Geometry
Chapter 12 Answers
Astronomy and Geometric Models
Geometry can be used to find out about celestial bodies other than the earth. Even though a human has never actually set foot on Mars, there are many existing pictures of the planet's surface. Features on Mars are identified using a latitude and longitude grid like that on Earth. In this lesson, you will use longitude and latitude markings on Mars to figure out distances on the planet. You will also think about which units of measurement are best to measure which types of distances.
Part B, Longitude and Latitude
1. Go to the Martian
Hemisphere Web site to see four separate views of Mars
projected into "point perspective." What does "point
perspective" mean in these images?
[It is a view similar to what one would
see from a spacecraft.]
a. For this exercise, you will concentrate on the Schiaparelli and Syrtis Major mosaics. What are the latitude and longitude of the center of each of these images?
[Schiaparelli: latitude -3°, longitude 343°; Syrtis Major: latitude 0°, longitude 310°]b. What are the latitude and longitude limits of each of these mosaics?
[Schiaparelli: latitude -60° to 60°, longitude 260° to 30°; Syrtis Major: latitude -60° to 60°, longitude 260° to 350°]c. Which of these mosaics covers a greater surface area of the planet? How many degrees of longitude along Mars' equator does this image cover?
[Schiaparelli covers 130° of longitude along the equator.]d. Go to the Mars Web site to find information which will allow you to find the circumference of Mars at the equator. Calculate this value to the nearest kilometer.
[21344 km]e. How many kilometers along Mars' equator does the Schiaparelli image cover?
[7708 km]f. Assuming that both the Schiaparelli and Syrtis Major mosaics are centered on the equator, what is the distance in kilometers between the centers of these images?
[1957 km]
Part C, Measurement in Astronomy
2. Return to the Mars Web site to answer the following question.
a. Find out approximately how far Mars is from the Sun in both kilometers and astronomical units (A.U.).
[227,940,000 km or 1.52 AU]b. According to your textbook, a light-year is approximately 9.5 trillion kilometers. What is the distance between the Sun and Mars in light-years?
[0.000024 light-years]c. Which of these three units is the best for measuring the distance between the Sun and Mars?
[Answers may vary.]
3. Mars has two moons, or satellites.
a. Find the names of these two moons and their circumferences at their equators in kilometers.
[Phobos has a circumference of 69 km and Deimos has a circumference of 38 km.]b. What are their circumferences in astronomic units? (According to your text,
)
[Phobos: 0.00000046 AU; Deimos: 0.00000025 AU]c. In general, when is it most appropriate to use kilometers? Astronomical units? Light-years?
[Answers may vary. A possible answer would be that you use kilometers when measuring distances on a planet, astronomical units when measuring distances between planets and light-years when measuring distances between stars and across galaxies.]
Euclid's geometry postulates for planes model relatively small areas fairly well. For this reason, we can assume his postulates are true when making calculations such as those necessary to build houses and design containers. However, his postulates do not apply to many astronomical measurements. In this lesson, you will work with a non-Euclidean geometry called spherical geometry using the planet Venus as a model.
Part A, Euclidean Geometry and
Part B, Non-Euclidean Geometry
According to your text, in the spherical model of Riemannian
geometry, a plane is a sphere and lines are great circles on
the sphere. For this lesson, assume that Venus is a perfect sphere.
1. Go to the Venus
Web site to see a list of images of Venus. For many of the images,
the location is given by latitude and longitude lines like those
on earth. Look at the image of the volcanoes at 20° North
by 357° and the image at 35° South by 357° towards
the bottom of the list. Is there a Riemannian line between these
two points? If so, give the name of the line. If not, explain
why not.
[Yes, the 357° longitude line is
a Riemannian line in spherical geometry.]
a. Go to the Second planet Web site and read the information about Venus. Then find the shortest distance along the planet's surface between the the two volcanoes mentioned in 1.
[5809 km]b. Is the distance you found in 1a shorter or longer than the Euclidean distance between these two points?
[Longer. The Euclidean distance is a straight line segment while the Riemannian distance is an arc of a great circle.]c. Is there a great circle on the surface of Venus that goes through the Golubkina crater and the volcanoes you looked at in the beginning of this problem? Is there a line of longitude that goes through these two points?
[Yes, there is a great circle that goes through the points. However, there is not a line of longitude.]
2. Does Euclid's Straight-Line Postulate hold in this spherical
version of Riemannian geometry? If so, explain why. If not, give
a counterexample.
[No. For a counterexample, take the
poles of Venus. All of the longitude lines go through both poles.
Hence, these two points actually have an infinite number of lines
that go through them.]