Main Page Binary PCK Hands-On Lesson (MATLAB)
## Table of Contents
       Binary PCK Hands-On Lesson (MATLAB)
          Overview
          Note About HTML Links
          References
             Tutorials
             Required Readings
             The Permuted Index
             Mice API Documentation
          Kernels Used
          Mice Modules Used
       Moon rotation (mrotat)
          Task Statement
          Learning Goals
          Approach
          Solution
             Solution Meta-Kernel
             Solution Source Code
             Solution Sample Output
       Earth rotation (erotat)
          Task Statement
          Learning Goals
          Approach
          Solution
             Solution Meta-Kernel
             Solution Source Code
             Solution Sample Output
Top # Binary PCK Hands-On Lesson (MATLAB) May 21, 2018 Top ## Overview In this lesson you will develop two programs that demonstrate geometric computations using ``high-accuracy'' Earth and Moon binary PCKs. The programs also demonstrate use of frame kernels and SPK files normally used together with these high-accuracy PCKs. Top ## Note About HTML Links The HTML version of this lesson contains links pointing to various HTML documents provided with the Toolkit. All of these links are relative and, in order to function, require this document to be in a certain location in the Toolkit HTML documentation directory tree. In order for the links to be resolved, if not done already by installing the lessons package under the Toolkit's ``doc/html'' directory, create a subdirectory called ``lessons'' under the ``doc/html'' directory of the ``mice/'' tree and copy this document to that subdirectory before loading it into a Web browser. Top ## References This section lists SPICE documents referred to in this lesson. Of these documents, the ``Tutorials'' contains the highest level descriptions with the least number of details while the ``Required Reading'' documents contain much more detailed specifications. The most complete specifications are provided in the ``API Documentation''. Top ### Tutorials The following SPICE tutorials serve as references for the discussions in this lesson:
       Name              Lesson steps/functions it describes
       ----------------  -----------------------------------------------
       Frames            Moon rotation, Earth rotation
       PCK               Moon rotation, Earth rotation
       ``High Accuracy
       Orientation and
       Body-Fixed
       frames for Moon
       and Earth''
       (backup)          Moon rotation, Earth rotation
These tutorials are available from the NAIF ftp server at JPL:
       http://naif.jpl.nasa.gov/naif/tutorials.html
Top ### Required Readings The Required Reading documents are provided with the Toolkit and are located under the ``mice/doc'' directory in the Mice installation tree.
       Name             Lesson steps/functions that it describes
       ---------------  -----------------------------------------
       frames.req       Using reference frames
       pck.req          Obtaining planetary constants data
       spk.req          Obtaining ephemeris data
       time.req         Time conversion
       mice.req         The Mice API
Top ### The Permuted Index Another useful document distributed with the Toolkit is the permuted index. This is located under the ``mice/doc'' directory in the MATLAB installation tree. This text document provides a simple mechanism by which users can discover which Mice functions perform functions of interest, as well as the names of the source files that contain these functions.
Top ### Mice API Documentation A Mice routine's specification is found in the HTML API documentation page located under ``mice/doc/html/mice''. For example, the document
       mice/doc/html/mice/cspice_str2et.html
describes the cspice_str2et routine. Top ## Kernels Used The following kernels are used in examples provided in this lesson:
       #  FILE NAME                      TYPE DESCRIPTION
       -- ------------------------------ ---- ------------------------------
       1  naif0008.tls                   LSK  Generic LSK
       2  de414_2000_2020.bsp            SPK  Solar System Ephemeris
       3  moon_060721.tf                 FK   Lunar FK
       4  pck00008.tpc                   PCK  NAIF text PCK
       5  moon_pa_de403_1950-2198.bpc    PCK  Moon binary PCK
       6  earthstns_itrf93_050714.bsp    SPK  DSN station Ephemeris
       7  earth_topo_050714.tf           FK   Earth topocentric FK
       8  earth_000101_070725_070503.bpc PCK  Earth binary PCK
These SPICE kernels are included in the lesson package available from the NAIF server at JPL:
       ftp://naif.jpl.nasa.gov/pub/naif/toolkit_docs/Lessons/
Top ## Mice Modules Used This section provides a complete list of the functions and kernels that are suggested for usage in each of the exercises in this lesson. (You may wish to not look at this list unless/until you ``get stuck'' while working on your own.)
       CHAPTER EXERCISE   FUNCTIONS      NON-VOID       KERNELS
       ------- ---------  -------------  -------------  ----------
          1    mrotat     cspice_furnsh  cspice_str2et  1-5
                          cspice_kclear  cspice_spkpos
                                         cspice_reclat
                                         cspice_dpr
                                         cspice_vsep
                                         cspice_subpnt
                                         cspice_vdist

