SpaceBanter.com - View Single Post - formula for astronomical midnight at Greenwich, in UT?

**Greg Neill[_6_]** · September 22nd 10, 12:34 PM posted to sci.astro.amateur,uk.sci.astronomy

wrote:
On Sep 22, 3:05 am, William Hamblen
wrote:

On Tue, 21 Sep 2010 09:00:24 -0700 (PDT),
wrote:

Hi, I mentioned this in another thread but thought I'd give it a
thread of its own.

Can someone help with a formula for astronomical midnight at
Greenwich, in UT, given the Julian Day Number? I.e. the UT of the
first astronomical midnight (lower culmination of the apparent Sun)
following JDN = x.

Ideally I need an accuracy of a few seconds.

You want to get Jean Meeus' book on Astronomical Algorthims. The
answer is from chapter 11.

T = (JD - 2452545.0)/36525

mean sidereal time in seconds = 24110.64841 + 8640184.812866*T +
0.093104*T^2 - 0.0000062 * T^3.

This works only for 0 h UT.

Hi and many thanks Greg and Bud.

I've now got hold of Jean Meeus's book - what an amazing source! Going
by my limited but hopefully growing understanding, I think the
equation of time may be the way to go. (I'm not sure whether nutation
needs to be taken into account, to get an accuracy of a few seconds
rather than 0.01 seconds or so).

There is also the following formula with accuracy of around half a
minute:
E (in minutes) = 9.87 sin 2B - 7.53 cos B - 1.5 sinB
where B (in degrees) = (360/365) * (N - 81) (in degrees)
and N is day number, counted from B=1 at 1 Jan

This seems to work to give astronomical midnight in UT when subtracted
from 00:00, although the accuracy is an order of magnitude less than I
need.

It's a simple curve fit to the equation of time for a
particular epoch. Because the perihelion of the Earth's
orbit shifts over time, the "shape" of the E of T curve
changes over time too, and the accuracy will not be
maintained as one strays from the epoch.

Once I plug the formulae from chap.27 of Meeus into an equation, would
it be OK simply to subtract from UT like this, or would this lose too
much accuracy?

On p.173 Meeus also gives an equation (27.3) which seems to be a
version of this but with more accuracy, although it's unclear how much
more.

It should be pretty good (probably within a few seconds I'd
guess), since it calculates the parameters for the curve
fit from determinations of the current orbital parameters.

(BTW I'm doing this only for the Greenwich meridian; longitude
correction is simple).

Bud - am I right to think that if I used the formula you posted for
GMST, I'd need to keep a table of the UT of the vernal equinox each
year? Or am I barking up completely the wrong tree? Excuse my newbie
ignorance, but I don't understand how to use that formula to get
astronomical midnight on any particular day of the year, in UT.

That is one way to pin down the UT versus Dynamical Time
relationship for a given year. Another is to keep a
table of Delta-T values, or calculate Delta-T values from
a curve fit. Future values of Delta-T can really only
be roughly predicted, since it depends upon many dynamical
factors affecting Earth rotation which accumulate over
time. See Meeus chapter 9 (chapter 10 in the newer
editions).