Lunar Laser Ranging: Measurements, Analysis, and Contribution to the Reference Systems Technical

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ua ae agn measurements Ranging Laser Lunar acceleration tidal the 1 of improvement correction the the and ecliptic, longitude dynamical Paris in the constant the of precession 1998, orientation the Moon. the In of the as value of such than 1976 physics. 2002 IAU remarkable longer in fundamental contribution the brought interval this obtained to and In still has time results observations. geophysics technique is some LLR a of This emphasize dynamics, (LLR) analysis covers we performed system Ranging observations improved. has (POLAC) Laser solar regularly Center LLR Analysis is Lunar Lunar of science, Observatory measurements the set lunar of today, complete in accuracy and, The results the deployed and France. been years have and 30 reflectors USA more in 4 operating date this Since FranceAbstract. Paris, 75014, l’Observatoire, 8630/CNRS, de UMR Référence Av. SYstème - de 61 - (SYRTE) Temps-Espace Paris de Observatoire Francou G. and Chapront systems J. reference and the analysis, to measurements, contribution Ranging: Laser Lunar n2 uy16,tefis ua eetrwspae nteMo ufc yAol 1mission. 11 Apollo by surface Moon the on placed was reflector lunar first the 1969, July 21 On hc h inlhsbe ot h te eetr r tl operating still are reflectors from other 1 4 carried Lunakhod the which 1). Except lost, (Fig. (1973) been normally has 21 2. signal au- and Luna Soviet the 1 and two which and Lunakhod (1970) 15), American reflectors, 17 Apollo two French-built Luna and by 14 missions, placed (Apollo tomatic 1971 been in have missions the arrays Apollo to reflector sent more pulse Afterwards, laser a from returned Sea Observatory Moon. photons McDonald the later detecting in weeks array in few the a reflector succeeded and on first 1969 the reflector July down in Tranquillity a put these of and between Armstrong distance Neil Earth the the points. of measurement two on equivalent an transmitter is time a which travel Moon, between round-trip light the the determining in of consists telemetry laser Lunar eet nietlgtbc oispito rgn(i.2). (Fig. origin of cube point corner its each to 15). 11-cm straightforward: back Apollo of is light for arrays cubes incident reflector 300 corner reflects of and faced design 14 triangularly The and 14 edge. on have 11 mounted arrays Apollo cubes Lunakhod for corner The (100 diameter panel 3.8-cm aluminium of an consist arrays Apollo The i.1 itiuino h eetr ntelnrsurface lunar the on reflectors the of Distribution 1. Fig. Technical IERS Note 97 No. 34 98 No. 34 IERS Note Technical ua ae agn:maueet,aayi,adcnrbto oterfrnesystems reference the to contribution and analysis, measurements, Ranging: Laser Lunar i.4 EG ae agn tto Paeud aen Caus- Calern, de (Plateau station ranging France) Laser sols, CERGA 4. Fig. Texas) Davis, (Fort station ranging Laser McDonald 3. Fig. 14 Apollo by placed array reflector the of Photography 2. Fig. .Carn n .Francou G. and Chapront J. J. Chapront and G. Francou IERS Lunar Laser Ranging: measurements, analysis, and contribution to the reference systems Technical

Note No. 34

Since 1970 there were few LLR observing stations. The first one was McDonald Observatory (2.7-m telescope) near Fort Davis, Texas (USA). It was fully dedicated to lunar ranging and ceased operation in 1985 after maintaining routine activities for more than 15 years. The transition was made in the mid-1980s to the McDonald Laser Ranging Station (0.76-m telescope) on two sites (Saddle and Mt. Fowlkes): MLRS1 (1983-1988) and MLRS2 (since 1988) which share lunar and artificial satellite ranging facilities (Fig. 3). In the 1980s, two other stations have carried out Lunar Laser Rang- ing. The Haleakala Observatory on Maui, Hawaii (USA) produced high quality data over a few years around 1990. Since 1982, the CERGA station (Centre d’Etudes et de Recherche en Géodynamique et Astronomie) has been operating at the ’Observatoire de Côted’Azur’ (OCA) on the ’Plateau de Calern’ near Grasse (France), with a 1.54-m Cassegrain telescope which replaced its Rubis laser by a YAG in 1987 (Fig. 4). Occa- sionally some other artificial-satellite stations have performed success- fully LLR observations such as in Australia and in Germany, but all the data used for the analyses come from the 3 observatories: McDonald, Haleakala and CERGA. Today, just two observatories are still operating: McDonald (MLRS2) and CERGA. The principle of Lunar Laser Ranging is to fire laser pulses towards the target reflectors on the Moon, to receive back localised and recognizable signals and to measure the duration of the round-trip travel of the light. If the concept is elementary, it is in fact a real technical challenge. At first, the performances of the observations depend on the quality of the time measurements. On average the duration of the round-trip flight is 2.5 seconds varying according to the lunar distance, the mean distance Earth-Moon being 385000 km. If one aims at a precision of 1 cm for the separation between transmitter and reflector, one needs an accuracy of the order of 0.1 nanosecond (10−10 second) in the measurement of the round-trip travel duration. The measure of the time is based on a very stable high frequency signal generated by a caesium atomic clock whose the frequency accuracy is better than 10−12 yielding an uncertainty below 10 picoseconds (10−11 second) over the journey of the light. But several factors affect the precision of the measurements. The atmosphere induces a time delay which is difficult to estimate rigorously, probably between 50 and 100 picoseconds; it depends on temperature, pressure and humidity. The libration of the Moon makes an oscillation of the orientation of the array which produces, in worse cases, an un- biased scatter in distance of few centimeters (about 200 picoseconds in the time of flight). There are other sources of uncertainties such as the photodiodes connected to the start timer or the return detector. The CERGA team has tested several approaches and has converged towards widths of about 200 picoseconds. And to improve the chance to trap- ping the right return photon several kinds of filtering are performed to eliminate the noisy photons as much as possible. In these conditions, the precision of a measurement of the round-trip travel time of ’one photon’ is of the order of 3 cm. With the most unfavourable librations this value can grow up to about 5 cm. Nevertheless, the main problem stands elsewhere. The outgoing beam has a divergence of 3” to 4” after crossing the Earth atmosphere so that the size of the light spot on the Moon is about 7 km in diameter, which means that only one photon out of 109 impacts a reflector array. Besides the reflected beam has a significant angular divergence (12”) due to the diffraction by the pupil of the reflector yielding a spot nearly 25 km

