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    Re: Equation of Time Simplified?
    From: Richard B. Langley
    Date: 2013 Mar 11, 14:29 -0300

    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)
    > 
    > View and reply to this message: http://fer3.com/arc/m2.aspx?i=122773
    > 
    
    -----------------------------------------------------------------------------
    | Richard B. Langley                            E-mail: lang---.ca         |
    | Geodetic Research Laboratory                  Web: http://www.unb.ca/GGE/ |
    | Dept. of Geodesy and Geomatics Engineering    Phone:    +1 506 453-5142   |
    | University of New Brunswick                   Fax:      +1 506 453-4943   |
    | Fredericton, N.B., Canada  E3B 5A3                                        |
    |        Fredericton?  Where's that?  See: http://www.fredericton.ca/       |
    -----------------------------------------------------------------------------
    
    

       
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