          2    erotat     cspice_furnsh  cspice_str2et  1-2,4,6-8
                          cspice_kclear  cspice_spkpos
                                         cspice_reclat
                                         cspice_dpr
                                         cspice_vsep
                                         cspice_spd
                                         cspice_timout
                                         cspice_pxform
                                         cspice_twopi
                                         cspice_subslr
                                         cspice_vdist
Refer to the Mice HTML API documentation pages located under ``mice/doc/html/mice'' for detailed interface specifications of these functions. Top # Moon rotation (mrotat) Top ## Task Statement Write a program that performs the following computations: `1.` Convert the time string 2007 JAN 1 00:00:00 UTC to a double precision number representing seconds past J2000 TDB. `` In the following instructions, we'll call the result of this computation ET. `2.` Compute the apparent position of the Earth as seen from the Moon in the IAU_MOON reference frame at the epoch ET. Use light time and stellar aberration corrections. Use cspice_reclat to compute the planetocentric longitude and latitude of the Earth position vector; display these coordinates in degrees. `3.` Repeat the computation of step 2 using the MOON_ME reference frame. Display the results as above. `4.` Compute the angular separation of the position vectors found in steps 2 and 3. Display the result in degrees. `5.` Repeat the computation of step 2 using the MOON_PA reference frame. Display the results as above. `6.` Compute the angular separation of the position vectors found in steps 3 and 5 (these vectors are expressed in the MOON_ME and MOON_PA frames). Display the result in degrees. `7.` Compute the apparent sub-Earth point on the Moon at ET, expressed in the MOON_ME reference frame and using light time and stellar aberration corrections. Convert the sub-Earth point to latitudinal coordinates using cspice_reclat. Display the longitude and latitude of the sub-Earth point in degrees. `8.` Repeat step 7, now using the MOON_PA frame. `9.` Compute the distance between the two sub-Earth points found above in steps 7 and 8. Display the result in kilometers. Top ## Learning Goals Familiarity with SPICE kernels required to obtain high-accuracy orientation of the Moon. Understanding the differences between results obtained using low and high-accuracy Moon orientation data. Understanding the difference between the MOON_ME and MOON_PA frames. Top ## Approach The following ``tips'' may simplify the solution process. `--` Examine the SPICE kernels provided with this lesson. Use BRIEF to find coverage periods of SPK kernels and binary PCKs. Use COMMNT to view the comment areas of binary PCKs. Examine text kernels, in particular text kernel comments, using a text editor or browser. `--` Decide which SPICE kernels are necessary. Prepare a meta-kernel listing the kernels and load it into the program. `--` Consult the above list titled ``Mice Modules Used'' to see which routines are needed. `--` The computational steps listed above should be followed in the order shown. You may find it useful to consult the permuted index, the Mice HTML API documentation pages, and the tutorials titled ``PCK'' and ``High Accuracy Orientation and Body-Fixed frames for Moon and Earth.'' Top ## Solution Top ### Solution Meta-Kernel The meta-kernel we created for the solution to this exercise is named 'mrotat.tm'. Its contents follow:
       KPL/MK

       Meta-kernel for the ``Moon Rotation'' task in the Binary PCK
       Hands On Lesson.

