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    Re: Eqn. of time
    From: Frank Reed CT
    Date: 2005 Jan 12, 18:46 EST
    Bill, you wrote:
    "Are the resulting times at Greenwich?"
     
    Yes. You saw, I think, that the fourth step in my earlier post was to test a case for 0h UT ("zero hours Universal Time" -- in other words, midnight at Greenwich for the day just beginning). Actually, there is a small wrinkle to worry about. The time variable in these calculations is actually dynamical time ("real" time) while UT wiggles around a bit because of the uneven rotation of the Earth. The difference amounts to about a minute right now, and it will not affect the outcome of  the EqT calculation so you don't need to worry about it in practice.

    And:
    "To calculate 2006 it appears one could:
    1. Continue the current spreadsheet, with 1 Jan, 2006 DT as 366
    2. Use a new Sun mean longitude and anomaly for Jan 1 2006 (which may, or
    may not be 280.7506 + 0.9856481*366 and 357.7244+0.9856003*366
    respectively)."
     
    Yep. Either will work. But the second choice is perhaps pointless extra work. The spreadsheet software can handle running sequences of serial dates very easily. Try this (I'm assuming you're using Microsoft Excel, but every spreadsheet has some similar technique): type the number 38353 in cell A1. Now put the formula "=A1+1" in cell A2. Then fill down (Edit menu, Fill, Down) from there for a thousand cells. You should end up with the formula "=A999+1" in cell A1000 and similarly for every cell in between. Nothing fancy so far: you've got a serial list of numbers. Next select all those cells (highlight them) and go to the "Format" menu. Choose "Cells" and then in the dialog that appears on the "Number" tab, select "Date". This tells the spreadsheet software to display these serial numbers as dates. The first cell should now "apparently" show "January 1, 2005" while it's still a number internally. Notice that the month rollovers, year changes, even leap years occur automatically. Next, in cell B1, enter the formula "=A1-38353" and fill down from there for a thousand cells. But this time reformat all the cells in column B as "General" Numbers. This is exactly that "DT" quantity that we need to calculate everything else. Now you can go ahead and put the formulas for L, M, and EqT in succeeding columns just as before. If you want to look at a different stretch of time, just change the single number in cell A1. It's all automatic. There's lots of other neat tricks you can do with date formatting and date functions in spreadsheets. It's worth poking around in the Help for Excel to learn more.

    And you wrote:
    "I have been unable to determine by inspection what the Sun's mean longitude
    actually means or how it is calculated for say, Jan. 1, 2005.  Can you shed
    some light on that for me please?"
     
    It's the Sun's mean ecliptic longitude. It's zero when the Sun is at the vernal equinox and increases steadily at the rate of 360/365.25 degrees per day. If you look at a star chart that shows the ecliptic with dates along it for the Sun's position you can see that you have to go about 280 degrees around to get from the "First Point of Aries" around to that spot in Sagittarius where the Sun is found on the first of the year.

    And:
    "3.  Same questions for the Sun's mean anomaly."
     
    The mean anomaly is an orbital parameter. It is similar to mean longitude but measured from the Sun's perihelion instead of from the equinox. You probably remember that perihelion occurs on Jan. 4 which is why the mean anomaly is just shy of 360 degrees on Jan.1, 2005.
     
    These two parameters, mean longitude and mean anomaly, are sufficient to calculate EqT at this level of accuracy because the equation of time has two basic causes. First, the Sun actually travels faster across the sky when we are at perihelion. That means that the Sun "gets ahead" of its mean position during some parts of the year and "falls behind" at other times. Second, because the ecliptic is tilted, the mean Sun's steady motion is divided into portions which are north-south and east-west. When the Sun is near the solstices (horizontal parts of the ecliptic on a star map), its full motion is parallel to the equator. But near the equinoxes only a fraction of the motion is parallel to the equator. There are lots of interesting details and animations on this topic at a web site which has been around almost since the w.w.w. got started. It is www.analemma.com.

    -FER
    42.0N 87.7W, or 41.4N 72.1W.
    www.HistoricalAtlas.com/lunars
       
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