Transcript Construction of the Non-Rigid Earth Rotation Series

Investigation of the short periodic
terms of the rigid and non-rigid
Earth rotation series
V.V. Pashkevich
Central (Pulkovo) Astronomical Observatory
of the Russian Academy of Science
St.Petersburg
Space Research Centre of the Polish Academy of Sciences
Warszawa
2008
Aims:
Investigation of the short periodic terms (diurnal and subdiurnal) in a long time new high-precision series of the rigid
(S9000A) and non-rigid Earth rotation (SN9000A).
Choice of the optimal spectral analysis scheme for this
investigation.
Method
1. To update the previous version of the high-precision
numerical solution of the rigid Earth rotation (V.V.Pashkevich,
G.I.Eroshkin and A.Brzezinski, 2004), (V.V.Pashkevich and
G.I.Eroshkin, Proceedings of “Journees 2004”) by using the
integration step of 1 day and the interpolation step of 0.1 day.
All calculations have been carried out with the quadruple
precision (Real 16) on PC Intel® Core™2 Duo E6600 (2.4GHz) .
We considered only the Kinematical (relativistic) case.
Method (continued)
2. The initial conditions have been calculated from the model
SMART97 (P.Bretagnon, G.Francou, P.Rocher,
J.L.Simon,1998). The discrepancies between the numerical
solution and the semi-analytical solution SMART97 were
expressed in Euler angles over 2000 years with 0.1 day spacing.
3. Investigation of the discrepancies has been carried out by the
least squares (LSQ) and by the spectral analysis (SA) algorithms
(V.V.Pashkevich and G.I.Eroshkin, Proc. “Journees 2005”).
4. The high-precision rigid Earth rotation series S9000
(V.V.Pashkevich and G.I.Eroshkin, 2005 ) has been update by
including the short periodic terms (diurnal and sub-diurnal).
5. The new high-precision non-rigid Earth rotation series
(SN9000A) has been constructed. This series is expressed in the
function of the three Euler angles and containing the short
periodic terms. We applied the method of P.Bretagnon et al.
(1999) and the transfer function of P. M. Mathews et al.
(2002).
Part I:
The rigid Earth rotation
Fig.1 Numerical solution for the rigid Earth rotation minus solution
SMART97 in the longitude of the ascending node of the Earth equator.
the computer system Parsytec CCe20 case
Secular terms of…
smart97 (as)
the personal computer Intel® Core™2 Duo case
Secular terms of…
smart97 (as)
d (as)
d (as)
7.00
50384564881.3693 T
2
- 107194853.5817 T
- 1143646.1500 T
3
4
1328317.7356 T
- 9396.2895 T
5
9.51
- 206.50 T
50384564881.3693 T
2
- 3451.30 T
3
1125.00 T
4
- 788.00 T
5
- 57.50 T
6
- 3415.00 T
- 107194853.5817 T
- 1143646.1500 T
2
3
4
1328317.7356 T
- 9396.2895 T
5
- 211.16 T
2
- 3481.61 T
3
1142.76 T
4
- 712.75 T
- 71.86 T
5
6
- 3464.88 T
Fig.2 Numerical solution for the rigid Earth rotation minus solution
SMART97 in the angle of the proper rotation of the Earth.
the computer system Parsytec CCe20 case
Secular terms of…
smart97 (as)
Secular terms of…
smart97 (as)
d(as)
1009658226149.3691
6.58
474660027824506304.0000 T
2
- 98437693.3264 T
3
- 1217008.3291 T
4
1409526.4062 T
5
- 9175.8967 T
the personal computer Intel® Core™2 Duo case
d (as)
1009658226149.3691
99598.30 T
2
- 7182.30 T
3
1066.80 T
4
- 750.00 T
5
- 30.30 T
6
- 3676.00 T
8.90
474660027824506304.0000 T
2
- 98437693.3264 T
3
- 1217008.3291 T
4
1409526.4062 T
5
- 9175.8967 T
-236.95 T
2
-7235.36 T
3
524.24 T
4
-684.84 T
5
-40.93 T
6
-3717.78 T
Fig.3 Numerical solution for the rigid Earth rotation minus solution
SMART97 in the inclination angle.
the computer system Parsytec CCe20 case
the personal computer Intel® Core™2 Duo case
Secular terms of…
Secular terms of…
 smart97(as)
d(as)
 smart97(as)
d(as)
84381409000.0000
1.42
84381409000.0000
1.59
- 265011.2586 T
- 96.61 T
2
5127634.2488 T
3
- 7727159.4229 T
4
- 4916.7335 T
5
33292.5474 T
- 265011.2586 T
2
- 353.10 T
3
771.50 T
4
- 84.50 T
5
- 86.00 T
6
- 247.50 T
- 97.64 T
2
5127634.2488 T
3
- 7727159.4229 T
4
- 4916.7335 T
5
33292.5474 T
2
- 355.10 T
