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Geometric Event Finding Hands-On Lesson, using MPO (FORTRAN)

Table of Contents 


   Geometric Event Finding Hands-On Lesson, using MPO (FORTRAN)
      Overview
      Note About HTML Links
      References
         Tutorials
         Required Readings
         The Permuted Index
         API Documentation
      Kernels Used
      SPICE Modules Used

   Find View Periods
      Task Statement
      Learning Goals
      Approach
         Solution steps
      Solution
         Solution Meta-Kernel
         Solution Code
         Solution Sample Output

   Find Times when Target is Visible
      Task Statement
      Learning Goals
      Approach
         Solution steps
      Solution
         Solution Meta-Kernel
         Solution Code
         Solution Sample Output

   Extra Credit
      Task statements
      Solutions


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Geometric Event Finding Hands-On Lesson, using MPO (FORTRAN)




March 01, 2023



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Overview



This lesson illustrates how the Geometry Finder (GF) subsystem of the SPICE Toolkit can be used to find time intervals when specified geometric conditions are satisfied.

In this lesson the student is asked to construct a program that finds the time intervals, within a specified time range, when the BepiColombo Mercury Planetary Orbiter (MPO) is visible from ESA's deep space station in New Norcia. Possible occultation of the spacecraft by Mercury is to be considered.



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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 ``toolkit/'' tree and copy this document to that subdirectory before loading it into a Web browser.



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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''.

In some cases the lesson explanations also refer to the information provided in the meta-data area of the kernels used in the lesson examples. It is especially true in case of the FK and IK files, which often contain comprehensive descriptions of the frames, instrument FOVs, etc. Since both FK and IK are text kernels, the information provided in them can be viewed using any text editor, while the meta information provided in binary kernels -- SPKs and CKs -- can be viewed using ``commnt'' or ``spacit'' utility programs located in ``toolkit/exe'' of Toolkit installation tree.



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Tutorials



The following SPICE tutorials serve as references for the discussions in this lesson: 

 
   Name              Lesson steps/routines it describes
   ----------------  -----------------------------------------------
   Time              Time Conversion
   SCLK and LSK      Time Conversion
   SPK               Obtaining Ephemeris Data
   Frames            Reference Frames
   Using Frames      Reference Frames
   PCK               Planetary Constants Data
   Lunar-Earth PCK   Lunar and Earth Orientation Data
   GF                The SPICE Geometry Finder (GF) subsystem
   DSK               Detailed Target Shape (Topography) Data
These tutorials are available from the NAIF server at JPL: 

   https://naif.jpl.nasa.gov/naif/tutorials.html


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Required Readings



The Required Reading documents are provided with the Toolkit and are located under the ``toolkit/doc'' directory in the SPICE Toolkit installation tree. 

   Name             Lesson steps/routines that it describes
   ---------------  -----------------------------------------
   cells.req        Cell/window initialization
   frames.req       Using reference frames
   gf.req           The SPICE geometry finder (GF) subsystem
   kernel.req       Loading SPICE kernels
   naif_ids.req     Body and reference frame names
   pck.req          Obtaining planetary constants data
   spk.req          Computing positions and velocities
   time.req         UTC to ET time conversion
   windows.req      The SPICE window data type


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The Permuted Index



Another useful document distributed with the Toolkit is the permuted index. It is located under the ``toolkit/doc'' directory in the FORTRAN installation tree.

This text document provides a simple mechanism by which users can discover which SPICE routines perform functions of interest, as well as the names of the source files that contain these routines. This is particularly useful for FORTRAN programmers because some of the routines are entry points; the names of these routines do not translate directly into the name of the respective source files that contain them.



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API Documentation



The most detailed specification of a given SPICE FORTRAN routine is contained in the header section of its source code. The source code is distributed with the Toolkit and is located under the ``toolkit/src/spicelib'' path.

For example the path of the source code of the STR2ET routine is 

   toolkit/src/spicelib/str2et.for
Since some of the FORTRAN routines are entry points they may be part of a source file that has different name. The ``Permuted Index'' document mentioned above can be used to locate the name of their source file.



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Kernels Used



The following kernels are used in examples provided in this lesson: 

   1.  Solar System Ephemeris SPK, subsetted to cover only the time
       range of interest:
 
          de432s.bsp
 
   2.  ESA stations SPK:
 
          estrack_v04.bsp
 
   3.  ESA stations frame definitions:
 
          estrack_v04.tf
 
   4.  EARTH_FIXED/ITRF93 frame connection:
 
          earthfixeditrf93.tf
 
   5.  Binary PCK for Earth:
 
          earth_070425_370426_predict.bpc
 
   6.  BepiColombo MPO Spacecraft Trajectory SPK, subsetted to cover
       only the time range of interest:
 
          bc_mpo_mlt_50037_20260314_20280529_v05.bsp
 
   7.  Generic LSK:
 
          naif0012.tls
 
   8.  Generic PCK:
 
          pck00011.tpc
 
   9.  Low-resolution Mercury DSK:
 
          mercury_lowres.bds
These SPICE kernels are included in the lesson package.



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SPICE Modules Used



This section provides a complete list of the routines 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   ROUTINES   FUNCTIONS  KERNELS
   ------- ---------  ---------  ---------  ----------
      1    viewpr     FURNSH     RPD        1-8
                      TOSTDO     WNCARD
                      REPMC
                      REPMF
                      STR2ET
                      TIMOUT
                      SSIZED
                      WNINSD
                      GFPOSC
                      WNFETD
 
      2    visibl     FURNSH     RPD        1-9
                      TOSTDO     WNCARD
                      REPMC
                      REPMF
                      STR2ET
                      TIMOUT
                      SSIZED
                      WNINSD
                      GFPOSC
                      GFOCLT
                      WNDIFD
                      WNFETD
 
           extra (*)  GFDIST                1,6-8
                      KCLEAR
 
 
      (*) Additional APIs and kernels used in Extra Credit tasks.
Refer to the headers of the various routines listed above, as detailed interface specifications are provided with the source code.



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Find View Periods






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Task Statement



Write a program that finds the set of time intervals, within the time range 

   2027 JAN 03 TDB
   2027 JAN 06 TDB
when the BepiColombo Mercury Planetary Orbiter (MPO) is visible from ESA's New Norcia station. These time intervals are frequently called ``view periods.''

The spacecraft is considered visible if its apparent position (that is, its position corrected for light time and stellar aberration) has elevation of at least 6 degrees in the topocentric reference frame NEW_NORCIA_TOPO. In this exercise, we ignore the possibility of occultation of the spacecraft by Mercury.

Use a search step size that ensures that no view periods of duration 5 minutes or longer will be missed by the search.

Display the start and stop times of these intervals using TDB calendar dates and millisecond precision.