99 100 No. 34 ua ae agn data Ranging Laser Lunar 2 IERS Note Technical ua ae agn:maueet,aayi,adcnrbto oterfrnesystems reference the to contribution and analysis, measurements, Ranging: Laser Lunar ih o EG n - o LS.Ti nomto sueu nlunar in useful is per information Points’ This ’Normal for MLRS2. 8 for shows are 3-4 there 10 and average CERGA night Figure for In per night Points’ 1988-2001. ’Normal station. period of CERGA number the the at during of dur- 2000 distribution except the in noise stations heavy as both of eclipses because Moon Moon observations). Full Quarter ing (morning at First observation Quarter the no Third is around obser- There the grouped no and are with month. observations) They lunation lunar (evening per Moon. days a New 25 within around about observations vation for out of 8 carried distribution are and the Observations CERGA shows for 9 20 Figure around CERGA maximum for a Point’ with ’Normal MLRS2. MLRS2 per 1988-2001. for returns period 15 50 the and during respectively of 2000), Point’ is distribution ’Normal (in the average gives per observations The 8 returns 1000 Figure of about year. number per is maximum the 700 the stations around 2001; being both average and MLRS2 for the 1988 the data between for dark) annual Points’ (in of ’Normal CERGA of and distribution light) reflectors time (in the other shows ma- the 2. 7 the with Lunakhod Figure compared to and ) 14 contributes Apollo (80% 15 11, observations Apollo Apollo the reflector of between The part data Ranging jor data. Laser reflectors. ranging Lunar lunar the several of of Since the distribution 70% the produced Points’. gives has 6 ’Normal station Figure of CERGA the hundreds the with several 1990s the provided Moon Haleakala station, Hawaii, the McDonald, laser Mc- Maui, American range the second on to a stations: years, 1990 station Around early LLR only 15 telescope. 3 2.7-m the the was During the Observatory 1970. of Donald since contribution in CERGA mm the and Haleakala 5-10 shows to 5 equivalent Figure is which picosec- time, 30-60 travel distance. of round-trip one-way precision’ the by internal in provided ’instrumental Points’ onds sub-centimetric an ’Normal have at the Today, CERGA aiming 1992. the project in a initiated station, was laser precision lunar CERGA level. the 3-cm At to standard 1985, dropped the after system, measurements timing device, early the YAG better the in a a uncertainty in and by widths cm replaced pulse was 10-15 smaller much laser around with ’station-reflector’ Rubis better distance the getting one-way When gradually the 1980s. im- and in noticeably 1970s cm the been 25 in has around was measurements Ranging, inter- It Laser the the Lunar proved. of the of of quality parameter middle beginning instrumental one the the the in a Since centered essentially observations. time the is of travel It val round-trip the returns. on time statis- fitted individual on built the Points’ from ’Normal of tics consists data observational actual The second. per a pulses the 10 seconds, at of 10 rate pulse every typical or per a pulses, with mJ pulses 100 10 every producing 300 to system return equivalent about laser one is of is average it beam an of gives nanometers; the out 532 of detected of energy wavelength is the photon 1987), one (since just factors, other 10 for allowing Finally, intercepted. is photons the ako h at.Wt -ee eecp,ol rcin10 fraction a only telescope, 1-meter a With Earth. the on back 20 mte nteiiilple ihteYGlsro h CERGA the of laser YAG the With pulse. initial the in emitted .Carn n .Francou G. and Chapront J. 18 htn.That photons. − 9 of J. Chapront and G. Francou IERS Lunar Laser Ranging: measurements, analysis, and contribution to the reference systems Technical

Note No. 34

McDonald Lunakhod 2 Apollo 11 Tel 2.7m 2% 9% CERGA Yag 23% 46%

McDonald MLRS1 3% Apollo 14 9% Apollo 15 McDonald 80% MLRS2 CERGA 17% Rubis Haleakala 8% 3%

Fig. 5. Distribution of ’Normal Points’ per LLR Fig. 6. Distribution of ’Normal Points’ per Lunar station Reﬂector

900 250 800 CERGA MLRS2 700 200 600 CERGA MLRS2 500 150 400 100 300 F r e q u n c y

N o r m a l P i n t s 200 50 100 0 0

R e t u r n s / N o r m a l P o i n t

Fig. 7. Distribution of LLR ’Normal Points’ Fig. 8. Distribution of Returns per ’Normal over(1988-2001) Point’

700 250 CERGA MLRS 600 200 500 CERGA MLRS 150 400

300 100

200 F r e q u n c y

N o r m a l p i n t s 50 100

0 0 0 3 6 9 12 15 18 21 24 27 30 0 2 4 6 8 10 12 14 16 18 20 A g e o f t h e M o o n ( d a y ) N o r m a l P o i n t s / N i g h t

Fig. 9. Distribution of LLR ’Normal Points’ Fig. 10. Distribution of LLR ’Normal Points’ per within the lunar month Night

