Procedure

   SpiceDouble dvsep_c (ConstSpiceDouble s1[6], ConstSpiceDouble s2[6] ) 

Abstract

 
   Calculate the time derivative of the separation angle between
   two input states, S1 and S2.
 

Required_Reading

 
   None. 
 

Keywords

 
   GEOMETRY
 

Brief_I/O

 
   VARIABLE  I/O  DESCRIPTION 
   --------  ---  -------------------------------------------------- 
   s1         I   State vector of the first body 
   s2         I   State vector of the second  body 
 

Detailed_Input

 
   s1         the state vector of the first target body as seen from 
              the observer 
 
   s2         the state vector of the second target body as seen from 
              the observer 
 
   An implicit assumption exists that both states lie in the same 
   refrence frame with the same observer for the same epoch. If this 
   is not the case, the numerical result has no meaning. 
 

Detailed_Output

 
   The function returns the double precision value of the time derivative 
   of the angular separation between S1 and S2.
 

Parameters

 
    None. 
 

Exceptions

   1) The error SPICE(NUMERICOVERFLOW) signals if the inputs S1, S2
      define states with an angular separation rate ~ DPMAX().

   2) If called in RETURN mode, the return has value 0.

   3) Linear dependent position components of S1 and S1 constitutes
      a non-error exception. The function returns 0 for this case.
 

Files

 
    None. 
 

Particulars

 

   In this discussion, the notation

      < V1, V2 >

   indicates the dot product of vectors V1 and V2. The notation

      V1 x V2

   indicates the cross product of vectors V1 and V2.

   To start out, note that we need consider only unit vectors,
   since the angular separation of any two non-zero vectors
   equals the angular separation of the corresponding unit vectors.
   Call these vectors U1 and U2; let their velocities be V1 and V2.

   For unit vectors having angular separation

      THETA

   the identity

      || U1 x U1 || = ||U1|| * ||U2|| * sin(THETA)                (1)

   reduces to

      || U1 x U2 || = sin(THETA)                                  (2)

   and the identity

      | < U1, U2 > | = || U1 || * || U2 || * cos(THETA)           (3)

   reduces to

      | < U1, U2 > | = cos(THETA)                                 (4)

   Since THETA is an angular separation, THETA is in the range

      0 : Pi

   Then letting s be +1 if cos(THETA) > 0 and -1 if cos(THETA) < 0, 
     we have for any value of THETA other than 0 or Pi


                                2          1/2
      cos(THETA) = s * ( 1 - sin (THETA)  )                       (5)

   or  

                                2          1/2
      < U1, U2 > = s * ( 1 - sin (THETA)  )                       (6)


   At this point, for any value of THETA other than 0 or Pi,
   we can differentiate both sides with respect to time (T) 
   to obtain

                                                    2        -1/2
      < U1, V2 > + < V1, U2 > =    s * (1/2)(1 - sin (THETA))

                                 * (-2) sin(THETA)*cos(THETA)

                                 * d(THETA)/dT                   (7a)


   Using equation (5), and noting that s = 1/s, we can cancel 
   the cosine terms on the right hand side

                                                    -1
      < U1, V2 > + < V1, U2 > =    (1/2)(cos(THETA))

                                 * (-2) sin(THETA)*cos(THETA)

                                 * d(THETA)/dT                   (7b)

   With (7b) reducing to

      < U1, V2 > + < V1, U2 > = - sin(THETA) * d(THETA)/dT        (8)

   Using equation (2) and switching sides, we obtain

      || U1 x U2 || * d(THETA)/dT  =  - < U1, V2 > - < V1, U2 >   (9)

   or, provided U1 and U2 are linearly independent,
     
      d(THETA)/dT = ( - < U1, V2 > - < V1, U2 > ) / ||U1 x U2||  (10)

   Note for times when U1 and U2 have angular separation 0 or Pi
   radians, the derivative of angular separation with respect to
   time doesn't exist. (Consider the graph of angular separation
   with respect to time; typically the graph is roughly v-shaped at
   the singular points.)
 

Examples

   None. 

Restrictions

   None. 

Literature_References

   None. 

Author_and_Institution

   E.D. Wright    (JPL) 

Version

   -CSPICE Version 1.0.0, 09-MAR-2009 (EDW) (NJB)

Index_Entries

 
   time derivative of angular separation 
 

Link to routine dvsep_c source file dvsep_c.c