Lab 2: Angles and other needed math (or the history of astronomy)

Transcription

1 Astronomy 101 Name(s): Lab 2: Angles and other needed math (or the history of astronomy) Purpose: This lab is an overview of much of the math skills you will need for this course. As I hope you will see in this exercise, the development of astronomy parallels the development of sophisticated math, from arithmetic to trigonometry (hey, a use for trig!); in essence, it is the story of human intellectual growth. If you are not comfortable with some or all of the math tools that follow, you have a couple of resources available to you: me and the wonderful tutors at the Math Learning Center (ED 1845B check the bookmark for hours of availability). The tutors can certainly help you with the mechanical solving of a math problem, but, as knowledgeable as they are, they (for the most part) have not had astronomy, so please don t expect them to be able to explain the astronomical ideas for that, please talk to me! Introduction: The care and feeding of angles Because early astronomers did not know the distances to objects in the sky, they used angles to describe the positions of these objects. Angles were measured between objects or between an object and a fixed point, such as the horizon. Angles are also used to measure the apparent size of an object. Angles are measured in degrees of arc, or degrees for short. It was short step from there to dividing the sky into segments for easy reference; one way (for instance, the Sumerians did this) was to divide the apparent path of the sun through the sky during the year into twelve equal parts and assign a constellation to represent each part (the collection of all twelve is called the Zodiac). 1. Given what you know about circles, how many degrees does each Zodiacal constellation subtend? (Assume the constellations represent equal areas of the sky)

2 Figure 2.1 shows angles defined by two line segments (called radii, which is plural for radius) joined at a point called the vertex of the angle. For an angle of 360, these ends of these segments sweep out an area of a full circle, and the vertex is the center of the circle. The circumference, or length around the circle, equals 2pr where r is the radius of the circle. 2. a. Calculate the circumference of a circle of radius 1 (don't worry about length units). Then calculate the ratio of the circumference of this circle divided by the number of degrees in this circle (see figure 2.1d). For any angle less than 360, the length of the section of the circle between the radii is called the arc (its length is called, not surprisingly, the arc length). We will use the letter s to stand for the arc length. b. Consider a half-circle (also called a semicircle) of radius 1. Calculate the ratio of s divided by the number of degrees in a half-circle (see figure 2.1c). c. Consider a quarter-circle (also called a quadrant) of radius 1. Calculate the ratio of s divided by the number of degrees in a quarter-circle (see figure 2.1b). d. Let's generalize. If you've done problems 12a through 12c correctly, what is the ratio of any angle's arc length s divided by the number of degrees in the angle? In fact, we can set up an equation which describes the relationship between an angle a and the arc length s for an arc of radius r: 2pr 360 = s a This ratio equals the ratios calculated in problems 2a through 2d. Solving for s, you get: (eq. 2.2) s ~ a r where ~ means "approximately equal"

3 Unfortunately, most angles used in astronomy are much smaller than even one degree, so as precision of instruments improved, subdivisions of degrees were introduced. Since longitude (the degrees east or west of the prime meridian) was measured by using a chronometer (a very accurate clock), divisions of the clock became divisions of angles. Thus, one degree (1 ) was subdivided into sixty arcminutes (60 ). Each minute was further subdivided into sixty arcseconds (60 ). 3. a. The average angular diameter of the Sun is Express this to the nearest arcminute. b. The mean angular diameter of the Moon is Express this as a decimal degree figure. (extra credit) Express the angular size of the Moon as an arcminutes/arcseconds figure (i.e., = ). Please show the steps of the conversion. 4. a. A total solar eclipse occurs when the Moon s disk completely obscures the Sun (this will happen Friday over New Zealand and parts of Central America like usual, we won t even see a partial eclipse). An annular solar eclipse occurs when a ring (hence annular ) of the Sun s edge is visible around the Moon s disk even at the height of the eclipse. So is the typical solar eclipse total or annular? b. Draw the Sun, part of the orbit of the Earth around the Sun and the Moon around the Earth to show how any total eclipses can occur. Hint:

4 One neat aspect of equation 2.2 is that it can be used to compute astronomical distances. The equation (remember it is an approximation) works for small a angles only, as shown in figure Using equation 2.2, the information from problem 3a and the Earth-Moon average distance of 384,000 km, calculate the diameter of the Moon in kilometers. Show your equation setup! 6. The Snoqualmie tunnel was a key linkage in the Northwest rail system, connecting Seattle with Chicago. Even today, you can ride through the 2.2 mile tunnel on a bicycle; it is perfectly straight, so you can see the other end as a distant point of light. But wait a minute! Is it really a point? Calculate the apparent angular size (in degrees use equation 2.2) of the far tunnel opening if you are standing just inside the tunnel. Assume that the tunnel is a 20-foot diameter tube and you will need the conversion factor 5280 feet = 1 mile. 7. Venus is the closest planet to us; Jupiter is the biggest planet. Rhetoric question: which is more important in determining angular size: planet size or distance from Earth? Use the information in Appendix A of the text (page 376) and equation 2.2 to determine the average angular size of Venus and the average angular size of Jupiter, as seen from Earth. Hint: to determine the distance, you will have to do a subtraction.

5 8. The apparent angular size of the moon you already know from question 3b. Order the moon, Venus, Jupiter and the far end of the Snoqualmie tunnel in order of decreasing angular size. Scientific notation 9. Convert the following numbers into scientific notation: a. 227,900,000 km b m 10. Perform the following calculations (a calculator is strongly recommended): a. ( )( ) = b. ( )/( ) = The motion of planets Planets that are closer to the Sun than the Earth are called inferior planets (e.g., Mercury); planets that are further are called superior planets. During this month, the planets Venus, Mars and Jupiter are in conjunction (occupy the same part of the sky) with the Moon. A believer of geocentrism would draw the following picture to explain the positions of the planets during this time.

6 11. He challenges you, as a heliocentrist, to come up with a drawing that shows how the same conjunction could be achieved with the planets orbiting the Sun. Using the heliocentric system, draw a picture (from the viewpoint of the North Celestial Pole, similar to the picture above) of the solar system and how the planets and the Moon must be arranged so that the conjunction can take place. 12. To settle debate about geocentrism and heliocentrism, you and he go outside and observe Venus for many days. You both agree that the furthest Venus gets away from the Sun is 47 (this angular measurement is called Venus s greatest western or eastern elongation, depending on which side of the Sun it s on). With which system is this observation consistent? Draw a solar-system-from-above sketch that demonstrates your point.

7 The history of astronomy the size of the Earth Eratosthenes, a Egyptian astronomer of the 3 rd century BC, saw that on June 21 (the summer solstice), there was no shadow in a well in the town of Syene; in other words, the Sun was at zenith. At the same time, in his hometown of Alexandria, he saw that there was a shadow cast by an obelisk; in other words, the Sun was not exactly overhead. By measuring the angle of the sunlight, using the top of the obelisk s shadow, he determined that the Sun was 7.2 off of directly overhead (see the diagram). 13. If Syene and Alexandria are separated by 800 km, what is the Earth s circumference? Hint: you will need to set up a proportion. 14. Therefore, what is the Earth s diameter? How does this compare to the actual figure? The Earth-Moon distance Aristarchus, an astronomer in the Greek island of Samos in the 3 rd century BC, observed that the Earth s shadow was 8/3 the size of the Moon. He determined this by carefully observing a total lunar eclipse and how long it took the Moon to pass completely through the Earth s shadow. 15. Determine the diameter of the Moon in kilometers just from the result of question 14 and the number Aristarchus determined.

8 16. What assumption do you have to make about the Earth s shadow, and what other assumption does that lead to about the distance to the Sun? 17. Now that you know the diameter of the Moon, calculate the distance to the Moon, using only the fact that the Moon is 0.5 in angular diameter and the narrow triangle technique used earlier. How does this compare to the actual figure (check appendix A)? The Earth-Sun distance From the previous part of the exercise, you know that when the moon is in its first quarter phase, the Moon, the Earth and the Sun form a right triangle, with the Earth at the right angle corner of the triangle (see the diagram). Aristarchus was among the first astronomers to realize that the Moon, the Earth and the Sun do not form a perfect right triangle and that, in fact, the Sun-Earth-Moon angle was a little less to be exact.

9 He used the following formula for the cosine of the Sun-Earth-Moon angle: cosine (Sun - Earth - Moon angle) = Moon - Earth distan ce (adjacent side) Sun - Earth dis tan ce (hypotenuse) 18. Rewrite the equation so that the Sun-Earth distance is alone on the left side. 19. If the Sun-Earth-Moon angle was actually exactly 90, what would the Sun- Earth distance be? (As a check, make sure the cosine of 90 = 0) 20. Using the actual value of , calculate the Sun-Earth distance in kilometers. How does this compare to 1 AU?