Hour angle

In astronomy and celestial navigation, the hour angle is a vital measurement used to locate objects in the sky. It is defined as a dihedral angle, which is an angle between two planes. One plane is the meridian plane, which contains Earth's axis and the zenith, or the point directly above an observer. The other plane is the hour circle, which contains Earth's axis and a specific object of interest.

By using this angle alongside declination, astronomers can fully specify an object's location within the equatorial coordinate system. This system acts like a map for the entire celestial sphere.

Measuring the hour angle involves specific mathematical conventions. It can be expressed in degrees, in units of time, or in rotations. If measured in time, 24 hours is exactly equal to 360 degrees.

The direction of the angle is also important for tracking movement. An angle can be negative if it is east of the meridian plane. It is expressed as positive if it is west of the meridian plane. Some systems use positive westward measurements from 0° all the way to 360°.

These different ways of writing the number ensure that navigators and scientists can communicate clearly.

Understanding the movement of an object requires looking at its relationship to the meridian. The meridian is the imaginary line that passes directly overhead. When an object's hour angle is zero, it is sitting exactly on the meridian. If the hour angle is negative, such as between -180° and 0°, the object is approaching the meridian. Conversely, a positive hour angle between 0° and 180° indicates the object is moving away from the meridian.

This simple rule helps observers predict when a star will reach its highest point in the sky.

There are several distinct types of hour angles used depending on the observer's needs. In celestial navigation, experts use the Greenwich hour angle (GHA), which measures westward from the prime meridian. They also use the local hour angle (LHA), which is measured from the observer's own local meridian.

Another important type is the sidereal hour angle (SHA). The SHA measures the angular distance west of the first point of Aries, or the March equinox. Navigators often find these specific values published in nautical almanacs to help them find their way at sea.

Calculating the local hour angle (LHA) involves a specific relationship with right ascension. The LHA of an object is found by subtracting the object's right ascension from the local sidereal time (LST).

It can also be calculated using Greenwich sidereal time (GST) plus the observer's longitude, then subtracting the right ascension. Right ascension is often written in a format called sexagesimal hours-minutes-seconds (HH:MM:SS). Because the Earth rotates about 365.2564 times in a sidereal year, stars appear to move differently than the Sun. A star's hour angle increases by about 1.0027 per hour. This means it takes only 59 minutes and 50.17 seconds for the hour angle to increase by one full hour.

The Sun has its own unique measurement called the solar hour angle. This is an expression of time expressed in angular units, usually degrees, measured from solar noon. At solar noon, the solar hour angle is zero degrees. Times before noon are expressed as negative degrees, while times after noon are positive.

For example, at 10:30 AM local apparent time, the solar hour angle is -22.5°. This is because it is 1.5 hours before noon, and each hour accounts for 15 degrees.

While the solar hour angle increases by one hour per hour on average, it is not perfectly steady. This variation is caused by something called the equation of time. In mid-September, a solar day is about 22 seconds shorter than 24 hours. During this time, the solar hour angle increases by 1.00025 hours every hour. In late December, the situation reverses, and a solar day is about 28 seconds longer.

In that period, the angle increases by only 0.99968 hours per hour. Scientists also use the cosine of the hour angle to calculate the solar zenith angle. Because the cosine of a negative angle is the same as the cosine of a positive angle, the Sun reaches the same altitude at 11:00 AM as it does at 1:00 PM.

The sidereal hour angle (SHA) provides a stable way to track celestial bodies. For a star, the SHA varies by less than a minute of arc per year because of a process called precession.

However, the SHA of a planet can change significantly from one night to the next. This difference highlights how different objects move through our solar system and the wider galaxy. By mastering these angles, humans have been able to map the heavens and navigate the oceans for centuries.