space clock and mea-surement of the stability of remote ground clocks overcontinental distances, itself having a time stability in the a r X i v : . [ a s t r o - ph . E P ] F e b FIG. 1: Ground to space time transfer with T2L2FIG. 2: Global synoptic of the T2L2 instrument on Jason-2together with the LRA and the DORIS USO range of 1 ps over 1000 s.The principle is based on light pulses propagation be-tween ground laser stations and a satellite equiped witha photo detection system and a time-tagging unit (fig.1). The space instrument uses a Laser Ranging Array(LRA) which is also used for some laser ranging pur-poses and the Doris ultra-stable oscillator (USO) funda-mentally used by the DORIS positioning system (fig. 2).The USO is linked to the time tagging unit. It is theonboard reference clock of the project.A laser station at ground emits short asynchronouslaser pulses towards the satellite. The LRA returns afraction of the received photons back to the station. Todescribe roughly the main idea of the principle of themeasurements, the station records the start ( t start ) andreturn ( t return ) times of each light pulse. The T2L2 pay-load records the arrival time ( t board ) in the temporal ref-erence frame of the onboard oscillator. These data areregularly downloaded every 2 hours to the ground via aregular microwave communication link.For a given light pulse emitted from ground, the timeoffset θ GS between ground clock G and space clock Sis deduced from the measurement triplets t start , t board , t return with the following time equation: θ GS = ( t start + t return ) / − t board + (cid:15), (1)where (cid:15) is a corrective term, including atmospheric effectsfor instance. The time transfer between several laser sta-tions at ground (A, B, C, ...) is then deduced from thedifference between each ground to space time transfer.However, a formulae like (1) is based on some inad-equate physical concepts from the start. Obviously, itinvolves only a rough notion of time, leaving in the darkthe fact that different proper times have to be consideredin the problem, as it is well known from the relativisticgravity theory. In fact, (1) is correct as long as the new-tonian description of (space-)time correctly describes thephysics. However, at the level of precision reached by ex-periments from years ago, it is well known it is no longerthe case. Of course, it is possible to put the so-called”relativistic corrections” into the (cid:15) corrective term in (1).However, it is better to reformulate the problem directlyin the relativistic framework, this way ensuring that allthe ”relativistic effects” will be consistently taken intoaccount at the considered order.Depending on the distance between the laser groundstations (roughly 6000 km for Jason2) T2L2 can be oper-ated in a common view mode or in a non-common viewmode. In common view configuration, with two laserranging stations A and B firing towards the satellite si-multaneously, the noise of the on-board oscillator has tobe considered over time interval equal to the delay be-tween consecutive laser pulses. In such a way, the noisecoming from the onboard oscillator can be considerednegligible in the global error budget. In a non-commonview mode, the temporal information is carried by thesatellite local oscillator over the distance separating thetwo ground stations visibility and the USOs noise be-comes an important part. B. T2L2 on JASON2
Jason-2 is a French-American follow-on mission toJason-1 and Topex/Poseidon. Conducted by NASA andCNES, its goal is to study the internal structure and dy-namics of ocean currents mainly by radar altimetry. Forthe needs of precise determination of the satellite orbit,three independent positioning systems are also included:a Doris transponder, a GPS receiver and the LRA (LaserRanging Array) retroreflector. The satellite was placed ina 1,336 km orbit with 66 o inclination by a Delta launcher.The time interval between two passes varies from 2 to 14hours with an average duration of about 1000 s. TheT2L2 instrumentation contains two main subsystems: ofa photo detection module and an event timer (fig. 3).The photo detection unit is made with 2 avalanche photodiodes. One is working in a special non-linear Geigermode for precise chronometry. The other works in lin-ear gain mode in order to trigger the whole detectionchain and to measure the received optical energy and the FIG. 3: Synoptic of the T2L2 space instrument continuous noise flux originating in the earth albedo. Tominimize the false detection rate, the detection thresholdmay be adjusted either by remote control or automati-cally as a function of the continuous flux measurement.Some elements of the photo detection unit are locatedoutside the main Jason-2 payload on a boom which sup-port also the LRA. These elements are oriented towardthe earth surface. The rest of the detection is, togetherwith event timer, inside an unique package located insidethe Jason-2 payload. All this equipment has a mass of10 kg and a power consumption of 42 W.
