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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).

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## 36th COSPAR Scientific Assembly Beijing, China, 16 – 23 July 2006

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**36th COSPAR Scientific AssemblyBeijing, 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: david.lucchesi@ifsi-roma.inaf.it 2) Istituto di Scienza e Tecnologie della Informazione ISTI/CNR Via Moruzzi, 1, 56124 Pisa, Italy**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.**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).**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): • precise orbit determination (POD); • orbital residuals determination (ORD); • Statistical analysis; • accurate modelling;**Table of Contents**Orbital residuals determination (ORD): the new method; ORD: the new method proof and the Lense-Thirring effect; Application to the secular effects; Application to the periodic effects; ORD, unmodelled effects and background gravity model; Conclusions;**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): 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. Vector of parameters to be determined Observation error of the i-th observation**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 The meaning of orbital residuals: The computed element (C) Models implemented in the orbital analysis of LAGEOS satellites with GEODYN II**The meaning of orbital residuals: The observed**element (O) Of course, this is only an idealway 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 orbitalelements used to define the orbit orientation and satellite position in space. Hence, we need a practicalway 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**The meaning of orbital residuals: 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 Courtesy of R. Peron**The meaning of orbital residuals: 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 adjustedelements of the long–arc with the previously determined dailyelements.**Long arc**Daily values 0 0 15 15 30 30 45 45 time time Residuals The meaning of orbital residuals: The usual way 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.**The meaning of orbital residuals: 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): we first subdivide the satellite orbit analysis in arcs of 15 days time span (arc length); 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; 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); 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;**The meaning of orbital residuals: 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 notmodelled 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).**The meaning of orbital residuals: 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.**X(t)**X2 X1 Arc-1 Arc-2 Arc-3 t t t t The meaning of orbital residuals: 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.**X(t)**X2 X1 Arc-1 Arc-2 Arc-3 t t t t The meaning of orbital residuals: The new method That is, the continuouslinefits 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 hasnotbeenincluded or is partly unknown. Therefore, the difference Arc-2 minus Arc-1 represents the secular and long–period orbital residual in the element X. 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.**X(t)**X2 X1 Arc-1 Arc-2 Arc-3 t t t t The meaning of orbital residuals: 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:**Table of Contents**Orbital residuals determination (ORD): the new method; ORD: the new method proof and the Lense-Thirring effect; Application to the secular effects; Application to the periodic effects; ORD, unmodelled effects and background gravity model; Conclusions;**ORD: The analytical proof**(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). • 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 5.3109m/s2, again much smaller than the monopole term. • Under this approximation the differential equations for the osculating orbital elements can be treated following the perturbation theory.**Perturbation**ORD: The analytical proof If represents the vector of the orbital elements as a function of time, the corresponding differential equations can be written as: (1) where 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. The solution is: (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).**ORD: The analytical proof**(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: (3) 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): (4) where Y0(t) represents the evolution of the reference orbital element.**Orbital element**X(t) Y0(t) Y(t) t0 t1 t 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). The dot–dashed red line gives the evolution of the reference element Y0(t). Finally the dashed blue line represents the observations Y(t).**(4)**ORD: The analytical proof Eq. (5) gives, as a first approximation, the relation between the two cited elements: (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: (6) valid for a small Dt = t1 – t0 and with:**Adjusted element**True element (6) Perturbation ORD: The analytical proof 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: (7) that is: (8) Our generic perturbation may be written in terms of a secular effect plus a periodical effect and a systematic effect: (9) Introducing Eq. (9) into Eq. (8) we obtain: (10)**ORD: The analytical proof**(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: (11) Then substituting Eq. (10) into Eq. (11) we get: (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.**t0**t1 t2 X(t) X2 X1 Arc-1 Arc-2 Arc-3 t t t t ORD: The analytical proof (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: (11) Then substituting Eq. (10) into Eq. (11) we get: (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.