Solar and Stellar Declination and Altitude

The terms declination and altitude refer to two angles that an observer would see in reference to a star, including our home star, the Sun. Click on the image at the left to see it in full size.The declination angle refers to the angle the star makes with respect to the center of the Earth and the Earth's Equatorial Plane. In the image, the Sun's declination is shown as a yellow dotted line, and the angle is called out as alpha.For reference purposes, the angle phi sub L refers to the observer's latitude, and it is indicated by a white arc. The angle that the polestar, Polaris makes is the angle between the line called out as "Your Westerly Horizon" and the blue line ending in the North Pole. Simple geometric arguments reveal that the angle the polestar makes with an observer's northern horizon is the same as the observer's latitude.The angle that a star, including the Sun, makes with the southern or equatorial horizon at local noon, the stellar altitude, is a little more complicated to calculate. The formula is "90 degrees minus the observer's latitude, plus the star's declination". For the image shown, the Sun's declination, or the solar declination, is 23.5 degrees, because the day of the drawing is the Summer Solstice, when the Sun shines directly over the Tropic of Cancer, latitude 23.5 degrees. So for an observer in Chicago, where the latitude is about 42 degrees, the Sun's altitude, or solar altitude, or the angle the Sun makes with the equatorial horizon at local noon, is "90 minus 42 plus 23.5" or 71.5 degrees. This is the highest angle the Sun will make at local noon in Chicago.

Please note that the solar declination varies between plus 23.5 degrees and minus 23.5 degrees, so the smallest solar altitude an observer in Chicago would experience is "90 degrees minus 42 degrees minus 23.5 degrees", or 24.5 degrees. This solar altitude occurs in Chicago on the Winter Solstice.

Notice also that simple geometric arguments reveal that the angle made between the Sun's celestial path (shown as a solid yellow line), the point of impingement (shown as a orange dot), and the vertical up from that point, an angle here called the sunset angle gamma, is the same as the observer's latitude, i.e., gamma is the same as phi sub L. See this web page for further discussion of the sunset angle.

One other thing: from an alt-azimuth perspective, the azimuthal angle where the Sun sets on the Summer Solstice is 300 degrees, 30 degrees more than cardinal west (azimuth 270 degrees). So on the Summer Solstice, the shadows cast by vertical objects at sunset should be 30 degrees south of cardinal east.

For homework purposes, some problems ask what the stellar altitude is for stars with declination zero degrees, i.e., stars shining directly over the Equator. Using the formula above, the angle would be the complement of the observer's latitude, which for Chicago is "90 degrees minus 42 degrees" or 48 degrees.