C. T2L2 Objectives
The objectives of the T2L2 experiment on Jason-2 arethreefold: • Validation of optical time transfer, including thevalidation of the experiment, its time stability andaccuracy. It should further allow to demonstrateone-way laser ranging • Scientific applications concerning time and fre-quency metrology allowing the calibration of ra-diofrequency time transfer (GPS and Two-Way),fundamental physics with the measurement of lightspeed anisotropy and alpha fine structure constant,Earth observation and very long baseline interfer-ometry (VLBI). • Characterization of the on-board Doris oscilla-tor, especially above the South Atlantic Anomaly(SAA).
D. First results
T2L2 relies on the international laser ranging networkthat permanently contributes to the International Earth Reference System (terrestrial frame and Earth’s rota-tion), and to the traking of several scientific space mis-sions. Since 1998, this network has been organised asan International Service (named ILRS), as it was thecase for the GPS and VLBI geodetic space techniques.Among 35 Satellite Laser Ranging (SLR) stations, 17of them provide full rate ranging data on Jason-2 thatare downloaded to the European Data Center (EDC inMuechen, Germany) regularly; the data files are up-loaded by the T2L2 Scientific Mission Center, locatedin Grasse, France. Among these stations, 10 of them usethe new data format that permits to extract the starttime of laser pulses at the ps level, whereas the othersuse the current format. This data format was fitted forlaser ranging only and use a maximum of 12 digits torecord the start epoch of the laser pulses (correspondingto 100 ns).Several steps are necessary to proceed together groundSLR and on board T2L2 data sets (the T2L2 data filesare uploaded from CNES, Toulouse to the Mission Centerevery day with a time delay of 1-2 days): • selection of data relatively to the SLR stations andsatellite passes, • extraction of measurement triplets from both datasets, • estimation of instrumental corrections, • determination of the time of light travel betweenthe SLR station and the T2L2 space instrument, • computation of the ground to space time transfer.
1. Data selection
The first step of the data treatment consists in deter-mining the absolute frequency offset and the phase delaybetween the space clock and the UTC time scale, alongthe time. The goal is to compute the on board date ofeach detected optical event (i.e., arrival time of emittedlaser pulses) in a time scale that is identical to UTC atthe picoseconde level. Because the second step of thedata treatment (Triplets extraction), is based on a directcomparison of the dates of T2L2 data and ground SLRdata.The T2L2 space instrument being connected to theGPS of the Jason-2 plateform, we use Pulse Per Second(PPS) that are permanently emitted by this system at ± µ sec and then recorded by T2L2 in the space clocktime scale. That permits to establish a phase link andan absolute frequency offset between the space clock andthe UTC-GPS time scale, respectively to 0.3 µ sec and to2-3.10 − Hz (over 5000 sec).
2. Triplets extraction
The second step of the data treatment consists in rec-ognizing from all the events recorded by T2L2 those cor-responding to emitted laser pulses. Because the eventscoming from the SLR stations are blended together, itis necessary to find from which station a given event iscoming up. On the other hand, due to the combination ofseveral parameters such as (i) the low energy level used bycurrent SLR systems (around 50-100 mJ and even lower),(ii) the energy dispersion due to atmospheric effects, (iii)and the aperture size of the T2L2 optical module rela-tively to the diameter of the laser beam, all the emittedlaser pulses don’t have a corresponding on board event.As a statistical result, T2L2 is detecting more events fromSLR stations that use a higher energy level for their laserpulses (there is a factor 2 to 3 between the stations).
3. Instrumental corrections
Several instrumental corrections must be estimatedboth for the ground and space segments. At ground, theaccuracy of the range is obtained by a calibration pro-cess which uses a calibration target located at a knownposition by a classical survey at the mm level. At thesatellite, several parameters have to be taken into ac-count as (i) the geometrical delay between the referencepoints of the LRA and the T2L2 optical module, (ii) thetime walk compensation of the photo detection modulewhich is sensitive to the number of received photons, (iii)the event timer calibration based on an internal measure-ment process.All these corrections are computed at the ps level,using the available mechanical, optical and electronicalmeasurements that were determined for all Jason-2 sci-entific equipments before its launch. In addition, thestellar sensors of the satellite are providing the attitudeinformations needed to compute the geometrical effects.
