The rules: Collaborative work is encouraged. This homework can be done in consultation with your fellow classmates, the AI, or the professor. However, everyone must submit their own solutions to get credit, and all help should be acknowledged (a single sentence mentioning the others in your study group is sufficient). Show your work.
1. Just the facts [5 pts]
(a) [1 pts] Where are you standing (on earth) if the stars do not
rise or set?
(b) [2 pts] During the entire year, roughly what percentage of the
celestial sphere can you see from the equator?
(c) [2 pts]
What are the vernal and autumnal equinoxes? What are
the summer and winter solstices?
2. How far is that? [20 pts]
(a) [5 pts] Aristarchus of Samos was able to calculate the ratio of the Sun/Moon distances (from Earth) in the 3rd century BC based on the observed angle between the Sun and the Moon:
For a measured angle of 87o, what is the ratio between Dsun and Dmoon? Aristarchus's measurement was somewhat imprecise; today, the accepted value for the sun-earth-moon angle at quarter moon is 89.86o. Using the more recent value, what is the ratio between Dsun and Dmoon?
(b) [5 pts] Around 200 BC, Eratosthenes was able to calculate the
radius of the Earth based on the report that at noon on the
first day of summer the sun was directly overhead in the city
of Syene (no shadows were seen in the deep wells of Syene)
while he observed
an angle of 7.2o at Alexandria. The two cities are
directly north-south of each other, and are separated by 5000 stadia
(1 stadia ~ 0.168 km). Calculate the radius of
the Earth using this data.
(c) [5 pts] The diameter of the sun is 1.4 x 1011 cm, and the distance to the nearest star, Proxima Centauri, is 4.2 light years. Suppose you want to build an exact scale model of the Sun and Proxima Centauri, and you are using a tennis ball 5 cm in diameter to represent the Sun. In your scale model, how far away would Proxima Centauri be from the Sun? Give your answer in kilometers, and use scientific notation.
(d) [5 pts] On December 11, 2000, the planet Venus was at a distance of 0.951 AU from the Earth. The diameter of Venus is 12104 km. What was the angular size of Venus as seen from Earth on December 11, 2000? Give your answer in arcminutes.
3. Celestial Sphere [25 pts]
(a) [5 pts] Draw the view of the celestial sphere from Bloomington. Label the North Celestial Pole (NCP), the altitude of the NCP, zenith, horizon, and meridian.
(b) [5 pts] Suppose that you live in Juneau, Alaska (longitude 134.4W, latitude N58.3). What is the elevation of the Sun above the southern horizon at midday at the time of the winter solstice?
(c) [5 pts] In the northern hemisphere, houses are designed to have "southern exposure," that is, with the largest windows on the southern side of the house. But in the southern hemisphere, houses are designed to have "northern exposure." Why are houses designed this way, and why is there a difference between the hemispheres?
(d) [5 pts] Right Ascension is measured in hours, minutes, and seconds. Because 24 hours of right ascension takes you all the way around the celestial equator, it follows that 24 h = 360o. What is the angle in the sky (measured in degrees) between a star with R. A. = 8h 0m 0s, Decl = 0o 0' 0" and a second star with R.A. = 11h 20m 0s, Decl = 0o 0' 0"?
(e) [5 pts] What is the right ascension of a star that is on the meridian at midnight at the time of the autumnal equinox? Explain.