| gfpa_c |
|
Table of contents
Procedure
gfpa_c ( GF, phase angle search )
void gfpa_c ( ConstSpiceChar * target,
ConstSpiceChar * illmn,
ConstSpiceChar * abcorr,
ConstSpiceChar * obsrvr,
ConstSpiceChar * relate,
SpiceDouble refval,
SpiceDouble adjust,
SpiceDouble step,
SpiceInt nintvls,
SpiceCell * cnfine,
SpiceCell * result )
AbstractDetermine time intervals for which a specified constraint on the phase angle between an illumination source, a target, and observer body centers is met. Required_ReadingGF NAIF_IDS SPK TIME WINDOWS KeywordsEPHEMERIS EVENT GEOMETRY SEARCH WINDOW Brief_I/O
VARIABLE I/O DESCRIPTION
-------- --- --------------------------------------------------
SPICE_GF_CNVTOL
P Default convergence tolerance.
target I Name of the target body.
illmn I Name of the illuminating body.
abcorr I Aberration correction flag.
obsrvr I Name of the observing body.
relate I Relational operator.
refval I Reference value.
adjust I Adjustment value for absolute extrema searches.
step I Step size used for locating extrema and roots.
nintvls I Workspace window interval count.
cnfine I-O SPICE window to which the search is confined.
result O SPICE window containing results.
Detailed_Input
target is the name of a target body. Optionally, you may supply
a string containing the integer ID code for the object.
For example both "MOON" and "301" are legitimate strings
that indicate the Moon is the target body.
Case and leading or trailing blanks are not significant
in the string `target'.
illmn is the name of the illuminating body. This will normally
be "SUN" but the algorithm can use any ephemeris object.
Case and leading or trailing blanks are not significant
in the string `illmn'.
abcorr is the description of the aberration corrections to apply
to the state evaluations to account for one-way light
time and stellar aberration.
This routine accepts only reception mode aberration
corrections. See the header of spkezr_c for a detailed
description of the aberration correction options. For
convenience, the allowed aberration options are listed
below:
"NONE" Apply no correction. Returns the "true"
geometric state.
"LT" "Reception" case: correct for
one-way light time using a Newtonian
formulation.
"LT+S" "Reception" case: correct for
one-way light time and stellar
aberration using a Newtonian
formulation.
"CN" "Reception" case: converged
Newtonian light time correction.
"CN+S" "Reception" case: converged
Newtonian light time and stellar
aberration corrections.
Case and leading or trailing blanks are not significant
in the string `abcorr'.
obsrvr is the name of an observing body. Optionally, you may
supply a string containing the integer ID code for the
object. For example both "MOON" and "301" are legitimate
strings that indicate the Moon is the observer.
Case and leading or trailing blanks are not significant
in the string `obsrvr'.
relate is a relational operator used to define a constraint on a
specified phase angle. The result window found by this
routine indicates the time intervals where the constraint
is satisfied. Supported values of `relate' and
corresponding meanings are shown below:
">" The phase angle value is greater than the
reference value `refval'.
"=" The phase angle value is equal to the
reference value `refval'.
"<" The phase angle value is less than the
reference value `refval'.
"ABSMAX" The phase angle value is at an absolute
maximum.
"ABSMIN" The phase angle value is at an absolute
minimum.
"LOCMAX" The phase angle value is at a local
maximum.
"LOCMIN" The phase angle value is at a local
minimum.
`relate' may be used to specify an "adjusted" absolute
extremum constraint: this requires the phase angle to be
within a specified offset relative to an absolute
extremum. The argument `adjust' (described below) is used
to specify this offset.
Local extrema are considered to exist only in the
interiors of the intervals comprising the confinement
window: a local extremum cannot exist at a boundary
point of the confinement window.
Case and leading or trailing blanks are not
significant in the string `relate'.
refval is the double precision reference value used together
with the argument `relate' to define an equality or
inequality to be satisfied by the phase angle. See the
discussion of `relate' above for further information.
The units of `refval' are radians.
adjust is a double precision value used to modify searches for
absolute extrema: when `relate' is set to "ABSMAX" or
"ABSMIN" and `adjust' is set to a positive value, gfpa_c
finds times when the phase angle is within `adjust' radians
of the specified extreme value.
