36th cospar scientific assembly beijing, china, 16 – 23 july 2006

Istituto Nazionale di AstrofisicaIstituto di Fisica dello Spazio Interplanetario

Beijing, July 21 - 2006David M. Lucchesi

36th COSPAR Scientific AssemblyBeijing, China, 16 – 23 July 2006

David M. Lucchesi (1,2)

The LAGEOS satellites orbital residuals determination and the way

to extract gravitational and non–gravitational unmodelled perturbing

effects

1) Istituto di Fisica dello Spazio Interplanetario IFSI/INAF

Via Fosso del Cavaliere, 100, 00133 Roma, ItalyEmail: [email protected]

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.



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):

I. precise orbit determination (POD); II. orbital residuals determination (ORD); III. Statistical analysis;IV. accurate modelling;

Preamble



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;



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.

The meaning of orbital residuals

ijj j

iii OP

P

CCO

,, xxP Vector of parameters to be determined

Observation error of the i-th observationiO



The meaning of orbital residuals: The computed element (C)

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

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 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



The meaning of orbital residuals: GEODYN II range residuals

Accuracy in the data reduction

From January 3, 1993

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.

LAGEOS range residuals (RMS)

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 adjusted elements of the long–arc with the previously determined daily elements.



time0 15 30 45

The meaning of orbital residuals: The usual way

Daily values

Long arc

time0 15 30 45

Residuals

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):

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;




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).




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.




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.

X1

X2

X(t)

t

Arc-1 Arc-2 Arc-3

ttt

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.




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.

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.

X1

X2

X(t)

t

Arc-1 Arc-2 Arc-3

ttt




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:

nnn XSS

XSS

XS

1

212

11

ttt

ttt

t

nn 1

12

1

X1

X2

X(t)

t

Arc-1 Arc-2 Arc-3

ttt



Table of Contents






6. Conclusions;



ORD: The analytical proof

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 5.3109 m/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.

11107 monopole

NGP

9102 monopole

GP

(Lucchesi and Balmino, Plan. Space Sci., 54, 2006)




If represents the vector of the orbital elements as a function of time, the corresponding differential equations can be written as:

Y

YHYHY

10

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.

0H

tytytYtY 22

10

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).

(1)

(2)

The solution is:Perturbation



orderfirstyY

HYHy

orderzerothYHY

Y

10

011

000

0


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:

tytytYtY 22

10 (2)

(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):

tytYtY 10 (4)

where Y0(t) represents the evolution of the reference orbital element.



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).

Orbital element

tt0 t1

X(t)

Y0(t)

Y(t)



Eq. (5) gives, as a first approximation, the relation between the two cited elements:


i

i

YXX

XY

tYtYtX

ii

i

0

0

00

(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:

tyXtXtyXY

tYtXtY ii

i

110

0

valid for a small t = t1 – t0 and with:

(6)

OYtYi

00 )(

tytYtY 10 (4)



1

0

21

t

t

i dtXty

1

0

1

1t

t

i dttyt

X

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:


that is:

(7)

(8)

Our generic perturbation may be written in terms of a secular effect plus a periodical effect and a systematic effect:

CtBtAty cos1(9)

Introducing Eq. (9) into Eq. (8) we obtain:

Ctt

t

t

Btt

AX i

2

cos

2

2sin

21010

(10)

tyXtXtY i 1 (6)True element

Perturbation

Adjusted element

iYXX ii 0




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:

Ctt

t

t

Btt

AX i

2

cos

2

2sin

21010

(10)

ii

XXtXtXX 121112

Then substituting Eq. (10) into Eq. (11) we get:

1

2

sin

2

2sin

tt

t

tBtAX

(11)

(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.




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:

Ctt

t

t

Btt

AX i

2

cos

2

2sin

21010

(10)

ii

XXtXtXX 121112

Then substituting Eq. (10) into Eq. (11) we get:

1

2

sin

2

2sin

tt

t

tBtAX

(11)

(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.

X1

X2

X(t)

t

Arc-1 Arc-2 Arc-3

tttt0 t1 t2



1

2

sin

2

2sin

tt

t

BAt

X

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:


CtBtAty cos1

1

2

sin

2

2sin

tt

t

tBtAX

(12)

(13)



11 sin1

tBAtydt

d

t

X

t

1

2

2sin

2

t

t


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:

(14)

1

2

sin

2

2sin

tt

t

BAt

X

(13)

Obviously, the right hand sides of Eqs. (14) and (13) coincide if:

(15)

that is if is small, or equivalently:

where T represents the period of the disturbing effect.

