3. Celestial Cycles |
Reading: Chapter 3 - expecially pages 47-57 Chapter S1 - pages 72-91 Review: Chapter 1 - particularly Section 1.5 |
Where ever you stand on the Earth (except for at North and South poles - we will come back to this later), the stars RISE in the EAST and SET in the WEST. That is, the APPARENT motion of the stars is moving east to west. The ACTUAL motion is that the Earth is spinning on its axis. The stars are fixed (at least on the time scale of the Earth's 24 hour daily spin). For this session we will be concerned about the APPARENT motion of the Sun, stars and Moon to an observer standing on Earth. These APPARENT motions are due to the fact that the Earth is spinning on its axis, that the Earth is orbiting the Sun and that the Moon is orbiting the Earth - the ACTUAL motions. The ACTUAL motions are the ones we would observe if we could step of the Earth, zoom out and look at the Earth - Moon - Sun system from afar.
In session 1. Our Place in the Universe we saw how the apparent motion of the stars is primarily due to the Earth spinning once per day on its north-south axis. Remind yourself of the meaning of the terms: celestial sphere, Celestial Equator, North Celestial Pole and South Celestial Pole. On the time scale of a day the Earth's orbital motion around the Sun is small. Thus, the Sun's apparent motion across the sky is similar to the motion of all the other stars - rising in the East, setting in the West.
| Home Experiment: In this experiment you can use a piece of string with a weight on the bottom as a simple sundial, and measure the motion of the Sun. |
(1) From your window you see the Sun pass behind a nearby tall building at 3.15pm. At what time the next day should you look out of the same window to see the Sun move behind the building? What is the name that we call the rotation period of the Earth "relative to the Sun"? (2) If you note that a particular star crosses the meridian at 9pm one night, what time will the same star cross the meridian the next night? What is the name that we call the rotation period of the Earth "relative to the stars"? |
The difference in duration of the solar and the sidereal day is due to the motion of the Earth (the spaceship from which we observe the sky) in its orbit around the Sun. As you know, the Earth takes 1 year (about 365 days) to orbit the Sun. First, let's think about the APPARENT motion of the Sun due to the ACTUAL motion of the Earth over the course of a year. Check back to the first session for the meaning of the terms: zodiac, ecliptic.
(3) What do we call the line that passes through the signs of the zodiac? (4) What celestial object follows this line? (5) Why do people talk about "winter constellations" or "fall constellations", etc.? |
The ecliptic is tilted with respect to the celestial equator. This is due to the fact that the Earth's axis is tilted 23.5° with respect to the plane of its orbit. From the perspective of a stationary Earth, it would appear that the ecliptic is tilted 23.5° with respect to the projection of the Earth's equator.
Note that the Earth's orbit is very close to circular. The distance of
the Earth from the Sun only varies by 4% over the year. Furthermore, the
Earth is closest to the Sun in January - which is summer for the southern
hemisphere but winter for the north.
With an Earth tilted 23.5° with respect to the ecliptic, the Sun will appear at different elevations, or angle above the horizon, at different times of the year. This variation in the elevation of the Sun over the year is the cause of the seasons.
The elevation of the Sun varies at different times of the year in Boulder. Polaris is always 40° above the northern horizon in Boulder (same as Boulder's latitude). The celestial equator is also always fixed at 50° above the southern horizon. During the course of the year, the Sun will move 23.5° to the north of and to the south of the celestial equator. It will be furthest north during the summer solstice; it will be furthest south during the winter solstice.
(6a) The Sun is higher in the sky in summer than in the winter. Are shadows longer or shorter in winter? (b) If you are in the southern hemisphere (say, in Australia), in which month of the year is the Sun highest in the sky? Is the Sun to the north or south of the zenith when you are in Australia? (c) The tilt of the Earth's spin axis has changed over time. When the Earth's tilt was less, say 10°, how much does the Sun's elevation change over the year? Would you expect the seasonal differences in temperature to be more or less? |
| Sundial Home Experiment If you live near the Boulder campus or have access to a sundial, you can do an exercise that leads you apparent motion of the Sun. |
Figure S1.1 shows some of the constellations projected onto the celestial sphere. Notice the zodiacal signs Pisces and Aries and the dotted line of the ecliptic. The ecliptic in the diagram marks the path that the Sun takes in the sky. If we were to view the Solar System from the outside, the ecliptic would actually be the plane of the orbit of the Earth about the Sun (see Figure 1.19). From our normal perspective from the Earth however, the ecliptic appears as the path that the Sun takes over the year through the constellations of the zodiac.
Remember that the Sun appears to be in different constellations of the zodiac at different times of the year because the Earth is actually orbiting the Sun. In Figure S1.1 of the celestial sphere, this would correspond to the Sun moving along the ecliptic, making a complete circuit once a year. Remember that the celestial sphere encompasses the entire sky!