If `adjust' is non-zero and a search for an absolute
minimum `min' is performed, the result window contains
time intervals when the phase angle has values between
`min' and min+adjust.
If the search is for an absolute maximum `max', the
corresponding range is from max-adjust to `max'.
`adjust' is not used for searches for local extrema,
equality or inequality conditions.
step is the step size to be used in the search. `step' must
be shorter than any maximal time interval on which the
specified phase angle function is monotone increasing or
decreasing. That is, if the confinement window is
partitioned into alternating intervals on which the
phase angle function is either monotone increasing or
decreasing, `step' must be shorter than any of these
intervals.
However, `step' must not be *too* short, or the search
will take an unreasonable amount of time.
The choice of `step' affects the completeness but not
the precision of solutions found by this routine; the
precision is controlled by the convergence tolerance.
See the discussion of the parameter SPICE_GF_CNVTOL for
details.
`step' has units of TDB seconds.
nintvls is an integer parameter specifying the number of intervals
that can be accommodated by each of the dynamically allocated
workspace windows used internally by this routine.
In many cases, it's not necessary to compute an accurate
estimate of how many intervals are needed; rather, the user
can pick a size considerably larger than what's really
required.
However, since excessively large arrays can prevent
applications from compiling, linking, or running properly,
sometimes `nintvls' must be set according to the actual
workspace requirement. A rule of thumb for the number of
intervals needed is
nintvls = 2*n + ( m / step )
where
n is the number of intervals in the confinement
window.
m is the measure of the confinement window, in units
of seconds.
step is the search step size in seconds.
cnfine is a double precision SPICE window that confines the time
period over which the specified search is conducted.
`cnfine' may consist of a single interval or a collection
of intervals.
In some cases the confinement window can be used to
greatly reduce the time period that must be searched
for the desired solution. See the -Particulars section
below for further discussion.
The endpoints of the time intervals comprising `cnfine' are
interpreted as seconds past J2000 TDB.
See the -Examples section below for a code example
that shows how to create a confinement window.
In some cases the observer's state may be computed at
times outside of `cnfine' by as much as 2 seconds. See
-Particulars for details.
`cnfine' must be declared as a double precision SpiceCell.
CSPICE provides the following macro, which declares and
initializes the cell
SPICEDOUBLE_CELL ( cnfine, CNFINESZ );
where CNFINESZ is the maximum capacity of `cnfine'.
Detailed_Output
cnfine is the input confinement window, updated if necessary so the
control area of its data array indicates the window's size
and cardinality. The window data are unchanged.
result is the SPICE window of intervals, contained within the
confinement window `cnfine', on which the specified phase
angle constraint is satisfied.
`result' must be declared and initialized with sufficient
size to capture the full set of time intervals within the
search region on which the specified condition is satisfied.
If `result' is non-empty on input, its contents will be
discarded before gfpa_c conducts its search.
The endpoints of the time intervals comprising `result' are
interpreted as seconds past J2000 TDB.
If the search is for local extrema, or for absolute
extrema with `adjust' set to zero, then normally each
interval of `result' will be a singleton: the left and
right endpoints of each interval will be identical.
If no times within the confinement window satisfy the
search criteria, `result' will be returned with a
cardinality of zero.
`result' must be declared as a double precision SpiceCell.
CSPICE provides the following macro, which declares and
initializes the cell
SPICEDOUBLE_CELL ( result, RESULTSZ );
where RESULTSZ is the maximum capacity of `result'.
Parameters
SPICE_GF_CNVTOL
is the convergence tolerance used for finding endpoints
of the intervals comprising the result window.
SPICE_GF_CNVTOL is used to determine when binary
searches for roots should terminate: when a root is
bracketed within an interval of length SPICE_GF_CNVTOL,
the root is considered to have been found.
The accuracy, as opposed to precision, of roots found by
this routine depends on the accuracy of the input data.
In most cases, the accuracy of solutions will be
inferior to their precision.
SPICE_GF_CNVTOL is declared in the header file
SpiceGF.h.
Exceptions
1) In order for this routine to produce correct results,
the step size must be appropriate for the problem at hand.
Step sizes that are too large may cause this routine to miss
roots; step sizes that are too small may cause this routine
to run unacceptably slowly and in some cases, find spurious
roots.
This routine does not diagnose invalid step sizes, except that
if the step size is non-positive, an error is signaled by a
routine in the call tree of this routine.
2) Due to numerical errors, in particular,
- truncation error in time values
- finite tolerance value
- errors in computed geometric quantities
it is *normal* for the condition of interest to not always be
satisfied near the endpoints of the intervals comprising the
`result' window. One technique to handle such a situation,
slightly contract `result' using the window routine wncond_c.
3) If the number of intervals `nintvls' is less than 1, the error
SPICE(VALUEOUTOFRANGE) is signaled.
4) If result window, `result', is not at least 2 and an even value,
the error SPICE(INVALIDDIMENSION) is signaled by a routine in
the call tree of this routine is signaled.
5) If `result' has insufficient capacity to contain the
number of intervals on which the specified angle condition
is met, an error is signaled by a routine in the call
tree of this routine.
6) If an error (typically cell overflow) occurs during
window arithmetic, the error is signaled by a routine
in the call tree of this routine.
7) If the relational operator `relate' is not recognized, an
error is signaled by a routine in the call tree of this
routine.
8) If `adjust' is negative, an error is signaled by a routine in
the call tree of this routine.
9) If `adjust' has a non-zero value when `relate' has any value other
than "ABSMIN" or "ABSMAX", an error is signaled by a routine
in the call tree of this routine.
10) If any of the input body names, `target', `illmn', `obsrvr', do
not map to NAIF ID codes, an error is signaled by a routine
in the call tree of this routine.
11) If the input body names, `target', `illmn', `obsrvr', are not
distinct, an error is signaled by a routine in the call
tree of this routine.
12) If required ephemerides or other kernel data are not
available, an error is signaled by a routine in the call tree
of this routine.
13) If the aberration correction specifier contains an
unrecognized value, an error is signaled by a routine in the
call tree of this routine.
14) If a transmit mode aberration correction is requested, an
error is signaled by a routine in the call tree of this
routine.
15) If any of the `target', `illmn', `abcorr', `obsrvr' or
`relate' input string pointers is null, the error
SPICE(NULLPOINTER) is signaled.
16) If any of the `target', `illmn', `abcorr', `obsrvr' or
`relate' input strings has zero length, the error
SPICE(EMPTYSTRING) is signaled.
17) If any the `cnfine' or `result' cell arguments has a type
other than SpiceDouble, the error SPICE(TYPEMISMATCH) is
signaled.
18) If memory cannot be allocated to create the temporary variable
required for the execution of the underlying Fortran routine,
the error SPICE(MALLOCFAILED) is signaled.
Files
Appropriate SPK and PCK kernels must be loaded by the calling
program before this routine is called.
The following data are required:
- SPK data: the calling application must load ephemeris data
for the targets, observer, and any intermediate objects in
a chain connecting the targets and observer that cover the
time period specified by the window `cnfine'. If aberration
corrections are used, the states of target and observer
relative to the solar system barycenter must be calculable
from the available ephemeris data. Typically ephemeris data
are made available by loading one or more SPK files using
furnsh_c.
- In some cases the observer's state may be computed at times
outside of `cnfine' by as much as 2 seconds; data required to
compute this state must be provided by loaded kernels. See
-Particulars for details.
Kernel data are normally loaded once per program run, NOT every
time this routine is called.
Particulars
illmn OBS
illmn as seen * /
from TARG at | /
et - lt. | /
>|..../< phase angle
| /
. | /
. | /
. * TARG as seen from OBS
SEP . TARG at `et'
. /
/
*
This routine determines if the caller-specified constraint
condition on the geometric event (phase angle) is satisfied for
any time intervals within the confinement window `cnfine'. If one
or more such time intervals exist, those intervals are added
to the `result' window.
This routine provides a simpler, but less flexible interface
than does the routine gfevnt_c for conducting searches for
illuminator-target-observer phase angle value events.
Applications that require support for progress reporting,
interrupt handling, non-default step or refinement functions
should call gfevnt_c rather than this routine.
Below we discuss in greater detail aspects of this routine's
solution process that are relevant to correct and efficient
use of this routine in user applications.
The Search Process
==================
Regardless of the type of constraint selected by the caller, this
routine starts the search for solutions by determining the time
periods, within the confinement window, over which the
phase angle function is monotone increasing and monotone
decreasing. Each of these time periods is represented by a SPICE
window. Having found these windows, all of the phase angle
function's local extrema within the confinement window are known.
