investigation of the short periodic terms of the rigid and non-rigid earth rotation series v.v....
TRANSCRIPT
Investigation of the short periodic terms of the rigid and non-rigid
Earth rotation series V.V. Pashkevich
Central (Pulkovo) Astronomical Observatoryof the Russian Academy of Science
St.PetersburgSpace Research Centre of the Polish Academy of Sciences
Warszawa
2008
Investigation of the short periodic terms (diurnal and sub-diurnal) 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.
Method1. 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.
Aims:
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 the personal computer Intel® Core™2 Duo case
Secular terms of… Secular terms of…
smart97 (as) d (as) smart97 (as) d (as)
7.00 9.51
50384564881.3693 T - 206.50 T 50384564881.3693 T - 211.16 T
- 107194853.5817 T2
- 3451.30 T2
- 107194853.5817 T2
- 3481.61 T2
- 1143646.1500 T3
1125.00 T3
- 1143646.1500 T3
1142.76 T3
1328317.7356 T4
- 788.00 T4
1328317.7356 T4
- 712.75 T4
- 9396.2895 T5
- 57.50 T5
- 9396.2895 T5
- 71.86 T5
- 3415.00 T6
- 3464.88 T6
Fig.2 Numerical solution for the rigid Earth rotation minus solution SMART97 in the angle of the proper rotation of the Earth.
Secular terms of… Secular terms of…
smart97 (as) d(as) smart97 (as) d (as)
1009658226149.3691 6.58 1009658226149.3691 8.90
474660027824506304.0000 T 99598.30 T 474660027824506304.0000 T -236.95 T
- 98437693.3264 T2
- 7182.30 T2
- 98437693.3264 T2
-7235.36 T2
- 1217008.3291 T3
1066.80 T3
- 1217008.3291 T3
524.24 T3
1409526.4062 T4
- 750.00 T4
1409526.4062 T4
-684.84 T4
- 9175.8967 T5
- 30.30 T5
- 9175.8967 T5
-40.93 T5
- 3676.00 T6
-3717.78 T6
the computer system Parsytec CCe20 case the personal computer Intel® Core™2 Duo case
Fig.3 Numerical solution for the rigid Earth rotation minus solution SMART97 in the inclination angle.
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 - 265011.2586 T - 97.64 T
5127634.2488 T2
- 353.10 T2
5127634.2488 T2
- 355.10 T2
- 7727159.4229 T3
771.50 T3
- 7727159.4229 T3
775.68 T3
- 4916.7335 T4
- 84.50 T4
- 4916.7335 T4
- 79.67 T4
33292.5474 T5
- 86.00 T5
33292.5474 T5
- 89.51 T5
- 247.50 T6
-250.65 T6
the computer system Parsytec CCe20 case the personal computer Intel® Core™2 Duo case
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
D after removal of the secular trend
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| > ||
till the end of the spectrum
Set of nutation terms of SMART97
Removal of this harmonic from the D and
from the Spectrum
YesNo
Spectral Analysis (SA) A L G O R I T H M – 1 for removal of the periodical terms from the Discrepancies (D)
… Construction of the new nutation series New high-
precision series
S9000A1
D after removal of the secular trend
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| > ||
till the end of the spectrum
Set of nutation terms of SMART97
Removal of this harmonic from the D and
from the Spectrum
YesNo
Spectral Analysis (SA) A L G O R I T H M – 2for removal of the periodical terms from the Discrepancies (D)
… Construction of the new nutation series
Construction of the new D-2 and apply the algorithm - 1
New high-precision
series S9000A
Fig.5. Spectra of the discrepancies between numerical solutions and SMART97 for the proper rotation angle.
for only long periodical terms for all periodical terms ALGORITHM – 1the computer system Parsytec CCe20 case 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 for all periodical terms ALGORITHM – 1the computer system Parsytec CCe20 case the personal computer Intel® Core™2 Duo case
the computer system Parsytec CCe20 case the personal computer Intel® Core™2 Duo case for ALGORITHM – 1
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.
Fig.7. Spectra of the discrepancies between numerical solutions and SMART97 for the proper rotation angle. for only long periodical terms ALGORITHM – 2 for all periodical terms ALGORITHM – 11-iteration
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
The personal computer Intel® Core™2 Duo case
for only long periodical terms ALGORITHM – 2 for all periodical terms ALGORITHM – 11-iteration
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) ALGORITHM – 1 ( for all periodical terms)
The end of the
for all periodical terms ALGORITHM – 2 for all periodical terms ALGORITHM – 12-iteration fig.7
Fig.9. Spectra of the discrepancies between numerical solutions and SMART97 for the proper rotation angle.
