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Omar Reis wrote-

>I mentioned the DeltaT because of  M. Dorl calculation below:
>
>> So, I get a time of 11:44:00 TDT dT = 95.4 for the
>> 11/18/1580 opposition TDT JDate 2298474.98889.
>
>I had previous problems with the DeltaT.
>For the date 1580, using J. Meeus AA formula:
>  DeltaT=50.6+T*(67.5+T*22.5)      (T in centuries since J2000)
>I got DeltaT=164, which differs from M. Dorl dT=95.4.

>As F. Reed said in another message, the DeltaT is a difficult value.
>I'm not sure how the formula for early DeltaT values in
>Meeus book was constructed. Methods for the DeltaT
>calculation from the same book are already diverging just 20
>years from publication. How someone could estimate what
>the value of DeltaT was 500 years ago scapes me.
>
>Omar

==============

Response from George,

There seems to be a lot of confusion about this matter, not Omar's fault.

I have Meeus' Astronomical Algorithms 2nd ed., 1998.

This has table 10A giving delta-T for various years from 1620, for which he
quotes +121 sec, to 1998. Unfortunately, this table says nothing about the
errors involved. Meeus' Table 10A differs significantly (for dates around
1620) from the numbers given in my 1984 Astronomical Almanac, which gives
+98 sec for 1621. Meeus states that the Astronomical Almanac values have
been "corrected" according to a new value for tidal acceleration of the
Moon, given in note S055, issued by the Bureau des Longitudes, Paris, in
1997, by Chapront et al. The tables agree, for more recent dates.

Meeus (1998) also gives equation 10.2 giving delta-T "from +948 to + 1600,
and after the year +2000". in seconds, as-
102 + 102 t + 25.3 t-squared,  (Meeus eq 10.2, on page 78)

where t is the time in centuries from year 2000, so for 1600, t = -4.0

This is quite different from the expression Omar quoted, as

 "DeltaT=50.6+T*(67.5+T*22.5)      (T in centuries since J2000) "

se perhaps we have different editions. Has that equation 10.2 (1998) been
subsequently revised in a later edition of Meeus? Otherwise, where did Omar
get it from?.

The two formulae give rather different results for the year 1600, the last
year that Meeus quotes for the period of validity of his equation 10.2
(1998).

Eq. 10.2 (1998) gives delta-t of +99 sec for the year 1600,which is very
out-of line with the trend in his table 4.2 of 121 sec in 1620 , decreasing
about 4 sec per year

The expression Omar quotes gives 146 sec for 1600, which fits in much
better with the trend of table 4.2.

As for late 1580, for which t = -4.19 centuries, eq 4.2 (1998) predicts 118
sec for Michael Dorl's date, whereas with Omar's formula he and I would
agree on 163 or 164 sec.

So where does Michael Dorl's value for delta-t in 1580, and where does
Omar's equation, come from?

=====================

The truth of the matter is that nobody really has much idea exactly what
delta-T was, back in 1580, or even 1620. It represents the discrepancy
between a precisely uniform time-scale (now defined by atomic time, until
something better supersedes it) defined according to the exact length of
the year 1900.0, and the non-uniform time-scale defined by the varying
rotation of the Earth.

In the days before atomic time, we had Ephemeris Time (more or less the
same thing). Only if we could measure time in exactly uniform intervals
could the predictions of celestial dynamics give the right answers. And so
Ephemeris Time was introduced, as a sort of fudge-factor to make the
predictions of the ephemeris fit the observed motions, because Earth
rotations were not a sufficiently uniform timekeeper. It was rather more
than a fudge-factor, though, because it showed up the physical reality of
the variations in Earth rotation speed.

The deviations of phenomena from the predictions show up most in the motion
of the Moon, which moves against the star background much faster than
anything else does.

To determine delta-T at a past era, say Tycho's in 1580, what was needed
was a recorded  observation which depended on the precise position of the
Moon (such as an eclipse or an occultation) and for which the precise
time-of-day (or night) had been noted. Observational astronomy was only
just reawakening in the West in Tycho's time, and timekeeping was, for him,
particularly uncertain. Did Tycho make any such useful observations? I
doubt it: if he did, there would be values for delta-T deduced for his era,
well before the 1620's where the table 10A starts.

Accurate timing to relate the time of an observation to, say, local
apparent noon, wasn't available until the development of the pendulum in
the later 1600's by Huyghens and Hooke.

I've read none of the papers describing how the historical analysis for
delta-T has been done, but my guess is at follows-

Collect together useful observations going back into time as far as
possible, which include ancient Arab and Chinese records, and even
Babylonian. For example, an account that the Moon rose or set during an
eclipse could give a useful, though imprecise, value for delta-T.

Analyse the resulting curve into two main components. The first, the tidal
slowing of the Earth, would be reasonably constant, and would result in a
parabolic variation of delta-T with time, which would be predictable,
future or past. The second, mainly due to internal motion within the Earth,
would have no long-term component, but a big short-term wander. That part
would be unpredictable, future or past. Because these random, short-term,
fluctuations are so significant, there must be quite a lot of guesswork in
the analysis.

George.

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