Absolute extrema then can be found very easily.
Within any interval of these "monotone" windows, there will be at
most one solution of any equality constraint. Since the boundary
of the solution set for any inequality constraint is contained in
the union of
- the set of points where an equality constraint is met
- the boundary points of the confinement window
the solutions of both equality and inequality constraints can be
found easily once the monotone windows have been found.
Step Size
=========
The monotone windows (described above) are found using a two-step
search process. Each interval of the confinement window is
searched as follows: first, the input step size is used to
determine the time separation at which the sign of the rate of
change of phase angle will be sampled. Starting at
the left endpoint of an interval, samples will be taken at each
step. If a change of sign is found, a root has been bracketed; at
that point, the time at which the time derivative of the
phase angle is zero can be found by a refinement process, for
example, using a binary search.
Note that the optimal choice of step size depends on the lengths
of the intervals over which the phase angle function is monotone:
the step size should be shorter than the shortest of these
intervals (within the confinement window).
The optimal step size is *not* necessarily related to the lengths
of the intervals comprising the result window. For example, if
the shortest monotone interval has length 10 days, and if the
shortest result window interval has length 5 minutes, a step size
of 9.9 days is still adequate to find all of the intervals in the
result window. In situations like this, the technique of using
monotone windows yields a dramatic efficiency improvement over a
state-based search that simply tests at each step whether the
specified constraint is satisfied. The latter type of search can
miss solution intervals if the step size is longer than the
shortest solution interval.
Having some knowledge of the relative geometry of the target,
illumination source, and observer can be a valuable aid in
picking a reasonable step size. In general, the user can
compensate for lack of such knowledge by picking a very short
step size; the cost is increased computation time.
Note that the step size is not related to the precision with which
the endpoints of the intervals of the result window are computed.
That precision level is controlled by the convergence tolerance.
Convergence Tolerance
=====================
As described above, the root-finding process used by this routine
involves first bracketing roots and then using a search process
to locate them. "Roots" are both times when local extrema are
attained and times when the geometric quantity function is equal
to a reference value. All endpoints of the intervals comprising
the result window are either endpoints of intervals of the
confinement window or roots.
Once a root has been bracketed, a refinement process is used to
narrow down the time interval within which the root must lie.
This refinement process terminates when the location of the root
has been determined to within an error margin called the
"convergence tolerance." The default convergence tolerance
used by this routine is set by the parameter SPICE_GF_CNVTOL (defined
in SpiceGF.h).
The value of SPICE_GF_CNVTOL is set to a "tight" value so that the
tolerance doesn't become the limiting factor in the accuracy of
solutions found by this routine. In general the accuracy of input
data will be the limiting factor.
The user may change the convergence tolerance from the default
SPICE_GF_CNVTOL value by calling the routine gfstol_c, e.g.
gfstol_c ( tolerance value );
Call gfstol_c prior to calling this routine. All subsequent
searches will use the updated tolerance value.
Setting the tolerance tighter than SPICE_GF_CNVTOL is unlikely to be
useful, since the results are unlikely to be more accurate.
Making the tolerance looser will speed up searches somewhat,
since a few convergence steps will be omitted. However, in most
cases, the step size is likely to have a much greater effect
on processing time than would the convergence tolerance.
The Confinement Window
======================
The simplest use of the confinement window is to specify a time
interval within which a solution is sought. However, the
confinement window can, in some cases, be used to make searches
more efficient. Sometimes it's possible to do an efficient search
to reduce the size of the time period over which a relatively
slow search of interest must be performed.
Certain types of searches require the state of the observer,
relative to the solar system barycenter, to be computed at times
slightly outside the confinement window `cnfine'. The time window
that is actually used is the result of "expanding" `cnfine' by a
specified amount "T": each time interval of `cnfine' is expanded by
shifting the interval's left endpoint to the left and the right
endpoint to the right by T seconds. Any overlapping intervals are
merged. (The input argument `cnfine' is not modified.)
The window expansions listed below are additive: if both
conditions apply, the window expansion amount is the sum of the
individual amounts.
- If a search uses an equality constraint, the time window
over which the state of the observer is computed is expanded
by 1 second at both ends of all of the time intervals
comprising the window over which the search is conducted.
