FIND ANGULAR RATE

Syntax

   FIND @name ANGULAR RATE
 
        (3:3){            OF       @body
             |            FROM     @body
             | @options | RELATIVE @body }
 
        (1:1){ GREATER THAN           @number
             | LESS THAN              @number
             | EQUAL TO               @number
             | BETWEEN  @number AND   @number
             | ABSOLUTE MINIMUM PLUS  @number(0:)
             | ABSOLUTE MAXIMUM MINUS @number(0:)
             | ABSOLUTE MINIMUM
             | ABSOLUTE MAXIMUM
             | LOCAL    MINIMUM
             | LOCAL    MAXIMUM  }
 
        (0:1){ WITHIN @name }
 
        STEP SIZZE @number(0:)

Description

This command determines intervals during which the angular rate OF a target body as seen FROM an observing body satisfies a specified constraint.

The command can also be used to locate intervals when the angular rate of the target body RELATIVE to a second body as seen from an observing body satisfies a specified constraint.

In both cases, angular rate is defined to be the norm of a vector that represents the angular motion of the target body. The vector is defined as follows.

Informal definition

First consider the case where no reference body (RELATIVE) is specified. Imagine that the observing body is at the center of a sphere, the orientation of which is fixed relative to some inertial frame.

The direction vector from the observer to the apparent position of the target body intersects the sphere, tracing out a curve on the sphere as a function of time. The angular rate of the target body is the angular rate at which the point of intersection moves across the surface of the sphere.

Now consider the case where a reference body is specified. The sphere is no longer fixed with respect to an inertial frame. At any instant, the x-axis of a frame tied to the sphere is defined by the direction from the observer to the apparent position of the reference body. The apparent velocity vector of the reference body lies in the equator of the sphere.

As in the previous case, the direction vector from the observer to the apparent position of the target body intersects the sphere, tracing out a curve on the sphere as a function of time, and the angular rate of the target body is the angular rate at which the point of intersection moves across the surface of the sphere. However, because the sphere is rotating with respect to inertial space, it is a different curve, and a different rate.

Formal definition

First consider the case where no reference body (RELATIVE) is specified. Let P and V be the apparent position and velocity vectors of the target body T as seen from the observing body. There exists a vector W(T), called the angular velocity vector of T, such that the non-radial component of V is given by the cross product

   W(T) x P

Let U be the unit vector parallel to P. Then the norm of

   W(T) x U

is the the angular rate of the target body.

Now consider the case where a reference body R is specified. Let W(R) be the angular velocity vector of the reference body (defined similarly to W(T) above). Then the norm of

   ( W(T) - W(R) ) x U

is the angular rate of the target body relative to the reference body.

Examples

Example 1

In the following example, the FIND ANGULAR RATE command is used to locate intervals during which the motion of an asteroid against the background sky is at a local minimum.

   FIND STILL ANGULAR RATE OF ASTEROID
        FROM EARTH LOCAL MINIMUM
        STEP SIZE GRANULARITY;

Example 2

In the following example, the FIND ANGULAR RATE command is used to locate intervals when the relative motion of Io and Europa is less than 0.05 arc seconds per second. The search is restricted to intervals (determined previously) when the satellites are within 60 arc seconds of each other.

   IMPORT IO_EUROPA_CLOSE AS CLOSE;
 
   FIND SLOW ANGULAR RATE OF IO
        RELATIVE EUROPA  FROM EARTH
        LESS THAN 0.05 ARCSECONDS/SECOND
        WITHIN CLOSE
        STEP SIZE GRANULARITY;

Notes

The definition of relative angular rate is not symmetric. That is,

   FIND W ANGULAR RATE OF BODY_1 RELATIVE BODY_2 FROM BODY_3 ...

and

   FIND W ANGULAR RATE OF BODY_2 RELATIVE BODY_1 FROM BODY_3 ...

are not equivalent commands. However, when two objects are close enough to be observed simultaneously (as in the second example above), the asymmetry is negligible.