 Amsat-UK's Oscar News, 1991 Feb No. 87 p 16-17

# 1991, 1992, 1993 ..... AND ALL THAT

### by James Miller G3RUH

```
19xx January is that time of year for the annual pleas, by users of
ancient software, that they've run out of "sidereal time data".

This quantity is not some arcane kludge divined by sorcerers from the
entrails of slide rules, but a simple quantity that describes the
Terrestrial longitude of the Celestial 0 deg longitude at 0000 utc.

Some satellite tracking software needs this for 19xx Jan 0.0.  Look at the
following table:

YEAR     GHAA(0) deg      GMST rev
----------------------------------------
1989      99.636681       0.276769         This means that at the
1990      99.397970       0.276105         beginning of 1989, the
1991      99.159257       0.275442         Celestial 0 deg of longitude
1992      98.920546       0.274779         will be over 99.6 deg Earth
longitude (Mexico City).
1993      99.667481       0.276854         As the Earth rotates it
1994      99.428769       0.276191         moves West at a rate of
1995      99.190057       0.275528         approx. 15 deg/hour.
1996      98.951346       0.274865

1997      99.698282       0.276940
1998      99.459570       0.276277
1999      99.220859       0.275613
2000      98.982147       0.274950
-------------------------------------

There are some obvious rhythms in it.  The angular measure (Greenwich Hour
Angle Aries Jan 0.0). is roughly 99 degrees.  It decreases by 0.238711
degrees for three years, and then increases by 0.746936 after the fourth,
which you will notice is a leap year.  So, you can always get next year's
value from last year's.

Where do -0.238711 and +0.746936 come from?  Easy! The Earth rotates
360. 985647367 degrees per day; 360 degrees on its own axis, plus the
little bit (0.9856 deg) because it revolves around the Sun too.  So in a
leap year of 366 days it turns through 366 * 360.985647367 = 132120.746936
degrees, which is just 367 revolutions + 0.746936 deg.  In a 365 day year
it's 366 rev - 0.238711 deg.

OK, so now you want a formula for all this.   Mechanising the rhythm just
described gives:

GHAA(0) = 99.6367 - 0.2387*(YR-1989) + 0.9856*INT((YR-1989)/4)   degrees

where GHAA(0) is the Greenwich hour angle for year YR Jan 00, 0000utc, and
YR = is four digits, e.g. 1991, 1992 etc.

Divide this value by 360 if you want it in revolutions.

This is perfectly adequate for satellite work.  But perfectionists will
want to account for the fact that the year gets longer as the Earth slows
down.  By Astronomical convention, the adopted formula is  (Ref 1):

GMST = 24110.54841 + TU*(8640184.812866 + TU*(0.093104 + TU*(-6.2E-6)))
seconds

For degrees, divide by 240, for revolutions divide by 86400.

Tu is the interval of time, (measured in Julian centuries of 36525
days of universal time), elapsed since the epoch 2000 Jan 1.5
i.e TU = (JD - 2451545.0)/36525

GMST is Greenwich Mean Sidereal Time.  (Astronomers  are historically stuck
with using units of time for longitude.   The rest of us are stuck with
degrees. 24 hrs = 360 deg).

Now, this is pretty tedious to evaluate, so I've given a short program to

I want to make it clear that modern tracking software doesn't need any of
this mumbo-jumbo.  It will have the celestial longitude initialised as 99.6
degrees on 1989 Jan 0.0 and can calculate any future value quite simply
from this.

The writers of software that ask you to put in a value for each
year from the table must have regretted ever since.  PLEASE give them a
break - work it out for yourself!

Ref 1: Aoki S., Guinot B., Null G.H., Kinoshita H., McCarthy D.D.,
Seidelmann P.K.;   Astron. Astrophysics, 105, 359.  1982

10 T\$="B.GHAA(0)": REM     GMST of 0000 utc Calculator.  Based on
20 REM                   1984 Astronomical  Almanac Pages B6 & S13.
30 REM
50 REM
60 REM                 (C)1991 J R Miller G3RUH
70 REM
80 REM GMST =24110.54841 + TU*(8640184.812866+TU*(0.093104+TU*(-6.2E-6)))
90 REM (seconds).
100 REM
110 REM  where Tu is the interval of time, (measured in Julian centuries
120 REM  of 36525 days of universal time), elapsed since the epoch
130 REM  2000 Jan 1.5   i.e TU = (JD - 2451545.0)/36525
140 REM
150 FI%=&80F: FF%=&2060F: REM  Print formats;  #########   #.######
160 :
170 PRINT"Program ";T\$;"    GHAA for Year yyyy Jan 0.0     (C)1991 G3RUH"
180 PRINT
190 PRINT"           YEAR      GHAA deg        GMST rev"
200 PRINT"         -------------------------------------"
210 FOR YR = 1989 TO 2000
220   DD = FND0(YR)-(FND0(2000)+1.5):         REM Days since 2000 Jan 1.5
230   TU = DD/36525
240   G1 = TU*(184.812866+TU*(0.093104+TU*(-6.2E-6)))/86400 :REM Small part
250   G2 = TU*100                                           :REM Big part
260   G3 = 0.2790572733                              :REM 24110.54841/86400
270   GMST = G1+G2+G3: GMST = GMST-INT(GMST): GHAA = GMST*360
280   @%=FI%: PRINT YR;: @%=FF%: PRINT GHAA,GMST
290   NEXT
300 STOP
310 DEF FND0(YR) =INT((YR-1)*365.25)

```

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