101 102 No. 34 IERS Note Technical ua ae agn:maueet,aayi,adcnrbto oterfrnesystems reference the to contribution and analysis, measurements, Ranging: Laser Lunar t eo o-xasiels fthem. enumer- of have We list 1970 domain. non-exhaustive since scientific a large collected below a ate data to these Ranging contributions determine Laser significant to given Lunar reflector of thousands per The and night need per we Points’ Generally ’Normal parameters. latitude). 4 of or parameters (variation orientation 3 VOL Earth and the UT0-UTC of (EOP) estimations provide which analysis • • • • • npriua oiinn ftednmclrfrnefaei the in section. frame last the reference in dynamical discussed the is allows topic of It This positioning system. ICRS. dynamical a a particular intrinsically in defines motion a lunar mode, The wobble a systems Reference libration: mode). longitude of a modes is and fre- Libration, 3 mode libration Forced a (and precession - total modeled, frequencies Libration Total the libration 3 difference: Once forced exhibits the the of motion. and analysis observations quency of libration the equations free by The the known evolution. in Moon’s implicit the of parameter is knowledge the the of for determination mental a in result low inertia, of number moment param- rameter Love principal lunar the harmonics, effects: the of gravitational of of estimate degree kinds analysis an different the derives two Through one to eters, librations. submitted free is and rotation forced in a Moon induces The selenophysics which and action, science lunar Lunar longitude. the lunar the (Love to of dissipation due acceleration tidal delay) secular Earth time the of of domain and A evaluation numbers precise nutation. in a con- terms is It long-periodic preces- excellence few the of technique. of knowledge and SLR constant better a sion with to coefficients combined VLBI, the to when complementary and potential tributes, mention mass Earth shall improvement lunar We the the the of of system. to determination Earth-Moon contributor the particular the main in of a parameters also various is of technique LLR The Geodynamics Ori- dissipation Earth energy of the Moon. determination of the better modeling of a a finally, Moon), and and the Parameters, Earth entation the on longer on a tides motion and body (plate accuracy solid range solutions improved additional more. (an some or data span), the 2 time to factor due a mainly by is reduced It been grav- have in parameters uncertainties physics the itational determinations, previous grav- to solar (the compared the G 1996, of of change and of constant) rate itational the of parameters determina- boundaries a precise PPN Correlatively, principle, the equivalence the of of tion verification the concerns It physics Gravitational librations) free parallel. and in (forced elaborated (ELP) models been solutions have Libration analytical integrations constructed. and numerical been JPL) have Several the at followed measurements. has Ephemerides Moon LLR (Development the the of in motion orbital accuracy the of improvement The Ephemerides Q L aaaecmie ihlnrobtn aeltsand satellites orbiting lunar with combined are data LLR . β and J γ 2 n h edtcprecession. geodetic the and , aebe lodrvd Since derived. also been have C/MR .Carn n .Francou G. and Chapront J. κ 2 n ispto pa- dissipation and 2 hc sfunda- is which J. Chapront and G. Francou IERS Lunar Laser Ranging: measurements, analysis, and contribution to the reference systems Technical

Note No. 34

3 Lunar Laser Ranging data analysis: evaluation of the residuals

LLR data analysis has to be performed rigorously in the frame of General Relativity. We use below the common language of classical mechanics for sake of simplicity, without changing the nature of the physical problem. The LLR stations provide the ’Normal Points’, which are used as ’observed values’. A ’Normal Point’ consists in ∆to, the duration of the round trip travel of the light in atomic time (TAI, Temps Atomique International) and t1, the date of the starting pulse in Universal Time Coordinated (UTC). The other data supplied are the wavelength of the pulse, the number of returns, the estimated uncertainty of measurements, the signal/noise rate and the temperature, pressure and humidity of the atmosphere.

The principle of the analysis is to compare the ’observed value’ ∆to with a ’computed value’ ∆tc and to deduce the residual: ρ = ∆t0 − ∆tc.

The ’computed value’ ∆tc is determined in two steps as follows: from the LLR station transmitter O at time t1 to the lunar reflector R at time t2, and then from R at time t2 to O at time t3. The propagation time has to be corrected taking into account several effects such as the relativistic curvature of the light beam or the influ- ence of the troposphere. Relativistic changes of time scales have to be performed in order to express the various components in the same time scale. In the frame of the General Relativity theory, the computed ∆tc depends on the barycentric positions of T, L and S, respectively the center of mass of the Earth, of the Moon and of the Sun; B is the barycenter of the solar system. Hence, as shown on Figure 11, the computation involves the following coordinates in the same celestial barycentric reference system:

• the coordinates of TL (Earth-Moon) provided by a lunar ephemeris, and the barycentric coordinates of the Earth-Moon barycenter BG provided by a planetary ephemeris, • the coordinates LR (lunar reﬂector) which requires an ephemeris of the lunar librations and the selenocentric coordinates of the reﬂectors, • the coordinates of TO (observing station) which requires a precise knowledge of the Earth rotation (precession, nutation, polar motion, and Universal Time UT through the sidereal time), of the coordinates of the station in a terrestrial reference system, and of relativistic corrections.

The value of ∆tc is given by:

∆tc = [t3 − ∆T1(t3)] − [t1 − ∆T1(t1)] 1 t = t + |BR(t ) − BO(t )| + ∆T + ∆T 3 2 c 2 3 3 4 1 t = t + |BR(t ) − BO(t )| + ∆T + ∆T 2 1 c 2 1 3 4

where c is the velocity of the light, ∆T1 is a relativistic correction on the time scales, ∆T3 is the time contribution due to the gravitational curvature of light beam and ∆T4 is the atmospheric delay. The correction ∆T2, not mentioned above is the diﬀerence between Barycentric Dynamical Time (TDB) and Terrestrial Time (TT).