**ORD: The analytical proof**(12) Concerning the long–period effects we generally obtain — with respect to the perturbation expression (Eq. (9)) — 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: (13)**ORD: The analytical proof**(13) In our determination of the residuals (previous Section) we stated that with the difference between the twoarcselement we obtain the rate in the element residual, that is: (14) Obviously, the right hand sides of Eqs. (14) and (13) coincide if: (15) (16) that is if is small, or equivalently: where T represents the period of the disturbing effect.**ORD: The analytical proof**(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. (15) (16) We also notice that the phase of the rate is conserved in this approach.**ORD: The analytical proof**All the periodic effects with period T such that: k = integer 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 long-period 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). Hence Eq. (13) acts like a filter, which keeps the long–period effects almost unmodified (if ), while the short period effects are rejected if the time span t is sufficiently long, .**Istituto di Fisica dello Spazio Interplanetario**Istituto Nazionale di Astrofisica Table of Contents Orbital residuals determination (ORD): the new method; ORD: the new method proof and the Lense-Thirring effect; Application to the secular effects; Application to the periodic effects; ORD, unmodelled effects and background gravity model; Conclusions;**X(t)**X2 X1 Arc-1 Arc-2 Arc-3 t t t t 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 . Ciufolini, Nuovo Cimento (1996) We therefore need to compute the following orbital residuals combination: and add over the consecutive arcs differences.**Application to the secular effects: the Lense–Thirring**effect Ciufolini, Chieppa, Lucchesi, Vespe, (1997): Ciufolini, Lucchesi, Vespe, Mandiello, (1996): JGM-3 3.1–year JGM-3 2.2–year The plot has been obtained after fitting and removing 13 tidal signals and also the inclination residuals. The plot has been obtained after fitting and removing 10 periodical signals.**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).**Application to the secular effects: the Lense–Thirring**effect Ciufolini, Pavlis, Peron and Lucchesi, (2002): 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. EGM96 7.3–year**X(t)**X2 X1 Arc-1 Arc-2 Arc-3 t t t t Application to the secular effects: the Lense–Thirring effect LAGEOS and LAGEOS II satellites node–node combination: CHAMP and GRACE Cancels J2 and solve for . We therefore need to compute the following orbital residuals combination: and add over the consecutive arcs differences.**EIGEN2S**9–year I 0.545II (mas) 600 400 200 0 EIGEN-GRACE02S 11–year (mas) 0 2 4 6 8 10 12 years Application to the secular effects: the Lense–Thirring effect Lucchesi, Adv. Space Res., 2004 After the removal of 6 periodic signals Ciufolini & Pavlis, 2004, Letters to Nature without the removal of periodic signals**Table of Contents**Orbital residuals determination (ORD): the new method; ORD: the new method proof and the Lense-Thirring effect; Application to the secular effects; Application to the periodic effects; ORD, unmodelled effects and background gravity model; Conclusions;**Incident**Earth Sun Light 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: Rubincam, 1988, 1990; Rubincam et al., 1997; Slabinski 1988, 1997; Afonso et al., 1989; Farinella et al., 1990; Scharroo et al., 1991, Farinella and Vokrouhlický, 1996; Métris et al., 1997; Lucchesi, 2001, 2002; Lucchesi et al., 2004;**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: Eccentricity vector excitations; Perigee rate; Nodal rate;**Application to the periodic effects: the Yarkovsky–Schach**effect Eccentricity vector excitations: long–period effects where Sx, Sy and Sz are the equatorial components of the satellite spin–vector and represents the ecliptic obliquity.**e**Application to the periodic effects: the Yarkovsky–Schach effect Z Orbital plane Equatorial plane I Y X Ascending Node direction**Application to the periodic effects: the Yarkovsky–Schach**effect Argument of perigee rate: long–period effects where the quantities h1 … h4 and k1 … k6 are functions of the satellite spin–axis components, the satellite inclination and ecliptic obliquity.**Application to the periodic effects: the Yarkovsky–Schach**effect Ascending node longitude rate: long–period effects where Fis due to the dependency from the physical shadow function, represents the mean motion times the retroreflectors thermal inertia.**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: are not satisfied. 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‘’.**Spectral line**Period (days) Angular rate (rad/day) 953 226 365 6.59103 27.80103 17.21103 0.99929 0.98744 0.99517 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: As we can see the amplitude reduction is negligible, less than 1.3% in its maximum discrepancy.**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 0.99678 0.99996 0.993260.97912 0.98989 0.99854 Application to the periodic effects: the Yarkovsky–Schach effect LAGEOS II argument of perigee rate: As we can see the amplitude reduction is negligible, about 2% in its maximum discrepancy.**Application to the periodic effects: the Yarkovsky–Schach**effect LAGEOS II argument of perigee rate: Numerical simulation Lucchesi, 2002 Spectral analysis over 5 years Most important lines: 665 days 252 days**Spectral line**Period (days) Angular rate (rad/day) 113 183 139 55.60103 34.33103 45.20103 0.95051 0.98089 0.96707 Application to the periodic effects: the Yarkovsky–Schach effect LAGEOS II ascending node longitude rate: 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.**Application to the periodic effects: the Yarkovsky–Schach**effect LAGEOS II perigee rate residuals: EGM96 Residuals in LAGEOS II perigee rate (mas/yr) over a time span of about 7.8 years starting from January 1993. The rms of the residuals is about 3372 mas/yr. These residuals have been obtained modelling the LAGEOS II orbit with the GEODYN II dynamical model, which does notinclude the solar Yarkovsky–Schach effect. The EGM96 gravity field solution model has been used.

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