Hour angle

In astronomy and celestial navigation, the hour angle is one of the coordinates used in the equatorial coordinate system to give the position of a point on the celestial sphere. The hour angle of a point is the angle between two planes: one containing the Earth's axis and the zenith (the meridian plane), and the other containing the Earth's axis and the given point. The angle may be expressed as negative east of the meridian plane and positive west of the meridian plane, or as positive westward from zero to 360 degrees. The angle may be measured in degrees or in time, with 24 hours equalling 360 degrees exactly. In celestial navigation, the convention is to measure in degrees westward from the prime meridian (GHA), the local meridian (LHA) or the first point of Aries (SHA).

The hour angle is paired with the declination to fully specify the position of a point on the celestial sphere.

Relation with the right ascension

The hour angle (HA) of an object is equal to the difference between the current local sidereal time (LST) and the right ascension (α) of that object:

HA_object = LST - α_object

Thus, the object's hour angle indicates how much sidereal time has passed since the object was on the local meridian. It is also the angular distance between the object and the meridian, measured in sidereal hours (1 hour = 15 degrees). For example, if an object has an hour angle of 2.5 hours, it crossed the local meridian 2.5 sidereal hours ago (i.e., hours measured using sidereal time), and is currently 37.5 degrees west of the meridian. Negative hour angles indicate the time until the next transit across the local meridian. Of course, an hour angle of zero means the object is currently on the local meridian.

Solar hour angle

The hour angle (h or ha) of a point on the Earth's surface is the angle through which the earth would turn to bring the meridian of the point directly under the sun. The earth is rotating, so this angular displacement represents time.

So in observing the sun from earth, the solar hour angle is an expression of time, expressed in angular measurement, usually degrees, from the solar noon.

At solar noon, at the observer's longitude on earth, the hour angle is 0.000 degrees with the time before solar noon expressed as negative degrees, and the local time after solar noon expressed as positive degrees.

The hour angle is the angular displacement of the sun east or west of the local meridian due to rotation of the earth on its axis at 15° per hour with morning being negative and afternoon being positive. For example, at 10:30 AM local apparent time the hour angle is -22.5° (15° per hour times 1.5 hours before noon).

The cosine of the hour angle (cos(h)) becomes an easy computation tool in determining the cosine term for the computation of the angle of the sun's altitude (or the complementary zenith angle) at any time during the day. At solar noon, h=0.000 so of course, cos(h)=1, and before and after solar noon the cos(± h) term = the same value for morning (negative hour angle) or afternoon (positive hour angle), i.e. the sun is at the same altitude in the sky at both 11:00AM and 1:00PM solar time, etc.

Sidereal hour angle

The sidereal hour angle of a body on the celestial sphere is its angular distance west of the vernal equinox generally measured in degrees. The SHA of a star changes slowly, and the SHA of a planet doesn't change fast, so SHA is a convenient way to list their positions in an almanac. SHA is often used in celestial navigation and navigational astronomy.