From: werme@alliant.com (Ric Werme)
Newsgroups: rec.org.mensa
Subject: The Equation of Time
Date: 29 Jul 92 13:30:47 GMT
Organization: Alliant Computer Systems Corp.

I usually post this annually in sci.astro early in December in an attempt to
head of the annual question "Why is the earliest sunset weeks before the
latest sunrise?"  This has a lot of stuff Mensans might like to stash away
even though it isn't useful in Trivial Pursuit.  There is a little
astronomical jargon here that I don't define.  The meaning can be gleaned
from the context or a decent dictionary or encyclopedia.  If you still have
trouble, I can post a clarification.

In the northern temperate zone, the earliest sunset of the year occurs a
couple weeks before the winter solstice, and the latest sunrise occurs a
couple weeks after the solstice.  (Stranger things happen at the poles and
near the equator.)  The reason for this involves the Equation of Time.

Noon used to be when the sun crossed the meridian, the line going from the
North horizon, through the zenith (the point overhead) and down to the
south horizon.  The difficulties this caused with train schedules led to
standard time and the Earth's time zones.  I don't know who came up with
daylight time, but double daylight was installed during WWII as a conservation
measure.  Along with the transition from telling time with sundials to
telling time with clocks, people realized that the time from one noon to the
next was not constant.  Part of the jargon invented to deal with the situation
is "sun time", the time a sundial would tell, and "mean time", the time an
accurate clock would tell.  The varying part of the error has come to be
called the "Equation of Time", even though it is usually described as a
graph of error vs date, or as error vs. the sun's declination, or just as
a table.

It looks like:
                                                           **** +16:25
                                                        ***   ***
                                                       **       **
                                                      **          *
                                                     *             *
                                                   **               *
                                                  **                 *
                       ******** +3:43            **                  **
  Jan   Feb   Mar    **       ***     Jul   Aug **                    **
|-----|-----|-----|**---|-----|-**--|-----|----*|-----|-----|-----|----*|
*                 ** Apr  May  Jun***        ***  Sep   Oct   Nov   Dec *
**               **                 ****   ***
 **            **                      ***** -6:28
  **          **
   **       ***
    ****  ***
       ****  -14:16

The resolution is two minutes per line and five days per column.  The numbers
are the maxima and minima in minutes:seconds.  For the above, I used the
equation:

	E = E1 + E2, where
	E1 = -7.64 * sin(d - 2)
	E2 =  9.86 * sin(2*(d - 80)), where
	d is in days and sin() expects days (convert to rads by d*2*pi/365.24)

Please note this is a crude approximation and leaves out lots of detail.
In particular, the functions are not really sinusoids. They are
periodic, so they can be described by fourier transforms.  All I've done
is used the first term, which is accurate to a few seconds.  Just don't
try to reverse engineer the theory from my equation!

The E1 component is due to the eccentricity of the Earth's orbit.  The d-2
term reflects the perihelion early in January.  When the Earth is near
perihelion, its angular velocity with respect to the sun is higher than
average and the Sun's motion in the sky lags a bit.

The E2 component is due to the tilt of the Earth's axis.  The d-80 term
reflects the vernal equinox about 80 days into the year (31 in Jan., ~28 in
Feb., ~21 in March).  If the Earth were in a circular orbit, each sidereal day
the Sun's subsolar point would just under one degree.  When the sun was on the
equator, its motion would be 23.44 degrees off the equator, so the
longitudinal component would be cos(23.44).  When the sun was on the Tropic of
Cancer, then it would be moving at right angles to longitude lines, but they'd
be closer together and the sun would make up the lost time.  The effect is
exactly analoguous to the problem a universal joint has going through a large
angle.

An interesting thing about this component is what happens with increasing
tilt.  The real equation involves transforming the motion of the sun in the
ecliptic (which would be constant were the Earth's orbit a circle) into the
apparent motion relative the equator.  This transformation is:

	Q = atan( cos(tilt) * tan(C) ), where
	C is the position on the ecliptic (degrees from the spring equinox)
	Q is the equivalent motion along the equator
	tilt is 23.44 degrees, the tilt of the Earth.

cos(23.44) = 0.917, so Q and C don't differ by much.  However, the cosine
falls off rapidly with greater angles, and a greater tilt would result in
a highly disproportionate increase in its effect.

Is any of this worth remembering?  Beats me.  I got interested for reasons
too complex to go into here.  The points I like are:

1) If you apply the equation of time to sundial time, the result is remarkably
   accurate for a well designed, properly installed sundial.
2) The rate of change is greatest around the beginning of winter (both E1 and
   E2 have negative slopes) so the delay between earliest sunset and
   latest sunrise is greater than between earliest sunrise and latest sunset.
3) The effects the eccentricity and the Earth's tilt are comparable, with
   the tilt being the greater.
4) On the equator, the equation of time is the only significant adjustment
   in the time of sunrise, sunset, and the meridian passage.  This leads to
   one earliest sunrise, but two local minima.
5) I plotted the equation of time with minutes as a function of date.  If you
   plot it with minutes on the X axis and the sun's declination on the Y axis,
   you'll get an analemma.

Further references:
Mills, HR "Positional Astronomy and Astro-Navigation made Easy",
Stanley Thornes (Publishers) Ltd, ISBN 0-85950-062-4
   This is designed for people with calculators.  Mills doesn't carry the
   theory as far as I'd like, but the drawings and practical applications
   make this my main source of information.
Bate, RR et al "Fundamentals of Astrodynamics", Dover 1971 ISBN 0-486-60061-0
   Heavy on the theory, designed as an Air Force text, so there's a sizable
   emphasis on orbit determination, rendevous, etc.  In one of my postings a
   while back, I forgot to include an anecdote from here on why orbits are
   ellipses.  I should quote that whole section someday.
Menzel, Donald & Pasachoff, Jay "A Field Guide to the Stars and Planets",
(Vol. 15 of the Peterson Field Guide Series), Houghton Mifflin 1983,
ISBN 0-395-34835-8
   Color plate 71 is that great "natural" analemma photogragh by Dennis di
   Cicco, page 417 has a analemma with the dates marked along the figure 8.
Meeus, Jean "Astronomical Formulae for Calculators", Willman-Bell, Inc. 1982
ISBN 0-943396-01-8
   All numbers (like for the coeficients I skipped).  All facts, no theory.
   Velikovskyites: The eccentricity of the Earth's orbit is:
	e = 0.01675104 - 0.0000418T - 0.000000126T^2
   Where T is centuries since 1900.  I don't know the corresponding equation
   for Venus, but I'm sure a similar equation applies.  I don't know if this
   is accurate for more than 10 Kiloyears.
See also: The Old Farmer's Almanac, Byte Magazine (a few years back),
   publications of the US Naval Observatory.
-- 
| A pride of lions               | Eric J Werme                 |
| A gaggle of geese		 | 77 Tater St			|
| An odd lot of programmers      | Mont Vernon NH  03057	|
| A Constitution of Libertarians | Phone: 603-673-3993		|
Newsgroups: rec.org.mensa