       The names and contents of the kernels referenced by this
       meta-kernel are as follows:

       File name                    Contents
       ---------------------------  ------------------------------------
       naif0008.tls                 Generic LSK
       de414_2000_2020.bsp          Solar System Ephemeris
       moon_060721.tf               Lunar FK
       pck00008.tpc                 NAIF text PCK
       moon_pa_de403_1950-2198.bpc  Moon binary PCK

       \begindata

          KERNELS_TO_LOAD = ( 'kernels/lsk/naif0008.tls'
                              'kernels/spk/de414_2000_2020.bsp'
                              'kernels/fk/moon_060721.tf'
                              'kernels/pck/pck00008.tpc'
                              'kernels/pck/moon_pa_de403_1950-2198.bpc' )
       \begintext
Top ### Solution Source Code A sample solution to the problem follows:
       %
       % Binary PCK lesson: Moon rotation
       %
       function mrotat()

       %
       % Local Parameters
       %
       METAKR = 'mrotat.tm';

       %
       % Load the kernels this program requires.
       %
       cspice_furnsh( METAKR );

       %
       % Convert our UTC string to seconds past J2000 TDB.
       %
       timstr = '2007 jan 1 00:00:00';
       et     = cspice_str2et ( timstr );

       %
       % Look up the apparent position of the Earth relative
       % to the Moon's center in the IAU_MOON frame at ET.
       %
       [imoonv, ltime] = cspice_spkpos( 'earth', et, 'iau_moon', ...
                                        'lt+s', 'moon'              );
       %
       % Express the Earth direction in terms of longitude
       % and latitude in the IAU_MOON frame.
       %
       [r, lon, lat] =  cspice_reclat( imoonv );

       fprintf ( '\n%s\n', ...
                 'Moon-Earth direction using low accuracy' )
       fprintf ( '%s\n', 'PCK and IAU_MOON frame:' )
       fprintf ( '%s%15.6f\n', 'Earth lon (deg):        ', ...
                        lon*cspice_dpr() )
       fprintf ( '%s%15.6f\n', 'Earth lat (deg):        ', ...
                        lat*cspice_dpr() )

       %
       % Look up the apparent position of the Earth relative
       % to the Moon's center in the MOON_ME frame at ET.
       %
       [mmoonv, ltime] = cspice_spkpos( 'earth', et, 'moon_me', ...
                                        'lt+s', 'moon'              );
       %
       % Express the Earth direction in terms of longitude
       % and latitude in the MOON_ME frame.
       %
       [r, lon, lat] =  cspice_reclat( mmoonv );

       fprintf ( '\n%s\n', ...
                 'Moon-Earth direction using high accuracy' )
       fprintf ( '%s\n', 'PCK and MOON_ME frame:' )
       fprintf ( '%s%15.6f\n', 'Earth lon (deg):        ', ...
                        lon*cspice_dpr() )
       fprintf ( '%s%15.6f\n', 'Earth lat (deg):        ', ...
                        lat*cspice_dpr() )

       %
       % Find the angular separation of the Earth position
       % vectors in degrees.
       %
       sep = cspice_dpr * cspice_vsep( imoonv, mmoonv );

       fprintf ( '\nFor IAU_MOON vs MOON_ME frames:\n' )
       fprintf ( '%s%15.6f\n', ...
                 'Moon-Earth vector separation angle (deg):     ', ...
                 sep )

