latitude

1. Thinking and calibrating	3. Find latitude from Polaris altitude (finish as much as possible week 1)
2. Makeshift sextant and kamal	4. Find latitude from Solar altitude and declination (finish this week 2)

ACTIVITY 1. THINK FIRST: Imagine that you are shipwrecked or lost with no equipment. If you could find a way to measure the height (or altitude) of objects in the sky, how could you use them to orient yourself? Be specific: What objects' altitudes or locations in the sky would you like to know?

What would this tell you about your location on Earth?

THEN DO: You already have one set of tools to measure locations in the sky - your own body.

* Recall how you calibrated your hands and fingers in our first workshop. Use the charts on 151 to check the calibration of your fingers and handspan.
Hold your fingers up to the stars (before class as part of your homework, when the weather clears).
Compare them to celestial spacings (e.g. Fig. 11-2, p.151) to find out their angular size, at arm's length (from Kaufmann, Universe).

* Also calibrate your wink, to find the angular parallax between your two eyes. Use the method on p.155 in Burch.

	angular size (degrees)		angular size (degrees)
pinky nail		width of base of hand
pointer width		outstretched thumb-to-pointer
thumb width		outstretched hand
parallax to outstretched finger (WINK)

Compare the angular size of your fingers, hands, and wink with those of classmates. What are the ranges? How exact (or uncertain) are these measurements?

When you get to class, we will choose a proxy for Polaris for everyone to measure, if the sky is not clear enough to measure the actual Polaris. Try to forget whatever you may already know about Polaris' actual altitude, to better simulate a shipwreck situation.

First, find the altitude of (proxy-) Polaris with your body. How far off could your measurement be?
Polaris altitude from hands: ___________________ Uncertainty in your measurement with hands: ________________

ACTIVITY 2. If you were shipwrecked or lost with only basic materials, here are two simple instruments you could use to measure the altitude of objects in sky. Half the class will build kamals, half will build makeshift sextants. If you have time next week, you might build both. (If you have the opportunity to learn to use a Sextant in the future, you can take more preceise measurements.)

A. Build and use a makeshift kamal as directed in Burch p150. You need:
* a centimeter ruler
* string
* holes in your ruler (use Zita's drill, carefully. Please, no injuries.)

Use it to measure the angular height, or altitude, of a reference object to be chosen together in class.
Polaris altitude from kamal: __________ Uncertainty in your kamal measurement: _________

B. Build and use a makeshift sextant as in Burch p154, using:
* a protractor for angle measurements (you could make one of these from scratch if you had to)
* a straw for your sight tube (tape it to the flat edge of the protractor)
* a string weighted with a nut for your plumb bob.

Use it to measure the angular height of Polaris as drawn on the blackboard. (Stand at the opposite wall.)
Polaris altitude from sextant: __________ Uncertainty in your sextant measurement: _________

Do these instruments give you more precision than your hands? Which measurement do you feel is best? How good is it?

ACTIVITY 3. You can find your latitude precisely, given an altitude measurement.
First, use your best Polaris measurement above, and correct it as directed in Burch Ch.11.2.

Your Latitude = H_s - Dip - Refraction + Polaris correction, where
H_s = apparent altitude of Polaris = _______
Dip = correction if your eye is not lying on the ground = 1' Sqrt[ Height of Eye (in feet)] = ________
(For the purposes of this exercise, assume your eye is 16 feet above sea level - what could you be standing on in that case?)

Refraction correction (RC) for light bent up by thick atmosphere at the horizon (this applies for measurements outside, at a distance, as for stars or the Sun)
RC = 60' / H_s (for H_s > 6 degrees) or
RC = a larger number (<34.5') found on Figure 11-7, p.157 (for H_s < 6^degrees).
Refraction correction = __________

Polaris correction = perpendicular distance from Polaris to the true North Pole, a maximum of 48', found by noting the direction of Polaris with respect to an imaginary line that intersects true N Pole (drawn from the ends of Casseopeia and Ursa Major), as in Figure 11-8, p.159.
Borrow Zita's crossed sticks to measure the angle you need for the Polaris correction, then return them so the next group can use them.
(Stand anywhere for this measurement.)
Polaris correction = __________

These several small corrections matter, because if you measure Polaris' altitude off 30' (half a degree), your latitude will be off 30 miles. That could make the difference between seeing the clouds or birds of a lone island in the morning, or missing it.

Add (or subtract) your corrections appropriately to find your
Latitude from Polaris measurement = ___________ Uncertainty in your latitude = ____________

ACTIVITY 4. You can also calculate your latitude with a solar altitude measurement (Burch Ch.11.7)

* know the declination of the Sun (know the date)
* measure the sun's altitude at noon (then calculate its zenith distance = 90^degrees - altitude = distance of the Sun from zenith, the top of the sky)
* Find your Latitude from the Sun's declination and altitude

A. First, let's figure out how the Sun's altitude depends on the date (or its declination) and our location (latitude).

Recall the latitudes of a few key places on Earth: N Pole______; Olympia _____; Tropic of Cancer _______; Equator ______; Tropic of Capricorn ______

Recall that the Sun's declination is its angle from the celestial equator: the Sun's declination is its celestial latitude. This depends only on the date. The Sun stands directly overhead at the Earth latitude that equals the Sun's declination, on a given date.
When does the Sun rise and set due east and west? ______________________
Where is the Sun directly overhead at noon on those days? _________________
What is the relation between the Sun's path (the ecliptic) and the celestial equator on those days? _____________.
What is the angular difference between the Sun's position and the celestial equator on those days? _____________
This is the Sun's declination at equinox:: 0 degrees.

