How would I find the angle of the rising sun?

Factual Questions
1

I would like to know the easiest way to figure out where the sun will rise on any given day at my house. Well actually only on a couple of days, mainly the solstices. I figure that I can take a couple of sticks and do it, but I’m sure by now there is a mathmatical way to do it.

I am around 34°21’N 098°59’W, though I want to know how to do it myself. Also if there are any good web sites, I don’t know what to look for or I would have, or books on the subject of the movment of the sun/moon.

Though I hope I get good weather next week so I can stand out in the cold and put some sticks in the ground.

2

The US Naval Observatory has some information on this, and has forms to do the calculation for you.

The “Equation of Time” (google that) is kind of a pain to work with, but some people like that kind of thing.

3

On the solstices, the sun rises due east and sets due west. More generally, if we call the angle from north of the sunrise beta, then the angle of sunrise is given by

cos(beta) = sin(delta) / cos (lambda)

where delta is the declination of the sun on the day in question (positive = N, negative = S) and lambda is your latitude. If the right-hand side of the equation ends up greater than 1, then that means that the sun doesn’t set that particular day; if it’s less than -1, then the sun doesn’t rise. The declination of the sun is essentially how far north or south the sun is from the celestial equator. You can look up the declination of the sun for any day of the year at this website.

If all you’re concerned about is a formula you can use, then stop reading. What follows is a derivation of the above equation:

To derive this equation we need to use spherical trigonometry. Draw the celestial hemisphere and label three points: P, the North Celestial Pole; Z, the zenith; and S, the location of sunset. We know that the length of arc PZ is 90 - lambda; the length of arc PS is 90 - delta; and the length of arc ZS is 90. The angle we want to find is beta, the angle opposite the arc PS. So we can use the law of cosines for spherical triangles to say that

cos(90 - delta) = cos(90 - lambda) cos(90) + sin (90 - lambda) sin(90) cos(beta)

Applying standard trigonometric identities yields the above formula.

Oh, and by the way, the derivation these kinds of things is called Positional Astronomy.

Hope all of this helps.

4

I think you mean, on the equinoxes.

5

Gah! Of course I do. Thanks for the correction.

6

I am confused about where your house is. Is it at Fredrick Municipal Airport in Kansas, or in “Location: Maryland”?

7

How the hell did you know I used an airport for the lat and lon? Well I made a huge mistake, I pulled my IFR book and looked up Frederick Muni, and didn’t even think about the lat/lon being that far off and it never dawned on me that there could be two Frederick Munis. The correct ones should be 39°25’N 77°22’W. That’ll teach me to read better.

:wally

8

Mapquest has an option to enter lat & lon for position. Your coordinates were smack dab in the middle of a rural Kansas airport. With the resolution of your coordinates being roughly +/- half a mile, your house must have been on airport grounds.