3
775.68 T
4
- 79.67 T
5
- 89.51 T
6
-250.65 T
Fig.4. Differences between the numerical solutions and SMART97
after the formal removal of secular trends.
the computer system Parsytec CCe20 case
the personal computer Intel® Core™2 Duo case
Spectral Analysis (SA) A L G O R I T H M – 1
for removal of the periodical terms from the Discrepancies (D)
D after removal of the
secular trend
Set of nutation terms of
SMART97
LSQ method computes the amplitudes
of the spectrum of D for the all periodical terms
LSQ method determines the amplitude and
phase of the largest from the residual harmonics
if |Am| > ||
No
Yes
till the end of
the spectrum
… Construction of the
new nutation series
Removal of this harmonic
from the D and
from the Spectrum
New highprecision
series
S9000A1
Spectral Analysis (SA) A L G O R I T H M – 2
for removal of the periodical terms from the Discrepancies (D)
D after removal of the
secular trend
Set of nutation terms of
SMART97
LSQ method computes the amplitudes
of the spectrum of D for the long periodical terms
LSQ method determines the amplitude and
phase of the largest from the residual harmonics
if |Am| > ||
No
Yes
till the end of
the spectrum
… Construction of the
new nutation series
Removal of this harmonic
from the D and
from the Spectrum
Construction of
the new D-2 and
apply the
algorithm - 1
New highprecision
series
S9000A
Fig.5. Spectra of the discrepancies between numerical solutions
and SMART97 for the proper rotation angle.
for only long periodical terms
the computer system Parsytec CCe20 case
for all periodical terms ALGORITHM – 1
the personal computer Intel® Core™2 Duo case
Fig.5. Spectra of the discrepancies between numerical solutions
and SMART97 for the proper rotation angle. DETAIL
for only long periodical terms
the computer system Parsytec CCe20 case
for all periodical terms ALGORITHM – 1
the personal computer Intel® Core™2 Duo case
Fig.6. The numerical solution - 2 NS-2 (NS-2A1) of the rigid Earth
rotation minus S9000 (S9000A1) after formal removal of the secular
trends in the proper rotation angle.
the computer system Parsytec CCe20 case
the personal computer Intel® Core™2 Duo case
for ALGORITHM – 1
The personal computer Intel® Core™2 Duo case
Fig.7. Spectra of the discrepancies between numerical solutions
and SMART97 for the proper rotation angle.
for only long periodical terms ALGORITHM – 2
1-iteration
for all periodical terms ALGORITHM – 1
The personal computer Intel® Core™2 Duo case
Fig.7. Spectra of the discrepancies between numerical solutions
and SMART97 for the proper rotation angle. DETAIL
for only long periodical terms ALGORITHM – 2
1-iteration
for all periodical terms ALGORITHM – 1
Fig.8. The numerical solution – 2 NS-2A2 (NS-2A1 ) of the rigid Earth
rotation minus S9000A2 (S9000A1) after formal removal of the secular
trends in the proper rotation angle. The personal computer Intel® Core™2 Duo case
The end of 1-iteration of the
ALGORITHM – 2 (for only long periodical terms)
The end of the
ALGORITHM – 1 ( for all periodical terms)
The personal computer Intel® Core™2 Duo case
Fig.9. Spectra of the discrepancies between numerical solutions
and SMART97 for the proper rotation angle.
for all periodical terms ALGORITHM – 2
2-iteration
for all periodical terms ALGORITHM – 1
fig.7
The personal computer Intel® Core™2 Duo case
Fig.9. Spectra of the discrepancies between numerical solutions
and SMART97 for the proper rotation angle. DETAIL
for all periodical terms ALGORITHM – 2
2-iteration
for all periodical terms ALGORITHM – 1
fig.7
Fig.10. The numerical solution – 3(2) NS-3(NS-2A1) of the rigid Earth
rotation minus S9000A (S9000A1) after formal removal of the secular
trends in the proper rotation angle. The personal computer Intel® Core™2 Duo case
for all periodical terms ALGORITHM – 2
2-iteration
for all periodical terms ALGORITHM – 1
fig.8
Part II:
The non-rigid Earth rotation
TRANSFER FUNCTION
P.M. Mathews, T.A. Herring, B.A. Buffett, 2002:
4