103 104 No. 34 IERS Note Technical ua ae agn:maueet,aayi,adcnrbto oterfrnesystems reference the to contribution and analysis, measurements, Ranging: Laser Lunar with: where to the close are function there linear Therefore transformations: (TDB). a successive Time by Dynamical replaced Barycentric is former the TCB of Coordinate Practically variety Barycentric (TCB). in great performed Moon, Time a are and computations Earth consider ephemeris of The tides to best solid ... delay, has at atmospheric represent one motions, plate to and models: model complex complete is the observations libration to Actually, ephemerides, the R solar in rotation. and reflector systems lunar Earth various the and provided: the of are to positions and (BRS) the O system which reference station barycentric of the the computations of relate the coordinates in barycentric needed the are transformations coordinates Several tr peeie lya motn oeo h ua ointhrough motion lunar masses. the planetary plan- particular on the in role and parameters, important unit various an astronomical play the relativistic ephemerides of and etary value longitude, that the lunar dissipation Indirectly the tidal of parameters. system, acceleration Earth-Moon of secular the coefficients the of orbits, produces mass terrestrial LLR field, and to gravity a fitted lunar the be on the can depend of them and elements of Many J2000.0 observations: of parameters. physical frame BG of ecliptic barycentre number dynamical Earth-Moon large the the of to and referred TL are Moon the of ephemerides The Earth-Moon T, B Earth system solar (Observer-Reflector), the OR of beam Barycenter G, Light barycenter 11. Fig. h lblfruaino ∆ of formulation global ∆ The correction relativistic ∆ the computation of the means of end the At O(t ) 1 TC UT m T T ∆ and → t c O(t ) L BO BR AI T = m R(t ) 3 L 1 +2∆ c ( ( 2 t t | r epcieyteErhadMo masses. Moon and Earth the respectively are → = ) = ) BR T T T ( BG BG 3 t 2∆ + 2 ) = ( ( − t t t AI T + ) + ) c BO T is: 4 m m − ( t 32 + B c t T T 1 ∆ m m stafre rmTBt A by TAI to TDB from tranformed is ) + + | T T L T + . 1 1 m 184 m . ( t 1 c L L 3 s ) | L T L T BR − → ( ( ∆ t ( t DB T + ) + ) t .Carn n .Francou G. and Chapront J. T 2 1 ) T ( − O T LR t 1 = BO ) G ( ( t T T t ) ) ( t 3 ∆ + ) | O L T 2 R J. Chapront and G. Francou IERS Lunar Laser Ranging: measurements, analysis, and contribution to the reference systems Technical

Note No. 34

The ephemeris of the reflectors LR is dependent of the reflector coordinates in a selenocentric frame defined by the principal axes of inertia of the Moon. They also depend on the lunar rotation which is provided by a theory of the forced libration and the free libration constrained by the elastic deformation of the Moon. The Love numbers and the value of the moments of inertia are fitted parameters. It is worth noticing that a coupling effect exists between the lunar libration and circulation, which is very sensitive to the harmonic coefficients of the lunar potential, in particular in the secular motions of the node and of the perigee. The selenocentric frame is linked to the dynamical ecliptic frame of J2000.0 by the transformation:

◦ Rz(−W1 + 180 ) × M(p1, p2, τ)

where Rz is the rotation around the z-axis, W1 is the mean lunar longitude and M is the matrix fonction of the libration variables p1, p2, τ at the time t2. The ephemeris of the station TO are primarily deﬁned in the Interna- tional Terrestrial Reference Frame (ITRF) and are subject to various corrections due to the Earth deformations: terrestrial and oceanic tides and pressure anomaly. For expressing the components of TO in the dynamical ecliptic frame of J2000.0 we have to apply the following transformations:

ITRF → equatorial frame of date → equatorial frame of J2000.0 → dynamical ecliptic frame of J2000.0

The ﬁrst transformation involves the Earth rotation parameters (EOP): the polar motion (xp, yp) and UT1-UTC used through the sidereal time θ as a function of UT1. The second transformation is obtained by the matrix of precession-nutation P × N. The last transformation is performed for referring the station in the dynamical equinox and ecliptic frame deﬁned by the lunar and planetary ephemerides with the position angles: (obliquity in J2000.0) and φ (arc separating the inertial equinox and the origin of the right ascensions on the equator of J2000.0) (see Fig. 13). Hence, the total transformation is:

−1 −1 Rx() × Rz(φ) × P × N × Rz(−θ) × Rx(yp) × Ry(xp)

where Rx, Ry and Rz are respectively the rotations around the x-axis, y-axis and z-axis.

4 Lunar Laser Ranging data analysis: dynamical model and ﬁtted parameters

Two diﬀerent approaches are possible to represent the planetary and lunar motions: the numerical integrations and the analytical theories. The Jet Propulsion Laboratory (JPL) has produced very precise ephemerides of the Moon and planets by numerical integration: the Develop- ment Ephemerides such as DE200, DE403, or DE405. The most recent integration DE405/LE405 covers 6000 years. It has been ﬁtted to observations and oriented onto the ICRF; it represents presently the most elaborated ephemeris, taking into account the most precise observations