       %
       % Look up the apparent position of the Earth relative
       % to the Moon's center in the MOON_PA frame at ET.
       %
       [pmoonv, ltime] = cspice_spkpos( 'earth', et, 'moon_pa', ...
                                        'lt+s', 'moon'              );
       %
       % Express the Earth direction in terms of longitude
       % and latitude in the MOON_PA frame.
       %
       [r, lon, lat] =  cspice_reclat( pmoonv );

       fprintf ( '\n%s\n', ...
                 'Moon-Earth direction using high accuracy' )
       fprintf ( '%s\n', 'PCK and MOON_PA frame:' )
       fprintf ( '%s%15.6f\n', 'Earth lon (deg):        ', ...
                        lon*cspice_dpr() )
       fprintf ( '%s%15.6f\n', 'Earth lat (deg):        ', ...
                        lat*cspice_dpr() )

       %
       % Find the angular separation of the Earth position
       % vectors in degrees.
       %
       sep = cspice_dpr * cspice_vsep( pmoonv, mmoonv );

       fprintf ( '\nFor MOON_PA vs MOON_ME frames:\n' )
       fprintf ( '%s%15.6f\n\n', ...
                 'Moon-Earth vector separation angle (deg):     ', ...
                 sep )

       %
       % Find the apparent sub-Earth point on the Moon at
       % ET using the Moon body-fixed frame MOON_ME.
       %
       [msub, trgepc, srfvec] = ...
              cspice_subpnt( 'near point: ellipsoid', ...
                             'moon', et, 'moon_me', 'lt+s', 'earth' );

       %
       % Display the sub-Earth point in latitudinal coordinates.
       %
       [r, lon, lat] = cspice_reclat( msub );

       fprintf ( [ 'Sub-Earth point on Moon using high accuracy\n',...
            'PCK and MOON_ME frame:\n' ] )
       fprintf ( 'Sub-Earth lon (deg):   %15.6f\n', lon*cspice_dpr )
       fprintf ( 'Sub-Earth lat (deg):   %15.6f\n', lat*cspice_dpr )
       fprintf ( '\n' )

       %
       % Find the apparent sub-Earth point on the Moon at
       % ET using the Moon body-fixed frame MOON_PA.
       %
       [psub, trgepc, srfvec] = ...
              cspice_subpnt( 'near point: ellipsoid', ...
                             'moon', et, 'moon_pa', 'lt+s', 'earth' );

       %
       % Display the sub-Earth point in latitudinal coordinates.
       %
       [r, lon, lat] = cspice_reclat( psub );

       fprintf ( [ 'Sub-Earth point on Moon using high accuracy\n',...
            'PCK and MOON_PA frame:\n' ] )
       fprintf ( 'Sub-Earth lon (deg):   %15.6f\n', lon*cspice_dpr )
       fprintf ( 'Sub-Earth lat (deg):   %15.6f\n', lat*cspice_dpr )
       fprintf ( '\n' )

       %
       % Find the distance between the sub-Earth points in km.
       %
       dist = cspice_vdist( msub, psub );

       fprintf ( [ 'Distance between sub-Earth points (km): ', ...
                   '%15.6f\n' ],  dist  )

       %
       % Unload all kernels.
       %
       cspice_kclear;

       %
       % End of function mrotat
       %
Top ### Solution Sample Output Execute the program:
       Moon-Earth direction using low accuracy
       PCK and IAU_MOON frame:
       Earth lon (deg):               3.613102
       Earth lat (deg):              -6.438342

       Moon-Earth direction using high accuracy
       PCK and MOON_ME frame:
       Earth lon (deg):               3.611229
       Earth lat (deg):              -6.439501

       For IAU_MOON vs MOON_ME frames:
       Moon-Earth vector separation angle (deg):            0.002194

       Moon-Earth direction using high accuracy
       PCK and MOON_PA frame:
       Earth lon (deg):               3.593319
       Earth lat (deg):              -6.417582

       For MOON_PA vs MOON_ME frames:
       Moon-Earth vector separation angle (deg):            0.028235

       Sub-Earth point on Moon using high accuracy
       PCK and MOON_ME frame:
       Sub-Earth lon (deg):          3.611419
       Sub-Earth lat (deg):         -6.439501

