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Appendix A: Longitudes and Latitudes of Cities Around the World

Appendix A: Longitudes and Latitudes of Cities Around the World

Appendix A: and of Cities Around the World

City Anchorage 149540W61130N Athens 23430E37580N Auckland, NZ 174440E36500S Beijing 116240E39550N Berlin 13230E52310N Buenos Aires 58230W34360S Cairo 31140E30030N Colombo 79510E6560N Dakar 17270W14420N Hong Kong 114100E22170N Honolulu 157500W21190N Istanbul 28570E41010N Jerusalem 35130E31470N Johannesburg 28030E26120S Lima 77020W12030S London 0080W51300N Los Angeles 118150W34030N Mexico City 99080W19260N Moscow 37370E55450N New Delhi 77130E28370N New York 73560W40400N Paris 2210E48510N 43120W22550S Santiago, Chile 70400W33270S Singapore 103500E1170N Sydney 151130E33520S Tehran 51250E35420N Tokyo 139420E35410N

© Springer International Publishing AG 2017 217 S. Kwok, Our Place in the Universe, DOI 10.1007/978-3-319-54172-3 Appendix B: Astronomical Measurements

Ancient astronomers typically made two types of measurements: position and brightness. Position refers to the angular position of a celestial object on the celestial sphere. Since our view of the sky is two-dimensional, we use the unit of angles to assign the positions of stars. The Babylonian concept of a is based on the fact that 1 has 365 days. Since 365 is close to the nice number 360 which can be divided by 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, etc., astronomers adopted 360 degrees as a full circle and this Babylonian unit is still in use today. Again, since 60 is a good number, we divide a degree into 60 arc , and an arc into 60 arc . To get an idea of how large these units are, a one-centimeter coin placed at a distance of 1 km will have an angular size of 2 arc seconds, so one arc is a very small separation indeed. Since both position and brightness change with , the measured quantities are tied to time. The natural units of time were and year. In this book, we discussed the various definitions of these two units (solar day, sidereal day, tropical year, etc.). A solar day is divided into subunits of 24 , each is divided into 60 minutes, and each minute into 60 seconds. A tropical year, which our is based on, is 365.2422 days. Efforts to make measurements on the third dimension, distance, were restricted to those of the , the , and, much later in modern , the stars. For objects in the Solar System, a commonly used unit is the astronomical unit (AU), the distance between the and the Sun. The brightness of stars is measured in magnitudes, which is an inverse logarith- mic scale. A brighter star has a smaller magnitude. A star that is 100 times brighter has a magnitude value of 5 smaller. Specifically, the brightness ratio of two stars with a magnitude difference of m is 10(–0.4m). For example, if star A has a magnitude of 2, and star B has a magnitude of 4, then star A is brighter than star B by a factor of 10À0.4(2À4) ¼ 6.3.

© Springer International Publishing AG 2017 219 S. Kwok, Our Place in the Universe, DOI 10.1007/978-3-319-54172-3 220 Appendix B: Astronomical Measurements

Some of the brightest stars that can be seen in the are (α CMa, À1.46 in visual magnitude), Canopus (α Car, À0.72 mag), Arcturus (α Boo, À0.04 mag), Vega (α Lyr, 0.03 mag), Altair (α Aql, 0.77 mag), and Antares (α Sco, 0.96 mag). Appendix C: How Long Does It Take for the Sun to Rise and Set?

A popular activity for someone on vacation in a resort location such as Hawaii or Phuket is to the Sun setting in the ocean. Watching a brilliant red Sun slowly descend into the green sea from the blue sky can be an amazing experience. This is more dramatic in a near-tropical location because the Sun sets nearly vertically. Although sunset seems to occur quickly, it is long enough for us to enjoy the experience and accurately time it. One can measure the actual time of the Sun’s descent. Go to a western view point where you can see the horizon. Use a stop watch to time the between the lowest edge of the Sun touching to the horizon to the that it is fully submerged. You can then compare this time with the theoretical expectation. We experience sunrise and sunset because of the rotation of the Earth. It takes 24 hours for the Earth to make one revolution, so the rotational rate of the Earth is 360/24 hours, or 15 degrees per hour, or 4 minutes to cover one degree. We also know that the Sun has an angular size of about half a degree. The time that it takes the Earth to rotate through half of a degree is therefore 2 minutes. However, when one goes to higher latitudes, the time of sunset is no longer given by this theoretical value. The Sun does not set vertically, but instead moves increasingly horizontally with increasing latitude (Fig. 5.2). This means that it will take longer for the upper edge of the Sun to submerge below the horizon. At the latitude of 45, the setting time is 2 minutes and 44 seconds. If one goes to the on June 21, it will take forever for the Sun to set! Exercise: Take a stop watch or use the stop-watch on your cell phone to time the duration of sunset at your latitude.

© Springer International Publishing AG 2017 221 S. Kwok, Our Place in the Universe, DOI 10.1007/978-3-319-54172-3 Appendix D: How Long Is a Day?

The concept of “day” is a natural one. On Earth, we experience a cycle of day and night and we organize our activities (work and sleep) around this cycle. In ancient times, once the Sun was down, human activities were severely curtailed due to our limited ability to see our surroundings. Since the introduction of artificial lighting about 100 ago, the divide between night and day has blurred somewhat but is still a significant part of our lives. How do we measure the beginning and end of this cycle? Since the length of time between the sunrise and sunset varies with the and geographical latitudes, our concept of “one day” must incorporate the total length of time occupied by day and night. Both eastern and western civilizations recognized that the most logical way to define “one day” was the time between noon and next noon (for the definition of noon, see Chap. 2). This definition worked well up to a point. The fact is that this “day”, which astronomers call the “apparent solar day”, can vary by as much as 16 minutes at different times of the year. There are two reasons. One is that the Earth does not follow a circular orbit around the Sun, it is closest to the Sun in January and farthest away in July. This results in a higher orbital velocity in January than in July (Kepler’s second law). On the day that the Earth is closest to the Sun (January 2), the Earth is orbiting at 30.4 km/s, in contrast to 29.4 km/s at the farthest point on July 4. Since the Earth has to turn an extra amount to face the Sun in January in order to compensate for the larger angular distance traveled, the apparent solar day is longer in January than in July. The second reason is that the Earth’s rotational axis is not at right angles to the orbital plane of revolution around the Sun. Together, these two effects cause the length of day to be irregular. Since a non-uniform day is obviously not desirable, astronomers devised a mean solar day that keeps the length of day constant throughout the year. The that we keep today has its origin in solar motion but is not strictly tied to it.

© Springer International Publishing AG 2017 223 S. Kwok, Our Place in the Universe, DOI 10.1007/978-3-319-54172-3 Appendix E: What Time Is Noon?

The obvious answer to this question is “when my watch says 12 o’clock” but this is incorrect. In fact, for most locations on Earth, noon almost never occurs at 12 o’clock. We know that the Sun rises every day, climbs from the horizon, makes its way across the sky, and sets on the opposite horizon. The moment when the Sun is highest in the sky is what we call noon (Chap. 2). By convention, we call this the 12th hour, and the 0th hour is called midnight. Our ancestors already knew that they could determine noon with a . At noon, the shadow of the Sun on the sundial is the shortest. Since the Sun rises in Sydney before it rises in Tokyo or Beijing, clearly the time of noon depends on location. In fact, if you are located on the and walk 460 m east, noon will arrive one second earlier. Now let us imagine that we start a journey at noon and travel west. If we travel fast enough (15 degrees of longitude per hour), the Sun will always be at the same highest position. In other words, time is standing still! In fact, some of you may have already experienced this during your airplane travels. When you fly from Bangkok to London during day time, the day seems never-ending. This can be really confusing for the traveler. Just imagine if you were traveling by train and every train station at a different longitude was on a different clock setting. Every town would have a different time, and it would be difficult to make an appointment. The solution was the system of standard time adopted in 1883, where 24 time zones were set centered on 0, 15, 30, etc. degrees of longitude. Every location 7.5 degrees of longitude east and west around these locations used the same time. If you traveled 15 degrees longitude, you changed your watch by exactly one hour, and not by minutes or seconds. In this way, we settled for convenience rather than accuracy. Rome’s longitude is 12E, New Delhi’sis77E, Hong Kong’s is 114E, Honolulu’s is 158W, Vancouver’s is 123W, and New York’sis74W. Since none of these longitudes are multiples of 15, noon never occurs at exactly 12 o’clock in these cities.

