Transcript High-precision numerical analysis of the rigid Earth

Improved Algorithm for the
Construction of the Non-Rigid
Earth Rotation Series
V.V. Pashkevich
Central (Pulkovo) Astronomical Observatory
of Russian Academy of Science
St.Petersburg
Space Research Centre of Polish Academy of Sciences
Warszawa
2006
The aim of the research is:
Obtainment of exactly algorithm for the
Construction of the Non-Rigid Earth Rotation
Series, which expressed in the function of the
three Euler angles.
Expressions for Euler angles:
   p   

   p    ,

   p   

p

5
 1t  ...   5 t 


 p   0   1t  ...   5 t 
5
 p   0   1t  ...   5 t 
5
0  0
1 

     [ S jk sin( j 0   j 1t )   C jk cos( j 0   j 1t )] t 
j k 0

4

k 
     [ S jk sin( j 0   j 1t )   C jk cos( j 0   j 1t )] t 
j k 0

4

k
     [ S jk sin( j 0   j 1t )   C jk cos( j 0   j 1t )] t 

j k 0
4
k
Algorithm of Bretagnon et al. 1999
Rigid Earth angular velocity vector:
p R   R sin  R sin  R   R cos  R
q R   R sin  R cos  R   R sin  R
rR   R cos  R   R
Non-Rigid Earth angular velocity vector is obtained by:
 p NR
 iq N R  
 pR
 iq R  g ( )
rN R  rR or rN R  rR   ( IE R S 1996)
The derivatives of Euler angles for Non-Rigid Earth rotation:

NR
  p N R sin  N R  q N R cos  N R  / sin  N R
 N R  p N R cos  N R  q N R sin  N R
 N R  rN R   N R cos  N R
For example:
Iterations for solve of the Algorithm of Bretagnon et al. 1999
1 iteration

0
NR
 R  
1
NR
 p N R cos 
0
NR
and by C ascade m e tho d from

1
NR
 ( p N R sin 
0
NR
 q N R cos 
 q N R sin 
2  
1
NR
) / sin 
1
NR
0
NR
 N R  rN R   N R cos  N R and from
1
1
1
0
NR


2  
1
NR

H e re p N R , q N R , rN R  p N R ( ), q N R ( ), rN R ( )
, NR
1
For example:
Iterations for solve of the Algorithm of Bretagnon et al. 1999
n iteration

n
NR
 p N R cos 
n 1
NR
 q N R sin 
and by C a scade m etho d fr om

n
NR
 ( p N R sin 

n
NR
 rN R  
n
NR
n 1
NR
 q N R cos 
cos 
n
NR
n 1
NR
2  
n 1
NR
n
NR
) / sin 
an d from

n
NR

2  
n
NR
,
n
NR
Iterations is repeated while then absolute value of the difference
between K-1 and K iterations is more than some DEFINITE values 
H ere p N R , q N R , rN R  p N R ( ), q N R ( ), rN R ( )
Common
form for the periodic part of Euler angles:
m
x 
 [ Ak sin( c 0  c1t )  B k cos( c 0  c1t )] t
k
k 0
m
x 
 [( kA
k
 c1 B k 1 ) sin( c 0  c1t ) 
k 1
 ( kB k  c1 Ak 1 ) cos ( c 0  c1t )] t
k 1

 c1 [  B m sin( c 0  c1t )  Am cos( c 0  c1t )] t ,
m
H ere x    j ,   j ,  
kAk  c1 B k 1   k 1 

kB k  c1 Ak 1   k 1


H er e k  1, ..., m 

 c1 B m   m

c1 Am   m

j
Cascade method:
Am 
 m / c1


B m    m / c1


 Ak 1  (  k 1  kB k ) / c1 
B k 1  ( kAk   k 1 ) / c1 

H e re k  m , ... , 1

2 
Common form for the secular part of Euler angles:
m
x 

Bk t
k
k 0
m
x 

kB k t
k 1
,
k 1
H er e x  
pj
kAk   k 1
,  pj ,  pj


kB k   k 1

H ere k  1, ..., m 
Ak 
 k 1 / k


 B k   k 1 / k

H ere k  m , ... ,1 
3
A l g o r i t h m 2006
 ( t )    N R sin  0  i   N R ,
z ( t )    R sin  0  i   R
 ( t )  g ( ) z ( t ) , g ( )  g R E ( )  ig IM ( )
H ere   and    nutations in lon gitud e and in
obliq ut y , respectively, for the N on-R igid (N R ) and
R igid (R ) E arth , g ( )  T ransfer function,
   ( j 1 ),
j1
 frequency of the nuta tion argum ent
g (0 )  1

NR
4
sin  0  g R E   R sin  0  g IM   R 

  N R  g R E   R  g IM   R sin  0 
5
A l g o r i t h m 2006
rR   R   R cos  R ,
 NR  rNR   NR cos  NR
a ) rN R  rR ; b ) rN R  rR   (IE R S 1996 ).
case a)
 N R   R  ( R cos  R   N R cos  N R ) 
 p N R    N R   p R    R  (
 (
fro m
pR
pNR
   R ) cos  R 
 
 4    p   pR   pNR ;  p
NR
) cos  N R
  pR   pNR
and from  1    0   p R 0   p N R 0
  p N R   p R   p (cos  R  cos  N R )

   N R    R  (   R co s  R    N R cos  N R )
  N R and from
2
and
 3    NR
6
Comment
Expressions (5), (6) and some expressions in the algorithm of
Bretagnon et al. 1999 are used for each component of the
periodic terms.
NOTE that t=0 for the arguments in all expressions:

   [ S j 0 sin  j 0   C j 0 cos  j 0 ]

j


   0   [ S j 0 sin  j 0   C j 0 cos  j 0 ] 
j

   0   [ S j 0 sin  j 0   C j 0 cos  j 0 ] 

j

CONCLUSION
• Improved Algorithm for the Construction
of the Non-Rigid Earth Rotation Series,
which expressed in the function of the
three Euler angles, are obtained.
REFERENCES
1.
2.
3.
V.A..Brumberg, P.Bretagnon Kinematical Relativistic
Corrections for Earth’s Rotation Parameters // in Proc. of IAU
Colloquium 180, eds. K.Johnston, D. McCarthy, B. Luzum and
G. Kaplan, U.S. Naval Observatory, 2000, pp. 293–302.
P.Bretagnon and G.Francou Planetary theories in rectangular
and spherical variables //, Astronomy and Astrophysics, 202,
1988, pp. 309–-315.
P.Bretagnon, G.Francou, P.Rocher, J.L.Simon SMART97: A
new solution for the rotation of the rigid Earth // Astron.
Astrophys. , 1998, 329, 1, pp.
ACKNOWLEDGMENTS
The investigation was carried out at the
Central (Pulkovo) Astronomical
Observatory of Russian Academy
of Science
and the
Space Research Centre
of Polish Academy of Science,
under a financial support of the
Cooperation between Polish and
Russian Academies of Sciences,
Theme No 31.