       Sub-Earth point on Moon using high accuracy
       PCK and MOON_PA frame:
       Sub-Earth lon (deg):          3.593509
       Sub-Earth lat (deg):         -6.417582

       Distance between sub-Earth points (km):        0.856182
Top # Earth rotation (erotat) Top ## Task Statement Write a program that performs the following computations: `1.` Convert the time string 2007 JAN 1 00:00:00 UTC to a double precision number representing seconds past J2000 TDB. `` In the following instructions, we'll call the result of this computation ET. `2.` Compute the apparent position of the Moon as seen from the Earth in the IAU_EARTH reference frame at the epoch ET. Use light time and stellar aberration corrections. Display the planetocentric longitude and latitude of the Moon position vector in degrees. `3.` Repeat the first computation using the ITRF93 reference frame. Display the results as above. `4.` Compute the angular separation of the position vectors found the the previous two steps. Display the result in degrees. The following computations (steps 5-10) examine the cause of the angular offset found above, which is attributable to the rotation between the ITRF93 and IAU_EARTH frames. Steps 11 and up don't rely on the results of steps 5-10, so steps 5-10 may be safely skipped if they're not of interest to you. For each of the two epochs ET and ET + 100 days, examine the differences between the axes of the ITRF93 and IAU_EARTH frames using the following method: `5.` Convert the epoch of interest to a string in the format style ``2007-MAY-16 02:29:00.000 (UTC).'' Display this string. `6.` Look up the 3x3 position transformation matrix that converts vectors from the IAU_EARTH to the ITRF93 frame at the epoch of interest. We'll call the returned matrix RMAT. `7.` Extract the first row of RMAT into a 3-vector, which we'll call ITRFX. This is the X-axis of the ITRF93 frame expressed relative to the IAU_EARTH frame. `8.` Extract the third row of RMAT into a 3-vector, which we'll call ITRFZ. This is the Z-axis of the ITRF93 frame expressed relative to the IAU_EARTH frame. `9.` Compute the angular separation between the vector ITRFX and the X-axis (1, 0, 0) of the IAU_EARTH frame. Display the result in degrees. `10.` Compute the angular separation between the vector ITRFZ and the Z-axis (0, 0, 1) of the IAU_EARTH frame. Display the result in degrees. This is the end of the computations to be performed for the epochs ET and ET + 100 days. The following steps are part of a new computation. Find the azimuth and elevation of the apparent position of the Moon as seen from the DSN station DSS-13 by the following steps: `11.` Find the apparent position vector of the Moon relative to the DSN station DSS-13 in the topocentric reference frame DSS-13_TOPO at epoch ET. Use light time and stellar aberration corrections. `` For this step, you'll need to have loaded a station SPK file providing geocentric station position vectors, as well as a frame kernel specifying topocentric reference frames centered at the respective DSN stations. (Other kernels will be needed as well; you must choose these.) `12.` Convert the position vector to latitudinal coordinates. Use the routine cspice_reclat for this computation. `13.` Compute the Moon's azimuth and elevation as follows: azimuth is the negative of topocentric longitude and lies within the range 0-360 degrees; elevation is equal to the topocentric latitude. Display the results in degrees. The next computations demonstrate ``high-accuracy'' geometric computations using the Earth as the target body. These computations are *not* realistic; they are simply meant to demonstrate SPICE system features used for geometry computations involving the Earth as a target body. For example, the same basic techniques would be used to find the sub-solar point on the Earth as seen from an Earth-orbiting spacecraft. `14.` Compute the apparent sub-solar point on the Earth at ET, expressed relative to the IAU_EARTH reference frame, using light time and stellar aberration corrections and using the Sun as the observer. Convert the sub-solar point to latitudinal coordinates using cspice_reclat. Display the longitude and latitude of the sub-solar point in degrees. `15.` Repeat the sub-solar point computation described above, using the ITRF93 Earth body-fixed reference frame. Display the results as above. `16.` Compute the distance between the two sub-solar points found above. Display the result in kilometers. Top ## Learning Goals Familiarity with SPICE kernels required to obtain high-accuracy orientation of the Earth. Understanding the differences between results obtained using low and high-accuracy Earth orientation data. Understanding of topocentric frames and computation of target geometry relative to a surface location on the Earth. Knowledge of SPICE kernels required to support such computations. Top ## Approach The following ``tips'' may simplify the solution process. `--` Examine the SPICE kernels provided with this lesson. Use BRIEF to find coverage periods of SPK kernels and binary PCKs. Use COMMNT to view the comment areas of binary PCKs. Examine text kernels, in particular text kernel comments, using a text editor or browser. `--` Decide which SPICE kernels are necessary. Prepare a meta-kernel listing the kernels and load it into the program. `--` Consult the above list titled ``Mice Modules Used'' to see which routines are needed. Note the functions used to provide the values ``seconds per day,'' ``degrees per radian,'' and ``2 times Pi.'' `--` Examine the header of the function cspice_reclat. Note that this function may be used for coordinate conversions in situations where the input rectangular coordinates refer to any reference frame, not only a body-centered, body-fixed frame whose X-Y plane coincides with the body's equator. `--` The computational steps listed above should be followed in the order shown, but steps 5-10 may be omitted. You may find it useful to consult the permuted index, the Mice HTML API documentation pages, and the tutorials titled ``PCK'' and ``High Accuracy Orientation and Body-Fixed frames for Moon and Earth.'' Top ## Solution Top ### Solution Meta-Kernel The meta-kernel we created for the solution to this exercise is named 'erotat.tm'. Its contents follow:
       KPL/MK