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Learning Goals



Exposure to SPICE GF event finding routines. Familiarity with SPICE windows and routines that manipulate them. Exposure to SPICE time parsing and output formatting routines.



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Approach





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Solution steps



A possible solution could consist of the following steps:

Preparation:

    1. Review the SPICELIB cell and window Required Reading.

    2. Decide what SPICE kernels are necessary. Use the SPICE summary tool BRIEF to examine the coverage of the binary kernels and verify the availability of required data.

    3. Create a meta-kernel listing the SPICE kernels to be loaded. (Hint: consult a programming example tutorial, or the Introduction to Kernels tutorial, for a reminder of how to do this.)

    Name the meta-kernel 'viewpr.tm'.

Next, write a program that performs the following steps:

    1. Use FURNSH to load the meta-kernel.

    2. Declare confinement and output SPICELIB windows and initialize them using SSIZED.

    3. Insert the given time bounds into the confinement window using WNINSD.

    4. Select a step size for searching for visibility state transitions: in this case, each target rise or set event is a state transition.

    The step size must be large enough so the search proceeds with reasonable speed, but small enough so that no visibility transition events---that is, target rise or set events---are missed.

    5. Use the GF routine GFPOSC to find the window of times, within the confinement window, during which the BepiColombo MPO spacecraft is above the elevation limit as seen from ESA's New Norcia station, in the reference frame NEW_NORCIA_TOPO.

    Use light time and stellar aberration corrections for the apparent position of the spacecraft as seen from the station.

    6. Fetch and display the contents of the result window. Use WNFETD to extract from the result window the start and stop times of each time interval. Display each of the intervals in the result window as a pair of start and stop times. Express each time as a TDB calendar date using the routine TIMOUT.

    The SPICELIB ``replace marker'' routines REPxxx are useful for substituting values into output strings. Consult, for example, the header of REPMC.

You may find it useful to consult the references listed above. In particular, the header of the SPICE GF routine GFPOSC contains pertinent documentation.



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Solution





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Solution Meta-Kernel



The meta-kernel we created for the solution to this exercise is named 'viewpr.tm'. Its contents follow: 

   KPL/MK
 
      This is the meta-kernel used in the solution of the tasks in the
      Geometric Event Finding Hands On Lesson.
 
      The names and contents of the kernels referenced by this
      meta-kernel are as follows:
 
         1.  Solar System Ephemeris SPK, subsetted to cover only the
             time range of interest:
 
                de432s.bsp
 
         2.  ESA stations SPK:
 
                estrack_v04.bsp
 
         3.  ESA stations frame definitions:
 
                estrack_v04.tf
 
         4.  EARTH_FIXED/ITRF93 frame connection:
 
                earthfixeditrf93.tf
 
         5.  Binary PCK for Earth:
 
                earth_070425_370426_predict.bpc
 
         6.  BepiColombo MPO Spacecraft Trajectory SPK, subsetted to
             cover only the time range of interest:
 
                bc_mpo_mlt_50037_20260314_20280529_v05.bsp
 
         7.  Generic LSK:
 
                naif0012.tls
 
         8.  Generic PCK:
 
               pck00011.tpc
 
 
   \begindata
 
      KERNELS_TO_LOAD = (
 
         'kernels/spk/de432s.bsp'
         'kernels/spk/estrack_v04.bsp'
         'kernels/fk/estrack_v04.tf'
         'kernels/fk/earthfixeditrf93.tf'
         'kernels/pck/earth_070425_370426_predict.bpc'
         'kernels/lsk/naif0012.tls'
         'kernels/spk/bc_mpo_mlt_50037_20260314_20280529_v05.bsp'
         'kernels/pck/pck00011.tpc'
 
                        )
 
   \begintext


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Solution Code



The example program below shows one possible solution. 

         PROGRAM VIEWPR
         IMPLICIT NONE
   C
   C     Find and display the window of times when the BepiColombo
   C     MPO spacecraft is above a specified elevation limit in the
   C     topocentric reference frame of ESA's New Norcia station.
   C
 
   C
   C     SPICELIB functions
   C
         DOUBLE PRECISION      RPD
         INTEGER               WNCARD
 
   C
   C     Global GF parameters: we import from this file
   C     the definition of the GF workspace size parameter
   C
   C        NWMAX
   C
         INCLUDE 'gf.inc'
 
   C
   C     Local parameters
   C
   C     Format string for time output:
   C
         CHARACTER*(*)         TDBFMT
         PARAMETER           ( TDBFMT =
        .                    'YYYY MON DD HR:MN:SC.### (TDB) ::TDB' )
 
   C
   C     The meta-kernel:
   C
         CHARACTER*(*)         METAKR
         PARAMETER           ( METAKR = 'viewpr.tm' )
 
 
   C
   C     Maximum number of intervals in any window:
   C
         INTEGER               MAXIVL
         PARAMETER           ( MAXIVL = 1000 )
 
   C
   C     Maximum result window size:
   C
         INTEGER               MAXWIN
         PARAMETER           ( MAXWIN = 2 * MAXIVL )
 
   C
   C     SPICELIB cell bound:
   C
         INTEGER               LBCELL
         PARAMETER           ( LBCELL = -5 )
 
   C
   C     String length parameters:
   C
         INTEGER               CORLEN
         PARAMETER           ( CORLEN = 10 )
 
         INTEGER               FRNMLN
         PARAMETER           ( FRNMLN = 32 )
 
         INTEGER               LNSIZE
         PARAMETER           ( LNSIZE = 200 )
 
         INTEGER               NAMLEN
         PARAMETER           ( NAMLEN = 32 )
 
         INTEGER               OPLEN
         PARAMETER           ( OPLEN  = 50 )
 
         INTEGER               TIMLEN
         PARAMETER           ( TIMLEN = 50 )
 
   C
   C     Local variables
   C
         CHARACTER*(CORLEN)    ABCORR
         CHARACTER*(NAMLEN)    CRDSYS
         CHARACTER*(NAMLEN)    COORD
         CHARACTER*(LNSIZE)    LINE
         CHARACTER*(FRNMLN)    OBSFRM
         CHARACTER*(OPLEN)     RELATE
         CHARACTER*(NAMLEN)    SRFPT
         CHARACTER*(TIMLEN)    START
         CHARACTER*(TIMLEN)    STOP
         CHARACTER*(NAMLEN)    TARGET
         CHARACTER*(TIMLEN)    TIMSTR
         CHARACTER*(LNSIZE)    TITLE
 
         DOUBLE PRECISION      ADJUST
 
   C
   C     Confinement window used to store interval to be searched:
   C
         DOUBLE PRECISION      CNFINE ( LBCELL : MAXWIN )
         DOUBLE PRECISION      ELVLIM
         DOUBLE PRECISION      ETBEG
         DOUBLE PRECISION      ETEND
         DOUBLE PRECISION      INTBEG
         DOUBLE PRECISION      INTEND
         DOUBLE PRECISION      REVLIM
 