© Springer International Publishing AG 2017 225 S. Kwok, Our Place in the Universe, DOI 10.1007/978-3-319-54172-3 Appendix F: How Far Can We See?

Since the surface of the Earth is curved, we can see farther by standing at a higher point. Assuming that the observer is on top of a mountain of height h above the surface of the Earth, the farthest point he can see is point B, which is distance D away (Fig. F.1). From the Pythagoras theorem, we have

ðÞR þ h 2 ¼ D2 þ R2 pffiffiffiffiffiffiffiffi D ¼ 2Rh where R is the radius of the Earth. Taking R ¼ 6371 km, we have rffiffiffiffiffiffiffi h D ¼ 112:88 km km

If we stand on a beach with our eyes about 2 m above ground, our horizon extends to about 5 km. If we are on top of a mountain 1 km high, we can see as far as 113 km.

© Springer International Publishing AG 2017 227 S. Kwok, Our Place in the Universe, DOI 10.1007/978-3-319-54172-3 228 Appendix F: How Far Can We See?

Fig. F.1 One can see farther by standing at a higher point Appendix G: Decrease of the Obliquity of the

The obliquity of the ecliptic was originally defined as the angle between the ecliptic and the . A modern interpretation of the is the inclination of the rotation axis of the Earth relative to the axis perpendicular to the Earth’s orbital plane around the Sun (Chap. 17). We have learned that the obliquity of the ecliptic is responsible for the seasons, and defines the latitude of the and . However, the value of the obliquity is not constant; it has been decreasing with time. The current value of decline is about 47” per . This change is the result of the combined gravitational forces of the acting on the Earth. Since the value of the obliquity has been measured relatively accurately for 3000 years (Chap. 6), it is possible to predict the changes in this value using dynamical models. Figure G.1 shows the expected change in the obliquity of the ecliptic over 20,000 years. The decrease of the obliquity of the ecliptic is caused by a shift of the plane of the Earth’s orbit around the Sun as the result of gravitational perturbations by other planets. Although the obliquity has been decreasing over the last 10,000 years, on a larger time scale the obliquity will increase again.

© Springer International Publishing AG 2017 229 S. Kwok, Our Place in the Universe, DOI 10.1007/978-3-319-54172-3 230 Appendix G: Decrease of the Obliquity of the Ecliptic

Fig. G.1 The change in the obliquity of the ecliptic over Appendix H: Synodic and Sidereal Periods

Let us denote E as the sidereal period of Earth and P as the sidereal period of a  superior . The angular speed of the Earth ωE is therefore 360 /E and the  angular speed of the planet ωP is 360 /P. Let us begin at a time when the planet is in opposition, i.e., the Sun, the Earth, and the planet are all in a straight line (position 1 in Fig. H.1). After 1 year, the Earth has come back to the same point (position 2). Since a superior planet has a lower angular speed, it has only moved to position 2. In order for the planet to be in opposition again, Earth has to move an extra distance, to position 3. If the synodic period of the planet is S, then S is the time for the planet to move from position 1 to position 3. The time that it will take the Earth to go from position 2 to position 3 is therefore SÀE. The angle between two oppositions is ωP Â S, or ωE Â (SÀE). Therefore we have   360 360 S ¼ ðÞS À E P E or

1 ¼ 1 À 1 P E S

Since the synodic periods of planets have been well known since ancient times and we know that the sidereal period of the Earth is 1 year, the sidereal periods of planets can be derived from the above expression. For example, the synodic period of is 780 days. By taking the to be 365.256 days, we can calculate that the sidereal period of Mars is 686.93 days, or 1.88 years. The derivation for the expression for inferior planets can be done simply by exchanging E and P, as the Earth is outside the inferior planet’s orbit:

© Springer International Publishing AG 2017 231 S. Kwok, Our Place in the Universe, DOI 10.1007/978-3-319-54172-3 232 Appendix H: Synodic and Sidereal Periods

Fig. H.1 Relation between the synodic and sidereal periods

1 ¼ 1 þ 1 P E S

Venus has a synodic period of 584 days. From the above expression, we can derive ’ sidereal period of to be 225 days. Appendix I: Modern Evidence for the Roundness of the Earth

The ancient Greeks knew that the Earth was round as early as the fourth century B.C. (Chap. 5). Magellan’s circumnavigation physically proved it. Today, we can sail or fly around the world and have a first-hand experience of the Earth’s roundness. The ultimate proof that the Earth is round is a picture of the Earth from far away. In 1950, Fred Hoyle wrote in his book The Nature of the Universe : “Once a photograph of the Earth, taken from outside, is available, once the sheer isolation of the Earth becomes known, a new idea as powerful as any in history will be let loose.” This feat was accomplished on December 7, 1972 when the crew of the Apollo 17 spacecraft took a picture of the Earth on their way to the Moon at a distance of 45,000 km (Fig. I.1). This picture had a tremendous impact on the general public because it was the first time that we could see how small and fragile our world was.

© Springer International Publishing AG 2017 233 S. Kwok, Our Place in the Universe, DOI 10.1007/978-3-319-54172-3 234 Appendix I: Modern Evidence for the Roundness of the Earth

Fig. I.1 A picture of the Earth taken by the Apollo 17 spacecraft crew members on their way to the Moon on December 7, 1972. The Arabian Peninsula can be seen near the top and the continent at the bottom. The white patches near the middle are clouds. Photo credit: NASA Appendix J: Modern Evidence for the Rotation and Revolution of the Earth

We now interpret the diurnal motion of the stars to be the result of the self-rotation of the Earth. There are physical experiments that can be done to show that it is indeed the Earth rotating, not the stars. Although the Earth is rotating with a uniform angular speed, the physical speed of the Earth’s rotation varies with latitude. The rotation velocity is highest at the equator, decreases with increasing latitude, and is zero at the poles. A rocket launched from the due south along a single longitude will land west of its target because the earth is rotating. A projectile launched from the equator to the north already has an eastward component of the rotation. Since this eastward motion is larger than the rotation speed at a more northern latitude, the projectile will land to the east. This apparent deflection is called the Coriolis effect. The deflection is always to the right from the point of view of the thrower. In 1851, Bernard Foucault suspended a ball from the ceiling of the Pantheon in Paris. As the ball oscillated back and forth, it traced a pattern on sand on the ground. The plane of oscillation did not stay the same but rotated from east to west. This showed that the floor was turning under the pendulum. The direction of rotation of the plane is opposite to the Earth’s rotation in the northern hemisphere. This effect varies with latitude. On the equator, there is no rotation of the Earth under the pendulum, and there is no change in the plane of oscillation. The effect is largest at the pole where the plane will rotate westward once every sidereal day. At intermediate latitudes, a pendulum rotates with a period of 24 hours/sin ϕ, where ϕ is the latitude. At ϕ ¼ 50, sin ϕ ¼ 0.77, and the period is 31 hours. The period of the rotation of the Earth can be determined this way. A particle of mass m at latitude ϕ on Earth experiences two forces: a gravita- tional force GM/r2, toward the center of the Earth and a centripetal acceleration 2 outward of an magnitude ω (REcos ϕ), where ω is the angular rotation rate of the Earth. The effective gravity is therefore largest at the poles and smallest at the equator. Since the Earth is not totally rigid, this results in the Earth taking on an oblate shape. The measured radius of the Earth at the equator is 21 km larger than the radius at the poles. This degree of oblateness, about 1/300, is an indication that the Earth is rotating.