       Meta-kernel for the ``Earth Rotation'' task
       in the Binary PCK Hands On Lesson.

       The names and contents of the kernels referenced by this
       meta-kernel are as follows:

       File name                       Contents
       ------------------------------  ---------------------------------
       naif0008.tls                    Generic LSK
       de414_2000_2020.bsp             Solar System Ephemeris
       earthstns_itrf93_050714.bsp     DSN station Ephemeris
       earth_topo_050714.tf            Earth topocentric FK
       pck00008.tpc                    NAIF text PCK
       earth_000101_070725_070503.bpc  Earth binary PCK

       \begindata

       KERNELS_TO_LOAD = ( 'kernels/lsk/naif0008.tls'
                           'kernels/spk/de414_2000_2020.bsp'
                           'kernels/spk/earthstns_itrf93_050714.bsp'
                           'kernels/fk/earth_topo_050714.tf'
                           'kernels/pck/pck00008.tpc'
                           'kernels/pck/earth_000101_070725_070503.bpc' )

       \begintext
Top ### Solution Source Code A sample solution to the problem follows:
       %
       % Binary PCK lesson: Earth rotation
       %
       function erotat()

       %
       % Local Parameters
       %
       METAKR = 'erotat.tm';

       %
       % Other constants
       %
       x = [ 1.; 0.; 0. ];
       z = [ 0.; 0.; 1. ];

       %
       % Load the kernels this program requires.
       %
       cspice_furnsh( METAKR );

       %
       % Convert our UTC string to seconds past J2000 TDB.
       %
       timstr = '2007 jan 1 00:00:00';
       et     = cspice_str2et ( timstr );

       %
       % Look up the apparent position of the Moon relative
       % to the Earth's center in the IAU_EARTH frame at ET.
       %
       [lmoonv, ltime] = cspice_spkpos( 'moon', et, 'iau_earth', ...
                                        'lt+s', 'earth'              );
       %
       % Express the Moon direction in terms of longitude
       % and latitude in the IAU_EARTH frame.
       %
       [r, lon, lat] =  cspice_reclat( lmoonv );

