Sidereal day

A sidereal day is the duration that a planet puts to make a turn on itself compared to the star S, independently of its revolution around the Sun. In the case of the Ground a apparent sidereal day is defined as being the time interval separating two transit S successive from the vernal Point with the Méridien.

In Astronomy, one is interested only in the duration of rotation of the Earth relative to stars known as fixed and not to the Sun. One thus uses a scale of time which is only given the responsability to determine the time put by the Earth to carry out a rotation of 360 degrees compared to stars. This duration of rotation is thus called the sidereal day.

Time that we use in the daily life is based on the solar Temps . The fundamental unit of solar time is the solar Jour: it is the time interval separating two passages from the Sun with the meridian line, because of the rotation of the Ground. But, actually, the Earth does not carry out a rotation of 360 degrees in one day solar. Indeed the Earth moves around the Sun and in one day she traverses a little less degree on her orbit: 360/365,25 = 0,9856 degree per day. Thus of 24 hours the direction Ground-Sun changes approximately degree. Consequently the Earth must actually carry out a rotation of approximately 361 degrees so that the Sun returns to the meridian line and seems to have traversed 360 degrees in the sky. On average, a sidereal day is 4 minutes shorter than a solar day, because of the degree of terrestrial rotation additional in the solar day.

The duration of a Tropical year is of 365,2422 solar days, in other words during this year the Earth carries out the same number of revolutions on itself compared to the Sun . But in same time, compared to the stars , the Earth carries out on it even a turn moreover i.e. 366,2422 turns on itself; one tropical year thus lasts 366,2422 days sidereal and a sidereal day is worth: : 365,2422/366,2422 = 0,997269 solar day; that is to say 23:56 m 4,09s. In practice one uses also the average sidereal day which holds account in particular movements of the pole.

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A manner of making calculation is the following one:

Sidereal day: $T_ {sidereal} = \ frac {2 \ pi} {\ Omega}$ with $\ omega$ angular velocity of rotation of the Earth on it even. On the figure above, that corresponds to the time put by the Earth to go from position 1 to 2 on the terrestrial orbit.

Solar day: $T_ {solar} = \ frac {2 \ pi + \ alpha} {\ Omega}$ . Here one added an angle $\ alpha$ because on the figure one goes from position 1 to position 3, therefore the Earth rotated of more than $2 \ pi$ radians. This angle $\ alpha$ corresponds, seen Sun, with the angle traversed between positions 1 and 3 (or 1 and 2 because in practice the angle between positions 2 and 3 is about 10 seconds of arc).

However between positions 1 and 3 it runs out by definition 24:00 and the angle traversed by the Earth in one day is: $\ alpha = \ frac {2 \ pi} {365,25}$

One can then deduce $\ omega$ :

$T_ {solar} = 24 \ cdot 3600 = \ frac {2 \ pi} {\ Omega} + \ frac {2 \ pi} {365,25 \ Omega}$ thus $\ Omega = 7,292 \ cdot 10^ {- 5} rad \ cdot s^ {- 1}$

Finally one finds the duration of the sidereal day: $T_ {sidereal} = \ frac {2 \ pi} {\ Omega} = \ frac {2 \ pi} {7,292 \ cdot 10^ {- 5}} = 86164,09$ seconds. What corresponds to one duration 23 hours 56 minutes 4,09 seconds.

See too

sidereal Time

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