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# Problems

 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.    Teachers using this material in class may obtain a list of solutions by regular mail, by sending a personal request on school letterhead toDr. David P. Stern, Code 695, Goddard Space flight Center, Greenbelt, MD 20771, USA [1] 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? [1] 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? [1] 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? [2] Rudyard Kipling in his poem "The Road to Mandalay" (Mandalay is in Burma-Myanmar) wrote "On the Road to MandalayWhere the flyin' fishes playAn' the dawn comes up like thunderOuter 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? [3] 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? [3] 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. [5] (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?   (b) 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). [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? [6] 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? [6] 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? [6] 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")? [6] 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? [7] 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? [8] (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." [9b] (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? [10] 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? [10] 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? [10] 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:
 satellite T days r in 106 km Io 1.77 0.4214 Europa 3.55 0.6705 Ganymede 7.15 1.0695 Callisto 16.67 1.8812
Check Kepler's 3rd law by deriving the ratios of period squared to distance cubed.

1. [10]  (a) Kepler's 3rd law is T2 = K R3, where T is the orbital period of a planet, R its average distance from the Sun and K is some number, the same for all planets. Assume the orbits are all circles around the Sun.

The formula obviously says that if a planet is more distant from the Sun (R larger), it also takes a longer time to complete each orbit (T is larger too). Could it be that all planets move with the same speed V, but more distant ones take longer to complete each orbit because their orbits are longer--or do more distant planets also move more slowly?

Imagine two planetary systems with circular orbits, where at distance R a planet moves with velocity V and takes time T to complete one circuit. The systems obey different laws: in #1 Kepler's laws hold, in #2 all planets move with the same velocity V, no matter what the distance is. If we go to a planet with orbital radius 2R--are the orbital periods in both system also equal, or if not, in which system is the orbital period longer?

(b) (The solution of part a to be used here.)  Suppose that in some different universe, with different laws, planetary system #2 existed, in which all circular orbits had the same velocity V. How likely would it be that we could find a pair of circular orbits, one in each system, which shared the same distance R, orbital velocity V and orbital period T? Explain.

2. [10] The mean distance of Neptune from the Sun is 30.07 AU (=astronomical unit, means Earth-Sun distance), that of Pluto 39.4 AU. Are these two numbers connected? (Hint: Derive the ratio of the orbital periods!)

3. [10] The period of Comet Halley is approximately 75 years. Assume its perigee is at 0.5 AU from the Sun (1 AU or "astronomical unit" is the mean Earth-Sun distance). How many AU is is it from the Sun to its apogee? Does it get further from the Sun than the mean distance of Pluto, about 39 AU?

4. [12] A satellite in a circular orbit just above the surface of the Earth (r = 1 Re) would need 8 km/sec to stay in orbit. If a missile is sent at that same speed straight up, how high will it get?

Hints: (1) The semimajor axis of an orbit depends only on the launch energy.(2) The trajectory of an object tossed straight up may be viewed as an ellipse of zero width.

5. [12] The scientific satellite ISEE 1 had its perigee is at 1.2 Re, apogee at 23 Re. About how much slower do you think its motion was at apogee, compared to its perigee pass?

6. [12] Meteorites tend to fall more frequently in the afternoon, suggesting they overtake the Earth in its orbital motion. What can this tell about their origin?

7. [13]
• (a) A golf ball is launched at a 45° angle to the horizontal and reaches a distance of 50 meters. If v is its initial velocity, express the time t during which it is in the air. Neglect any air resistance.

• (b) Express the horizontal distance covered in terms of v and t.

• (c) Using the fact that the ball covered 50 meters, derive v and t

• (d) Astronaut Alan Shepard drove a golf ball on the Moon, where the acceleration of gravity is only g/6. If the the ball is launched at 45° as before, with the same velocity, how far would it get?

8. [13] Baseball players have caught baseballs tossed from the top of the Washington Monument in Washington, DC (window height about 550 ft. 1 ft = 30.5 cm). How does their speed compare with that of a professionally pitched baseball, which may hit 90 mph? (1 mile = 1.6 km approx.) Assume g = 10 and neglect air resistance.

9. [14] If a force F is resolved into the sum of two forces Fx and Fy perpendicular to each, the values of Fx and Fy are not uniquely determined. Explain why, and show that in all such cases, the sum of squares Fx2 + Fy2 is always the same.

10. [14] When resolving a vector AB into components AC and CB, we draw a rectangle (or parallelogram) ACBD, of which AB is the diagonal. In vector addition, then, AB = AC + CB. How would you express the other diagonal? (Hint: you can use a minus sign.)

11. [14] A 3-dimensional vector V has components of magnitudes (Vx Vy Vz) along three mutually perpendicular axes. If V is the magnitude of the vector sum, show that

V2 = Vx2 + Vy2 + Vz2

(Hint: use Pythagoras!).

12. [15} A "Wispa" bar of chocolate milk is eaten by a high school student weighing 44 kg. Assuming the body converts 20% of the energy to muscle power, approximately how high is the mountain the student can climb, given the energy of the chocolate bar? Take g = 10 m/sec2 .

