Index Page dvsep_c
A B C D E F G H I J K L M N O P Q R S T U V W X
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
       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

    Wed Jun  9 13:05:21 2010