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Re: Equation of Time Simplified?
From: Richard B. Langley
Date: 2013 Mar 11, 14:44 -0300

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On 2013-03-11, at 2:32 PM, Richard B. Langley wrote:

>
> 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
> > ----------------------------------------------------------------
> > NavList message boards and member settings: www.fer3.com/NavList
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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/       |
> -----------------------------------------------------------------------------
>
>
>
> View and reply to this message: http://fer3.com/arc/m2.aspx?i=122775
>

-----------------------------------------------------------------------------
| 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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