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Introduction to Astrometry 位置天文学入門

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Introduction to Astrometry 位置天文学入門 Powered By Docstoc
					    Rotation
   (Euclidean) Distance-Invariant
     X   2
                 x 
                       2

   Finite Rotation: Matrix representation
     X  Rx
   Orthogonality
     Rx      2
                     x R Rx  x
                           T       T                   2


    R R  I or R  R
            T                         -1           T
1999-2000            Lecture Notes on Astrometry
  Infinitesimal Rotational
  Displacement
     Antisymmetric Matrix
           0      z           y 
                                     
         z     0              x 
                 x                
           y                       0              x 
                                                    
     Vector Product                           θ   y 
        x  θ  x                                
                                                    z
1999-2000        Lecture Notes on Astrometry
      Finite Rotation
   Expressions: Matrix, Spinol, Quarternion
   Rotation = Matrix Operation

   Rot. Matrix = Set of
                                Z
                                      Y
  Basis Vectors (= Triad)

   R  e X eY e Z 
                                             X




1999-2000      Lecture Notes on Astrometry
  Euler’s Theorem
     Any Finite Rotation = 3 Basic Rotation
 R  R ijk  ,  ,    R k ( )R j (  )R i ( )

   R ,  , 
            ijk
                        1
                               R kji   , , 
     Euler angles: 3 Angles of Basic
      Rotations

1999-2000         Lecture Notes on Astrometry
     Basic Rotation
        Rotation around z-axis by angle 
                              Y
                      y                        P
                                                        x


R 3 ( )  R z ( )
                                                           X




   1999-2000              Lecture Notes on Astrometry
  Basic Rotation (contd.)
     Rotation around j-axis by angle 
            R j ( )
     Inverse Rotation

            R  
               j
                        1
                               R j  

1999-2000              Lecture Notes on Astrometry
  Basic Rotation Matrix
               cos                    sin       0
                                                   
   R 3 ( )    sin                  cos       0
               0                                   
                                              0   1
     Example: Equatorial – Ecliptic R1  
     Obliquity of Ecliptic 
1999-2000        Lecture Notes on Astrometry
  Basic Rotation Matrix (contd.)
     Small Angle Approximation
                        0  0
                                 
        R 3    I     0 0   I   e 3 
                        0 0 0
                                 
                                       
          R j  j   I    j e j  
                                       
             j                j        
1999-2000          Lecture Notes on Astrometry
  Angular Velocity
                 d j
      ω                ej
             j   dt
                                        
      R   R j  j   I    j e j  
                                        
            j                  j        
     dR      d j 
           
                      e j   ω  
                           
      dt     j  dt        
1999-2000          Lecture Notes on Astrometry
  Euler Rotation
     3x2x2 = 12 different combinations
     3-1-3 Sequence (= x-convention)
           Most popular (Euler angles)
           Used to describe rotational dynamics

      R 313  , ,    R 3  R1  R 3  


1999-2000             Lecture Notes on Astrometry
          Euler Angles (3-1-3)

                     C C  S C S            C S  S C C     S S 
                                                                            
R 313  , ,      S C  C C S          S S  C C C    C S 
                           S S                             S C    C 
                                                                            
   cos cos  sin  cos sin     cos sin   sin  cos cos sin  sin  
                                                                            
   sin  cos  cos cos sin   sin  sin   cos cos cos cos sin  
             sin  sin                      sin  cos           cos 
                                                                            


        1999-2000              Lecture Notes on Astrometry
                         Z
Euler Angles

                                                   P
                                    

    X                                   
                                              Y
                                    N


 1999-2000       Lecture Notes on Astrometry
  Demerit of 3-1-3 Sequence
     Degeneration in case of small angles

                             
                                
   R 313  , ,    I   0  
                             
                                
     Solution: 3-2-1-like Sequences
1999-2000        Lecture Notes on Astrometry
  3-2-3 Sequence
     y-convention: precession
            P  R 323   A , A , z A 
     Conic Rotation                        sin  cos  
         Rotation around
                                                        
       
                                       n   sin  sin  
        a fixed direction                   cos 
      R 323  ,  ,                                  

       I+ sin  n   1  cos   n  n 
1999-2000            Lecture Notes on Astrometry
     Other Sequences
   1-3-1: Nutation
N  R131  A ,  , A   
   2-1-3: Polar Motion + Sidereal Rotation
        WS  R 312 , y p , x p 
   1-2-3: Aerodynamics, Attitude Control
       Best Recommended
1999-2000         Lecture Notes on Astrometry
   Small Angle Rotation
R123  ,  ,    R 3 ( )R 2 (  )R1 ( )
   C C        C S  S  S C                C S  C  S S 
                                                                    
   S C      S S  S  C C               S S  C  C S 
   S                  C  S                          C  C       
                                                                   
       
       
 I    
       
       
 1999-2000             Lecture Notes on Astrometry
  Rotational Velocity
     X  Rx 
              dR
     V  Rv     x  Rv  x
              dt


1999-2000   Lecture Notes on Astrometry

				
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