Using the celestial sphere, it becomes quite easy to calculate the what the height of the sun at noon will be for any given day, if you know your latitude. To start off, I'll define a few terms.

Imagine the earth is encased in an arbitrarily large crystal sphere: this is the celestial sphere. The celestial equator is the circumfrencial line that would appear if the equator on Earth was expanded outward until it was resting on the celestial sphere. Another important circumfrence on the sphere is the ecliptic. This is the plane that most of the planets orbit in. If the rotational axis of the Earth was perfectly perpendicular to the ecliptic plane, then the celestial equator and the ecliptic line on the celestial sphere would be the same. However, the Earth's axis is angled 23.5 degrees away from perpendicular.

When looking up at the celestial equator on Earth, the peak of the arc it describes in the sky is directly related to your latitude. If you're standing on the terrestrial equator (latitude = 0), then the peak of the celestial equator is at an elevation of 90 degrees. If you're at the one of the poles (absolute latitude = 90), then the celestial equator's elevation is 0 degrees. The height of the celestial equator is always equal to 90 minus your absolute latitude.

The celestial equator is an important land (sky?) mark because its height when observed from Earth is always easy to calculate, and the postion of other celestial objects relative to it can also be calculated. On the two equinoxes, the plane of the ecliptic describes the same arc as the celestial sphere. During the solstices, the ecliptic varies from the celestial equator by 23.5 degrees, the amount by which the Earth's axis varies from the ecliptic. During the winter solstice, for those in the northern hemisphere, the height of the ecliptic will be 90 - your latitude - 23.5 .

Again, the height of the ecliptic is useful because all objects in its plane (i.e, the Sun), are found at that height. So if you know your latitude, and want to calculate the height of the Sun at noon during the summer solstice (and you live in the northern hemisphere), the elevation will be: 90 - your latitude + 23.5 degrees. Ta da!

"Aah," you say. "That's sort of interesting, but what would be a lot cooler would be if I could do that for any given day."

Well, that's not too much harder. Imagine taking some cosmic scissors and snipping the juncture between the ecliptic and celestial equator at the spring equinox. Now peel these two lines off the sphere, and lay them on a graph, with the celestial equator as the x-axis. If the y-axis is elevation, the ecliptic will form a sine wave, with an amplitude of 23.5 degrees. To calculate the difference between the ecliptic from the celestial equator, the function is (where x = number of days past the spring equinox):
Difference = 23.5 * ( sin ( x * 360 / 365)) Make sure you use degrees to calculate sin(x)!
The inside function ( x* 360 / 365) converts the day of the year to degrees. To find the actual height of the sun at noon, take the difference, and add it to the height of the celestial equator for your observing location. Voila!
Written with the northern hemisphere in mind... swap relevant parts for the southern hemisphere

For latitudes near the equator, realise that the value can get above 90 degrees. (This will occur between the equinoxes).

In this case, the actual height above the horizon will be 180 minus the calculated value, although the sun will now be in the north, not the south.

The equation can be extended to give the height of the sun above the horizon at any hour... (day is measured from January 1st, and hour is 0-24)

c = 90 - latitude The celestial sphere
altitude = -c*cos(360*hour/24)+23.5*sin(360*(day-80)/365)
If the result is negative, then you will be in some stage of twilight or night.

The equation assumes the following approximations: I have to acknowledge PhasedWeasel's writeup without which I wouldn't have had the inspiration to take it further.

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