- Horizontal coordinate system
The horizontal coordinate system is a
celestial coordinate systemthat uses the observer's local horizonas the fundamental plane. This conveniently divides the sky into the upper hemisphere that you can see, and the lower hemisphere that you cannot (because the Earth is in the way). The pole of the upper hemisphere is called the zenith. The pole of the lower hemisphere is called the nadir.
The horizontal coordinates are:
* altitude (Alt), sometimes referred to as elevation, that is the angle between the object and the observer's local horizon.
azimuth(Az), that is the angle of the object around the horizon, usually measured from the north point towards the east. In former times, it was common to refer to azimuth from the south, as it was then zero at the same time the hour angleof a starwas zero. This assumes, however, that the star (upper) culminates in the south, which is only true for most stars in the Northern Hemisphere.
The horizontal coordinate system is sometimes also called the az/el [http://www2.keck.hawaii.edu/inst/KSDs/40/html/ksd40-55.4.html] or Alt/Az coordinate system.
The horizontal coordinate system is fixed to the Earth, not the stars. Therefore, the altitude and azimuth of an object changes with time, as the object appears to drift across the sky. In addition, because the horizontal system is defined by your local horizon, the same object viewed from different locations on Earth at the same time will have different values of altitude and azimuth.
Horizontal coordinates are very useful for determining the rise and set times of an object in the sky. When an object's altitude is 0°, it is on the horizon, if at that moment its altitude is increasing, it is rising, if its altitude is decreasing it is setting. However all objects on the celestial sphere are subject to the
diurnal motion, which is always from east to west, so the inherent cumbersome determination whether altitude is increasing or decreasing can be easily found by considering the azimuth of the celestial object instead (referenced to North as 0°):
*if the azimuth is between 0° and 180° (north—east—south), it is rising.
*if the azimuth is between 180° and 360° (south—west—north), it is setting.There are the following special cases:
*At the north pole all directions are south, and at the south pole all directions are north, so the azimuth is undefined in both locations. A star (or any object with fixed
equatorial coordinates) has constant altitude, and therefore never rises or sets when viewed from either pole. The Sun, Moon, and planetscan rise or set over the span of a year when viewed from the poles because their right ascensions and declinations are constantly changing.
*At the equator objects on the celestial poles stay at fixed points on the horizon.
Note that the above considerations are strictly speaking true for the "geometric horizon" only: the horizon as it would appear for an observer on sea level on a perfect smooth Earth without atmosphere. In practice the "apparent horizon", which you see, has a negative altitude, which absolute value gets larger when you come higher, due to the curvature of the Earth. In addition the
atmospheric refractionadds another 0.5° to that value.
Transformation of coordinates
It is possible to convert from the
equatorial coordinate systemto the horizontal coordinate system and back, once the observer's geographic latitudeis known (+90° on the north pole, 0° on the equator, -90° on the south pole).
We will use for the azimuth, for the altitude.
We will use for the
declination, for the hour angleexpressed in degrees (If in hours, it needs to be multiplied by 15).
equatorial to horizontal
One may be tempted to 'simplify' the last two equations by dividing out the leaving one expression in only. But the tangent cannot distinguish between (for example) an azimuth of 45° and 225°. These two values are very different: they are opposite directions, NE and SW respectively. One can do this only when the quadrant in which the azimuth lies is already known.
If the calculation is done with an electronic pocket calculator, it is best not to use the functions "arcsin" and "arccos" when possible, because of their limited 180° only range, and also because of the low accuracy the former gets around ±90° and the latter around 0° and 180°. Most scientific calculators have a "rectangular to polar" (R→P) and "polar to rectangular" (P→R) function, which avoids that problem and gives us an extra sanity check as well.
The algorithm then becomes as follows.
*Calculate the terms right of the = sign of the 3 equations given above
*Apply a R→P conversion taking the as the X value and the as the Y value
*The angle part of the answer is the azimuth, an angle over the full range of 0° to 360° (or -180° to +180° etc)
*Apply a second R→P conversion taking the radius part of the last answer as the X and the of the first equation as the Y value
*The angle part of the answer is the altitude, an angle between -90° and +90°
*The radius part of the answer must be 1 exactly, or you have made an error.
horizontal to equatorial
The same quadrant considerations from the first set of formulas also hold for this set.
The position of the Sun
There are several ways to compute the apparent position of the Sun in horizontal coordinates.
Complete and accurate algorithms to obtain precise values can be found in
Jean Meeus's book "Astronomical Algorithms".
Instead a simple approximate algorithm is the following:
* the date of the year and the
timeof the day
* the observer's
latitude, longitudeand time zone
You have to compute:
* The Sun declination of the corresponding day of the year, which is given by the following formula:
where is the number of days spent since
* The true
hour anglethat is the angle which the earth should rotate to take the observer's location directly under the sun.
** Let "hh":"mm" be the time the observer reads on the clock.
** Merge the hours and the minutes in one variable = "hh" + "mm"/60 measured in hours.
** "hh":"mm" is the official time of the time zone, but it is different from the true local time of the observer's location. has to be corrected adding the quantity + (Longitude/15 - Time Zone), which is measured in hours and represents the difference of time between the true local time of the observer's location and the official time of the time zone.
** If it is summer and
Daylight Saving Timeis used, you have to subtract one hour in order to get Standard Time.
** The value of the
Equation of Timein that day has to be added. Since is measured in hours, the Equation of Time must be divided by 60 before being added.
** The hour angle can be now computed. In fact the angle which the earth should rotate to take the observer's location directly under the sun is given by the following expression: = (12 - ) * 15. Since is measured in hours and the speed of rotation of the earth 15 degrees per hour, is measured in degrees. If you need measured in radians you just have to multiply by the factor 2π/360.
* Use the Transformation of Coordinates to compute the apparent position of the Sun in horizontal coordinates.
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