Question

3a. A planet that keeps the same hemisphere pointed towards the Sun must rotate once per orbit in the prograde direction. Draw a diagram to demonstrate this fact. The rotational period in an inertial frame, the sidereal day, for such a planet is equal to the orbital period, the length of the solar day on such a planet is infinite (this is known as being tidally locked). b. If a planet rotated once per orbit in the retrograde direction, how many solar days would it have per orbit? c. Earth rotates in the prograde direction. How many times must Earth rotate per orbit for there to be 365.24 solar days per year? Verify your result by comparing the length of Earth's sidereal rotational period 23 hours and 56 minutes) with the length of a mean solar day (24 hours). d. Determine a general formula relating the lengths of the solar and sidereal days on a planet. Use your formula to compute the lengths of solar days on Mercury, Venus, and Mars. e. For a planet on an eccentric orbit, the length of either the solar day or the sidereal day varies on an annual cycle. Which one varies, and why?

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Answer 1

3a. the following diagram shows a tidally locked planet in orbit around sun, we can see that the angular rotation vector of the planet and the angular revolution vecotr of the planet around the sun both point in the same direction and hence the prograde nature of rotation

b. if it rotated in the reterograde direction, it would still complete on rotation in one year, and one revolution in one year
   but the hemisphere facing the sun intiially, will face sun again after half an year and again after one full year, so we will have 2 days and 2 nights per year ( revolution)

c. for 365.24 solar days
   in prograde motion, earth must revolve n times (approximately) for there to be so many days
   now sideareal rotation period = 23 hours and 56 minutes
   hence
   n*(23 hours 56 minutes) = 365.24*24 hours
   hence
   n = 366.25738 times

d. hence, general formula relating length of solar day, Ts
   sideareal day, T'
   n*T' = y*Ts
   where y is length of mean solar year and n is numebr of revolutions made per year

   using this and data from internet we can derive the day length in various planets to be as under
   Mercury = 176 earth days
   Venus = 116.75 days ( for reterograde rotation)
   Mars = 24 houts 40 minutes