2tt

T

(16)



11 sin1

tBAtydt

d

t

X

t

1

2

2sin

2

t

t


(14)

1

2

sin

2

2sin

tt

t

BAt

X

(13)

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)

We also notice that the phase of the rate is conserved in this approach.

t

T

(16)




0

2

2sin

2

t

t

2

tx

kTt k = integer

2

2

2sin

t

t

are exactly cancelled.

All the periodic effects with period T such that:

)13(sin

2

2sin

1

2

tt

t

BAt

X

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, .

tT

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).




Table of Contents






6. Conclusions;



Application to the secular effects: the Lense–Thirring effect

yrmaskk LTLageosIILageosIILageos 1.6021

LageosIILageosIILageos kk 21

PhysicsClassical

lativityGeneralLT 0

Re1

We therefore need to compute the following orbital residuals combination:

and add over the consecutive arcs differences.

X1

X2

X(t)

t

Arc-1 Arc-2 Arc-3

ttt

LAGEOS and LAGEOS II satellites node–node–perigee combination:

Ciufolini, Nuovo Cimento (1996)

Cancels J2 and J4 and solve for .



JGM-3

2.2–year

JGM-3

3.1–year

The plot has been obtained after fitting and removing 10 periodical signals.

2.01.1

The plot has been obtained after fitting and removing 13 tidal signals and also the inclination residuals.

2.03.1

Ciufolini, Chieppa, Lucchesi, Vespe, (1997):Ciufolini, Lucchesi, Vespe, Mandiello, (1996):





EGM-96

4–year

Ciufolini, Pavlis, Chieppa, Fernandes–Vieira, (1998):

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).

03.010.1




EGM96

7.3–year

Ciufolini, Pavlis, Peron and Lucchesi, (2002):

Four small periodic signalscorresponding 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.

02.000.1




yrmasC LTLageosIILageos 1.483

LageosIILageos C 3

LAGEOS and LAGEOS II satellites node–node combination: CHAMP and GRACE

X1

X2

X(t)

t

Arc-1 Arc-2 Arc-3

ttt

We therefore need to compute the following orbital residuals combination:

and add over the consecutive arcs differences.

PhysicsClassical

lativityGeneralLT 0

Re1

Cancels J2 and solve for .



0 1000 2000 3000 4000

0

100

200

300

400 LI + 0.546 LII

Nod

es c

ombi

natio

n (m

as)

Time (days)

yrmas4.08.47

yr

masLT 1.48 EIGEN2S

9–year

0 2 4 6 8 10 12 years

yr

masLT 2.48

I 0.545II (mas)

(m

as)

600

400

200

0

yrmas69.47

EIGEN-GRACE02S

11–year


Lucchesi, Adv. Space Res., 2004

Ciufolini & Pavlis, 2004, Letters to Nature

After the removal of 6 periodic signals

without the removal of periodic signals



Table of Contents






6. Conclusions;



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;

Incident

Sun Light

Earth

n

Ta

2

SAa YSYSˆcos)(

TTR

mcA o

irYS 32

9

16




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. Eccentricity vector excitations;

2. Perigee rate;

3. Nodal rate;




sinsincossin2cossin2

cossincoscoscos

cossincoscoscos

cossincoscossin

cossincoscossin

4

3

2

22

22

2

zzyzx

zxyxyx

zxyxyx

zyyx

zyyx

YS

SSSIISS

ISSSSISS

ISSSSISS

ISSSIS

ISSSIS

na

A

dt

dk

sincoscos

sincoscos

sincossin

sincossin

4

3

22

222

zyyx

zyyx

zxyxyx

yxzxyx

YS

SSSS

SSSS

SSSSSS

SSSSSS

na

A

dt

dh

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.




Orbital plane Equatorial plane

X Y

Z

e

I

Ascending Node direction

k

h

sin

cos

eh

ek

hke



66

55

2424

1313

4242

3131

2

coscos

sinsin

coscos

coscos

sinsin

sin

8

3

kk

kk

khkh

khkh

khkh

khsnkh

nae

A

dt

d YS


where the quantities h1 … h4 and k1 … k6 are functions of the satellite spin–axis components, the satellite inclination and ecliptic obliquity.

Argument of perigee rate: long–period effects



F

ISSS

ISS

ISSSSSS

ISSSSSS

ISSSIS

ISSSIS

Ina

A

dt

d

zzy

zx

zxzyyx

zxzyyx

zyyx

zyyx

YS

cossincossin

coscos

sinsincoscos

sinsincoscos

sinsincossinsin

sinsincossinsin

sin4

2

22

22

2

112

sinsin1

cos1cossin1cos1

1

F


Ascending node longitude rate: long–period effects

where F is due to the dependency from the physical shadow function, represents the mean motion times the retroreflectors thermal inertia.



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.


1

2

2sin

2

t

t

t

T

11 sin1

tBAtydt

d

t

X

t

1

2

sin

2

2sin

tt

t

BAt

X

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‘’.



