Question

1. At the solstices and the equinoxes, how does the right ascension of the Sun compare with the sidereal time at solar noon? Does this relationship hold on other days of the year?

2. On what day of the year is apparent solar time the same as sidereal time? Explain why this would not happen on another day of the year.

Answer 1

A
Star time, properly called sidereal time, is the hour angle of the Vernal Equinox. Because the Sun moves to the east along the ecliptic, the Sun takes longer to make a circuit of the sky on its daily path than does a star or the equinox, so the solar day is 4 minutes longer than the sidereal day. As a result, the sidereal clock gains 4 minutes (actually 3 minutes 56 seconds) per day over the solar clock, starting from the time of solar passage across the autumnal equinox on September 23, when the two are the same.

EQUINOXES AND SOLSTICES
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Because of rotation, the Earth bulges slightly at its equator. The gravity of the Moon and Sun acting on the bulge causes the Earth's rotation axis to wobble around the perpendicular to its orbit with a period of 26,000 years, the phenomenon called precession. Precession causes the celestial poles (the NCP and the SCP) to move in circles around the ecliptic poles (the NEP and SEP), which changes the pole stars.
In ancient Egyptian times, for example, the North Celestial Pole (NCP) pointed close to Thuban in Draco (the Dragon). Precession also causes the plane of the Earth's equator to wobble, which changes the directions of the equinoxes and solstices against the background stars (continuously altering the right ascensions and declinations of stars, though in a precisely known way).

In ancient times, the Vernal Equinox was in Aries, which is why that constellation tops the classical list. In classical times the Summer and Winter Solstices were in Cancer and Capricornus, hence the names "Tropic of Cancer" and "Tropic of Capricorn." Our astronomical roots are deep indeed.

As a result of precession, around 1990 the Summer Solstice crossed the modern boundary from Gemini to Taurus, which now technically holds the point. Because the Summer Solstice is closer to the classic figure of Gemini than it is to that of Taurus, and since Gemini (along with Pisces, Libra, and Sagittarius) quarters the ecliptic, Gemini is still traditionally taken as the Solstice's celestial home.

B
12:00 noon on 2000 January 1,1.002737909350795 s of mean sidereal time was equal to 1 s of UT1. The ratio varies slightly with time, reaching 1.002737909409795 after a century.