On Fri, 3 May 1996, Jack Aubert wrote:
> I would like to forward the following enquiry to all those dialiasts
> who know more than I do about celestial mechanics. It seems to me
> that the answer has something to do with the fact that the sun appears
> as a disk rather than a point in space, but I don't know enough to think
> this through logically. The question was posed by Rory Sellers
> ([EMAIL PROTECTED]). What exactly does he have to do to subscribe
> to the sundial list?
>
> >Many years ago a golf-playing uncle in Los Angeles asked me why it was
> >that the shortest day of the year was the winter solstice, but the sun
> >actually set earliest on about Dec. 13 (he had noticed that he could get
> >in an extra hole of golf on Dec. 21 -- I'm not making this up.)
> >
> >After a lot of research (there was no Web then!) I realized the answer
> >had to do with the equation of time, the difference between solar and
> >sideral days, and mean solar time (and no doubt you know more about this
> >than I do.)
> >
> >So I told my uncle this, and the family got down to accomplishing other
> >life-tasks. Unfortunately, I recently realized that this explanation
> >must be false. This came about by looking more closely at an analemma on
> >a globe. If I am not mistaken, the analemma describes not only the
> >difference between mean and clock-time (due to the eccentricity of the
> >earth's orbit being non-zero) but also the (I guess arbitrary) dates on
> >which our clock is "set." I.e. when the official timekeepers say that
> >the annual clock is zeroed. And here, I was shocked to see a) that this
> >date seems to be the winter solstice; and b) that I had never noticed
> >this before!
> >
> >You see, I always knew that the earth was moving fastest around
> >Christmas time (near perihelion) and so I figured it made sense that the
> >clock was furthest "out of whack" around this time. And sure enough,
> >consulting either an almanac or a St. Joseph's Aspirin calendar, one can
> >see that the sun indeed sets earliest on Dec. 13, not Dec. 21. BUT HOW
> >IS THIS POSSIBLE, IF THE CLOCK IS "ZEROED" ON THE SOLSTICE? Shouldn't
> >the equation of time being equal to zero on Dec. 21 mean that the sun
> >should set the earliest on that date, too?
> >
> >Please help! A family conundrum that I thought we had settled twenty
> >years ago is now bothering me!
I believe it is the equation of time that is responsible for the
difference. Let's consider the sun to be a point (taking into account the
sun's semi-diameter won't affect us too much). When the sun is on the
horizon (actually sunset occurs when the refracted upper limb of the sun
is on the horizon, but as I said we won't worry about this detail), the
local hour angle of the true sun is given by
cos(H) = -tan(lat) * tan(delta)
where lat is the latitude of the place and delta is the sun's declination.
The hour angle of the true sun is equal to the hour angle of the mean sun
plus the equation of time and the hour angle of the mean sun is equal to
mean solar time minus 12 hours. So,
H = H_m + E = MT -12h + E/60
where E is in minutes.
So, MT of sunset, in hours, is given by
MT = 12h - E/60 + acos(-tan(lat)*tan(delta))/180*12.
Now, consider first a point on the equator:
MT = 12h - E/60 + 6h = 18h - E/60.
Notice that this equation is independent of the sun's declination,
except via E. The earliest sunset will occur when E is largest. This
happens around 3 November.
If we re-do the calculation for Los Angles, with latitude of about 34
degrees, we find that the earliest sunset occurs around 6 December. Note
that we have to take into account both the time variation of E and the
sun's declination.
===============================================================================
Richard B. Langley Internet: [EMAIL PROTECTED] or
[EMAIL PROTECTED]
Geodetic Research Laboratory BITnet: [EMAIL PROTECTED] or
[EMAIL PROTECTED]
Dept. of Geodesy and Geomatics Engineering Phone: (506) 453-5142
University of New Brunswick FAX: (506) 453-4943
Fredericton, N.B., Canada E3B 5A3 Telex: 014-46202
Fredericton? Where's that? See: http://degaulle.hil.unb.ca/NB/fredericton.html
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