Consider the Sun's declination, or celestial latitude, on a few other days.
When is the Sun directly over the Tropic of Cancer? __________ On this date, the Sun's declination is equal to the Tropic of Cancer's latitude: ___
When is the Sun directly over the Tropic of Capricorn? __________ On this date, the Sun's declination is equal to the Tropic of Capricorn's latitude: ___
When is the Sun directly over Olympia? ___________ Does the Sun ever have a declination equal to Olympia's latitude? ____
Notice that the maximum declination of the Sun is + 23.5 degrees (N) and the minimum declination of the Sun is - 23.5 degrees (S).

The Sun's declination is the latitude at which the Sun is overhead on a given day.
For example, at Equinoxes, the Sun is overhead at the equator: zero latitude -> zero declination on those two days.
The Sun's declination depends on the date but not on your latitude.

Summarize your results in this table. Leave the last two columns blank for now; just fill in the Latitude and Declination columns.

Place and date	Latitude on Earth	Sun's Declination	estimate Sun's altitude	calculcate Sun's altitude
Olympia at equinox
Olympia at summer solstice
Olympia at winter solstice
Equator at equinox
Equator at summer solstice
Equator at winter solstice
Tropic of Cancer at equinox
Tropic of Cancer at summer solstice
Tropic of Cancer at winter solstice

Next, use your Solar Motion demonstrator to estimate the Sun's altitude (at noon - we are considering the maximum angle above the horizon for a given place and date) at each place and time. Fill in the third column: estimate Sun's altitude. Leave the last column blank for now.

B. Note the patterns that you see emerging. Briefly summarize what you have just observed about how the Sun's declination depends on:

place:

date:

Propose a relationship between latitude, Sun's declination, and Sun's latitude. How could you predict the Sun's altitude for a given place on a given date?

Try your proposed calculation on the cases in the table above, and on a few other places and dates, e.g. the North pole. Note your findings.

Do your calculations match your measurements on the Solar Motion Demonstrator? Discuss with classmates and your professor.

C. Latitude = sun's declination - sun's zenith distance, where zenith distance = (90-Sun's altitude) = distance of Sun from the zenith, or the top of the sky.
N.B.: Looking south to the Sun, Zenith distance is (-). Looking north to the Sun, Zenith distance is (+). (Burch p.171)

How does this equation compare with the relationship you induced in part B above? Solve the equation algebraically for Sun's altitude = ____________________

Use this relation to calculate the Sun's altitude and fill in the last column in your table above. How do your calculations compare to your estimates from your Solar Motion Demonstrator? Can you trust the equation?

D. You need to know the Sun's declination to calculate your latitude from the Sun's altitude and declination: You now have an equation relating these three things. You know how to measure the Sun's altitude using your hands, a sextant, or a kamal.
Now how to find the Sun's declination? What does the Sun's declination depend on? Place? ____ Date? ____

Recall the Sun's declination at equinox ____ (date = __________), summer solstice ____ (date = ____________), and winter solstice ____ (date = ____________)

What if the date was something in between? You can estimate that the Sun's declination on 1.May, for example, is about midway between its dec. on 22. Mar (equinox) and its dec. on 22.June (summer solstice).
Estimate the Sun's declination for 1.Feb.__________, 1.May __________, 1.Aug.__________, and 1.Nov.__________.

There are a couple of ways to find the Sun's declination more precisely, for a given day. The easiest way is to look it up on a chart or tide table.
Look up the Sun's declination for 1.Feb.__________, 1.May __________, 1.Aug.__________, and 1.Nov.__________.

OPTIONAL: If you don't have a tide table with the sun's declination, you can calculate it from the date.
Solar declination = Earth's tilt * cos (date angle) where
Earth's tilt = 23.45^degrees
date angle = 90^degrees S / (S+E) where
S = days to the nearest solstice
E = days to the nearest equinox
For example, the date angle = 0 on solstices and +/- 23.45^degrees on equinoxes. The cycle starts at June 21.

E. You can now find your latitude by (1) observing the Sun's altitude at noon, (2) looking up the Sun's declination for your date, and (3) calculating the latitude with the equation we found in part C above.

Test case: It is August 1 and, looking north, you measure the Sun's altitude to be 10^degrees at local noon. Where are you on Earth?
First, GUESS: northern hemisphere _____ OR southern hemisphere? _______
Guess your approximate latitude: _______

Hs = apparent Solar altitude = _______
Dip = _____
RC = _____
Solar radius = 16' (p.156). Assume you are using a kamal, therefore you are measuring the altitude to the top of the Sun. Declinations are given for the Sun's center, so do you need to add or subtract the Sun's radius? _____

Ho = actual Solar altitude = ________

zd = solar zenith distance = ________

Sun's declination today = ________

Your latitude = ________ Uncertainty in your latitude? _______

How does this compare with your guess?

How can this help you find your way?