4. Ground to space time transfer first results
The ground to space time transfers that have been se-lected use a set of SLR data from very performant groundstations that, in addition, are equipped by a HydrogenMaser or a Cesium clock as a permanent system of time.Thus, the present analysis takes into account data setsfrom Wettzell (Germany), Matera (Italy), Changchun(China), and Grasse (France) stations.Nevertheless, if all these selected SLR stations providedata with an optimal precision (range and start times),a noise is introduced in the time transfer by the time in-tervals measured at ground. It mainly comes from thequality of the laser beam (i.e., the pulse width), theground detection module (a photo detector), the opti-cal behaviour of the on board LRA, and the atmosphereeffects (two times); the current budget error is of 20 to 35 ps depending of each station. By taking into accountthe cinematics between the satellite and the ground sta-tion during a satellite pass, it is possible to compute asynthetic time of flight by averaging the row data locallyover an integration duration of a few tens of seconds.The noise is thus decreased by a factor 1 / √ N where N is the number of row data. If that computation is doneproperly from a data set to another it permits also toreduce the slight but significant differences between thelaser stations.The noise of the time of light travel being reduced to4-5 ps inside the time transfer, the time stability canbe represented by a computed time variance betweenroughly 1 s and 600 s. As an example, the time sta-bility of the Wettzell’s Hydrogen Maser compared to theT2L2’s DORIS quartz oscillator is 40 ps at 1 s and 7 psat 30 s. For time integration greater than 30 s this mea-surement is limited by the DORIS time stability whichis 5 ps at 30 s and 10 ps at 100 s.
5. Improvements
Now, we can routinely compute time transfers betweendifferent ground clocks and the space DORIS oscillator.Nevertheless, some work is still required to improve theinstrumental model (physical behaviour of the hardware)to the level of 1-2 ps at 30 s. As examples of this model,(1) the T2L2 photo detection module is sensitive to thenumber of incoming photons that should be properly cor-rected, and (2) there is still a data filtering to develop inorder to avoid Solar events that are detected by T2L2just 1-3 ns before the real pulse event.But the current estimation of the T2L2 performancesseem to be in a good accordance with the previous speci-fications of the overall project thus justifying the presentrelativistics equations to be applied.
III. THE SPACE-TIME MODEL USED FORTIME TRANSFER
In GR, photons follow null geodesics of the space-timemetric g µν . We write the metric under the form ds = g c dt + 2 g i cdtdx i + g ij dx i dx j , (2)where ct = x . In the IAU2000 recommendations, termsin c − are neglected in g ij , while they are taken into ac-count in both g and g i (with no contribution in g i )[9]. Since we are interested in light motion, dx ∼ cdt , con-sidering c − terms in g (and g i ) would be meaningless.Hence, the metric to be considered writes ds = (cid:18) − wc (cid:19) c dt − w i c cdtdx i + (cid:18) wc (cid:19) δ ij dx i dx j (3)where [9] w (cid:0) t, x k (cid:1) = G (cid:90) σ (cid:0) t, y k (cid:1) (cid:107) y − x (cid:107) d y (4) w i (cid:0) t, x k (cid:1) = G (cid:90) σ i (cid:0) t, y k (cid:1) (cid:107) y − x (cid:107) d yσ and σ i being the gravitational mass and mass currentrespectively. (cid:107) u (cid:107) is a short notation for √ u k u k . Hence, w corresponds to the potential in Newtonian gravity. Letus write w under the form w = w E + w ext (5) w E = w E,m + w E,J + w E,up where w E is the Earth