Hour angle

In astronomy and celestial navigation, the hour angle is one of the coordinates used in the equatorial coordinate system to give the direction 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 hour circle passing through the point).

As seen from above the Earth's north pole, a star's local hour angle (LHA) for an observer near New York (red). Also depicted are the star's right ascension and Greenwich hour angle (GHA), the local mean sidereal time (LMST) and Greenwich mean sidereal time (GMST). The symbol ʏ identifies the vernal equinox direction.

The angle may be expressed as negative east of the meridian plane and positive west of the meridian plane, or as positive westward from 0° to 360°. The angle may be measured in degrees or in time, with 24^h = 360° exactly.

In astronomy, hour angle is defined as the angular distance on the celestial sphere measured westward along the celestial equator from the meridian to the hour circle passing through a point.^[1] It may be given in degrees, time, or rotations depending on the application. In celestial navigation, the convention is to measure in degrees westward from the prime meridian (Greenwich hour angle, GHA), the local meridian (local hour angle, LHA) or the first point of Aries (sidereal hour angle, SHA).

The hour angle is paired with the declination to fully specify the direction of a point on the celestial sphere in the equatorial coordinate system.^[2]

Relation with the right ascension

The local hour angle (LHA) of an object in the observer's sky is

$\text{LHA}_{\text{object}} = {\text{LST}} - \alpha_{\text{object}}$ (If result is negative, add 360 degrees. If result is greater than 360, subtract 360 degrees.)

or

$\text{LHA}_{\text{object}} = {\text{GST}} - \lambda_{\text{observer}} - \alpha_{\text{object}}$

where LHA_object is the local hour angle of the object, LST is the local sidereal time, $\alpha_{\text{object}}$ is the object's right ascension, GST is Greenwich sidereal time and $\lambda_{\text{observer}}$ is the observer's longitude (positive west from the prime meridian).^[3] These angles can be measured in time (24 hours to a circle) or in degrees (360 degrees to a circle)— one or the other, not both.

Negative hour angles indicate the time until the next transit across the meridian; an hour angle of zero means the object is on the meridian.

Solar hour angle

Observing the sun from earth, the solar hour angle is an expression of time, expressed in angular measurement, usually degrees, from solar noon. At solar noon 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. For example, at 10:30 AM local apparent time the hour angle is -22.5° (15° per hour times 1.5 hours before noon).^[4]

The cosine of the hour angle (cos(h)) is used to calculate the solar zenith angle. At solar noon, h = 0.000 so 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 11:00AM and 1:00PM solar time, etc.^[5]

Sidereal hour angle

The sidereal hour angle (SHA) 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 very quickly, so SHA is a convenient way to list their positions in an almanac. SHA is often used in celestial navigation and navigational astronomy.

The sidereal hour angle is defined as 360° minus the right ascension.

See Also

• Clock position

Notes and references

1. U.S. Naval Observatory Nautical Almanac Office (1992). P. Kenneth Seidelmann, ed. Explanatory Supplement to the Astronomical Almanac. University Science Books, Mill Valley, CA. p. 729. ISBN 0-935702-68-7.
2. Explanatory Supplement (1992), p. 724.
3. Meeus, Jean (1991). Astronomical Algorithms. Willmann-Bell, Inc., Richmond, VA. p. 88. ISBN 0-943396-35-2.
4. Kreider, J. F. (2007). "Solar Energy Applications". Environmentally Conscious Alternative Energy Production. pp. 13–92. doi:10.1002/9780470209738.ch2. ISBN 9780470209738.
5. Schowengerdt, R. A. (2007). "Optical radiation models". Remote Sensing. pp. 45–88. doi:10.1016/B978-012369407-2/50005-X. ISBN 9780123694072.