- If a search uses stellar aberration corrections, the time
window over which the state of the observer is computed is
expanded as described above.
When light time corrections are used, expansion of the search
window also affects the set of times at which the light time-
corrected state of the target is computed.
In addition to the possible 2 second expansion of the search
window that occurs when both an equality constraint and stellar
aberration corrections are used, round-off error should be taken
into account when the need for data availability is analyzed.
Examples
The numerical results shown for this example may differ across
platforms. The results depend on the SPICE kernels used as
input, the compiler and supporting libraries, and the machine
specific arithmetic implementation.
1) Determine the time windows from December 1, 2006 UTC to
January 31, 2007 UTC for which the sun-moon-earth configuration
phase angle satisfies the relation conditions with respect to a
reference value of .57598845 radians (the phase angle at
January 1, 2007 00:00:00.000 UTC, 33.001707 degrees). Also
determine the time windows corresponding to the local maximum and
minimum phase angles, and the absolute maximum and minimum phase
angles during the search interval. The configuration defines the
sun as the illuminator, the moon as the target, and the earth as
the observer.
Use the meta-kernel shown below to load the required SPICE
kernels.
KPL/MK
File name: gfpa_ex1.tm
This meta-kernel is intended to support operation of SPICE
example programs. The kernels shown here should not be
assumed to contain adequate or correct versions of data
required by SPICE-based user applications.
In order for an application to use this meta-kernel, the
kernels referenced here must be present in the user's
current working directory.
The names and contents of the kernels referenced
by this meta-kernel are as follows:
File name Contents
--------- --------
de421.bsp Planetary ephemeris
pck00009.tpc Planet orientation and
radii
naif0009.tls Leapseconds
\begindata
KERNELS_TO_LOAD = ( 'de421.bsp',
'pck00009.tpc',
'naif0009.tls' )
\begintext
End of meta-kernel
Example code begins here.
/.
Program gfpa_ex1
./
#include <stdio.h>
#include "SpiceUsr.h"
#define TIMFMT "YYYY MON DD HR:MN:SC.###"
#define NINTVL 5000
#define TIMLEN 41
#define NLOOPS 7
int main()
{
/.
Local variables
./
SpiceChar begstr [ TIMLEN ];
SpiceChar endstr [ TIMLEN ];
SPICEDOUBLE_CELL ( cnfine, 2 );
SPICEDOUBLE_CELL ( result, NINTVL*2 );
SpiceDouble adjust;
SpiceDouble et0;
SpiceDouble et1;
SpiceDouble phaseq;
SpiceDouble refval;
SpiceDouble start;
SpiceDouble step;
SpiceDouble stop;
SpiceInt i;
SpiceInt j;
/.
Define the values for target, observer, illuminator, and
aberration correction.
./
ConstSpiceChar * target = "moon";
ConstSpiceChar * illmn = "sun";
ConstSpiceChar * abcorr = "lt+s";
ConstSpiceChar * obsrvr = "earth";
ConstSpiceChar * relate [NLOOPS] = { "=",
"<",
">",
"LOCMIN",
"ABSMIN",
"LOCMAX",
"ABSMAX",
};
/.
Load kernels.
./
furnsh_c ( "gfpa_ex1.tm" );
/.
Store the time bounds of our search interval in
the confinement window.
./
str2et_c ( "2006 DEC 01", &et0 );
str2et_c ( "2007 JAN 31", &et1 );
wninsd_c ( et0, et1, &cnfine );
/.
Search using a step size of 1 day (in units of seconds).
The reference value is 0.57598845 radians. We're not using the
adjustment feature, so we set ADJUST to zero.
./
step = spd_c();
refval = 0.57598845;
adjust = 0.0;
for ( j = 0; j < NLOOPS; j++ )
{
printf ( "Relation condition: %s\n", relate[j] );
/.
Perform the search. The SPICE window `result' contains
the set of times when the condition is met.
./
gfpa_c ( target, illmn, abcorr, obsrvr,
relate[j], refval, adjust, step,
NINTVL, &cnfine, &result );
/.
Display the results.
./
if ( wncard_c(&result) == 0 )
{
printf ( "Result window is empty.\n\n" );
}
else
{
for ( i = 0; i < wncard_c(&result); i++ )
{
/.
Fetch the endpoints of the Ith interval
of the result window.
