hp41programs

Precession

Precession for the HP-41


Overview
 

 1°)  Precession Parameters pA & eA
 2°)  Precession Parameters QA & PA
 3°)  Polynomial Approximations from 8000 B.C. to A.D. 12000
 4°)  Transformation of Coordinates

      a)  Ecliptic Coordinates
      b)  Equatorial<>Ecliptic Conversion
      c)  Equatorial Coordinates
 

-Several programs are listed in "Transformation of Coordinates for the HP-41" to take the precession into account.
-They use formulae given in reference [2]
-But these formulas are accurate only a few centuries from J2000.

-The following routines employ the formulae given in reference [1] which are valid 200 thousands of years from J2000
-The accuracy is the same as IAU2006 near J2000 but the errors can reach a few arcminutes 200000 years before or after J2000.
-So, we'll get a much better precision though the execution time will be increased too...
 

1°)  Precession Parameters pA & eA
 

-"PAEA" takes the time expressed in 10000 years from J2000 ( 2000/01/01 12h TT ) in register R00
  and returns the mean obliquity of the ecliptic eA in Y-register and the general precession in longitude pA in X-register, both in decimal degrees.
 

Data Registers:     •  R00 = T = time from J2000                          ( Register R00 is to be initialized before executing "PAEA" )

                                   unit = 10000 years = 3652500 days                ( -20 < T < +20 )

                                   R01 = pA  R02 = eA ,  R03-R04: temp
Flags: /
Subroutines: /
 
 

 01  LBL "PAEA"
 02  DEG
 03  RCL 00 
 04  360
 05  *
 06  STO 04        
 07  2.03
 08  /
 09  STO 03
 10  86.021161
 11  +
 12  COS
 13  47604764
 14  *
 15  STO 01
 16  RCL 03
 17  4.429265
 18  -
 19  COS
 20  10841074
 21  *
 22  STO 02
 23  RCL 04
 24  2.77
 25  /
 26  STO 03
 27  10.754166
 28  +
 29  COS
 30  54765036
 31  *
 32  ST+ 01
 33  RCL 03
 34  75.298375
 35  -
 36  COS		  37  19377033
 38  *
 39  ST+ 02
 40  RCL 04 
 41  3.06
 42  /
 43  STO 03        
 44  19.460619
 45  +
 46  COS
 47  63817823
 48  *
 49  ST- 01
 50  RCL 03
 51  69.576035
 52  -
 53  COS
 54  22485751
 55  *
 56  ST- 02
 57  RCL 04
 58  40.43
 59  /
 60  STO 03
 61  32.946452
 62  +
 63  COS
 64  123082089
 65  *
 66  ST+ 01
 67  RCL 03
 68  77.972005
 69  -
 70  COS
 71  84131084
 72  *		  73  ST- 02
 74  RCL 04
 75  2.8892
 76  /
 77  STO 03        
 78  8.939928
 79  -
 80  SIN
 81  280606764
 82  *
 83  ST+ 01
 84  RCL 03
 85  9.088533
 86  -
 87  COS
 88  99471091
 89  *
 90  ST- 02
 91  RCL 04
 92  4.1715
 93  /
 94  STO 03
 95  19.648463
 96  -
 97  COS
 98  346046807
 99  *
100  ST+ 01
101  RCL 03
102  64.673899
103  +
104  COS
105  58513547
106  *
107  ST- 02
108  RCL 04		 109  4.029
110  /
111  STO 03       
112  45.952
113  +
114  COS
115  342696586
116  *
117  ST- 01
118  RCL 03
119  3.4661186
120  +
121  SIN
122  247556156
123  *
124  ST- 02
125  RCL 04
126  5.3722
127  /
128  STO 03
129  40.8256734
130  +
131  COS
132  533629325
133  *
134  ST+ 01
135  RCL 03
136  49.72366
137  -
138  COS
139  163051596
140  *
141  ST+ 02
142  RCL 04
143  3.9615
144  /		 145  STO 03       
146  8.0052016
147  +
148  COS
149  897273066
150  *
151  ST- 01
152  RCL 03 
153  73.9666746
154  -
155  COS
156  249224635
157  *
158  ST- 02
159  RCL 04
160  4.099
161  /
162  STO 03
163  22.3842941
164  -
165  COS
166  2075345042
167  *
168  ST- 01
169  RCL 03
170  66.1437076
171  +
172  COS
173  517770288
174  *
175  ST+ 02
176  RCL 00
177  19742583
178  CHS
179  RCL 00
180  75278		 181  *
182  +
183  *
184   E9
185  STO 04       
186  ST/ 01
187  /
188  140.0847779
189  +
190  *
191  2.259449203
192  +
193  ST+ 01
194  CLX
195  30556
196  *
197  CHS
198  112194
199  -
200  *
201  10067903
202  +
203  *
204  RCL 02
205  +
206  RCL 04
207  /
208  23.34116842
209  +
210  STO 02
211  RCL 01
212  END