105 106 No. 34 IERS Note Technical ua ae agn:maueet,aayi,adcnrbto oterfrnesystems reference the to contribution and analysis, measurements, Ranging: Laser Lunar h ria aaeesaerfre oJ000udrteformulation: the under the J2000.0 of hereunder; to fits listed referred are The are S2001 Q parameters in observations. orbital parameters var- of rotational the the periods and among the orbital and weights of lunar of stations treatment distribution an observing statistical adequate with ious an realistic reduction particular a and of in and numerical program data, the with model the to nutation model in also libration up-to-date respect but the with complements, in S2001 analytical mainly in solutions: introduced previous were to improvements fitted and Several ELP S2001. from called issued is solution observations analytical LLR the of version last The given: be various can the examples important aside Three the putting of by analysis signature an same possible components. make the prac- to parameters having this is numerous beyond interest perturbations fit main But to the we straightforward. advantage, solution are Then tical the derivatives inte- of partial solution. numerical the form analytical a since analytical numerical an of the of precision the nu- use advantages the plus can has the the ELP - retains solely all and improve for gration solution. they once analytical fixed constants; - the orbital complements of complements the precision numerical on is The merical a not depend it reference. with is not Hence, a difference it do LLR. as by term, integration by ex- obtained short numerical required contains complements JPL the level but numerical On centimeter integration with the completed numerical parameters. reach a of to number as adequate large precise a as The since plicitely not elaborated followed. versions are different been its 1982, has and ELP, approach solution analytical analytical lunar an Observatory, the Paris planets, At the of asteroids. libration. integration including dy- the simultaneous Relativity, and refined a General Moon very is of a characteristic frame and main the LLR) Its in and VLBI model data, namical spacecraft Mars (Radar, etre ic J2000.0. since centuries • b) a) c) = h ecnrclnrobtlparameters orbital lunar geocentric the ftema ogtd,ma ogtdso eie n node), and perigee of longitudes mean longitude, mean the of Q oin aebe banda ucino h P parameters PPN the of these function of a and as expressions obtained Analytical per been tides). arcseconds have figures, motions 2 (plan- lunar about contributions and other of Earth from ets, separated node, be and to have perigee which century, lunar mean the sidereal the of in motions exist understand Relativity to to due contributions librations Important free the the mechanism. of isolate excitation separation underlying to the The essential lunar the is state. of modes equilibrium departure two an a is from Sun which Earth, position component by angular free attraction a to and due time- planets figure from and lunar arising the component on forced torques a varying of timelag. composed the is able and libration be The number to Love particular the in fundamental precisely tides, is fit from it to effects residuals planetary When sev- the LLR tides. of separate in and to sum phenomena figure the total Earth’s the is planets, observing longitude and Sun lunar the components: eral of acceleration secular The (0) γ + ( Q β (1) = t γ + nGnrlRltvt) htcnb hntested. then be can that Relativity), General in 1 = Q (2) t 2 + Q (3) t 3 + Q (4) t 4 W where 1 (0) , .Carn n .Francou G. and Chapront J. W t 2 (0) stetm nJulian in time the is , W 3 (0) (constants , ν = β J. Chapront and G. Francou IERS Lunar Laser Ranging: measurements, analysis, and contribution to the reference systems Technical

Note No. 34

(1) W1 , Γ and E (sidereal mean motion, constants for inclination and eccentricity);

• the heliocentric orbital parameters of the Earth-Moon barycenter T (0), and $0(0), (constants of the mean longitude and longitude of perihe- lion), n0 and e0 (sidereal mean motion and eccentricity);

(2) (1) (1) • the bias parameters W1 , W2 , W3 , (observed corrections to the computed coeﬃcient of the quadratic term of the lunar mean longi- (2) tude, and the computed mean motions of perigee and node). W1 (2,T ) yields an observed value of W1 , the tidal part of the coeﬃcient of the quadratic term of the mean longitude of the Moon (half tidal secular acceleration);

• the 6 libration parameters which are the amplitudes of the 3 main free libration terms and their respective arguments.

Fits of other parameters have been also performed in S2001 such as the position angles of the dynamical ecliptic frame with respect to various systems of axes (, φ, ψ), the correction to the IAU 1976 precession constant ∆p and the coordinates of the reflectors and stations. Note that a simultaneous fit of all the parameters produces much correlated values. Hence, the fits have been performed in several steps and tests have been made in order to check the stability of the results. Indeed, strong corre- lations that exist among some parameters may weaken the accuracy of our determinations; in particular, it is the case of the variables related to the reference frame and the positions of the stations. The correction to IAU 1976 precession constant ∆p and the obliquity rate are correlated with the velocities of the stations. The principal nutation term and ∆p are also difficult to separate. It is also the case for the selenocentric coordinates of the reflectors and the parameters of the free libration.

We have adopted the following strategy. First, we determine the whole set of parameters mentioned above except the positions and velocities of the stations. Then fixing the value φ (position angle, see Fig. 13), we add the positions of the stations to the whole set and make a new improvement. Next, we determine the velocities of the stations separately. Finally, fixing all the parameters, we perform a last analysis including ∆p and the principal term of the nutation in longitude. At each step of the process we check the coherence of the determinations; for example we verify that the introduction of the fitted values of the station coordinates does not change significantly the value of φ when the first step is reiterated.

Table 1 shows the evolution of the residuals obtained with S2001 for the distance between the LLR stations and the lunar reflectors expressed in centimeters. It gives the time distribution of the root mean square of post-fit residuals obtained with the data provided by the different instruments used since 1972. We note an important gain of precision, between the earliest observations and the recent ones.

The evolution in the quality of the observations is also illustrated in Figure 12 that gives the time distribution of r.m.s. of the residuals obtained with S2001, retaining solely the data provided since1988 by the 2 modern instruments: MLRS2 for McDonald and YAG for the CERGA.

107 108 No. 34 IERS Note Technical ua ae agn:maueet,aayi,adcnrbto oterfrnesystems reference the to contribution and analysis, measurements, Ranging: Laser Lunar al .LRrsdas iedsrbto fters(ntc) is N (unit:cm). rms the of Points’. distribution ’Normal time LLR of residuals: number LLR the 1. Table i.1.Rsdaso h bevtoso h prtn ttos Mc- stations: operating 2 the of observations CERGA the and of DONALD Residuals 12. Fig. EG 9719 . 1574 5.3 232 1987-1991 451 5.8 Yag 1987-1991 CERGA 6.3 MLRS2 and 1987-1990 MLRS1 1487 McDONALD HALEAKALA 43.5 1972-1986 CERGA MLRS1 and m 2.70 Telescope McDONALD instruments and

rms (centimeter) BEVTR ieS01N S2001 Time OBSERVATORY 1 2 3 4 5 6 7 8 1988 Rubis 1990 1992 9418 871165 18.7 1984-1986 9520 . 3273 2044 3.0 1669 3.9 586 1995-2001 1991-1995 3.3 4.6 1995-2001 1991-1995 990 1035 29.1 27.7 1980-1986 1976-1979 nevlrms Interval 1994 1996 CERGA McDONALD 1998 .Carn n .Francou G. and Chapront J. 2000 2002 J. Chapront and G. Francou IERS Lunar Laser Ranging: measurements, analysis, and contribution to the reference systems Technical

Note No. 34

5 Lunar Laser Ranging data analysis: reference system

5.1 Positioning the inertial mean ecliptic of J2000.0

The lunar solution is referred to a dynamical system and introduces the inertial mean ecliptic of J2000.0. With LLR analysis, we can adjust the position of this plane with respect to various equatorial systems. Figure 13 presents the inertial mean ecliptic of J2000.0 and the various equatorial systems (R) involved in our study:

• ICRS: International Celestial Reference System, • MCEP: Reference system linked to CEP (Celestial Ephemeris Pole), • JPL: Reference system deﬁned by a JPL numerical integration such as DE403 or DE405.

