

Instructions  close window 
You should enter as much information as possible into the Inputs section, then click on the Calculate button. The calculation results will then appear in the Outputs section. Here is a list of inputs and how to use them:
When you've entered all of the inputs, click on the Calculate button, and the results will appear in the Outputs section immediately. Note that you can click on the labels of any input or output field to learn more about it. 

Altitude Angle  close window 
The altitude angle (sometimes referred to as the "solar elevation angle") describes how high the sun appears in the sky. The angle is measured between an imaginary line between the observer and the sun and the horizontal plane the observer is standing on. The altitude angle is negative when the sun drops below the horizon. (In this graphic, replace "N" with "S" for observers in the Southern Hemisphere.)
The altitude angle is calculated as follows:
where:


Azimuth Angle  close window 
The solar azimuth angle is the angular distance between due South (see note below) and the projection of the line of sight to the sun on the ground. A positive solar azimuth angle indicates a position East of South, and a negative azimuth angle indicates West of South.
The azimuth angle is calculated as follows:
where:
The sign of the azimuth angle also needs to be made equal to the sign of the hour angle when using the above equation. Note that SunAngle now allows for North to optionally serve as the zeroazimuth instead of South. This can be selected in the inputs section above. Thanks to Oddbjorn Grandum for the updated azimuth angle calculation, which is accurate for results > 90 degrees, and to Terry Leier for additional refinements! 

Time Basis  close window 
There are two important ways of describing time when calculating sun angles. "Clock time" is the artificial time that we use in everyday life to standardize our time measurements. It allows people in different locations to use the same time or to easily convert time from one location to another. "Local solar time" (or simply "solar time") is the time according to the position of the sun in the sky relative to one specific location on the ground. In solar time, the sun is always due south (or north in the southern hemisphere) at exactly noon. This means that someone a few miles east or west of you will realize a slightly different solar time than you, although your clock time is probably the same. For the purpose of calculating local solar time, clock time must modified to compensate for three things: (1) the relationship between the local time zone annd the local longitude, (2) daylight savings time, and (3) the earth's slightlyirregular motion around the sun (corrected for using the equation of time). Local solar time (LSoT) is calculated as follows:
Where:
Note that if the site is east of the LSTM, the (LL  LSTM) factor should be a positive number, and if it is west it should be negative. The "4" in the equation is the quotient of 60 minutes of time and the 15 degrees of longitude that the earth rotates in that time (i.e., the earth rotates one degree every four minutes). To convert from LSoT to clock time, perform the reverse of this formula. In the input section of SunAngle, you should indicate whether the time you entered is based on clock time or solar time. The output section then includes both clock time and solar time for reference. 

Hour Angle  close window 
The hour angle is an expression describing the difference between local solar time and solar noon. Although it is calculated directly from measurements of time, it is expressed in angular units (degrees). The hour angle measures time before solar noon in terms of one degree for every four minutes, or fifteen per hour. Time after solar noon is expressed using a negative hour angle. Therefore, at two hours before solar noon the hour angle is 30 degrees, and at two hours after solar noon it is 30 degrees. This chart depicts the relationship between the hour angle and local solar time ("N" represents North but would be South in the Southern Hemisphere).
Note that the hour angle does not represent the altitude of the sun in the sky because it is a measurement of time. The sun rises and sets when its altitude is zero degrees, not necessarily when its hour angle is +/ 90 degrees. 

Declination  close window 
Declination is the angular distance of the sun north or south of the earth's equator.
The earth's equator is tilted 23.45 degrees with respect to the plane of the earth's orbit around the sun, so at various times during the year, as the earth orbits the sun, declination varies from 23.45 degrees north to 23.45 degrees south.
This gives rise to the seasons. Around December 21, the northern hemisphere of the earth is tilted 23.45 degrees away from the sun, which is the winter solstice for the northern hemisphere and the summer solstice for the southern hemisphere. Around June 21, the southern hemisphere is tilted 23.45 degrees away from the sun, which is the summer solstice for the northern hemisphere and winter solstice for the southern hemisphere. On March 21 and September 21 are the fall and spring equinoxes when the sun is passing directly over the equator. Note that the tropics of cancer and capricorn mark the maximum declination of the sun in each hemisphere. Declination is calculated with the following formula:
Where:
Note: SunAngle currently uses a more sophisticated algorithm for declination calculations than this method, but the formula on this page suffices for most applications. Please view the source code of this page if you'd like to review the algorithm actually being used. 

Equation of Time  close window 
The equation of time (EOT) is a formula used in the process of converting between solar time and clock time to compensate for the earth's elliptical orbit around the sun and its axial tilt. Essentially, the earth does not move perfectly smoothly in a perfectly circular orbit, so the EOT adjusts for that. Graphically, it appears as: For example, the EOT adjustment in midFebruary is about 14 minutes. So when converting clock time to local solar time, you'd subtract 14 minutes. When converting from local solar time to clock time, you'd add 14 minutes. The EOT can be approximated by the following formula:
Where:
Where:
Note: The SunAngle program currently uses a more sophisticated algorithm for EOT calculations, but the above formula is a decent approximation and much simpler. Note also that the EOT output is in hours, so please multiply by 60 if you'd like to obtain results in minutes. 

Sunrise / Sunset  close window 
The times of sunrise and sunset are calculated as the times of morning and evening that the sun is apparently at the horizon. In the outputs section of SunAngle, sunrise and sunset times are given in clock time. The sun would normally appear to be exactly on the horizon when its altitude angle is zero degrees, except that the atmosphere refracts sunlight when it's low in the sky, and the observer's elevation relative to surrounding terrain also impacts the apparent time of sunrise and sunset. The difference between the time of apparent sunrise or sunset and the time when the sun's altitude angle is zero is usually on the order of several minutes, so it's necessary to correct for these factors in order to obtain an accurate result. The times of sunrise and sunset are calculated by computing the hour angle when the altitude angle is a certain value, then converting the hour angle to clock time. The value used for the altitude angle when the sun is apparently on the horizon is:
If we then take the altitude angle equation:
where:
and solve for the hour angle, setting Al equal to the equation above that corrects for refraction and elevation, we get:
Note that "cos^{1}" represents the inverse cosine function. The hour angle is converted to a number of minutes by multiplying the angle by 4 minutes per degree of hour angle, then that number of minutes is subtracted from 12:00 noon for the time of sunrise and added to 12:00 noon for the time of sunset, both in local solar time (LSoT). LSoT is then converted into clock time by performing the reverse of the calculations described on the time basis page. For U.S. cities, there are tables of sunrise and sunset times available from the U.S. Naval Observatory Astronomical Applications Department. 

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