potential, written as the sum ofthe monopolar term w E,m , the oblateness term w E,J ,and the other terms w E,up , when developed in sphericalharmonics, and w ext the external potentials, caused bythe presence of the Moon and the Sun essentially. Theorders of magnitude of these different terms are given by w E,m c ∼ GM E rc ∼ − w E,J c ∼ J GM E rc R E r ∼ − w E,m ∼ − w E,up c ∼ − w E,J ∼ − w ext c ∼ x i x j ∂ i ∂ j U ∼ GMLc r L = GM E rc MM E (cid:16) rL (cid:17) where M E and R E are Earth’s mass and radius, and U the Newtonian potential for a remote body of mass M ,at distance L . It results that, for satellites at low alti-tude (say (cid:46) km ), the term w E,m induces correctionswhich can reach ∼ − s for the time transfer. The ef-fect of the J term can then be of the order of 10 − s .The following terms in the Earth potential developmentresults into corrections of order 10 − s . For both theSun and the Moon, w ext is of order 10 − .The time-space term in the metric (dragging effect) isof order 8 w i c ∼ Gc (cid:90) ρV i (cid:0) t, y k (cid:1) (cid:107) y − x (cid:107) d y (6)where ρ is the Earth’s density, and V the field of mattervelocity inside the Earth. Assuming a spherical rigidlyrotating Earth, with uniform density and angular veloc-ity ω , the expression (6) leads to, for a satellite locatedin the equatorial plane at altitude h (cid:13)(cid:13) w i (cid:13)(cid:13) c = 85 GM E R E c ωR E c (cid:18) R E R E + h (cid:19) ∼ . − (7)for a low altitude satellite. The resulting correction to thephoton flying time is then expected to be of the order of10 − s .In a foreseeable future, time transfer precision of order10 − s is expected. Hence, for low altitude satelites, onehas to consider the J term in Earth potential while allthe other terms may be discarded. IV. APPLICATIONA. The photon orbit
From the previous section, we will consider in this sec-tion the time transfer problem in a space-time describedby the metric ds = (cid:18) − wc (cid:19) c dt + (cid:18) γ wc (cid:19) δ ij dx i dx j (8)where γ is a PPN parameter including possible extensionsof GR, like scalar-tensor (ST) theories [10]. The potential w is given by w = − w + J ∼ w (9)with − w = GM E r and ∼ w = GM E R r (cid:18) − z r (cid:19) (10) R being a parameter of the order of the Earth radius.Since we are interested in photon propagation, one hasto solve the geodesic equation ddλ (cid:0) g αβ k β (cid:1) = 12 k µ k ν ∂ α g µν with k α = dx α dλ (11)( λ being an affine parameter). The wave quadri-vector k satisfies the null condition g αβ k α k β = 0 (12)expressing the motion occurs on light cones. The waychosen to solve equations (11) and (12), up to first orderin w , is the way chosen in [11], in which only the − w -termwas considered in the potential. The aim is to derivethe trajectory of the photon x i (cid:0) x (cid:1) , obtained after theelimination of the affine parameter. (This eliminationcan be made both after the integration of the equationsor at the level of the differential equation (11) itself.)Up to O ( c − ) terms, the result reads x iph (cid:0) t, n k , x kA ( t e ) (cid:1) = x i + n i ct + (1) x iph (cid:0) t, n k , x kA ( t e ) (cid:1) (13)where, using the short notation m = GM E /c x iph (cid:0) t, n k , x kA ( t e ) (cid:1) = 1 + γ mn i ln r − Λ r + Λ − (1 + γ ) m ξ i K r + (1 + γ ) J mR Ψ i ( t ) + Z i withΨ i ( t ) = (cid:20) − ξ i + 2 δ i ξ K + n n ξ i + 2 n n i ξ K + 4 ξ ξ ξ i K (cid:21) r + (cid:20) − n i + 2 δ i n K − n n n i K + n i ξ ξ − n ξ ξ i K (cid:21) Λ r + (cid:20) ξ i + 2 δ i ξ K − n n ξ i + n n i ξ K − ξ ξ ξ i K (cid:21) r + (cid:20) − n n n i + n i ξ ξ − n ξ ξ i K (cid:21) Λ r + (cid:20) − n n i ξ + 12 n n ξ i − ξ ξ ξ i K (cid:21) r (14)where x i is the position at t = 0 (the integration con-stants Z i being adjusted this way [12]). The three num-bers n i are also integration constants satisfying the nor-malization condition n i n i = 1 . (15)In the first order terms, one hasΛ = ct + n i x i (16) K = x i x i − (cid:0) n i x i (cid:1) (17) r = (cid:112) K + Λ (18) ξ i = x i − n i n k x k . (19)Let us remark that ξ i ξ i = K and n i ξ i = 0. B. Time transfer and useful formulae