 
    ( 651 bytes / SIZE 005 )
 
 

      STACK        INPUTS      OUTPUTS
           Y             /            eA
           X             /            pA

  with  pA = general precession in longitude  and eA = obliquity of the ecliptic ( in decimal degrees )

Example:    T = 0.8   ( i-e 10000/03/01  12h  TT )

  0.8   STO 00   XEQ "PAEA"  >>>>   pA = 113°4650921                     ---Execution time = 32s---
                                                X<>Y   eA =  22°65102781

Note:

-For T = 0  "PAEA"  gives   pA = -3 E-9  degree instead of exactly 0  ( roundoff-errors )
-If you want to get  pA = 0 , replace line 191  by  2.259449206 , this will not change the other results significantly.
 

2°)  Precession Parameters QA & PA
 

-"QAPA" takes the time expressed in 10000 years from J2000 ( 2000/01/01 12h TT ) in register R00
  and returns PA = Sin pA Sin ¶A  in Y-register and QA = Sin pA Cos ¶A  in X-register.
 

Data Registers:       •  R00 = T = time from J2000                               ( Register R00 is to be initialized before executing "QAPA" )

                                           unit = 10000 years = 3652500 days              ( -20 < T < +20 )

                                     R01 = QA  R02 = PA ,  R03-R04: temp
Flags: /
Subroutines: /
 
 

 01  LBL "QAPA"
 02  RCL 00 
 03  360
 04  *
 05  STO 04
 06  5.47
 07  /
 08  STO 03        
 09  12.750573
 10  -
 11  COS
 12  28803536
 13  *
 14  STO 01
 15  RCL 03
 16  69.122621
 17  +
 18  COS
 19  35964826
 20  *
 21  STO 02
 22  RCL 04
 23  8.82
 24  /
 25  STO 03
 26  9.938168
 27  +
 28  COS
 29  55921543		  30  *
 31  ST+ 01
 32  RCL 03        
 33  64.659178
 34  -
 35  COS
 36  56902710
 37  *
 38  ST- 02
 39  RCL 04
 40  6.22
 41  /
 42  STO 03
 43  24.635276
 44  -
 45  COS
 46  96811871
 47  *
 48  ST- 01
 49  RCL 03
 50  61.676072
 51  +
 52  COS
 53  105812918 
 54  *
 55  ST- 02
 56  RCL 04
 57  11.83
 58  /		  59  STO 03        
 60  11.161703
 61  -
 62  COS
 63  52772369
 64  *
 65  ST- 01
 66  RCL 03
 67  66.952667
 68  -
 69  COS
 70  55779703
 71  *
 72  ST+ 02
 73  RCL 04
 74  4.922
 75  /
 76  STO 03
 77  49.766163
 78  +
 79  COS
 80  153380984 
 81  *
 82  ST- 01
 83  RCL 03
 84  42.299904
 85  -
 86  COS
 87  155273469		  88  *
 89  ST+ 02
 90  RCL 04        
 91  16.2
 92  /
 93  STO 03
 94  37.887736
 95  +
 96  COS
 97  140670994
 98  *
 99  ST+ 01
100  RCL 03
101  34.7353008
102  -
103  COS
104  208729670
105  *
106  ST- 02
107  RCL 04
108  23.09
109  /
110  STO 03
111  12.6655261
112  +
113  COS
114  696470770
115  *
116  ST+ 01		 117  RCL 03       
118  .4167179
119  +
120  SIN
121  654152372
122  *
123  ST- 02
124  RCL 04 
125  7.0815
126  /
127  STO 03
128  7.0655556
129  +
130  SIN
131  1546147930
132  *
133  ST- 01
134  RCL 03
135  6.9380447
136  +
137  COS
138  1535340316
139  *
140  ST- 02
141  RCL 00
142  556
143  RCL 00
144  121389
145  *		 146  +
147  *
148  CHS
149  32471717
150  +
151  *
152  444690639
153  -
154  ST+ 01
155  CLX
156  28056
157  *
158  803139
159  -
160  *
161  3302778
162  -
163  *
164  1625446580
165  +
166  RCL 02        
167  +
168  PI
169  18 E10
170  /
171  ST* 01
172  *
173  STO 02
174  RCL 01
175  END

