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Interesting, deductive development.

When it comes down to it, virtually everything in celestial mechanics can be 
expressed using equations, albeit at times with limited precision, and/or 
extremely long ones. Some of the equations are intuitive, some less so.

The equation of time, E, can be written in equation form as (see Meeus, Chapter 27):

E = L_o - 0.0057183� - alpha + deltaPsi * cos (epsilon)

where:
L_o = sun's mean longitude (given by planetary ephemerides; also representative by an equation)
alpha = apparent right ascension of the sun
deltaPsi = nutation in longitude (given by the nutation theory and a very long equation)
epsilon = obliquity of the ecliptic (can be represented by a polynomial)

The Explanatory Supplement to the Astronomical Almanac, gives a low precision 
version of the equation of time as

E = -1.915� * sin(G) - 0.020� * sin (2G) + 2.466� * sin(2*lambda) - 0.053� * sin(4*lambda)

(nicely showing the annual and seasonal variations)

where:
G = 357.528� + 35999.050� * T
L = 280.460� + 36000.770� * T
lambda = L + 1.915� * sin(G) + 0.020� * sin(2G)
T = number of Julian centuries from J2000.

Wikipedia also gives some expressions for computing EoT:
http://en.wikipedia.org/wiki/Equation_of_time

-- Richard Langley

On 2013-03-11, at 1:49 PM, George Brandenburg wrote:

> I have always been intrigued by the Equation of Time, especially at the time 
of year when the earliest sunset and the latest sunrise occur a couple of 
weeks before and after the shortest day. When I took Frank's course a few 
years ago I learned how to use the EoT and what it looked like plotted as a 
function of time, and at some point I learned that it resulted both from the 
elliptical shape of the earth's orbit and the tilt of the earth's axis. But I 
had the impression that the EoT was the result of a fairly complicated 
calculation that couldn't be written as an actual equation. And a quick look 
through the Wikipedia entry for EoT didn't dispel this notion.
> 
> So recently a favorite pastime has been to try and visualize the effects 
that lead to this time shift and how they change over the course of the year. 
(It's actually been a great way to get to sleep after I go to bed!) I found I 
could come up with relatively simple explanations for the two contributions 
that only depended on basic geometry and physics. In fact the only part I 
couldn't work out in my head was the spherical geometry needed for the axis 
tilt effect - somehow this was never included in my education as a physicist. 
But a quick look in Bowditch provided the necessary spherical trig formula.
> 
> Once I had a formulas for both effects in mind I put them into excel 
together with a calculation of the EoT from NOAA, and I was pleased to see 
that they agreed almost perfectly! So my next exercise was to write up what I 
had done in case any other Cel Nav geek might be interested:-). I also 
generated a couple of figures to help explain what I was doing. The end 
result is attached, and yes it does conclude with a relatively simple formula 
for the EoT.
> 
> If this is of interest to any NavListers I'd love to hear your comments, criticisms, etc.
> 
> Cheers,
> George B
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> Attached File: 
> EoT-Simplified.pdf (no preview available)
> 
> : http://fer3.com/arc/m2.aspx?i=122773
> 

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| Richard B. Langley                            E-mail: lang@unb.ca         |
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