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 25 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 + 25) = 0.342. 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 - 25) = 0.940. 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).
To calculate the length of day during the winter solstice and summer solstice for the given planet with a tilt of 25 degrees, we need to consider the effect of the tilt on the angle that the light strikes the surface. The angle factor is equal to the cosine of this angle.
a) The winter solstice occurs when the tilt of the planet is angled away from the Sun, resulting in shorter days. To calculate the length of day during the winter solstice, we need to find the angle at which the light strikes the surface. Since the tilt of the planet is 25 degrees, and the Sun is directly overhead at noon on the equator, the angle at the winter solstice can be calculated as 45 degrees (90 - 25). The angle factor for the winter solstice is cos(45 + 25) = cos(70) ≈ 0.342.
b) The summer solstice occurs when the tilt of the planet is angled towards the Sun, resulting in longer days. Similarly, to calculate the length of day during the summer solstice, we need to find the angle at which the light strikes the surface. Since the tilt of the planet is 25 degrees, the angle at the summer solstice can be calculated as 45 degrees (90 - 25). The angle factor for the summer solstice is cos(45 - 25) = cos(20) ≈ 0.940.
c) To calculate the total amount of sunlight-hours on the winter solstice, we need to multiply the angle factor by 24 (the number of hours in a day). Therefore, the total amount of sunlight-hours on the winter solstice is 0.342 * 24 ≈ 8.23 hours.
d) Similarly, to calculate the total amount of sunlight-hours on the summer solstice, we multiply the angle factor by 24. Therefore, the total amount of sunlight-hours on the summer solstice is 0.940 * 24 ≈ 22.56 hours.
e) To find the ratio of sunlight-hours for the summer solstice to the winter solstice, we divide the total amount of sunlight-hours on the summer solstice by the total amount of sunlight-hours on the winter solstice.
This can be calculated as 22.56 / 8.23 ≈ 2.74, indicating that the summer solstice has approximately 2.74 times more sunlight-hours than the winter solstice.