CSPICE_DVSEP calculates the time derivative of the separation angle
s1 defining a SPICE state(s);
[6,n] = size(s1); double = class(s1)
s1 = (r1, dr1 ).
s2 defining a second SPICE state(s);
[6,n] = size(s2); double = class(s2)
s2 = (r2, dr2 ).
An implicit assumption exists that 's1' and 's2' are specified
in the same reference frame. If this is not the case, the numerical
result has no meaning.
dvsep = cspice_dvsep( s1, s2)
dvsep time derivative(s) of the angular separation between 's1' and
[1,n] = size(dvsep); double = class(dvsep)
'dvsep' returns with the same vectorization measure (N)
as 's1' and 's2'.
Any numerical results shown for this example may differ between
platforms as the results depend on the SPICE kernels used as input
and the machine specific arithmetic implementation.
% Load SPK, PCK, and LSK kernels, use a meta kernel for convenience.
cspice_furnsh( 'standard.tm' )
% An arbitrary time.
BEGSTR = 'JAN 1 2009';
et = cspice_str2et( BEGSTR );
% Calculate the state vectors sun to Moon, sun to earth at ET.
[statee, ltime] = cspice_spkezr('EARTH', et, 'J2000', 'NONE', 'SUN' );
[statem, ltime] = cspice_spkezr('MOON', et, 'J2000', 'NONE', 'SUN' );
% Calculate the time derivative of the angular separation of
% the earth and Moon as seen from the sun at ET.
dsept = cspice_dvsep( statee, statem );
fprintf( 'Time derivative of angular separation, rads/sec: %.10e\n', ...
% It's always good form to unload kernels after use,
% particularly in Matlab due to data persistence.
Time derivative of angular separation, rads/sec: 3.8121193603e-09
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
|| U1 x U1 || = ||U1|| * ||U2|| * sin(THETA) (1)
|| U1 x U2 || = sin(THETA) (2)
and the identity
| < U1, U2 > | = || U1 || * || U2 || * cos(THETA) (3)
| < 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
cos(THETA) = s * ( 1 - sin (THETA) ) (5)
< 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)
< 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
< 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.)
For important details concerning this module's function, please refer to
the CSPICE routine dvsep_c.
-Mice Version 1.0.0, 12-MAR-2012, EDW (JPL), SCK (JPL)
time derivative of angular separation