./
wnfetd_c ( &result, i, &start, &stop );
phaseq = phaseq_c ( start, target, illmn, obsrvr, abcorr );
timout_c ( start, TIMFMT, TIMLEN, begstr );
printf ( "Start time = %s %16.9f\n", begstr, phaseq );
phaseq = phaseq_c ( stop, target, illmn, obsrvr, abcorr );
timout_c ( stop, TIMFMT, TIMLEN, endstr );
printf ( "Stop time = %s %16.9f\n", endstr, phaseq );
}
printf("\n");
}
}
return ( 0 );
}
When this program was executed on a Mac/Intel/cc/64-bit
platform, the output was:
Relation condition: =
Start time = 2006 DEC 02 13:31:34.414 0.575988450
Stop time = 2006 DEC 02 13:31:34.414 0.575988450
Start time = 2006 DEC 07 14:07:55.470 0.575988450
Stop time = 2006 DEC 07 14:07:55.470 0.575988450
Start time = 2006 DEC 31 23:59:59.997 0.575988450
Stop time = 2006 DEC 31 23:59:59.997 0.575988450
Start time = 2007 JAN 06 08:16:25.512 0.575988450
Stop time = 2007 JAN 06 08:16:25.512 0.575988450
Start time = 2007 JAN 30 11:41:32.557 0.575988450
Stop time = 2007 JAN 30 11:41:32.557 0.575988450
Relation condition: <
Start time = 2006 DEC 02 13:31:34.414 0.575988450
Stop time = 2006 DEC 07 14:07:55.470 0.575988450
Start time = 2006 DEC 31 23:59:59.997 0.575988450
Stop time = 2007 JAN 06 08:16:25.512 0.575988450
Start time = 2007 JAN 30 11:41:32.557 0.575988450
Stop time = 2007 JAN 31 00:00:00.000 0.468279091
Relation condition: >
Start time = 2006 DEC 01 00:00:00.000 0.940714974
Stop time = 2006 DEC 02 13:31:34.414 0.575988450
Start time = 2006 DEC 07 14:07:55.470 0.575988450
Stop time = 2006 DEC 31 23:59:59.997 0.575988450
Start time = 2007 JAN 06 08:16:25.512 0.575988450
Stop time = 2007 JAN 30 11:41:32.557 0.575988450
Relation condition: LOCMIN
Start time = 2006 DEC 05 00:16:50.317 0.086121423
Stop time = 2006 DEC 05 00:16:50.317 0.086121423
Start time = 2007 JAN 03 14:18:31.977 0.079899769
Stop time = 2007 JAN 03 14:18:31.977 0.079899769
Relation condition: ABSMIN
Start time = 2007 JAN 03 14:18:31.977 0.079899769
Stop time = 2007 JAN 03 14:18:31.977 0.079899769
Relation condition: LOCMAX
Start time = 2006 DEC 20 14:09:10.392 3.055062862
Stop time = 2006 DEC 20 14:09:10.392 3.055062862
Start time = 2007 JAN 19 04:27:54.600 3.074603891
Stop time = 2007 JAN 19 04:27:54.600 3.074603891
Relation condition: ABSMAX
Start time = 2007 JAN 19 04:27:54.600 3.074603891
Stop time = 2007 JAN 19 04:27:54.600 3.074603891
Restrictions
1) The kernel files to be used by this routine must be loaded
(normally using the CSPICE routine furnsh_c) before this
routine is called.
Literature_ReferencesNone. Author_and_InstitutionN.J. Bachman (JPL) J. Diaz del Rio (ODC Space) E.D. Wright (JPL) Version
-CSPICE Version 1.1.0, 01-NOV-2021 (JDR) (EDW)
Updated short error message for consistency within CSPICE wrapper
interface: MALLOCFAILURE -> MALLOCFAILED.
Added use of ALLOC_CHECK_INTRA to check net null effect on
alloc count.
Updated header to describe use of expanded confinement window.
Edited the header to comply with NAIF standard.
Updated the description of "nintvls", "cnfine" and "result" arguments.
Replaced entry #8 by new entries #8 and #9, and added entry #11
in -Exceptions section.
-CSPICE Version 1.0.0, 15-JUL-2014 (EDW) (NJB)
Index_EntriesGF phase angle search |
Fri Dec 31 18:41:07 2021