I γ2000(R) is the ascending node of the inertial mean ecliptic of J2000.0 on the equator of (R), that is the ’inertial equinox of J2000.0’, and o(R) is the origin of right ascensions on the equator of (R). The position angles , φ and ψ are:

– (R), the inclination of the inertial mean ecliptic to (R), I – φ(R), the arc between o(R) and γ2000(R) on (R), I I – ψ(R), the arc between γ2000(ICRS) and γ2000(MCEP ) on the mean ecliptic of J2000.0.

Inertial mean ecliptic of J2000.0

γI (MCEP) ε 2000 (MCEP) Mean equator of J2000.0 o(MCEP)

I γ (JPL) ε(JPL) 2000 “Equator” JPL o(JPL) ε(ICRS) “Equator” ICRS γ I (ICRS) o(ICRS) 2000

Fig. 13. Position angles of the inertial mean ecliptic of J2000.0

Concerning the ICRS and MCEP reference system, two solutions ’S2001’ have been investigated. In both of them the position angles and φ are ﬁtted, but the transformations of the terrestrial coordinates of the stations to celestial ones are diﬀerent:

• in S2001(ICRS), the precession-nutation matrix P × N is computed via the conven- tional set of values recommended by the International Earth Rotation System (IERS), in particular the nutation corrections δ and δψ of the series EOP(C04) produced by the Earth Orientation Center,

109 110 No. 34 IERS Note Technical al .Psto fteieta enelpi fJ000wt epc ovrosequatorial various to respect with J2000.0 of ecliptic mean systems inertial the celestial of Position 2. Table CP0.40564 MCEP E0 0.40928 DE403 E0 0.40960 DE405 CS0.41100 ICRS R ua ae agn:maueet,aayi,adcnrbto oterfrnesystems reference the to contribution and analysis, measurements, Ranging: Laser Lunar − h CSeutr(e i.3 a edtrie sn h ausof values the using determined be can Fig.13) 2: (see Table equator ICRS the ICRS, in projection ascensions the right of and origin the between separation The ob- versions value Constants. previous recent ∆ Astronomical are notations with a on S2000 The (CEP) and and Pole S2001. analysis S1998 of measurements; Ephemeris LLR VLBI Celestial with by the computed tained of ICRS offsets to the respect gives numerical 3 JPL corresponding Table the the in arbi- in mentioned have used dates we observations the integrations. DE405, of the and function DE403 about in For literature MCEP. epoch the and observations chosen LLR ICRS of trarily of date case mean mentioned weighted the the epoch in to mean corresponds The table system. this reference in equatorial each for mined al ie h ieetvle ftepsto angles position the of values different the gives 2 Table 23 R • ± ± ± ± ◦ epc otesse endb h orsodn P numerical JPL corresponding the by defined with system J2000.0 ephemeris integration. of the JPL ecliptic any to mean to inertial respect compared the referring solution angles lunar the ∆ the provides constant way, precession same case, 1976 this the IAU in In nutation; the of polyno- to theory correction and solutions: the precession analytical the of by expressions provided mial is precession-nutation the S2001(MCEP), in nS01 ui:aceod.Teucranisaefra errors. formal are uncertainties The arcsecond). (unit: S2001. in 26 al .Ost fClsilEhmrsPl tJ000wt re- with J2000.0 at ∆ Pole S2001, Ephemeris in Celestial ICRS of to Offsets spect 3. Table netite r omlerrors. formal are uncertainties .00 -0.01460 -0.05542 0.00009 0.00005 .00 -0.05028 -0.05294 0.00001 0.00000 0 ehdSuc ∆ Source Method 21” LIIU00()-0.0049 (*) IAU2000 VLBI o o L OA 20 -0.0054 -0.0054 S2001 POLAC -0.0056 S2000 POLAC LLR S1998 POLAC LLR LLR ( ( ICRS ICRS * uuhm,20,Rpr nAtooia Constants. Astronomical on Report 2000, Fukushima, (*) ) ) o o 0 0 ( ( o DE DE 0 ( PL JP 0)=1 = 403) 0)=0 = 405) ψ φ ± ± ± ± .01 e 1994 Dec 0.0445 0.00015 0.00011 .00 0.0064 0.00480.00001 0.00001 fteoii frgtascensions right of origin the of ) 0 and 0 ψ . . and 9 7 sin mas mas 0 ψ r hs sdi A Report IAU in used those are ± ± ± ± sin 0 ( ( Epoch, Epoch, .03-0.0167 -0.0177 0.0003 -0.0173 0.0002 -0.0183 0.0002 0.0002 ± ± ± 0 ui:aceod.The arcsecond). (unit: .03Dc1994 Dec 0.0003 .03Jn1990 Jan 1985 Jan 0.0003 0.0004 .Carn n .Francou G. and Chapront J. 1985 1990 , p φ enEpoch Mean Jan Jan sas fitted. also is ψ and sin o ± ± ± ± ( o 1) 1) PL JP ( ψ 0.0005 0.0004 0.0004 0.0004 ICRS 0 deter- on ) ), J. Chapront and G. Francou IERS Lunar Laser Ranging: measurements, analysis, and contribution to the reference systems Technical