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.


953 226 365

6.59103 27.80103 17.21103

0.99929 0.98744 0.99517

Spectral line Period (days) Angular rate (rad/day) 2sin xx

LAGEOS II eccentricity vector excitations:2

tx

As we can see the amplitude reduction is negligible, less than 1.3% in its maximum discrepancy.



LAGEOS II argument of perigee rate:


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


As we can see the amplitude reduction is negligible, about 2% in its maximum discrepancy.

2

tx




LAGEOS II argument of perigee rate: Numerical simulation Lucchesi, 2002

0,000 0,002 0,004 0,006 0,008 0,010

0,0

0,1

0,2

0,3

0,4

0,5

155

1031365

315433

249685

Ampl

itude

S1/

2

(1/days)

Spectral analysis over 5 years

665 days

252 days

Most important lines:



LAGEOS II ascending node longitude rate:


2

2

2

113 183 139

55.60103 34.33103 45.20103

0.95051 0.98089 0.96707


2

tx

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.



LAGEOS II perigee rate residuals:

0 500 1000 1500 2000 2500 3000-10000

-5000

0

5000

10000 Residuals

LAG

EO

S II

per

igee

rate

(mas

/yr)

Time (days)


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 not include the solar Yarkovsky–Schach effect.

The EGM96 gravity field solution model has been used.

EGM96



0,00 0,01 0,02 0,03 0,04

0,0

0,1

0,2

0,3

0,4

0,5

309

252

665

Ampl

itude

: S1/

2

Frequency (1/days)



Spectral analysis over 7.8 yearsThe three main spectral lines:

are well known spectral lines that characterise the Yarkovsky–Schach effect in LAGEOS II perigee rate (Lucchesi, 2002).

665 days

252 days

309 days





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:

AYS = 103.5 pm/s2 for the amplitude

= 2113 s for the CCR thermal inertia

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.

0 500 1000 1500 2000 2500 3000-10000

-5000

0

5000

10000

Residuals YS (Lucchesi, 2002)

LAG

EO

S II

per

igee

rate

(mas

/yr)

Time (days)

EGM96




LAGEOS II perigee rate residuals: Fit for the Yarkovsky–Schach effect amplitude AYS

0 500 1000 1500 2000 2500 3000-10000

-5000

0

5000

10000

Residuals YS fit (Lucchesi et al., 2004) YS (Lucchesi, 2002)

LAG

EO

S II

per

igee

rate

(mas

/yr)

Time (days)

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.

Initial Yarkovsky–Schach parameters:

AYS = 103.5 pm/s2 for the amplitude

= 2113 s for the CCR thermal inertia

Final Yarkovsky–Schach amplitude:

AYS = 193.2 pm/s2

i.e., about 1.9 times the pre–fit value.

Lucchesi, Ciufolini, Andrés, Pavlis, Peron, Noomen and Currie, Plan. Space Science, 52, 2004

With EGM96 in GEODYN II software and the LOSSAM model in the independent numerical simulation (red and blue lines).

EGM96




0 500 1000 1500 2000 2500 3000-10000

-5000

0

5000

10000


LAG

EO

S II

per

igee

rate

(mas

/yr)

Time (days)

LAGEOS II perigee rate residuals: Fit for the Yarkovsky–Schach effect amplitude AYS

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.

No improvements have been obtained varying the thermal inertia of the satellite.

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.

Correlation 0.795



Real component rate residuals:

0 500 1000 1500 2000 2500 3000-150

-100

-50

0

50

100

150


Rea

l com

pone

nt ra

te (m

as/y

r)

Time (days)


No direct fit, but the same amplitude obtained from the perigee rate fit has been assumed, that is:

AYS = 193.2 pm/s2

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%

We have obtained a very good agreement between the orbital residuals and the numerical integration performed for the nominal Yarkovsky–Schach perturbing effect.





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.

-500 0 500 1000 1500 2000 2500 3000 3500

-10000

-5000

0

5000

10000

Residuals YS (Lucchesi et al., 2004)

LAG

EO

S II

per

igee

rate

(mas

/yr)

Time (days)

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.

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). Correlation 0.731

GGM01S



0 500 1000 1500 2000 2500 3000

-10000

-5000

0

5000

10000

EGM96 GGM01S

LAG

EO

S II

per

igee

rate

resi

dual

s (m

as/y

r)

Time (days)



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).

7.8 years comparison of LAGEOS II perigee rate residuals, determined with the “difference method”, with two different gravity fields solutions:

EGM96

GGM01S

EGM96 GGM01S



0 500 1000 1500 2000 2500 3000

-10000

-5000

0

5000

10000

EGM96 GGM01S

LAG

EO

S II

per

igee

rate

resi

dual

s (m

as/y

r)

Time (days)



Min

Max

Mean

rms

Correlation

8711.00

+9439.08

170.99

3371.60

0.80

8658.99

+8488.11

+349.24

3395.87

0.75

EGM96 GGM01S

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).