1. The time transfer
Let us now consider two photons emitted from the sta-tion A and received by the station B. Let dτ A and dτ B bethe proper time intervals between, respectively, the twoemissions and the two receptions, and dt A , dt B the corre-sponding coordinate time intervals. Let x iA , x iB be thepositions and v iA , v iB the coordinate velocities of A andB at the coordinate time corresponding to the emissionof the first photon by A.Since both A and B have velocities corresponding to or-bital free motions (for a satellite) or smaller (for a groundstation), the velocities of both are at best of the order of (cid:112) GM E /r , in such a way that the g i term in (3) con-tributes as a c − term in each station proper time. Theground station’s potential has to be modelized more pre-cisely than in (9) [4]. Hence we generally denote it by W and not by w which is its approximation (9) ( w is only suitable for a satellite and one can replace W by w inthis case in practical calculations). One has dτ A = dt A (cid:20) − W A c − v A c + O ( c − ) (cid:21) (20)and an analogous expression for dτ B . Since t B = t A + t ,where t is the time transfer (to be determined later), thisleads to dτ B dτ A = (cid:20) (cid:18) dtdt A (cid:19) (cid:21) × (cid:20) W A − W B c + v A − v B c + O ( c − ) (cid:21) . (21)The indexes 0 recall that the corresponding quantitieshave to be computed at the first photon emission time.The function t ( t e ) as a function of the emission time t e (= t A ) is implicitely obtained from the equation definingthe interception of the photon by the station B x iph (cid:0) t, n k , x kA ( t e ) (cid:1) = x iB (cid:0) t, x kB ( t e ) , v kB ( t e ) , ... (cid:1) (22)where one has explicitely written the dependence ofthe station B in initial (i.e. at the emission time) or-bital parameters. The function x iph (cid:0) t, n k , x kA ( t e ) (cid:1) is ex-plicitely given by (13), (14) up to first order. The func-tion x iB (cid:0) t, x kB ( t e ) , v kB ( t e ) , ... (cid:1) can be obtained from thegeodesic equation if the station B is a satellite while it isgiven by an Earth rotation model in the case of a groundbased station. Let us write these functions as x iph (cid:0) t, n k , x kA ( t e ) (cid:1) = x iA ( t e ) + n i ct + (1) x iph (cid:0) t, n k , x kA ( t e ) (cid:1) + (3 / x iph (cid:0) t, n k , x kA ( t e ) (cid:1) (23)and x iB (cid:0) t, x kB ( t e ) , v kB ( t e ) , ... (cid:1) = (24) x iB ( t e ) + tv iB ( t e ) + 12 t a iB ( t e ) + 16 t b iB ( t e )where (3 / x iph corresponds to the (unknown) 3 / g i term).The coefficients a iB ( t e ) and b iB ( t e ) correspond to the ac-celeration of the station B and its derivative at the emis-sion time. The four r.h.s. terms in (24) are respectivelyzeroth, first, second and third order terms in c − (since tv B = ct ( v B /c ), t a B = ( ct ) (cid:0) a B /c (cid:1) and so on).Solving (24) leads to ct = (cid:15) (0) θ + (1 / θ + (cid:15) (1) θ + O ( c − ) (25) n i = (cid:15) (0) η i + (1 / η i + (cid:15) (1) η i + O ( c − )where (cid:15) = ± and with (quantities at time t e ) (0) θ = d where d ≡ (cid:113)(cid:0) x iB − x iA (cid:1) (cid:0) x iB − x iA (cid:1) (0) η i = x iB − x iA d (1 / θ = d (0) η i v iB c (1 / η i = p ik v kB c where p