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Re: Time of meridian passage accuracy, Smart, and Cotter.
From: George Huxtable
Date: 2009 Sep 29, 11:37 +0100

```Douglas Denny misrepresents my words, in writing "You are too quick to
deride Smart's formula" My words, in [9966] were-.

"So my warning stands. Do not take the values from Smart equation 32, if you
want precise answers for Equation of Time. Instead, go to Meeus, where you
will find all the details to get it exactly right."

No "derision" there, A simple statement of fact.

He continued- "Meeus in fact suggests using it - and gives it as an
alternative to his first formula."

Yes-exactly as I had pointed out to him, when I wrote-
"That equation has also been reproduced in Meeus, as eq. 28.3. Meeus quotes
Smart's 1956 edition, presumably the copy that was on his shelf. I expect
it's also in the same form in Douglas Denny's 1977 edition of Smart, as it
applies always. Meeus gives a way to calculate precisely the "constants" in
that equation, as they are not actually constant, but vary slowly with
time."

Douglas continued- "Smart's equation can incorporate the modern constants
easily enough, and does so for the most important parameter (Mean longitude)
used in Smart's anyway."

Well, Sun's mean longitude is nothing more than a measure of time. It's the
other changing parameters that are important, as I have listed: obliquity,
eccentricity, longitude of perihelion. Smart doesn't tell you how to
recalculate the current values of those, but Meeus does.

It turned out, from Douglas' own words in [9963], where he wrote- "The mean
longitude of the Sun I used in Smart's formula was calculated with up to
date parameters for obliquity and nutation included.", that he had not
himself taken Smart's equations at face value, just in accordance with my
warning.  Instead, he took equation 29, which is true always, and updated
certain parameters to the present day, just as I stated was necessary to get
precise answers. Indeed, the accuracy of Smart's equations (from either
edition) can't be maintained to the present-day, without such updating.
Where were Douglas' new values taken from, then? Douglas mentioned only
obliquity and nutation, but in addition changes occur in eccentricity of the
Earth's orbit, and particularly in the longitude of perihelion, the
parameter to which EoT is most sensitive. All need to be allowed for, to
preserve precision long-term.

Now for a bit of real pedantry-

Although I wrote above that Smart's equation 29 is true always (when the
current parameters are plugged in), that is only approximately the case.
Perturbations, from the other planets and from the Moon, can shift the Sun's
GHA by (rarely) up to 0.2 arc-minutes. Such effects were ignored by Smart,
so his approach can hardly be described as "rigorous".  That explains why
calculating by his formula is so much simpler than Meeus' full-works, which
include many, many terms from Bretagnon. However, I suggest those
perturbations are small enough to allow we-mariners to discard them. And
then, Smart's predictions did not include the effects of nutation, which can
displace the Sun's GHA by up to about 0.3 arc-minutes.. Mariners often
ignore nutation; fair enough if they're aware of that approximation. I.too,
have ignored both effects in this discussion, because being cyclic
phenomena, they don't accumulate up. Unlike the secular changes that were
the object of my postings, which accumulate indefinitely.

George.

contact George Huxtable, at  george@hux.me.uk
or at +44 1865 820222 (from UK, 01865 820222)
or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.

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