Astronomical Ephemeris (6) for the HP-41

Overview
 

 1°)  3 Subroutines

  1-a)  An M-Code Routine: Kepler's Equation
  1-b)  2 Focal Programs

 2°)  Sidereal Time - Sun - Mercury - Venus - Mars
 3°)  Jupiter
 4°)  Saturn
 5°)  Uranus - Neptune
 6°)  Moon
 
 

-These programs calculate the position of the Sun, the major planets and the Moon.
-They may be found in a module here.

-For the Sun, Mercury, Venus and Mars, the time-span is +/- 10000 years from J2000, i-e the interval [ -8000 , +12000 ]

-With Jupiter, Saturn, Uranus, Neptune & the Moon, it's preferable to limit the interval to [ -4000 , +8000 ] ,
  the results will be much less accurate 10000 years from J2000.

-Anyway, the precision will remain low: errors about 1 or 2 arcminutes are quite possible.
-The best precision is probably achieved with the heliocentric coordinates of Mercury.

-The longitudes and latitudes are expressed in decimal degrees, the distances in astronomical units.

-They are referred to the mean ecliptic & equinox of the date:  so, the aberration, nutation and light-time are not taken into account.
 

Important Notes about Flags F01 & F04
 

-If flag F01 is clear, we get the geocentric coordinates of the planets.
-If flag F01 is set, the HP41 returns the heliocentric coordinates only.

-In both cases, for the Sun & the Moon, the coordinates are always geocentric.

-If flag F04 is clear, the dates must be expressed in the Gregorian calendar
-If flag F04 is set, the dates must be expressed in the Julian calendar
 

Registers
 

•  R00 = Time from 2000/01/01 0h TT ( unit = 10,000 Julian years )            ( Register R00 must be initialized before computing the coordinates )

 R01-R02 = Rectangular coordinates of the Sun,  X & Y

   -> Always XEQ "SUN+"  before the planets if you want the geocentric coordinates.

 R03 = Geocentric longitude
 R04 = Geocentric latitude
 R05 = Distance Earth-Planet

 R06 = Heliocentric longitude
 R07 = Heliocentric latitude
 R08 = Radius vector

 R09 tà R15: temp
 
 

1°)  3 Subroutines
 

     a) An M-Code Routine: Kepler's Equation
 

-The M-Code routine "VR" takes the eccentricity e < 1 in Y-register and the mean anomaly M in X-register
-It returns the true anomaly v in X-register and r/a in Y-register
-The eccentric anomaly E is also given in T-register

-The angles must be expressed in decimal degrees.
-There is no check for alpha data
 

092  "R"
016  "V"
071  ?NCXQ
05C  DEG
2A0  SETDEC
0F8  C=X
128  L=C
10E  A=C ALL
0B8  C=Y
05E  C= | C |
0E8  X=C
0AE  A<>C ALL
0A8  Y=C
04E  C
2BE
35C
050  =
21C
250
250
110  -10^(-6)
028  T=C
046  C=0 S&X
01D  C=
060  A+C
2EE  ?C#0 ALL
01B  JNC+03
2FE  ?C<0 ALL
01F  JC+03
0B5  ?NCGO
0A2  DATAERROR
0B8  C=Y                    loop
10E  A=C ALL
0F8  C=X
070  polar
3C4 rectangular
1D5
078  conversion
2BE  C=-C
00E  A
35C =
162  1
01D  C=
060  A+C
068  Z=C
0B0  C=N ALL
231  ?NCXQ
064  R-D
10E  A=C ALL
0B8  C=Y
2BE  C=-C
01D  C=
061  A+C
138  C=L
025  C=
060  AB+C
078  C=Z
269  C=
060  AB/C
070  N=C ALL
0B8  C=Y
025  C=
060  AB+C
0A8  Y=C
0B0  C=N ALL
05E  C=| C |
10E  A=C ALL
046  C
270  =
038  T
01D  C=
060  A+C
2FE  ?C<0
2B3  JNC-42d           goto loop
0F8  C=X
068  Z=C
0B8  C=Y
028  T=C
10E  A=C ALL
04E  C
35C  =
090  2
261  C=
060  A/C
070  C
209  =
048  tan(C°)
070  N=C ALL
078  C=Z
2BE  C=-C
00E  A
35C  =
162  1
01D  C=
060  A+C
05E  C=| C |
128  L=C
078  C=Z
00E  A
35C  =
162  1
01D  C=
060  A+C
0A8  Y=C
138  C=L
269  C=
060  AB/C
305  C=
060  sqrt(AB)
0B0  C=N ALL
13D  C=
060  AB*C
070  C
2A9  =
040  atan(C°)
10E  A=C ALL
01D  C=
060  A+C
0E8  X=C
070  C
1F1  =
048  cos(C°)
10E  A=C ALL
078  C=Z
135  C=
060  A*C
001  C=
060  AB+1
070  N=C ALL
0B8  C=Y
10E  A=C ALL
138  C=L
135  C=
060  A*C
0B0  C=N
269  C=
060  AB/C
0A8  Y=C
3E0  RTN

