S.A.S. #3 Apparent Tilt Of The Earth (w/respect to the Sun) Equation

(You might wish to view SAS #5 before or with this vid).

Here is a basic equation for the (APPARANT) tilt of the Earth with respect to the 0 degree equator latitude line [more correctly, it is the tilt of the equator with respect to the solar plane or orbit around the sun]. You might want to look at my last vid. also.

The upper half of the graph is for the standard Sine of the angle. (I drew the graph sort of clumsy with my hands, but it's supposed to be smoother and even, but you can get the idea.)

Note, for reference and for calculating what day of the year it is (beginning with March 21):

Day 1 = March 21 (where the tilt is 0 deg.)
Day 91.25 = June 21
Day 182.5 = Sept. 21
Day 273.75 = Dec. 21

Notice that 365 is very close to 360 (the angle commonly associated with a full circle), and the tilt equation is a good approximation at:

Angle = 23.5 Sin (Day from March 21)

The value of 23.5 is simply a multiplier to adjust the resulting "output" range from 0 to 23.5; instead of 0 to 1 as for the std. sine (abreviated sin on calculators) equation. 23.5 degrees is the maximum tilt of the Earth. The northern Earth will tilt 23.5 degrees towards the Sun in the summer, and will tilt 23.5 degrees away from the Sun in the winter.

Now for a more correct formula, we will break 365 days into a range of 360 linear (equally distanced-spaced) steps as required for the complete std. sine graph (which has 360 linear steps of the angle). To do this we see that : 360/365 = 0.986301369

.. and that this multiplier value (factor) will be used as a correction for the formula.

So in other words: 365 x 0.986301369 = 360

As a test to see if you are utilizing your calculator right, here is an example to enter:

Ex. Lets say it's April 21 and you want to know the current tilt of the Earth. Since April 21 is 30 days from March 21, these arguments [current data] used in the formula result to:

Tilt Angle = 23.5 sin (0.986301369 x 30)
Tilt Angle = 23.5 sin (29.58904107)
Tilt Angle = 23.5 (0.493775549)
Tilt Angle = 11.604 degrees

Notice [especially visualized on the graph] that at this date that about half of the max. tilt of 23.5 degrees has already occured. The tilt is getting slower in amt. per day and will take nearly two more months to reach 23.5 degrees.

Perhaps this calculation can be used to adjust a sundial so that the time throughout the year is more accurate as the earth tilts. For example, the sundial for New York City is tilted somewhere about 40 degrees on March 21 since that is the latitude of NYC [more correctly, it is the angle NYC is tilted with respect to the solar plane]. On June 21, it is summer and this angle has decreased by 23.5 degrees to a total of only 21.5 degrees. The sundial should only be tilted 21.5 degrees then for correct time. I guess the positions of the "hour lines" would have to be recalculated and positioined on the sundial. I think there is another less common type of sundial where the hour lines need not be recalculated and positioned. Perhaps the gnome's (or the angle pointer thing) angle can be adjusted for each day or week of the year instead of redrawing lines or "adjusting" for the time by adding or subtracting some value.

I understand that I could be wrong about the exact equation to utilize, particularly the Sine equation. It appears to me that the tilt is sinusoidal in nature, but could be some other "curve" such as a cycloid maby [maby some astronomer or meteorologist can let me know.. that is if they even know themselves]. I have checked the results with a website that displays the current tilt and it appears to be very close.

As an extra note, the part of the curve above the horizontal line, and below the horizontal line are symetrical, hence basically the same. Now, taking either the part above or below the horizontal line, the curve is symetrical to the extreme points [peaks of the curve]. Knowing this, you only need calculate the tilt for a quarter of the year or 91.25 days and you can then reuse the data if you make a list. You could also say you only need to calculate this part since the earth tilting is periodic or repetative .. and that is what the sin function can describe.

Here is a website with some beautiful photos taken with a special camera of the apparent change in the angle of the sun: http://www.newscientist.com/gallery/mg20026761900-solargraphs-show-half-a-yea...

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