© Springer International Publishing AG 2017 235 S. Kwok, Our Place in the Universe, DOI 10.1007/978-3-319-54172-3 236 Appendix J: Modern Evidence for the Rotation and Revolution of the Earth

The Coriolis effect, Foucault’s pendulum, and the oblateness of the Earth are the three pieces of physical evidence for the self-rotation of the Earth. In the , one can directly observe the Earth rotating, for example, by observing the different faces of the Earth over the course of an Earth day from the Moon. The inability of Tycho to detect the apparent changes in positions of stars (parallax) as the Earth revolved around the Sun was used as an argument against the heliocentric model. The negative results were strictly due to the very large distances to the stars. In 1838, over a period of 15 , Friedrich Bessel measured a parallax of 0.31 arc sec for 61 Cygni, which translated to a distance of 600,000 AU from the Sun to 61 Cygni. We all have experienced the effect of running in falling rain. When we stand still, rain seems to fall vertically but once we start walking or running, the rain seems to fall at an angle on our face. If the speed of light is finite (as in the case of the rain), then the direction of starlight falling on Earth will seem to tilt as the Earth revolves around the Sun. In 1729, James Bradley measured a tilt of the angle of stars near the ecliptic pole of 20.49 arc sec, which can be translated to an Earth orbital speed of 29.8 km/s, assuming the speed of light. This effect is known as aberration of starlight. The third direct piece of evidence for the revolution of the Earth is the Doppler effect, which states that the wavelength λ of light will change by Δλ ¼ λ(v/c), where v is the line of sight velocity of the moving object. As the Earth orbits around the Sun, a star on the ecliptic plane will experience the maximum amount of Doppler shift. The effect decreases with ecliptic latitude and vanishes for stars on the ecliptic pole. A star on the ecliptic will appear to move towards or away from the Earth as the Earth orbits the Sun. At the visual wavelength of λ ¼ 5000 Angstroms, an Earth’s orbital speed will give a shift of v/c ¼ (30 km/s)/(300,000 km/s) ¼ 10À4 times 5000 Angstroms, or 0.5 angstroms. Appendix K: Escape from Earth

We first consider the simple example of throwing a stone vertically upward from the surface of the Earth. The potential energy of the stone at any one time is Àmgh, where m is the mass of the stone, g is the gravitational acceleration at the surface of the Earth, and h is the height above the Earth’s surface. The kinetic energy of the stone is ½mv², where and v is the vertical speed. The total energy TE ¼ ½mv²Àmgh is conserved. When the stone is on the ground (h ¼ 0), TE ¼ ½mvi², where vi is the initial speed the stone is thrown. The maximum height reached by the stone (at which time v ¼ 0) is hmax ¼ vi²/2g. Since g ¼ 9.8 m/s², a stone thrown with speed of 14 m/s will reach a maximum height of 10 m before falling down. We can generalize to a rocket launch. Since gravity varies with distance to the center of the Earth, g is no longer constant. The general formula as derived by Newton is

v 2 m i À GMEm ¼À GMEm 2 RE RE þ hmax where G is the ,hmax is the height reached when v ¼ 0, and ME and RE are the mass and radius of the Earth, respectively. The initial speed needed for the rocket can reach hmax ¼1is rffiffiffiffiffiffiffiffiffiffiffiffiffi 2GME vesc ¼ RE or 11.2 km/s. This is known as the escape velocity of the Earth. To launch an artificial satellite, we need to first have a rocket that can take the satellite to a certain altitude (hmax), then use a second stage rocket to give the satellite the necessary horizontal speed to go into a circular orbit. For a satellite in a circular orbit at altitude h, it must have an orbital speed vo given by

© Springer International Publishing AG 2017 237 S. Kwok, Our Place in the Universe, DOI 10.1007/978-3-319-54172-3 238 Appendix K: Escape from Earth

v2 o ¼ GME 2 RE þ h ðÞR þ h rffiffiffiffiffiffiffiffiffiffiffiffiffiffiE GME vo ¼ RE þ h

The minimum orbital speed at the surface of the Earth is 7.9 km/s. At an altitude of 500 km, the horizontal speed necessary to achieve circular orbit is 7.6 km/s. If the injected horizontal speed is higher than vo, the satellite will go into an elliptical orbit with the launch point as the perigee. The semi-major axis (a) of the orbit is given by þ ¼ h RE a 2 2 À ðÞv=vo

½ When v ¼ 2 vo ¼ 1.44 vo, the semi-major axis becomes infinity. The satellite will no longer be in a bound orbit but will escape from the Earth in a parabolic orbit. We can reduce the required injection speed by taking advantage of the Earth’s self-rotation. The Earth is rotating at a rate of v ¼ 2πRE/day ¼ 0.46 km/s at the equator. It is therefore preferable to launch an Earth-orbiting satellite on the equator towards the east. Since most European countries are located at northern latitudes, the European Space Agency chose French Guiana (500 km north of the equator) as its launch site. To calculate the speed needed to launch a spacecraft from Earth to leave the Solar System, we can replace ME by Msun and RE by AU: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v ¼ 2GMsun esc AU

The required speed is 42.1 km/s. We can take advantage of the fact that the Earth is already revolving around the Sun at a speed of 29.6 km/s and launch the spacecraft along the direction of the Earth’s motion, reducing the speed needed to 12.5 km/s. Appendix L: Travel to the Planets

Contrary to popular belief, spacecraft do not require fuel for interplanetary travel. Fuel is required from the rocket to launch the spacecraft at sufficient speed to leave the Earth’s orbit, but after that the spacecraft travels on its own under free fall and no further fuel is required until it gets near the target. For a spacecraft to go from Earth to another planet, we never fly directly from the Earth to the planet because this would require an incredible amount of fuel. Instead, we inject the necessary horizontal (orbital) speed to put the spacecraft into an elliptical orbit with a trajectory that will intersect with the orbit of the planet. To go to Mars, for example, we would launch from Earth at perihelion and project the spacecraft to arrive at Mars at aphelion (Fig. L.1). In this case, the semi- major axis (a) of the orbit of the spacecraft is given by

¼ þ ¼ þ : 2a rperihelion r aphelion 1 1 524 or,a¼ 1.252 AU. The required injection speed from Earth (at the perihelion point of the spacecraft orbit) is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 v ¼ GM À sun AU a or v ¼ 33 km/s. The time for the spacecraft to reach Mars is given by Kepler’s3rd law: P2 ¼ a3. This gives an of 1.42 years. Since perigee to apogee is half an orbit, it takes 0.71 years to reach Mars. Since the sidereal period of Mars is 1.88 years, the spacecraft must be launched (0.71/1.88) Â 360¼ 136 behind its rendezvous point. This is why there are certain launch windows for journeys to Mars. Upon arrival, the speed of the spacecraft is

© Springer International Publishing AG 2017 239 S. Kwok, Our Place in the Universe, DOI 10.1007/978-3-319-54172-3 240 Appendix L: Travel to the Planets

Fig. L.1 The minimum energy orbit for a spacecraft to go from Earth to Mars. The blue and red circles are the orbits of the Earth and Mars respectively. The black line is the trajectory of the spacecraft, starting its elliptical orbit from its perihelion point at Earth and arriving at Mars at its aphelion point

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 v ¼ GM À sun 1:524AU a or v ¼ 21.5 km/s. Since the orbital velocity of Mars around the Sun is v ¼ (2π/1.88 yr) 1.524 AU ¼ 24 km/s, the spacecraft must arrive at its apogee first before Mars, so that Mars can catch up with it. Similar calculations show that it takes 2.7 years to reach Jupiter and 6 years to reach Saturn. Review Exercises

1. Does the Sun always rise in the East? If not, when does the Sun rise in due East? Is this different between Los Angeles and London? Indicate the general direction (E, NE, NW, etc.) of the rising Sun in Los Angeles on (i) January 1; (ii) August 15; and (iii) December 21. What is the direction of the rising Sun on January 1 and July 1 at the North Pole? 2. Ancient people were able to make measurements about the annual motion of the Sun by tracking daily changes in the trajectory of the Sun’s shadow made by a stick on flat ground. (i) Draw a diagram that shows the trajectory of such a shadow as seen in Chicago on the summer , the and the winter solstices. (ii) Carefully explain your diagram and what it suggests about the Sun’s annual motion and our sense of direction. 3. The following two observational facts were obvious to even the most casual observer in ancient times: (i) the Sun generally rises at different times and from different directions each day; and (ii) different stars are seen in different times of the year. What was the model that the Greeks constructed to quantitatively explain these facts? 4. Tom finds that his shadow at noon always points to the north. What can you say about Tom’s geographical location? 5. Every day the Sun rises in the eastern horizon and sometime later sets in the western horizon, only to re-emerge again later in the eastern horizon. If you were an observer 2500 years ago, how would you explain this fact? What model you would construct that can explain this behavior of the Sun? 6. What is the observational evidence that the Sun and the stars revolve daily around the same axis? 7. What evidence did our ancestors have for the fact that stars are always in the sky, even in the day time? 8. The polar axis upon which the stars turn is tilted at an angle of 38 from the horizon at Athens. Is there somewhere on Earth that the polar axis is perpen- dicular to the horizon?