( 139 words )
 
 

 
Example:

   0.2   ENTER^
   24    XEQ "VR"  >>>>   35°94864074
                             X<>Y   0.826226700

-And the eccentric anomaly in T-register is  E = 29°67281468

Notes:

-It may take several seconds...
-If e < 0 , it's replaced by -e
-If e = 1 or e > 1 , there will be a DATA ERROR message.
 

     b) 2 Focal Programs:  K & L
 

 "L"  calculates the variable µ = 205°99908944 T  and stores several multiples of µ
 "K" transforms the elliptic elements into heliocentric coordinates and geocentric coordinates ( if F01 is clear )
 
 
 

 
      ( 176 bytes / SIZE 016 )
 

2°)  Sidereal Time - Sun - Mercury - Venus - Mars
 
 

 "ST+" calls "J1" ( cf "Julian & Gregorian Calendars for the HP41 " )
 
 

 
       ( 1347 bytes / SIZE 016 )
 
 

 
Example:    Sidereal time on 12000/03/16 at 0h
 

   12000.0316  ENTER^
            0          XEQ "ST+"   >>>>   ST = 12h03m18s
 

  and  T = 1  is stored in R00
 

Note:

 12h03m18s is the mean sidereal time at 0h TT at the ephemeris meridian
 It is also the mean sidereal time at Greenwich at 0h UT

 The longitude of the ephemeris meridian = 1.002738 ( TT - UT )

 And TT-UT may be approximately estimated in seconds by the routine "TT-UT"

  12000.20  XEQ "TT-UT"  >>>>   TT - UT = 331617 seconds
 

The Sun
 
 
 

 
   XEQ "SUN+"  ( or R/S )   >>>>   l = 358°092 = R03
                                            RDN    b =  0 = R04
                                            RDN    r = 1.00213  AU = R05
 

And we have in R01-R02 the rectangular ecliptic coordinates of the Sun

   R01 = X = 1.00157 AU
   R02 = Y = -0.03337 AU

   ( Z = 0 to the low accuracy of these results )

Note:

-See below for another example in the interval  [ -4000 , +8000 ]
 

Mercury
 
 
 

 
   XEQ "ME+"  ( or R/S )   >>>>   l = 21°807 = R03
                                          RDN   b = -2°826 = R04
                                          RDN   r = 1.11352  AU = R05

And the heliocentric coordinates of Mercury in R06-R07-R08

    R06 = L = 86°028
    R07 = B = -6°994
    R08 = R = 0.45095 AU

Note:

-See below for another example in the interval  [ -4000 , +8000 ]
 

Venus
 
 

 
   XEQ "VE+"  ( or R/S )   >>>>   l = 336°333 = R03
                                          RDN   b = 1°143 = R04
                                          RDN   r = 1.54902  AU = R05

And the heliocentric coordinates in R06-R07-R08

    R06 = L = 305°321
    R07 = B = 2°453
    R08 = R = 0.72171 AU

Note:

-See below for another example in the interval  [ -4000 , +8000 ]
 

Mars
 
 

