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From: ntt@dciem.UUCP (Mark Brader)
Newsgroups: net.astro
Subject: Re: Solstice =/= Earliest Sunset?
Message-ID: <597@dciem.UUCP>
Date: Wed, 4-Jan-84 13:28:05 EST
Article-I.D.: dciem.597
Posted: Wed Jan  4 13:28:05 1984
Date-Received: Wed, 4-Jan-84 15:43:01 EST
References: <4696@decwrl.UUCP>, <2139@allegra.UUCP>, <3540@hp-pcd.UUCP>
Organization: NTT Systems Inc., Toronto, Canada
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It appears that the explanations given by hpfcla!ajs (Alan Silverstein)
and allegra!karn (Phil Karn) are both correct; I checked a couple of books
and saw explanations like this:

	...the days are not of equal length.  The earth's orbit is
	elliptical, not circular, and the earth moves more rapidly
	when near the sun and more slowly when farther away.  Also,
	the earth's axis of rotation is tipped relative to the orbit.
	These two phenomena cooperate to make the `apparent' sun first
	run ahead of and then lag behind its average position.  To
	avoid the inconvenience of changing the rate of our clocks
	from day to day, we employ `mean' instead of `apparent solar
	time' for practical use.
		--Donald H. Menzel in `A Field Guide to the Stars & Planets'

Incidentally, the fact that the orbit is elliptical has another effect
besides the variation of orbital speed; it means that the sun is not at
the center of the orbit, but is about 1.5 million miles from it, i.e.,
at one focus.

I don't really understand how these effects combine to give the actual
result, though, which is more complicated than Phil Karn said.  In fact,
when I think about it, I don't understand why the axial tilt matters at all,
except perhaps to modify the *amount* of the effects of the elliptical orbit.
Those effects alone, I should think, will give longest and shortest apparent
days (i.e. noon to noon, nothing to do with longest and shortest periods of
daylight, which depend only on the axial tilt) at the aphelion and perihelion,
which is to say, the equation of time should be changing most rapidly then.

The actual equation of time, from Menzel's book, however, exhibits FOUR, not
two, lobes in the year, and all of them are unequal.  A table is given, which
I will summarize now.  Again, `equation of time' means apparent time minus
mean time, and is the correction to *subtract* from the time on a sundial
to give the actual time (after allowing for longitude and daylight time!).

I start at Feb. 14 when the ET is at its minimum.  All the dates are
estimated because the table is every 5 days.  However, it is clear that
neither the solstice/equinox dates of about the 21st of Mar./June/Sep./Dec.
(based on axial tilt) nor the aphelion/perihelion dates of about the 3rd
of Jan./July (based on elliptical orbit) appear directly in the table.
In the first column I give days from Feb. 14.

	0(365)	Feb 14	ET = -14.3 min.	major minimum
	 61	Apr 16	ET = 0		(and increasing 14 seconds/day)
	 90	May 15	ET = +3.7 min.	lesser maximum
	121	Jun 15	ET = 0		(and decreasing 13 seconds/day)
	165	Jul 29	ET = -6.4 min.	lesser minimum
	200	Sep  2	ET = 0		(and increasing 19 seconds/day)
	265	Nov  6	ET = +16.3 min.	major maximum
	315	Dec 26	ET = 0		(and decreasing 30 seconds/day)

Mark Brader