Finally, we will look at a system that is in an orbit comparable to the Earth to see how the angle that the light strikes the surface and the length of day contribute to the total amount of energy that we receive from the Sun at the two solstices. Recall that the changing distance of the Earth gives a maximum difference of about 6% between the closest and farthest distances. We will find that the effects studied here are much larger than this.
Before continuing we need a little more information about the angle factor given in the formula above. It turns out that the angle factor is equal to the cosine of the angle that the light strikes the surface as measured from perpendicular. In this question, the angle factor will be calculated for you. However, it is still informative to see how that factor is calculated for an observer at a given latitude and with a given tilt of the planet.
Set the tilt of the planet to be 30 degrees.
a) What is the length of day during the winter solstice for this planet?
b) What is the length of day during the summer solstice for this planet?
c) The angle factor for the winter solstice is cos(45 + 30) = 0.259. What is the total amount of sunlight-hours on the winter solstice for this planet? (Be sure to show your work on all of the following questions.)
d) For the summer solstice the angle factor is equal to cos(45 - 30) = 0.966. What is the total amount of sunlight-hours for the summer solstice?
e) What is the ratio of the sunlight-hours for the summer solstice to the winter solstice (divide the larger number by the smaller). You should get an answer that is larger than 1.06 (the ratio due to the changing planet-Sun distance).
sunlight-hours = (angle factor)*(hours of daylight)
To answer these questions, we need to calculate the length of day for each solstice and use the given angle factor to find the total amount of sunlight hours.
a) To find the length of day during the winter solstice, we need to consider that the tilt of the planet is 30 degrees. During the winter solstice, the tilt of the planet means that the North Pole is tilted away from the Sun, resulting in shorter daylight hours. The length of the day can be calculated using the formula:
length of day = 24 - (tilt of the planet - observer's latitude)
Since the tilt of the planet is 30 degrees, let's assume an observer's latitude of 45 degrees for simplicity.
length of day = 24 - (30 - 45)
length of day = 24 - (-15)
length of day = 24 + 15
length of day = 39 hours
Therefore, the length of day during the winter solstice for this planet is 39 hours.
b) Similarly, to find the length of day during the summer solstice, we can use the same formula. However, during the summer solstice, the North Pole is tilted towards the Sun, resulting in longer daylight hours. Let's assume the same observer's latitude of 45 degrees.
length of day = 24 - (tilt of the planet + observer's latitude)
length of day = 24 - (30 + 45)
length of day = 24 - 75
length of day = -51 hours
Since the result is negative, it means that during the summer solstice, there is no nighttime and the Sun is continuously above the horizon. Therefore, the length of day during the summer solstice for this planet is 0 hours.
c) The given angle factor for the winter solstice is cos(45 + 30) = 0.259. To find the total amount of sunlight-hours on the winter solstice, we need to multiply the angle factor by the length of day calculated in part a.
sunlight-hours = (angle factor) * (hours of daylight)
sunlight-hours = 0.259 * 39
sunlight-hours = 10.101 hours
Therefore, the total amount of sunlight hours on the winter solstice for this planet is approximately 10.101 hours.
d) The given angle factor for the summer solstice is cos(45 - 30) = 0.966. To find the total amount of sunlight hours on the summer solstice, we again multiply the angle factor by the length of day calculated in part b.
sunlight-hours = (angle factor) * (hours of daylight)
sunlight-hours = 0.966 * 0
sunlight-hours = 0 hours
Since the length of day during the summer solstice is 0 hours, the total amount of sunlight hours is also 0.
e) To find the ratio of sunlight hours for the summer solstice to the winter solstice, we divide the larger number (winter solstice sunlight hours) by the smaller number (summer solstice sunlight hours).
ratio = (sunlight-hours for summer solstice) / (sunlight-hours for winter solstice)
ratio = 0 / 10.101
ratio = 0
Therefore, the ratio of sunlight hours for the summer solstice to the winter solstice is 0.