© Springer International Publishing AG 2017 241 S. Kwok, Our Place in the Universe, DOI 10.1007/978-3-319-54172-3 242 Review Exercises

9. Our everyday experience tells us that the Earth is a firmly fixed object. What were the arguments that led the Greeks to believe that the Earth was a free- floating object unattached to anything? 10. Describe how ancient Greeks learned that the Earth was round from observa- tions of (i) the Sun; (ii) the Moon; and (iii) the stars. 11. The length of day on the longest day of the year is different for different locations in the northern hemisphere. What is the maximum number of daylight hours and what is the minimum number of daylight hours on this day? Where are the corresponding locations? 12. May Ling takes a walk in a park in Beijing and found that the shadows of trees are very long at noon. She is puzzled why it is the case. Can you help her explain why trees have long shadows at noon? 13. The majority of Greek philosophers did not think that the Earth moved. Using arguments from Aristotelian physics and common sense, explain why this viewpoint remained persuasive until Galileo’s time. 14. Describe the basic observations of the Sun, the Moon, and the stars made by ancient civilizations. What are the qualitative and quantitative elements that they measured? How did ancient civilizations use these observations to con- struct ? 15. John’s plane crashed into the Pacific and he was washed ashore on a deserted island. He has no idea where he was. At night, he can see that the Pole Star is about 22 degrees above the horizon (Fig. 1). What is the latitude of this island? 16. A pilot flying in the Arctic is completely lost due to the endless white ground without any visible landmark. From his watch, he knows that the time is close to noon. How can he use the direction of the Sun to determine his orientation? 17. While going out on her daily walk, Mary observes that the shadow of the lamppost is about half of the height of the lamppost. What is the altitude of the Sun? 18. Refer to the star trail photo in Fig. 4.1. Discuss the pattern observed. What do you think will happen to the pattern if the observer travels to a higher latitude? 19. The motion of the stars follows a regular pattern about a fixed point in the sky known as the North celestial pole. State the two principal characteristics of this pattern and how long it takes to complete a cycle. 20. An observer finds stars are rising in the east along a path that is perpendicular to the horizon. What is the latitude of the observer? 21. At 8 pm one winter evening, Peter looked east and saw the bright star Sirius rising from the ocean horizon. If he wants to see Sirius rise from the sea two days later, what time does he have to look? 22. Assuming that stars are evenly distributed on the celestial sphere, from which location(s) on Earth would an observer see the most number of stars? From which location(s) on Earth would an observer see a minimum number of stars? 23. The philosopher Karl Popper calls Anaximander’s idea that the Earth is a free floating object in space “one of the boldest, most revolutionary, and most portentous ideas in the whole history of human thinking”. What evidence did Anaximander have to support his hypothesis? Review Exercises 243

Fig. 1 Locating your latitude by observing Polaris

24. A star is found to be visible above the horizon for exactly 12 hours. What is the declination of this star? 25. A star on the ecliptic is found to rise exactly in the east and sets exactly in the west. What is the right ascension of this star? 26. A star has declination of +30. For an observer in Seoul, Korea (latitude 37.6N), in what approximate directions will this star rise and set? Do these directions of rise and set change with the seasons? 27. Ancient people knew that what stellar constellation they could see depends on the seasons. Based on the location of Orion in the celestial sphere as shown in Fig. 6.3, which is it most likely for an observer in the Northern hemisphere to see Orion? 28. Why can’t you see your constellation on the night of your birthday? 29. The armillary sphere is a remarkably useful practical device to predict the motions of the Sun and the stars. What is the theoretical basis of the armillary sphere? What are the assumptions that go into the construction of this model? 30. From the model of the armillary sphere in Fig. 7.3, we can see that the horizon, equator, and the ecliptic are all oriented at different angles. What are the significance and implications of these inclinations? 31. What is wrong with the model of the armillary sphere shown in Fig. 2? What are the observational consequences if this model is true? 244 Review Exercises

Fig. 2 Picture of an armillary sphere in Venetian Macau. Photo by the author

32. If you see the setting in the western horizon, what time of day is it (give an approximate hour)? Can a Moon be seen during most of the daylight hours? If so, what is the phase of the Moon? 33. Jack took Jane out for a romantic drive at night. They parked the car at midnight to enjoy the view of the night sky. They saw the Moon was just setting in the western horizon. What is the phase of the Moon? Review Exercises 245

34. What is the evidence that the Moon does not shine on its own but is bright only because of reflected sunlight? 35. When and in which direction do you expect to first see the Moon after a new moon? 36. The side of the Moon that we cannot see from Earth is sometimes called the “dark side of the Moon”. Is this description correct? Is the far side of the Moon always dark? 37. Joan sits out in her balcony at 9 pm and sees the Moon slowly sinking towards the western horizon. She went out to the balcony again 3 nights later, where can she find the Moon? If she wants to see the moonset, what time should she be on her balcony? 38. Show that the Moon moves against the background of fixed stars at a rate of about 13 per day. 39. This picture of the crescent moon (Fig. 3) was taken in Vancouver, Canada. From the shape of the Moon, can you estimate what time of day it is? In what direction is the Moon? How about the approximate day of the lunar ? (hint: where is the Sun?) 40. How did the Greeks explain the difference between the synodic and sidereal periods of the Moon? 41. What are the two basic hypotheses that allow the Greek astronomers to develop a model of the Moon which can explain the changing phases and the rise and set locations of the Moon? 42. Before the invention of clock and other mechanical time keeping devices, how did people of 5000 years ago tell time of the day? Give some examples of methods used. 43. If lunar eclipse is due to Earth’s shadow, why is there not a lunar eclipse every full moon? 44. Running out of food in America, Christopher Columbus predicted that there would be a lunar eclipse on February 29, 1504, therefore greatly impressing the natives of Jamaica to give him supplies. What was the phase of the Moon on that day? 45. The ancient had a year that was 365 days long and was divided into 12 equal months of 30 days each, plus five extra days at the end of the year for celebrations. By comparing this calendar with the actual motion of the celestial objects and seasons on Earth, discuss the disadvantages of this calendar system. 46. was able to determine the length of the tropical year to be 365.2467 days, or within 0.0012% of the modern value. How could he be so accurate without modern instrumentation? 47. Why does the autumnal sometimes occur on September 22 and some- times on September 23? 48. From the internet, find out the date and time of the vernal equinox, summer , autumnal equinox and winter solstice for the current year. Calculate the lengths of the four seasons using this information. 246 Review Exercises

Fig. 3 The Moon is seen near the horizon in Vancouver, Canada. What time is it?

49. Babylonians were able to obtain an accurate value of the synodic month from the length of the year. Assuming that the length of the tropical year is 365.25 days and the is exact, what is the length of the synodic month? 50. By making use of a or a calendar with moon phases, determine the dates of the Easter Sunday for the next 2 years. 51. was born on Christmas day of 1642 in England. But in , his birth year was considered to be in the year 1643. What caused this difference? 52. If there were no obliquity of the ecliptic and therefore no seasons, would ancient cultures develop the concept of a year? If they did, how would they determine the length of a year? 53. What if the obliquity of the ecliptic is 90? What would be the effect on the seasons? What would the positions of the Sun be on the days of equinoxes and solstices for observers at (i) the north pole; (ii) mid-northern latitude; and (iii) on the equator? 54. England adopted the in 1751, 169 years after the Gregorian reform. How many days were there in the year of 1751 in England? 55. If the local noon (when the Sun is highest in the sky) occurs at 1800 hour (GMT), what is the longitude of the location? 56. The fact that different stellar constellations are seen in different times of the year (e.g., Cygnus in summer and Orion in winter) was obvious to even the most casual observer in ancient times. Given that stars can only be seen when the Sun cannot be seen, what does the changing visibility of the constellation tell us about the motion of the Sun? What was the model that the ancient Greeks use to explain this fact? Review Exercises 247