 
    ( 532 bytes / SIZE 005 )
 
 

      STACK        INPUTS      OUTPUTS
           Y             /            PA
           X             /            QA

     with  PA = Sin pA Sin ¶A  &  QA = Sin pA Cos ¶A

Example:    T = 0.8   ( i-e 10000/03/01  12h  TT )

  0.8   STO 00   XEQ "QAPA"  >>>>   QA = -0.015413403                     ---Execution time = 26s---
                                                X<>Y    PA =   0.006932629

 from which we deduce ( with R-P ASIN )  pA = 0°968386023  &  ¶A = 155°7827747

Notes:

-The HP-41 must be set in DEG mode.
-For T = 0, we get  QA =  2.4 E-11  &  PA = -1.7 E-11  instead of exactly 0.
 

3°)  Polynomial Approximations from 8000 B.C. to A.D. 12000
 

-If you plan to use these formulae between these 2 dates, polynomial approximations may also be used to get faster results.
-The following ones are valid for -1 < T < +1   (  I found them thanks to my HP-48 )
 

  pA = 139°688785 T + 3°070652 T2 + 0°011186 T3 - 0°646288 T4  - 0°023670 T5
          + 0°050722 T6 + 0°013594 T7 - 0°001690 T8 - 0°002000 T9 - 0°000003 T10

  eA = 23°4392794 - 1°301021 T - 0°000459 T2 + 0°553207 T3 - 0°021415 T4  - 0°068157 T5
          - 0°000784 T6 + 0°003674 T7 + 0°001093 T8 - 0°000081 T9 - 0°000141 T10

  ¶A = 174°874109 - 24°111006 T + 0°339983 T2 - 0°025435 T3 - 0°017068 T4  - 0°000671 T5
          + 0°000418 T6 + 0°000029 T7 - 0°000007 T8 + 0 T9 - 0°000002 T10

  pA = 1°305527 T - 0°093051 T2 - 0°034510 T3 + 0°002859 T4  + 0°000058 T5
          - 0°000016 T6 + 0°000003 T7 - 0°000002 T8
 

-Simply store T into R00 before executing these routines and you'll get  pA or  eA or  ¶A or  pA
 

Data Registers:       •  R00 = T = time from J2000               ( Register R00 is to be initialized before executing "PA" "EA" "PIA" or "PIa" )

                                     unit = 10000 years = 3652500 days            ( -1 < T < +1 )
Flags: /
Subroutines: /
 
 

 01  LBL "PA" 
 02  RCL 00        
 03  2 E3
 04  RCL 00 
 05  3
 06  *
 07  +
 08  *
 09  CHS
 10  1690
 11  -
 12  *
 13  13594
 14  +
 15  *
 16  50722
 17  +
 18  *
 19  23670
 20  -
 21  *
 22  646288
 23  -
 24  *
 25  11186
 26  +
 27  *
 28  3070652		  29  +
 30  *
 31  139688785
 32  +
 33  *
 34   E6
 35  /
 36  RTN
 37  LBL "EA"
 38  RCL 00
 39  81
 40  RCL 00        
 41  141
 42  *
 43  +
 44  *
 45  CHS
 46  1093
 47  +
 48  *
 49  3674
 50  +
 51  *
 52  784
 53  -
 54  *
 55  68157
 56  -		  57  *
 58  21415
 59  -
 60  *
 61  553207
 62  +
 63  *
 64  459
 65  -
 66  *
 67  1301021
 68  -
 69  *
 70   E6
 71  /
 72  23.4392794
 73  +
 74  RTN
 75  LBL "PIA"
 76  RCL 00 
 77  RCL 00        
 78  RCL 00
 79  2
 80  *
 81  *
 82  CHS
 83  7
 84  -		  85  *
 86  29
 87  +
 88  *
 89  418
 90  +
 91  *
 92  671
 93  -
 94  *
 95  17068
 96  -
 97  *
 98  25435
 99  -
100  *
101  339983
102  +
103  *
104  24111006
105  -
106  *
107  174874109 
108  +
109   E6               
110  /
111  RTN
112  LBL "PIa"		 113  RCL 00
114  3
115  RCL 00        
116  2
117  *
118  -
119  *
120  16
121  -
122  *
123  58
124  +
125  *
126  2859
127  +
128  *
129  34510
130  -
131  *
132  93051
133  -
134  *
135  1305527 
136  +
137  *
138   E6
139  /
140  END