ik = δ ik − (0) η i (0) η k (1) θ = 12 d (cid:18) δ kl + (0) η k (0) η l (cid:19) v kB v lB c + 12 d η k a kB c − (0) η k (1) x kph (cid:18) dc , (0) η m , x mA (cid:19) (1) η i = p ik (cid:20) d a kB c − d (1) x kph (cid:18) dc , (0) η m , x mA (cid:19)(cid:21) − (0) η i p kl v kB v lB c (cid:15) = + corresponds to a photon travelling from A to B( t A < t B ) while (cid:15) = − corresponds to a photon travellingfrom B to A ( t A > t B ).Considering now two emissions, at times t e and t e + dt e ,the transfer times are t and t + dt , while the ”directions”are n i and n i + dn i . Differentiating t b iB ( t e ) with respectto t e leads to a c − term, since B has at best a satellitelike velocity. Differentiating (22), (23), (24) and retainingonly terms up to c − leads to cn i − v iB − ta iB + ∂ (1) x iph ∂t − t b iB + ∂ (3 / x iph ∂t dt + v iA − v iB − ta iB − t b iB + v kA ∂ (1) x iph ∂x kA dt e + ctδ ik + ∂ (1) x iph ∂n k + ∂ (3 / x iph ∂n k dn k = 0 . (26)All the coefficients are taken at the emission time t e .Since both n i and n i + dn i are normalized, one has also n k dn k = 0 . (27)Solving the four equations (26), (27) gives the four quan-tities dt/dt e and dn k /dt e . In the time transfer problemwe are dealing with, only dt/dt e is required.Making the scalar product of (26) by n i , it turns out that, using (27) c + n i − v iB − ta iB + ∂ (1) x iph ∂t dt + n i v iA − v iB − ta iB − t b iB + v kA ∂ (1) x iph ∂x kA dt e + n i ∂ (1) x iph ∂n k + ∂ (3 / x iph ∂n k dn k = 0 (28)where the third order terms in the coefficient of dt havebeen discarded since there is no zeroth order term in thecoefficients of both dt e and dn k . In (28), dn k has to beknown up to the c − order only, since there is no zerothnor first order term in c − in its coefficient. This is givenby equation (26), written at the relevant order (cid:2) cn i − v iB (cid:3) dt + (cid:2) v iA − v iB (cid:3) dt e + ctdn i = 0 . (29)Inserting (29) into (28) to eliminate dn k leads to an equa-tion which can easily be solved in dt/dt e . It turns outthat (3 / x iph leads to terms of order c − or more. Hence,this term is not needed. Finally dtdt e = (1 / dtdt e + (1) dtdt e + (3 / dtdt e (30)with (1 / dtdt e = n i v iB − v iA c (31)(classical Doppler effect) (1) dtdt e = n i v iB − v iA c n k v kB c + n i ta iB c (32) (3 / dtdt e = n i v iB − v iA c (cid:18) n k v kB c (cid:19) + n i v iB − v iA c n k ta kB c + 12 n i t b iB c − n i v iB − v iA c n k c ∂ (1) x kph ∂t − n i v kA c ∂ (1) x iph ∂x kA + (cid:20) − n i ct v kB − v kA c + n i n k ct n l v lB − v lA c (cid:21) ∂ (1) x iph ∂n k . (33)This expression has now to be inserted in (21). However, v B and W B are considered at the reception time in (21)(where dt A = dt e ), while all the coefficients in (31)-(33)are considered at the emission time. It turns out v B ( t e + t ) = v B + 2 tv iB a iB + O ( c − ) (34) W B ( t e + t ) = W B + tv iB ∂ i W B + O ( c − ) (35)all the terms being considered at time t e when not pre-cised. One obtains finally (suppressing the zero indexes) dτ B dτ A = 1 + n i v iB − v iA c (36)+ v A − v B c + W A − W B c + n i v iB − v iA c n k v kB c + n i cta iB c + n i v iB − v iA c (cid:18) v A − v B c + W A − W B c (cid:19) − tv iB a iB + ∂ i W B c + (3 / dtdt e + O ( c − )where the dt/dt e terms of orders (1 /
2) and (1) are writtenexplicitely to exhibit first and second order terms, cor-responding to classical Doppler effect, special relativisticcontributions and Einstein’s gravitational effects. All the c − terms have been grouped on the third and fourth lineof equation (36). Remark that ∂ i W B in (35) and (36) cor-responds to a iB in the case of free fall motion (i.e. if B isa satellite).