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
The personal computer Intel® Core™2 Duo case
for all periodical terms ALGORITHM – 2 for all periodical terms ALGORITHM – 12-iteration 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 for all periodical terms ALGORITHM – 12-iteration fig.8
Part II:
The non-rigid Earth rotation
TRANSFER FUNCTION
4
0 01
( ; | ) 1 11
RR
R
e QT e e N Q
e s
P.M. Mathews, T.A. Herring, B.A. Buffett, 2002:
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
0 0
( 1) is frequency in cycles per sidereal day (cpsd)
with respect to inertial space
/(1 ) ( 1; | ) , 1 1
/(1 )
, elli
Here
where
pt
RR R
R
e eT e e N N
e e
e e
icities for rigid and non-rigid Earth, respectively.
4
0 1 0 10
4
0 1 0 10
4
0 1 0 10
[ sin( ) cos( )]
[ sin( ) cos( )]
[ sin( ) cos( )]
kS jk j j C jk j j
j k
kS jk j j C jk j j
j k
kS jk j j C jk j j
j k
t t t
t t t
t t t
0 0 2
51 5
50 1 5
50 1 5
...
, ...
...
pp
p p
p p
t t
t t
t t
Expressions for Euler angles:
sin sin cos
sin cos sin
cos
R R R R R R
R R R R R R
R R R R
p
q
r
sin cos / sin
cos sin
cos
NR NR NR NR NR NR
NR NR NR NR NR
NR NR NR NR
p q
p q
r
( ; | )
NR NR R R R
NR R
p iq p iq T e e
r r
Algorithm of Bretagnon et al. (1999)
1.The rigid Earth angular velocity vector:
3.The derivatives of Euler angles for the non-rigid Earth rotation:
2.The non-rigid Earth angular velocity vector is obtained by:
0 1 0 10
1 1 0 11
11 1 0 1
1 0 1 0 1
[ sin( ) cos( )]
[( )sin( )
( )cos
Here
( )]
[ sin( ) cos( )] ,
, ,
mk
k kk
m
k kk
kk k
mm m
j j j
x A c c t B c c t t
x kA c B c c t
kB c A c c t t
c B c c t A c c t t
x
1 1 1 1
1 1 1 1
1 1 1
1 1 1 1
1
/
/
1,..., ( ) / 3
( ) /
,...
Her
e ,H 1
e
re
k k k m m
k k k m m
k k k
m m k k k
m m
kA c B A c
kB c A B c
k m A kB c
c B B kA c
c A k m
Cascade method:
Common form of the periodic part of Euler angles:
0 1 0 0
1
1 0 0 1
1 1 1 1 1
and by from
and from
He
cos sin
3
( sin cos ) / sin
cos 3 ,
Cascade me
, , ( ), ( ), ( )
tho
re
d
NR R NR NR NR NR NR
NR
NR NR NR NR NR NR
NR NR NR NR NR NR
NR NR NR NR NR NR
p q
p q
r
p q r p q r
1-th iteration
Details: Iterative solution of the algorithm of Bretagnon et al. (1999)
(Pashkevich, 2008)
1 1
1 1
and by fr
cos sin
3
( sin cos ) / sin
cos 3 ,
, , ( ), ( ),
Ca om
an
scade metho
d from
Here ( )
d
n n nNR NR NR NR NR
nNR
n n n nNR NR NR NR NR NR
n n n n nNR NR NR NR NR NR
NR NR NR NR NR NR
p q
p q
r
p q r p q r
n-th iteration
Iterations are repeated until the absolute value of the difference between iterations K-1 and K exceedes some a priori adopted values
Details: Iterative solution of the algorithm of Bretagnon et al. 1999
(Pashkevich, 2008)
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- 2SMART97 S9000A
25.3564
25.3564
14.5558
14.5558
SMN
SN9000A
26.3775
26.3775
21.4672
21.4672
0.507904
λ3 +D- 2SMART97 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- 2SMART97 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 1. Comparison of different solutions for selected durnal and semidurnal periodical terms in the longitude of the ascending node of the Earth equator.
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- 2SMART97 S9000A
-5.7900
-5.7900
10.0863
10.0863
SMN
SN9000A
-8.5395
-8.5395
10.4933
10.4933
0.507904
λ3 +D- 2SMART97 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- 2SMART97 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:
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- 2SMART97 S9000A
-0.1279
-0.1279
-0.0728
-0.0728
SMN
SN9000A
0.8089
0.8089
6.2684
6.2684
0.507904
λ3 +D- 2SMART97 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- 2SMART97 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.
Results:Table 3. Comparison of different solutions for selected durnal and semidurnal periodical terms in the angle of the proper rotation of the Earth.
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.
R E F E R E N C E S
1. P.Bretagnon and G.Francou. Planetary theories in rectangular and spherical variables // Astronomy and Astrophys., 202, 1988, pp. 309–-315.
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A C K N O W L E D G M E N T S
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.