(7) Looking at Figure 1.19, find a constellation that is high in the sky around mid-night in (a) summer, (b) winter, (c) spring, and (d) fall. (8) Review of seasons: Fill in the blanks in the paragraph below using words from the list. Seasons occur on Earth because Earth's ______________ is ___________ with respect to the _____________ which causes the _____________ and the _____________ to cross at two points. These points are referred to as the ______________ and the ______________ equinox. When the sun, in its apparent path around the Earth, is at equinox, the lengths of ____________ and ____________ are ____________. These positions mark the beginning s of the seasons ____________ and _____________ respectively. On the longest day of the year, the sun is at the ______________ and reaches its ______________ point in the sky. On the shortest day of the year, the sun is at the ________________ and reaches its _______________ point in the sky. These two positions mark the beginnings of the seasons of _____________ and ______________
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Seasons Link:
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We can actually use the positions of stars in the sky to tell us our location on the Earth. The picture below shows how we can use the stars to tell our latitude, which is a measure in degrees of how far we are from the Equator on the Earth. If we are standing at the equator, we are at 0°. If we are standing at the North Pole, we are at 90° North; likewise for the South Pole, the latitude is 90° South. For us in Boulder, the latitude is 40°.
The star Polaris is located very close to the North Celestial Pole. As a result, as the Earth rotates about its axis in its diurnal cycle, all the stars appear to rotate about us in the sky except for Polaris which stays fixed in its position near the North Celestial Pole. This fact gives us the ability to figure out our latitude on the Earth by looking for Polaris: To measure latitude on the Earth, measure the angle between Polaris and the horizon.
Thus if you were to stand at the North Pole, the North Celestial Pole would appear directly overhead to you, and Polaris would be overhead as well. The angle between what's directly overhead and the horizon is a right angle, or 90°. Thus your latitude is 90° North.
If you were at the equator, Polaris would appear to be on the horizon towards the north. Their is thus no difference between its position and the horizon, that is, its angle is 0°. Thus we are at a latitude of 0°.
For us living in Boulder, if we were to go out in the night and measure the angle of Polaris above the horizon, we would find it to be 40°.
| (9) How would you measure your latitude if you were in Australia? Would you be able to see Polaris? What would you use instead of the North Celestial Pole to aid you in navigating? Is there a star close to the South Celestial Pole like there is a star near the North Celestial Pole? |
Longitude, an east-west measurement is a little harder. For latitude, we had the equator and the North and South Poles as reference points. We can talk about being some number of degrees north or south of the equator. For locations east or west, there isn't any obvious natural reference position.
As a result, we have invented a 0 degree longitude position. This is defined to be a line running north-south through Greenwich, England, which is near London. All geographic locales on the globe have longitudinal positions relative to Greenwich. Boulder happens to be 105° West.
So how do we tell how far east or west we are from Greenwich? Let's look at the following diagram:
We have to remember that the Earth is rotating about its axis and that the Sun appears overhead once every 24 hours at every place on the Earth. In the diagram above, we are looking down on the Earth's North pole. An arrow shows the direction of the Earth's rotation. The Sun is directly overhead in London (making it noon there), while in Boulder, the Sun hasn't even risen yet since Boulder is still on the nighttime side of the Earth.
In the next diagram, we are now 7 hours later. The Earth has rotated bringing Boulder into a position where the Sun is now overhead in Boulder. For Londoners, the Sun has just set since now London is on the nighttime side of the Earth.
Since we know the Earth has to rotate completely in 24 hours, and one
entire rotation means circling 360°, the rate in degrees that the Earth
rotates per hour is:
360 degrees 15 degrees
Rotation rate = ------------- = ------------
24 hours hour
Or 15° per hour. Since it took 7 hours before Boulder rotated to have the Sun overhead, then number of degrees that it rotated must have been:
15 degrees
Total amount rotated = 7 hours x ------------ = 105 degrees
hour
Boulder is thus 105° West of Greenwich. To summarize, what we did was to time the difference between when the Sun was overhead at Greenwich and when it was overhead in Boulder to figure out the longitude. Thus if you wanted to navigate while sailing across the ocean, you needed to have a good clock, since you needed to know the time in Greenwich.
(10a) What is the longitude of the International Date Line? (b) Hawaii is at a longitude of 157° . How many hours is Hawaii ahead or behind Boulder? |
| READING RECOMMENDATION: The story of the race to develop an accurate method of measuring longitude is very well told by Dava Sobel in an excellent little book called Longitude: the True Story of a Lone Genius Who Solved the Greatest Scientific Problem of His Time (1996, Penguin USA). The story is of John "Longitude" Harrison, who solved the problem of determining east-west position, something that both Newton and Galileo had failed to do, by inventing a clock that could withstand the pitching and rolling motions of a ship on the ocean, the temperature and humidity variations, and keep perfect time. I am afraid that the astronomers did not behave well on this occasion. |
There are angles like longitude and latitude on the celestial sphere
that describe the location of a star, such as Betelguese. These angles are
called Right Ascension and Declination.
We will not use these coordinates in this course but you should know that
astronomers use them to describe the locations of objects in the sky.
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Model answers to the comprehension questions.