   C
   C     STEPSZ is the step size, measured in seconds, used to search
   C     for times bracketing a state transition.
   C
         DOUBLE PRECISION      STEPSZ
 
   C
   C     Result window used to store rise/set times:
   C
         DOUBLE PRECISION      RISWIN ( LBCELL : MAXWIN )
 
   C
   C     Workspace array:
   C
         DOUBLE PRECISION      WORK   ( LBCELL : MAXWIN,  NWMAX )
 
 
         INTEGER               I
         INTEGER               WINSIZ
 
   C
   C     Load the meta-kernel.
   C
         CALL FURNSH ( METAKR )
 
   C
   C     Assign the inputs for our search.
   C
   C     Since we're interested in the apparent location of the
   C     target, we use light time and stellar aberration
   C     corrections.  We use the "converged Newtonian" form
   C     of the light time correction because this choice may
   C     increase the accuracy of the occultation times we'll
   C     compute using GFOCLT.
   C
         SRFPT  = 'NEW_NORCIA'
         OBSFRM = 'NEW_NORCIA_TOPO'
         TARGET = 'MPO'
         ABCORR = 'CN+S'
         START  = '2027 JAN 03 TDB'
         STOP   = '2027 JAN 06 TDB'
         ELVLIM =  6.0D0
   C
   C     The elevation limit above has units of degrees; we convert
   C     this value to radians for computation using SPICE routines.
   C     We'll store the equivalent value in radians in REVLIM.
   C
         REVLIM = RPD() * ELVLIM
 
   C
   C     Since SPICE doesn't directly support the AZ/EL coordinate
   C     system, we use the equivalent constraint
   C
   C        latitude > REVLIM
   C
   C     in the latitudinal coordinate system, where the reference
   C     frame is topocentric and is centered at the viewing location.
   C
         CRDSYS = 'LATITUDINAL'
         COORD  = 'LATITUDE'
         RELATE = '>'
 
   C
   C     The adjustment value only applies to absolute extrema
   C     searches; simply give it an initial value of zero
   C     for this inequality search.
   C
         ADJUST = 0.D0
 
   C
   C     STEPSZ is the step size, measured in seconds, used to search
   C     for times bracketing a state transition. Since we don't expect
   C     any events of interest to be shorter than five minutes, and
   C     since the separation between events is well over 5 minutes,
   C     we'll use this value as our step size. Units are seconds.
   C
         STEPSZ = 300.D0
 
   C
   C     Display a banner for the output report:
   C
         TITLE  = 'Inputs for target visibility search:'
         CALL TOSTDO ( ' '   )
         CALL TOSTDO ( TITLE )
         CALL TOSTDO ( ' '   )
 
         LINE   = '   Target                       = #'
         CALL REPMC  ( LINE, '#', TARGET, LINE )
         CALL TOSTDO ( LINE )
 
         LINE   = '   Observation surface location = #'
         CALL REPMC  ( LINE, '#', SRFPT,  LINE )
         CALL TOSTDO ( LINE )
 
         LINE   = '   Observer''s reference frame   = #'
         CALL REPMC  ( LINE, '#', OBSFRM,  LINE )
         CALL TOSTDO ( LINE )
 
         LINE   = '   Elevation limit (degrees)    = #'
         CALL REPMF ( LINE, '#', ELVLIM, 7, 'F', LINE )
         CALL TOSTDO ( LINE )
 
         LINE   = '   Aberration correction        = #'
         CALL REPMC  ( LINE, '#', ABCORR, LINE )
         CALL TOSTDO ( LINE )
 
         LINE   = '   Step size (seconds)          = #'
         CALL REPMF ( LINE, '#', STEPSZ, 9, 'F', LINE )
         CALL TOSTDO ( LINE )
 
   C
   C     Convert the start and stop times to ET.
   C
         CALL STR2ET ( START, ETBEG )
         CALL STR2ET ( STOP,  ETEND )
 
   C
   C     Display the search interval start and stop times
   C     using the format shown below.
   C
   C        2004 MAY 06 20:15:00.000 (TDB)
   C
         CALL TIMOUT ( ETBEG, TDBFMT, TIMSTR )
         LINE = '   Start time                   = '//TIMSTR
         CALL TOSTDO ( LINE )
 
         CALL TIMOUT ( ETEND, TDBFMT, TIMSTR )
         LINE = '   Stop time                    = '//TIMSTR
         CALL TOSTDO ( LINE )
 
         CALL TOSTDO ( ' ' )
 
   C
   C     Every SPICELIB window must have its size initialized.
   C
   C     Initialize the "confinement" window with the interval
   C     over which we'll conduct the search.
   C
         CALL SSIZED ( MAXWIN, CNFINE )
 
         CALL WNINSD ( ETBEG,  ETEND, CNFINE )
 
   C
   C     Initialize the result window; this window will contain
   C     the rise/set times found by our search.
   C
         CALL SSIZED ( MAXWIN, RISWIN )
 
   C
   C     In the call below, the workspace dimensions are
   C
   C        ( MAXWIN, NWMAX )
   C
   C     Now search for the time period, within our confinement
   C     window, during which the apparent target has elevation
   C     at least equal to the elevation limit.
   C
         CALL GFPOSC ( TARGET, OBSFRM, ABCORR, SRFPT,
        .              CRDSYS, COORD,  RELATE, REVLIM,
        .              ADJUST, STEPSZ, CNFINE, MAXWIN,
        .              NWMAX,  WORK,   RISWIN         )
 
   C
   C     The function WNCARD returns the number of intervals
   C     in a SPICE window.
   C
         WINSIZ = WNCARD( RISWIN )
 
         IF ( WINSIZ .EQ. 0 ) THEN
 
            WRITE (*,*) 'No events were found.'
 
         ELSE
   C
   C        Display the view periods.
   C
            LINE = 'Visibility times of # as seen from #:'
            CALL REPMC ( LINE, '#', TARGET, LINE )
            CALL REPMC ( LINE, '#', SRFPT,  LINE )
 
            CALL TOSTDO ( LINE )
            CALL TOSTDO ( ' '  )
 
            DO I = 1, WINSIZ
   C
   C           Fetch the start and stop times of the Ith interval
   C           from the search result window RISWIN.
   C
               CALL WNFETD ( RISWIN, I, INTBEG, INTEND )
 
   C
   C           Convert the rise time to a TDB calendar string.
   C
               CALL TIMOUT ( INTBEG, TDBFMT, TIMSTR )
 
   C
   C           Write the string to standard output.
   C
               IF ( I .EQ. 1 ) THEN
                  LINE = 'Visibility or window start time:  #'
               ELSE
                  LINE = 'Visibility start time:            #'
               END IF
 
               CALL REPMC ( LINE, '#', TIMSTR, LINE )
               CALL TOSTDO ( LINE )
 
   C
   C           Convert the set time to a TDB calendar string.
   C
               CALL TIMOUT ( INTEND, TDBFMT, TIMSTR )
   C
   C           Write the string to standard output.
   C
               IF ( I .EQ. WINSIZ ) THEN
                  LINE = 'Visibility or window stop time:   #'
               ELSE
                  LINE = 'Visibility stop time:             #'
               END IF
 
               CALL REPMC ( LINE, '#', TIMSTR, LINE )
               CALL TOSTDO ( LINE )
 
               CALL TOSTDO ( ' ' )
 
            END DO
 
         END IF
 
         END


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Solution Sample Output



Numerical results shown for this example may differ across platforms since the results depend on the SPICE kernels used as input and on the host platform's arithmetic implementation.

After compiling the program, execute it. The output is: 

 
   Inputs for target visibility search:
 
      Target                       = MPO
      Observation surface location = NEW_NORCIA
      Observer's reference frame   = NEW_NORCIA_TOPO
      Elevation limit (degrees)    = 6.000000
      Aberration correction        = CN+S
      Step size (seconds)          = 300.000000
      Start time                   = 2027 JAN 03 00:00:00.000 (TDB)
      Stop time                    = 2027 JAN 06 00:00:00.000 (TDB)
 
   Visibility times of MPO as seen from NEW_NORCIA:
 
   Visibility or window start time:  2027 JAN 03 00:00:00.000 (TDB)
   Visibility stop time:             2027 JAN 03 10:58:25.063 (TDB)
 
   Visibility start time:            2027 JAN 03 21:55:08.488 (TDB)
   Visibility stop time:             2027 JAN 04 11:01:14.279 (TDB)
 
   Visibility start time:            2027 JAN 04 21:58:41.333 (TDB)
   Visibility stop time:             2027 JAN 05 11:04:00.020 (TDB)
 
   Visibility start time:            2027 JAN 05 22:02:18.477 (TDB)
   Visibility or window stop time:   2027 JAN 06 00:00:00.000 (TDB)


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Find Times when Target is Visible






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Task Statement



Extend the program of the previous chapter to find times when the BepiColombo MPO orbiter is:

    -- Above the elevation limit in the NEW_NORCIA_TOPO topocentric reference frame.

    -- and is not occulted by Mercury

Finding time intervals that satisfy the second condition requires a search for occultations of the spacecraft by Mercury. Perform this search twice: once using an ellipsoidal shape model for Mercury, and once using a DSK shape model.

Compute the final results twice as well, using the results of both occultation searches.

For each of the two shape model cases, store the set of time intervals when the spacecraft is visible in a SPICELIB window. We'll call this the ``result window.''

Display each of the intervals in the result window as a pair of start and stop times. Express each time as a TDB calendar date using the same format as in the previous program.



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Learning Goals



Familiarity with the GF occultation finding routine GFOCLT. Experience with Digital Shape Kernel (DSK) shape models. Further experience with the SPICELIB window routines.



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Approach





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Solution steps



A possible solution would consist of the following steps:

    1. Use the meta-kernel from the previous chapter as the starting point. Add more kernels to it as needed.

    Name the meta-kernel 'visibl.tm'.

    2. Include the code from the program of the previous chapter in a new source file; modify this code to create the new program.

    3. Your program will need additional windows to capture the results of occultation searches performed using both ellipsoidal and DSK shape models. Additional windows will be needed to compute the set differences of the elevation search (``view period'') window and each of the occultation search windows. Further details are provided below.

    Declare additional output SPICELIB windows and initialize them using SSIZED.

    4. The remaining steps can be performed twice: once using an ellipsoidal shape model for Mercury, and once using a DSK Mercury shape model. Alternatively, two copies of the entire solution program can be created: one for each shape model.

    5. Search for occultations of the BepiColombo MPO orbiter as seen from New Norcia station using GFOCLT. Use as the confinement window for this search the result window from the elevation search performed by GFPOSC.

    Since occultations occur when the apparent BepiColombo MPO spacecraft position is behind the apparent figure of Mercury, light time correction must be performed for the occultation search. To improve accuracy of the occultation state determination, use ``converged Newtonian'' light time correction.

    6. Use the SPICELIB window subtraction routine WNDIFD to subtract the window of times when the spacecraft is occulted from the window of times when the spacecraft is above the elevation limit. The difference window is the final result.

    7. Modify the code to display the contents of the difference window.

This completes the assignment.



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Solution





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Solution Meta-Kernel



The meta-kernel we created for the solution to this exercise is named 'visibl.tm'. Its contents follow: 

   KPL/MK
 
      This is the meta-kernel used in the solution of the tasks in the
      Geometric Event Finding Hands On Lesson.
 
      The names and contents of the kernels referenced by this
      meta-kernel are as follows:
 
         1.  Solar System Ephemeris SPK, subsetted to cover only the
             time range of interest:
 
                de432s.bsp
 
         2.  ESA stations SPK:
 
                estrack_v04.bsp
 
         3.  ESA stations frame definitions:
 
                estrack_v04.tf
 
         4.  EARTH_FIXED/ITRF93 frame connection:
 
                earthfixeditrf93.tf
 
         5.  Binary PCK for Earth:
 
                earth_070425_370426_predict.bpc
 
         6.  BepiColombo MPO Spacecraft Trajectory SPK, subsetted to
             cover only the time range of interest:
 
                bc_mpo_mlt_50037_20260314_20280529_v05.bsp
 
         7.  Generic LSK:
 
                naif0012.tls
 
         8.  Generic PCK:
 
               pck00011.tpc
 
         9. Low-resolution Mercury DSK:
 
               mercury_lowres.bds
 
   \begindata
 
      KERNELS_TO_LOAD = (
 
         'kernels/spk/de432s.bsp'
         'kernels/spk/estrack_v04.bsp'
         'kernels/fk/estrack_v04.tf'
         'kernels/fk/earthfixeditrf93.tf'
         'kernels/pck/earth_070425_370426_predict.bpc'
         'kernels/lsk/naif0012.tls'
         'kernels/spk/bc_mpo_mlt_50037_20260314_20280529_v05.bsp'
         'kernels/pck/pck00011.tpc'
         'kernels/dsk/mercury_lowres.bds'
 
                        )
 
   \begintext


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Solution Code 



         PROGRAM VISIBL
         IMPLICIT NONE
   C
   C     Find and display the window of times when the BepiColombo
   C     MPO spacecraft is above a specified elevation limit in the
   C     topocentric reference frame of ESA's New Norcia station
   C     and is not occulted by Mercury.
   C
 
   C
   C     SPICELIB functions
   C
         DOUBLE PRECISION      RPD
         INTEGER               WNCARD
 
   C
   C     Global GF parameters: we import from this file
   C     the definition of the GF workspace size parameter
   C
   C        NWMAX
   C
   C     and the occultation parameters
   C
   C        OCCLN
   C        SHPLEN
   C
         INCLUDE 'gf.inc'
 
   C
   C     Local parameters
   C
   C     Format string for time output:
   C
         CHARACTER*(*)         TDBFMT
         PARAMETER           ( TDBFMT =
        .                    'YYYY MON DD HR:MN:SC.### TDB ::TDB' )
   C
   C     The meta-kernel:
   C
         CHARACTER*(*)         METAKR
         PARAMETER           ( METAKR = 'visibl.tm' )
 
   C
   C     Maximum number of events we can handle in our event set:
   C
         INTEGER               MAXEVT
         PARAMETER           ( MAXEVT = 1000 )
 
   C
   C     Maximum result window size:
   C
         INTEGER               MAXWIN
         PARAMETER           ( MAXWIN = 2 * MAXEVT )
 
   C
   C     SPICELIB cell bound:
   C
         INTEGER               LBCELL
         PARAMETER           ( LBCELL = -5 )
 
   C
   C     String length parameters:
   C
         INTEGER               BDNMLN
         PARAMETER           ( BDNMLN = 36 )
 
         INTEGER               CORLEN
         PARAMETER           ( CORLEN = 10 )
 
         INTEGER               FRNMLN
         PARAMETER           ( FRNMLN = 32 )
 
         INTEGER               LNSIZE
         PARAMETER           ( LNSIZE = 78 )
 
         INTEGER               MTHLEN
         PARAMETER           ( MTHLEN = 100 )
 
         INTEGER               NAMLEN
         PARAMETER           ( NAMLEN = 32 )
 
         INTEGER               OPLEN
         PARAMETER           ( OPLEN  = 50 )
 
         INTEGER               TIMLEN
         PARAMETER           ( TIMLEN = 28 )
 
   C
   C     Local variables
   C
         CHARACTER*(CORLEN)    ABCORR
         CHARACTER*(BDNMLN)    BACK
         CHARACTER*(FRNMLN)    BFRAME
         CHARACTER*(MTHLEN)    BSHAPE
         CHARACTER*(TIMLEN)    BTMSTR
         CHARACTER*(NAMLEN)    CRDSYS
         CHARACTER*(NAMLEN)    COORD
         CHARACTER*(TIMLEN)    ETMSTR
         CHARACTER*(FRNMLN)    FFRAME
         CHARACTER*(FRNMLN)    FRAME
         CHARACTER*(BDNMLN)    FRONT
         CHARACTER*(MTHLEN)    FSHAPE
         CHARACTER*(LNSIZE)    LINE
         CHARACTER*(OCLLN)     OCCTYP
         CHARACTER*(OPLEN)     RELATE
         CHARACTER*(NAMLEN)    SRFPT
         CHARACTER*(TIMLEN)    START
         CHARACTER*(TIMLEN)    STOP
         CHARACTER*(NAMLEN)    TARGET
         CHARACTER*(LNSIZE)    TITLE
 
         DOUBLE PRECISION      ADJUST
 
   C
   C     Confinement window used to store interval to be searched:
   C
         DOUBLE PRECISION      CNFINE ( LBCELL : MAXEVT )
         DOUBLE PRECISION      DINTBG
         DOUBLE PRECISION      DINTEN
         DOUBLE PRECISION      ELVLIM
         DOUBLE PRECISION      ETBEG
         DOUBLE PRECISION      ETEND
         DOUBLE PRECISION      INTBEG
         DOUBLE PRECISION      INTEND
         DOUBLE PRECISION      REVLIM
 
   C
   C     STEPSZ is the step size, measured in seconds, used to search
   C     for times bracketing a state transition.
   C
         DOUBLE PRECISION      STEPSZ
 
   C
   C     Result window used to store occultation times:
   C
         DOUBLE PRECISION      OCCWIN ( LBCELL : MAXWIN )
 
   C
   C     Result window used to store occultation times
   C     computed using DSK data:
   C
         DOUBLE PRECISION      DOCWIN ( LBCELL : MAXWIN )
 
   C
   C     Result window used to store rise/set times:
   C
         DOUBLE PRECISION      RISWIN ( LBCELL : MAXWIN )
 
   C
   C     Result windows used to store visibility periods:
   C
         DOUBLE PRECISION      DVISWN ( LBCELL : MAXWIN )
         DOUBLE PRECISION      EVISWN ( LBCELL : MAXWIN )
 
   C
   C     Workspace array:
   C
         DOUBLE PRECISION      WORK   ( LBCELL : MAXWIN,  NWMAX )
 
 
         INTEGER               I
         INTEGER               WINSIZ
 
   C
   C     Load the meta-kernel.
   C
         CALL FURNSH ( METAKR )
 
   C
   C     Assign the inputs for our search.
   C
         SRFPT  = 'NEW_NORCIA'
         FRAME  = 'NEW_NORCIA_TOPO'
         TARGET = 'MPO'
         ABCORR = 'CN+S'
         START  = '2027 JAN 03 TDB'
         STOP   = '2027 JAN 06 TDB'
         ELVLIM =  6.0D0
 
   C
   C     The elevation limit above has units of degrees; we convert
   C     this value to radians for computation using SPICE routines.
   C     We'll store the equivalent value in radians in REVLIM.
   C
         REVLIM = RPD() * ELVLIM
 
   C
   C     Since we're interested in the apparent location of the
   C     target, we use light time and stellar aberration
   C     corrections.
   C
   C     Since SPICE doesn't directly support the AZ/EL coordinate
   C     system, we use the equivalent constraint
   C
   C        latitude > REVLIM
   C
   C     in the latitudinal coordinate system, where the reference
   C     frame is topocentric and is centered at the viewing location.
   C
         CRDSYS = 'LATITUDINAL'
         COORD  = 'LATITUDE'
         RELATE = '>'
 
   C
   C     STEPSZ is the step size, measured in seconds, used to search
   C     for times bracketing a state transition. Since we don't expect
   C     any events of interest to be shorter than five minutes, and
   C     since the separation between events is well over 5 minutes,
   C     we'll use this value as our step size. Units are seconds.
   C
         STEPSZ = 300.D0
 
   C
   C     We model the target shape as a point. We either model the
   C     blocking body's shape as an ellipsoid, or we represent its shap
   e
   C     using actual topographic data. No body-fixed reference frame is
   C     required for the target since its orientation is not used.
   C
   C     We'll set the shape specifications just before our calls to
   C     the GF occultation finding routine GFOCLT.
   C
         BACK   = TARGET
         BSHAPE = 'POINT'
         BFRAME = ' '
         FRONT  = 'MERCURY'
         FFRAME = 'IAU_MERCURY'
 
   C
   C     The occultation type should be set to 'ANY' for a point
   C     target.
   C
         OCCTYP = 'ANY'
 
   C
   C     Display a banner for the output report:
   C
         TITLE  = 'Inputs for target visibility search:'
         CALL TOSTDO ( ' '   )
         CALL TOSTDO ( TITLE )
         CALL TOSTDO ( ' '   )
 
         LINE   = '   Target                       = #'
         CALL REPMC  ( LINE, '#', TARGET, LINE )
         CALL TOSTDO ( LINE )
 
         LINE   = '   Observation surface location = #'
         CALL REPMC  ( LINE, '#', SRFPT,  LINE )
         CALL TOSTDO ( LINE )
 
         LINE   = '   Observer''s reference frame   = #'
         CALL REPMC  ( LINE, '#', FRAME,  LINE )
         CALL TOSTDO ( LINE )
 
         LINE   = '   Blocking body                = #'
         CALL REPMC  ( LINE, '#', FRONT, LINE )
         CALL TOSTDO ( LINE )
 
         LINE   = '   Blocker''s reference frame    = #'
         CALL REPMC  ( LINE, '#', FFRAME,  LINE )
         CALL TOSTDO ( LINE )
 
         LINE   = '   Elevation limit (degrees)    = #'
         CALL REPMF ( LINE, '#', ELVLIM, 7, 'F', LINE )
         CALL TOSTDO ( LINE )
 
         LINE   = '   Aberration correction        = #'
         CALL REPMC  ( LINE, '#', ABCORR, LINE )
         CALL TOSTDO ( LINE )
 
         LINE   = '   Step size (seconds)          = #'
         CALL REPMF ( LINE, '#', STEPSZ, 9, 'F', LINE )
         CALL TOSTDO ( LINE )
 
   C
   C     Convert the start and stop times to ET.
   C
         CALL STR2ET ( START, ETBEG )
         CALL STR2ET ( STOP,  ETEND )
 
   C
   C     Display the search interval start and stop times
   C     using the format shown below.
   C
   C        2004 MAY 06 20:15:00.000 (TDB)
   C
         CALL TIMOUT ( ETBEG, TDBFMT, BTMSTR )
         LINE = '   Start time                   = '//BTMSTR
         CALL TOSTDO ( LINE )
 
         CALL TIMOUT ( ETEND, TDBFMT, ETMSTR )
         LINE = '   Stop time                    = '//ETMSTR
         CALL TOSTDO ( LINE )
 
         CALL TOSTDO ( ' ' )
 
   C
   C     Every SPICELIB window must have its size initialized.
   C
   C     Initialize the "confinement" window with the interval
   C     over which we'll conduct the search.
   C
         CALL SSIZED ( MAXWIN, CNFINE )
 
         CALL WNINSD ( ETBEG,  ETEND, CNFINE )
 
   C
   C     Initialize the result window; this window will contain
   C     the rise/set times found by our search.
   C
         CALL SSIZED ( MAXWIN, RISWIN )
 
   C
   C     Initialize the occultation and visibility windows.
   C
         CALL SSIZED ( MAXWIN, OCCWIN )
         CALL SSIZED ( MAXWIN, DOCWIN )
         CALL SSIZED ( MAXWIN, EVISWN )
         CALL SSIZED ( MAXWIN, DVISWN )
 
   C
   C     The adjustment value only applies to absolute extrema
   C     searches;  simply give it an initial value of zero
   C     for this inequality search.
   C
         ADJUST = 0.D0
 
   C
   C     Note that the workspace dimensions are ( MAXWIN, NWMAX ).
   C
   C     Now search for the time period, within our confinement
   C     window, during which the apparent target has elevation
   C     at least equal to the elevation limit.
   C
         CALL GFPOSC ( TARGET, FRAME,  ABCORR, SRFPT,
        .              CRDSYS, COORD,  RELATE, REVLIM,
        .              ADJUST, STEPSZ, CNFINE, MAXWIN,
        .              NWMAX,  WORK,   RISWIN         )
 
   C
   C     Now find the times when the apparent target is above
   C     the elevation limit and is not occulted by the
   C     blocking body (Mercury). We'll find the times when the target
   C     is above the elevation limit and *is* occulted, then subtract
   C     that window from the view period window RISWIN found above.
   C
   C     For this occultation search, we can use RISWIN as
   C     the confinement window because we're not interested in
   C     occultations that occur when the target is below the
   C     elevation limit.
   C
   C     Find occultations within the view period window.
   C
         FSHAPE = 'ELLIPSOID'
 
         WRITE (*,*) 'Searching using ellipsoid target shape model...'
 
         CALL GFOCLT ( OCCTYP, FRONT,  FSHAPE, FFRAME,
        .              BACK,   BSHAPE, BFRAME, ABCORR,
        .              SRFPT,  STEPSZ, RISWIN, OCCWIN )
         WRITE (*,*) 'Done.'
 
   C
   C     Subtract the occultation window from the view period
   C     window: this yields the time periods when the target
   C     is visible.
   C
         CALL WNDIFD ( RISWIN, OCCWIN, EVISWN )
 
   C
   C     Repeat the search using low-resolution DSK data
   C     for the front body.
   C
         FSHAPE = 'DSK/UNPRIORITIZED'
 
         WRITE (*,*) 'Searching using DSK target shape model...'
 
         CALL GFOCLT ( OCCTYP, FRONT,  FSHAPE, FFRAME,
        .              BACK,   BSHAPE, BFRAME, ABCORR,
        .              SRFPT,  STEPSZ, RISWIN, DOCWIN )
         WRITE (*,*) 'Done.'
 
         CALL WNDIFD ( RISWIN, DOCWIN, DVISWN )
 
   C
   C     The function WNCARD returns the number of intervals
   C     in a SPICE window.
   C
         WINSIZ = WNCARD( EVISWN )
 
         IF ( WINSIZ .EQ. 0 ) THEN
 
            WRITE (*,*) 'No events were found.'
 