57. From Pythagoras to Hipparchus, Greek Astronomers were able to come up with a mechanical model of the Earth, Sun and Stars that is summarized by the armillary sphere. What are the fundamental features of this model? Discuss a few techniques used in this to expand our understanding of the universe. Please cite specific experiments. What are some of the limitations of this model that were known to the Greeks? 58. The fact that the four seasons are unequal was known to the ancient people. What geometric model did the Greeks use to explain this fact? What is the modern explanation of the unequal seasons? 59. John lives in Boston. On one November evening, he saw the planet Venus high above the horizon. What is the approximate direction (E, W, NW, etc.) of Venus? 60. From the maximum elongation of Venus, estimate how long before sunrise or after sunset one can observe Venus. 61. By observing planets (e.g., Mars) night after night, it is easy to plot their paths through the constellations. However, we cannot see stars during the day and therefore we cannot use stars as a reference to measure the movement of the Sun through the stars. How did ancient astronomers determine the position of the ecliptic in a star chart? 62. The polar radius of the Earth is 6356.75 km. How far one has to walk north- south to see the altitude of the Pole Star change by 1? 63. From the sidereal periods of the planets given in Table 16.1, calculate the synodic period of Earth when observed from Mars. 64. List some observational facts that show the motions of Mars, Jupiter, and Saturn are connected to the Sun. 65. Upon which principal model were most of the ancient cosmologies based? List its main features and explain, giving at least four reasons, why it was so persuasive. 66. used three mathematical constructions, the eccentric, epicycle and equant to model the motion of the planets. Draw figures that clearly illustrate each of these constructions. 67. Give a detailed comparison between Ptolemy’s and Copernicus’s model of planetary motion and discuss their relative advantages and disadvantages. Can either claim to be closer to the truth? 68. Discuss the connections between the Copernican model and the Aristotelian/ Ptolemaic models and why Copernicus’ work was considered problematic in his own day. 69. An astronaut has landed on the near side of the Moon and has a full view of the Earth. How will the astronaut’s view of the Earth change over the course of 1 month? 70. The planet Uranus was discovered in 1781 by William Herschel. It was the first planet discovered after the five ancient planets. When the orbit of Uranus was observed, it was found that it has a synodic period of close to 1 year (Fig. 4). What is the reason for this? 248 Review Exercises

Fig. 4 The path of Uranus along the ecliptic showing the retrograde motions at approximately yearly intervals. Star charts generated by Starry Night Pro. Version 6.4.3 © copyright Imaginova Corp. All rights reserved. www.StarryNight.com

71. The Greek astronomers did not have the ability to measure the distances to the planets but yet they knew the approximate order of planets. How did they do that? 72. If the radius of the Earth is 6400 km, find how far one has to go along the equator to have time change by one second? Laboratory Exercises

These laboratory exercises are designed to give students experiences in the direct observations of celestial objects. Most of these exercises could have been performed by naked eye observers ago. However, given the light and air pollution in urban areas as well as the busy schedules of current generations, students may find it difficult to perform these observations. The availability of computer software that simulates motions of celestial objects at any time as observed from any place allows us to re-live the experiences of ancient observers. The following exercises can be performed either by actual observations or using commercial planetarium software.

A. Observations of Sunrise

Aim: to experience how simple observations of the direction of sunrise reveal its variation in a regular pattern over the course of a year, and to explain how this pattern provides evidence for a spherical Earth and a variation in the speed of the Sun around the Earth. Exercises: Observe sunrise at your local location once every 14 days over the period of a year. Record the azimuth (i.e. the angular on the horizon) and the time of sunrise. (1) Plot (a) the azimuth of sunrise against date (Graph 1). (b) the time of sunrise against date (Graph 2). (2) From Graph 1, determine the dates when the Sun rises at its southernmost position (defined as Day 1), directly in the east (Days 2 and 4), and at its northernmost position (Day 3) for your location. [Hint: You may need to

© Springer International Publishing AG 2017 249 S. Kwok, Our Place in the Universe, DOI 10.1007/978-3-319-54172-3 250 Laboratory Exercises

observe more frequently at certain points, add these observations to the data and the curve.] (3) Repeat Step 1 above for two other locations at different latitudes, preferably one north of and one south of your location. Plot the azimuth of sunrise of for these two locations in Graph 1, and plot the time of sunrise of these locations in Graph 2.

Questions and Discussion (1) Count the number of days between Day 1 and Day 2. Also count the number of days between Day 3 and Day 4. Discuss the significance of these numbers. (2) From Graph 1, calculate the difference in angular position between the north- ernmost and the southernmost rising Sun for your local location. Also roughly estimate the differences for the other two locations. What can you say about these differences? Why do you think this is the case? (3) From Graph 2, compare the shapes of the sunrise time curves for the three locations. Discuss the possible causes of the shapes.

B. Observation of Moonset and Moon Phase

Aim: to find out the variation of direction of moonset Exercise: (1) From an observing location that has a clear view to the west, preferably with a background of flat land or sea, mark the time of the Moon when it hits the horizon. Start from a date shortly after a new Moon. Repeat this measurement once every two or three days over the period of a month. On a piece of graph paper, plot the time of moonset against the date. Also label the phase of the Moon for each measurement. (2) Mark the position of moonset with a cell phone equipped with an electronic compass. Make measurements once every two or three days over the period of 1 month. (3) Plot the azimuth of the setting Moon as a function of time. (4) As the Moon goes through its phases over a month, note the direction of the bright side of the Moon. Compare this to the direction (or inferred direction) of the Sun. Laboratory Exercises 251

C. Synodic Month

Aim: to determine the length of the synodic month Exercise (1) From your local location, find the day that the crescent moon is first visible in the western horizon after sunset. (2) Repeat this search for the next 12 months (3) Count the number of days between each of these first sightings. Average the result to find the length of the synodic month.

D. The Metonic Cycle

Aim: to confirm the Metonic cycle Exercise (1) Beginning with the year 2000, find the date of the first new moon of the year for the following 20 years (2) Find the length of the period that the pattern of the dates of the first new moon of the year repeat themselves (3) Predict the date of the first new moon of the year for the year 2050

E. The Geminus Calendar

Aim: to appreciate the difficulty of designing a calendar that reconciles the solar cycle and the moon phase Exercise (1) Based on the principle of Geminus, design a calendar system of 125 30-day months and 110 29-day months (2) Specifically, when will the switch between 30-day and 29-day months occur?

Discussion (1) What are the desirable and undesirable features of your calendar system? (2) What is the problem with our present system of more-or-less alternating 30 and 31 day months? 252 Laboratory Exercises

(3) What is the problem of the of alternating 29 and 30 day months?

F. Observations of Venus

Aims (1) To learn what kinds of observations of Venus that ancient civilizations could and could not make. (2) To reveal how observations of Venus can be used as evidence for the helio- centric model of Copernicus.

Exercise (1) Observe Venus from the location of your home town once every 15 days either at sunrise or at sunset for one complete cycle between maximum angular separations with the Sun along ecliptic. For each observation, write down (1) the position of Venus on the ecliptic, i.e. its ecliptic longitude (assume the deviation of Venus from the ecliptic is small, i.e. its ecliptic latitude can be neglected), and (2) the angular separation between Venus and the Sun. Also record whether Venus is seen at sunrise or at sunset for each observation. Notes: If Venus is in front of the Sun (i.e. Venus is towards east of the Sun) on the ecliptic, treat the angular separation as positive. Otherwise, treat it as negative. (2) Plot (a) the angular separation between Venus and the Sun against date (Graph 1); (b) the position of Venus on the ecliptic against date (Graph 2). (3) By making more frequent observations wherever necessary, identify the day (s) on which: (a) Venus disappears and reappears. (If the angular separation between Venus and the Sun is within 7, the sky will be too bright for Venus to be seen.) (b) the angular separations between Venus and the Sun are the greatest. (c) Venus is stationary on the ecliptic. (4) Zoom in on Venus by setting the field of view to 2 arc minutes. Observe its phases on each of the dates that you have identified in question 3. Print screen to show the phases in order of the dates with the size of Venus in proportion.

