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 5.3109 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  k2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
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  0.545II (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 cosI   S y2 cos   S y S z sin   cosI  



sin     S x2 cosI   S y2 cos   S y S z sin   cosI  



3
A
dk

 2  YS cos    S x S y cosI   S x S y cos   S x S z sin   cosI   
dt
4na 

cos    S x S y cosI   S x S y cos   S x S z sin   cosI   
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
4na sin I  cos    S x S y  S y S z cos   S x S z sin   sin I   
cos S S cosI  

x z


sin   S S cos   S 2 sin   cosI 

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
6.59103
27.80103
17.21103
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
14.06103
1.48103
20.33103
35.90103
24.93103
9.45103
  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
55.60103
34.33103
45.20103
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 
CA
M  R2
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 (CA)
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  eT 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