13. [20] If T1 is the orbital period around Earth at a radial distance 1.1 Re, and T2 the orbital period around the Moon at 1.1 Rm (Rm = the Moon's radius), which is bigger, and by how much? Assume that on the Moon the acceleration of gravity is 1/6 g, and that Rm = 0.273 Re.

14. [21] The Earth moves around the Sun in an orbit that is approximately a circle of 150,000,000 km radius with a velocity of about 30 km/sec. An object falling near the surface of the Earth has an acceleration of about g = 10 m/s2. If an object were to be placed in the Earth orbit but with no velocity relative to the Sun, it would fall sunward. How would its acceleration compare to g? (The ratio of accelerations is also the ratio of the Sun's attraction at the Earth's orbit to that of Earth's gravity near the surface of the Earth.)

15. [21] Suppose a space probe escaped the Earth's gravity, but it still shares the Earth's orbital motion around the Sun, in a near-circle at 30 km/sec. We then fire an on-board rocket to give it an opposite velocity of 30 km/sec, so that its net velocity is zero and it falls down to the Sun.

How would you find the time T needed for reaching the Sun? (Ignore melting on the way!). Calculate T, if you can.

Hint: Can Kepler's 3rd law help?

16. [21a] The satellites of the Global Positioning System (see sect. #29d) are in 12-hour orbits. If the orbits are circular, what is their distance from the center of Earth?

17. [22a] In problem (9) it was pointed out that a rocket launched east from Cape Canaveral needs less thrust than one launched southward, because it already has the velocity given to it by the spin of the Earth, which equals a few 100s of meters/sec. A rocket launched westward similarly needs more thrust, by the same amount
Do airliners flying eastwards and westwards similarly experience a difference due to the Earth's rotation?

18. [23] A string of length L, with a weight m at its end, hangs from a rotating hook, which causes it to rotate with a period T (like some amusement-park carrousels, whose cars are suspended by long chains). As the string rotates, it describes a cone, and it forms an angle a with the vertical direction. Express a (or its sine or cosine).

Hint: In a rotating frame, the string makes a constant angle a with the vertical, under the action of the centrifugal force and gravity. Each of these forces can be resolved into components along the string and perpendicular to it.

The components along the string just keep the string stretched. However, if the string is in equilibrium in the rotating frame, the perpendicular components must cancel each other, i.e. be equal: if either were stronger, the string would move in its direction and change the angle a.

19. [24] Jules Verne in his book "From the Earth to the Moon" claimed that for passengers on a spaceship passing from Earth to the Moon, the "down" direction reversed when they passed from the region where the Earth's gravity was stronger than the Moon's to the one where the Moon's began to dominates, with "zero g" at the point where both were equal. What is wrong with this idea?

20. [24] Before the satellite age, someone suggested we were actually living inside a spinning hollow sphere, and what we thought was gravity was really the centrifugal force. How many arguments could you suggest against that theory?

21. [27] Rocket engines are cooled by fuel and oxidizer (e.g. liquid oxygen) circulating in pipes along their hot parts, before being burned. What do you think needs the cooling most--the combustion chamber or the wide "bell" through which the gases exit?

22. [30] The SHARP projectile weighs 10-20 kg. Why does the gun need recoil wagons, and why do you think the one behind the auxiliary barrel is 10 times heavier than the other one?

23. [34] Suppose you are in a space at the L2 point of the Sun-Earth system. You look in the direction of Earth: what do you see?
You may assume that the width of the Earth is 3.5 times that of the Moon (see problem 15), that the Moon is 60 Earth radii from the center of Earth and that, as seen from Earth, it is equal in size to the Sun.

24. [34] The Earth-Moon also has Lagrangian points L1 and L2. Its L2 point is on the opposite side of the Moon, about same distance as L1. Is this L2 point a good place to monitor the hidden side of the Moon--e.g. for nuclear test ban violations?

25. [34] Mars has surface gravity 0.39 g or about 3.9 m/s2, radius r = 3332 km and a rotation period of 24 hr. 37.38 min.

(a)What is the orbital velocity at distance r?
(b)What is the escape velocity from the surface?
(c) For communication, astronauts on Mars may use a synchronous satellite. At what distance R (in Mars radii) would it orbit? (Use a calculator with cube roots or 1/3 powers.)

#### ...and just for fun

Get hold of a map of the Moon and see if you can find craters named after personalities you met here. Some of the larger ones: Tycho (distinguished by bright streaks that radiate from it), Ptolemy ("Ptolemaeus"), Copernicus, Kepler, Aristarchus, Hipparchus, Erathosthenes.

Names were bestowed in the 17th century, and latecomers had to make do with left-overs: the craters Newton and Cavendish are at the southern edge of the visible disk, Goddard and Lagrange too are near the edge. Also, "Galilaei" is a small undistinguished crater (because of Galileo's banishment?), Meton and Pythagoras are on the edge, near the northern pole However, since the Russians were the first to observe the rear side of the Moon, a prominent crater there bears the name of Tsiolkovsky.

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Author and Curator:   Dr. David P. Stern
Mail to Dr.Stern:   stargaze("at" symbol)phy6.org .

Last updated 25 November 2001

# Above is background material for archival reference only.

NASA Official: Adam Szabo

Curators: Robert Candey, Alex Young, Tamara Kovalick