[time-nuts] Sidereal time

Magnus Danielson magnus at rubidium.dyndns.org
Sat Jan 16 02:47:45 UTC 2010

```J. Forster wrote:
> That's the point I was making earlier.
>
> Most telescopes have a FOV of at least 15 arc-minutes. You only need to
> get the guide stars into the field and go from there.
>
> Also, a telescope's pointing can be off in BOTH RA and Dec. Dec has
> nothing to do with siderial time.

While displaying Hour Minutes and Seconds of apparent local sidereal
time may be fun, the actual need is to calculate an angle in degrees and
minutes for which the object of interest position can be converted into
suitable pointing angle.

The simplest approximation can make use of the fact that on 365,25
normal days, there is 366,25 sidreal days. The error of that
approximation is 366,25/366,2425/365,25*365,2425 - 1 = -5,6E-8 days/day
or -1,211 arcmin per day or -0,44 arcmin per year. The Gregorian
correction was used for comparision value rather than a tabulated value,
but I was lazy to get a quick back-off-envelope type of result.

My point being that fairly simple approximate "gears" could be used to
give a good-enought result such that remaining drift can be compensated
using regular observation. Pointing towards known fix-stars for
calibration of local position, local pointing error and clock offset
would end up as a single correction factor of pointing angle correction.

The only thing one wants is that date, time and position sets the local
sidreal time close enought for manual correction to be a matter of minor

To convert the day (D) of a year into a sidreal day (DS) one gets
DS = D*366,25/365,25 = D + D/365,25

For hours we would use the relation HS = 24*DS, D = DI + H/24 and used
modulo 24

HS = 24*D + 24*D/365,25 = 24*DI + H + 24*D/365,25 = H + 24*D/365,25

Thus, the time of day is adjusted with the date, but there is no need to
calculate the full number of seconds. Similarly may the time of day be
converted to degrees.

AS = 360*DS mod 360 = 360*D + D*360/365,25 mod 360
= 360*DI + 15*H + DI*360/365,25 + H*15/365,25 mod 360
= 15*H + H*15/365,25 + DI*360/365,25 mod 360

H is then broken into HI, MI and SI for normal wall-clock
representation. This approximate convertion on the back-of-envelope
level has silently ignored the phase error, but retracing to the USNO
webpage should allow a more thorough calculation of a suitable offset.

The remaining mod 360 operation needs to handle the addition of three
0-360 degree ranges, so the range only needs to extend over 1080 degrees.

From the above formula it becomes apparent that the pointing needs an

So, a very coarse calculation could be good enought. A fairly trivial
calculation with a correction from date and fine-correction for hour and
minute may provide enought pointing precission. It is trivial to correct
for local offset from the UTC time using the GPS position.

Should be doable even for a tiny processor. Should be not too hard to
include into a motor control to keep the scope pointed to the right point.

Cheers,
Magnus

```