Mean solar day is the time interval between two successive noon w… - Vidyadip

Question

Mean solar day is the time interval between two successive noon when sun passes through zenith point (meridian). Sidereal day is the time interval between two successive transit of a distant star through the zenith point (meridian). By drawing appropriate diagram showing earth’s spin and orbital motion, show that mean solar day is four minutes longer than the sidereal day. In other words, distant stars would rise 4 minutes early every successive day.

✓

Answer

According to the diagram alongside, when the earth revolves about its polar axis in one sidereal day, it also moves from E to E’ around the sun due to translational motion and the point P’ is at P’’.
When the Earth still rotates through an angle about its axis to complete one solar day until the point P’ is at P”, again facing the sun.
Earth advances in its orbit by approximately 1° every day, i.e. in 24 hours. Then, it will have to rotate by 361° (which we define as 1 day) to have the sun at zenith point again,

$\therefore$ Time taken to traverse $1^0=\frac{24\text{h}}{361^0}\times1^0\frac{24\times60\times6\text{s}}{361}=239.3\text{s}$
i.e., 239.3s = 3 min 59.3s ≈ 4 min
Hence, distante stars would rise 4 minutes early every successive day.

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