Discussion From this exercise, describe what you have learnt about the cycle of Venus. Laboratory Exercises 253

G. Synodic Period of Mars

Aim: to determine the synodic period of Mars Exercise: using a planetarium software, look up the positions of the Sun and Mars at your current time. Find next five consecutive occurrences of Mars in opposition. Determine the length of the synodic period of Mars to the best accuracy possible. Glossary

A.D. Anno Domini. Year in the after the birth of Christ. 1 A.D. immediately follows 1 B.C. There is no year zero. Recently, the term Common Era (C.E.) is used to avoid the religious connotations. Altitude The angle measured along the great circle perpendicular to the horizon. It is measured from the horizon, positive to the zenith (90 degrees), and negative to the nadir (À90 degrees). It is sometimes called the elevation. Ante meridiem a.m., the period of a solar day before the Sun crosses the local celestial . Aphelion A point in a planet’s orbit which is farthest from the Sun Arc minute One sixtieth of a degree. See Degree. Arc second One sixtieth of an arc minute. See Degree. Autumnal equinox Spatial definition: The intersection of the ecliptic and the celestial equator where the Sun goes from positive to negative declination. Temporal definition: date on which the Sun crosses the celestial equator moving southward, occurring on or near September 22. Azimuth Angle measured on the horizon with north as the zero point, through east, south, and west. Since east is 90 degrees around the horizon from north, its azimuth is 90 degrees, that of south and west are respectively 180 degrees and 270 degrees. B.C. The year Before Christ. The year 1 B.C. is immediately followed by year 1 A.D. There is no year zero. Recently, the term Before Common Era (B.C.E.) is used to avoid the religious connotations. Celestial equator The projection of Earth’s equator onto the celestial sphere. Celestial pole Projection of Earth’s north or onto the celestial sphere. Celestial sphere An imaginary sphere surrounding Earth to which all stars were once considered to be attached. Conjunction Orbital configuration in which a planet lies in the same direction as the Sun, as seen from Earth. The value of elongation at conjunction is 0. Day (1) The time when the Sun is above horizon. Opposite of night. (2) The length of time from noon to noon, see Solar day and Mean solar day. Deferent The large circle upon which the center of the epicycle moves (continued)

© Springer International Publishing AG 2017 255 S. Kwok, Our Place in the Universe, DOI 10.1007/978-3-319-54172-3 256 Glossary

Declination The elevation of the point on the celestial sphere from the plane of the celestial equator. The celestial equator therefore has declination 0, and the north and south celestial poles have declinations 90 and À90, respectively. It is the equivalent of latitude on the celestial sphere. Degree () The unit of angular measure defined such that an entire rotation is 360 degrees. This unit dates back to the Babylonians, who used a base 60 number system. The number 360 likely arose from the Babylonian year, which was composed of 360 days (12 months of 30 days each). The degree is subdivided into 60 arc minutes per degree, and 60 arc seconds per arc minute. 1 ¼ 600 and 10 ¼ 6000. Diurnal motion The daily motion of the Sun and the stars East One of the two intersection points of the celestial equator and the horizon. If we face north, east is 90 degrees to the right. Ecliptic The path of the Sun on the celestial sphere over the course of a year. Elongation Angular separation between a planet and the Sun. The angle between the line joining the planet and Earth and the line joining the Sun and Earth. Epicycle A circle whose center is on the boundary of another circle. Geocentric Earth-centered Gnomon A vertical stick used to measure the length and direction of the Sun’s shadow. When a star rises on the eastern horizon before sunrise for the first time following a solar conjunction Heliacal setting The occasion when a star is seen for the last time to set in the west after the Sun in the evening sky. Heliocentric Sun-centered Horizon The maximum visible extent of the horizontal plane on which an observer stands. Hour (1) The measure defined such that the period is one twenty- fourth of a mean solar day. The hour is subdivided into 60 minutes per hour, and 60 seconds per minute. 1h ¼ 60m and 1m ¼ 60s. (2) The unit of measure of right ascension representing 15 degrees, or one twenty-fourth of a great circle. 1h ¼ 15,1m ¼ 150, and 1s ¼ 1500. Hence, 100 ¼ 0.0667s. Inferior conjunction Orbital configuration in which an inferior planet lies closest to Earth. Inferior planet Planets whose orbits lie between Earth and the Sun, i.e. and Venus. Law An empirical relationship between two or more observable quantities. Year in which an additional day is inserted into the calendar in order to keep the synchronized with the length of the tropical year. There is a leap year in every 4 years except in years which are multiples of 100, but not multiples of 400. This makes a total of 146,097 days in every 400 years, or 365.2425 days per year which is close to the 365.2422 days in a tropical year. The most recent modern corrections are that the years which are multiples of 4000 will not be leap years so that there are 7,304,845 days in 20,000 years, which is 365.24225 days per year. Local celestial The great circle that passes through the north and south celestial poles meridian and the zenith and nadir. (continued) Glossary 257

Mean solar day Average length of time from one noon to the next, taken over the course of a year. Meridian An imaginary line on the celestial sphere through the north and south celestial poles, passing directly overhead at a given location. A meridian is a line of constant longitude. Minute (1) The unit of time, one sixtieth of an hour. See Hour definition 1. (2) The unit of angle, one sixtieth of a degree. See Degree. (3) The unit of angle, one sixtieth of an hour. See Hour definition 2. Month See Synodic month. Night The time when the Sun is below the horizon. Opposite of day. See day. Noon The instance when the Sun is the highest in the sky in one day. It is also the time when the Sun crosses the local (celestial) meridian. North Found by locating the north celestial pole and dropping a vertical line from it to the horizon. Obliquity of the The angle between the plane of the ecliptic relative the celestial equator. ecliptic A modern interpretation of this term is the tilt of the earth’s rotation axis relative to the orbital plane of the Earth around the Sun. Opposition Orbital configuration in which a planet lies in the opposite direction from the Sun, as seen from Earth. The value of elongation at opposition is 180 degrees. Parallax The change in the relative positions of stars as the result of the changing position of the observer. Perihelion The point in a planet’s orbit which is closest to the Sun Planetary Planets coming together sharing similar apparent positions in the sky conjunction Planetary transit Orbital configuration in which an inferior planet is observed to pass directly in front of the Sun. Pole star The bright star closest to the celestial pole. Currently the Pole Star in the north is Polaris. There is no pole star in the south. Polyhedron Three-dimensional solids with flat polygonal faces and straight edges. Post meridiem p.m., the local (celestial) meridian, after noon. The slow gyration of the rotation axis of the rotation axis of the Earth relative to the ecliptic polar axis as the result of external gravitational influence. It makes the vernal equinox drift slowly around the zodiac. This drift is in clockwise direction as seen over the north ecliptic pole. Principle An idea that is assumed to be universal. For example, the principle of relativity refers to the idea that all motions are relative. Quadrature Orbital configuration in which a planet is at 90 degrees from the Sun, as seen from Earth. Elongation is 90 degrees. Right ascension The arc of the celestial equator measured eastward (anti-clockwise as viewed over the north celestial pole) from the vernal equinox to the foot of the great circle passing through the celestial poles and a given point on the celestial sphere, expressed in hours. It is the equivalent of longitude on the celestial sphere. The zero point (0 hour) is the position of the Sun at the vernal equinox. Scientific notation The expression of a number in the form a  10p, where p is an integer called the “order of magnitude”. For example, the scientific notation of 101325 is 1.01325  105. The order of magnitude is 5. (continued) 258 Glossary

Second (1) The unit of time, one sixtieth of a minute. See Hour definition 1. (2) The unit of angle, one sixtieth of an arc minute. See Degree and Hour definition 2. Sidereal day The time between successive risings of a given star, or the time for a star to pass the celestial meridian on successive nights. One sidereal day ¼ 23h56m4.091s which is roughly 4 minutes shorter than a solar day. Sidereal month Time required for the Moon to complete one trip around the celestial sphere (27.32166 days, 27d7h43m11.5s). Sidereal orbital See Sidereal period. period Sidereal period (of a Time required for a planet to complete one cycle around the Sun and planet) return to the starting position on the orbit. It is also called the sidereal orbital period. Sidereal year Time required for the constellations to complete one cycle around the sky and return to their starting points, as seen from a given point on Earth. Earth’s orbital period around the Sun is 1 sidereal year (365.256 mean solar days), or 20 minutes longer than the tropical year because of precession. Solar day The period of time between one noon and the next. South The opposite direction (180 degrees) from north. Spatial definition: point on the ecliptic where the Sun is at its northern- most point above the celestial equator Temporal definition: the date on which the Sun reverses direction from going north to going south, occurring on or near June 21. Superior conjunction Orbital configuration in which an inferior planet lies farthest from Earth (on the opposite side of the Sun). Superior planet Planets whose orbits lie outside that of Earth, i.e. Mars, Jupiter, Saturn, Neptune and Uranus. Synodic month Time required for the Moon to complete a full cycle of phases (29.53059 days). Synodic period Time required for a planet to return to the same apparent position relative to the Sun, e.g., from opposition to opposition, or from inferior conjunction to inferior conjunction. Region on Earth in which all keep the same time, regardless of the precise position of the Sun in the sky, for consistency in travel and communications. The standard time zones have been adopted around the world since 1884. The time of each zone is defined to be the local mean of the central longitude of the zone except some curving of the time zone boundaries is introduced to cater for the non-straight bound- of some countries. Transit See Planetary transit. Tropic of Cancer The northern most latitude that the Sun can be seen directly overhead. For the year 2013 it is at the latitude of 232601400N. Tropic of Capricorn The southern most latitude that the Sun can be seen directly overhead. For the year 2013, it is at latitude 232601500S. The geographical region between latitudes of 23.5 N and 23.5 S, between the Tropic of Cancer and Tropic of Capricorn. (continued) Glossary 259

