Question

5. The difference between a solar day on Earth and a sidereal day is about 4 minutes. If the Earth orbited the Sun at 0.5 AU, what would the difference be? (Assume the Earth’s orbit is circular and the rotation rate is the same.)

Please explain clearly and provide explanations for how you solve this so I can learn. Thank you.

Answer 1

The sidereal day is w.r.t. to a distant fixed star and solar day to the sun. Earth revolves around the sun with time period 365.25 days and completes 360 degrees. So each day it covers 0.9856 degrees (roughly 1 degree) which are equivalent to 4 mins (Earth rotates about .25 degrees (360/(24*60)) in one minute so 1 degree in 4 mins). However, the distant star is at the same location so that the angle between the line joining the sun and earth, and distant star and earth is 0.9856 degrees. So we get sidereal day 4 min faster than the solar day.

Now, if we reduce the orbital radius to half, the time period is also reduced to half i.e. 365/2 = 182.5 days (assuming velocity is the same). In this case, each day the earth will cover 360/182.5 = 1.973 degree (roughly 2 degrees) which is equivalent to 8 mins (since rotation rate is the same). So the difference between a sidereal day and solar day is 8 mins.