Appendix A: Longitudes and Latitudes of Cities Around the World City Longitude Latitude 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 Rio de Janeiro 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 degree is based on the fact that 1 year 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 minutes, and an arc minute into 60 arc seconds. 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 second is a very small separation indeed. Since both position and brightness change with time, the measured quantities are tied to time. The natural units of time were day 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 hours, each hour is divided into 60 minutes, and each minute into 60 seconds. A tropical year, which our calendar is based on, is 365.2422 days. Efforts to make measurements on the third dimension, distance, were restricted to those of the Sun, the Moon, and, much later in modern times, the stars. For objects in the Solar System, a commonly used unit is the astronomical unit (AU), the distance between the Earth 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 northern hemisphere are Sirius (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 watch 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 duration between the lowest edge of the Sun touching to the horizon to the moment 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 Arctic Circle 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 years 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 seasons 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 clock 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 sundial. 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 equator 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.