David M. Lucchesi - Istituto Nazionale di Fisica Nucleare
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David M. Lucchesi - Istituto Nazionale di Fisica Nucleare
Istituto di Fisica dello Spazio Interplanetario Istituto Nazionale di Astrofisica 36th COSPAR Scientific Assembly Beijing, China, 16 – 23 July 2006 The LAGEOS satellites orbital residuals determination and the way to extract gravitational and non–gravitational unmodelled perturbing effects David M. Lucchesi (1,2) 1) Istituto di Fisica dello Spazio Interplanetario IFSI/INAF Via Fosso del Cavaliere, 100, 00133 Roma, Italy Email: [email protected] 2) Istituto di Scienza e Tecnologie della Informazione ISTI/CNR Via Moruzzi, 1, 56124 Pisa, Italy David M. Lucchesi Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario Istituto Nazionale di Astrofisica Preamble Long–arc analysis of the orbit of geodetic satellites (LAGEOS) is a useful way to extract relevant information concerning the Earth structure, as well as to test relativistic gravity in Earth’s surroundings: • • • • • • Gravity field determination (both static and time dependent parts); Tides (both solid and ocean); Earth’s rotation (Xp,Yp, LOD, UT1); Plate tectonics and regional crustal deformations; …; Relativistic measurements (Lense–Thirring (LT) effect); … all this thanks i) to the Satellite Laser Ranging Technique (SLR) (with an accuracy of about 1 cm in range and a few mm precision in the normal points formation); ii) and the good modelling of the orbit of LAGEOS satellites. David M. Lucchesi Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario Istituto Nazionale di Astrofisica Preamble The physical information is concentrated in the satellite orbital residuals, that must be extracted from the orbital elements determined during a precise orbit determination (POD) procedure. The orbital residuals represent a powerful tool to obtain information on poorly modelled forces, or to detect new disturbing effects due to force terms missing in the dynamical model used for the satellite orbit simulation and differential correction procedure. However, the physical information we are interested to, especially in the case of tiny relativistic predictions, is biased both by observational errors and unmodelled (or mismodelled) gravitational and non– gravitational perturbations (NGP). David M. Lucchesi Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario Istituto Nazionale di Astrofisica Preamble In the case of the two LAGEOS satellites orbital residuals, several unmodelled long–period gravitational effects, mainly related with the time variations of Earth’s zonal harmonic coefficients, are superimposed with unmodelled NGP due to thermal thrust effects and the asymmetric reflectivity from the satellites surface. The way to extract the relevant physical information in a reliable way represents a challenge which involves (at the same time): I. II. III. IV. precise orbit determination (POD); orbital residuals determination (ORD); Statistical analysis; accurate modelling; David M. Lucchesi Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario Istituto Nazionale di Astrofisica Table of Contents 1. Orbital residuals determination (ORD): the new method; 2. ORD: the new method proof and the Lense-Thirring effect; 3. Application to the secular effects; 4. Application to the periodic effects; 5. ORD, unmodelled effects and background gravity model; 6. Conclusions; David M. Lucchesi Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario Istituto Nazionale di Astrofisica The meaning of orbital residuals In general, by residual we mean the difference (O – C) between the observed value (O) of a given orbital element, and its computed value (C): Oi Ci j Ci Pj Oi Pj P x , x , Vector of parameters to be determined Oi Observation error of the i-th observation The computed element is determined—at a fixed epoch—from the dynamical model included in the orbit determination and analysis software employed for the orbit simulation and propagation. The observed value of the orbital element is the one obtained from the observations, i.e., by the tracking system used for the satellite acquisition at the same epoch of the computed value. David M. Lucchesi Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario The meaning of orbital residuals: Istituto Nazionale di Astrofisica The computed element (C) Models implemented in the orbital analysis of LAGEOS satellites with GEODYN II Geopotential (static part) JGM–3; EGM–96; CHAMP; GRACE; Geopotential (tides) Ray GOT99.2; Lunisolar + Planetary Perturbations JPL ephemerides DE–403; General relativistic corrections PPN; Direct solar radiation pressure cannonball model; Albedo radiation pressure Knocke–Rubincam model; Earth–Yarkovsky effect Rubincam 1987 – 1990 model; Spin–axis evolution Farinella et al., 1996 model; Stations position ITRF2000; Ocean loading Scherneck model (with GOT99.2 tides); Polar motion IERS (estimated); Earth rotation VLBI + GPS David M. Lucchesi Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario The meaning of orbital residuals: Istituto Nazionale di Astrofisica The observed element (O) Of course, this is only an ideal way to define the residuals. Indeed, from the tracking system we usually obtain the satellite distance with respect to the stations which carry out the observations, and not the orbital elements used to define the orbit orientation and satellite position in space. Hence, we need a practical way to obtain the residuals, which retains the same meaning of the difference (O – C). Normal points with a precision of a few millimeters from the ILRS in the case of the LAGEOS–type satellites David M. Lucchesi Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario The meaning of orbital residuals: Istituto Nazionale di Astrofisica GEODYN II range residuals Accuracy in the data reduction LAGEOS range residuals (RMS) The mean RMS is about 2 – 3 cm in range and decreasing in time. This means that “real data” are scattered around the fitted orbit in such a way this orbit is at most 2 or 3 cm away from the “true” one with the 67% level of confidence. From January 3, 1993 David M. Lucchesi Courtesy of R. Peron Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario The meaning of orbital residuals: Istituto Nazionale di Astrofisica The usual way The usual way is to take Keplerian elements as a data type; so we can take short–arc Keplerian elements and directly fit them with a single long–orbit–arc and evaluate the misclosure in the long–arc modelling directly. That is to say, we can take tracking data over daily intervals and fit them with a force model as complete as possible, say at a 1 cm accuracy (rms) level. We then take the single set of elements at epoch and build a data set of these daily values. Then we can fit these daily values, for instance every 15 days, with a longer arc and then obtain the difference between the adjusted elements of the long–arc with the previously determined daily elements. David M. Lucchesi Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario Istituto Nazionale di Astrofisica The meaning of orbital residuals: The usual way Long arc Daily values 0 15 30 45 time 45 time Residuals 0 15 30 This difference is a measure of unmodelled long–period force model effects. The feature of this procedure is its simplicity but it is also time consuming. David M. Lucchesi Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario The meaning of orbital residuals: Istituto Nazionale di Astrofisica The new method In our derivation of the relativistic Lense–Thirring precession, to obtain the residuals of the Keplerian elements we instead followed the subsequent method (Lucchesi 1995 in Ciufolini et al., 1996): 1. we first subdivide the satellite orbit analysis in arcs of 15 days time span (arc length); 2. the couple of consecutive arcs are chosen in such a way to overlap in time for a small fraction (equal to 1 day) of their time span, in order that the consecutive residuals are determined with a 14 days periodicity; 3. the orbital elements of each arc are adjusted by GEODYN II to best–fit the observational SLR data; all known force models are included in the process (except the Lense–Thirring effect if it is to be recovered); 4. we then take the difference between the orbital elements close to the beginning of each 15–day arc and the orbital elements (corresponding to the same epoch) close to the end of the previous 15–day arc; David M. Lucchesi Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario The meaning of orbital residuals: Istituto Nazionale di Astrofisica The new method It is clear that the orbital elements differences computed in step 4 represent the satellite orbital residuals due to the uncertainties in the dynamical model, or to any effect not modelled at all. The arc length has been chosen in order to avoid stroboscopic effects in the residuals determination. Indeed, 15 days correspond to a large number of orbital revolutions of the LAGEOS satellites around the Earth. We used 15 days arcs in our analysis of the Lense–Thirring effect because during this time span the accumulated secular effect on the LAGEOS satellites node (about 1 mas) is of the same order–of–magnitude as the accuracy in the SLR measurements (about 0.5 mas on the satellites node total precession for a 3 cm accuracy in range). David M. Lucchesi Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario The meaning of orbital residuals: Istituto Nazionale di Astrofisica The new method One more advantage of such a method to obtain the orbital residuals with respect to other techniques, is that the systematic errors common to the consecutive arcs are avoided thanks to the difference between the arcs elements. Furthermore, since with the described method the residuals are determined by taking the difference between two sets of orbital elements that have been estimated and adjusted over the arc length, they express, in reality, the variation of the Keplerian elements over the arc length. In other words, these differences, after division by the time interval t between consecutive differences (14 days in our analyses) are the residuals in the orbital elements rates. David M. Lucchesi Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario The meaning of orbital residuals: Istituto Nazionale di Astrofisica The new method In the Figure we schematically compare the ‘’true‘’ temporal evolution of a generic orbital element (dashed line) with the corresponding element adjusted (continuous line) over the orbital arc length. The dashed line represents the time evolution of the element X assumed to play the true evolution due to all the disturbing effects acting on the satellite orbit. The continuous (horizontal) lines are representative of the adjustment of the orbital element over the consecutive arcs corresponding to a t time span (14 days in the case of the Lense–Thirring effect analysis). The quantities X1 and X2 represent the variations of the element due to the mismodelling of the perturbation. David M. Lucchesi X(t) X2 X1 Arc-1 Arc-2 Arc-3 t t t t Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario The meaning of orbital residuals: Istituto Nazionale di Astrofisica The new method That is, the continuous line fits the orbital data but it is not able to ‘’follow‘’ them (dashed line) correctly because in the dynamical model used in the orbit analysis–and– simulation a given perturbation has not been included or is partly unknown. Therefore, the difference Arc-2 minus Arc-1 represents the secular and long–period orbital residual in the element X. X(t) Of course, as we can see from the Figure, this difference represents the variation X of the orbital element due (mainly) to the disturbing effect not included in the dynamical model during the orbit analysis. Hence the quantity X/t represents the rate in the orbital residual. David M. Lucchesi X2 X1 Arc-1 Arc-2 Arc-3 t t t t Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario The meaning of orbital residuals: Istituto Nazionale di Astrofisica The new method From the Figure it is also clear why the systematic errors are avoided with the suggested method. Suppose the existence of a systematic error common to both arcs (say a constant error due to some coefficient or to some wrong calibration), this produces the same vertical shift of the two continuous lines but it will leave unchanged their difference. Finally, in order to obtain the secular/long–period effects from the set of orbital elements differences Xi, we simply need to add—over the consecutive arcs—the various residuals obtained with the ‘’difference–method‘’, that is: X(t) X2 X1 S1 X 1 S 2 S1 X 2 S n S n 1 X n David M. Lucchesi t1 t 2 t1 t t n t n 1 t Arc-1 Arc-2 Arc-3 t t t t Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario Istituto Nazionale di Astrofisica Table of Contents 1. Orbital residuals determination (ORD): the new method; 2. ORD: the new method proof and the Lense-Thirring effect; 3. Application to the secular effects; 4. Application to the periodic effects; 5. ORD, unmodelled effects and background gravity model; 6. Conclusions; David M. Lucchesi Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario ORD: The analytical proof Istituto Nazionale di Astrofisica (Lucchesi and Balmino, Plan. Space Sci., 54, 2006) We start observing that we are dealing with small perturbations with respect to the Earth’s monopole term. Indeed, the main gravitational acceleration on LAGEOS satellites is about 2.8 m/s2 while the accelerations produced by the main unmodelled non–gravitational perturbation (the solar Yarkovsky–Schach effect) is about 200 pm/s2 (Métris et al., 1997; Lucchesi, 2002; Lucchesi et al., 2004). NGP monopole 7 10 11 Concerning the gravitational perturbations, the largest effect is produced by the uncertainty in the Earth’s GM (where G represents the gravitational constant and M the Earth’s mass), corresponding to an acceleration of about 5.3109 m/s2, again much smaller than the monopole term. GP 2 10 9 monopole Under this approximation the differential equations for the osculating orbital elements can be treated following the perturbation theory. David M. Lucchesi Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario Istituto Nazionale di Astrofisica ORD: The analytical proof If Y represents the vector of the orbital elements as a function of time, the corresponding differential equations can be written as: Y H 0 Y H1 Y (1) where H 0 corresponds to the reference model (used in the reference orbit), while the unknown or unmodelled perturbation is given by the second term with being a small parameter. Perturbation The solution is: Y t Y0 t y1 t 2 y2 t (2) expanded as a power series of the small parameter . Because we are dealing with small perturbations we can neglect the second–order effect represented by the third term on the right side of equation (2). David M. Lucchesi Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario Istituto Nazionale di Astrofisica ORD: The analytical proof Y t Y0 t y1 t 2 y2 t (2) Hence the second term represents the perturbation on the reference orbital element. Computing the time derivative of Eq. (2) and substituting into Eq. (1) we obtain: Y H Y 0 0 0 H 0 y H Y 1 0 1 Y zeroth order Y0 y1 (3) first order For sake of simplicity let us drop the vector notation (or restrict to just one orbital element Y). The relationship between the “true” element and the reference one is simply given by Eq. (4): Y t Y0 t y1 t (4) where Y0(t) represents the evolution of the reference orbital element. David M. Lucchesi Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario Istituto Nazionale di Astrofisica ORD: The analytical proof Of course, there is a difference between this (reference) orbital element, which is related to the propagation (by numerical integration) of the orbital element over the arc length, and the adjusted orbital element X introduced in the previous Section. The latter is obtained through a fit of the SLR data using GEODYN II with all perturbation models, except the one we are looking for. What about the relationship between X(t) and Y0(t)? In the Figure we see how they work. The continuous black line represents the time evolution, over 1–arc length, of the adjusted element X(t). Orbital element X(t) Y0(t) Y(t) The dot–dashed red line gives the evolution of the reference element Y0(t). Finally the dashed blue line represents the observations Y(t). David M. Lucchesi t0 t1 t Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario ORD: The analytical proof Istituto Nazionale di Astrofisica Y t Y0 t y1 t (4) Eq. (5) gives, as a first approximation, the relation between the two cited elements: Y0 t X t Y t X i 0 Y 0i X X Y i 0i i (5) the lower index i refers to the initial conditions at the beginning of the arc (epoch t0). Therefore, from Eqs. (4) and (5) we obtain: Y t X t Y0 t X i y1 t X t X i y1 t (6) Y0i valid for a small t = t1 – t0 and with: Y0 (t ) Y0i O David M. Lucchesi Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario True element ORD: The analytical proof Istituto Nazionale di Astrofisica Adjusted element Y t X t X i y1 t X i X i Y0 (6) i The quantity Xi must be related to the perturbation y1(t) in order to minimise the difference between Y(t) and X(t), i.e., we need to minimise the quantity: Perturbation t1 y1 t X i dt 2 (7) t0 that is: t1 1 X i y1 t dt t t (8) 0 Our generic perturbation may be written in terms of a secular effect plus a periodical effect and a systematic effect: y1 t A t B cos t C (9) Introducing Eq. (9) into Eq. (8) we obtain: t sin 2 t0 t1 t t X i A cos 0 1 C B t 2 2 2 David M. Lucchesi (10) Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario Istituto Nazionale di Astrofisica ORD: The analytical proof t sin 2 t0 t1 t t X i A cos 0 1 C B t 2 2 2 (10) Now, in order to determine the orbital residual, we take the difference between the orbital elements of two consecutive arcs as underlined in the previous Section. With t2 – t1 = t1 – t0 = t, where t1 is the epoch of the difference, we obtain: X X 2 t1 X 1 t1 X 2i X 1i (11) Then substituting Eq. (10) into Eq. (11) we get: 2 t sin 2 sin t X A t B t 1 t 2 (12) This shows that the secular term is preserved and the systematic effect has been removed; therefore the proposed method is very good for the determination of the secular effects. David M. Lucchesi Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario X(t) Istituto Nazionale di Astrofisica ORD: The analytical proof t X2 sin 2 t0 t1 t t X i A cos 0 1 C B X t 2 2 1 2 Arc-1 (10) Arc-2 Arc-3 Now, in order to determine the orbital residual, we take the difference between the t t t t orbital elements of two consecutive arcs as underlined in the previous Section. t t t 0 1 2 With t2 – t1 = t1 – t0 = t, where t1 is the epoch of the difference, we obtain: X X 2 t1 X 1 t1 X 2i X 1i (11) Then substituting Eq. (10) into Eq. (11) we get: 2 t sin 2 sin t X A t B t 1 t 2 (12) This shows that the secular term is preserved and the systematic effect has been removed; therefore the proposed method is very good for the determination of the secular effects. David M. Lucchesi Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario Istituto Nazionale di Astrofisica ORD: The analytical proof 2 t sin 2 sin t X A t B t 1 t 2 (12) Concerning the long–period effects we generally obtain — with respect to the perturbation expression (Eq. (9)) — y1 t A t B cos t C an amplitude reduction with respect to the initial value B plus a phase shift of /2. If we divide by t we obtain the rate in the residual: 2 t sin 2 X sin t A B 1 t t 2 David M. Lucchesi (13) Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario Istituto Nazionale di Astrofisica ORD: The analytical proof 2 t sin 2 X sin t A B 1 t t 2 (13) In our determination of the residuals (previous Section) we stated that with the difference between the two arcs element we obtain the rate in the element residual, that is: X d y1 t A B sin t1 t dt t1 (14) Obviously, the right hand sides of Eqs. (14) and (13) coincide if: 2 t sin 2 1 t 2 that is if t 2 is small, or equivalently: (15) T t (16) where T represents the period of the disturbing effect. David M. Lucchesi Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario Istituto Nazionale di Astrofisica ORD: The analytical proof 2 t sin 2 X sin t A B 1 t t 2 X d y1 t A B sin t1 t dt t1 (13) (14) Therefore, given a generic perturbation with angular frequency , the ‘’difference– method‘’ correctly reproduces the orbital elements residuals—their rate more precisely—provided that conditions (15) or (16) are satisfied. 2 t sin 2 1 t 2 T t (15) (16) We also notice that the phase of the rate is conserved in this approach. David M. Lucchesi Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario ORD: The analytical proof Istituto Nazionale di Astrofisica t sin 2 t 2 2 All the periodic effects with period T such that: 2 t kT k = integer t sin 2 0 t 2 x are exactly cancelled. That is, a particular choice of the arc length t will allow us to cancel specifics periodic effects shorter than t. This also means that with a convenient choose of the arc length the ‘’difference– method‘’ automatically gives us the longperiod effects removing the short–period ones. Indeed, t=14 days corresponds to an integer number of the LAGEOS satellites orbits, k=89 for LAGEOS orbital period (13,526 s) and k=91 for LAGEOS II orbital period (13,350 s). David M. Lucchesi t 2 2 t sin 2 X sin t A B 1 t t 2 (13) Hence Eq. (13) acts like a filter, which keeps the long–period effects almost unmodified (if T t), while the short period effects are rejected if the time span t is sufficiently long, . Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario Istituto Nazionale di Astrofisica Table of Contents 1. Orbital residuals determination (ORD): the new method; 2. ORD: the new method proof and the Lense-Thirring effect; 3. Application to the secular effects; 4. Application to the periodic effects; 5. ORD, unmodelled effects and background gravity model; 6. Conclusions; David M. Lucchesi Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario Istituto Nazionale di Astrofisica Application to the secular effects: the Lense–Thirring effect LAGEOS and LAGEOS II satellites node–node–perigee combination: Cancels J2 and J4 and solve for . Lageos k1 LageosII k 2 LageosII LT 60 .1 mas yr Ciufolini, Nuovo Cimento (1996) LT 1 General Re lativity 0 Classical Physics X(t) We therefore need to compute the following orbital residuals combination: Lageos k1 LageosII k2LageosII and add over the consecutive arcs differences. David M. Lucchesi X2 X1 Arc-1 Arc-2 Arc-3 t t t t Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario Istituto Nazionale di Astrofisica Application to the secular effects: the Lense–Thirring effect Ciufolini, Chieppa, Lucchesi, Vespe, (1997): Ciufolini, Lucchesi, Vespe, Mandiello, (1996): JGM-3 JGM-3 2.2–year 3.1–year The plot has been obtained after fitting and removing 13 tidal signals and also the inclination residuals. 1.3 0.2 David M. Lucchesi The plot has been obtained after fitting and removing 10 periodical signals. 1.1 0.2 Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario Istituto Nazionale di Astrofisica Application to the secular effects: the Lense–Thirring effect Ciufolini, Pavlis, Chieppa, Fernandes–Vieira, (1998): EGM-96 4–year They fitted (together with a straight line) and removed four small periodic signals, corresponding to: LAGEOS and LAGEOS II nodes periodicity (1050 and 575 days), LAGEOS II perigee period (810 days), and the year periodicity (365 days). 1.10 0.03 David M. Lucchesi Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario Istituto Nazionale di Astrofisica Application to the secular effects: the Lense–Thirring effect Ciufolini, Pavlis, Peron and Lucchesi, (2002): EGM96 7.3–year Four small periodic signals corresponding to: LAGEOS and LAGEOS II nodes periodicity (1050 and 575 days), LAGEOS II perigee period (810 days), and the year periodicity (365 days), have been fitted (together with a straight line) and removed with some non–gravitational signals. 1.00 0.02 David M. Lucchesi Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario Istituto Nazionale di Astrofisica Application to the secular effects: the Lense–Thirring effect LAGEOS and LAGEOS II satellites node–node combination: CHAMP and GRACE Lageos C3 LageosII LT 48.1 mas yr LT Cancels J2 and solve for . 1 General Re lativity 0 Classical Physics X(t) We therefore need to compute the following orbital residuals combination: Lageos C3 LageosII and add over the consecutive arcs differences. David M. Lucchesi X2 X1 Arc-1 Arc-2 Arc-3 t t t t Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario Istituto Nazionale di Astrofisica Application to the secular effects: the Lense–Thirring effect Lucchesi, Adv. Space Res., 2004 LI + 0.546 LII 300 LT 200 After the removal of 6 periodic signals EIGEN2S mas 48.1 yr 9–year Ciufolini & Pavlis, 2004, Letters to Nature 100 47.8 0.4 mas yr 0 0 1000 2000 3000 Time (days) without the removal of periodic signals I 0.545II (mas) 600 4000 (mas) Nodes combination (mas) 400 LT 48.2 400 mas yr EIGEN-GRACE02S 11–year 200 47.9 6 mas yr 0 0 2 4 6 8 10 12 years David M. Lucchesi Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario Istituto Nazionale di Astrofisica Table of Contents 1. Orbital residuals determination (ORD): the new method; 2. ORD: the new method proof and the Lense-Thirring effect; 3. Application to the secular effects; 4. Application to the periodic effects; 5. ORD, unmodelled effects and background gravity model; 6. Conclusions; David M. Lucchesi Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario Istituto Nazionale di Astrofisica Application to the periodic effects: the Yarkovsky–Schach effect In the case of the LAGEOS satellites, the most important periodic non–gravitational perturbation not yet included in the orbit determination software (now included in GEODYN II NASA official version) is the Yarkovsky–Schach effect: 16 ir 2 3 aYS AYS ( ) cos Sˆ AYS R To T Rubincam, 1988, 1990; 9 mc Rubincam et al., 1997; Slabinski 1988, 1997; Afonso et al., 1989; Incident Farinella et al., 1990; Scharroo et al., 1991, Farinella and Vokrouhlický, 1996; Earth Sun Light Métris et al., 1997; Lucchesi, 2001, 2002; Lucchesi et al., 2004; David M. Lucchesi a 2T n Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario Istituto Nazionale di Astrofisica Application to the periodic effects: the Yarkovsky–Schach effect It is therefore interesting to see what happens for the fit of the Yarkovsky–Schach effect from LAGEOS satellites orbital residuals. Here we show the results for the following elements: 1. 2. 3. Eccentricity vector excitations; Perigee rate; Nodal rate; David M. Lucchesi Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario Istituto Nazionale di Astrofisica Application to the periodic effects: the Yarkovsky–Schach effect Eccentricity vector excitations: long–period effects sin S x2 cosI S y2 cos S y S z sin cosI sin S x2 cosI S y2 cos S y S z sin cosI 3 A dk 2 YS cos S x S y cosI S x S y cos S x S z sin cosI dt 4na cos S x S y cosI S x S y cos S x S z sin cosI 2S S sin I cos 2 sin I S S cos S 2 sin sin y z z x z sin S x S y cos S x S z sin S x S y 3 AYS sin S x S y S x S y cos S x S z sin dh 2 dt 4na cos S x2 S y2 cos S y S z sin 2 2 cos S x S y cos S y S z sin where Sx, Sy and Sz are the equatorial components of the satellite spin–vector and represents the ecliptic obliquity. David M. Lucchesi Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario Istituto Nazionale di Astrofisica Application to the periodic effects: the Yarkovsky–Schach effect Z Orbital plane h I e k h k e cos h e sin Equatorial plane e k X Y Ascending Node direction David M. Lucchesi Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario Istituto Nazionale di Astrofisica Application to the periodic effects: the Yarkovsky–Schach effect Argument of perigee rate: long–period effects sin h1 k3 sn h1 k3 sin h2 k 4 sin h2 k 4 3 A cos h3 k1 cos h3 k1 d 2 YS dt 8nae cos h4 k 2 cos h4 k 2 sin k5 sin k5 cos k 6 cos k 6 where the quantities h1 … h4 and k1 … k6 are functions of the satellite spin–axis components, the satellite inclination and ecliptic obliquity. David M. Lucchesi Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario Istituto Nazionale di Astrofisica Application to the periodic effects: the Yarkovsky–Schach effect Ascending node longitude rate: long–period effects sin S x2 sin I S y2 cos S y S z sin sin I sin S x2 sin I S y2 cos S y S z sin sin I AYS d cos S x S y S y S z cos S x S z sin sin I 2 F dt 4na sin I cos S x S y S y S z cos S x S z sin sin I cos S S cosI x z sin S S cos S 2 sin cosI y z z F 1 cos 1 sin cos 1 1 cos 1 sin sin 1 2 1 where F is due to the dependency from the physical shadow function, represents the mean motion times the retroreflectors thermal inertia. David M. Lucchesi Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario Istituto Nazionale di Astrofisica Application to the periodic effects: the Yarkovsky–Schach effect Concerning the periodic long–term perturbing effects on the satellite elements, the orbital residuals rate determined with the ‘’difference–method‘’ may give a wrong result if the conditions: 2 t sin T 2 1 t t are not satisfied. 2 In this case the residuals will be indeed affected by an amplitude reduction. This condition is related to the periodicity of a given perturbation (T) and to the arc length (t). In particular, the lower the periodicity T of a given component the larger will the amplitude reduction be with the ‘’difference–method‘’. David M. Lucchesi 2 t sin 2 X sin t A B 1 t t 2 X d y1 t A B sin t1 t dt t1 Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario Istituto Nazionale di Astrofisica Application to the periodic effects: the Yarkovsky–Schach effect Our point here is to verify if this perturbation can be derived correctly from the LAGEOS satellites residuals or if some caution must be taken because of amplitude reduction in one or more of the periodic components that characterise the effect. LAGEOS II eccentricity vector excitations: Spectral line Period (days) Angular rate (rad/day) 953 226 365 6.59103 27.80103 17.21103 x t 2 sin x x 2 0.99929 0.98744 0.99517 As we can see the amplitude reduction is negligible, less than 1.3% in its maximum discrepancy. David M. Lucchesi Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario Istituto Nazionale di Astrofisica Application to the periodic effects: the Yarkovsky–Schach effect x LAGEOS II argument of perigee rate: Spectral line Period (days) Angular rate (rad/day) 447 4244 309 175 252 665 14.06103 1.48103 20.33103 35.90103 24.93103 9.45103 t 2 sin x x 2 0.99678 0.99996 0.99326 0.97912 0.98989 0.99854 As we can see the amplitude reduction is negligible, about 2% in its maximum discrepancy. David M. Lucchesi Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario Istituto Nazionale di Astrofisica Application to the periodic effects: the Yarkovsky–Schach effect LAGEOS II argument of perigee rate: Numerical simulation Lucchesi, 2002 Spectral analysis over 5 years 0,5 685 Most important lines: 249 Amplitude S1/2 0,4 0,3 665 days 252 days 0,2 0,1 433 1031 315 365 155 0,0 0,000 0,002 0,004 0,006 0,008 0,010 (1/days) David M. Lucchesi Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario Istituto Nazionale di Astrofisica Application to the periodic effects: the Yarkovsky–Schach effect LAGEOS II ascending node longitude rate: Spectral line Period (days) Angular rate (rad/day) 2 2 2 113 183 139 55.60103 34.33103 45.20103 x t 2 sin x x 2 0.95051 0.98089 0.96707 As we can see the amplitude reduction is very small, less than 5% in its maximum discrepancy. However, the impact of the Yarkovsky–Schach effect on the nodal rate is very small. David M. Lucchesi Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario Istituto Nazionale di Astrofisica Application to the periodic effects: the Yarkovsky–Schach effect LAGEOS II perigee rate residuals: EGM96 Residuals LAGEOS II perigee rate (mas/yr) 10000 The rms of the residuals is about 3372 mas/yr. 5000 0 -5000 -10000 Residuals in LAGEOS II perigee rate (mas/yr) over a time span of about 7.8 years starting from January 1993. 