How to Calculate The Sun's Altitude

How to Calculate The Sun's Altitude
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You may have heard in your travels that at noon, the sun is "directly overhead" in the sky. Unless you happen to be at or north of the Arctic Circle, this is technically never the case. Not only that, but unless you live at Earth's equator, the highest position above the horizon reaches each day ​​that is, the sun's altitude ​​ varies slightly from day to day over the course of the year.

The sun's altitude in degrees depends on two factors: your distance from the equator and the date.

Step 1: Get Situated

Your latitude is a number between 0 degrees (if you live at the equator) and 90 degrees (if you live at the North or South pole). Most people in the United States are between 25 degrees north latitude and 45 degrees north latitude. Because the Earth's circumference is about 25,000 miles and there are 360 degrees in a circle, each degree of latitude works out to a little less than 70 miles.

If you don't know your latitude, visit the NASA Latitude/Longitude Finder (see Resources) and enter your location. For example, Boston, Massachusetts, USA is at 42.36 degrees north latitude.

Step 2: Determine the Sun's Equinox Altitude

The Earth is tilted 23.5 degrees from a line perpendicular to its plane of rotation, like a spinning top that has begun to wobble. This is what causes the seasons, and is also the reason the sun's highest altitude varies. On about March 22 or 23 and again on about September 22 or 23, Earth passes through an equinox -- Latin for "equal night." On these two days, the Earth gets 12 hours or light and 12 hours of darkness, and the sun climbs to an altitude equal to:


in degrees above the horizon. In the case of Boston, then, this is:

\text{altitude}=90-42.36 = 47.64

47.64 degrees above the horizon, which is just over halfway to the zenith (the point directly overhead).

Step 3.  Determine the Sun's Solstice Altitudes

Starting on the vernal equinox in the northern hemisphere on March 22 or 23, the first day of spring, the amount of time the Earth spends in light continues to increase, and the sun climbs to a progressively higher point each day. After three months, on June 22 or so, the summer solstice, the first day of summer and the so-called "longest day of the year," arrives. Because of the 23.5-degree tilt mentioned above, the sun at noon in Boston is now:

(90 - 42.36 + 23.5) = 71.14

71.14 degrees above the horizon. This is about 80 percent of the way from the horizon to the zenith.

Six months later, on December 22 or 23, the autumnal equinox has come and gone and the winter solstice arrives. On this day, the first day of winter and the so-called "shortest day of the year," the situation from summer is reversed, and the sun only reaches an altitude of:

(90 - 42.36 - 23.5) = 24.1

24.1 degrees. This is just over one-fourth of the distance from the horizon to the zenith.

Step 4: Factor in the Declination for Today

The variation owing to the Earth's tilt is called the declination. It is a positive number in the spring and summer and a negative number in the fall and winter, varying between the values of 23.5 and -23.5 degrees.

The equation for calculating the altitude above the horizon on any given day is:


In our initial examples, on the equinoxes, ​D​ was zero and was therefore not explicitly included.

To determine the declination for today and the sun's altitude, you can use the NOAA Solar Calculator or the Kiesan Calculator, both online. If you don't have access to one of these, you can make a decent guess as long as you know the date and your approximate latitude. For example, if it's early May and you're in Miami, Florida, you know that the sun's declination is about halfway between 0 and 23.5 because spring is half over, and your latitude is about 25 degrees. Therefore, you can estimate that the sun will climb to an elevation of about:

(90 - 25 + 11.5) = 76.5 \text{ degrees}

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