The problems below are all related to "Stargazers to Starships." They are arranged in the order of the relevant sections, whose numbers are given in brackets [ ]. Re denotes Earth radius and additional problems are posted here. |
Teachers using this material in class may obtain a list of solutions by regular mail, by sending a personal request on school letterhead to
Dr. David P. Stern, 31 Lakeside Drive, Greenbelt, MD 20770, USA
-  Suppose you look down on the solar system from somewhere north of it (from the direction of the star Polaris). You note that the Earth orbits around the Sun in a counterclockwise direction. If you assume the Earth is fixed and the Sun moves ("apparent motion of the Sun")--does the Sun circle the Earth clockwise or counterclockwise?
-  You have a telescope, mounted on an equatorial axis, with a clockwork to track the stars. It has crosshairs and a scale going through the middle of your image.
You suspect that the positions of stars near the horizon are shifted by refraction of light through the atmosphere. (Air refracts light much less than water or glass--but light from a star near the horizon must pass through a very thick layer.) How can you check this out, and measure the effect if it exists?
-  You are in a lifeboat boat close to the equator, somewhere south of Hawaii. The pole star is too close to the horizonto be seen, but Orion is in the sky, rather close to the horizon, too, and you know that the 3 conspicuous stars in a line, forming Orion's "belt," straddle the celestial equator. How do you find where north is?
-  Rudyard Kipling in his poem "The Road to Mandalay" (Mandalay is in Burma-Myanmar) wrote
"On the Road to Mandalay
Where the flyin' fishes play
An' the dawn comes up like thunder
Outer China 'cross the Bay
- Is sunrise any faster in the tropics--or actually slower--or else, latitude really makes no difference? Explain.
- You are on a seashore in the tropics, watching sunset. If the bending of light in the atmosphere is neglected, and the visual size of the sun's disk is half a degree in diameter, how much time (approximately) passes from the moment the disk just touches the horizon to when the disk disappears completely?
- [2a] Can a sundial work correctly if its gnomon casts its shadow not on a horizontal surface but on a vertical one, e.g. the wall of a house? Explain.
- [2a] Suppose you have built a really big sundial, big enough to have divisions for minutes between the hour lines. You have corrected it for your position in your time zone and are taking the equation of time into account. What else may affect its accuracy?
-  At high latitudes, close to the pole--Alaska, Canada, Scandinavia etc.--the Sun is never far from the horizon. In the summer it moves around the horizon and may be visible 18, 20 or even 24 hours of the day. In the wintertime the Sun rises only for a short time, or in regions near the pole, not at all.
To what extent does the Moon act that way?
-  People watching the Moon from the US see the eyes of the "Man in the Moon" above the Moon's middle and his mouth below the middle. Do people in southern Argentina see it the same way, or upside down? Explain.
-  (a)
A polar satellite, in a low Earth orbit passing over both poles, makes 16 orbits each day. Viewed from Earth, how far apart in longitude are its consecutive passes over the equator?
The Space Shuttle has a low Earth orbit inclined by about 30° to the equator. How far apart are its consecutive passes over the equator? (sin30°=0.5).
-  The war between Japan and the US started in 1941 when Japanese warplanes bombed, at almost the same time, US bases on the Phillipine islands and at Pearl Harbor on Hawaii. History books tell that Pearl Harbor was attacked on December 7, 1941, while the Phillipines were attacked on December 8. How can that be?
- [5a] This problem concerns example (2) in the section on navigation, about the position of the noontime Sun at the time of the summer solstice (21 June). A formula there states that the angle a south of the zenith, at which the Sun at noon crosses the north-south direction at any latitude l, equals on that day
a = l - e
where e=23.5° is the inclination angle by which the Earth's axis deviates from the direction perpendicular to the ecliptic.
What happens if l is smaller than e?
-  A desk calendar has two cubes, next to each other on a shelf, to mark the day of the month---from 01, 02, 03.... to ...29, 30, 31. By rearranging the cubes, the owner of the calendar can always display the proper number of the date. What numerals should be on the faces of each cube, if the numeral "6" can also spell "9" when placed upside-down?
-  At a typical location on Earth, how many moonrises occur in a year?
