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Solar Lights: Longitude And Latitude Problems Part 1? (12/3/2011)

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?

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 Earths 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?
Could Hipparchus have used a sundial to check if the eclipses at the Hellespont and in Alexandria reached their peak at the same time?

A sundial obviously wont 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?

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 Earths 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?

Calculate the size (in degrees) of the angle ACB or ACB 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.
How many km equal a parsec? A light year? Take the radius of the Earths 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!).

(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 oce

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