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Index Page dvsep_c
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Procedure Abstract Required_Reading Keywords Brief_I/O Detailed_Input | Detailed_Output Parameters Exceptions Files Particulars Examples | Restrictions Literature_References Author_and_Institution Version Index_Entries
#### Procedure
       dvsep_c ( Time derivative of separation angle )

       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,
       s2          are, respectively, the state vector of the first and
                   second target bodies as seen from the observer

                   An implicit assumption exists that both states lie in
                   the same reference 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)  If numeric overflow and underflow cases are detected, an error
           is signaled by a routine in the call tree of this routine.

       2)  If called in "RETURN" mode, the function returns 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
       N.J. Bachman        (JPL)
       J. Diaz del Rio     (ODC Space)
       E.D. Wright         (JPL)
#### Version
       -CSPICE Version 1.0.1, 13-AUG-2021 (JDR)

           Edited the header to comply with NAIF standard.

       -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

    Fri Dec 31 18:41:05 2021