eR   


Q  
T ( ; e | eR ) 
N 0 1  1     Q0  

 eR  1 
 1   s  

Here (  1) is frequency in cycles per sidereal day (cpsd)
with respect to inertial space
e /(1  e)
T ( 1; e | eR )  N 0 , where N 0 
1
eR /(1  eR )
1
eR , e  ellipticities for rigid and non-rigid Earth, respectively.
Geophisical model includes the following effects:
electromagnetic coupling, ocean tides, mantle inelasticity,
atmospheric influence, change in the global Earth
dynamical flattening and in the core flattening
Expressions for Euler angles:
   p   

   p    ,

   p   
p 
 1t  ...   5t 5 

 p   0  1t  ...  5t 
5
 p   0  1t  ...   5t 
5
0  0
 2

   [ S jk sin( j 0  j1t )  C jk cos( j 0  j1t )] t 
j k 0

4

k 
   [S jk sin( j 0  j1t )  C jk cos( j 0  j1t )] t 
j k 0

4

   [ S jk sin( j 0  j1t )  C jk cos( j 0  j1t )] t k 

j k 0
4
k
Algorithm of Bretagnon et al. (1999)
1.The rigid Earth angular velocity vector:
pR   R sin  R sin  R   R cos R
qR   R sin  R cos R   R sin  R
rR   R cos  R   R
2.The non-rigid Earth angular velocity vector is obtained by:
 pNR  iqNR    pR  iqR  T ( ; e | eR )
rNR  rR
3.The derivatives of Euler angles for the non-rigid Earth rotation:
 NR   pNR sin  NR  qNR cos  NR  / sin NR
 NR  pNR cos  NR  qNR sin  NR
 NR  rNR  NR cos NR
Common
form of the periodic part of Euler angles:
m
x   [ Ak sin(c0  c1t )  Bk cos(c0  c1t )] t k
k 0
m
x   [(kAk  c1Bk 1 )sin(c0  c1t ) 
k 1
 (kBk  c1 Ak 1 ) cos(c0  c1t )] t k 1 
 c1[ Bm sin(c0  c1t )  Am cos(c0  c1t )] t m ,
Here x   j ,  j ,  j Cascade method:
kAk  c1 Bk 1   k 1 
Am   m / c1



kBk  c1 Ak 1   k 1 
Bm   m / c1


Here k  1,..., m   Ak 1  (  k 1  kBk ) / c1 

c1 Bm   m
Bk 1  (kAk   k 1 ) / c1 


c1 Am   m


Here k  m,...,1
3
Details:
Iterative solution of the algorithm of Bretagnon et al. (1999)
(Pashkevich, 2008)
1-th iteration