Tropical year The time interval between one vernal equinox and the next. It is approximately 365.2422 mean solar days. In terms of hours and minutes, it is 365 days, 5 hours, 48 minutes, 45.19 seconds. Tropical period the amount of time for a planet to go once around the ecliptic Local mean solar time at Greenwich (0 longitude). West One of the intersections of the celestial equator and the horizon. If we face north, west is 90 degrees to the left. Winter solstice Spatial definition: point on the ecliptic where the Sun is at its southern- most point below the celestial equator Temporal definition: the date on which the Sun reverses direction from going south to going north, occurring on or near December 21. Vernal equinox Spatial definition: one of the two intersections of the ecliptic and the celestial equator when the Sun passes from negative to positive decli- nation. Temporal definition: date on which the Sun crosses the celestial equator moving northward, occurring on or near March 21. Year See tropical year Zodiac The 12 constellations on the ecliptic. With modern constellation boundaries defined by the International Astronomical Union (IAU) in 1930, the ecliptic also goes through the modern constellation of Ophiuchus. Zodiac signs The ecliptic is divided into 12 equal zones, each is assigned a sign, in the order of Aires, Pisces, Aquarius, Capricornus, Sagittarius, Scorpius, , Virgo, Leo, Cancer, Gemini, and Taurus. Further Reading

Aveni, A. 1993, Ancient Astronomers, St. Remy Press Aveni, A. 1999, Stairways to the Stars, Wiley Aveni, A. 2002, Conversing with the Planets, University Press of Colorado Campion, N. 2009, A History of Volume II: the medieval and modern worlds, Bloomsbury Academic Chan, K.H. 2007, Chinese Ancient Star Maps, Hong Kong Space Museum publications Chen, C.Y. 1995, Early Chinese Work in Natural : a re-examination of the physics of motion, acoustics, astronomy and scientific thought, Hong Kong University Press Couprie, D.L. 2011, Heaven and Earth in Ancient Greek Cosmology, Springer Crowe, M.J. 2001, Theories of the world from antiquity to the Copernican revolution (2nd edition), Dover Danielson, Dennis Richard (ed.) 2000, The Book of the Cosmos, Persus Evans, J. 1998, The History and Practice of Ancient Astronomy, Oxford University Press Ferguson, K. 2002 Tycho and Kepler: the unlikely partnership that forever changed our under- standing of the heavens, Walker Books Fowles, G.R. 1962, Analytical Mechanics, Holt, Rinehart and Winston Gingerich, O. 1997, The Eye of Heaven: Ptolemy, Copernicus, Kepler, Sprinter Gingerich, O. 2005, The Book Nobody Read: Chasing the Revolutions of , Walker & Co Heilbron, J.L. 2010, Galileo, Oxford University Press Hoskin, M. 1997, Cambridge Illustrated , Cambridge University Press Hoyle, F. 1973, Nicolaus Copernicus, Harper & Row Kelley, D.H., & Milone, E.F. 2011, Exploring Ancient Skies: a survey of ancient and cultural astronomy, Springer. Kaler, J. 1996, The Ever Changing Sky: a guide to the celestial sphere, Cambridge University Press Koestler, A. 1959, The Sleepwalkers, Hutchinson Krupp, E.C. 1983, Echoes of the Ancient Skies: the astronomy of lost civilizations, Harper & Row. Kuhn, T.S. 1957, The Copernican Revolution, Harvard University Press Leverington, D. 2003, Babylon to Voyager and Beyond: a history of planetary astronomy, Cambridge University Press Motz, L. & Duveen, A. 1968, Essentials of Astronomy, Wadsworth Selin, H. 2000, Astronomy Across Cultures, Kluwer Thurston, H. 1994, Early Astronomy, Springer Walker, C. 1996, Astronomy Before the Telescope, British Museum Press. Yip, Chee-kuen, 2001, Moving Stars and Changing Scenes, Hong Kong Science Museum Zeilik, M., & Gregory, S.A. 1998, Introductory Astronomy & Astrophysics, Brooks/Cole

© Springer International Publishing AG 2017 261 S. Kwok, Our Place in the Universe, DOI 10.1007/978-3-319-54172-3 Index

A Aurora, 1 Abu Raihan al-Biruni, 132 Autumnal equinox, 12, 56, 62, 120 Acceleration, 202, 206 Azimuth, 22, 39 Action-in-a-distance, 209 Afonso VI, 153 Agriculture, 2, 11 B Alexander the Great, 128, 145 Bandwagon effect, 157 Alexandria, 33, 128, 129, 138, 145 Bible, 152 Almagest, 113, 138, 149, 153, 154, 159, 177 Big Horn Medicine Wheel, 7 Al-Maʾmun, 131, 151 Bishop of Warmia, 159 Altair, 57 Bruno, G., 181 Altitude, 12, 22, 49 Anaxagoras, 74, 111, 143 Anaximander, 31, 34, 111, 112, 143 C Anaximenes, 111 Calculus, 205 Anglican church, 90 Calendar, 7, 26, 83, 116, 124 Annual motion, 14, 53, 112, 114 , 161 Ante meridiem (AM), 21 Callanish Stones, 7, 73 Aphelion, 190 , 117 Apollonius of Perga, 137, 141, 146, 191 Canopus, 127 Apollonius of Rhodes, 128 Caracol temple, 7 Apparition, 98 Cardinal Schonberg,€ 167 Archimedes, 113, 145 Cathedral of Frombork, 159 Arctic Circle, 42, 63 Celestial equator, 42, 56, 58, 83, 120, 172 Aristarchus, 78, 79, 113, 114, 116, 131, 145, Celestial meridian, 44 154, 161 Celestial navigation, 45, 58 Aristotle, 127, 137, 143, 149, 152, 153, 179, Celestial poles, 26, 83 194, 201, 202 Celestial sphere, 17, 21, 56, 109, 137, 143, 147, Armillary sphere, 63, 65, 67, 128 178 Artificial satellites, 207 Central force, 206 Astrolabe, 151 Centrifugal force, 207 Astrology, 108 Chiche´n Itza´,7 Astronomical unit, 131, 151, 167, 192 Chinese calendar, 88, 119 Athens, 29, 33, 112, 113, 128, 144 Christianity, 146, 157, 182

© Springer International Publishing AG 2017 263 S. Kwok, Our Place in the Universe, DOI 10.1007/978-3-319-54172-3 264 Index