 
    ( 301 bytes / SIZE 001 )
 
 

             STACK             INPUTS          OUTPUTS
                 X                   /   pA or eA or ¶A or pA

   All these angles in decimal degrees  with  -1 < T < +1  in R00

Examples:

    •  T = 0.987  STO 00

   XEQ "PA"    >>>>  pA = 140°295412
   XEQ "EA"   >>>>   eA =  22°605889
  XEQ "PIA"   >>>>   ¶A = 151°366870
  XEQ "PIa"   >>>>    pA =  1°167480

    •  T = -0.987  STO 00

   XEQ "PA"    >>>>  pA = -135°448671
   XEQ "EA"   >>>>   eA =  24°231402
  XEQ "PIA"   >>>>   ¶A = 199°012112
  XEQ "PIa"   >>>>    pA =  -1°343381

Notes:

-If you compare the results given by "QAPA" followed by  R-P when  T < 0,  pA is positive and  ¶A differs from the above value by 180°
-But this does not change the outputs of the programs hereunder.

-The differences between the outputs of these routines and "QAPA" or "PAEA" are smaller than 0"01 provided  | T |  doesn't exceed 1
-Don't use these expressions outside the interval  [-1,+1]
 

4°)  Transformation of Coordinates
 

     a)  Ecliptic Coordinates
 

-"PREC" uses the routines listed in paragraphs 1 & 2 to perform the transformations.
 

Data Registers:         R00 = T = time from J2000    ,    unit = 10000 years = 3652500 julian days            ( -20 < T < +20 )

                                    R01 thru R09: temp
Flag:  F00
Subroutines:

   "J1" or "J2" or "J0" ( cf "Julian & Gregorian Calendars for the HP-41" )
   "QAPA" & "PAEA" ( cf §1 and §2 )
   "EE"  ( cf "Transformation of coordinates for the HP-41" )
 
 

 01  LBL "PREC" 
 02  DEG
 03  HR
 04  STO 05        
 05  X<>Y
 06  HR
 07  STO 06 
 08  R^
 09  STO 07
 10  R^		  11  SF 00
 12  LBL 00
 13  X<> 07
 14  XEQ "J2"
 15  .5
 16  -
 17  3652500
 18  /
 19  STO 00        
 20  XEQ "QAPA"		  21  R-P
 22  ASIN
 23  STO 09
 24  X<>Y
 25  STO 08        
 26  ST- 05
 27  XEQ "PAEA"
 28  FS? 00
 29  ST- 05
 30  X<> 09		  31  FS? 00
 32  CHS
 33  RCL 06
 34  RCL 05        
 35  XEQ "EE" 
 36  RCL 08
 37  +
 38  STO 05 
 39  X<>Y
 40  STO 06		  41  FS?C 00
 42  GTO 00 
 43 HMS
 44  X<>Y
 45  RCL 09        
 46  +
 47  360
 48  MOD
 49  HMS
 50  END

 
    ( 93 bytes / SIZE 010 )
 
 

        STACK          INPUTS        OUTPUTS
             T    YYYY.MNDD1              /
             Z    YYYY.MNDD2              /
             Y        b1 ( ° . ' " )         b2 ( ° . ' " )
             X        l1  ( ° . ' " )         l2  ( ° . ' " )

    where   l  = mean ecliptic longitudes , b  = mean ecliptic latitudes

Example1:

-On  -8000/01/01 0h TT ( gregorian calendar, -8000 = 8001 B.C. )   l1 = 167°23'45"  and b1 = -12°34'56"
-Calculate the mean ecliptic longitude & latitude of this fictitious star on  200000/01/01 0h TT

