CHAPTER 1 THE CYCLES OF THE SKY

1 Copyright 2020 © McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.

CHAPTER 1 THE CYCLES OF THE SKY

Lecture Suggestions

Planetarium software may be helpful for this chapter. An orrery (mechanical model of the Sun and Earth) or even just an approximation of one can also help illustrate various motions and that the constellations change with the seasons. Set it on a table in the front of the room and then mark off constellations on the walls with chalk or paper decorations. Moving the model Earth around the model Sun then allows students to see how the stars visibly change and how the Sun “moves” through the Zodiac. In effect, this turns the room into a simple planetarium.A flashlight with a fat beam shows the importance of angle to seasonal heating.When directed directly at the wall the energy is concentrated in a small area. When shined obliquely at the wall, the beam covers a larger area, implying less concentration of heat and therefore a lower temperature. It’s also important to include the idea that the day is longer in the summer, which is sometimes overlooked.A bright light source and tennis balls, basketballs, volleyballs or even golf balls can demonstrate features of eclipses and phases of the moon.

Answers to Thought Questions

• If you were standing on Earth’s equator, looking due north you would see the north

celestial pole on the horizon (and the south celestial pole on the horizon, looking due south).You cannot see the north celestial pole from Australia (it’s below the horizon), only the south celestial pole.

2. SKETCH FOR STUDENTS

• The main astronomical reason why there are 12 zodiacal signs is that the Sun appears

to move about 30 degrees per month across the background stars (360 / 30 = 12).

• SKETCH FOR STUDENTS. At the horizon, setting or rising stars move

perpendicularly to the horizon (so they are useful for East-West navigation). At the north pole, stars more or less do not set, they just circle in the sky. At a mid-latitude, the stars make an angle with the horizon. In the Northern Hemisphere, as they set they also move more toward the north. Extremely schematically: setting stars looking west: at the equator, | | |; at a mid-latitude, Northern Hemisphere: \ \ \; at the North Pole, ---.

• When it is winter in New York, the Northern Hemisphere is tilted away from the Sun;

therefore at that time the Southern Hemisphere is tilted towards the Sun and it’s summer in Australia. Although Paris is partway around the world from New York, it’s at about the same latitude and it is also winter there. The part of the sky that you see at night is the part of the sky away from the Sun, so everywhere on Earth sees the same “half” of the sky at night during a 24 hour period, as Earth rotates viewers into nighttime: all three locations should be able to see Orion, which straddles the celestial equator.

(Explorations Introduction to Astronomy, 9e Thomas Arny, Stephen Schneider) (Solution Manual) 1 / 4

Chapter 1 The Cycles of the Sky 2 Copyright 2020 © McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.

• If Earth’s orbit had no tilt, there would be no variation in the angle of sunlight over the

year, nor would there be variation in the length of day—conditions would be somewhat like the equinox all the time and a 12 hour day everywhere, every day. There would be a slight variation in temperature over the year with higher temperatures in January based on the small change in Earth’s distance from the Sun, but this effect would be very small (clearly it does not affect the seasons induced by the tilt very much). Northern and Southern hemispheres would experience these weak seasons at the same time.

• The position of sunrise along the eastern horizon changes during the year because

Earth’s axis (and correspondingly, the celestial equator) is tilted at 23.5 degrees to the plane of its orbit (the ecliptic) and Earth maintains this same tilt throughout the year. At the equinoxes (March 21 and Sept. 23), the Sun lies on the celestial equator. Because the celestial equator cuts the horizon at the east and west points, the Sun will rise and set due east and due west, respectively. In winter, the tilt of Earth results in the Sun rising north of east and setting north of west, and in winter, the Sun rising south of east and setting south of west. At the winter solstice (Dec. 21), the Sun lies 23.5 degrees south of the celestial equator on the sky. It will therefore rise the most to the south of the East on the horizon and set the most to the south of West that year. At the summer solstice (June 21), the Sun lies 23.5 degrees north of the celestial equator. It will therefore rise the most north of East and set the most North of West.

• We have time zones to keep our local time in approximate alignment with solar time,

and to standardize time between different parts of countries and the world—using exact local solar time everywhere would be just as confusing as using one set of hours for the whole world. The sketch can show how it’s solar noon on one part of earth (the Sun is highest in the sky) and a very different time elsewhere (the Sun would be high or low in the sky), or just show a close-up of the difference between the local time in one time zone (say noon) and an adjacent time zone (say 1 pm).

• Some possible ideas: (One or two of these or related ideas should be sufficient).

