hp41programs

binarystar Binary Stars for the HP-41
 

Overview

-This program calculates the polar coordinates ( r ; µ ) of a binary star  ( S1 ; S2 )
 and the eccentricity e' of the apparent orbit, as seen from the Earth.
 

                   North
                       |
            S2       |
              \        |
                \  µ  |
               r  \    |                                     r  =  the angular distance = S1S2
                    \  |                                     µ  =  the position angle = ( S1N ; S2N )
                      \|
                      S1
 

Program Listing
 

Data Registers:       When the program stops,   R00 = e' =  eccentricity of the apparent orbit

    -Registers R08 & R09 are used for temporary data storage.          Registers R01 thru R07 are to be initialized before executing "BSTAR"

                            •   R01 = P = period of revolution ( expressed in years )
                            •   R02 = T = time of perihelion passage ( given as a year and decimals )
                            •   R03 = e = eccentricity
                            •   R04 = a = semimajor axis ( in arcseconds )
                            •   R05 = i  = inclination of the orbit
                            •   R06 = OMEGA = position angle of the ascending node
                            •   R07 = omega = longitude of the periastron.

Flags: /
Subroutines: /

-Lines 52 to 66 solve Kepler's equation.
-The iterations stop when the difference between 2 successive approximations is smaller than 0.001°  ( line 64 )
 
 

  01  LBL "BSTAR"
  02  DEG
  03  STO 00
  04  RCL 03
  05  RCL 07
  06  COS
  07  *
  08  X^2
  09  STO 08
  10  RCL 05
  11  COS
  12  X^2
  13  ST* Y
  14  -
  15  X<> 08
  16  RCL 03
  17  X^2
  18  -
  19  1
  20  +
  21  RCL 05
  22  COS
  23  RCL 07		   24  RCL 03
  25  P-R
  26  *
  27  *
  28  ST+ X
  29  X^2
  30  RCL 08         
  31  ST- Z
  32  R^
  33  +
  34  X^2
  35  +
  36  SQRT
  37  ENTER^
  38  ST+ Y
  39  R^
  40  +
  41  /
  42  SQRT
  43  X<> 00
  44  RCL 02
  45  -
  46  360		   47  *
  48  RCL 01
  49  /
  50  STO 08
  51  STO 09         
  52  LBL 01
  53  RCL 09
  54  ENTER^
  55  SIN
  56  RCL 03
  57  R-D
  58  *
  59  RCL 08
  60  +
  61  STO 09
  62  -
  63  ABS
  64   E-3
  65  X<=Y?
  66  GTO 01
  67  RCL 09
  68  2
  69  /		   70  1
  71  RCL 03         
  72  ST+ Y
  73  ASIN
  74  COS
  75  /
  76  P-R
  77  LASTX
  78  /
  79  R-P
  80  X<>Y
  81  ST+ X
  82  RCL 07
  83  +
  84  1
  85  P-R
  86  STO 08
  87  X<>Y
  88  RCL 05
  89  COS
  90  *
  91  X<>Y
  92  R-P		   93  CLX
  94  RCL 06         
  95  +
  96  1
  97  RCL 09
  98  COS
  99  RCL 03
100  *
101  -
102  RCL 04
103  *
104  RCL 08
105  *
106  RCL Y
107  RCL 06
108  -
109  COS
110  /
111  END

 
    ( 135 bytes / SIZE 010 )
 
 

      STACK        INPUTS      OUTPUTS
           Y            /            µ
           X       YYYY.yy             r

 
Example:   The elements of  Gamma Virginis are:

       P = 168.68 years              a = 3.697"             OMEGA =  36.9°
       T = 2005.13                      i = 148.0°               omega   = 256.5°
       e = 0.885

-Calculate  r  and  µ  for the epoch  2010.25    ( ~ 2010/04/01 )

-Store these 7 numbers in registers R01 thru R07  ( in the order  P ,  T ,  e  ,  a  ,  i  ,  OMEGA  ,  omega  )

    2010.25  XEQ "BSTAR"  >>>>   r =  1.544"           ( in 33 seconds )
                                            X<>Y   µ =  19.66°         ( and the apparent eccentricity  e' = 0.844 in R00 )

Notes:

-If you don't need to compute the apparent eccentricity e' , delete lines 03 to 43.
-Execution time may tend to infinity as e tends to 1.

-Here are the elements of a few binary stars. Right Ascensions and declinations are referred to J2000
-m1 & m2 are the magnitudes.
 
 

   STARS   Right Asc        Decl        m1        m2         P         T         e         a         i   OMEGA    omega
   eta Cas   0h49m05s  +57°49'00"        3.7        7.5       480.0    1889.6      0.497     11.994       34.8     278.4    268.6
  alpha Psc   2h02m03s   +2°45'49"        4.3        5.3     933.050   2098.643      0.696      4.000      120.9       23.3    225.4
  alpha For   3h12m04s  -28°59'21"        3.8        7.0       269.0    1947.0      0.730      4.000        81.0     117.0      43.0
    Sirius   6h45m09s  -16°42'58"      -1.4         8.4      50.090   1894.130      0.592      7.500      136.5       44.6    147.3
   Castor   7h34m36s  +31°53'19"        2.0        2.9     467.000   1958.000      0.343      6.805      114.5       41.3    249.5
  Procyon   7h39m18s   +5°13'39"        0.4      10.8      40.650   1927.600      0.400      4.548       35.7     284.3    269.8
 gamma Leo 10h19m58s  +19°50'31"        2.6        3.8     618.557   1743.320      0.843      2.505       36.4     143.2    162.5
   xi Uma 11h18m11s  +31°31'45"        4.3        4.8       59.840   1995.050      0.412      2.533      121.2     100.9    126.8
 gamma Vir 12h41m40s   -1°26'58"        3.6        3.7     168.680   2005.130      0.885      3.697      148.0      36.9    256.5
  zeta Uma 13h23m55s  +54°55'31"        1.6        2.1       36.5    1994.8        0.6       0.80        93.0      32.0    287.0
   xi  Boo 14h51m24s  +19°06'02"        4.8        6.9     151.505   1909.361      0.512      4.904      140.0     168.1     23.9
  eta CrB 15h23m12s  +30°17'18"        5.0        5.3      41.585   1933.721      0.262      0.868       59.0     203.2     38.4
 sigma CrB 16h14m42s  +33°51'31"        5.7        6.7     888.989   1826.949      0.761      5.927       31.8      16.9     72.2
  zeta Her 16h41m17s  +31°36'07"        2.8        5.5       34.45    1967.7       0.46       1.33      131.0      50.0    111.0
   70 Oph 18h05m27s   +2°30'09"        4.3        6.0       88.38   1895.94      0.499      4.554      121.2     302.1     14.0
gamma CrA 19h06m25s  -37°03'46"        4.2        4.3     121.760   2000.640      0.320      1.896      149.6      50.3    349.0
  beta Cyg 19h30m43s  +27°57'35"        3.2        5.3      96.840   2010.270      0.719      0.586      118.0      98.8     77.2
   61 Cyg 21h06m54s  +38°44'58"        5.6        6.3     722.000   1689.700      0.401     24.650       51.8     172.3    158.0
  zeta Aqr 22h28m50s   -0°01'12"        3.7        3.9      587.18    1973.39      0.396      3.847      138.2     129.8    255.9

 

References:

[1]  Jean Meeus , "Astronomical Algorithms" - Willmann-Bell  -  ISBN 0-943396-61-1
[2]  Washington Double Star Catalogue.