   -8000.0101   ENTER^
 200000.0101  ENTER^
       -12.3456   ENTER^
       167.2345  XEQ "PREC"   >>>>     191.534869                   ---Execution time = 131s---
                                                X<>Y    -12.194353

-Thus,   l2 = 191°53'48"69  &  b2 = -12°19'43"53

Example2:

-Same inputs but the second date =  12000/01/01 0h TT

   -8000.0101   ENTER^
   12000.0101  ENTER^
       -12.3456   ENTER^
       167.2345  XEQ "PREC"   >>>>       87.070354                    ---Execution time = 131s---
                                                X<>Y    -14.250176

-Thus,   l2 = 87°07'03"54  &  b2 = -14°25'01"76

Note:

-In the second example, we can use the polynomial approximations and the following program to get faster results, almost as accurate.
-This variant employs the polynomials given in paragraph 3.
 

Data Registers:         R00 = T = time from J2000    ,    unit = 10000 years = 3652500 julian days            ( -1 < T < +1 )

                                    R01 thru R05: temp
Flag:  F00
Subroutines:

   "J1" or "J2" or "J0" ( cf "Julian & Gregorian Calendars for the HP-41" )
   "PIA" "PIa" & "PA" ( cf §3 )
   "EE"  ( cf "Transformation of coordinates for the HP-41" )
 
 

 01  LBL "PREC"
 02  DEG
 03  HR
 04  STO 01        
 05  X<>Y
 06  HR
 07  STO 02
 08  R^
 09  STO 03
 10  R^		  11  SF 00
 12  LBL 00
 13  X<> 03
 14  XEQ "J2"
 15  .5
 16  -
 17  3652500
 18  /
 19  STO 00        
 20  XEQ "PIA" 		  21  STO 04 
 22  ST- 01
 23  XEQ "PA"
 24  STO 05       
 25  FS? 00
 26  ST- 01
 27  XEQ "PIa" 
 28  FS? 00
 29  CHS
 30  RCL 02		  31  RCL 01 
 32  XEQ "EE" 
 33  RCL 04
 34  +
 35  STO 01        
 36  X<>Y
 37  STO 02
 38  FS?C 00
 39  GTO 00
 40  HMS		  41  X<>Y
 42  RCL 05        
 43  +
 44  360
 45  MOD
 46  HMS
 47  END

 
    ( 90 bytes / SIZE 006 )
 
 

        STACK          INPUTS        OUTPUTS
             T    YYYY.MNDD1                /
             Z    YYYY.MNDD2                /
             Y        b1 ( ° . ' " )         b2 ( ° . ' " )
             X        l1  ( ° . ' " )         l2  ( ° . ' " )

    where   l  = mean ecliptic longitudes , b  = mean ecliptic latitudes

Example:

-On  -8000/01/01 0h TT ( gregorian calendar, -8000 = 8001 B.C. )   l1 = 167°23'45"  and b1 = -12°34'56"
-Calculate the mean ecliptic longitude & latitude of this fictitious star on  12000/01/01 0h TT

   -8000.0101   ENTER^
   12000.0101  ENTER^
       -12.3456   ENTER^
       167.2345  XEQ "PREC"   >>>>      87.070353                    ---Execution time = 27s---
                                                X<>Y    -14.250176

-Thus,   l2 = 87°07'03"53  &  b2 = -14°25'01"76

Note:

-The differences with the version above are only of the order of  0"01
 

     b)  Equatorial <> Ecliptic Conversion
 
 

Data Registers:         R00 = T = time from J2000    ,    unit = 10000 years = 3652500 julian days            ( -20 < T < +20 )

                                    R01 thru R06: temp
Flags:  /
Subroutines:

 "J1" or "J2" or "J0" ( cf "Julian & Gregorian Calendars for the HP-41" )
 "PAEA"  ( §1 )
 "EE"  ( cf "Transformation of coordinates for the HP-41" )
 
 