         ELSE
   C
   C        Display the visibility time periods.
   C
            WRITE (*,*) ' '
            LINE = 'Visibility start and stop times of '
        .   //     '# as seen from #'
            CALL REPMC ( LINE, '#', TARGET, LINE )
            CALL REPMC ( LINE, '#', SRFPT,  LINE )
 
            WRITE (*, '(A)') LINE
            WRITE (*, '(A)') 'using both ellipsoidal and DSK '
        .   //               'target shape models:'
            WRITE (*,*) ' '
 
            DO I = 1, WINSIZ
   C
   C           Fetch the start and stop times of the Ith interval
   C           from the ellipsoid search result window EVISWN.
   C
               CALL WNFETD ( EVISWN, I, INTBEG, INTEND )
   C
   C           Convert the start and stop times to TDB calendar
   C           strings. Write the results.
   C
               CALL TIMOUT ( INTBEG, TDBFMT, BTMSTR )
               CALL TIMOUT ( INTEND, TDBFMT, ETMSTR )
 
               WRITE (*,'('' Ell: '',A, '' : '', A)' ) BTMSTR, ETMSTR
   C
   C           Fetch the start and stop times of the Ith interval
   C           from the DSK search result window DVISWN.
   C
               CALL WNFETD ( DVISWN, I, DINTBG, DINTEN )
   C
   C           Convert the start and stop times to TDB calendar
   C           strings. Write the results.
   C
               CALL TIMOUT ( DINTBG, TDBFMT, BTMSTR )
               CALL TIMOUT ( DINTEN, TDBFMT, ETMSTR )
 
               WRITE (*,'('' DSK: '',A, '' : '', A)' ) BTMSTR, ETMSTR
               WRITE (*,*) ' '
 
            END DO
 
         END IF
 
         END


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Solution Sample Output



Numerical results shown for this example may differ across platforms since the results depend on the SPICE kernels used as input and on the host platform's arithmetic implementation.

After compiling the program, execute it. The output is: 

 
   Inputs for target visibility search:
 
      Target                       = MPO
      Observation surface location = NEW_NORCIA
      Observer's reference frame   = NEW_NORCIA_TOPO
      Blocking body                = MERCURY
      Blocker's reference frame    = IAU_MERCURY
      Elevation limit (degrees)    = 6.000000
      Aberration correction        = CN+S
      Step size (seconds)          = 300.000000
      Start time                   = 2027 JAN 03 00:00:00.000 TDB
      Stop time                    = 2027 JAN 06 00:00:00.000 TDB
 
    Searching using ellipsoid target shape model...
    Done.
    Searching using DSK target shape model...
    Done.
 
   Visibility start and stop times of MPO as seen from NEW_NORCIA
   using both ellipsoidal and DSK target shape models:
 
    Ell: 2027 JAN 03 00:00:00.000 TDB : 2027 JAN 03 01:28:03.419 TDB
    DSK: 2027 JAN 03 00:00:00.000 TDB : 2027 JAN 03 01:28:03.202 TDB
 
    Ell: 2027 JAN 03 02:00:42.993 TDB : 2027 JAN 03 03:49:50.750 TDB
    DSK: 2027 JAN 03 02:00:43.226 TDB : 2027 JAN 03 03:49:50.589 TDB
 
    Ell: 2027 JAN 03 04:22:20.803 TDB : 2027 JAN 03 06:11:38.050 TDB
    DSK: 2027 JAN 03 04:22:21.167 TDB : 2027 JAN 03 06:11:37.927 TDB
 
    Ell: 2027 JAN 03 06:43:58.528 TDB : 2027 JAN 03 08:33:25.499 TDB
    DSK: 2027 JAN 03 06:43:58.803 TDB : 2027 JAN 03 08:33:25.452 TDB
 
    Ell: 2027 JAN 03 09:05:36.070 TDB : 2027 JAN 03 10:55:12.991 TDB
    DSK: 2027 JAN 03 09:05:36.483 TDB : 2027 JAN 03 10:55:13.005 TDB
 
    Ell: 2027 JAN 03 21:55:08.488 TDB : 2027 JAN 03 22:44:11.490 TDB
    DSK: 2027 JAN 03 21:55:08.488 TDB : 2027 JAN 03 22:44:11.836 TDB
 
    Ell: 2027 JAN 03 23:15:19.552 TDB : 2027 JAN 04 01:05:59.339 TDB
    DSK: 2027 JAN 03 23:15:20.564 TDB : 2027 JAN 04 01:05:59.788 TDB
 
    Ell: 2027 JAN 04 01:36:56.572 TDB : 2027 JAN 04 03:27:47.253 TDB
    DSK: 2027 JAN 04 01:36:56.903 TDB : 2027 JAN 04 03:27:47.794 TDB
 
    Ell: 2027 JAN 04 03:58:33.411 TDB : 2027 JAN 04 05:49:35.238 TDB
    DSK: 2027 JAN 04 03:58:33.685 TDB : 2027 JAN 04 05:49:35.857 TDB
 
    Ell: 2027 JAN 04 06:20:10.165 TDB : 2027 JAN 04 08:11:23.310 TDB
    DSK: 2027 JAN 04 06:20:10.819 TDB : 2027 JAN 04 08:11:23.843 TDB
 
    Ell: 2027 JAN 04 08:41:46.813 TDB : 2027 JAN 04 10:33:11.480 TDB
    DSK: 2027 JAN 04 08:41:47.399 TDB : 2027 JAN 04 10:33:12.291 TDB
 
    Ell: 2027 JAN 04 21:58:41.333 TDB : 2027 JAN 04 22:22:13.911 TDB
    DSK: 2027 JAN 04 21:58:41.333 TDB : 2027 JAN 04 22:22:13.969 TDB
 
    Ell: 2027 JAN 04 22:51:24.368 TDB : 2027 JAN 05 00:44:02.576 TDB
    DSK: 2027 JAN 04 22:51:24.088 TDB : 2027 JAN 05 00:44:02.498 TDB
 
    Ell: 2027 JAN 05 01:13:00.256 TDB : 2027 JAN 05 03:05:51.406 TDB
    DSK: 2027 JAN 05 01:13:00.056 TDB : 2027 JAN 05 03:05:51.377 TDB
 
    Ell: 2027 JAN 05 03:34:36.025 TDB : 2027 JAN 05 05:27:40.260 TDB
    DSK: 2027 JAN 05 03:34:36.194 TDB : 2027 JAN 05 05:27:40.400 TDB
 
    Ell: 2027 JAN 05 05:56:11.727 TDB : 2027 JAN 05 07:49:29.298 TDB
    DSK: 2027 JAN 05 05:56:11.995 TDB : 2027 JAN 05 07:49:29.743 TDB
 
    Ell: 2027 JAN 05 08:17:47.213 TDB : 2027 JAN 05 10:11:18.377 TDB
    DSK: 2027 JAN 05 08:17:47.173 TDB : 2027 JAN 05 10:11:18.893 TDB
 
    Ell: 2027 JAN 05 10:39:22.574 TDB : 2027 JAN 05 11:04:00.020 TDB
    DSK: 2027 JAN 05 10:39:22.690 TDB : 2027 JAN 05 11:04:00.020 TDB
 
    Ell: 2027 JAN 05 22:27:17.183 TDB : 2027 JAN 06 00:00:00.000 TDB
    DSK: 2027 JAN 05 22:27:17.436 TDB : 2027 JAN 06 00:00:00.000 TDB


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Extra Credit




In this ``extra credit'' section you will be presented with more complex tasks, aimed at improving your understanding of the geometry event finding subsystem and particularly the GFPOSC and GFDIST routines.

