# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**Re: Equation of Time Simplified?**

**From:**Richard B. Langley

**Date:**2013 Mar 11, 14:44 -0300

If your e-mail program displays some of the text in the previous message (reproduced below) as a question mark, interpret that to mean the degrees symbol. 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 > > Members may optionally receive posts by email. > > To cancel email delivery, send a message to NoMail[at]fer3.com > > ---------------------------------------------------------------- > > > > 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/ | -----------------------------------------------------------------------------