0
NR
 R  
1
NR
 pNR cos 
0
NR
 qNR sin 
and by Cascade method from  3  

1
NR
 ( pNR sin 

1
NR
 rNR 
1
NR
0
NR
1
NR
 qNR cos  ) / sin 
cos 
0
NR
1
NR
1
NR
0
NR


and from  3  
1
NR
,

Here pNR , qNR , rNR  pNR ( ), qNR ( ), rNR ( )
1
NR
Details:
Iterative solution of the algorithm of Bretagnon et al. 1999
(Pashkevich, 2008)
n-th iteration

n
NR
 pNR cos 
n 1
NR
 qNR sin 
n 1
NR
n
and by Cascade method from  3   NR


n
NR
 ( pNR sin 

n
NR
 rNR 
n
NR
n 1
NR
 qNR cos 
cos 
n
NR
n 1
NR
) / sin 
n
NR
and from  3  

n
NR
,
n
NR
Iterations are repeated until the absolute value of the difference
between iterations K-1 and K exceedes some a priori adopted values 
Here pNR , qNR , rNR  pNR ( ), qNR ( ), rNR ( )
Results:
Table 1. Comparison of different solutions for selected durnal and
semidurnal periodical terms in the longitude of the ascending
node of the Earth equator.
Period (days),
Argument
Solution
rigid Earth
rotation
Ψ(sin)
μas
Ψ(cos)
μas
Solution
non-rigid
Earth rotation
Ψ(sin)
μas
Ψ(cos)
μas
1.03505
λ3 +D- 
SMART97
S9000A
-34.8213
-34.8213
4.2714
4.2714
SMN
SN9000A
-33.6604
-33.6604
1.5930
1.5930
1.03474
3λ3 +3D-2F- 
SMART97
S9000A
-0.0133
-0.0806
0.0016
-0.0612
SMN
SN9000A
-0.0225
-0.0898
-0.0100
-0.0728
1.00000
λ3 - 
SMART97
S9000A
-0.1996
-0.1996
0.0275
0.0275
SMN
SN9000A
-0.1969
-0.1969
0.0005
0.0005
0.99758
λ3 +D-l- 
SMART97
S9000A
-19.8544
-19.8521
2.4906
2.4941
SMN
SN9000A
-19.8316
-19.8293
-1.0929
-1.0894
0.96215
λ3 +D+ 
SMART97
S9000A
38.1283
38.1283
4.6945
4.6945
SMN
SN9000A
36.3795
36.3795
2.1620
2.1620
0.527517
3λ3 +3D-2 
SMART97
S9000A
-0.1783
-0.1783
0.2578
0.2578
SMN
SN9000A
-0.3211
-0.3211
0.4389
0.4389
1997; BMS- Bretagnon et al., 1999; MHB- Mathews et al., 2002,SMN=SMART97+MHB; SN9000A=S9000A+MHB.
Results:
Table 1. Comparison of different solutions for selected durnal and
semidurnal periodical terms in the longitude of the ascending
node of the Earth equator.
Period (days),
Argument
Solution
rigid Earth
rotation
Ψ(sin)
μas
Ψ(cos)
μas
Solution
non-rigid
Earth rotation
Ψ(sin)
μas
Ψ(cos)
μas
0.51753
2λ3 +2D- 2
SMART97
S9000A
25.3564
25.3564
14.5558
14.5558
SMN
SN9000A
26.3775
26.3775
21.4672
21.4672
0.507904
λ3 +D- 2
SMART97
S9000A
-0.2193
-0.2193
0.3213
0.3213
SMN
SN9000A
-0.3258
-0.3258
0.3536
0.3536
0.50000
2λ3-2 
SMART97
S9000A
10.6324
10.6324
6.0997
6.0997
SMN
SN9000A
11.2675
11.2675
9.2705
9.2705
0.49863
2
SMART97
S9000A
31.8523
31.8523
-18.2836
-18.2836
SMN
SN9000A
33.8043
33.8043
-27.8520
-27.8520
0.49860
λ3 +D-F- 2
SMART97
S9000A
4.3198
4.3198
2.4797
2.4797
SMN
SN9000A
4.5837
4.5837
3.7774
3.7774
0.49795
λ3+2 
SMART97
S9000A
-0.1957
-0.1957
-0.2142
-0.2142
SMN
SN9000A
-0.2732
-0.2732
-0.2130
-0.2130
1997; BMS- Bretagnon et al., 1999; MHB- Mathews et al., 2002,SMN=SMART97+MHB; SN9000A=S9000A+MHB.
