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Lesson Plan #8     http://www.phy6.org/Stargaze/Lnavigat.htm

(5a) Navigation 

This lesson covers basic methods for finding one's position on Earth. Latitude can be deduced from the height above the horizon of the pole star or of the noontime Sun, while longitude requires an accurate clock giving universal time.

Part of a high school course on astronomy, Newtonian mechanics and spaceflight
by David P. Stern

This lesson plan supplements: "Navigation," section #5a: on disk Snavigat.htm, on the web

"From Stargazers to Starships" home page and index: on disk Sintro.htm, on the web

Goals: The student will

  • Know that on a clear night one can find one's local latitude λ by observing the elevation above the horizon of the celestial pole (within 0.5 degree of the pole star), an angle which should be equal to λ.

  • Know that in the daytime one can find one's local latitude λ by observing the elevation angle of the Sun above the horizon during its noontime passage south of the observer.

  • Know also that the elevation of the noontime Sun equals λ at equinox, λ +23.5° at the summer solstice and λ–23.5° at the winter solstice. Tables exist for extracting it on other days.

  • Understand how finding one's local longitude requires comparing local time (e.g. local noon, when the Sun is at its highest above the horizon), with universal time, (UT). Also that UT may be obtained from an accurate clock, possibly calibrated by a time signal received by radio.

  • Have an appreciation for the problems of time-keeping, e.g that a pendulum clock will not work well aboard a swaying ship, but a balance-wheel clock may do so, if adjusted for temperature.

Terms: Global Positioning System (GPS), sextant, chronometer

Stories and extras: A verse from "Sea Fever" by John Masefield. The existence and use of the Global Positioning System The story of finding longitude and John Harrison's chronometers. The story of Nansen and his "Fram" expedition, and how he let his chronometer run down. Also the story of Robert Wood devising a crude navigation instrument to overcome wartime secrecy and deduce their ship's position in mid-ocean

Start class by setting up the background:

    Today's astronomy is "pure research," aimed at extending our understanding and exploring of the universe. But in the days of the sailing ships, it was a very practical field--even one of strategic importance.

    Between 1500 and 1800, when trading monopolies on spices, tea, chinaware, silk and other precious goods were hotly contested, and the navies of Spain, Britain, France, Holland and other countries competed for mastery of the oceans--in those days, sea-captains needed astronomy for finding their way at sea.

    Great Britain established and supported the Royal Observatory in Greenwich (outside London), headed by the "Astronomer Royal," while the French king created a rival institution, the Observatory of Paris. Sea officers learned to measure the positions of the Sun and of stars, and to use them in determining position in mid-ocean.

    To measure latitude measuring (or deducing) the highest point in the Sun's daily motion across the sky gave the required information. We will come to that.   (Note: An early instrument for such measurements, the cross staff, is described elsewhere in "Stargazers.").

    To measure longitude, however, navigators needed to accurately know the time. Some used a small telescope to observe eclipses of the moons of Jupiter, whose times could be tabulated in advance, enabling them to set their clocks and thus determine their longitude. This method worked well on land, allowing geographers to accurately derive latitudes and longitudes on land and helping map the known world.

    It was much harder in mid-ocean, however. The problem was finally solved by John Harrison, a British clockmakers, who constructed the first "chronometers," clocks accurate enough for the job. (They were about as accurate as modern wrist-watches, which count the vibrations of a tiny quartz crystal; but in the 1700s, such accuracy was at the cutting edge of technology.) Anyone looking for more of the story may read the book [put on blackboard] "Longitude" by Dava Sobel, or look up on the web: The Discovery of Longitude by Jonathan Medwin.

    This brief survey can only give a very quick overview of methods used in navigation.

Then go over section 5a of "Stargazers." The questions below may be used in the presentation, the review afterwards or both

-- How can the position of the Pole Star help you find your latitude λ at night?
    The elevation angle of the celestial pole above the northern horizon is also the latitude angle latitude λ of the observer. The Pole Star, to a good approximation, marks the celestial pole.