These ``extra credit'' tasks are provided as task statements, and unlike the regular tasks, no approach or solution source code is provided. In the next section, you will find the numeric solutions to the questions asked in these tasks.



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Task statements



    1. Write a program that finds the times, within the time range 

         2027 JAN 03 TDB
         2027 JAN 04 TDB
    when the BepiColombo Mercury Planetary Orbiter (MPO) crosses Mercury' equator. Display the results using TDB calendar dates and millisecond precision.

    2. Write a program that finds the times, within the time range 

         2027 JAN 03 TDB
         2027 JAN 04 TDB
    when the BepiColombo Mercury Planetary Orbiter (MPO) is at periapsis. Display the results using TDB calendar dates and millisecond precision.

    3. Write a program that finds the times, within the time range 

         2027 JAN 03 TDB
         2027 JAN 04 TDB
    when the BepiColombo Mercury Planetary Orbiter (MPO) is at apoapsis. Display the results using TDB calendar dates and millisecond precision.



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Solutions



    1. Solution for the equator crossing search, using GFPOSC for the BepiColombo Mercury Planetary Orbiter (MPO) spacecraft latitude in the Mercury body-fixed frame equal to 0 degrees: 

 
   Inputs for equator crossing search:
 
      Target                       = MPO
      Observer                     = MERCURY
      Observer's reference frame   = IAU_MERCURY
      Latitude limit (degrees)     = 0.000000
      Aberration correction        = NONE
      Step size (seconds)          = 300.000000
      Start time                   = 2027 JAN 03 00:00:00.000 (TDB)
      Stop time                    = 2027 JAN 04 00:00:00.000 (TDB)
 
   MPO MERCURY equator crossing times:
 
    Equator crossing or start time:  2027 JAN 03 00:21:02.744 (TDB)
    Equator crossing time:           2027 JAN 03 01:34:39.885 (TDB)
    Equator crossing time:           2027 JAN 03 02:42:45.780 (TDB)
    Equator crossing time:           2027 JAN 03 03:56:22.917 (TDB)
    Equator crossing time:           2027 JAN 03 05:04:28.849 (TDB)
    Equator crossing time:           2027 JAN 03 06:18:06.007 (TDB)
    Equator crossing time:           2027 JAN 03 07:26:11.888 (TDB)
    Equator crossing time:           2027 JAN 03 08:39:49.081 (TDB)
    Equator crossing time:           2027 JAN 03 09:47:54.901 (TDB)
    Equator crossing time:           2027 JAN 03 11:01:32.084 (TDB)
    Equator crossing time:           2027 JAN 03 12:09:37.930 (TDB)
    Equator crossing time:           2027 JAN 03 13:23:15.135 (TDB)
    Equator crossing time:           2027 JAN 03 14:31:20.939 (TDB)
    Equator crossing time:           2027 JAN 03 15:44:58.128 (TDB)
    Equator crossing time:           2027 JAN 03 16:53:03.959 (TDB)
    Equator crossing time:           2027 JAN 03 18:06:41.171 (TDB)
    Equator crossing time:           2027 JAN 03 19:14:46.962 (TDB)
    Equator crossing time:           2027 JAN 03 20:28:24.150 (TDB)
    Equator crossing time:           2027 JAN 03 21:36:29.935 (TDB)
    Equator crossing time:           2027 JAN 03 22:50:07.114 (TDB)
    Equator crossing or stop time:   2027 JAN 03 23:58:12.944 (TDB)
    2. Solution for the periapsis search, using GFDIST for the BepiColombo Mercury Planetary Orbiter (MPO) spacecraft distance from Mercury at a local minimum: 

 
   Inputs for periapsis search:
 
      Target                       = MPO
      Observer                     = MERCURY
      Aberration correction        = NONE
      Step size (seconds)          = 300.000000
      Start time                   = 2027 JAN 03 00:00:00.000 (TDB)
      Stop time                    = 2027 JAN 04 00:00:00.000 (TDB)
 
   MPO periapsis times:
 
    Periapsis or start time:         2027 JAN 03 00:18:33.937 (TDB)
    Periapsis time:                  2027 JAN 03 02:40:16.998 (TDB)
    Periapsis time:                  2027 JAN 03 05:01:59.964 (TDB)
    Periapsis time:                  2027 JAN 03 07:23:43.026 (TDB)
    Periapsis time:                  2027 JAN 03 09:45:25.991 (TDB)
    Periapsis time:                  2027 JAN 03 12:07:09.042 (TDB)
    Periapsis time:                  2027 JAN 03 14:28:52.095 (TDB)
    Periapsis time:                  2027 JAN 03 16:50:35.082 (TDB)
    Periapsis time:                  2027 JAN 03 19:12:18.042 (TDB)
    Periapsis time:                  2027 JAN 03 21:34:01.097 (TDB)
    Periapsis or end time:           2027 JAN 03 23:55:44.079 (TDB)
    3. Solution for the apoapsis search, using GFDIST for the BepiColombo Mercury Planetary Orbiter (MPO) spacecraft distance from Mercury at a local maximum: 

 
   Inputs for apoapsis search:
 
      Target                       = MPO
      Observer                     = MERCURY
      Aberration correction        = NONE
      Step size (seconds)          = 300.000000
      Start time                   = 2027 JAN 03 00:00:00.000 (TDB)
      Stop time                    = 2027 JAN 04 00:00:00.000 (TDB)
 
   MPO apoapsis times:
 
    Apoapsis or start time:          2027 JAN 03 01:29:25.529 (TDB)
    Apoapsis time:                   2027 JAN 03 03:51:08.495 (TDB)
    Apoapsis time:                   2027 JAN 03 06:12:51.561 (TDB)
    Apoapsis time:                   2027 JAN 03 08:34:34.611 (TDB)
    Apoapsis time:                   2027 JAN 03 10:56:17.595 (TDB)
    Apoapsis time:                   2027 JAN 03 13:18:00.653 (TDB)
    Apoapsis time:                   2027 JAN 03 15:39:43.611 (TDB)
    Apoapsis time:                   2027 JAN 03 18:01:26.677 (TDB)
    Apoapsis time:                   2027 JAN 03 20:23:09.638 (TDB)
    Apoapsis or stop time:           2027 JAN 03 22:44:52.618 (TDB)