Results:
Table 2. Comparison of different solutions for selected durnal and
semidurnal periodical terms in the inclination angle.
Period (days),
Argument
Solution
rigid Earth
rotation
θ (sin)
μas
θ (cos)
μas
Solution
non-rigid
Earth rotation
θ (sin)
μas
θ (cos)
μas
1.03505
λ3 +D- 
SMART97
S9000A
-1.6400
-1.6400
-13.3708
-13.3708
SMN
SN9000A
-0.5515
-0.5515
-12.6886
-12.6886
1.03474
3λ3 +3D-2F- 
SMART97
S9000A
-0.0010
0.0073
-0.0085
-0.0065
SMN
SN9000A
-0.0044
0.0039
-0.0006
0.0014
1.00000
λ3 - 
SMART97
S9000A
-0.0109
-0.0109
-0.0790
-0.0790
SMN
SN9000A
-0.0001
-0.0001
-0.0777
-0.0777
0.99758
λ3 +D-l- 
SMART97
S9000A
-0.9877
-0.9891
-7.8729
-7.8720
SMN
SN9000A
0.4389
0.4375
-7.8522
-7.8513
0.96215
λ3 +D+ 
SMART97
S9000A
1.8625
1.8625
-15.1270
-15.1270
SMN
SN9000A
0.8531
0.8531
-14.4120
-14.4120
0.527517
3λ3 +3D-2 
SMART97
S9000A
0.0201
0.0201
0.0131
0.0131
SMN
SN9000A
0.0915
0.0915
0.0471
0.0471
1997; BMS- Bretagnon et al., 1999; MHB- Mathews et al., 2002,SMN=SMART97+MHB; SN9000A=S9000A+MHB.
Results:
Table 2. Comparison of different solutions for selected durnal and
semidurnal periodical terms in the inclination angle.
Period (days),
Argument
Solution
rigid Earth
rotation
θ (sin)
μas
θ (cos)
μas
Solution
non-rigid
Earth rotation
θ (sin)
μas
θ (cos)
μas
0.51753
2λ3 +2D- 2
SMART97
S9000A
-5.7900
-5.7900
10.0863
10.0863
SMN
SN9000A
-8.5395
-8.5395
10.4933
10.4933
0.507904
λ3 +D- 2
SMART97
S9000A
-0.1128
-0.1128
-0.0770
-0.0770
SMN
SN9000A
-0.1071
-0.1071
-0.1077
-0.1077
0.50000
2λ3-2 
SMART97
S9000A
-2.4263
-2.4263
4.2293
4.2293
SMN
SN9000A
-3.6876
-3.6876
4.4822
4.4822
0.49863
2
SMART97
S9000A
-7.2728
-7.2728
-12.6702
-12.6702
SMN
SN9000A
-11.0792
-11.0792
-13.4478
-13.4478
0.49860
λ3 +D-F- 2
SMART97
S9000A
-0.9867
-0.9867
1.7190
1.7190
SMN
SN9000A
-1.5033
-1.5033
1.8250
1.8250
0.49795
λ3+2 
SMART97
S9000A
-0.0852
-0.0852
0.0778
0.0778
SMN
SN9000A
-0.0847
-0.0847
0.1086
0.1086
1997; BMS- Bretagnon et al., 1999; MHB- Mathews et al., 2002,SMN=SMART97+MHB; SN9000A=S9000A+MHB.
Results:
Table 3. Comparison of different solutions for selected durnal and
semidurnal periodical terms in the angle of the proper rotation
of the Earth.
Period (days),
Argument
Solution
rigid Earth
rotation
φ (sin)
μas
φ (cos)
μas
Solution
non-rigid
Earth rotation
φ (sin)
μas
φ (cos)
μas
1.03505
λ3 +D- 
SMART97
S9000A
-32.2917
-32.2917
3.9612
3.9612
SMN
SN9000A
-31.2266
-31.2266
1.5038
1.5038
1.03474
3λ3 +3D-2F- 
SMART97
S9000A
-0.0129
-0.0754
0.0016
-0.0564
SMN
SN9000A
-0.0213
-0.0838
-0.0090
-0.0670
1.00000
λ3 - 
SMART97
S9000A
-0.1834
-0.1834
0.0253
0.0253
SMN
SN9000A
-0.1809
-0.1809
0.0005
0.0005
0.99758
λ3 +D-l- 
SMART97
S9000A
-18.2334
-18.2314
2.2873
2.2908
SMN
SN9000A
-18.2125
-18.2105
-1.0005
-0.9974
0.96215