If the question arises: we can also calculate the small difference between the positions of the pole and the pole star, and take it into account

-- Suppose the date is the equinox, March 21 or September 22. How can the noontime Sun give you your latitude λ?
    The noontime Sun's altitude above the southern horizon is 90–λ (degree of latitude) degrees

-- How would you measure that angle of the noonday Sun?
    By an instrument called a sextant, using it "to shoot the Sun." [Before the sextant was invented, by the cross staff.]

-- Suppose the date is the winter solstice, December 21. How can the noontime Sun give you your latitude λ?
    The altitude of the Sun above the horizon is (90 – 23.5) – latitude λ.

-- What if this formula for latitude in the Northern Hemisphere gives a negative number?
    Your location is so far north that the Sun never rises--it is midwinter, the dark part of the year, and you are inside the polar circle, λ larger than 66°!

-- Suppose the date is the summer solstice, June 21. How can the noontime Sun give you your latitude λ.
    The noontime Sun's altitude above the horizon is (90°+ 23.5°)– λ.

-- What if this formula gives an angle greater than 90 degrees?
    The formula gives the angle from the southern horizon. If that angle is larger than 90°, the noontime Sun passes north of you! This can only happen if your latitude λ is less than 23.5°. That means you are close to the equator (closer than anywhere in the continental US). You would be getting close to the Southern Hemisphere, where the noontime Sun usually passes north of the observer.

Mention to the class that for other dates, formulas and tables exist which derive the local latitude from the height above the horizon of the noontime Sun.

-- Besides observations of the Sun, what additional information is needed to measure longitude?
    To determine longitude, we need the accurate time.

-- What is a chronometer?
    An accurate clock used by seafarers until recently, for determining longitude

--How did Robert Wood, sailing across the North Atlantic in World War 1, determine his ship's latitude, even though it was kept secret?

    His companion Colpitts made a crude instrument out of sticks of wood and measured with it the elevation of the pole star above the horizon. That gave the latitude (very nearly).

--How did Robert Wood determine his ship's secret longitude, knowing the date was that of the Fall equinox, and having noted the time of sunset on the ship's clock? The ship was sailing east, from America to England, and its clock was presumably set (by time zones) a known number of hours ahead of the time at its port of departure.

(The teacher might fill in the details below)

    On the equinox, sunset is exactly 6 hours after noon--at the ship, as well as at its point of departure.
        Suppose that by the clock, when the Sun set at the ship, the time at the port of departure was 3:28 pm. The Earth rotates 360° in 24 hours, which comes to 360/24 = 15° per hour. Wood knew that it would only set 2:32 = 2.5333 hours later at the port of departure. During that time, the Earth would rotate

    2.5333 × 15° = 38°

    Therefore the longitude of the ship was 38° further east than the longitude of the port from which it started, which could be looked up on an atlas.

(Robert Wood was known both for his ingenious observations and for his sense of humor. In his lab on a farm in East Hampton he built a long spectroscope, an instrument for analyzing the fine structure of the color of light, from which a lot of information on the emitting atoms and molecules could be derived.

    The light was channeled inside a 42-foot wooden tube, about 6 inches across, but spiders got inside and spun their webs. So Wood cleaned the tube: he opened the door at one end, shoved the family cat into it ("not without a struggle"), then quickly shut the door again. The poor cat had no choice but to run to the other end, in the process brushing away the cobwebs.)

--Who was Fridtjof Nansen?

Have a student find out about Nansen and report on it before the class.

[In addition to leading the "Fram" expedition, Nansen also crossed Greenland on skis, alone, was active in establishing the independence of Norway (from Sweden) and earned the Nobel peace prize for his work in resettling refugees from a bitter war between Greece and Turkey in the years after World War I.]

--How can a radio help in determining longitude?
    Even if your clock does not have the high accuracy of a chronometer, it can give you the time with sufficient accuracy for determining your position, if you use radio signals to reset it every day.

-- What is the Global Positioning System (GPS)?
    A system of 24 satellites continually broadcasting their positions. Simple receivers can detect those signals, and by combining the signals from 3 or more satellites, determine their distances. From that they derive their own positions with great accuracy.

Has anyone used or seen a GPS receiver? What is it like? What does it show? (It can show latitude and longitude, your changing position on a small map, and much more.)

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Author and Curator:   Dr. David P. Stern
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Last updated: 28 August 2004

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