-You can see some phases during the day, so the geometry is incorrect for the phase to be a shadow.-You can determine the Sun-Earth-Moon angle is not 180 degrees during most phases.-Lunar eclipses (shadow on the Moon) do occur, and only during the 180 degree/full moon alignment.-The curvature of the terminator during Moon phases is not consistent from phase to phase—if it was Earth’s shadow, it would always be the same shape. The changing terminator shape is consistent with a partially illuminated sphere.-The radius of curvature of the terminator for phases does not match the radius of the shadow of Earth during an eclipse.-Eclipses happen over a period of hours, while the phases change slowly over weeks, suggesting they are not caused by the same thing.

• In this case, the sidereal month would remain 27.3 days as the periodic alignment

with the stars would not change, but the solar month would be shorter because the Moon will reach new moon before re-aligning with the stars instead of after. The redrawn figure 2 / 4

Chapter 1 The Cycles of the Sky 3 Copyright 2020 © McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.of 1.14 should indicate that this is the case (the right part of the diagram will happen in opposite order).

Answers to Problems

• 360 degrees / 24 hours = 15 degrees/hr. A simple problem but a good number for

students to know.

• The equinox (Sept. 22) latitude of the Sun will be 90° - 55° = 35°. The highest angle =

35° + 23.5° = 58.5°, lowest = 35° - 23.5° = 11.5°.

• MAKE SKETCH FOR STUDENTS. Use the phases diagram and label the time on

different parts of Earth; then draw horizon lines tangent to Earth to the waxing crescent moon to determine the time it is 1 st visible rising the east is at about 9 AM and it sets in the west at about 9 PM.

• The Moon’s sidereal period is 27.3 days. Therefore, it moves…

360° / 27.3 days = 13.2°/day.Earth must rotate an extra 13.2° to “catch up” to the Moon. Earth rotates about 15°/hr (Problem 1), so the Moon rises about 13.2°/(15°/hr) = 0.88 hr ≈ 53 minutes later each day.

• Moon’s draconic orbital period is 27.2122 days = P.

Moon’s synodic period is 29.5306 days = S.

242  P = 6,585.3524 days 223  S = 6,585.3238 days

242  P = 6585.3524 days /(365.24 days/yr)= 18.03 years.The result gives a pattern of solar and lunar eclipses which repeat over about 18 years, but shift in location across Earth because there is not an even number of solar days in a year.

• Circumference of moon’s orbit = C = 2  384,400 km = 2.42  10

6 km V = C / Period = 2.42  10 6 km / (27.322 d  24 h/d) = 3680 km/h Earth’s diameter = 6400 km  2 = 12,800 km Time = D / V = 12,800 km / (3680 km/h) = 3.5 hours (approx) Since Earth only moves a small distance around its orbit in a few hours of time, the time for the Moon to travel through Earth’s shadow is the dominant effect in determining the duration of the lunar eclipse. (See also next question).

• We would need to decide exactly when to start and when to stop counting—the above

estimate considers the leading edge of the moon going from one side of Earth to the other, but we should probably add the Moon’s diameter to the distance traveled so we 3 / 4

Chapter 1 The Cycles of the Sky 4 Copyright 2020 © McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.move all the way out of Earth’s shadow. This would increase the time (by about diameter of Moon / diameter of Earth ~ 30%). However, Earth’s shadow is not exactly the size of Earth’s diameter—it is a bit smaller, and that would decrease the time (a noticeable correction). As the whole system moves around the Sun, there is also a slight shift in the shadows; since Earth and Moon are moving in the same direction this will lengthen the shadow passage very, very slightly.

• The approach to the estimation is still fine: the Moon is still moving fast enough that

any effect on the shadows from the Earth-Moon system moving around the Sun is negligible for our purposes. We do need to treat the radius as a ~ 200 km shadow passing overhead; 200 km / (3680 km/hr) = 0.05 h = 3.26 minutes.

Answers to Test Yourself

• (d) The north celestial pole is directly overhead.

• (b) Altitude = latitude = 30°.

• (a), (c), (d) Sun, Moon, stars would rise in West, rotate around Polaris clockwise.

• (d) longer day and more concentrated Sun

• (d) Above the arctic circle, or below the Antarctic circle, the daylight varies from 0 to

24 hours over the year.

• (b) Waning gibbous. Must be same moon phase seen all over Earth.

• (c) Full moon rises at 6pm.

• (b) between first quarter (rises at noon) and full (rises at 6pm)

• (c) 2 weeks

• (c) Moon covering sun

• (b) bigger moon, more eclipses, but not every month.

Sun Moon Rsun Rmoon d

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