Christmas, 89, 90 Eudoxus of Cnidus, 17, 84, 114 Circumnavigation, 133 Evening star, 96, 101 Circumpolar, 57 Clay tablets, 111 Columbus, C., 44, 134, 135 F Comets, 2, 15, 108, 185 , 75, 109, 150, 179 Confucius, 133, 152, 182 Friction, 202, 205 Conic section, 141, 192 Conjunction, 101, 161, 188 Constantine I, 146 G Constellations, 2 Galileo, 181, 197, 205 Copernicus, 113, 131, 153, 159, 167, 170, 173, Geminus of Rhodes, 88 177, 179, 182, 194, 198 Geocentric model, 166, 169, 172, 198 Cosmology, 17, 108, 201 Geometry, 111 Council of Nicaea, 89 Gerard of Cremona, 153 Giese, T., 167 Gnomon, 18, 22, 33, 47 D Goethe, 181 Day, 3, 19, 83 Gravitational mass, 209 Declination, 56, 57, 67 Gravity, 206 Deferent, 137, 141, 149, 169 Greece, 111 Dias, B., 153 Gregorian calendar, 89, 90, 205 Dingqi 定氣, 119 Guo Shoujing, 67, 88 Diurnal motion, 53, 68, 74, 83, 112, 137 Gutenberg, J., 153 Dresden Codex, 96 Duke Cosimo II, 198 Dynamics, 202 H Harrison, J., 46 Heliacal rising, 52, 84, 125 E Heliocentric model, 113, 161, 166, 167, 169, Earth, size of, 129, 131 172, 174, 177, 190 Easter, 89, 160 Hellenistic period, 145 Eccentric, 118, 125, 139, 169, 190, 194 Herakleides of Pontus, 113, 115, 134, 154, Eccentricity, 161 161, 179 Ecliptic, 50, 56, 64, 65, 75, 77, 83, 93, 101, Hesiod, 85 114, 118, 120, 156, 172 Hesperus, 96, 112 Ecliptic pole, 57, 172 Hicetas of Syracuse, 113, 161, 179 Egyptian system, 115 Hipparchus, 78, 86, 115, 117, 120, 137, 146, Einstein, A., 209 154, 166, 173, 185, 188 Ellipse, 141, 191, 193, 195 Homer, 84 Elongation, 95, 154, 161, 165 Hooke, R., 205 Emperor Rudolph II, 188, 193 Horizon, 3, 12, 17, 26, 29, 37, 47, 49, 61, 64, Emperor Theodosius I, 146 127, 178 Epicycles, 118, 125, 137, 139, 149, 154, 161, Horoscopes, 193 166, 167, 169–171, 177, 193, 194 House of Wisdom, 131 Equant, 139, 149, 154, 161, 167, 169, 190 Hyperbola, 141 Equator, 14, 38, 63 Equatorial system, 56, 67 Equinoxes, 19 I Eratosthenes, 128, 129, 131, 134, 151 Index of Forbidden Books, 200 Ether, 144, 205, 208 Inertia, 205 Euclid, 145 Inertial force, 205 Index 265

Inertial mass, 209 Midnight, 74 Inferior conjunction, 161 Milky Way, 2 Inferior planets, 93, 105, 154, 161 Moon, 2, 71, 154, 197, 206 Inner planets, 93 distance to, 79, 117, 131 Intercalary month, 87, 88 far side of, 80 International Astronomical Union, 123 phases of, 71 Inverse-square law, 206 rise, 72 , 87 size of, 79 Morning star, 96, 98 Music, 112 J Jewish calendar, 88 Jian yi, 67 N Julian calendar, 84, 87, 89, 160, 205 Nadir, 38 Julian the Apostate, 154 Natural motion, 145 Julius Caesar, 84 Navigation, 46 Jupiter, 108 Nebra Sky Disk, 5 Jupiter, of, 198 Newton, I., 205 Night, 3 Noon, 12, 19, 46, 74 K North, 18, 26 Kepler, J., 188 North celestial pole, 26, 49, 56, 63, 120 Kepler, planetary laws of, 190, 191, 206 North Pole, 63 King Christian IV, 188 Novae, 2 King Frederick II, 187 Kochab, 120 Koestler, A., 170, 181 O Kuhn, T., 68 Obliquity of the ecliptic, 55, 59, 86, 125, 129, 151 Opposition, 95, 101, 106, 109, 161 L Orthodox Church, 90 Lactantius, 146 Osiander, A., 167 Lascaux Cave, 4 Outer planets, 95, 114 Latitude, 38, 41, 44, 57 Leap year, 87 Li Zhi Zao, 88 P Library of Alexandria, 113, 128, 151 Parabola, 141, 202 Lippershey, H., 197 Parallax, 188 Longitude, 39, 41 Parmenides of Elea, 111 Lunar eclipse, 37, 77, 116, 127 Perihelion, 190 Phosphorus, 96, 112 Pingqi 平氣, 119 M Planetary orbits, size of, 165, 177 Magellan, 134 Planets, 2, 93 Makahiki, 85 motion of, 137, 149 Mars, 100, 190 Plato, 112, 114, 137, 143 Mercury, 95, 114, 115, 163, 170 Pleiades, 5, 85 Meridian, 20, 25, 49, 63 Polar axis, 29, 59, 63, 65, 120 Mesopotamia, 4, 96 Polaris, 26, 46, 120 Metaphysics, 145, 149 Pole Star, 25 Meteors, 2, 15, 108 Polyhedra, 189 , 87 Polynesians, 2, 14 Metonic cycle, 88 Polytheism, 146 266 Index

Pope Gregory XIII, 89 Strabo, 134 Pope John Paul II, 200 Sub-lunary, 144, 180, 186 Pope Leo X, 160 summer solstice, 12, 14, 20, 34, 63, 98, Pope Paul III, 167 116, 119 Pope, Urban VIII, 200 Sun, 2, 11, 195, 198 Post meridiem (PM), 21 distance to, 130 Precession of the equinox, 120, 123–125, size of, 130 173, 188 Sundial, 18 , 39 Sunrise, 5–7, 11–13, 26, 65, 71, 73, 85, Prograde motion, 101, 154 95, 152 Projectiles, 202 Sunspots, 198 Ptolemy, 41, 113, 115, 118, 138, 140, 149, 154, Superior conjunction, 161 161, 166, 169, 170, 177 Superior planets, 95, 101, 154, 161 Pyramids, 111 Super-lunary, 144, 180, 185 Pythagoras, 96, 112, 113, 127, 143, 154 Syene, 129 Synodic month, 71, 78, 87, 88, 115 Synodic period, 75, 101, 104, 105, 162, 163 R Ramadan, 87 Reductionism, 210 T Reflecting telescope, 205 Telescope, 197 Renaissance, 153, 159 Terrenz, J., 88 Retrograde motion, 101, 103, 106, 109, 139, Thales of Miletus, 17, 111, 143 141, 154, 159, 164 Thuban, 120 Rheticus, G.J., 167 Tides, 2 Rho, G., 88 Time zones, 90 Right ascension, 56, 67 Timocharis, 120 Tower of Pisa, 201 Tropic of Cancer, 14, 63, 64, 123 S Tropic of Capricorn, 14, 63, 65, 123 Saros, 78 Tropical period, 104, 105, 114, 150, 174 Saturn, 108 Tropical year, 86, 88, 89, 103, 115, 116, 124 Scaliger, J., 92 Trundholm Sun chariot, 6 Schall von Bell, J.A., 88 , 111 Seasonal markers, 119 Twilight, 23 Seasons, 2, 4, 14, 83, 113, 125, 172 Two-sphere universe, 61–63, 68, 137, 178 origin of, 55 , 185, 188, 190, 201 Seven luminaries, 95 Tychonic system, 188, 201 Sidereal day, 83, 173 Sidereal month, 58, 75, 173 Sidereal period, 162, 192 U Sidereal year, 124 UFO, 96 Sirius, 26, 57, 84, 85, 125, 127 Unevenness of the seasons, 117, 118, 125, 137 Solar day, 83, 173 Universe, size of, 150, 179 Solar eclipse, 77 Uraniborg, 187 South, 18 South celestial pole, 27, 42, 56, 63 Southern Cross, 27, 124 V Spica, 120 Vatican, 90, 200 Stars, distance to, 181 Vega, 57, 120 St. Thomas Aquinas, 146, 152 Venus, 7, 95, 96, 112, 114, 115, 162, 170 Stjerneborg, 188 phase of, 198 Stonehenge, 7 Verbiest, F., 67, 89 Index 267

Vernal equinox, 13, 62, 64, 84, 86, 89, 119, Y 120, 123 Yang Guang Xian, 89, 133 Violent motion, 145, 208 Year, length of, 84, 86, 89, 116 Vision, 90 Yi Shin, 133

W Z , 95 Zenith, 17, 20, 38, 44 Winter solstice, 12, 14, 20, 34, 88, 89 Zodiac, 51, 75, 83, 93, 123 Zodiac signs, 64, 123

X Xu Guangqi, 88

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