Horizontal coordinate system

The horizontal coordinate system is a celestial coordinate system that uses the observer's local horizon as the fundamental plane. This conveniently divides the sky into the upper hemisphere that you can see, and the lower hemisphere that you can't (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), 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 (measured from the North point, toward the East).

The horizontal coordinate system is sometimes also called the 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 either rising (if its azimuth is less than 180°) or setting (if its azimuth is greater than 180°).

Transformation of Coordinates

It is possible to pass from the equatorial coordinate system to the horizontal coordinate system.

Let <math>\delta<math> be the declination, <math>H<math> the hour angle, <math>\phi<math> the observer's latitude.

The equations of the transformation are:

<math>\sin Alt = \sin \phi \cdot \sin \delta + \cos \phi \cdot \cos \delta \cdot \cos H<math>

<math>\cos Az = \frac{\cos \phi \cdot \sin \delta - \sin \phi \cdot \cos \delta \cdot \cos H}{\cos Alt}<math>

Use the inverse trigonometric functions to get the values of the coordinates.

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:

Given:

• the date of the year and the time of the day
• the observer's latitude, longitude and time zone

You have to compute:

• The Sun Declination of the corrisponding Day of the Year, which is given by the following formula:

<math>\delta = -23.45^\circ \cdot \cos \left ( \frac{360}{365} \cdot \left ( N + 10 \right ) \right )<math>

where <math>N<math> is the number of days spent since January 1.

Here cos operates on degrees.

• The true hour angle that 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 <math>T<math> = 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. <math>T<math> 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 Time is used, you have to subtract one hour in order to get Standard Time.
  • The value of the Equation of Time in that day has to be added. Since <math>T<math> 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: <math>H<math> = (12 - <math>T<math>) * 15. Since <math>T<math> is measured in hours and the speed of rotation of the earth 15 degrees per hour, <math>H<math> is measured in degrees. If you need <math>H<math> 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.

---

This article's initial version originated from 'Jason Harris' Astroinfo which comes along with KStars, a Desktop Planetarium for Linux/KDE. See http://edu.kde.org/kstars/index.phtml