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).



One more application:

The anomalous J2 behaviour (1998)

Cox and Chao, Science 297, 2002

Around 1998 J2 reversed its decreasing trend and started increasing.

At present no theoretical explanation of this effect.

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.

22

RM

ACJ

1112 106.2 yrJ

dt

d

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.



One more application: The “difference–method” and LAGEOS residuals

The anomalous J2 behaviour (1998)

0 1000 2000 3000 4000-300

-200

-100

0

100

200

300

400

LAGEOS nodal rate (mas/yr)

Nod

al ra

te (m

as/y

r)

Time (days)

0 1000 2000 3000 4000

0

100

200

300

400

500 LAGEOS node (mas)

Nod

e (m

as)

Time (days)

1998

1998

222

2

1

cos

2

3J

e

I

a

RnClass

The analysis using EGM96 (as well as other gravity fields) has been performed by Ciufolini, Pavlis and Peron (New Astronomy, 11, 2006).

EGM96

EGM96



Table of Contents






6. Conclusions;



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.




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:

dt

dI

dt

deR

a

r

nadt

d

dt

dIu

efTfR

nae

e

dt

d

frIH

W

dt

d

frH

W

dt

dI

ufTfRna

e

dt

de

fRfTeTendt

da

cos12

cossin1

1sincos

1

sinsin

cos

coscossin1

sincos1

2

2'

2

2

2

2

GAUSS equations

wWtTrRAcc ˆˆˆ

R = radial acceleration

T = transversal acceleration

W = out–of–plane acceleration

(when the perturbing force is generic)




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 :

EGM96

7.3–year

yrmaskk LTLageosIILageosIILageos 1.6021

J2 and J4 cancelled

WTRA

fAfAAA

emp

CSemp

,,

cossin0

Empirical Accels:

A few examples:

Lense-Thirring effect




-500 0 500 1000 1500 2000 2500 3000 3500

-20

0

20

40

60

80

Com

bine

d no

des

(mas

)

Time (days)

Combined nodes

-500 0 500 1000 1500 2000 2500 3000 3500

-700

-600

-500

-400

-300

-200

-100

0

100

node

(m

as)

Time (days)

LAGEOS II node - EGM96

-500 0 500 1000 1500 2000 2500 3000 3500-50

0

50

100

150

200

250

300

350

400

node

(m

as)

Time (days)

LAGEOS node - EGM96

yrmasC LTLageosIILageos 1.483

Only J2 cancelled

Bad combination because of J4 error




-500 0 500 1000 1500 2000 2500 3000 3500-10000

-8000

-6000

-4000

-2000

0

2000

4000

6000

8000

10000

perig

ee r

ate

(mas

/yr)

Time (days)

EGM96 0 accels EGM96 5 accels

4

5,398x10-7

5,399x10-7

5,400x10-7

EGM96 EIGEN2S

C(4

,0)

Degree

-500 0 500 1000 1500 2000 2500 3000 3500

-20

0

20

40

60

80

Com

bine

d no

des

(mas

)

Time (days)

Combined nodes

2 4 6

0,0

1,0x10-10

2,0x10-10

3,0x10-10

4,0x10-10

5,0x10-10

EGM96 EIGEN2S Difference

Coe

ffici

ents

Err

ors

Degree

Only J2 cancelled

Bad combination because of J4 error

Empirical Accels:

LAGEOS II




Reliability of the models and empirical accelerations: GGM01SA 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).

-500 0 500 1000 1500 2000 2500 3000 3500

-1400

-1200

-1000

-800

-600

-400

-200

0

200

400

600

800

1000

1200

1400

1600

1800

LAG

EO

S II

nod

al r

ate

(mas

/yr)

Time (days)

EGM96 GGM01S EIGEN2S

-500 0 500 1000 1500 2000 2500 3000 3500

0

100

200

300

400

500

Com

bine

d no

des

(mas

)

Time (days)

GGM01S EIGEN2S

2.11

4

5,3970x10-7

5,3980x10-7

5,3990x10-7

5,4000x10-7

5,4010x10-7

5,4020x10-7 EGM96 EIGEN2S GGM01S EIGEN-GRACE02S

C(4

,0)

Degree



Table of Contents






6. 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–ThirringLense–Thirring precession measurement;

We have justified the new method (difference–methoddifference–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;

Conclusions



The main results obtained can be summarised as follows:

Conclusions

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;



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;



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 …



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)

Conclusions

finis

36th cospar scientific assembly beijing, china, 16 – 23 july 2006

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