2. Useful formulae
Let us consider measurements made from a ground-based station (G), emitting a photon towards a satellite(S). This photon is reflected at (S) and come back to (G).To each event (emission, reflexion, return) is associateda proper time measured on the involved clock. Let τ − G bethe proper time on (G) at emission, τ S the proper timeon (S) at reflexion, and τ + G the proper time on (G) whenthe photon comes back. Two problems may be tackled :- the ”distance problem”, involving τ + G − τ − G ;- the ”offset problem”, involving (cid:0) τ + G + τ − G (cid:1) / − τ S .Obviously, the (gravitational) theory doesn’t give thevalues of this last quantity, but allows to calculate dτ + G and dτ S as soon as dτ − G and the orbits of the ground-based and the onboard clocks are known. Reciprocally,the measurements of these quantities lead to constraintson both the orbits and the real (unperfect) clocks usedin the time-measurement process.Hence, let us define the ”distance problem” and ”offsetproblem” quantities (observables) by, respectively DP = ddτ − G (cid:0) τ + G − τ − G (cid:1) OP = ddτ − G (cid:18) τ + G + τ − G − τ S (cid:19) . For exhaustivity, let us point out the ratio of the groundemitted ν − G and the ground received ν + G frequencies issimply related to DP by ν − G ν + G = dτ + G dτ − G = 1 + DP.
Remark that since both (G) and (S) data are available,these quantities can be evaluated as functions of both τ − G , τ + G or τ S . However, in relevant applications, the (G)clock has a precision sensitively better than the (S) one.Hence, it is well-suited using the (G) clock as referencetime.
3. Explicit expressions of DP and OP Let us calculate DP and OP as functions of the in-volved quantities at the emission ( (cid:15) = − ) or reception( (cid:15) = +) time t (cid:15)G : DP ( t (cid:15)G ) = dτ + G dτ − G ( t (cid:15)G ) − OP ( t (cid:15)G ) = 12 + 12 dτ + G dτ − G ( t (cid:15)G ) − dτ S dτ − G ( t (cid:15)G ) . (38)In the relevant applications, the ground based station isthe Earth. It is legitimate to consider that, during thetime transfer, (1) the Earth is rigid and uniformly rotat-ing, (2) the Earth gravitational field only is acting (seediscussion in section III). Hence, one has, from (20) dτ + G dτ − G = dt + G dt − G + O (cid:0) c − (cid:1) . This shows the shift between the emitted and comingback frequencies of the photon doesn’t require an Earthmodel (for both potential and rotation). The interpre-tation is obvious, but this doesn’t explicitely appear informulae available in the Blanchet et al paper [4].Let us put X i = x iG − x iS (39) V i = v iG − v iS A i = a iG − a iS B i = b iG − b iS D = √ X i X i N i = X i /DP ik = δ ik − N i N k One finds, for the involved quantities dτ + G dτ − G ( t (cid:15)G ) = 1 + c − / D (cid:15) + c − D (cid:15) + c − / D (cid:15) dτ S dτ − G ( t (cid:15)G ) = 1 + c − / E (cid:15) + c − E (cid:15) + c − / E (cid:15) with (1 / D (cid:15) = 2 N i V i (1) D (cid:15) = 2 (1 + (cid:15) ) (cid:0) N i V i (cid:1) − (cid:15)V i V i − (cid:15)DN i A i (3 / D (cid:15) = N i V i (cid:104) v kG v kG + (cid:0) N k v kG (cid:1) (cid:105) +2 N i v iS P kl v kG V l − (cid:0) N i V i (cid:1) N k v kS + D (cid:2) v kG a kG + 2 N k v kG N l a lG + P ik V i a kG + 3 V i A i (cid:3) − (cid:2) (3 + 4 (cid:15) ) N i V i + 2 N i v iG (cid:3) (cid:2) DN i A i + P ik V i V k (cid:3) + D N i (cid:2) b iG + B i (cid:3) − D P ik V i c x kph − N i V i N k c ∂ (1) x kph c∂t − D N k P il V i c ∂ (1) x kph ∂n l − N k v lS c ∂ (1) x kph ∂x lS and (1 / E (cid:15) = N i V i (1) E (cid:15) = − (cid:18)