       fprintf ( '\n%s\n', ...
                 'Earth-Moon direction using low accuracy' )
       fprintf ( '%s\n', 'PCK and IAU_EARTH frame:' )
       fprintf ( '%s%15.6f\n', 'Moon lon (deg):        ', ...
                               lon*cspice_dpr() )
       fprintf ( '%s%15.6f\n', 'Moon lat (deg):        ', ...
                               lat*cspice_dpr() )
       %
       % Look up the apparent position of the Moon relative
       % to the Earth's center in the ITRF93 frame at ET.
       %
       [hmoonv, ltime ] = cspice_spkpos( 'moon', et, 'itrf93', ...
                                         'lt+s', 'earth'              );
       %
       % Express the Moon direction in terms of longitude
       % and latitude in the ITRF93 frame.
       %
       [r, lon, lat] =  cspice_reclat( hmoonv );

       fprintf ( '\n%s\n', ...
                 'Earth-Moon direction using high accuracy' )
       fprintf ( '%s\n', 'PCK and ITRF93 frame:' )
       fprintf ( '%s%15.6f\n', 'Moon lon (deg):        ', ...
                               lon*cspice_dpr() )
       fprintf ( '%s%15.6f\n', 'Moon lat (deg):        ', ...
                               lat*cspice_dpr() )
       fprintf ( '\n' )

       %
       % Find the angular separation of the Moon position
       % vectors in degrees.
       %
       sep = cspice_dpr * cspice_vsep( lmoonv, hmoonv );

       fprintf ( '%s%15.6f\n\n', ...
                 'Earth-Moon vector separation angle (deg):     ', ...
                 sep )

       %
       % Next, express the +Z and +X axes of the ITRF93 frame in
       % the IAU_EARTH frame. We'll do this for two times: ET
       % and ET + 100 days.
       %
       for i = 1 : 2

          %
          % Set the time, expressing the time delta in
          % seconds.
          %
          t = et + (i-1)*cspice_spd*100;

          %
          % Convert the TDB time t to a string for output.
          %
          outstr = ...
             cspice_timout ( t, 'YYYY-MON-DD HR:MN:SC.### (UTC)' );

          fprintf ( 'Epoch: %s\n', outstr )

          %
          % Find the rotation matrix for conversion of position
          % vectors from the IAU_EARTH to the ITRF93 frame.
          %
          rmat = cspice_pxform ( 'iau_earth', 'itrf93', t );

          itrfx = rmat(1,:)';
          itrfz = rmat(3,:)';

          %
          % Display the angular offsets of the ITRF93 +X and +Z
          % axes from their IAU_EARTH counterparts.
          %
          sep = cspice_vsep ( itrfx, x );

          fprintf ( '%s%13.6f\n', ...
                    [ 'ITRF93 - IAU_EARTH +X axis ', ...
                      'separation angle (deg): '    ], ...
                    sep * cspice_dpr()                     );

          sep = cspice_vsep ( itrfz, z );

          fprintf ( '%s%13.6f\n\n', ...
                    [ 'ITRF93 - IAU_EARTH +Z axis ', ...
                      'separation angle (deg): '    ], ...
                    sep * cspice_dpr()                     );
       end

       %
       % Find the azimuth and elevation of the apparent
       % position of the Moon in the local topocentric
       % reference frame at the DSN station DSS-13.
       % First look up the Moon's position relative to the
       % station in that frame.
       %
       [topov, ltime] = cspice_spkpos( 'moon', et, 'dss-13_topo', ...
                                       'lt+s', 'dss-13'               );

       %
       % Express the station-Moon direction in terms of
       % longitude and latitude in the DSS-13_TOPO frame.
       %
       [r, lon, lat]= cspice_reclat( topov );

       %
       % Convert to azimuth/elevation.
       %
       az = -lon;

       if  az < 0
          az = az + cspice_twopi;
       end

       el = lat;

       fprintf ( [ 'DSS-13-Moon az/el using high accuracy ', ...
                   'PCK and DSS-13_TOPO frame:\n' ]                )
       fprintf ( 'Moon Az (deg):         %15.6f\n', az*cspice_dpr )
       fprintf ( 'Moon El (deg):         %15.6f\n', el*cspice_dpr )
       fprintf ( '\n' );