0 500 1000 1500 Time (days) David M. Lucchesi 2000 2500 3000 These residuals have been obtained modelling the LAGEOS II orbit with the GEODYN II dynamical model, which does not include the solar Yarkovsky– Schach effect. The EGM96 gravity field solution model has been used. Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario Istituto Nazionale di Astrofisica Application to the periodic effects: the Yarkovsky–Schach effect LAGEOS II perigee rate residuals: Spectral analysis over 7.8 years 0,5 The three main spectral lines: 0,4 252 665 days 309 days 0,3 0,2 309 252 days Amplitude: S1/2 665 0,1 are well known spectral lines that characterise the Yarkovsky–Schach effect in LAGEOS II perigee rate (Lucchesi, 2002). David M. Lucchesi 0,0 0,00 0,01 0,02 0,03 0,04 Frequency (1/days) Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario Istituto Nazionale di Astrofisica Application to the periodic effects: the Yarkovsky–Schach effect LAGEOS II perigee rate residuals: LAGEOS II perigee rate (mas/yr) 10000 Residuals YS (Lucchesi, 2002) EGM96 5000 Yarkovsky–Schach effect as in Lucchesi 2002 but with the LOSSAM model for the satellite spin–axis evolution (Andrés et al., 2004). Yarkovsky–Schach parameters: 0 AYS = 103.5 pm/s2 for the amplitude -5000 -10000 = 2113 s for the CCR thermal inertia 0 500 1000 1500 2000 2500 3000 Time (days) As we can see, the numerical simulation of the Yarkovsky–Schach effect on the perigee rate well reproduces the satellite perigee rate residuals determined from the 7.8 years analysis of LAGEOS II orbit. This means that the Yarkovsky–Schach thermal effect strongly influences the satellite perigee rate residuals with its characteristic signatures. David M. Lucchesi Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario Istituto Nazionale di Astrofisica Application to the periodic effects: the Yarkovsky–Schach effect LAGEOS II perigee rate residuals: Fit for the Yarkovsky–Schach effect amplitude AYS LAGEOS II perigee rate (mas/yr) 10000 Residuals YS fit (Lucchesi et al., 2004) YS (Lucchesi, 2002) EGM96 5000 The plot (red line) represents the best–fit we obtained for the Yarkovsky–Schach perturbation assuming that this is the only disturbing effect influencing the LAGEOS II argument of perigee. 0 Initial Yarkovsky–Schach parameters: AYS = 103.5 pm/s2 for the amplitude -5000 -10000 = 2113 s for the CCR thermal inertia 0 500 1000 1500 Time (days) 2000 2500 3000 Final Yarkovsky–Schach amplitude: With EGM96 in GEODYN II software and the AYS = 193.2 pm/s2 LOSSAM model in the independent numerical i.e., about 1.9 times the pre–fit value. simulation (red and blue lines). Lucchesi, Ciufolini, Andrés, Pavlis, Peron, Noomen and Currie, Plan. Space Science, 52, 2004 David M. Lucchesi Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario Istituto Nazionale di Astrofisica Application to the periodic effects: the Yarkovsky–Schach effect LAGEOS II perigee rate residuals: Fit for the Yarkovsky–Schach effect amplitude AYS Residuals YS fit (Lucchesi et al., 2004) YS (Lucchesi, 2002) LAGEOS II perigee rate (mas/yr) 10000 5000 0 No improvements have been obtained varying the thermal inertia of the satellite. -5000 -10000 This result reduces the rms of the post–fit residuals to a value of about 2029 mas/yr, corresponding to a fractional reduction of about 40% with respect to the initial value. Correlation 0.795 0 500 1000 1500 2000 2500 3000 Time (days) The independence of the fit rms from the thermal inertia is due to the independence of the perigee rate expression from this characteristic time, see Lucchesi (2002) and also Métris et al. (1997). Indeed, while the semimajor axis, inclination and nodal rates depend on both the CCR thermal inertia and the amplitude of the perturbative effect, the perigee rate is a function of the Yarkovsky–Schach effect amplitude only. David M. Lucchesi Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario Istituto Nazionale di Astrofisica Application to the periodic effects: the Yarkovsky–Schach effect Real component rate residuals: Residuals YS fit (Lucchesi et al., 2004) YS (Lucchesi, 2002) Real component rate (mas/yr) 150 100 AYS = 193.2 pm/s2 50 0 -50 -100 -150 No direct fit, but the same amplitude obtained from the perigee rate fit has been assumed, that is: 0 500 1000 1500 2000 2500 3000 We have obtained a very good agreement between the orbital residuals and the numerical integration performed for the nominal Yarkovsky–Schach perturbing effect. Time (days) The long-term oscillations of the effect—characterised by the strong yearly periodicity— are clearly visible in the orbital residuals, see also Lucchesi (2002). The pre–fit rms was about 63 mas/yr, while the post–fit value is about 32 mas/yr with a fractional reduction of about 49% David M. Lucchesi Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario Istituto Nazionale di Astrofisica Application to the periodic effects: the Yarkovsky–Schach effect LAGEOS II perigee rate residuals: Finally we investigated the sensitivity of the Yarkovsky–Schach perturbations recovery by our method, to the reference gravity field model used, since this is the major source of disturbances on the LAGEOS orbits. We did it by using another model, here GGM01S, recently computed from the GRACE twin satellites mission. Residuals The Yarkovsky–Schach effect amplitude has not been adjusted, but it is just the one fitted to the observations with EGM96 (Lucchesi et al., 2004). 10000 LAGEOS II perigee rate (mas/yr) The satellite residuals, in mas/yr, have been obtained from an analysis of about 8.9 years of LAGEOS II orbital data, starting from January 1993, using the GGM01S gravity field model in the GEODYN II software. YS (Lucchesi et al., 2004) GGM01S 5000 0 -5000 -10000 -500 Correlation 0.731 0 500 1000 1500 2000 2500 3000 3500 Time (days) David M. Lucchesi Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario Istituto Nazionale di Astrofisica Application to the periodic effects: the Yarkovsky–Schach effect LAGEOS II perigee rate residuals: 7.8 years comparison of LAGEOS II perigee rate residuals, determined with the “difference method”, with two different gravity fields solutions: EGM96 GGM01S LAGEOS II perigee rate residuals (mas/yr) In the Figure we compare directly, and on the same period of 7.8 years, the LAGEOS II perigee rate residuals obtained with our method using on one hand the EGM96 gravity field model (continuous line) in the whole process and on the other hand the GGM01S model (dotted line). EGM96 GGM01S 10000 EGM96 GGM01S 5000 0 -5000 -10000 0 500 1000 1500 2000 2500 3000 Time (days) David M. Lucchesi Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario Istituto Nazionale di Astrofisica Application to the periodic effects: the Yarkovsky–Schach effect Statistics of the differences between the LAGEOS II perigee rate residuals obtained with our method and with two different gravity field models: EGM96 and GGM01S, over the common period of 7.8 years (the values are in mas/yr). LAGEOS II perigee rate residuals (mas/yr) LAGEOS II perigee rate residuals: EGM96 GGM01S 10000 5000 0 EGM96 GGM01S -5000 -10000 0 500 1000 1500 2000 2500 3000 Time (days) The correlations are between the determined residuals and the independent fit obtained using the Yarkovsky–Schach perturbation over a 7.8 years period (Lucchesi et al., 2004). David M. Lucchesi Min 8711.00 8658.99 Max +9439.08 +8488.11 Mean 170.99 +349.24 rms 3371.60 3395.87 Correlation 0.80 0.75 Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario Istituto Nazionale di Astrofisica One more application: The anomalous J2 behaviour (1998) Around 1998 J2 reversed its decreasing trend and started increasing. At present no theoretical explanation of this effect. d J 2 2.6 10 11 yr 1 dt Cox and Chao, Science 297, 2002 J 2 CA M R2 David M. Lucchesi Deleflie et al., 2003 (Advances in Geosciences) have been able to prove that this anomalous behaviour cannot be due to a correlation with the 18.6 years solid tide. The previous trend is due mainly to the slow rebound of the polar caps after the end of the last glaciation. The ice melting spread out mass from the poles regions, diminishing the (CA) difference between the moments of inertia; the crust responds to the new load with a delay. Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario Istituto Nazionale di Astrofisica One more application: The “difference–method” and LAGEOS residuals The anomalous J2 behaviour (1998) 400 The analysis using EGM96 (as well as other gravity fields) has been performed by Ciufolini, Pavlis and Peron (New Astronomy, 11, 2006). LAGEOS nodal rate (mas/yr) 300 EGM96 100 0 -100 500 -200 400 -300 0 1000 2000 Time (days) 1998 2 Class 3 R cos I J 2 n 2 a 1 e2 2 3000 Node (mas) Nodal rate (mas/yr) 200 LAGEOS node (mas) 1998 300 4000 200 100 EGM96 0 0 1000 2000 3000 4000 Time (days) David M. Lucchesi Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario Istituto Nazionale di Astrofisica Table of Contents 1. Orbital residuals determination (ORD): the new method; 2. ORD: the new method proof and the Lense-Thirring effect; 3. Application to the secular effects; 4. Application to the periodic effects; 5. ORD, unmodelled effects and background gravity model; 6. Conclusions; David M. Lucchesi Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario Istituto Nazionale di Astrofisica ORD, unmodelled effects and background gravity model The orbital residuals represent a powerful tool to obtain information on poorly modelled forces, or to detect new disturbing effects due to force terms missing in the dynamical model used for the satellite orbit simulation and differential correction procedure. However, once the residuals have been determined, we must be very careful in order to estimate the magnitude and the behaviour of the unmodelled effects: 1. the unmodelled effects are mixed; 2. they may have similar signatures (correlations …); 3. reliability of the models implemented in the software for the POD; 4. use of empirical accelerations during the POD; 5. …; In the case of the two LAGEOS orbital residuals, several unmodelled long– period gravitational effects, mainly related with tides and the time variations of Earth’s zonal harmonic coefficients, are superimposed with unmodelled NGP due to thermal thrust effects and the asymmetric reflectivity from the satellites surface. David M. Lucchesi Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario Istituto Nazionale di Astrofisica ORD, unmodelled effects and background gravity model In order to bypass such problems, we need to look to different elements when estimating the parameters of a given unmodelled effect and also to different satellites (hopefully with same POD). GAUSS equations may help us: GAUSS equations (when the perturbing force is generic) da 2 T eT cos f R sin f dt n 1 e 2 de 1 e2 R sin f T cos f cos u dt na dI W r cos f dt H d W r sin f dt H sin I d 1 e2 1 cos I d R cos f T sin f sin u dt nae dt 1 e2 Acc Rrˆ Ttˆ Wwˆ R = radial acceleration T = transversal acceleration W = out–of–plane acceleration d ' 2 r d d R 1 e2 cos I dt na a dt dt David M. Lucchesi Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario Istituto Nazionale di Astrofisica ORD, unmodelled effects and background gravity model A few examples: Reliability of the models and empirical accelerations: EGM96 In the measurement of the LT effect with JGM3 and EGM96 the LAGEOS satellites nodes were combined with LAGEOS II perigee in order to cancel the uncertainties in J2 and J4 and solve for the LT effect parameter : Lageos k1 LageosII k 2 LageosII LT 60 .1 mas yr EGM96 7.3–year Lense-Thirring effect David M. Lucchesi J2 and J4 cancelled Empirical Accels: Aemp A0 AS sin f AC cos f Aemp R, T ,W Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario Istituto Nazionale di Astrofisica ORD, unmodelled effects and background gravity model LAGEOS II node - EGM96 100 LAGEOS node - EGM96 400 0 350 300 -200 node (mas) node (mas) -100 -300 -400 250 200 150 100 -500 50 -600 0 -700 -500 0 500 1000 1500 2000 2500 3000 3500 Time (days) -50 -500 0 500 1000 1500 2000 2500 3000 3500 Time (days) 80 Combined nodes (mas) 60 Combined nodes Lageos C3 LageosII LT 48.1 mas yr 40 Only J2 cancelled 20 0 -20 -500 0 500 1000 1500 2000 2500 3000 3500 Bad combination because of J4 error Time (days) David M. Lucchesi Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario Istituto Nazionale di Astrofisica ORD, unmodelled effects and background gravity model 80 60 -7 5,400x10 40 C(4,0) Combined nodes (mas) EGM96 EIGEN2S Combined nodes 20 -7 5,399x10 Only J2 cancelled 0 -7 5,398x10 Bad combination because of J4 error -20 -500 0 500 1000 1500 2000 2500 3000 3500 4 Time (days) -10 10000 8000 Degree 5,0x10 LAGEOS II EGM96 0 accels EGM96 5 accels -10 6000 4,0x10 4000 Coefficients Errors perigee rate (mas/yr) EGM96 EIGEN2S Difference 2000 0 -2000 -4000 -10 2,0x10 -10 -6000 1,0x10 Empirical Accels: -8000 -10000 -500 -10 3,0x10 0 500 1000 1500 2000 Time (days) David M. Lucchesi 2500 3000 0,0 3500 2 4 6 Degree Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario Istituto Nazionale di Astrofisica ORD, unmodelled effects and background gravity model GGM01S EIGEN2S Combined nodes (mas) 1400 1200 1000 800 600 400 200 0 -200 -400 -600 -800 -1000 -1200 -1400 -500 0 500 1000 1500 2000 2500 3000 -7 5,4010x10 Time (days) 500 1 .2 400 1 300 200 100 EGM96 0 EIGEN2S GGM01S -500 0 EIGEN-GRACE02S -7 5,4020x10 3500 GGM01S EIGEN2S 500 1000 1500 2000 2500 3000 3500 Time (days) -7 5,4000x10 C(4,0) LAGEOS II nodal rate (mas/yr) 1800 1600 Reliability of the models and empirical accelerations: GGM01S A shift is present on LAGEOS satellites nodal rate when comparing different gravity field models. There is a 20% deviation for the Lense-Thirring effect measurement with respect to the relativistic prediction (due to the larger J4). EGM96 -7 5,3990x10 -7 5,3980x10 -7 5,3970x10 4 David M. Lucchesi Degree Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario Istituto Nazionale di Astrofisica Table of Contents 1. Orbital residuals determination (ORD): the new method; 2. ORD: the new method proof and the Lense-Thirring effect; 3. Application to the secular effects; 4. Application to the periodic effects; 5. ORD, unmodelled effects and background gravity model; 6. Conclusions; David M. Lucchesi Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario Istituto Nazionale di Astrofisica Conclusions This new method of determination of the satellite orbital residuals has been quoted in the literature since 1996 to determine the LAGEOS satellites orbital residuals in the case of the relativistic Lense–Thirring precession measurement; We have justified the new method (difference–method) both practically and analytically; The method has been proved to work correctly in the case of the secular effects recovery; In the case of the periodic effects some caution must instead be taken under some conditions, but the method works very well for the estimate of the unmodelled long–period effects; David M. Lucchesi Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario Istituto Nazionale di Astrofisica Conclusions The main results obtained can be summarised as follows: 1. the method is based on the difference between the satellite orbital elements belonging to two consecutive arcs of 15 days length (a one day overlap reduces the time interval between differences to 14 days), instead of a single long–arc which would fit daily values of predetermined elements, as usually done; 2. the difference value is a measure of the misclosure in the element rate and not in the element itself; 3. with regard to the secular effects, the arc length depends on the entity of the secular effect to be determined in relation with the accuracy in the range observations of the tracking system. Moreover, concerning the arc length, caution must be considered in order to avoid the possibility of stroboscopic effects in the computed residuals if we instead take too short arcs; David M. Lucchesi Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario Istituto Nazionale di Astrofisica Conclusions 4. the analytical study has proved that the unmodelled secular effects are determined very well with the introduced method, without loosing any information, while the constant (systematic) errors are removed with the differencing procedure; 5. concerning the periodic effects, the analytical study has shown that the phase of the effects is conserved (in the rate), but some amplitude reduction exists if some condition is not satisfied. This amplitude reduction must be considered case by case, in order to see if it is negligible or not. Anyway, each reduced amplitude may in principle be corrected a–posteriori by an ad hoc analysis; 6. we applied successfully the method to the determination of the secular perturbation produced by the Lense–Thirring effect when combining the nodes of LAGEOS and LAGEOS II; David M. Lucchesi Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario Istituto Nazionale di Astrofisica Conclusions 7. in the case of the analysed Yarkovsky–Schach effect, clearly visible in LAGEOS II orbital residuals, we proved that the “difference– method” could be well used to fit the effect parameters, more precisely the amplitude; in this case the amplitude reduction is negligible for each periodic component of the non–gravitational effect; The ‘’difference–method‘’ for the orbital residuals determination is therefore a useful tool in satellite geodesy for the study of the poorly modelled or unmodelled gravitational and non–gravitational effects resulting in secular and/or long–period perturbations. In particular we are now able to remove the unmodelled Yarkovsky–Schach effect from the orbital residuals of the LAGEOS satellites and look at other subtle perturbations. However, caution must be devoted to such operations … David M. Lucchesi Beijing, July 21 - 2006 Istituto di Fisica dello Spazio Interplanetario Istituto Nazionale di Astrofisica Conclusions This presentation has been mainly based on the work: The LAGEOS satellites orbital residuals determination and the Lense–Thirring effect measurement by David M. Lucchesi and Georges Balmino Planetary Space Science, 54, 581–593 (2006) finis David M. Lucchesi Beijing, July 21 - 2006