12 + (cid:15) (cid:19) V i V i + (1 + (cid:15) ) (cid:104)(cid:0) N i V i (cid:1) − DN i a iG (cid:105) + (cid:15)DN i a iS + W G − W S (3 / E (cid:15) = N i v iS V k v kS + 12 N i V i (cid:104) v kG v kG − (cid:0) N k v kS (cid:1) (cid:105) − (cid:15) ) N i V i P kl V k V l + D (cid:20) − V i a iS + (1 + (cid:15) ) V i (cid:0) a iG − a iS (cid:1)(cid:21) + D (cid:20) − N i V i N k a kS + N i v iS N k a kS (cid:21) − (cid:15) ) DN i V i N k A k + D (cid:20) − N i b iS + (1 + (cid:15) ) N i b iG (cid:21) + N i V i ( W G − W S ) + (cid:15)Dv iS ∂ i W S − D P ik V i c x kph − N i V i N k c ∂ (1) x kph c∂t − D N k P il V i c ∂ (1) x kph ∂n l − N k v lS c ∂ (1) x kph ∂x lS The quantities
D, N, v G , v S , a G , a S , b G , b S are taken atthe time t (cid:15)G , while the photon first order term (1) x ph and its derivatives have to be computed for t = + D/c , n = +∆ x/D and x S (at time t (cid:15)G ) for both (cid:15) = ± . Let usrecall that c x ph and ct are zeroth order quantities.From these expressions, it turns out that OP ( t (cid:15)G ) = c − OP (cid:15) + c − / OP (cid:15) with (1) OP (cid:15) = 12 V i V i + DN i a iG − W G + W S (3 / OP (cid:15) = − N i V i (cid:0) N k v kS (cid:1) − V i V i N k V k + (cid:0) N i V i (cid:1) + D (cid:20) v iG a iG − (cid:15)V i (cid:0) A i + a iG (cid:1) + 12 N i V i N k a kS (cid:21) − (cid:15)D N i b iG − N i V i ( W G − W S ) − (cid:15)Dv iS ∂ i W S . V. CONCLUSION AND DISCUSSION
In this paper, one has proposed formulae involvedin two way time transfer problems in a form directlyadapted for practical uses. All these quantities are ex-pressed as functions of quantities defined at a unique(emission or reception) ground-based time. Consideringforeseeable increase of clock’s accuracy, the formulae aregiven including the J contribution, but can be restrictedto the spherical case making J = 0, which is sufficientfor the time being applications.In this paper, the chosen unique time is a ground timebecause ground based clocks have generally precisionssensitively better than on board clocks. However, in thecase where one needs to make some time transfer betweentwo different ground based stations, distant in such a waythere is no spacecraft in common view, the link will in-volve an on board clock, with the resulting limitation inprecision. Let t and t be the times of the links betweenthe spacecraft and the first and second ground stations.Since the spacecraft motion is considered during the timeinterval [ t , t ], the crossed cdtdx i term in (2) is a c − order term. Hence, the onboard proper time formula re-duces to (cid:18) dτdt (cid:19) = 1 + 2 wc − v c + O ( c − )leading to the spacecraft proper time interval∆ τ ( t , t ) = t − t + 1 c (cid:90) t t (cid:18) w − v (cid:19) dt + O ( c − ) . Let us enforce the fact that, since the time interval [ t , t ]is finite, the orbit used in the previous r.h.s. member inte-gral cannot be developped around the emission time, andshould even include all the perturbations acting on thesatellite’s motion. On the other hand, a classical descrip-tion of this orbit is sufficient, since relativistic correctionsgenerate c − contributions to ∆ τ . Acknowledgments
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