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Given that Venus’ sidereal rotation period is 243.01 days and its orbital period is 224.70 days, develop a for?

• Given that Venus’ sidereal rotation period is 243.01 days and its orbital period is 224.70 days, develop a formula and use it to show that a solar day on Venus lasts 116.8 days. (Hint: the formula relating Venus’ solar day to its sidereal rotation period and orbital period is similar to the first formula in Box 4-1. Draw a picture! It’s easiest to work things out from the perspective of a hypothetical Venusian observer. The accompanying picture shows the Sun moving counter-clockwise (CCW) around Venus (from the Venusian observer’s point of view) while Venus rotates clockwise (CW). Start things out with the Sun at position 1 due overhead at noon (Venus time). Our hypothetical Venusian observer rotates CW at a rate thru an angle φ in a Venusian solar day where TV (= 243.01 days) is the rotational period of Venus. Meanwhile, the Sun revolves (apparently from the Venusian point of view) around Venus at a rate thru an angle 360 - φ in the same time where P (= 224.70 days) is the sidereal period of Venus around the Sun. After one Venusian solar day (which we’ll call S) elapses, Venus will have rotated through the angle and the Sun will be in position 2 (see figure) relative to the background stars, having rotated through the angle . The observer will again see the Sun directly overhead. Now, do the math to calculate the length of the solar day … HINT: Let’s examine this problem from the point of view of someone on Venus. From this point of view, the Sun orbits Venus counter-clockwise. At noon, our hypothetical Venusian would see the Sun directly overhead … Now, if Venus DID NOT ROTATE, then the Sun would have to travel 360 degrees completely around Venus in order to appear overhead on the next day, i.e., the solar day on Venus would equal its ORBITAL PERIOD! Let’s call that time, TO. But Venus does rotate backwards … clockwise … or retrograde. So it will take LESS time than 1 orbital period for the Sun to appear overhead again, because while it’s moving forward, counter-clockwise, around Venus … Venus, itself, is rotating backwards or clockwise. Let’s call the time it takes for the Sun to appear overhead again, S, the time of 1 solar day. Venus has rotated backwards during a time S thru an angle of 360 S/TR, while the Sun has moved forward thru an angle 360 S/TO, until the Sun appears overhead again. What is the sum of these 2 angles? If you can answer that, you can find an equation and solve it for S.

• Answer:

  Venus’ sidereal rotation period is 243.01 days and its orbital period is 224.70 days The Sun moves counterclockwise 360° in one Venus year or 360°/224.70 = 1.60° per "Earth" day.due to Venus' revolution Venus rotates clockwise, its day is 243.01 "Earth" days long, In one "Earth" day the Sun moves counterclockwise 360°/243.01 = 1.48° per "Earth" day. The sum of the two effects is 3.08°, so Venus' solar day is 360/3.08 = 116.9 "Earth days" 1/revolution period) +1/( sidereal period) = 1/ (solar day) 1/(224.70) +1/(243.01) = 1/(solar day) solar day = 116.75 "Earth days"