Note No. 34

Note that the numerical integration DE405 uses very precise VLBI observations made between 1989 and 1994 and is oriented onto the ICRS. The angle ψ(R) is the separation between the inertial equinox of J2000.0 I in ICRS, γ2000(ICRS), and the inertial equinox of J2000.0 in the ref- I erence system (R), γ2000(R). It depends of the expression of the lunar mean longitude of the Moon W1 which has the following secular expan- sion: (0) (1) (2) 2 (3) 3 (4) 4 W1 = W1 + W1 t + W1 t + W1 t + W1 t where t is the time in Julian centuries since J2000.0. (0) (1) W1 is the constant term (mean longitude at J2000.0), W1 is the (2) sidereal mean motion (also noted ν) and W1 is the total half-secular acceleration of the Moon. Hence, ψ(R) can be determined by:

ψ(R) = W1(ICRS) − W1(R) =

(0) (0) [W1 (ICRS) − W1 (R)] +

(1) (1) [W1 (ICRS) − W1 (R)]t +

(2) (2) 2 [W1 (ICRS) − W1 (R)]t where t is the time at a mean epoch, reckoned in centuries from J2000.0 (see Table 2).

5.2 Tidal acceleration

Figure 14 shows the evolution of the correction to the 3 components of (0) (1) (2) W1: ∆W1 , ∆W1 and ∆W1 with the variation of the upper limit of the time span covered by the fit. In other words, the graph represents (i) the different values of ∆W1 with: i = 0, 1, 2 for intermediate solutions in which the characteristics of the fit are the same as in the solution S2001, except the time intervals of the LLR observations that we have successively limited to equidistant dates between 1996 and 2001. We can notice in particular the convergence of the acceleration. This is mainly explained by the evolution of the fitted value of the tidal part of the acceleration when using more and more recent LLR observations. The tidal component of the secular acceleration of the Moon’s longitude is a fundamental parameter, which expresses the dissipation of energy in the Earth-Moon system. It is due to a misalignment of the bulge of the Earth relative to the Earth-Moon direction, which exerts a secular torque. It produces a secular negative acceleration in the lunar longitude of approximately −25.8”/cy2 and correspondingly a decrease in the Earth’s rotation rate (or an increase of the length of day). Another con- sequence is the displacement of the Moon that corresponds to an increase of the Earth-Moon distance of 3.8 cm/year. Table 4 gives a list of determinations of the tidal secular acceleration of the lunar longitude since 1939 and provided by several types of observations: occultations, eclipses and LLR. The most recent values have been obtained with LLR observations. We note for this type of determination a significant improvement of the precision when increasing the number of observations and their accuracy. It is also worth noticing that the most recent determination around −25.86”/cy2 comes closer to the value of Morrison & Ward (1975) obtained with optical observations over a very long time span.

111 112 No. 34 IERS Note Technical ua ae agn:maueet,aayi,adcnrbto oterfrnesystems reference the to contribution and analysis, measurements, Ranging: Laser Lunar i.1.Eouino h orcin otesclrcmoet fthe of components secular the to corrections the longitude lunar of mean Evolution 14. Fig. -0.01 0.00 0.01 0.02 0.03 0.04 0.05 1996 al .Tdlaclrto ftelnrma mean mean lunar arcsecond/cy the (unit: of acceleration longitude. Tidal 4. Table c a observations: of Type S2001) (Solution S2000) (Solution S1998) (Solution al. Touz´e et Chapront al. et Dickey al. et Newhall Williams Publication and Dickey al. et Value Dickey al. et Ferrari Mulholland & Calame Muller Ward & Morrison Cohen & Oesterwinter Jones Spencer Authors ogtd nvrosJLehmrds(unit: ephemerides mean JPL lunar the various arcsecond/cy of in acceleration Tidal longitude 5. Table LLR, : Occultations, : ∆ P peei au Publication Value ephemeris JPL w 1 (1) E0 2.2 1998 1995 1990 1982 -25.826 -25.580 -25.625 -23.895 DE405 DE403 DE245 DE200 1997 b ("/cy) d L n ua orbiter Lunar and LLR : c c 2 d Sads,18,19,1998) 1995, 1982, (Standish, ) ∆ a c w b c c c 1998 Eclipses, : 1 a (2) ∆ c ("/cy w c c a 1 (0) 2 2.21997 1988 -25.62 -24.90 1975 -38 2.5 2002 2000 1999 -25.858 -25.836 1994 -25.78 1982 1982 -25.88 1980 1978 -25.10 1976 -23.8 1975 -23.8 -24.6 1939 -30 -26 -22 ) (") 1999 2 ) .Carn n .Francou G. and Chapront J. 2000 2001 J. Chapront and G. Francou IERS Lunar Laser Ranging: measurements, analysis, and contribution to the reference systems Technical

Note No. 34

Table 5 gives the intrinsic values of the tidal acceleration in various JPL numerical integrations. The diﬀerence between the last JPL value (DE405) and our determination in S2001 gives an idea of the present uncertainty and allows to ensure nowadays a realistic precision of better than 0.03”/cy2 in the knowledge of this fundamental lunar parameter. Note that the values presented here as ’JPL values’ do not appear explic- itly in the list of parameters provided by JPL. We have deduced them from the comparisons of the solution S2001 to the JPL ephemerides.

5.3 Precession constant

In the solution S2001(MCEP), which is linked to the Celestial Ephemeris Pole (CEP), the precession constant has been ﬁtted and the correction ∆p to the IAU 1976 constant is equal to −0.3364”/cy. We observed that this correction was noticeably divergent from the values of this correction obtained by previous solutions such as S1998 or S2000.

