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This is done by measuring the instant when the star passes through the local meridian.
It is different from a solar day, which is measured by the passage of the Sun across the local meridian.
Of course, an hour angle of zero means the object is currently on the local meridian.
It is based on the apparent solar day, the interval between two successive returns of the Sun to the local meridian.
Because it is fixed to the local horizon, stars will appear to drift past the local meridian as the earth spins.
The local meridian is the plane containing the direction to the zenith and the direction to the celestial pole.
At the moment when the vernal equinox crosses the local meridian, Local Apparent Sidereal Time is 00:00.
A more constant day can be defined by the Sun passing through the local meridian, which happens at local noon (upper culmination) or midnight (lower culmination).
Once that is determined, the values are applied to a spherical triangle, and the angle from the local meridian to the required direction of Mecca can be determined.
Prior to the global adoption of Standard Time, numerous Prime Meridians were decreed by various countries where time was defined by local noon (thereby, local meridian).
It is important not to confuse sidereal hour angle with the astronomical concept of hour angle, which measures angular distance of an object westward from the local meridian.
In the Northern Hemisphere, Orion's Belt is best visible in the night sky during the month of January around 9:00 pm, when it is approximately around the local meridian.
The great circle defining the meridian is divided into the local meridian (containing the zenith and terminated by the celestial poles) and the antimeridian (opposite half containing the nadir).
The meridian circle is an instrument for timing of the passage of stars across the local meridian, an event known as a transit, while at the same time measuring their angular distance from the nadir.
Standard time is the result of synchronizing clocks in different geographical locations within a time zone to the same time rather than using the local meridian as in local mean time or solar time.
Strictly speaking, the zenith is only 'approximately' contained in the local Meridian (astronomy) plane because the latter is defined in terms of the rotational characteristics of the celestial body, not in terms of its gravitational field.
You had to be able to compare the time at which the Sun, or better still, another star (the sun is a star) was directly above the local meridian, as compared with the time it was directly,above another meridian.
Also known as the Axial system, it divides the celestial equator in twelve 30 sectors (starting at the local meridian) and projects them on to the ecliptic along the equally spaced meridians (great circles containing the North and South poles).
Apparent solar time (or true or real solar time) is the time indicated by the Sun on a sundial (or measured by its transit over a preferred local meridian), while mean solar time is the average as indicated by well-regulated clocks.