 
   XEQ "MA+"  ( or R/S )   >>>>   l = 22°325 = R03
                                          RDN   b = -1°234 = R04
                                          RDN   r = 2.4584  AU = R05

And the heliocentric coordinates in R06-R07-R08

    R06 = L = 37°243
    R07 = B = -1°898
    R08 = R = 1.5987 AU

Note:

-See below for another example in the interval  [ -4000 , +8000 ]
 

3°)  Jupiter
 
 
 
 

 
        ( 744 bytes / SIZE 016 )
 
 

 
Example:   the same date & time
 

   XEQ "JU+"  ( or R/S )   >>>>   l = 189°481 = R03
                                          RDN   b = -0°299 = R04
                                          RDN   r = 3.907  AU = R05

And the heliocentric coordinates in R06-R07-R08

    R06 = L = 187°163             here, the error is about 0°1 in the longitude !
    R07 = B = -0°239
    R08 = R = 4.893 AU

Note:

-See below for another example in the interval  [ -4000 , +8000 ]
 

4°)  Saturn
 
 
 
 

 
      ( 1415 bytes / SIZE 016 )
 
 

 
Example:
 

   XEQ "SA+"  ( or R/S )   >>>>   l = 5°851 = R03
                                          RDN   b = 0°391 = R04
                                          RDN   r = 10.431  AU = R05

And the heliocentric coordinates in R06-R07-R08

    R06 = L = 6°673             the error is about 0°18 in the longitude !
    R07 = B = 0°432
    R08 = R = 9.439 AU

Note:

-See below for another example in the interval  [ -4000 , +8000 ]
 

5°)  Uranus - Neptune
 
 
 
 

 
        ( 1586 bytes / SIZE 016 )
 
 

 
Example:
 

 Uranus
 

   XEQ "UR+"  ( or R/S )   >>>>   l = 104°268 = R03
                                          RDN   b = -0°723 = R04
                                          RDN   r = 19.575  AU = R05

And the heliocentric coordinates in R06-R07-R08

    R06 = L = 107°044             error is only 0°015 in the longitude, by chance !
    R07 = B = -0°712
    R08 = R = 19.878 AU

Note:

-See below for another example in the interval  [ -4000 , +8000 ]
 

Neptune
 

   XEQ "NE+"  ( or R/S )   >>>>   l = 336°107 = R03
                                          RDN   b = 0°762 = R04
                                          RDN   r = 31.240  AU = R05

And the heliocentric coordinates in R06-R07-R08

    R06 = L = 335°398              error is about 0°05 in the longitude.
    R07 = B = 0°785
    R08 = R = 30.313 AU

Note:

-See below for another example in the interval  [ -4000 , +8000 ]
 

6°)  The Moon
 
 
 

 
     ( 1017 bytes / SIZE 016 )
 
 

 
Example:
 

   XEQ "MO+"  ( or R/S )   >>>>   l = 21°139 = R03       "SOLEX"  gives  21°126
                                          RDN   b = -2°567 = R04
                                          RDN   r = 0.002475  AU = R05

Note:

-Here is another example in the interval  [ -4000 , +8000 ]

-With T = 0.6 STO 00   ( i-e 8000/02/15  0h  TT )  we get the following heliocentric coordinates L  B  R ( geocentric for the Sun & the Moon ):
-L & B are expressed indecimal degrees, R in Astronomical Units

-The results given by "Solex" - which uses DE431 - are also given for comparisons.
 
 

References:
 

[1] Jacques Laskar - "Secular terms of Classical Planetary Theories using the results of General Theory" - A&A 157, 59-70 ( 1986 )
[2] Planetary theory TOP 2013 - Jean-Louis Simon ftp://ftp.imcce.fr/pub/ephem/planets/top2013/
[3] S M Kudryavtsev - "Long-term harmonic development of lunar ephemeris" - A&A 471, 1069-1075 ( 2007 )
[4] J Vondrak, N Capitaine, P Wallace - "New precession expression, valid for a long tme-span" - A&A 534, A22 ( 2011 )
[5] An excellent software by Aldo vitagliano, "Solex" which may be downloaded freely here