       %
       % Find the sub-solar point on the Earth at ET using the
       % Earth body-fixed frame IAU_EARTH. Treat the Sun as
       % the observer.
       %
       [ lsub, trgepc, srfvec ] =                   ...
           cspice_subslr( 'near point: ellipsoid',  ...
                          'earth', et, 'iau_earth', 'lt+s', 'sun' );

       %
       % Display the sub-point in latitudinal coordinates.
       %
       [r, lon, lat] = cspice_reclat( lsub );

       fprintf ( [ 'Sub-Solar point on Earth using low accuracy\n',...
                   'PCK and IAU_EARTH frame:\n' ] )
       fprintf ( 'Sub-Solar lon (deg):   %15.6f\n', lon*cspice_dpr )
       fprintf ( 'Sub-Solar lat (deg):   %15.6f\n', lat*cspice_dpr )
       fprintf ( '\n' )

       %
       % Find the sub-solar point on the Earth at ET using the
       % Earth body-fixed frame ITRF93. Treat the Sun as
       % the observer.
       %
       [ hsub, trgepc, srfvec ] =                   ...
           cspice_subslr( 'near point: ellipsoid',  ...
                          'earth', et, 'itrf93', 'lt+s', 'sun' );

       %
       % Display the sub-point in latitudinal coordinates.
       %
       [r, lon, lat] = cspice_reclat( hsub );

       fprintf ( [ 'Sub-Solar point on Earth using high accuracy\n',...
                   'PCK and ITRF93 frame:\n' ] )
       fprintf ( 'Sub-Solar lon (deg):   %15.6f\n', lon*cspice_dpr )
       fprintf ( 'Sub-Solar lat (deg):   %15.6f\n', lat*cspice_dpr )
       fprintf ( '\n' )

       %
       % Find the distance between the sub-solar point
       % vectors in km.
       %
       dist = cspice_vdist( lsub, hsub );

       fprintf ( [ 'Distance between sub-solar points (km): ', ...
                   '%15.6f\n' ],  dist  )

       %
       % Unload all kernels.
       %
       cspice_kclear;

       %
       % End of function erotat
       %
Top ### Solution Sample Output Execute the program:
       Earth-Moon direction using low accuracy
       PCK and IAU_EARTH frame:
       Moon lon (deg):             -35.496272
       Moon lat (deg):              26.416959

       Earth-Moon direction using high accuracy
       PCK and ITRF93 frame:
       Moon lon (deg):             -35.554286
       Moon lat (deg):              26.419156

       Earth-Moon vector separation angle (deg):            0.052002

       Epoch: 2007-JAN-01 00:00:00.000 (UTC)
       ITRF93 - IAU_EARTH +X axis separation angle (deg):      0.057677
       ITRF93 - IAU_EARTH +Z axis separation angle (deg):      0.002326

       Epoch: 2007-APR-10 23:59:59.998 (UTC)
       ITRF93 - IAU_EARTH +X axis separation angle (deg):      0.057787
       ITRF93 - IAU_EARTH +Z axis separation angle (deg):      0.002458

       DSS-13-Moon az/el using high accuracy PCK and DSS-13_TOPO frame:
       Moon Az (deg):               72.169006
       Moon El (deg):               20.689488

       Sub-Solar point on Earth using low accuracy
       PCK and IAU_EARTH frame:
       Sub-Solar lon (deg):       -177.100531
       Sub-Solar lat (deg):        -22.910377

       Sub-Solar point on Earth using high accuracy
       PCK and ITRF93 frame:
       Sub-Solar lon (deg):       -177.157874
       Sub-Solar lat (deg):        -22.912593

       Distance between sub-solar points (km):        5.881861