λ3 +D+ 
SMART97
S9000A
35.0904
35.0904
4.3205
4.3205
SMN
SN9000A
33.4859
33.4859
1.9970
1.9970
0.527517
3λ3 +3D-2 
SMART97
S9000A
-0.1117
-0.1117
0.1630
0.1630
SMN
SN9000A
-0.2427
-0.2427
0.3291
0.3291
1997; BMS- Bretagnon et al., 1999; MHB- Mathews et al., 2002,SMN=SMART97+MHB; SN9000A=S9000A+MHB.
Results:
Table 3. Comparison of different solutions for selected durnal and
semidurnal periodical terms in the angle of the proper rotation
of the Earth.
Period (days),
Argument
Solution
rigid Earth
rotation
φ (sin)
μas
φ (cos)
μas
Solution
non-rigid
Earth rotation
φ (sin)
μas
φ (cos)
μas
0.51753
2λ3 +2D- 2
SMART97
S9000A
-0.1279
-0.1279
-0.0728
-0.0728
SMN
SN9000A
0.8089
0.8089
6.2684
6.2684
0.507904
λ3 +D- 2
SMART97
S9000A
-0.2471
-0.2471
0.3617
0.3617
SMN
SN9000A
-0.3448
-0.3448
0.3914
0.3914
0.50000
2λ3-2 
SMART97
S9000A
-0.4033
-0.4033
-0.2352
-0.2352
SMN
SN9000A
0.1794
0.1794
2.6739
2.6739
0.49863
2
SMART97
S9000A
31.9704
31.9704
-18.3513
-18.3513
SMN
SN9000A
33.7614
33.7614
-27.1303
-27.1303
0.49860
λ3 +D-F- 2
SMART97
S9000A
4.7817
4.7817
2.7448
2.7448
SMN
SN9000A
5.0238
5.0238
3.9354
3.9354
0.49795
λ3+2 
SMART97
S9000A
-0.1964
-0.1964
-0.2148
-0.2148
SMN
SN9000A
-0.2675
-0.2675
-0.2137
-0.2137
1997; BMS- Bretagnon et al., 1999; MHB- Mathews et al., 2002,SMN=SMART97+MHB; SN9000A=S9000A+MHB.
Kinematical solution of the rigid Earth rotation= Dynamical
solution of the rigid Earth rotation + Geodetics corrections
Kinematical solution of the non-rigid Earth rotation= Kinematical
solution of the rigid Earth rotation + TRANSFER FUNCTION
CONCLUSION
•
•
•
The optimal spectral analysis scheme for this investigation
(with respect to the accuracy) has been determined. It is
algorithm-2, which used two iterations. The first iteration
investigated the discrepancies for only the long periodical
harmonics, while the second iteration investigated the
discrepancies for all harmonics.
The new version the high-precision rigid Earth rotation
series S9000A has been constructed. It contains the short
periodic terms (diurnal and sub-diurnal).
The high-precision non-rigid Earth rotation series SN9000A
(containing the short periodic terms) has been constructed.
This series is expressed in the function of the three Euler
angles with respect to the fixed ecliptic plane and equinox
J2000.0 and is dynamically adequate to the ephemerides
DE404/LE404 over 2000 years.
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ACKNOWLEDGMENTS
The investigation was carried out at the
Central (Pulkovo) Astronomical
Observatory of the Russian Academy
of Sciences
and at the
Space Research Centre
of the Polish Academy of Sciences,
under a financial support of the
agreement cooperation between
the Polish and Russian Academies
of Sciences, Theme No 31.