-0.28

-0.29 ∆ ∆ 1p- 2p -0.30

-0.31

-0.32 ∆ 1p -0.33

arcsecond/century -0.34

-0.35

-0.36 1996 1997 1998 1999 2000 2001

Fig. 15. Evolution of the correction to the IAU 1976 precession constant with the time span covered by the ﬁt: ∆1p for S2001(MCEP) and ∆2p for S2001(ICRS)

Figure 15 illustrates the evolution of ∆p with the variation of the upper limit of the time span covered by the fit with the same division in ’slices of time’ as we did previously in Figure 14 for the components of the mean lunar longitude. We note ∆1p the evolution of ∆p in solution S2001(MCEP). The ’accidental jump’ that we observe for 1997 corresponds to an offset of 0.7 nanosecond in the CERGA measurements which has been partly corrected in our analysis. In fact, the relevance of this graph is to put in evidence a residual ∆2p which also arises when we put ∆p in the list of the fitted parameters of the solution S2001(ICRS). We note in particular that the difference ∆1p − ∆2p is almost constant on the time interval [1996-2001]. S2001(MCEP) and S2001(ICRS) use the same observations, which are the main source of errors, and the same models, except for the motion of the reference frame due to precession and nutation. If we assume that the series EOP(C04) δ and δψ used in S2001(ICRS) and based on VLBI

113 114 No. 34 ilorpyadReferences and Bibliography Conclusion 6 IERS Note Technical ua ae agn:maueet,aayi,adcnrbto oterfrnesystems reference the to contribution and analysis, measurements, Ranging: Laser Lunar Calame Boucher work. analysis LLR the Chapront-Touzé in M. contributor and main contributions a fruitful is its who We for points. Mignard normal F. and observations thank LLR the providing for here McDONALD presented results Acknowledgements: the account into will constants taking 2002). the data Francou, version and LLR this (Chapront in to and of fitted solution theory ELP/MPP02, be previous A perturbations, the analytical Moon. planetary replace the the new will improve of including motions to motion CERGA. rotational lunar is the and the by analysis control orbital provided our of the are of role of they solution a goal soon also first as assume the data CERGA we Nevertheless, LLR the analysis, of of LLR validation staff the and LLR Besides lunar the Observatory with Paris 1998. the cooperation since of in team works the The center between analysis analysis. residuals margin the LLR of a the values the by still an and obtained is data shows the there measurement of longitude. So, accuracy ’instrumental’ LLR present lunar picoseconds. individual 30-60 the the an around of for for CERGA, error and acceleration noted the constant secular also at precession tidal is Presently, the the accuracy to of correction in other determination the and gain of the ICRS This estimation of the the positioning to systems. the respect reference in with involved equatorial frame angles reference provide the ecliptic to of dynamical allowed have determination as- model years better Earth-Moon last a refined 10 more the a during with measurements sociated LLR the of improvement The mas/year. 0.03 nu- determinations than of VLBI smaller theory and General separation LLR the a IAU with of 2002). the nicely value al., converge in the et (Herring or presented MHB2000 2000) value tation (Fukushima, the 2000 in analysis: Assembly VLBI precession by 1976 IAU obtained to correction the ∆ of value constant last the that shows 6 Table nbte siaeo h orcin oteIU17 rcsinconstant precession 1976 IAU the to ∆ corrections than the of estimate ∆ better corrections two ∆ an the residual between the differences we the that Hence, matrix, hypothesis precession-nutation the make the can to ideally contributes observations, rscn/y.Teucranisaefra errors. ∆ formal constant are precession 1976 uncertainties IAU The the arcsecond/cy). to Correction 6. Table ehdSuc ∆ Source Method LIMB00(ern,20)-0.2997 -0.297 2002) (Herring, 2000) (Fukushima, MHB2000 IAU2000 VLBI VLBI L 20 Carn,20)-0.302 2002) (Chapront, S2001 LLR oe2,CnrlBra fIR,Osraor eParis de Observatoire IERS, of Bureau Central 20, Note 1 p . uhladJ . 98 cec,19 3. 199, Science, 1978, D., J. Mulholland O., . laiiZ,FislM,SladP,19,IR Technical IERS 1996, P., Sillard M., Feissel Z., Altamimi C., hscreto ean rudtevalue the around remains correction This . p bandb L r eycoet eetdeterminations recent to close very are LLR by obtained eaegaeu oLRsaso EG and CERGA of staffs LLR to grateful are We 2 p sas nlddi ∆ in included also is .Carn n .Francou G. and Chapront J. p − ∆ = 0 . ± ± ± 302” p 1 p 0.0008 0.004 0.003 − p /cy ∆ (unit: 2 . p gives 1 p . J. Chapront and G. Francou IERS Lunar Laser Ranging: measurements, analysis, and contribution to the reference systems Technical

Note No. 34

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115 116 No. 34 IERS Note Technical ua ae agn:maueet,aayi,adcnrbto oterfrnesystems reference the to contribution and analysis, measurements, Ranging: Laser Lunar Standish Spencer Simon, Williams Williams Williams Standish Standish Standish rno,G,Lsa,J,19,Ato.Atohs 8,663 282, Astrophys. Astron. 1994, J., Laskar, G., Francou, 4 1077-80. 44, 1439. 27, Sciences, Planetary 6730-6739. 12, 53, D.C. Washington Eds., Pasadena. 312.F-98-048, IOM DE405/LE405, Pasadena. 314.10-127, IOM, DE403/LE403, .L,Beann . hpot . Chapront-Touz´e, M., J., Chapront, P., Bretagnon, L., J. oe . 99 otl oie o.Ato.Sc,9,541. 99, Soc., Astron. Roy. Notices Monthly 1939, H., Jones .M,18,Ato.Atohs,14 297-302. 114, Astrophys., Astron. 1982, M., E. .M,20,i A olqim10 .J ontne al. et Johnston J. K. 180, Colloquium IAU in 2000, M., E. Ephemerides, Lunar and Planetary JPL 1998, M., E. ephemerides, Lunar and Planetary JPL 1995, M., E. .G,NwalX . ikyJ . 98 lnt pc Sci., Space Planet. 1998, O., J. Dickey X., Lunar X. 1996, Newhall O., G., J. J. Dickey F., C. Yoder X., X. D, Newhall Review G., Physical J. 1996, O., J. Dickey X., X. Newhall G., J. .Carn n .Francou G. and Chapront J.