Sidereal Day
The planet is positioned in a sidereal day to arrive in 2
and a solar day to arrive in 3
A sidereal day is the time it takes a planet to go around on itself in relation to the stars , regardless of its revolution around the Sun. In the case of Earth , an apparent sidereal day is defined as the time interval between two transits successive vernal equinox at the meridian.
In astronomy , we are interested only in the length of Earth's rotation with respect to so-called fixed stars, not the Sun.
So we use a time scale which is responsible only for determining the time it takes Earth to rotate 360 degrees relative to the stars. This period of rotation is the stellar day.
The sidereal day is different from the day star, due to the movement of precession of the vernal equinox.
The time that we use in everyday life is based on the solar time. The fundamental unit is the solar time solar day : it is the time interval between two passages of the sun at meridian, due to the rotation of the Earth. But in reality, the Earth does not rotate 360 degrees in a solar day. Indeed, the Earth moves around the sun and one day she runs a little less than one degree in its orbit: 360 / 365.25 = 0.9856 degree per day. So in 24 hours, the Earth-Sun direction changes by about one degree. Consequently, the Earth has actually rotated about 361 degrees so that the Sun returns to the meridian and appears to have traveled 360 degrees in the sky. On average, a sidereal day is 4 minutes shorter than a solar day , due to the additional degree of rotation of the Earth in the solar day.
The duration of a tropical year is 365.2422 days of sun, that during this year the Earth does the same number of turns in on itself from the Sun. But at the same time, relative to the stars, the Earth makes one revolution on itself is more to say 366.2422 turns on itself, a tropical year thus lasts 366.2422 days and one day sidereal sidereal is: 365.2422 / 366.2422 = 0.997 269 solar day, or 23h 56m 4.09 sec In practice, it also uses the sidereal day average that takes into account in particular the movements of the pole.
One way of doing the calculation is as follows:
Sidereal day: 
with the angular velocity of rotation of the Earth itself. On the figure above, this is the time taken for the Earth to go from position 1 to 2 of Earth's orbit.
Solar day: 
. Here we added an angle in the figure because it goes from position 1 to position 3, so the Earth has turned on itself over 2 radians. This angle is seen from the Sun, the angle traveled between positions 1 and 3 (or 1 and 2 because in practice the angle between positions 2 and 3 is about 10 arc seconds).
Gold between 1 and 3 positions it takes by definition 24 and the angle through which the Earth in one day is: 
We can deduce :
So 
Finally we find the length of sidereal day: 
seconds. This corresponds to a period 23 hours 56 minutes 4.09 seconds.
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