 01  LBL "EQECL"
 02  DEG
 03  HR
 04  15
 05  *
 06  STO 05          
 07  X<>Y
 08  HR
 09  STO 06
 10  R^
 11  XEQ "J2"		  12  3652500
 13  /
 14  STO 00
 15  XEQ "PAEA" 
 16  CLX
 17  RCL 06          
 18  RCL 05
 19  XEQ "EE"
 20  X<>Y
 21  HMS
 22  X<>Y		  23  360
 24  MOD
 25  HMS
 26  RTN
 27  LBL "ECLEQ"
 28  DEG
 29  HR
 30  STO 05          
 31  RDN
 32  HR
 33  STO 06		  34  X<>Y
 35  XEQ "J2"
 36  3652500
 37  /
 38  STO 00
 39  XEQ "PAEA" 
 40  X<>Y
 41  CHS
 42  RCL 06          
 43  RCL 05
 44  XEQ "EE"		  45  X<>Y
 46  HMS
 47  X<>Y 
 48  15
 49  /
 50  24
 51  MOD            
 52  HMS
 53  END

 
     ( 110 bytes / SIZE 007 )
 
 

        STACK         INPUT1       OUTPUT1          INPUT2         OUTPUT2
             Z    YYYY.MNDD            eA     YYYY.MNDD             -eA
             Y       decl ( ° . ' " )         b ( ° . ' " )         b ( ° . ' " )       decl ( ° . ' " )
             X      RA (hh.mnss)         l ( ° . ' " )          l ( ° . ' " )      RA (hh.mnss)

  where   RA = right-ascension , decl = declination , l  = ecliptic longitude , b  = ecliptic latitude

Example1:     On  2134/04/04 at 0h  if  Right-Ascension = 12h34m56s  and  Declination =  41°16'57"  ( mean coordinates )

  10000.0707  ENTER^
        41.1657  ENTER^
        12.3456  XEQ "EQECL"  >>>>  l  =  168°40'20"09    X<>Y    b  =  40°45'17"52                ---Execution time = 38s---

Example2:

  10000.0707        ENTER^
        40.451752    ENTER^
      168.402009    XEQ "ECLEQ"  >>>>   RA = 12h34m56s00    X<>Y   Decl = 41°16'57"00                ---Execution time = 38s---

Note:

-If -1 < T < +1 , lines 39-40 & 15-16 may be replaced by  XEQ "EA"
 

     c)  Equatorial Coordinates
 

-"PREQ"  calls  "EQECL" then "PREC" and finally "ECLEQ"
 

Data Registers:         R00 = T = time from J2000    ,    unit = 10000 years = 3652500 julian days            ( -20 < T < +20 )

                                    R01 thru R11: temp
Flag:  F00
Subroutines:   "EQECL"  "PREC"  ( 1st version )  "ECLEQ"
 
 

 01  LBL "PREQ"
 02  R^
 03  STO 10
 04  X<> T
 05  STO 11
 06  RDN
 07  XEQ "EQECL"
 08  RCL 10
 09  RCL 11
 10  R^
 11  R^
 12  XEQ "PREC"
 13  RCL 11
 14  X<> Z
 15  X<>Y
 16  XEQ "ECLEQ"
 17  END

 
( 45 bytes / SIZE 012 )
 
 

        STACK          INPUTS        OUTPUTS
             T    YYYY.MNDD1                /
             Z    YYYY.MNDD2                /
             Y      Decl1 ( ° . ' " )       Decl2 ( ° . ' " )
             X      RA1  ( ° . ' " )       RA2  ( ° . ' " )

    where   RA = right-ascension , Decl = declination

Example:

-On  -8000/01/01 0h TT ( gregorian calendar, -8000 = 8001 B.C. )   RA1 = 12h34m56s  and  Decl1 = 41°16'57"
-Calculate the right-ascension & declination of this fictitious star on  12000/01/01 0h TT

   -8000.0101   ENTER^
   12000.0101  ENTER^
        41.1657   ENTER^
        12.3456   XEQ "PREQ"   >>>>         5.312210                    ---Execution time = 3m29s---
                                                X<>Y       61.041700

-So,  RA2 = 5h31m22s10  &  Decl2 = 61°04'17"00

Notes:

-In fact, "PAEA" is called 4 times but 2 times would be enough and execution time could be reduced...

-One can use other precession parameters to get these results more easily
-They are given in reference [1] which also contains the precession matrices ... and so on ... all valid   +/- 200,000 years from J2000.0
 

References:

[1]   J. Vondrak, N. Capitaine, P. Wallace - "New precession expressions, valid for long time intervals" - Astronomy & Astrophysics 534, A22 ( 2011 )
[2]   Circular n° 179 http://aa.usno.navy.mil/publications/docs/circular_179.html