Hint: The Moon circles the Earth in the same direction as the Earth spins. Imagine a weightless string connecting the Earth and the Moon. As the Earth rotates, the string gets wound up around it, but being perfectly stretchable, it never tears but always continues to bridge the distance between the two bodies.
After one year, how many times is the string wrapped around the Earth?
-  A synchronous satellite keeps its position above the same spot on Earth. Is its period 24 hours or 23 hrs. 56.07 min ("star day")?
-  In the calendar of the Maya Indians, living in Yucatan (around latitude 20 North), special attention was given to the "zenial days" when the noontime Sun was exactly overhead ("at the zenith"). At what dates of the year (approximately) were those days?
-  In one of the eclipses of 1999 the Moon is unable to cover the entire Sun. In the middle of the eclipse zone, where one would expect a total eclipse, a narrow ring of light remains, extending all the way around the dark disk of the Moon. Not knowing anything more about that eclipse, in what part of the year would you think it is most likely to be?
(a) The radius of the Earth is 6371 km. What is the velocity, in meters/sec, of a point on the surface of the Earth, at the equator?
(b) When a rocket is launched, it starts not with velocity zero, but with the rotation velocity which the Earth gives it. Thus if a rocket is launched eastward, it requires a smaller boost (and if westward, a larger one) to achieve orbit. Cape Canaveral is at latitude 28.5 north, cos(28.5°) = 0.8788: how many meters/sec. do we gain the the cape, by launching a rocket eastward?
If orbital velocity is 8 km/sec, what percentage of it do we gain.
(One important reason the main US launch facility was placed in Cape Canaveral was the ability to launch eastward over the ocean).
- (a) [8b] Could Hipparchus have used a sundial to check if the eclipses at the Hellespont and in Alexandria reached their peak at the same time?
(b) [8c] A sundial obviously won't work at night, but could Hipparchus have used an instrument tracking the positions of the stars (the way a sundial tracks the position of the Sun) to tell the duration of a lunar eclipse?
(c) [8c] Let the duration of a lunar eclipse be the time between the moment the Moon goes completely dark to the moment it begins to be uncovered; it is visible, of course, all over the Earth's night side.
Similarly, the duration of a solar eclipse would be the time between the beginning of totality anywhere on Earth and the end of totality anywhere (at a different location!). What would you think lasts longer, and why: the longest lunar eclipse or the longest solar eclipse?
- [8c] Calculate the size (in degrees) of the angle ACB or A'CB' in the drawing of section (8c), i.e. the angles between the lines from your left and right eyes to your outstretched thumb. Assume that the approximate rule, that AC and BC are 10 times the distance AB, holds exactly. Rather than using trigonometry, you may view the distance AB as part of a large circle.
- [8c] How many km equal a parsec? A light year? Take the radius of the Earth's orbit as 300 million km, the velocity of light as 300,000 km/sec.
(This calculation is best done using the scientific notation for large numbers. You may know the phrase "astronomical number" for a number that is very, very, very big--this might well be where the term originated!).
- [9a] Express the observational result on the position of the half-moon (the way Aristarchus believed it was), using the terms "parallax" and "baseline."
(a) If Aristarchus had continued to observed a lunar eclipses, he might have concluded that the width of the Earth shadow was not twice the width of the Moon but 2.5 times that width. Using such a more accurate observation, how many Moon diameters would equal the width of the Earth?
(b) In the drawing of section (9b), suppose we were in a spaceship near point C during a total eclipse of the Moon. What would we see?
-  Tycho's nova had right ascension RA = 0 h, 22 m, declination d = 63° 53'. Look up a star chart--in which constellation did it occur?
-  Section #8b, about using a total solar eclipse to estimate the distance of the Moon, includes a map of the eclipse of August 11, 1999. The path of totality across the Black Sea is shown, as are samples of the region of totality at selected times. You will notice that region is nearly circular.
However, on a map of the complete path of totality (which by the way is available at the web site cited there), you will find that as you follow that path, the patch of totality becomes more and more elliptical and elongated. By the time the eclipse ends, at sunset in India, the patch is a rather lengthy ellipse. Why? And why do you suppose the duration of the eclipse is shorter there?
-  From a handbook, the periods T in days and the distances r in millions of kilometers, for the 4 main satellites of Jupiter (known as the "Galilean satellites" since Galileo discovered them) are: